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Cybernetics for Systems Engineers – DRAFT F

Cybernetics for Systems Engineers (DRAFT)

1 Contents1 Introduction.................................................................................................................................12

2 Cybernetics: Basics......................................................................................................................15

2.1 What is Cybernetics?...........................................................................................................15

2.2 Information..........................................................................................................................15

2.3 Transformation....................................................................................................................17

2.4 Algebraic Description...........................................................................................................19

2.5 The Determinate Machine with Input..................................................................................21

2.6 Coupling Systems.................................................................................................................22

2.7 Feedback..............................................................................................................................23

2.8 Immediate Effect and Reducibility.......................................................................................24

2.9 Stability................................................................................................................................24

2.10 Equilibrium in Part and Whole.............................................................................................25

2.11 Isomorphism and Homomorphism......................................................................................25

3 Cybernetics: Variety.....................................................................................................................27

3.1 Introduction.........................................................................................................................27

3.2 Constraint............................................................................................................................27

3.3 Incessant Transmission........................................................................................................31

3.4 Entropy................................................................................................................................32

4 Cybernetics: Regulation...............................................................................................................35

4.1 Requisite Variety..................................................................................................................35

4.2 Error Controlled Regulation.................................................................................................37

5 Practical Finite State Machine Control.........................................................................................39

5.1 Introduction.........................................................................................................................39

5.2 The Relay and Logic Functions.............................................................................................39

5.2.1 Example : Capacitor Discharge Unit.............................................................................41

5.2.2 Reversing a DC Motor..................................................................................................44

5.2.3 Alarm Circuit................................................................................................................45

5.3 Conclusions..........................................................................................................................46

6 Background Calculus....................................................................................................................47

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6.1 Basic Ideas...........................................................................................................................47

6.1.1 Introduction.................................................................................................................47

6.1.2 Areas Bounded by Straight Lines..................................................................................48

6.1.3 Coordinate Geometry..................................................................................................51

6.2 Areas....................................................................................................................................52

6.2.1 Curved Boundary.........................................................................................................54

6.2.2 Mathematical Induction...............................................................................................56

6.2.3 Higher Order Polynomials............................................................................................57

6.2.4 The Circle.....................................................................................................................59

6.2.5 Concluding Comment...................................................................................................61

6.3 Some Applications...............................................................................................................62

6.3.1 Introduction.................................................................................................................62

6.3.2 Uniform Acceleration...................................................................................................62

6.3.3 Volumes of revolution..................................................................................................65

6.4 Centres of Gravity................................................................................................................67

6.5 Second Moment of Area......................................................................................................69

6.5.1 Moment of inertia of a Disc.........................................................................................70

6.6 Engineer’s Bending Theory..................................................................................................71

6.7 Ship Stability........................................................................................................................74

6.8 Fractional Powers................................................................................................................76

6.9 The Usual Starting Point......................................................................................................79

6.9.1 Introduction.................................................................................................................79

6.10 Applications.........................................................................................................................81

6.10.1 Newton’s Method........................................................................................................81

6.10.2 Maxima and Minima....................................................................................................83

6.10.3 Body under Gravity......................................................................................................84

6.10.4 Maximum Area............................................................................................................85

6.10.5 Concluding Comments.................................................................................................85

6.11 Higher Order Derivatives.....................................................................................................86

6.11.1 Introduction.................................................................................................................86

6.11.2 Radius of Curvature Approximation.............................................................................86

6.12 More Exotic Functions.........................................................................................................88

6.12.2 The Integral of the Reciprocal Function.......................................................................89

6.13 Searching for a Solution.......................................................................................................90

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6.13.1 Finding the Magic Number...........................................................................................92

6.13.2 Cooling of a Bowl of Soup............................................................................................93

6.13.3 Capacitor Discharge.....................................................................................................94

6.13.4 Concluding Comments.................................................................................................95

6.14 Onwards and Upwards........................................................................................................95

6.15 Trigonometrical Functions...................................................................................................96

6.15.1 Introduction.................................................................................................................96

6.15.2 Angles..........................................................................................................................96

6.15.3 The Full Set of Trigonometric Functions.......................................................................98

6.15.4 Relationhips Between Trigonometric Functions........................................................100

6.15.5 Special Cases..............................................................................................................101

6.15.6 Functions of Sums and Differences of Angles............................................................102

6.15.7 Derivatives of Trigonometric Functions.....................................................................103

6.16 Binomial Theorem.............................................................................................................105

6.17 A Few Tricks.......................................................................................................................106

6.18 Differential Equations........................................................................................................107

6.18.1 Introduction...............................................................................................................107

6.18.2 First Order Equation...................................................................................................107

6.18.3 Integration by Parts...................................................................................................112

6.18.4 More Exotic Forms.....................................................................................................115

6.19 The Lanchester Equation of Combat..................................................................................117

6.20 Impulse Response..............................................................................................................120

6.20.1 Second, and Higher Order..........................................................................................122

6.20.2 Resonant Circuit.........................................................................................................125

6.20.3 Pin Jointed Strut.........................................................................................................125

6.20.4 Mass – Spring – Damper............................................................................................127

6.20.5 Higher Order Systems................................................................................................130

6.21 Conclusions........................................................................................................................130

7 More Advanced Calculus...........................................................................................................132

7.1 Introduction.......................................................................................................................132

7.2 Arc Length and Surface areas of Revolution......................................................................132

7.2.1 Why Bother?..............................................................................................................134

7.3 The Catenary......................................................................................................................134

7.3.1 Comment...................................................................................................................138

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7.4 About Suspension Bridges.................................................................................................139

7.5 Arc Length of an Ellipse......................................................................................................141

7.5.1 Towards a better solution..........................................................................................144

7.5.2 About Errors...............................................................................................................147

7.6 Convergence......................................................................................................................147

7.7 Concluding Comment........................................................................................................150

8 The Calculus of Variations..........................................................................................................151

8.1.1 Application to Maximum Area...................................................................................152

8.1.2 Comment...................................................................................................................155

9 Cybernetics and Automatic Control...........................................................................................157

9.1 Introduction.......................................................................................................................157

9.2 Implications for System Specification................................................................................157

9.3 On Feedback......................................................................................................................158

9.4 About Intelligence..............................................................................................................159

9.5 Automatic Control.............................................................................................................161

10 Stability..................................................................................................................................162

10.1 Introduction.......................................................................................................................162

10.2 Additional Background.......................................................................................................162

10.3 Matrix Algebra...................................................................................................................164

10.3.1 Definition and Basic Operations.................................................................................164

10.3.2 Additional Operations................................................................................................165

10.3.3 The Identity Matrix and the Matrix Inverse...............................................................166

10.3.4 Checking for Linear Independence - Determinants....................................................167

10.4 State Space Representation...............................................................................................169

10.4.1 ‘Sign’ of a Matrix........................................................................................................170

10.4.2 Eigenvalues and Eigenvectors....................................................................................171

10.4.3 Calculation of Eigenvalues.........................................................................................173

10.5 Stability – a Different Perspective......................................................................................174

11 Examples................................................................................................................................177

11.1 Introduction.......................................................................................................................177

11.2 Vibrating String..................................................................................................................177

11.3 Directional Stability of a Road Vehicle...............................................................................179

11.4 Hunting of a Railway Wheelset..........................................................................................182

12 Linearisation..........................................................................................................................187

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12.1 Introduction.......................................................................................................................187

12.2 Stability of a Circular Orbit.................................................................................................187

12.3 Rigid Body Motion.............................................................................................................188

12.3.1 Nomenclature............................................................................................................189

12.3.2 Translation Equation..................................................................................................190

12.3.3 Rotation Equations.....................................................................................................192

12.4 Perturbation forms............................................................................................................194

12.4.1 Cartesian Rocket – Steady Flight................................................................................194

12.4.2 Cartesian Rocket – High Angle of Attack....................................................................195

12.4.3 Spinning Projectile.....................................................................................................196

12.4.4 Spacecraft..................................................................................................................197

12.4.5 Aircraft.......................................................................................................................198

12.4.6 Wider Application......................................................................................................198

12.5 Partial Derivatives..............................................................................................................199

12.5.1 The Symmetrical Rocket Re-visited............................................................................200

12.5.2 Stability of a Spinning Shell........................................................................................201

12.6 A Few More Tricks.............................................................................................................202

12.6.1 Spinning Shell Conditions for Stability.......................................................................205

12.6.2 The Aeroplane............................................................................................................206

12.6.3 The Brennan Monorail...............................................................................................209

12.6.4 Equation of Motion....................................................................................................210

12.6.5 Approximate Factorisation.........................................................................................211

12.6.6 Brennan’s Solution.....................................................................................................211

12.6.7 Cornering...................................................................................................................212

12.6.8 Side Loads..................................................................................................................213

12.7 Concluding Comments.......................................................................................................213

13 Modifying the System Behaviour...........................................................................................214

13.1 Introduction.......................................................................................................................214

13.2 The Laplace Transform.......................................................................................................215

13.3 Transforms of Common Functions.....................................................................................216

13.3.1 Derivatives and Integrals............................................................................................216

13.3.2 Exponential Forms.....................................................................................................217

13.3.3 Polynomials................................................................................................................218

13.4 Impluse Function and the Z Transform.............................................................................218

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13.5 Laplace Transform Solution of Non-Homogenous Equations............................................219

14 Impulse Response Methods – Evan’s Root Locus..................................................................222

14.1 Introduction.......................................................................................................................222

14.2 Example: Missile Autopilot................................................................................................223

14.3 General Transfer Function.................................................................................................226

14.4 Rules for Constructing the Root Locus...............................................................................227

14.4.1 Real Axis.....................................................................................................................227

14.4.2 Asymptotes................................................................................................................228

14.4.3 Exit and Arrival Directions..........................................................................................229

14.4.4 Break Away Points.....................................................................................................229

14.4.5 Value of Gain..............................................................................................................230

14.4.6 Comment on Rules.....................................................................................................230

14.5 Autopilot Revisited............................................................................................................231

14.5.1 Numerical Example....................................................................................................231

14.5.2 Initial Selection of Servo Pole.....................................................................................233

14.5.3 Adding Compensation................................................................................................235

14.5.4 Concluding Comments...............................................................................................237

14.6 Gyro Monorail....................................................................................................................238

14.6.1 Representative Parameters.......................................................................................238

14.6.2 Root Locus.................................................................................................................239

14.7 Comments on Root Locus Examples..................................................................................242

14.8 The Line of Sight Tracker....................................................................................................242

14.9 Tracking and the Final Value Theorem...............................................................................244

14.10 The Butterworth Pole Pattern............................................................................................245

14.11 Status of Root Locus Methods...........................................................................................246

15 Representation of Compensators..........................................................................................247

15.1 Introduction.......................................................................................................................247

15.2 Modelling of Continuous Systems......................................................................................247

15.3 Sampled Data Systems.......................................................................................................250

15.4 Zero Order Hold.................................................................................................................251

15.5 Sampled Data Stability.......................................................................................................251

15.6 Illustrative Example...........................................................................................................252

15.7 Sampled Data Design.........................................................................................................256


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16 Frequency Domain Description..............................................................................................258

16.1 Introduction.......................................................................................................................258

16.2 Sinusoidal Response..........................................................................................................258

16.3 Stability..............................................................................................................................260

16.4 Stability Margins................................................................................................................261

16.5 Gain and Phase Margin......................................................................................................263

16.6 Closed Loop Response from the Harmonic Locus..............................................................263

16.7 The Monorail Example.......................................................................................................265

16.8 Sampled Data Systems.......................................................................................................268

16.9 Common z transforms.......................................................................................................269

16.9.1 Sine and Cosine..........................................................................................................269

16.10 Frequency Warping............................................................................................................271

16.11 Decibels.............................................................................................................................271

16.12 Signal Shaping in the Frequency Domain – Bode Plots......................................................272

16.12.1 Second Order Transfer Functions and Resonance..................................................274

16.13 Gain and Phase Margins from Bode Plots..........................................................................276

16.14 Phase Unwrapping.............................................................................................................277

16.15 The Nichol’s Chart..............................................................................................................277

16.16 A Word of Warning............................................................................................................278


17 Time Varying Systems............................................................................................................280

17.1 Introduction.......................................................................................................................280

17.2 Missile Homing..................................................................................................................281

17.3 Significance of Sight Line Spin............................................................................................283

17.4 Pursuit Solution.................................................................................................................283

17.5 Seeking a More Satisfactory Navigation Law.....................................................................285

17.5.1 Lateral Acceleration History.......................................................................................286

17.6 Blind Range........................................................................................................................287

17.7 Sensitivity...........................................................................................................................289

17.7.1 Block Diagram Adjoint................................................................................................291

17.8 Example: Homing Loop with First Order Lag......................................................................292


18 Rapid Non-Linearities.............................................................................................................302

18.1 Introduction.......................................................................................................................302

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18.2 Describing the Distorted Signal..........................................................................................303

18.3 Fourier Series Curve Fit......................................................................................................303

18.4 The Describing Function....................................................................................................305

18.5 Application of Describing Function to the Harmonic Locus...............................................307

18.6 Bang-Bang Control.............................................................................................................307

18.7 Schilovski Monorail............................................................................................................309

18.8 More General Servo Considerations..................................................................................310

18.9 The Phase Plane.................................................................................................................312

18.9.1 Dominant Modes.......................................................................................................312

18.9.2 Representation of a Second Order Plant....................................................................313

18.9.3 Unforced Plant...........................................................................................................314

18.10 Introducing Control............................................................................................................316

18.11 Hierarchical System Considerations...................................................................................319

18.12 Scanning System................................................................................................................320

18.12.1 Outer Loop.............................................................................................................323


19 The Representation of Noise.................................................................................................326

19.1 Introduction.......................................................................................................................326

19.2 Correlation.........................................................................................................................326

19.2.1 The Same, Or Different?.............................................................................................326

19.2.2 Pattern Matching.......................................................................................................328

19.3 Auto Correlation................................................................................................................331

19.4 Characterising Random Processes.....................................................................................333

19.4.1 White Noise...............................................................................................................334

19.4.2 System Response to White Noise..............................................................................335

19.4.3 Implications for Time-Varying Systems......................................................................337

19.5 Frequency Domain.............................................................................................................338

19.5.1 The Fourier Transform...............................................................................................338

19.5.2 Application to Noise...................................................................................................339

19.6 Application to a Tracking System.......................................................................................341

20 Line of Sight Missile Guidance...............................................................................................343

20.1 Introduction.......................................................................................................................343

20.2 Line of Sight Variants.........................................................................................................343

20.3 Analysis..............................................................................................................................344

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20.4 Comments.........................................................................................................................346

21 Multiple Outputs...................................................................................................................347

21.1 Introduction.......................................................................................................................347

21.2 Nested Loops.....................................................................................................................347

21.3 Pole Placement..................................................................................................................350

21.4 Algebraic Controllability.....................................................................................................351

21.4.1 Monorail....................................................................................................................353

21.4.2 Road Vehicle..............................................................................................................353

21.5 Comment on Pole-placement............................................................................................355

21.6 The Luenberger Observer..................................................................................................357

21.7 The Separation Theorem...................................................................................................358

21.8 Reduced Order Observers..................................................................................................359


22 Respecting System Limits – Optimal Control.........................................................................362

22.1 Introduction.......................................................................................................................362

22.2 Optimal Control.................................................................................................................362

22.3 Optimisation Background..................................................................................................364

22.3.1 Principle of Optimality...............................................................................................364

22.3.2 Application to the Control Problem...........................................................................366

22.4 Gyro Monorail....................................................................................................................368

22.5 The Optimal Observer........................................................................................................370

22.5.1 Covariance Equation..................................................................................................371

22.5.2 The Observer..............................................................................................................373

22.5.3 Tracker Example.........................................................................................................374

22.5.4 Alpha beta Filter.........................................................................................................375

22.6 Comment...........................................................................................................................375

23 Multiple Inputs......................................................................................................................377

23.1 Introduction.......................................................................................................................377

23.2 Eigenstructure Assignment................................................................................................378

23.3 Using Full State Feedback..................................................................................................378

23.3.1 Basic Method.............................................................................................................378

23.3.2 Fitting the Eigenvectors.............................................................................................379

23.3.3 Calculation of Gains...................................................................................................381

23.3.4 Comment on Stability................................................................................................382

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23.4 Loop Methods....................................................................................................................383

23.4.1 General Considerations..............................................................................................383

23.4.2 Nyquist Criterion........................................................................................................384

23.5 Sequential Loop Closing.....................................................................................................385

23.6 Diagonalisation Methods...................................................................................................386

23.6.1 ‘Obvious’ Approach....................................................................................................386

23.6.2 Commutative Compensators.....................................................................................386

23.7 Comment...........................................................................................................................386

24 Closed Loop Methods............................................................................................................388

24.1 Introduction.......................................................................................................................388

24.2 The H-infinity Norm...........................................................................................................389

24.3 MIMO Signals.....................................................................................................................390

24.4 Closed Loop Characterisation............................................................................................392

24.4.1 Tracker Example.........................................................................................................393

24.4.2 Comment...................................................................................................................393

24.4.3 Input Output Relationships........................................................................................394

24.4.4 Weighted Sensitivity..................................................................................................395

24.5 Stacked Requirements.......................................................................................................396

24.5.1 Robustness.................................................................................................................397

24.5.2 Phase in MIMO Systems............................................................................................397


25 Catastrophe and Chaos..........................................................................................................400

25.1 Introduction.......................................................................................................................400

25.2 Catastrophe.......................................................................................................................400

25.3 The ‘Fold’...........................................................................................................................401

25.4 The ‘Cusp’..........................................................................................................................402

25.5 Comment...........................................................................................................................403

25.6 Chaos.................................................................................................................................404

25.6.1 Adding Some Dynamics..............................................................................................404

25.6.2 Attractors...................................................................................................................406

25.7 The Lorenz Equation..........................................................................................................407

25.7.1 Explicit Solution..........................................................................................................409

25.8 The Logistics Equation.......................................................................................................410

25.9 Bluffer’s Guide...................................................................................................................412

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25.9.1 Basin of Attraction.....................................................................................................412

25.9.2 Bifurcation.................................................................................................................412

25.9.3 Cobweb Plot...............................................................................................................412

25.9.4 Feigenbaum Number.................................................................................................412

25.9.5 Liapunov Exponent....................................................................................................412

25.9.6 Poincaré Section........................................................................................................413

25.9.7 Symmetry Breaking....................................................................................................413


26 Artificial Intelligence and Cybernetics....................................................................................414

26.1 Introduction.......................................................................................................................414

26.2 Intelligent Knowledge Based Systems................................................................................414

26.3 Fuzzy Logic.........................................................................................................................415

26.4 Neural Nets........................................................................................................................417


27 Conclusions............................................................................................................................419

27.1 Introduction.......................................................................................................................419

27.2 Hierarchy...........................................................................................................................419

27.3 Intelligence........................................................................................................................420

27.4 Seeking Null.......................................................................................................................420

27.5 And Finally.........................................................................................................................421

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1 IntroductionIt is only fair to the reader to start with sufficient information to decide whether to proceed.

The word ‘system’ as used today is most frequently meant in the sense implied by ‘systems analyst’, a specialist concerned with mapping technically naive user’s requirements into computer code. Before the pre-eminence of software engineering, it had another meaning, as defined by the likes of Norbert Wiener, William Ross Ashby, Harry Nyquist, and Claude Shannon to describe a subject which was intended for the highly numerate, in order to gain a scientific understanding of system behaviour.

Without passing further comment on the works which expound the modern innumerate approach, I will merely comment that this is not such a work.

In order to discriminate between the two usages, I shall refer to the former as ‘systems engineering’ and the latter I shall call ‘cybernetics’ from the Greek word for ‘helmsman’.

To supplement the management/organizational characterization of the system, we require a means of characterizing components of system elements which do not exist, but are postulated as potential means of achieving improved system performance. This requires the methods of cybernetics.

The methods which work, and have found practical application over the years belong to a body of knowledge called ‘control theory’. In recent years, for various reasons, this has become perceived as an arcane specialisation, which is unfortunate because it constitutes a universal means of system description and analysis, which should form part of every engineer’s toolkit, regardless of specialisation.

The book is unusual in appealing to the broad knowledge and experience of the practitioner, rather than trying to present the subject as self-contained. Like all systems approaches, it is useless within itself; it’s value becomes apparent in its application to real problems. With that in mind, the text presents a number of practical problems to illustrate the methods used.

The approach is eclectic, taking examples from all branches of engineering, to illustrate the universal application of cybernetics to all self-correcting systems, regardless of their actual physical implementation.

As far as possible, the concepts are explained in plain English, so hopefully a reasonable understanding of the subject matter may be gained from the text alone.

The early chapters present the fundamental ideas more as a philosophical discussion, than pragmatically useful methods. However, the reader is advised to get to grips with the Principle of Requisite Entropy, on which all valid systems thinking, mathematical or not, is based, yet judging by the decrees of some so-called ‘experts’, is not as widely understood as it should be.

This is followed by a brief presentation on the practical implementation of finite state controllers, leading to the conclusion that a problem formulation which enables hardware behaviour to be taken into account is to be preferred to general software development methods. This theme of the

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constraint imposed by the real world forms the basis of the remainder of the book, but requires significantly greater mathematical knowledge.

We then proceed with the background mathematics needed to understand the remainder of the book. This is no more than would be covered in the first year of a typical engineering course. It is included because many line engineers may not have had call to use their mathematics in support of their largely administrative and managerial duties, and a reminder is in order. The mathematically adept may safely skip these chapters. The presentation differs from the more usual approach in that we start with integral calculus, which is immediately applicable to engineering problems, and differentiation is introduced as its inverse. This is expected to appeal to the natural engineer, who has little inclination to discuss the number of angels which can dance on a pinhead.

The main body of the book then considers the natural behaviour of dynamic systems, covering the methods used to uncover the causes of instability, and the identification of critical parameters. Many systems issues are resolved cheaply and simply by changes to the plant parameters, without introducing control at all. We emphasise the need to understand the dynamic behaviour of the plant before considering introducing artificial means of improving its behaviour. All the clues we need to design our closed loop system arise from this understanding of the basic plant dynamics.

After the basic stability has been considered, we cover the body of knowledge which has become known as ‘classical control’. These methods relate the open loop plant behaviour to that of the closed loop, so that the effect of compensators introduced within the loop may be predicted. Central to this presentation are the Laplace and Z transforms, which enable the behaviour of complex networks of dynamic elements to be predicted readily.

These methods are concerned dominantly with stability, although tracking behaviour is also considered. It becomes evident that stability alone is insufficient, we need to introduce constraints of noise and saturation to decide the characteristics of the compensation we may employ. To this end we investigate the effects of non-linearities on stability using describing functions and phase-plane methods.

Before proceeding to characterise noise, we take a detour into the world of terminal controllers, which naturally introduces the method of adjoint simulation. The ideas introduced here are needed to understand the effects of noise impulses applied to a linear system using a time domain description. The concept of white noise is introduced, and from it the means of predicting the accuracy of tracking from the noise sources within the loop.

All the ‘classical’ approaches are intended for single input, single output systems. With the reduced cost of sensors in recent years, most closed loop systems have multiple outputs, and we revert back to the state-space descriptions used to describe our basic plant dynamics. This introduces pole placement and the idea of an observer, as a special type of compensator which models the plant dynamics. The separation theorem is invoked to show that the dynamics of the controller do not affect those of the observer, so the two can be designed in isolation. These methods work directly with the closed loop plant.

The state space methods are extended to consider controllers whose states are limited in range (typically by saturation), and observers whose accuracy is limited by noise (the fixed gain Kalman

13


filter). The fundamental limitation of these approaches, that the separation theorem is unlikely to apply in practice, is identified.

Moving on from muti-output systems, we proceed to consider systems which are multi-input as well as multi-output. In view of the cost of servos, such systems are unlikely to be common. Usually we deal with multiple de-coupled single-input/single-output systems. However, these are recognised as potentially difficult to design, and considerable intellectual effort has been invested in them over the past thirty years. Most of the recent work lies beyond the scope of the current book, which is not intended for post graduate studies into control. Some techniques which the author has found practically useful are presented. However, the most important question to be resolved before embarking on a MIMO controller; is why is the plant so horribly cross-coupled in the first place?

The dominant method, that of H∞ ‘robust’ control, or the ideas behind it, are presented briefly, but the absence of examples which were not contrived for the sole purpose of illustrating the method, is rather a handicap when presenting it to an audience of pragmatic engineers.

The remainder of the book reverts back to the philosophical discussion of the opening chapters, except that the exposition is more speculative. These sections may be skipped unless the ideas presented are of particular interest to the reader.

Well, that completes the introduction, it should be evident whether or not the book is for you. If not, I thank you for your time. If so, do I hear the satisfying ring of a cash register?

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2 Cybernetics: Basics

2.1 What is Cybernetics?Cybernetics is a theory of machines, but does not deal with the gears or circuit boards of physical machines, but in how machines behave independently of their physical implementation. In essence, it is the study of machines in the abstract.

As it deals with behaviour, rather than implementation, the ‘what’ of the system function, rather than the ‘how’, it would be reasonable to use it as the basis for system specification, particularly as the resulting definitions of behaviour are concise, compact and unambiguous.

The approach deals with ‘black boxes’, that is we deal with how entities behave, or are required to behave, not how they are implemented in any specific instance. We find that even the most complex behaviours can be characterized with extremely simple system models. Indeed, it is shown that much of the complexity of ‘complex’ systems serves to simplify behaviour. This is in contrast to the ‘white box’ approach characteristic of more mainstream systems engineering, where internal structure and component functions must be known, in order for the system to be specified.

It follows that the methods of cybernetics, in containing no pre-conceptions as to how behaviours are achieved ,furnishes the techniques most appropriate for system specification, yet paradoxically, their strong emphasis on mathematical formulation, appears to have relegated them to the ‘implementation’ stages of system development, in the minds of many who call themselves ‘systems engineers’.

2.2 InformationIn the author’s opinion, the best introduction to cybernetics was written in the mid-Twentieth Century by William Ross Ashby, and is entitled, not surprisingly; ‘An Introduction to Cybernetics’. This book has been out of print for many years, but it is well worth seeking it out. What follows summarises his ideas, but with nowhere near the same quality of prose.

For consistency with modern systems engineering practice I have attempted to introduce the subject using UML (Unified Modelling Language) diagrams, as these are becoming the de-facto standard, as a means of capturing basic system function. Unfortunately, it soon came apparent that cybernetic ideas cannot be expressed using them, as they are too application-oriented for the level of abstraction required.

We are told that with the internet, we have access to more information than has ever been possible in the past. Well, it all depends on how you define ‘information’. We have the basis of information when we can change the state of the environment, for example tying a knot in a handkerchief to remind us to do something. Whenever we alter our surroundings, we may ascribe meaning to the changes we introduce. We code our abstract thoughts as configurations of matter in the physical world.

The action of forces of nature on the configuration of matter may introduce changes, so that it no longer conveys the same idea, e.g. an animal may accidentally trample the stack of stones.

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No matter how we encode it any physical medium can become corrupted, there will always be noise present. We usually distinguish a valid configuration from the corrupted ones by having a limited number of valid configurations, which have a low probability of occurring by chance.

The commonest form of encoding is through speech. Of all the potential sounds and sequences of sounds the voice is capable of making, only a tiny part of the potential repertoire is used to convey messages.

Claude Shannon proved that there exists a coding scheme which will convey an error-free message regardless of the noise present in the communication channel. It is a reasonable guess that spoken language isn’t too far off this ideal encoding for the noise sources in existence when language first emerged. Whether it remains ideal in the industrial age is a moot point.

The simplest form of encoding is to ensure the signal level is higher than the noise level. Ultimately, the noise limits the difference between adjacent signal amplitudes, introducing a quantisation error, even when the signal is not digital in nature.

More typically, a set of symbols would be defined as a sequence of signal levels in time (e.g. Morse code symbols), or in space (printed characters). We know that the set of permissible sequences is limited for sequences conveying information. So a sequence of letters taken from the Latin alphabet in which the letter X is as common as the letter E, is not likely to convey English language messages.

Similarly we could check to see whether the combinations of letters formed English words.

We could repeat this process by checking the frequency of words, but ultimately all we are doing is checking that the message is valid English. We are not able, through this process, to decide whether the message is true, or indeed conveys any meaning.

It is in the information-theoretic sense described above, that we claim there is an information explosion. The reality is, searching for useful information is like picking up a weak signal on a radio receiver – turning up the volume just increases the noise. The internet has given us a spectacular increase in uninformed opinion, misinformation and utter nonsense, with valid information becoming ever more elusive. The needle is the same size as ever, but the haystack has increased by orders of magnitude.

Information theory infers that information is being conveyed if the statistical distribution of the symbols in a message is different from a uniform distribution in which all symbols are equally likely. The distribution implies the conveyance of information, without specifying its exact form. For this reason the information is characterised as a function of the probabilities of each symbol occurring, called the entropy of the message.

The entropy is defined in such a way that if a message is passed through a process the output entropy is equal to the sum of the process entropy and that of the original message. As an example, in order to improve the match between a message and a communications channel (which must have capacity to transmit every symbol with equal probability), the message may pass through some form of coder. At the other end, a decoder, matched to the coder, so by implication has same entropy, re-creates the original message. This matching of entropies illustrates a fundamental principle of cybernetics, called the principle of requisite entropy.

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A system which remains in the same state indefinitely is not particularly interesting, so cybernetics concerns itself with the evolution of system states with respect to time, or some other independent variable. The most important results can be derived using what is called a finite state machine formulation.

The machine has a number of states which we can give English language labels, or symbols (e.g; letters, pictograms, or numbers ). These are precisely the same as a set of symbols we might use to code information. We assume that the transitions between states take place instantaneously.

We describe our system as a set of symbols, and we can, over a long period of time determine the probability of the machine being in each of its states at any particular time. Thus we can characterize the machine’s evolution as an entropy, in much the same way as we can define the entropy of a messages made up of symbols taken from a set.

However, the statistical behaviour of a finite state machine introduces excessive complication, which will obscure the concepts I am trying to introduce. The most important ideas can be understood, assuming the machine to be deterministic, rather than statistical (stochastic) in nature.

Admittedly, systems which actually transition neatly between states in this manner are rarely, if ever, to be found in nature. Our primary system consideration is whether the states can in fact be reached and maintained. With a digital computer this issue is hidden in the design of the circuits, in more general systems, we cannot be certain that this will be the case.

2.3 TransformationImagine yourself in an environmentally-controlled greenhouse, and you notice the following:

The temperature falls

The carbon dioxide concentration reduces

The humidity falls

You conclude that somebody has opened the door.

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The UML state chart for this behaviour shows ‘Open door’ as the condition which causes the transitions from the states before the door was opened to some (unspecified) time later. The implication is that the disturbance caused by leaving the door open exceeds the capability of the environmental control system to maintain regulation, so that a new (unsatisfactory) equilibrium condition results.

Alternatively we may consider ‘Open door’ to be an operator, and we may denote it with an arrow . We take one of the effects, say the first, and write it as a compact expression:

AB

Where A is shorthand for ‘warm’ is the ‘operand’, the arrow implies an influence, B is shorthand for ‘cool’, is what Ross Ashby calls the ‘transition‘. The total set of transitions consists of a vector of operands which become a vector of transitions. The set of transitions is called a ‘transformation‘.

The UML state chart could represent this by defining a pair of compound states which include the individual states:

The UML description necessarily contains labels associating the diagrams with specific entities, which is fine for describing specific systems, but of little help in seeking universal principles which apply to all systems. For this reason, the transition arrow label will be omitted from the state diagrams and states will be labelled by letters, without reference to any specific system.

This example is an ‘open’ transformation, because the transitions and operands are from different sets, if the transform has the same set of symbols in the operand and transition sets, it is called a closed transformation.

All the above transforms are single valued. Other types are possible, for example:

AB or D

BA

CB or C

DD

Is not single valued. In principle, the system may start and end on any state, so we have omitted the start and end node from the state chart. The alternative is to present 16 practically identical diagrams, which seems an inefficient means of presenting a relatively simple idea.

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Another important type of transformation is ‘one-to-one‘. This is single valued, so that each operand yields a unique transition, but in addition, each transition indicates a unique operand. A transformation which is single valued, but not one-to-one is called ‘many to one’.

To complete the set of possible transforms, we have the identity which transforms the operand to itself.

The transitions in UML state diagrams usually correspond to a defined condition, rather than the natural evolution of the system over a single fixed time interval. For this reason they do not represent an efficient means of expressing system behaviour, beyond the execution of computer code.

Activity diagrams, sequence diagrams and collaboration charts, likewise are doubtlessly invaluable for defining how code should be organised and executed, but outside the realm of software engineering UML appears somewhat over-sold.

2.4 Algebraic Description

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The arguments of this and subsequent sections, are more compactly illustrated by using a matrix representation of the transitions. The first row is the start state, whilst the second row is the final state. The presence of transformations which are not single valued raises the difficulty that more than one matrix is needed. When this occurs, it is evident that more information is needed to determine the system evolution.

For one to one, or many to one transitions, we have sufficient information in our transform definition to determine how the system behaves when the transform is repeated.

We refer to Tn as the nth power of T, i.e. the result of applying T n times.

More generally, we have the product, or composition of two different transforms operating on the same set of operands:

Consider now a more complicated transformation:

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By starting at any state and following the arrows, the repeated transformation ends up cycling around a set of states indefinitely. If any state transitions back to itself, the transformation will stop once it is reached. The separate regions of the kinematic graph are known as ‘basins‘. By implication, the system either converges on to its final state or cycles around a set of states in a basin.

Unfortunately, this benign behaviour is only a consequence of the finite state machine assumption. In practice it is very likely that there will be states which the machine can reach which are not in the finite state machine model of the system. The belief that a finite state machine will be adequate outside of a controlled environment is a dangerous delusion.

As an example; consider a strategy system which has a gambit for every anticipated move the opponent may make. There is nothing to stop the opponent from making a move which was not anticipated, and for which no counter was anticipated.

The implicit assumption of the finite state machine, that the set of disturbances to which it is subjected is closed, limits its operation to extremely benign controlled environments. A system which is capable of surviving a real environment, with its open set of disturbances, presents a more challenging problem.

2.5 The Determinate Machine with InputA determinate machine is any process which behaves as a closed single-valued transformation.

By ‘single value’ we do not necessarily imply a single number, but an operand in a transformation. Each operand in the previous section was a label covering sets of attributes; a particular combination of data and/or behaviour which we called a ‘state‘. Since each state is assigned a unique label, the ‘single-valued ‘requirement implies that the machine’s transition from one state to another is strictly one-to-one.

We have defined our deterministic cybernetic machine by the transforms which characterise its behaviour. The set of operands are states, which themselves may be transforms. Thus we can define a transform U whose operands are transforms T1, T2 and T3. Applying U has the effect, for example:

T1T2

T2T2

T3T1

In addition to our requirement for single valued transformations, this discussion requires all transformations to be closed. The three transformations in U could have distinct sets of operands, in which case, U is merely a ‘big’ transform, no different in nature from its component transforms. The more interesting case is when all three component transforms have the same set of operands.

For example, we assume all three component have a common set of operands; A,B,C,D:

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The three transformations may be considered three ways in which the same machine can be made to behave. That it is the same machine is implicit in the common operands (states).

If we refer to ‘change’ in the context of such a machine, it is change in behaviour which is of interest, rather than the transformations associated with each individual behaviour. Those with programming experience will recognise the idea of a subroutine call in assembler, or a function /procedure call in higher level languages. The subscript 1,2 or 3 applied to the individual transformations indicates which behaviour is selected at any time, and will be called a ’parameter’.

A real machine, which can be described as a set of closed single-valued transformations will be called a machine with input. The parameter specifying which behaviour is selected is called the ‘input’.

2.6 Coupling SystemsSuppose a machine P is to be connected to another, R. This is possible if the parameters of R are functions of the states of P. We define the two machines by their sets of transformations. R will have three, P only one.

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P R

P R


We construct a relation, which is a single valued transformation relating the state of P to the parameter values of R. We assume also that the two machines transforms are synchronised to a common timescale. An example of the coupling relationship could be:

State of P i j k

Value of parameter 2 3 2

The two machines now form a new machine of completely determined behaviour. The set of states is equal to the number of combinations of the 3 states of P and 4 states of R, and each individual state is a vector quantity, e.g. (a,i). Taking these compound states, a kinematic graph may be drawn which has behaviour which is different from both P and R.

If machine P had more than one transform, it could accept input from R, so that state of each machine would determine the transform of the other. Again, the result is a new machine with its 12 states, and completely new behaviour. Its behaviour is determined by considering its time evolution from a start state, rather than directly from the coupling scheme.

The coupling is achieved by specifying what function the input parameters of one machine are to be of the other machines states. Evidently, the parts may be coupled in different ways to form a whole. In other words knowledge of the behaviour of the parts of a machine is inadequate to predict the behaviour of the whole. This is an important result of cybernetics, and explains the emphasis on system behaviour.

2.7 FeedbackIn the previous section P and R were joined such that P affected R but R did not affect P. In this relationship, P is said to dominate R:

In order to avoid confusion with transformations, boxes have been drawn around the machine identifiers to indicate that we are talking about relationships between machines.

The more interesting case arises when R also affects P.

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This circularity of action is frequently referred to as ‘feedback’. However, as will be shown , feedback is a particular kind of coupling between machines which has the effect of driving both to a state of equilibrium.

2.8 Immediate Effect and ReducibilityWe say a variable has an immediate effect on a transformation, if it affects the result during a single update interval. In practice, this is how transformations are characterised, and the workings of the machine deduced. A variable u which has no immediate effect on variable y, may change the value of x, which does, so that after two update steps, u appears to affect y. Other variables may influence y through longer sequences, and these will be referred to as ‘ultimate effects’. Once the set of all immediate effects for all transforms has been derived, ultimate effects may be determined. If a variable has no ultimate effect on another, the second is said to be independent of the first.

In a system consisting of a pair of machines with feedback, the parts of each may have an immediate effect on the other. The feedback may be present for ultimate effects, but as far as immediate effects are concerned, the action may be one way, in which case we deduce that the system is less richly connected internally. It is possible for the complexity of connection to be reduced until the two machines have no immediate effect on each other. This permits the behaviour of each machine to be determined independently of the other, and hence the system is termed ‘reducible’.

2.9 StabilityAs long as we are dealing with finite state machines, the worst behaviour we can expect is a cycling through states in a basin. Ross Ashby overcame this difficulty by considering that for an organism interacting with its environment, one of those states would be ‘death’, which effectively stops further cycling through states.

The problem with finite state machines when we consider self correction, is there is no representation of the process of change itself. We can trigger events to change states, and we can include a process called ‘adaptation’ amongst the available transforms, which must be triggered by events. If the only state which triggers adaptation is ‘death’, the adaptation will be too late.

In order to cover the process of adaptation, Ross Ashby cheated by introducing an additional state, triggered when the essential variables (those associated with survival) exceed certain bounds. This is just a quantisation of the ‘death’ state, implying that the quantities on which adaptation depends must be quantised to at least two levels.

Instability may be defined as the condition where the essential variables are out of bounds for a sufficiently long period for the organism to die. In a mechanical system we have upper bounds on all variables describing the system, where beams break, gear teeth strip, tyres slip, wings stall, gyros topple, measurements saturate – there are myriad and usually fairly obvious ways of defining catastrophic failure of artificial systems. Disaster is normally easily recognisable; it is preventing it from happening which is the difficult task.

This is a more general definition of instability, where the system deviates to some point of no return. It is a property of the entire system and not any particular component.

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2.10 Equilibrium in Part and WholeA property of coupled systems is that equilibrium of the whole is only possible if all the parts are in equilibrium in the conditions applied by the other parts. Effectively, each part has a veto on the equilibrium condition.

Suppose a system is composed of two parts, A and B, which are coupled together. The state of the whole is a vector made up of the state of A and the state of B. Equilibrium requires this vector to remain constant in time, which is only possible if the components remain constant in time, i.e. both A and B must be in equilibrium under the inputs from each other.

It follows that no state (of the whole) can be a state of equilibrium unless it is acceptable to every one of the component parts.

2.11 Isomorphism and HomomorphismIsomorphic means ‘similar in pattern’. A map and the countryside it represents are isomorphic. The streamlines around a faired shape form a pattern which is the same as the electric field in a non-conducting medium, about a metal object of the same shape.

In studying systems we require models which are isomorphic with the real system. In general, this is impossible, the best we can do is specify the aspects of system behaviour which interest us and construct a model which is an adequate isomorphism of that aspect. We would not use the dynamics model used to design a motor car suspension to determine the layout of the production line to produce the components. Although the same physical objects are represented, our cybernetic requirements differ.

More frequently we are concerned with a reduced set of system attributes.

What we usually use is a ‘homomorphism’ of the machine of interest, which is a machine having the behaviours of interest. This is achieved when the one-to-one mapping of the states of the isomorphism is replaced with a many-to-one mapping.

Consider the machine:

M↓ a b c d e

i b a b c a

j a b c b c

k a b b e d

l b c a e e

The bottom right box contains states e and d, whilst all other entries are a,b,c. We can imagine circumstances where the differences between states a,b,c are irrelevant to our current model of the

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system. Likewise, d and e could be treated as the same: So we can define a many to one mapping of the original five states to a machine of only two states:

N↓

g h

α g g

β g h

The parameters i and j are replaced with α, k and l with β, and the states g and h correspond to separate groupings of the original states.

Machine N is equivalent to a simplified version of M. Or, if two machines are so related that a many to one transformation can be found that, applied to one of the machines, gives a machine which is isomorphic with the other, then the simpler of the two is a homomorphism of the more complex.

The important point is that although the homomorphism is only a partial description it is complete in itself. What we are doing is associating states, which at a simpler level of representation are indistinguishable. The homomorphism is not simply an approximation, and hence simply wrong, it is correct in its representation of the features of interest.

The autopilot designer of a guided weapon may insist that every detail of construction, and every disturbance, be represented as fully as possible. The performance modeller only requires at most a second order lag representation of the airframe response, whilst the battle modeller is happy with coverage diagrams, hit probabilities and flight times. The representation at each level is derived from lower level analysis, but is complete in itself.

With modern levels of processing, the tendency is to use the autopilot designer’s model at the battle level, for fear that omission of all the extraneous detail may risk ignoring the potential for ‘emergent’ properties. If the entities concerned had been studied at the detailed level, this fear would be unfounded.

Note that our ‘high level’ system description can only arise by simplifying more detailed descriptions. To work from the ‘top level’ down requires the spontaneous creation of information, which is a direct contravention of the Second Law of Thermodynamics. Either we must abandon the idea of having no pre-conceptions, or we must characterise our system from the bottom up, before we have sufficient information to commence the ‘top down’ requirements analysis prescribed by modern systems engineering practice.

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3 Cybernetics: Variety

3.1 IntroductionIn this chapter we revisit the idea of entropy introduced earlier, but extend the ideas to consider the implications of an organism (represented as a deterministic machine) with its environment.

Ideas of the previous section considered the functioning of a deterministic machine, which once defined by configuring its inputs followed a fixed behaviour. When considering regulation and control, however, we need to consider the broader set of what it might do, when ‘might’ is given an exact definition.

We used the term ‘dynamic range’ to characterise the finite selection of potential signals, more generally we refer to a set of possibilities which the signal may adopt at any sample. For example, if the sequence is:

c,b,c,a,c,c,a,b,c,b,b,a

it contains twelve elements but only three distinct elements. In this discrete signal case, we say that the signal has a variety of 3. We can measure variety either as the number of options, or as the logarithm to base 2 of the variety, in which case it is measured in bits.

3.2 ConstraintAn important concept in cybernetics is that of constraint. It is a relation between two sets, and occurs when the variety that exists under one condition is less than exists under another. In an all-male gentleman’s club (an anachronism in the 21st Century) the variety in the sexes represented has dropped from the 1 bit of the human race to 0, indicating that a constraint is present.

Laws of Nature impose constraints on potential observations; we do not expect a planet to spontaneously skip to the opposite side of the Solar System within a few seconds.

The World is therefore extremely rich in constraints. Indeed, the whole concept of a deterministic machine, comes from recognising that sequences of behaviours show a particular form of constraint. Were the machine output to show no constraint, the observer would conclude that the device was something akin to a roulette wheel, and intrinsically unpredictable.

The effect of a machine on variety can be illustrated with the example transformation:

A→B

B→C

C→C

The operand may take on three values, whilst the transform has only two, the variety drops from 3 to 2 on applying the machine to a sequence of operands. Evidently, no loss of variety implies a one-to-one transformation. The machine, by applying a constraint, reduces variety.

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The variety in a machine cannot increase, although it usually decreases, but what happens if we have a machine with input?

Since behaviour depends both on the state as well as the input parameter, we shall consider a set of machines, all subjected to the same input but differing regarding their start state. If we observe the variety of states over the set of machines, we shall see it fall to some minimum. When the variety has reached its minimum the input parameter is changed. The change will change the machine’s graph from one form to another, as (for a machine having states A …F:

From: A C to: A→B C

↓ ↓ ↑ ↑

D ← B E ← F D→ E F

Under the first input, all machines started at A,B or D would go to D, and those started at C,E or F would go to E. The variety after some time would fall to 2 states. When the input is changed, all machines would then end up at B. The second input causes a further fall in variety. Changing the parameter value to a set of identical machines cannot increase the set’s variety. The continued application of inputs also reduces the dependency of the final state on the initial conditions. This is what Ross-Ashby refers to as the law of experience.

Our initial considerations of variety were concerned with efficient coding of messages to make maximum use of channel capacity and/or storage, in this chapter we will consider encoding as the effect the machine has on a message of known variety.

‘Encoding’ as used here refers to the transformation of a message to render it compatible with the medium. The message could use sound waves in the air, electrical impulses in a wire, positions of the arms of a semaphore, the smoke of a fire, reflections of the Sun off a mirror, - the list is practically endless. The physical means varies, yet something is preserved which does not depend on the material processes involved.

The operation of the coding is to transform the messages Mi to their corresponding codes C. It is a transformation process, and may therefore be represented as a deterministic machine. However, we note that all messages undergo a one-to-one transformation, so that the machine preserves variety. Indeed, if the coding were multi-stage, each stage must preserve variety if the original message is to be recoverable.

Assume we have a machine with input that can be in some state S i, where the number of states is finite. It also has a set of parameters that can take, at each moment, one of the values P j each of which determines a transformation in the S’s.

For simplicity, suppose the machine M can take four states: A,B,C and D; and the parameters can adopt three values: Q,R and S.

M↓ A B C D

Q C C A B

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R A C B B

S B D C D

The complete behaviour of the machine is presented in the above table.

If the machine starts in state B, and the input follows the sequence R Q R S S Q R R Q S R, the output will follow B C A A B D B C B C C B. There is no difficulty, given the machine and its initial state, in deducing the output sequence from the input sequence.

Having coded the sequence the next issue is whether it can be decoded. The shortest message is a sequence of two values, of which there are nine possibilities:

QQ, QR, QS, RQ, RR, RS, SQ, SR, SS

Starting with the machine in state A, we have as output:

CA, CB, CC, AC, AA, AB, BC, BC, BB

The sequence BC repeats, so that there are only 8 possibilities at the output. Starting at B:

CA, CB, CC, CA, …

Here CA repeats.

With C as the starting point:

AC, AA, AB, BC, BC, …

Again BC repeats. Starting with D:

BC, BC,…

The output sequence in all cases has lower variety than the input sequence, it follows that the inverse to the encoder does not exist.

It is easy to see that if, in each column of the table, every state had been different then every transition would have indicated a unique value of the parameter; so we would be able to decode any sequence of states emitted by the machine. The converse is also true; if we can decode any sequence of states each transition must determine a unique value of the parameter, and thus the states in the columns must all be different. This is the requirement for a perfect coder.

Given that we have designed the coder according to the above principles, how do we design a machine to perform the decoding? i.e. how do we recover the original sequence?

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We note that it is the transition which yields information about the original input, and not the actual value of the coder’s output. We want a machine which accepts transitions as input, but this involves two states which do not exist at the same sample interval. Consider the machine with input (D )represented by the above table:

If the start state is r, the input sequence Q S S R Q S becomes: r q s s r q s, which is the same as the input delayed by a single sample interval. A number of such machines will delay the input for as long as is required.

The new coder has the vices of the previous one corrected, with only single value transformations:

The inverter is the transformation:

The input sequence to the decoder is the vector

(state of delayer, state of coder)

By introducing a delayer, which enables transitions to be detected, provided the coder does not lose variety, an automatic inverter can always be built.

We could imagine the input to the coder to originate from a deterministic machine. The variety of the input cannot exceed 3 regardless of the structure of the machine. The coder itself has 4 states, hence a variety of 4, so the variety at the output from the coder is 12. Without the delay, the variety

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of the output stream is only 4, hence it is impossible to reconstruct the input signal. With the delay, the variety of the input to the decoder is 12, and a potential solution exists.

More generally, the output from a sequence of machines cannot exceed the product of the varieties of each, or expressed in logarithmic terms the output variety cannot exceed the sum of the two varieties. We note however, that there are 16 potential pairs of four letters, but not all are selected (AC, for example). Knowing the quantity of variety of the two processes is insufficient to predict how they will actually change.

Another way of considering the problem is to think of the coder as a communication channel sending the input message; it cannot transmit a variety greater than 4 at each sample time, but can potentially transmit a variety of 16 in two sample times. Expressed logarithmically, the capacity of the coder, treating it as a channel is log24 = 2 bits per sample. In general:

C=log2N

Where N is the number of distinct values the channel can take.

3.3 Incessant TransmissionWhilst the finite state deterministic machine is useful for illustrating basic principles, they settle down in a relatively short time and are not representative of the more general interchange of information between machines or between an organism and its environment. We must consider a more comprehensive form of machine and transformation – the non-determinate.

Instead of transforming to a particular new state, we may go to some one of the possible states, the selection of the particular state being made by a process that gives each state a constant probability. Such a transformation, and the set of trajectories it produces, is called ‘stochastic’.

We can express a deterministic single value transformation by its individual transitions:

A→B

B→A

C→C

Another way of representing it is as a matrix of probabilities:

↓ A B C

A 0 1 0

B 1 0 0

C 0 0 1

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Since the transformation is deterministic, the probabilities have values 0 or 1. The more general case deals with transition probabilities:

↓ A B C

A 0 0.9 0.1

B 0.9 0 0

C 0.1 0.1 0.9

The protocol to construct the table of transition probabilities is the same as for the deterministic case - the number of times each transition takes place is recorded and each tally is divided by the total number of transitions from each state.

The resulting sequence of states is known as a Markov chain.

The transition matrix probabilities show that some form of constraint is acting, because all outcomes are not equally likely. An unconstrained stochastic process is one for which all transitions are equally likely, so that all elements of the transition matrix are equal.

3.4 EntropyThe discussion so far has used the word ‘variety’ to describe the diversity of choice of a symbol set, where the symbol set corresponds to a signal over a continuous domain, we used the term ‘dynamic range’. But the variety is insufficient to describe the propagation of information.

Consider the communication with a prisoner in a cell, an accomplice does not have direct access, but may provide him with a cup of coffee via the warden. Now the coffee presents many opportunities for coded communication, shape of cup, black or white, hot or cold, with or without sugar. So to prevent the possibility of a message being concealed behind this apparently charitable act, if it is black, the warden will add milk, similarly will add sugar and will not hand it to the prisoner until it is cold. Furthermore, he will transfer the contents of the proffered cup to a standard mug from the canteen, and will tell the prisoner that he added the milk and sugar.

The state of the coffee is always the same one of the possible options, but because the variability is cut to one, no information can be exchanged.

In our coding example, a decoder could not be constructed considering the output signals individually, information about the input stream required consecutive samples to be considered. The extra information added to the input stream by the coder was the internal working of the coder, decoding required this information to be removed from the output sequence to reveal the input sequence. More importantly, whatever information was conveyed in the original input was conveyed with a different symbol set, which only needed at least the same variety as the input set. If the symbol set had lower variety, it would be impossible to avoid repeating symbols in the

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columns of the coder’s transition matrix, so a perfect coder would be impossible. This is an example of the Law of Requisite Variety, which will be covered in detail in later sections.

Information then does not depend on the symbols used, nor is it specific to the physical medium used to convey it. It is evident that more can be conveyed per unit time if the variety of the symbol set is large. A symbol set with only one entry cannot convey information. This is the fundamental reason for the logarithmic measure of variety; using it, the single symbol had zero variety.

Also the minimum amount of information which can be conveyed is when the differences between any two signals is not obscured by noise. Since we are always, in one form or another comparing two entities; noise and signal, it is reasonable to employ a logarithm to base 2 and deal with information in bits.

Variety characterises the potential of a symbol set to convey information, yet actual messages made up of the symbols are unlikely to use all of them. A message consisting of the same symbol repeated indefinitely has no variety at all, yet a message which uses a different symbol from the set at each sample has the same variety as the symbol set itself.

Information is a property of the diversity of symbols in a sequence. If we are dealing with a Markov chain, we can characterise the symbol set by the probability of each symbol occurring.

Suppose we have two distinct messages a,b, with probabilities of occurrence pa, and pb. The options are mutually exclusive as only one symbol may be selected at a time. Also:

pa+pb=1

If we consider a channel which has N potential levels and partition it into two intervals say 0 to n and n to N, and draw samples from a uniform random sequence of width N, whenever the sequence is above n, we add it to b, if below, we add it to a. In this way we ensure the two messages are distinct:

pa=nN, pb=

N−nN

Note: we could have partitioned N any way we choose provided the total intervals corresponding to each message are proportional to the probabilities. Since each message uses a reduced set of the potential symbols it cannot have the complete variety of the symbol set. The variety of a is:

log 2n= log2 pa+ log2N

The variety conveyed by both messages is, therefore:

V=pa log2 pa+ pb log2 pb+( pa+ pb) log2N

Or:

V=pa log2 pa+ pb log2 pb+ log2N

If the channel has 2 symbols each drawn with equal probability:

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V=−0 .5 log22−0 .5 log22+ log22=0

If one symbol is repeated indefinitely, this formulation would put N=1, and again the variety would be zero. The variety which we have calculated is that of the message plus that of the channel, much as we did in the coder example. The variety of the channel is log2N, regardless of the value of N, so the variety of the set of messages which can be conveyed is:

V= pa log2 pa+ pb log2 pb

The above reasoning may be extended to any size symbol set, also the negative variety obtained from taking logarithms of fractions introduces an inconvenient minus sign. Finally, it is normal to use the symbol H to denote the variety of a stochastic signal:

H=−∑ p i log2 pi

In this form, it is called ‘entropy’, which has the merit of fewer syllables than ‘stochastic variety’, but tends to be confused with the entropy of thermodynamics. There is a relationship between the two entropies but it lies well outside the scope of the current discussion.

In the interests of clarity, the deterministic counterpart of entropy i.e. ‘variety’ will be used as far as possible.

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4 Cybernetics: Regulation

4.1 Requisite Variety Regulation is the process of ensuring that the states of a system remain within permissible bounds, as the organism interacts with its environment. I.e, states such as ‘death’ are to be avoided.

The action of the regulator could be considered the resolution of a conflict between the environment and the organism which we hope the regulator wins. With that in mind, consider a game in which one player, R (regulator) is trying to reach an outcome ‘a’, and the other player, D (disturbance) is trying to prevent it.

D moves first, playing gambit 1,2 or 3, and then R responds by playing either of α, β or γ. The outcomes for each pair of gambits is summarised in the following table:

α β γ

1 b a c

2 a c b

3 c b a

In this particular case, R can always win, as each row contains an option whose outcome will be ‘a’.

If R acts according to the transformation:

1→β

2→α

3→γ

The outcome will always be ‘a’.

A less favourable game is presented in the following:

α β γ δ

1 b d a a

2 a d a d

3 d a a a

4 d b a b

5 d a b d

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Evidently, if ‘a’ is the target, R can always win. If ‘b’ is the target, R can only win if D plays 1,4 or 5. If the target is ‘c’, R’s position is hopeless.

In order to restrict the consideration to the non-trivial, we shall consider only cases where R must make a move in response to D, i.e. there will be no repeated entries in the game matrix columns, enabling R simply to repeat the previous move. Thus each move of D must be countered by a move by R. As an example of such a game consider the following table:

α β γ

1 f f k

2 k e f

3 m k a

4 b b b

5 c q c

6 h h m

7 j d d

8 a p j

9 l n h

R’s set of responses to each of D’s moves converts the game into a transformation. This is single-valued because R may only make one move at a time. An example might be:

1→γ

2→α

3→β

:

9→α

This transform uniquely specifies the outcome:

1→k

2→k

3→k

:

9→l

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We see that the variety of the set of outcomes cannot be less than D’s variety minus R’s variety

If R always plays the same move, the outcome will have variety equal to the variety in D’s moves. If R uses two moves, the outcomes can be reduced to half the potential outcomes, if three moves, to one third, and so on.

Therefore, if the variety is to be reduced to a specified level, R’s variety must be increased to at least the appropriate minimum. Expressing the varieties logarithmically, we have for the variety of the outcome Vo:

V o≥V D−V R

Where the subscripts denote D and R.

This is the Law of Requisite Variety.

The variety of a regulator must be at least equal to that of the disturbance if the variety of the essential variables is to be kept at zero.

This can be expressed, for stochastic processes, in terms of entropies.

H (E )≥H (D )+H D (R )−H (R )

Where HD(R) is the entropy in R, given D. If the relationship between R and D is deterministic, HD(R)=0, and the expression is the same as above, but in terms of entropy. More typically, D and R play simultaneously, and the strategy consists of selecting the probabilities of a random play which on average achieves R’s requirements.. The entropy of the outcome cannot be less than the difference in entropy of the disturbance and the regulation. In this more general case, we could refer to the Law of Requisite Entropy.

Except in the most benign of environments, an organism will be subjected to disturbances which will threaten, if it fails to perform some regulatory function, to drive the essential variables outside their proper range of values. If regulation is to be successful, it must be related to the disturbance via a relationship akin to the games described above, which summarise the outcome of the interaction of the regulator and the disturbance.

Perfect regulation matches the organism’s entropy to that of the environment, resulting in zero entropy. The less perfect the regulation, the more entropy the system will exhibit.

There is a widespread myth that complexity of behaviour increases with system complexity ( i.e. component count). The principle of requisite entropy shows this to be complete nonsense. We do not add servos and guidance system to an unguided rocket to render its behaviour less predictable, we do so to make it more predictable, to the extent that it hits the target.

4.2 Error Controlled RegulationDesigning a controller to deal with a closed set of disturbances is simply a matter of listing the disturbances and devising strategies to deal with them, as and when they were detected. Life would indeed be simple if the environment presented us with a closed set of disturbances.

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Real environments present us with open sets of disturbances, so our organism must adapt to changing circumstances or die. The Law of Requisite Entropy provides us with the clue as to how this can be achieved, and indeed how it has been achieved in Nature.

The organism may not have sufficient a priori knowledge of the environment to construct a strategy from first principles, but it will know whether its current strategy works or not. By acting on the degree of success or failure of a trial strategy, the organism may survive environmental change, perhaps not as an individual, but as a species, depending on how effective the process of formulating new strategies is.

Adaptation to an open environment requires the ability to detect when the system entropy is reduced near zero. What is needed is a null-seeking system. Feedback is the adjustment of the organism’s entropy based on the difference between the desired system entropy and its actual value.

Ross Ashby demonstrated that adaptation by random selection would work if there were few changes within in given timescale. The ability to handle ever greater numbers of changes within a fixed timescale forms the basis of an eclectic measure of intelligence.

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5 Practical Finite State Machine Control

5.1 IntroductionThe earliest automata devised by human beings were essentially finite state machines implemented in clockwork. A toy music box, steam engine valve gear, fairground organs, battleship gun loaders, capstan lathes, the Jacquard loom, to mention but a few, are cases where human effort was replaced by a mechanical system for performing highly repetitive, although sometimes very complex, tasks.

Over the years, cams, gears and mechanical links have been replaced by electromagnetic relays and finally by digital processors. The convenience and flexibility of the digital processor, over the limitations of earlier technologies has led to a significant increase in the amount of this type of automation.

From the principle of requisite variety, such systems can only operate in a controlled environment, where the set of disturbances is more or less closed. Such environments are commonplace – they are called factories.

The matrix description of the finite state machine used to explain the fundamentals of cybernetics in the previous three chapters is too abstract to be of practical value in this context.

As we shall see, the UML/high level language (such as C) of the software engineer is not the preferred approach either. To understand why this is so, we need to know a little about a typical industrial programmable logic controller (PLC). However, the method applies to all finite state machine controllers, regardless of their physical implementation, otherwise it would have no place in this book.

5.2 The Relay and Logic FunctionsA relay is a device which deals with all-or nothing signals. In its most abstract form it is a device in a low power consumption circuit (electrical, compressed air hydraulic, or just as a software entity), which when activated, switches the flow in a high power circuit on and off. An electrical relay consists of a coil of wire on an iron core which attracts an armature when the power is on, causing contacts to separate or meet. How it is implemented is of no concern. The relay, whether implemented as software or hardware, is the fundamental building block of this type of system.

The convention for representing relays is presented in Figure 5-1. A pair of normally open contacts is represented as a pair of spaced vertical lines, similar to the conventional representation of a capacitor. A pair of normally closed contacts is the same but with a diagonal line drawn through it. The controlling ‘coil’ is drawn as a circle. The contacts are associated with a specific coil by giving them the same letter identifier. For example, when coil A is activated the normally open contacts A close and the normally closed contacts A open.

Using this representation, we can implement the three logic functions, AND, OR and NOT, from which all logic functions can be derived.

39


Figure 5-1: Basic Symbol Set

The NOT function is a set of normally closed contacts, so that when the coil is energised by the input, the circuit opens. AND consists of wiring the normally open contacts from each input in series, and OR is obtained by connecting them in parallel. Thinking in terms of opening and closing contacts is a very natural way to think of circuit function, but by identifying the serial and parallel paths through the control circuit it is a simple matter to map them into logic functions which once would have been actually implemented using logic integrated circuits.

Figure 5-2 : Logic Symbol Equivalents

40


5.2.1 Example : Capacitor Discharge UnitA model railway is required to operate the points automatically on detecting an approaching train. The points motor consists of a solenoid which is usually energised manually by applying a currently transiently via a switch. This provides an impulse which causes the points to move.

The part of the problem which concerns us is when the train is approaching from a branch, with the points normally set for free movement along the straight. A detector of some kind (e.g. a reed switch in the track which detects a magnet under the train) is triggered by the approaching train. Evidently, if the points do not change, there will be an accident.

Figure 5-3: Model Railway Example

Suppose we have a capacitor with sufficient charge to operate the solenoid which changes the points. Detecting the approaching train could then close a pair of contacts causing the capacitor to discharge through the solenoid. That would be fine, apart from the fact that once the train passes the detector the contacts will open again, and there might not be enough energy in the partial discharge to change the points.

What we need is a means of keeping the contacts closed until the capacitor is fully discharged.

41


The simplest means of achieving this is to place normally open contacts in parallel with the switch. One the relay is triggered it will close its own circuit and remain on even though the initial trigger is removed. This is called a latch circuit.

The implicit assumption in UML state diagrams is that this latching action is always present. This would be the case for digital computers which invariably have latched outputs, but a systems approach cannot pre-judge the hardware solution.

The latch causes the solenoid to apply its full impulse, ensuring the points change.

The train traverses the junction, requiring the points to be returned to their initial state.

This is achieved with a second detector and latching relay which closes the points.

That seems OK except for two potential problems – how did the capacitors get charged in the first place? Also, leaving the system with both capacitors discharged and the points set straight does not help the next train coming from the branch.

42


Evidently, when we latch one relay, we must unlatch the other, so there will be a normally closed contact pair driven by the other relay in series with the contacts causing the latching. This returns the system to its original state when the train passes the second detector.

The resulting circuit is called an SR – latch (for set-reset). The same circuit expressed as logic functions is as shown in Figure 5-4. This may look neater, but we would have to write out the truth table to convince ourselves that it would work, whilst the relay-based representation is more readily understood.

Figure 5-4: Implementation of SR-Latch as Logic Functions

We see that the SR latch looks like two rungs of a ladder between the power line and the earth. More complex systems have many rungs running concurrently, hence this system characterisation has become known as ‘ladder logic’.

The final issue is that of charging the capacitors Each relay requires a ‘normally closed’ set of contacts to connect its capacitor to the power supply. In practice, the capacitor cannot be charged

43


directly from the power supply, as the charging current would initially be very high, sufficient to damage whatever is used to implement the ‘normally closed’ contacts. A ballast resistor would be included in this path to limit the charging current, but this has been omitted for clarity.

We haven’t finished yet. Suppose a train on the straight track passes the detector. It will cause the capacitor to discharge and latch the relay in the discharge position. Worse still, it will inhibit the latching of the relay on the ‘branch’ side. In order to avoid this, further sensors are needed to detect the presence of the train in the straight section, which inhibit the operation of the discharge unit for the time it takes to traverse the junction.

The resulting system is by no means robust, as it relies on its own correct operation to determine its state; it is control by decree. It is preferable to sense the position of the points explicitly, so that the system more closely resembles a feedback controller rather than a set of open loop controllers in close formation.

5.2.2 Reversing a DC MotorOperating a DC motor seems a similar problem to the capacitor discharge unit problem with forward and reverse push buttons operating an SR latch. It is evident from the ladder diagram that there is potential for a direct short circuit across the power supply.

No switching process is instantaneous and there remains a possibility of a transient as the power supply is briefly shorted out. This might be sufficient time to weld relay contacts or blow power transistors, but also may introduce a spurious spike into other control lines of the system, causing spurious switching and unpredictable behaviour. In practice we should expect the power supply to be switched out during the polarity reversal.

In fact, if the motor is coupled to a high inertia load, reversing the polarity of the applied voltage may give rise to very high currents until it has spun down. Delays may be included to give the motor time to slow down before the voltage is reversed.

44


5.2.3 Alarm CircuitIt would appear reasonable to design an alarm circuit with a number of activating switches in parallel, all connected to the warning device (e.g. a siren). This is not as sensible as it first appears because it relies on the connection between the switches and the siren remaining intact. However, the most likely mode of failure is expected to be an open circuit.

A break in the wire could lead to one of the alarm triggers failing, or if the wire connecting the common wire to the siren is broken, none of the triggers will work.

It is important to ensure that as far as possible the system should be ‘fail safe’, and relying on the integrity of the cabling is bad practice. A better design would connect the trigger contacts as normally closed in series providing power to a relay whose normally closed contacts are in series

45


with the alarm. If any of the triggers breaks the circuit, or if there is an accidental break, the relay will switch off and the contacts will close, setting off the siren.

5.3 ConclusionsThe examples presented illustrate just how inadequate a system description based on logic alone can be. The types of fault which arise from failure to take real component behaviour into account are usually show-stoppers, and not mere ‘implementation details’ as is too often claimed.

The logic representation used in UML assumes an ideal behaviour of the hardware which is not consistent with the real world. Divergences from the ideal behaviour often give rise to system critical issues which are not readily identifiable from the state charts. The desired logic is seldom a source of problems in real systems. Difficulties typically arise because the mathematical logic is only an approximation to the real world, and a formulation is needed which can take account of this discrepancy between ideal and actual behaviour.

Even for finite state machine controllers, it is necessary to have a formulation which enables the effects of real world hardware to be taken into account, and mitigating measures taken. For this reason all commercial programmable logic controllers (PLCs) are usually programmed using ladder logic, with software to convert the pictorial system description into code.

Evidently we must take real hardware behaviour into account, even when our system is nominally a finite state machine. How much less can we afford to ignore the intrinsic behaviour of less forgiving systems.

46


6 Background Calculus

6.1 Basic Ideas

6.1.1 Introduction Dealing with complex environments with open sets of disturbances is a more daunting prospect than controlling systems which are amenable to simple logic, and we need to extend our mathematical tool set accordingly.

The engineering temperament is characterised by a pragmatic view of the world. Motivation to learn is predicated on an awareness of the potential usefulness of the material. This introduces the difficulty that the most useful techniques tend to have obscure beginnings, which may test the patience of the natural engineer.

At the same time, the engineer is expected to have a certain curiosity as to why the world is the way it is. For example, rather than view the bizarre contraptions which emerged during the infancy of a new technology (e.g. early attempts at aircraft or motor cars) merely as objects of mirth, the engineer will consider why the particular solutions were adopted, and what their merits were.

Figure 6-5: We Calculate Area by Tesselating the Plane with Squares

At the elementary level, we are motivated largely by the handling of money to learn arithmetic. Also we can see a potential use for algebra in calculation of areas and volumes. These enable us to estimate quantities of paint, numbers of floor tiles or whether an object will float. So it is from the practical question of finding areas that we begin our discussion here.

47


6.1.2 Areas Bounded by Straight LinesI expect practically every reader to know all the results in this section, but there is method in my madness, so I hope you will not feel too insulted in being presented with elementary results which you might consider beneath your dignity. In my experience there is no shortage of individuals who can use results, but I am frequently shocked to discover how rarely the underlying principles are actually understood. Being able to manipulate algebra does not in itself constitute understanding.

My intention is to introduce the idea of extrapolating from a finite set size to an infinite set, right from the beginning.

I have included some revision boxes in this section, which are essential background to what follows.

When presented with a physical quantity, we seek a unit to measure it in. When it comes to area, we begin with plane surfaces, so we base our unit on a shape which will tessellate (tile) a plane surface. The candidates are triangles, squares or hexagons. Conventionally, we use the square. The reader is welcome to derive the basic ideas from equilateral triangles or regular hexagons – it could be interesting and amusing.

We can only talk about curved surfaces once we consider vanishingly small plane surfaces, which we shall do presently.

48


Figure 6-6: If the Edge is not a Multiple of the Unit, the number of squares becomes large, so multiply length by breadth to get the same answer

In order to demonstrate the (almost trivial) result that the area of a rectangle is equal to length times breadth, we had to show that the formula may legitimately be applied to real numbers, whilst our basic definition of the unit of area applies only to integers.

If we continued counting squares the problem would become ever more tedious as we are forced to take smaller squares. In the extreme, if the sides were irrational multiples of the unit length, there would be an infinite number of vanishingly small squares. However, by multiplying length by breadth we can find the area without such potential difficulties. This process of finding a way to sum up an infinite number of infinitesimal pieces is, in a nut shell, what calculus is all about.

49


Figure 6-7: Once We Can Deal with an Infinte number of Infinitesmal Squares we are no longer limited to Rectangles.

The areas of parallelograms and triangles follow from the area of a rectangle, from elementary geometry, using congruent triangles.

We can show the same result by what appears to be a laborious method, but it is one which can be extended to less tractable shapes.

One result, which is used frequently in the following sections is the fact that:

x0=1

Regardless of the value of x.

To appreciate this, apply the rule that we add indices when we multiply:

x0=xn x−n= xn

xn=1

Regardless of the values of x or n.

Also, all angles are in radians. There is something natural in defining the unit of angle as the rotation of a wheel which rolls a distance equal to its radius, as opposed to dividing a circle arbitrarily into 360 divisions.

50


Figure 6-8: Define Angles as the Arc Length Divided by the Radius, or the Distance Rolled Divided by the Radius

6.1.3 Coordinate GeometryI assume that the reader is familiar with the idea that a straight line may be represented by the formula:

y=mx+c

Where any vertical coordinate y may be calculated from the horizontal coordinate x.

Figure 6-9 : Equation of a Straight LIne

The line has slope ‘m’ and intercept ‘c’ on the vertical axis.

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Using this representation, we can calculate the area of a triangle ABC by calculating the areas under the line segments AC and CB, and subtracting the area under AB.

Figure 6-10 : Define a Triangle by the Coordinates of its Corners

The coordinate geometry approach is extended to other geometric shapes, which would give us the potential to calculate the areas of figures bounded by curved edges.

6.2 AreasThe calculation requires us to calculate the area below the curve for the set of curves forming the edges of the figure. Hence the problem reduces to one of calculating the area under the curve.

52


Figure 6-11: AS the Number of Strips Becomes Infinite, this Approximation to the Area becomes Exact

The general straight line:

y=mx+c

Intersects the x axis (y=0) at –c/m.

We shall consider the area under the line from this point to x=a. The height of the triangle at this point is:

y=ma+c

The area under the curve is known for this simple case; it should be:

12 (a+ cm )(ma+c )=m

2 (a+ cm )2

Rather than apply the formula, we divide the figure into rectangular strips, each of width w. Each strip has area:

S=wy

An approximation to the area is obtained by adding up the areas of the strips, the more, and narrower the strips, the more accurate will be the estimate of area.

Numbering the strips 1 to n, the x coordinate of the strip number i is:

x i=iw−cm

The height of this strip is:

y=miw+c−c=miw

The area of this strip is:

Si=miw2

The area under the line is found by adding all n strips:

A=mw2∑1

n

i

(Read the Greek symbol, sigma ∑ as the ‘sum from the lower value to the upper value’, in this case from 1 to n)

In other words, we can find the area if we know the sum of the series;

1+2+3+...+n

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This is a familiar arithmetic series. The sum is found by writing the series backwards and summing with the original sequence:

2∑1

n

i=(n+1 )+(n−1+2 )+ (n−2+3 )⋯+ (n+1 )

Collecting up terms:

∑1

n

i=n(n+1 )2

The area becomes:

A=n(n+1 )2

mw2=n (n+1 )2n2 (a+ cm )

2

This may be written:

A=m2 (a+ cm )

2

+ m2n2 (a+ cm)

2

As n becomes larger the second term on the right hand side becomes smaller. As n becomes infinite this term becomes identically zero, and we are left with the correct equation for the area under the straight line.

This approach allows us to consider what will happen when the number of strips becomes infinite.

If the bounding edge is a curve, rather than the straight line, the method remains the same, we end up with a term in which the number of strips appears in both numerator and denominator, and other terms which become zero as n becomes infinite.

6.2.1 Curved BoundaryAs an example of a curved boundary, consider a contour governed by the equation:

y=x2

Before proceeding, consider the simpler case of a straight line through the origin, bounded by the vertical line x=z. The equation is:

y=x

As before, the width of the strips is:

w= zn

The height of the ith strip is:

y i=iw

The area is:

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Si=i w2= in2z2

Summing over the n strips as before yields an area:

A=12z2

An even simpler case is the rectangular region under the horizontal line y=1, between the vertical lines x=0 and x=z.

The area is, by inspection:

A=z

Figure 6-12 : We can Approximate Any Curve by an Enormous Number of Tiny Strips

The new curve:

y=x2

is called a parabola. Following exactly the same procedure as for the straight line, we find the area of the ith strip as:

55


Si=i2w3= i

2

n3 z3

The problem reduces to finding the sum of the squares of integers, then neglecting the terms which become ever smaller as the number of strips increases to infinity.

We notice that the sum of n terms, each equal to 1 is equal to n, and the sum of the integers up to n is given by n(n+1)/2. These contain powers of n which are one greater than the power to which each of the terms is raised (in the first case the terms are raised to the power 0, resulting in n raised to the power of 1, in the second case, the terms are raised to the power 1, resulting in a formula with terms in n2).

Whatever the formula for the sum of the squares of integers, it cannot include powers of n greater than 3. If it did the estimate of the area would increase without bounds as the number of strips increased, which is clearly ridiculous, as the actual area is clearly finite.

The same may be said of higher order curves, for y=xm we need to sum im, and the area of each strip contains a term 1/nm+1. Consequently the formula for the sum of integers raised to the mth power cannot contain terms higher than m+1.

Using this as a clue, we deduce that the formula for the sum of the squares of integers will not contain terms of higher order than n3.

We guess the solution is of the form:

∑0

n

i2=an3+bn2+cn+d

Where a,b and c and d are constants. The series is from zero to n, which, since the first term is zero, is exactly the same as running the series from 1 to n, except that if n=0, the series must sum to 0, so d must be zero.

If this formula is correct for a series of length n, it will be true for a series of length n+1, and also for one of length n-1.

For a single term:

a+b+c=1

For a sequence of length 2, we have:

8a+4b+2c=5

For a sequence of length 3, we have:

27a+9b+3c=14

These three equations may be solved for a,b and c, yielding an expression for the sum of the squares of integers:

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a=13, b=1

2, c=1

6

As far as finding the sum of an infinite number of strips is concerned, only ‘a’ is relevant, because the terms in n2 and n tend to zero when divided by n3.

6.2.2 Mathematical InductionWe have found a formula, which we know works for n=0 to n=3, but there is no reason to believe it will work for any other value of n.

In order to demonstrate that the formula is correct, we show that it works for a sequence of length n, and for one of length n+1. If the difference in these sums is equal to the (n+1)th term of our series, i.e (n+1)2, we can say that for arbitrary n the formula applies also to n+1, so if it holds true for, say, n=1, 2 or 3, it will be true for all n.

If the formula is correct; the expression:

13((n+1 )3−n3 )+ 1

2( (n+1 )2−n2 )+ 1

6 ( (n+1 )−n )

Should be equal to

(n+1 )2=n2+2n+1

Evaluating the expression shows that the two are, in fact equal, so we can conclude that the sum of the squares of the integers is given by:

∑1

n

i2=n3

3+ n

2

2+ n6

From which it follows that the area under the parabola is:

A= z3

3

This is one-third the area of the rectangle containing it.

6.2.3 Higher Order PolynomialsWe could follow the same procedure for curves of the form:

y=x p

Where p is a positive integer.

If we summarise the results so far, as in the following table, a pattern is becoming apparent. It seems for this small sample at least:

A= z p+1

p+1

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P y A

0 1 z

1 x z2/2

2 x2 z3/3

We have seen that the process of finding the area under the curve amounts to summing the integers raised to the appropriate powers. The formulae for the sum contains terms of order one greater than the power of the terms of the series, this is the reason why the index in the expression for the area is one higher than that for the formula for the curve. The general formula for the sum takes the form:

∑0

n

ip=anp+1+bnp+c np−1⋯

Also:

∑0

n+1

i p=a (n+1 )p+ 1+b (n+1 )p+c (n+1 )p−1⋯

The difference between these is (n+1)p. Taking the difference term by term, we have:

For the term having ‘a’ as coefficient:

a ( (n+1 )p+1−np+1 )

The first term will contain n multiplied by itself p+1 times, i.e. np+1, all the other terms will contain lower powers on n, so the term in np+1 cancels in the expansion.

By the same reasoning, the term in np cancels in the expression having b as coefficient, so that the remainder of the expressions are polynomials of degree np-1 or lower. The polynomial multiplying ‘a’ contains terms of up to np and is the only one containing terms of this order.

We need to know what the second term in the expansion of a polynomial is in order to determine the coefficient of np.

In order to proceed, we need to be able to calculate the coefficients of a polynomial when it is expanded term by term.

Consider a few low order polynomial expansions:

(n+1 )0=n0+0×1

(n+1 )1=n1+1×1

(n+1 )2=n2+2n+1

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(n+1 )3=n3+3n2+3n+1

It looks as if the coefficient on nx-1 in the expansion of (n+1)x is just x. Let’s see if this is true for any value of x.

Expanding (n+1)x we know that there can only be one term in nx. We wish to know how many terms there will be in nx-1. We may factorise the expansion, assuming the coefficient of the second term in the expansion is equal to the degree of the polynomial:

(n+1 ) (nx−1+ ( x−1 )nx−2⋯ )=nx+ ( x−1 )nx−1+nx−1⋯

The coefficient of nx-1 is therefore equal to x. So if our hypothesis is true for x, it is also true for x-1. As it is true for x=0,1,2 and 3 it must be true for all values of x.

So it follows that the actual term in np is (p+1)np. All other coefficients are associated with polynomials in n of order less than p.

The (n+1)th term, contains the term np, so that by equating the coefficients of np on both sides of the equation, we have, for a curve defined by xp, the value of the only significant coefficient in the calculation of the area is:

a= 1p+1

So for any curve defined by a polynomial consisting of positive integer powers of x, we have no need to run through the rigmarole of finding the sums of infinite series, we just apply the relationship:

x p→ z p+1

p+1

Now graphs may be used to present data other than the shape of a physical boundary. Practically anything can be represented as a graph, and the area under the graph usually has great significance. The discussion was started with the simple application to physical areas to provide an early indication that this most certainly is not ‘useless knowledge’.

This process of finding areas under curves by summing an infinite number of pieces (in this case vertical strips) is given the name ‘integration’. The conventional way of representing it is by means of the old English s (for sum). The result is called the ‘integral’ of the argument, and is written:

∫0

z

x pdx= zp+1

p+1

The ‘dx’ is equivalent to the width of the strip, w, as the number of strips becomes infinite.

(say’ the integral from zero to z of xp dx’).

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6.2.4 The CircleBy far the commonest curve we are likely to meet is the circle, which unfortunately cannot be represented as the sum of a number of polynomials. If we restrict ourselves to positive values of x and y, we can represent a quadrant of a circle with the formula:

y=√r2−x2

Where r is the circle radius.

If we can find the area of this quadrant, the area of the circle follows by quadrupling the result. The range of values x can take lie in the range 0 to r, so dividing this range into n strips, we have the width of each strip:

w= rn

The height of the ith strip is:

y i=r √1−( iwr )2=r √1−( in )2The area of the ith strip is:

Si=r2

n √1−( in )2The calculation requires us to evaluate the following sum:

∑1

n 1n √1−( in )2

Tricky!

We could evaluate it for ever increasing values of n until the error in consecutive iterations becomes small. This is tedious to attempt manually, but should only take a few seconds on a computer.

The sum converges very slowly, as it takes the computer 25000 terms to reach a value of 3.1412 for π. There are more efficient algorithms for calculating π, but this is good enough for most purposes.

Looking at the geometry, it is evident that vertical strips are unlikely to provide an accurate approximation to the curve as x approaches r. Taking the area below a region between the verticals through x=0 and x=r/√2 , will allow us to calculate the area of the curved sector, by subtracting the area of a square of side r/√2 from the area under the curve. The r/√2 comes from the x coordinate of the intersection of a line at 45 with the circle. The required sum for this region now becomes:

∑1

n 1√2n √1−1

2 ( in )2

This takes 2000 terms to produce a value for π which is correct to four decimal places.

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Figure 6-13 : 1/8th Is easier to Calculate than 1/4 of the CIrcle

I hope the natural curiosity of the reader will motivate him or her to think up more efficient alternatives to those presented here.

We can calculate the perimeter of the circle from the ring (annulus) formed by the region between two concentric circles. Taking the radius of the inner circle as r-d, we have for the area:

A=π (r2−(r−d )2 )=2 πrd−πd2

As the ring tends to zero thickness, i.e. d→0, its area tends to the length of the perimeter P multiplied by the width:

Pd=2πrd−d2

If d=0, this yields the result for the circumference of a circle:

P=2 πr

The area of an ellipse follows a similar procedure. An ellipse is essentially a squashed circle, so that the equation in the positive quadrant is:

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y=b√1−( xa )2Where a is half the length of the major axis and b is half the length of the minor axis. Dividing the semi major axis into n strips, the area of the ith strip becomes:

Si=ba1n √1−( in )2

We have just shown that the rather nasty series obtained from the function of i sums to π4 , so the

area of the ellipse is given by:

A=πab

We cannot find the perimeter of the ellipse by considering the region between a smaller and a larger ellipse, because in this case the gap between the two curves is not constant.

6.2.5 Concluding CommentWe can describe a large number of curves with polynomials, i.e. expressions of the form:

y=an xn+an−1 x

n−1+an−2 xn−2⋯ a0

Where the ai are constants. The boundaries of the area concerned may be defined in this way and the areas under them found by integrating the polynomial term by term. The area of the enclosed region may then be found by adding and subtracting areas as appropriate to the particular boundary.

The importance of integration, and calculus in general, is that it deals with things which are changing continuously; the curves are smooth, not discrete steps, and this is how phenomena in nature (at least at the macroscopic scale relevant to most human experience) behaves. It is an extremely efficient means of counting an infinite number of squares.

Furthermore, any quantity may be plotted on a graph; the method is not limited to just finding the areas of plane figures bound by curves, it can be used to provide insights into any phenomenon which can be plotted on a graph.

As we can define the geometry as algebraic equations, and may then use integration to find areas, there is no need to actually draw the graph. Indeed, the method may be extended to functions of two variables, which require three dimensional surfaces to represent them. The methods apply to functions of more than two variables which cannot be represented as a three dimensional object at all.

6.3 Some Applications

6.3.1 IntroductionSome readers, who may have encountered calculus before, may be surprised at the order in which I have chosen to present the subject. Most texts begin with the inverse process of integration which is

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called differentiation. It is my belief that this, more usual, presentation is the reason calculus is popularly perceived as ‘hard’.

I begin with integration because it is immediately applicable to real world problems which actually interest the student. Few things irritate the engineer’s mentality more than wasting vast amounts of time and effort on methods which have no apparent practical application, particularly if they intrude on the traditional engineer’s pursuits of philandering and getting drunk.

Having the ability to calculate areas bounded by curves isn’t a bad addition to the engineer’s toolkit, but integration equips us with the ability to do so much more.

6.3.2 Uniform AccelerationNewton’s Second Law tells us that if we apply a force to a body, it will accelerate at a rate proportional to the magnitude of the force. The constant of proportionality is called the inertial mass of the body. This defines mass (not directly measurable) in terms of force (which is measurable) and acceleration (also measurable), rather than some circular definition involving vague and confusing statements about ‘the amount of matter’ in the body.

In practice, we compare the gravitational forces (weights) acting on different bodies to determine mass. A standard lump of matter, held in some august institution, forms the basis of this comparison, and is, by international agreement, said to have the unit mass.

Since weighing is much simpler than measuring force and acceleration simultaneously, and the consequent measurement is potentially much more accurate, practically all systems of units define force in terms of mass and acceleration. This tendency to view mass as a tightly bound property of the body leads us straight into conceptual difficulties when Special Relativity decrees that the mass of a body increases as it approaches the speed of light. The definition in terms of force and acceleration raises no such conceptual difficulties.

However we look at it, a constant force on a body gives rise to a constant acceleration, at least provided the speeds involved are small compared with that of light, which is usually the case.

Plotting a constant acceleration a having duration T results in a rectangular plot in the acceleration/time graph.

Now the change in speed in a short time δt (say ‘delta tee’) is:

δv=aδt

(read this as ‘delta vee equals a delta tee’).

The change in speed over the period T is found by summing all these small changes in speed, in other words:

v−u=∫0

T

adt

Where u is the initial speed and v the final speed. Now applying the formula for integration:

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a∫0

z

x pdx=a zp+1

p+1

Figure 6-14 : Accelerating Particle

In this case p=0, so the change in speed is given by:

v−u=aT

Or:

v=u+aT

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The change in position in a short time interval depends on the speed at the time the interval is considered. From the above formula, it is evident that the speed isn’t constant but varies linearly with time. Defining v(t) as the speed at any time between 0 ant T, we have:

v (t )=u+at

(the (t) indicates that the quantity depends on time)

This is now a function of time, rather than simply the final value. Plotting v(t) against time yields a straight line. Denoting the distance travelled by s, we have:

s (t )−s (0 )=∫0

t

(u+at )dt=ut+12a t2

If the upper limit of the integral is the independent variable, the integral becomes a function of the independent variable (in this case time). If it is a constant, the integral evaluates to a number, such as the area under the curve.

By starting from the acceleration, and applying the integration formula twice, we have derived expressions for speed and position as functions of time.

6.3.3 Volumes of revolution

Figure 6-15: Calculation of the Volume of a Body of Revolution

If we take a curve and rotate it about the x-axis we generate a surface of a three dimensional body. The curve is called the generator of the surface. For example, the straight line through the origin:

y=ax

will generate a conical surface.

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Taking thin slices through the body of thickness δx (this is just another way of writing w for the width of strips; it’s the same thing only consistent with accepted calculus jargon), the volume of an individual slice is given by:

δV=π y2 δx

So the volume between the vertical planes passing through x1 and x2 is given by adding all these volumes together:

V=π∫x1

x2

y2dx

The straight line gives us the volume of a cone of height h:

V=π∫0

h

a2 x2dx= π3a2h3

But a×h is equal to the radius of the base (substitute h for x in the equation for the line to see this). The volume of the cone is therefore given by:

V=13π r2h

Or, one-third the base area times the height.

Another useful example is the sphere:

y2=r 2−x2

For the hemisphere x=0 to x=r, the volume is:

V=π∫0

r

(r2−x2 )dx=π [rx− x33 ]0

r

=23 π r

3

Note the use of square brackets to evaluate the integral, before substituting the upper and lower bounds of the integral into it. The upper and lower bounds are usually called the limits of integration.

The volume of the sphere is obviously twice this value:

V= 43π r3

If we subtract from this the volume of a spherical void of radius t units less than the original sphere, the resulting spherical shell has volume:

V= 43π (r3−(r−t )3 )=4

3π (3 r2 t−3 r t 2+t3 )

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If the thickness is small compared with the radius, the second order and higher terms in t may be ignored. Also, as the thickness tends to zero, the volume tends to the surface area S multiplied by the thickness:

S t=4 π r2 t

So that as t→0, the surface area becomes:

S=4 π r2

Note; this method of finding the surface area only works with shapes like the sphere for which the thickness of the shell is uniform, and thickness is measured perpendicular to the surface. In the case of a cone, the thickness of the shell will be t if the inner cone is displaced along the x axis by an amount:

d= tr √r

2+h2

This void has volume:

V= π3a2 (h−d )3

The volume of the shell is,:

St ≈ π3a2 (h3−h3+3h2d−3hd2+d3 )

Ignoring high order terms in d:

St ≈π r 2d=πr √r2+h2tAs t→0, the surface area of the cone, ignoring the base, becomes:

S=πr √r2+h2

Note the approximation becomes more accurate, the smaller t becomes. When t is zero, the expression is exact.

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6.4 Centres of Gravity

Figure 6-16 : Definition of Centre of Mass

The centre of gravity of a body is the point through which its weight appears to act. For the sake of clarity, we restrict the discussion to solid bodies of uniform density throughout so that mass is directly proportional to volume.

The turning moment about the origin of a body of volume V is equal to:

M=ρV x

Where ρ (rho) is the weight per unit volume of the material of the body and x is the distance of the centre of gravity from the origin.

The turning moment of a transverse section of the body at distance x from the origin of thickness δx is given by:

δM=ρAxδx

Where A is the area of the section. The total turning moment is found by summing over all these sections, so that:

ρV x=ρ∫0

L

A (x ) xdx

L is the position of the point on the body furthest from the origin.

The density cancels out, and hence is irrelevant to the calculation. The cross section is a function of the distance from the origin. The centre of gravity is therefore found from:

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x=∫0

L

A ( x ) x dx

V

As an example, a cone cross sectional area, is presented in the previous section as:

A ( x )=π a2 x2

The volume was found to be π3a2h3 where h is the height of the cone.

∫0

h

A ( x ) xdx=∫0

h

π a2 x3dx= 14π a2h4

So the centre of gravity is:

x=34h

Below the apex, or a quarter of the height from the base.

For a hemisphere, it has been shown that:

V=23π r3

The cross section is given by:

A ( x )=π (r2−x2 )

The moment of area is:

∫0

r

π (r2 x−x3 )dx=π [ r2 x22 −x4

4 ]0

r

=π4 r

4

The position of the centre of gravity is, therefore:

x=38r

6.5 Second Moment of AreaThe centre of gravity is found by calculating the first moment of area with respect to a reference point. If the reference point is the centre of gravity, a force applied there will cause the body to move in translation only, and will not cause the body to rotate.

In general, an applied force cannot be expected to act through the centre of gravity, and consequently it applies a turning moment to the body. The moment is resisted by an inertial torque proportional to the angular acceleration, analogous to the inertial resistance the body has to translational acceleration by virtue of its mass.

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Figure 6-17 : Moment of Inertia about an Axis Perpendicular to the Page

The inertial moment arises because the angular acceleration of the body as a whole causes the individual particles of the body to accelerate linearly. The amount of linear acceleration is proportional to the distance of the particle from the axis of rotation, and the moment of the force produced by this acceleration is also proportional to the distance from the axis. The contribution of the particle to the inertial torque is equal to the angular acceleration multipled by the particle mass multiplied by the square of the distance of the particle from the axis.

The inertial resistance of the whole body to the applied torque is found by summing the contributions of all the particles of the body. This results in a quantity which depends on the mass and shape of the body which characterises its inertial resistance to angular acceleration. This quantity is called the moment of inertia with respect to the axis of rotation. It is also known as the second moment of area.

The second moment of area also characterises the ability of a beam cross-section to withstand bending moments, with the fibres furthest from the centre of the section contributing most to the bending stiffness, and those nearest the centre contributing least.

Second moments of area also arise in the analysis of ship stability, as the parts of the hull furthest from the centre line contribute most to the righting moment when the ship heels over.

All of these applications, and doubtless many more, require us to calculate the product of infinitesimal areas with the square of their distance from the axis of interest, and sum them over the entire body.

An example may clarify this concept. Consider a particle of mass m mounted on a rigid bar, length r, of negligible mass, pivoted to rotate freely about its other end, the pivot itself is prevented from translational motion, so maintains its position fixed at all times . The frictional torque in the pivot is considered negligible. The assembly experiences a constant angular acceleration of α

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(alpha)radians/sec/sec, and we need to find the magnitude of the torque needed to sustain the motion.

The instantaneous translational acceleration of the particle is αr, so the inertial force, by Newton’s Second Law is mαr, and the moment of this force about the pivot is mr2α. By analogy with Newton’s Second Law, in which force equals mass times acceleration, we say that the torque equals the moment of inertia times the angular acceleration. The moment of inertia of a single particle offset from the axis of rotation is therefore:

I=mr2

Note that although we are dealing with rotational motion, we derive the torque by considering only translational motion of the particle. If the spinning of the particle introduced any new effects, they would not be predictable from the standard equations of motion for a rigid body.

The moment of inertia is to angular motion what mass is to translational motion.

6.5.1 Moment of inertia of a Disc

Figure 6-18 : Moment of Inertia of a Uniform Solid Disc

We shall consider the moment of inertia of a circular disc, radius R and thickness t, rotating about an axis perpendicular to the plane of the paper. The mass of a ring of radius r and width δr is given by:

δm=ρ2πrt δr

Where ρ (rho) is the density of the material. Note that the δ (delta) in front of the symbol implies the quantity is ‘small’ in this context, and we can anticipate that the problem reduces to finding out what happens when such quantities approach zero.

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All the mass of this ring is distance r from the axis of rotation, so it will have moment of inertia:

δI=ρ2π r3 tδr

So the moment of inertia of the disc is found by summing over an infinite number of concentric rings, whose widths tend to zero.

I=ρt (2π∫0

R

r3dr)The term in brackets is the second moment of area.

Evaluating the integral:

I=ρtπ R4

2

But the mass of the disc is ρtπ R2, so the moment of inertia may be written:

I=12mR2

The disc behaves like a ring which has all its mass concentrated at a radius:

r g=R√2

Where rg is known as the radius of gyration. This is a useful concept, because it is much easier to measure the radius of gyration of an arbitrarily shaped body than it is to estimate it by calculation.

With practice it becomes possible to estimate the radius of gyration by eye.

6.6 Engineer’s Bending TheoryIf we consider a section of beam loaded so that it bends into a curve, we see that for the case shown, the lower surface of the beam is in tension, while the upper surface is in compression. If the radius of curvature of the deformation is large compared with the thickness of the beam, it is reasonable to assume that the deformation varies linearly across the depth of the beam. From Hooke’s Law, the stress at any point in the cross section is proportional to the strain, i.e. the deformation. It follows that the stress also varies linearly across the section.

The elastic moment resisting the bending moment due to the applied load may be found by summing the moments due to the internal stresses across the depth of the beam.

Consider a rectangular beam of breadth b and depth d. For consistency with the rest of this chapter we will take the vertical direction through the beam cross section as the x-axis, so the section is on its side, with the moment applied in the horizontal plane.

The origin is taken at the centre of the beam section, because this is where the stress is believed to be zero.

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The elastic force on a narrow strip of width (actually height) δx is given by:

δF=σ ( x )bδx

Where σ is the stress at this depth (stress is force per unit area, as the size of the area tends to zero).

This will generate a moment about the y axis of:

δM=xδF

In order to proceed, we need an expression for the distribution of stress.

Figure 6-19 : Small Angle Approximations

The length of beam of interest is distorted into a circular arc, which has radius R at the neutral axis (i.e. where the distortion is zero). The angle subtended at the centre of the circle is θ (theta) radians. The length of a section distance x below the neutral axis is:

l (x )=(R+x )θ

So the strain ε (epsilon) on this section is:

ε= (R+x )θ−RθRθ

= xR

According to Hooke’s Law, the stress is proportional to the strain:

σ=εE

Where E is called the modulus of elasticity of the material, or Young’s modulus.

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Figure 6-20 : Beam Bends Locally into a Circular Arc, with Linear Stress Distribution Across the Depth

The moment of the elemental strip is, therefore:

δM= ERb x2 δx

So the moment is found by summing over all strips and taking the width of the strips to zero.

M= ERb∫−d2

d2

x2dx=ERb [ x33 ]−b2

b2 =E

R ( b d3

12 )The term in brackets is the second moment of area, usually written I. The formula for the bending moment is given by:

MI= ER

The maximum stress is at the top or bottom of the beam, which is taken as a distance y from the neutral axis. The maximum stress is given by:

σ max=E εmax=ERy

The complete formula for engineer’s bending theory, becomes:

σy=MI= ER

Bearing in mind the fact that σ is the Greek letter s, this lends itself to the rude, but highly memorable mnemonic; ‘Screw Your Mother In the Engine Room’.

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6.7 Ship Stability

Figure 6-21 : Righting Moment for a Ship Heeled Over

If a ship heels over, one side becomes immersed more deeply in the water, whilst the other side is lifted out of the water. By Archimedes Principle, the immersed side experiences an increased upthrust whilst the weight of the ship on the other side is no longer completely supported by buoyancy forces.

For a symmetrical waterline planform, The extra up thrust at a point x from the longitudinal axis is the same as the excess weight on the other side, and the ship experiences a righting moment. The magnitude of the righting moment is calculated by summing the righting moments due to an infinite number of vanishingly small longitudinal slices.

As an illustrative example, consider a waterline section consisting of a parabolic arc.

The equation for this curve is:

x=w2 (1−(2 yL )

2) Where w is the width, and L the length of the waterline section.

This introduces a difficulty because so far we have only dealt with integer powers of x. This equation evidently introduces a square root.

The ship is heeled over at an angle φ (phi) so that a longitudinal slice distance x from the centre line is immersed to a depth φx. The volume of water displaced is φxyδx, so the upthrust is φρxyδx, where ρ is the water density.

The righting moment due to the infinitesimal longitudinal slice and the corresponding slice which is lifted out of the water on the other side is:

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δM=−2 ρϕ x2 yδx

The negative sign indicates that the righting moment is in the opposite direction to the angle of heel.

The total righting moment is:

M=−4 ρϕ∫0

w2

x2 ydx

The integral is the second moment of area of the waterline section about the centre line. So that in general the righting moment is given by:

M=ρϕI

Where I is the second moment of area of the waterline section.

In the example, i ntegrating with respect to x will introduce fractional powers of x, which we have yet to consider . In order to avoid the problem, we shall integrate with respect to y.

We have for the waterline section:

x=w2 (1−(2 yL )

2) We found from the analysis of the rectangular beam cross section that the second moment of area of a rectangle about an axis through its centroid is:

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I rectangle=bd3

12

The second moment of area for a transverse slice of the waterline section, of length δy is, therefore:

δI=(2x )3

12δy=w

3

12 (1−(2 yL )2)

3

δy

So I is found by summing over an infinite number of infinitesimal slices:

I=w3

12∫−L2

L2

(1−3( 2 yL )2

+3 ( 2 yL )4

−(2 yL )6)dy

From which the second moment of area is:

I= 4105

w3L

So the wider the ship, the larger the righting moment. But the wider the ship, the more power is required to propel it.

The astute reader will probably have noticed that the second moment of area consists of a numerical factor multiplied by the area of the enclosing rectangle, multiplied by the square of the dimension perpendicular to the axis. It is awareness of the expected size of the numerical factor that forms part of the engineer’s intuition as to whether a design ‘looks right’.

The excessive influence of academic mathematicians and commercial numerical analysis packages has largely supplanted the true engineer’s intuitive grasp of the problem. I am unrepentant in trying to bring it back.

In order to float at all, the average density of the ship must be much less than that of the water, so that the centre of gravity is expected to lie above the waterline. When the ship heels over, the moment of the weight about the centre line of the waterline section tends to cause the ship to heel further, so the moment preventing capsizing is the difference between the hydrostatic righting moment and the moment of weight:

M c=−( ρI−Wh )ϕ

Where W is the displacement and h the height of the centre of gravity above the waterline. The roll stability is usually characterised by considering the rolling moment to arise completely from the weight of the ship suspended distance hm below the waterline, where:

W hm=ρI−Wh

This distance is called the height of the metacentre, and unlike the quantities in the original roll equation, it is readily measured by forcing the ship to heel over by adding offset loads and then measuring the resulting angle of heel.

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However, requiring the ship to be built before its stability can be assessed is hardly satisfactory. Some means is required to estimate the metacentric height long before construction begins.

6.8 Fractional PowersThe ship stability problem introduced a requirement to integrate fractional powers of x, which were overcome by integrating the expression for second moment of area with respect to y. In effect, the second moments of area of infinitesimal lateral slices of the ship were taken, and then summed to get the second moment of area of the entire ship.

Recall that the area under a parabola y=x2is one third the area of the rectangle containing it. The curve x= y2 is also a parabola, but it is the area between the curve and the y-axis which is one third the area of the enclosing rectangle, the area below the curve must therefore be equal to the remaining two thirds of the enclosing rectangle.

The enclosing rectangle has area x×y, or x √x=x32 , so the area under the curve √ x=x

12 is:

∫0

z

x12 dx=2

3z32

This result is consistent with the general formula for positive integer powers of x. However, this could be coincidence.

In general, the area of the enclosing rectangle is xy, and this must be equal to the integral of the function with respect to y plus the integral of the same function with respect to x.

∫0

a

x dy=ab−∫0

b

ydx

So, if x is raised to a power which is a reciprocal of an integer, the integral can always be evaluated because it will be an integer power of y.

The general formula applies to powers which are reciprocals of positive integers. What about more general fractions?

Suppose we wish to evaluate:

∫a

b

xn ydx

When

x= ym

Both n and m are integers. This is the same as evaluating:

∫a

b

xn+ 1mdx

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We solved this in the ship stability example by integrating with respect to y.

If x is increased by an amount, δx, then in order to stay on the curve y must also change by an amount δy, such that:

x+δx=( y+δy )m

We have found that all terms higher that the first in infinitesimal quantities become zero as we sum over an infinite number of them. So we need only to expand the polynomial in y to the first order term in δy:

x+δx ≈ ym+m ym−1δy

Eliminating x from both sides:

δx=m ym−1δy

The integral becomes:

m∫c

d

xn ymdy

Where the limits are the values of y corresponding to the limit values for x i.e.:

c=a1m , d=b

1m

Writing the integral entirely in terms of y:

m∫c

d

ym (n+1 )dy=[ m(m (n+1 )+1 )

ym (n+1)+1]cd

Substituting for x:

∫a

b

xn+ 1mdx=[ 1

n+ 1m+1xn+ 1m +1]

a

b

This is the same result as would have been obtained by applying the general formula to the original integral.

The general formula therefore applies to powers which are improper fractions of the form (n+ 1m )

This implies that square roots, cube roots, etc may be integrated any number of times using the standard formula.

The general fractional index is:

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y=xnm

Let

z=x1m

The increment in z corresponding to the increment δx is found from the first order expansion of

( z+δz )m :

z+mzm−1 δz ≈ δx

Hence:

∫a

b

ydx=m∫c

d

zn+m−1dz= mn+m

[zn+m ]cd= 1nm+1

[ x nm+1]ab

This is the same result as would be obtained from the general formula, so it is concluded that the formula derived for positive integers applies to any positive power of x.

Hopefully, by now the reader is no longer daunted by the sight of an integral sign, and sufficient examples have been presented to indicate the utility of the technique of integration to a wide range of problems which would otherwise be intractable.

All this capability is at the reader’s fingertips by applying a little ingenuity and the formula;

∫a

b

x pdx=[ x p+1p+1 ]ab

6.9 The Usual Starting Point

6.9.1 IntroductionWe have introduced a method of dealing with problems which involve continuously changing quantities and demonstrated its utility with a number of examples. The more curious reader may wonder if the process of integration has an inverse process.

For example, if we have an expression for distance travelled as a function of time, can we derive the speed at any time, and the acceleration from it?

The simple answer is yes, the inverse process exists. It is called differentiation, and is the more usual starting point for texts on calculus. However, it is hard to find examples to motivate interest in the subject matter, which are not clearly contrived for the purpose, and it introduces the concept of a limit which is inherently more difficult to grasp than the idea of a sum over a huge number of tiny strips.

It is for these reasons,(i.e. that the student is presented with a difficult to understand concept, and has very little to indicate that there is any benefit in coming to grips with it anyway) that I have chosen to introduce the subject with integration.

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The standard notation for differentiation is illustrated as follows:

If:

∫a

b

z dx=[ y ]ab

Then:

dydx=z

Where dydx is called the derivative of y with respect to x.

From the formula for integration, it is evident that if:

y=x p

Then:

dydx=px p−1

That is all very well, but what does this mean in terms of functions plotted on a graph?

We start from the simplest of plots, the straight line:

y=mx+c

Applying the formula for differentiation:

dydx=m

This is the slope of the line.

If the plot is a curve, its slope will vary continuously. Differentiation yields the slope of the tangent at the point of interest.

We can calculate an approximate value for the slope at x by incrementing x slightly (by δx) and finding the corresponding increment in y (δy). Dividing the small change in y by the small change in x yields an estimate of the slope of the curve at x.

Consider:

y=xn

Increment x slightly:

y+δy=( x+δx )n

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We recall, from summing an infinite number of strips that:

( x+δx )n=xn+nxn−1δx+…

The remaining terms contain squares and higher powers of the increment , so we can anticpate that when we reduce the increment to zero, these terms disappear.

The y and xn terms on either side of the expression are equal and so cancel out, leaving an expression for δy:

δy=n xn−1 δx+⋯

The slope is given approximately by:

δyδx=n xn−1+terms∈δx

Now, if δx=0, so does δy, and we are left with the indeterminate result 00 .

This is the conceptual obstacle we must overcome. It was evident in the case of integration that the more and finer the strips, the more nearly they filled the area, so that with an infinite number, the formula produced became an exact answer, rather than a very close approximation.

Figure 6-22 : Differentiation finds the Tangent to the Curve

What we say is that as we make δx arbitrarily small, the estimate of the slope becomes ever closer to

nxn-1. Under these conditions we write the ratio of two small but finite changes in y and x as dydx ,

rather than δyδx . This is written as:

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dydx= limδ x→0

δyδx

Read this as ‘the limit as δx tends to zero’.

‘lim’ denotes the limiting value where the ratio still remains finite. Just as taking the increment to zero in the case of integration yielded an exact answer, taking it to zero when differentiating yields an exact answer for the slope.

To the critical mind this smacks a bit of Emperor’s New Clothes, which is not the best way to introduce a subject which has no immediately obvious applications.

I suggest that this is the reason calculus is popularly considered ‘hard’. Few engineers have much patience with arcane discussions about the number of angels which can dance on a pinhead.

So whereas integration calculates areas under curves, differentiation finds the slopes of tangents to curves.

6.10 Applications

6.10.1 Newton’s MethodQuite often the Laws of physics present us with equations which are not amenable to elementary methods, such as the formula for a quadratic equation. The problem is to find a solution to an equation which takes the form:

f ( x )=0

This is the intersection of the function:

y=f ( x )

And the x axis (y=0).

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Figure 6-23 : Successive Approximations

If we start at a rough solution, found for example, by sketching the curve, we can take the tangent to the curve as a straight line approximation in the vicinity of the solution. This straight line fit may be used to get another, hopefully better, approximation to the required solution, although it is not guaranteed that the method will always converge on to the solution.

Given an approximate solution xi, the tangent intersects the x-axis at the next iteration x i+1. From the geometry of the curve:

x i+1=x i−y (x i)dydx x=x i

Note if we are dealing with the slope at a particular point, rather than deriving the general formula for the slope, it is usual to indicate the fact with a subscript, as shown.

As an example, suppose we are seeking a root of the cubic equation:

x3+3x2−4 x+2= y=0

The derivative is found by applying the formula for differentiating powers of x term by term:

dydx=3 x2+6 x−4

Starting with the approximation x=-5.0, the progress of the algorithm can be seen in the following table.

Iteration

x Y

1 -5.0000 -28.0000

2 -4.3171 -5.2782

3 -4.1141 -0.4015

4 -4.0960 -0.0031

5 -4.0958 0.0000

6 -4.0958 0.0000

In this case the solution is found to an accuracy of four decimal places in five iterations.

6.10.2 Maxima and MinimaIf we plot a polynomial of degree higher than one, we may notice that in a particular region, y increases with increasing x but further on, y decreases with increasing x, or the slope starts positive and ends up negative. Alternatively, the slope may start negative and become positive.

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Figure 6-24 : Maxima and Minima Correspond to Turning Points

At some point the slope is zero, and this corresponds to a point on the curve where the y has its maximum value (in the first case) or its minimum value (in the second case).

So if:

y= f ( x )

These turning points are found where:

dydx=0

This doesn’t tell us what type of turning point it is (maximum or minimum), or whether the turning point is an absolute maximum (minimum) of the function, i.e. a supremum (infimum), when the function has many turning points.

A quick sketch of the function in the region of interest will resolve these issues, but we shall see how to distinguish maxima from minima purely analytically, presently.

6.10.3 Body under GravityRecall that we derived the expression for distance moved under constant acceleration as an illustration of the integration process:

x=ut+12a t2

In this case the independent variable t is time.

A body which is sufficiently dense is not expected to be influenced by air resistance, especially if the speeds are low, so this equation is a reasonable approximation to the motion of say a cricket ball thrown vertically upwards. The ball is subjected to its weight, which manifests itself as a constant

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acceleration towards the Earth this acceleration is conventionally denoted g. We shall take the positive direction as upwards, so the gravitational acceleration in the above equation will be denoted by –g.

The height h of the ball above the release point is then given by:

h=ut−12g t 2

where u is the speed the ball is thrown upwards with.

A question which comes to mind is; what is the maximum height the ball will reach?

Differentiating:

dhdt=u−¿

So the turning point is given by:

dhdt=0 ,∨t=u

g

The maximum height is found by substituting this value of time into the equation for height:

h=u( ug )−12g ( ug )

2

=12u2

g

6.10.4 Maximum Area

Figure 6-25: Find Proportions for Maximum Area

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Suppose we have a boundary of fixed length, say L, and we wish to find out the proportions of the rectangle which will enclose the maximum area. Let one side of the rectangle be x, so the adjacent side d must be:

d= L−2x2

The area of the rectangle is length times breadth:

A=xd=x L2−x2

The turning point is given by:

dAdx

=0= L2−2x

From which it follows that x=L4 and d=

L4 , and the rectangle which encloses the maximum area is a

square.

6.10.5 Concluding CommentsThe calculation of turning points has obvious application in problems where we are seeking the best use of resources.

The examples show that the application of calculus to ‘given’ problems does not appear particularly difficult; the hard part is invariably formulating real world problems in a form where they are amenable to mathematical analysis. This requires understanding of which terms can be ignored, which can be treated, near enough, as tractable mathematical expressions, and for what range of values of system parameters the current approximation remains adequately valid.

It is this step of converting problems into a form to which the methods of mathematics can be applied which is alien to the pure mathematician, but is the essence of the mathematically-inclined engineer’s task. The resulting displacement of the engineer by the pure mathematician, even in university engineering faculties, cannot be considered a healthy development, for this reason.

6.11 Higher Order Derivatives

6.11.1 IntroductionWhen a problem requires us to integrate an expression twice, we simply repeat the integral sign. However, most of the interesting problems requiring calculus involve dependencies on derivatives of order higher than just the first, so it is convenient to indicate the order of a derivative in the notation.

The derivative of:

dydx

Is written:

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ddx ( dydx )=d

2 yd x2

Differentiating n times is denoted:

dn yd xn

The following sections introduce some examples to illustrate this point.

6.11.2 Radius of Curvature ApproximationIf we move a small distance δs around the circumference of a circle, the tangent direction changes by an amount δθ (delta theta). This is the same angle as the radius rotates at the same time, so if δθ is in radians:

δθ= δsr

The radius of curvature at a point on a more general curve is equal to that of a circular arc which is tangential to the curve. This circular arc is called an ’osculating curve’ , which literally means ‘kissing’, conveying the notion that the two curves are intimately related.

The reciprocal of the radius of curvature is called simply the ‘curvature’ and is given by taking the limit of:

dθds= limδs→0

δθδs=1r

In order to use this relationship for general curves, we need to explore more functions than positive

powers of x. However, if the slope dydx is small in the region of interest (e.g. less than 5 ) we may

make the approximations:

δs ≈δx

δθ ≈ dydx

So the curvature is, approximately:

1r≈ d

2 yd x2

Near a turning point, the slope is near zero, so this approximation is valid. Now positive curvature implies the slope increases with x, so the turning point is a minimum. Negative curvature implies the slope decreases (i.e. becomes more negative) with increasing x, so the turning point must be a maximum.

Having found a turning point by differentiating the function and equating the resulting expression to zero, we find out if it is a maximum or minimum by differentiating again and substitute the value of x

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at the turning point. If the result is negative, the turning point is a maximum, if positive, it is a minimum.

Applying this to the cricket ball example:

dhdt=u−¿

d2hd t2

=−g

This is negative, so the solution does indeed correspond to a maximum value for h.

Another circumstance in which the slope is expected to be small is the lateral deflection of a loaded beam.

From the formula for engineer’s bending theory, we have for the curvature:

1r≈ d

2 yd x2

= 1EIM

If we have an expression for bending moment as a function of distance along the beam, we can solve for the deflection.

As an example consider a beam of length L, rigidly mounted at one end and loaded with weight W at its free end.

The bending moment is WL at the root, reducing linearly to zero at the tip:

M=W (L−x )

The deflection pattern may be found by integrating the bending equation twice.

dydx−

dydx x=0

=WEI [Lx− x

2

2 ]0x

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Figure 6-26 : Loaded Cantilever

The slope at the root is zero, so the equation for the slope of the beam is:

dydx=W L2

EI ( xL−12 ( xL )

2)The deflection is:

y− y (0 )=W L2

EI [ 12 x2L−16x3

L2 ]0

x

The deflection at the tip is:

y (L )=W L3

3 EI

6.12 More Exotic FunctionsSo far we have only been able to work with functions which are linear sums of positive powers of the independent variable, typically x.

6.12.1.1 Negative Powers The most obvious gap in the available set of functions is negative powers of x.

If:

y=x−n

xn y=1

Taking increments in x and y:

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( x+δx )n ( y+δy )=1

Expanding to first order of small quantities:

xn y+ny xn−1δx+xnδy=1

Collecting up terms:

xn δy=−nx−1δx

Taking this to the limit as x→0:

dydx=−n x−n−1

This is precisely the same equation as for positive powers, which implies the integration formula works for negative powers. The formula is:

∫a

b

x pdx= x p+1

p+1

If p=-1, the formula yields a value of infinity, so that:

∫a

b 1xdx

Is a special case.

6.12.2 The Integral of the Reciprocal FunctionIt appears our formula for integration works for every power of x apart from -1. We could hope that this particular integral doesn’t crop up very often in practice. Unfortunately, that is a forlorn hope, as the most important phenomena of all are governed by equations of his form.

6.12.2.1 The Tsiolovski Rocket EquationTake for example, the acceleration of a rocket. Matter is accelerated in the rocket motor, and ejected out of the nozzle, the force accelerating the exhaust matter also acts on the rocket itself, according to Newton’s Third Law. If we consider an element of the exhaust material of mass δm, accelerated to the exhaust velocity ue, it will impart an impulse of magnitude:

δI=ue δm

To the rocket. An impulse is a force acting for a very short length of time, so the continuous force acting on the rocket is:

T=dIdt=ue

dmdt

Where T denotes thrust.

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The matter which is ejected originated from the propellant on board the rocket, so the rocket is losing mass at a rate given by this expression.

According to Newton’s Second Law, the rocket will accelerate at a rate proportional to the thrust:

m dudt=ue

dmdt

The change in speed over finite time is given by:

u−u (m0 )=ue∫m0

m 1mdm

6.12.2.2 The Breguet Range EquationThe fuel consumption of a jet propelled aircraft is proportional to the thrust under a particular set of flight conditions. For the cruise, these flight conditions are sufficiently constant for the rate of fuel consumption to be directly proportional to the thrust:

dWdt

=−cT

Where c is a constant, characterising the propulsion, and is called the thrust specific fuel consumption, T is thrust, W the weight of the aircraft and t is time.

The aerodynamic efficiency of the aircraft is characterised by its lift to drag ratio. For steady level flight, thrust equals drag and lift equals weight:

dWdt

=−c( DL )W

In order to calculate the endurance, we need to evaluate:

t=−1c ( LD )∫W 0

W 1WdW

In order to calculate range, we need the rate of fuel consumption per unit distance.

dWdx

=dWdt

dtdx= 1UdWdt

Where U is the cruising speed. The result is an integral of the reciprocal of weight, as for the endurance equation. Note: a fast aircraft of inferior fuel consumption and aerodynamic efficiency than a slower one, consumes less fuel because the flight time is shorter. Hence global ranges require jet aircraft, which is counter-intuitive because they are less efficient than their propeller driven alternatives.

6.13 Searching for a SolutionThese two examples indicate the kind of problem in which this apparently intractable integral raises its head. They are characterised by the rate of change of a quantity being proportional to the

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quantity itself. Any self-correcting system must have this feature, so we can anticipate that the entire edifice of control engineering must rest on this one equation. Rather than obscure, this particular equation, and the set of functions associated with it, is inconveniently ubiquitous.

We proceed as before and glean what we can from summing strips of finite width.

Evidently, we are courting disaster if we try to find the value of the integral from zero to an upper value, as the function becomes infinite as x approaches zero. We must therefore evaluate it between two finite limits, say a and b.

The width of an individual strip is:

w= (b−a )n

The x coordinate of the ith strip is:

x i=a+ iw

The area of the ith strip is:

Si=w

a+iw=1n

( ba−1)1+ in ( ba−1)

This cannot be written as a constant multiplying a function of i, as in previous examples. The sum

depends on ba as well as on i. The integral is therefore some function of

ba , which we shall call

f ( ba )The integral between the two limits is also found by substituting the limits in the function and taking the difference:

∫a

b 1xdx=[ f (x ) ]a

b=[ f (b )−f (a ) ]

It follows that the function has the property:

f (b )−f (a )=f ( ba )In the dim and distant past, before the days of calculators, a function with this property would have been recognised immediately. Thankfully, the days of manual calculation are over, and we can concentrate on the more challenging task of formulating real world problems in mathematical form.

Let

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a=c y (a ) ,b=c y (b) , ba=cy( ba )

Where c is a constant. Dividing the expression for b in the above list by that for a, we have:

ba= c

y (b )

c y (a )=c( y (b )− y (a ))

The function y appears to have the property we are looking for. It is the power a certain base number must be raised to in order to yield the value of the independent variable. It is called the logarithm, in this case to base c. This is written:

x=c y

y=logc x

We have shown that the solution takes the form of a logarithm, but as yet we do not know the base to which it is referred.

Evidently, when y=0, x=1 regardless of the base used, and when y=1, x=c, the base of the logarithm generated by the integral.

6.13.1 Finding the Magic Number.We can find the base of the logarithms by evaluating the integral numerically from x=1 to a range of values and find the value of x for which y=1.

There are a number of methods which can be used. The obvious approach is to emulate our calculation of infinite sums by taking thin rectangular strips. In practice, this requires a very large number of samples. Fewer strips are needed if they fit the curve better than a rectangle. An obvious choice is to replace the rectangular strips with trapezoidal strips, which leads, believe it or not, to the trapezoidal rule:

A=w( 12 ( y1+ yn )+∑2

n−1

yi) This may still require too many samples for a manual calculation, but should be good enough for a computer program.

Performing the calculation yields a value of about 2.72 for the base of the logarithm obtained from integrating 1/x. This base is denoted e, and like π, it is a fundamental mathematical constant.

Since it has been derived from analysis, rather than an arbitrary number convention, e is referred to as the base for ‘natural’ logarithms, and the logarithm to base e is usually written:

log e x=ln x

The inverse function is then:

y=ex

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(Read this as ‘e to the x’). This is called the exponential function. The derivative may be found as follows:

ln y=x

1y=dxdy

And since the derivative is no more than one infinitesmal quantity divided by another, it is valid to take the reciprocal of both sides:

dydx= y=ex

So the exponential function has the property that it is unchanged by integration or differentiation. In addition and subtraction, the number 0 is probably the most important because it does not change the value of the answer, likewise, the number 1 has particular significance in multiplication and division, because it does not change the answer. The exponential function has a similar importance in calculus.

We notice that as x increases, the rate of change of x and all higher derivatives also increase, so that the value of y increases at an ever increasing rate, this is called exponential growth.

More interestingly, if x is negative, y decays at an ever reducing rate, so that it approaches the x axis but never actually reaches it. A curve of this form is called an asymptote to the x–axis.

y=e−x

ln y=−x

dydx=− y

This equation illustrates feedback in its simplest form. The quantity of interest, which usually needs to be kept near zero determines its own rate of change. If the feedback is positive, the quantity diverges in time without bound. If negative it tends to zero if disturbed.

6.13.2 Cooling of a Bowl of SoupAs an illustration, consider the archetypal boring activity of watching soup go cold. Watching paint dry could be described in much the same way.

According to Newton’s law of cooling the rate at which the temperature drops is proportional to the difference in temperature between the soup and its surroundings.

Let the temperature of the surroundings be Ts and the temperature of the soup be T. Call the temperature difference θ (theta).

We have:

θ=T s−T

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Differentiating with respect to time (t):

dθdt=−dT

dt

From the law of cooling:

dTdt=k (T s−T )

Where k is a constant depending on such factors as the size and shape of the bowl, the amount of soup in it and whether the surrounding air is stationary or flowing past it.

The equation becomes:

dθdt=−kθ

Integrating:

∫θ0

θ 1θdθ=−kt

ln ( θθ0 )=−ktθ=θ0e

−kt

The intelligent person can find something of interest in even the most mundane of phenomena. It is the blasé fool who is bored by anything and everything.

6.13.3 Capacitor DischargeA capacitor is a device for storing electric charge. The amount of charge which can be stored depends on the voltage applied across the device terminals, and the value of the capacitor.

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Figure 6-27 : Capacitor/Resistor (CR) Circuit

The charge is given by:

q=CV

Where q is the charge, V the electrical potential difference across the terminals and C the value of the capacitance. In SI units, the charge is in coulombs, the potential difference in volts, and the capacitance in Farads. Practical values of capaicitance are usually in the nanoFarad to microfarad range.

A capacitor of capacitance C is charged up to V0 volts, the switch is closed connecting the capacitor to resistor R. The current i through the resistor is given by Ohm’s Law:

i=VR

Where R is the value of the resistor.

Current is the rate of flow of charge, so:

i=dqdt=−C dV

dt

The negative sign implies the charge is flowing out of the capacitor.

The voltage across the capacitor once the switch is closed is governed by:

dVdt

=−1CR

V

This is of the same form as the soup temperature equation, so:

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V=V 0e−tCR

6.13.4 Concluding CommentsWe have considered problems from space science, aeronautics, heat and electronics, to illustrate the point that exponential functions are to be found everywhere in nature. Many more phenomena may be modelled as approximations to exponential functions.

It is the business of the engineer to find a mathematical description which is adequate for the real world problem of interest. New systems raise new and different problems, if they are expected to be superior to last years’ model, so new formulations will always be necessary which are unlikely to be available within existing applications.

6.14 Onwards and UpwardsWe now have formulas for differentiating and integrating a function which is a linear sum of powers of the independent variable.

If we can represent a function as a polynomial, we should be able to apply our standard formula to differentiating it and integrating it.

Suppose we can fit a function, near enough, to a polynomial of order n. The nth derivative will be a constant, so it will have the same value wherever we choose to evaluate it. Choose for example x=z:

dn yd xn

=( dn yd xn )x=zIntegrating:

dn−1 yd xn−1=( dn−1 y

d xn−1 )x=z

+x ( dn yd xn )x=zIf we choose z=0, the nth derivative will not interfere with the n-1th or lower order derivatives, likewise the n-1th derivative will not interfere with the lower derivatives and so on.

If we carry on integrating down to terms in x0, we end up with a polynomial approximation to the function:

y= y (0 )+( dydx )x=0x+1

2 ( d2 yd x2 )x=0

x2⋯+ 1n ! ( d

n yd xn )x=0

Where the exclamation denotes a factorial; n(n-1)(n-2)...×2×1

The highest order derivative may be as high as we please.

The curve fit (known as a Taylor series expansion) requires us to differentiate the function. However, integration is usually the more difficult task, and this furnishes a method of dealing with particularly awkward functions, as the series may be integrated term by term.

Recall that all derivatives of ex are equal to ex, and e0=1, the Taylor series expansion for ex becomes:

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ex=1+x+ x2

2+ x3

3×2⋯+ x

n

n!

The numerical value for e is found by setting x=1:

e=∑0

∞ 1i !

Or:

e=1+ 11+ 12×1

+ 13×2×1

+ 14×3×2×1

⋯

This converges quickly on to the value 2.718, to three places of decimals.

6.15 Trigonometrical Functions

6.15.1 IntroductionThis section is rather long and many readers will have covered the material, I suggest referring back to it only if the material of later sections becomes incomprehensible. However, consistent with a policy of producing a self contained text which does not reference material not already covered, I shall start from first principles. After all, you paid for this, so you might as well get your money’s worth.

6.15.2 AnglesFrom the beginning, I warned the reader that we shall measure angles using radians.

If we roll a wheel a distance equal to its radius, any spoke on the wheel rotates through one radian. Alternatively, a radian is the angle subtended at the centre of a circle by an arc of length equal to the radius. Take your pick which definition you prefer; they are equivalent. However, the latter, more usual definition, smacks a bit of obfuscation for my own personal preferences.

We don’t always measure an angle this way. When we specify a gradient of a hill, for example, we quote it in terms of the distance we need to travel in order to rise one unit, e.g. 1 in 20 means we must travel 20 metres in order to climb one metre.

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Figure 6-28 : Gradient of a Path

There is usually a pointless argument as to whether the ‘distance travelled’ is the horizontal distance or the distance along the slope. Now, if the gradient were greater than one in three, we would either do something to reduce it, or choose an alternative route.

There are very few, if any, practical cases where the gradient will exceed one in three. Within this limitation, if we take the distance travelled as the radius of the circle (r), there is hardly any difference between the arc length (s) and the difference in height (h). There is also hardly any difference between the horizontal distance (x) and the radius.

So for typical gradients:

sr≈ hx≈ hr

So for small angles (less than 1/3rd of a radian) the measures are, near enough, the same.

Suppose now we extend the gradient approach to defining the angle such that we can apply it to larger angles. We run into two difficulties. The first is one of convention. Since at larger angles x≠r, we must specify whether the gradient refers to distance travelled along the slope, or to horizontal distance travelled. The second problem is the arc length (s) deviates considerably from h as the angle increases.

The two ways of measuring gradient are evidently functions of the angle itself. Rather than give them intelligible names (e.g. call h/r ‘gradient’ and h/x ‘slope’) the world of mathematics brings wonderful clarity to the problem by calling the ratio of h/r the ‘sine’ of the angle, and ‘h/x’, the ‘tangent’ of the angle. Well, those are the names chosen for the functions, they are universally accepted, if obscure, so we have no choice but to use them here.

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Figure 6-29: Angles Become Large - We Need Trigonometry

The sine and tangent are two functions of an angle which relate it to the ratio of two perpendicular linear measurements. They are called ‘trigonometric’ functions, I believe named after the trigon or triangular harp of the ancient Greeks, because, I guess, of their association with a right angled triangle.

6.15.3 The Full Set of Trigonometric FunctionsIf we draw a right angled triangle, and call the angle shown θ (theta), denote the side opposite it ‘o’, the side adjacent to it as ‘a’, and the remaining side (the hypotenuse) ‘h’. From the previous discussion:

The sine of the angle is written:

sin θ=oh

The tangent of the angle is:

tanθ=oa

The inverse functions are written:

tan−1x=θ

There is a third possible ratio of the sides of the triangle, which is the ratio of the adjacent to the hypotenuse, this is called the ‘cosine’ , indicating a relationship with the sine function which is not immediately obvious:

cosθ=ah

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This is just a man-made naming convention which cannot be derived from first principles, so we have no alternative but to memorise the function names. The commonest mnemonic is to think of a South Sea island paradise and give it the name SOHCAHTOA, to remember that Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse and Tangent is Opposite over Adjacent.

If we find ourselves having to memorise excessive numbers of ‘facts’ rather than derive ideas from more fundamental ideas, we must suspect that the associated body of ‘knowledge’ is man-made convention rather than principles which govern reality. As Henri Poincaré put it; ‘facts do not constitute knowledge any more than a pile of bricks constitutes a house’.

Knowledge consists of understanding how facts inter-relate, so that we can predict the outcome of circumstances which have not yet been encountered. It is the closest thing we have to a practical gift of prophesy.

I have always considered suspect the requirement of religious fanatics to memorise verbatim their respective holy books, without applying reason to what is actually contained in them.

In addition to these three basic trigonometric functions, their reciprocals have also been given names.

The reciprocal of the cosine (NOT the sine) is given the name ‘secant’, the reciprocal of the sine is called the ‘cosecant’, just to avoid any possibility of confusion, but the reciprocal of the tangent is called the ‘cotangent’. These names are needed because we have used the index -1, usually used to denote a reciprocal, to mean an inverse function in this case.

These are written:

secθ= 1cosθ

csc θ= 1sin θ

cot θ= 1tan θ

The reciprocal of any trigonometric function is itself a defined trigonometric function. For this reason a superscript -1, is not used to signify a reciprocal, but to indicate the inverse of the function, e.g:

x=sin θ

Implies:

θ=sin−1 x

and similarly for the inverses of all the other trigonometric functions.

An alternative notation attaches the prefix ‘arc’ to the function name:

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If: x=sin θ

Then: θ=arc sin x

This latter notation, despite being less ambiguous, is not as common as the superscript -1 notation.

6.15.4 Relationhips Between Trigonometric FunctionsIf we take the hypotenuse of the triangle to be unit length, it follows that the opposite side will have length sinθ and the adjacent will have length cosθ. We have immediately:

tanθ= sin θcosθ

I have assumed the reader is familiar with Pythagoras Theorem. If not, I apologise for using it earlier in the calculation of the surface of a cone, and probably causing unnecessary confusion.

My preferred proof considers a right angled triangle with adjacent side length x and opposite side of length y, the length of the hypotenuse is r.

First we must express right angles in radians for consistency with the rest of the text: A right angle is a quarter of a turn of a wheel. We have seen from the derivation of the circumference of a circle that a full turn will advance the wheel by 2π times the radius, so a full turn is 2π radians. A half turn (180 degrees) is π radians, and a quarter turn is π/2 radians.

The internal angles of a triangle therefore add up to π radians. The angles of the triangle are

therefore θ,π2 and

π2−θ.

Figure 6-30: Re-Arrange Four Trianles Inside a Square to Demonstrate Pythagoras' Theorem

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We start by drawing a square of side (x+y). Four triangles may be arranged as shown within the square . The hypotenuses of the four triangles form a quadrilateral. Examining the angle at a corner of this rectangle yields the result:

φ=π−(θ+( π2 )−θ)= π2Since all its sides are equal in length (=r), and the corners are right angles, the enclosed figure is a square of side r. The area of the square containing the four triangles is four times the area of one triangle plus the area of this square made up of the hypotenuses of the four triangles:

( x+ y )2=4( xy2 )+r2From which:

x2+ y2=r2

Or, in the usual archaic language: the square on the hypotenuse equals the sum of the squares on the other two sides.

For our triangle of unit length hypoteneuse we have, as a relationship between sine and cosine:

sin2θ+cos2θ=1

Also:

1+t an2θ=sec2θ

And:

1+cot2θ=csc2θ

6.15.5 Special CasesThere are a few cases where the trigonometric functions may be found by inspection.

Consider first the case θ=45 (π/4 radians). The triangle is isosceles. If the shorter sides are of length b units, the hypotenuse is √b2+b2=√2b

It follows that:

sin( π4 )=cos( π4 )= 1√2

And:

tan( π4 )=1

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Another interesting case is θ=60 (π/3 radians) This is half an equilateral triangle. If the hypotenuse is b, the adjacent side is b/2 and the opposite side is:

o=√b2−( b2 )2=√32 b

From the geometry of the triangle, this also yields the trig. functions for 30 (π/6):

sin π3=cos π

6=√3

2

cos π3=sin π

6=1

2

tan π3=cot π

6=√3

6.15.6 Functions of Sums and Differences of AnglesThe sine, cosine and tangent are readily available as keystrokes on a scientific calculator. The inquisitive reader will, no doubt, wonder how this is done. The Taylor series expansion of the previous section implies that if we can differentiate these functions, we can produce a polynomial series which will enable us to calculate them for specific values of the argument.

In order to differentiate, say a sine function, we need to be able to deal with expressions of the form:

y+δy=sin ( x+δx )

In order to evaluate this, we must be able to evaluate the sine of the sum of a pair of angles in terms of functions of the two angles taken separately.

e.g. we need to evaluate expressions of the form:

sin (θ+ϕ ) , cos (θ+ϕ )

Construct a right angled triangle ABC,such that angle CAB=θ. Draw a line at angle φ to AC and draw a line perpendicular to AC passing through C, meeting this line at D. AD is of length secφ.

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Figure 6-31 : Trigonometric Functions of Sums of Angles

Draw a line perpendicular to AB, passing through D and intersecting AB at N. Then

sin (θ+ϕ )=DNAD

=DN cosϕ

The intersection of AC and DN is point E. Since AEN and DEC are vertically opposite, they are equal, so that angle EDC=angle EAN=θ. Hence:

DC=AD sinϕ=sec ϕsinϕ

DE=DC secθ=sec θ tan ϕ

Also:

EN=AE sin θ=( AC−EC ) sin θ=(1−DC tan θ ) sinθ

Or:

EN=(1−tan θ tan ϕ )sin θ

Hence, using the Pythagoras relationship between sine and cosine:

DN=DE+EN=secθ tan ϕ+sin θ−sec θ (1−cos2θ ) tan ϕ

Or:

DN=sin θ+cosθ tan ϕ

The required result for the sine of the sum is:

sin (θ+ϕ )=sin θ cosϕ+cosθ sin ϕ

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The cosine is:

cos (θ+ϕ )= ANAD

=AE cosθ cos ϕ=(1−tanθ tan ϕ )cos θ cosϕ

Or:

cos (θ+ϕ )=cosθ cosϕ−sinθ sinϕ

And:

tan (θ+ϕ )= sin (θ+ϕ)cos (θ+ϕ )

¿ sinθ cosϕ+cosθ sin ϕcosθcosϕ−sin θ sinϕ

Dividing top and bottom by cosθcos φ :

tan (θ+ϕ )= tan θ+ tan ϕ1−tanθ tan ϕ

We are now in a position to consider the derivatives of these functions.

6.15.7 Derivatives of Trigonometric FunctionsWe proceed as before by considering the change in the value of the function corresponding to a small change in the value of the independent variable. The derivative is what remains as these changes tend to zero.

For the sine function:

y=sin x

y+δy=sin ( x+δx )=sin xcos δx+cos x sin δx

Recall our discussion about hill gradients, that if the angle is small, the horizontal distance is almost equal to the distance along the slope, so that the cosine of a small angle is very nearly 1, and the sine of the angle is very close to the angle expressed in radians. These approximations become ever more exact as the angle approaches zero, so we are justified in writing:

y+δy=sin x+cos x δx

Eliminating y:

δy=cos x δx

In the limit:

dydx=cos x

The same procedure, using the same small angle approximations, yields for the cosine:

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ddx

(cos x )=−sin x

From the derivatives of the sine and cosine, it is evident that the second derivative must be:

d2

d x2(sin x )=−sin x

d2

d x2(cos x )=−cos x

Like the exponential function, these functions imply the associated system contains negative feedback; i.e. a tendency to correct itself.

The tangent is slightly more complicated:

y=tan x

y+δy= tan x+δx1−δx tan x

To first order of small quantities:

y− y tan x δx+δy= tan x+δx

Or:

δy=(1+ tan2 x )δx

The Pythagoras relationship for the tangent function yields:

δy=sec2 x δx

The derivative is, therefore:

dydx=sec2 x

The same approach works for the inverse function. Let:

y=tan−1 x

tan y=x

(1+ tan2 y )δy=δx

dydx= 11+x2

This is of the form:

y= y ( z )

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Where:

z=z ( x )

This form is very common, and the derivative is found from:

dydx=dydzdzdx

This can be shown by taking finite increments in x,y and z and considering the limits.

In the case of the derivative of tan-1x:

dydx=1z, ddz (dydx )=−1

z2

Where:

z=1+x2

It follows that:

d2 yd x2

= −2x(1+x2 )2

6.16 Binomial TheoremConsider now the Taylor series expansion of the binomial expression:

y= (1+x )n

From the reasoning of the previous section:

dydx=n (1+ x )n−1 dx

dx=n (1+x )n−1

So that:

( dydx )x=0=n

Repeated differentiation yields:

( dk yd xk )x=0

=n (n−1 )… (n−k+1 )= n !(n−k ) !

The kth term of the Taylor series expansion is therefore:

1k ! ( d

k yd xk )x=0

xk= n!(n−k )!k !

xk

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This result is known as the Binomial Theorem. We have shown from our derivation of the areas under curves, that the formula applies to fractional, as well as negative values of n.

6.17 A Few TricksGiven sufficient ingenuity and patience, the simple formula for differentiation will get there in the end. If all else fails, the derivative may be found by finite increments in the variables. For many functions that is extremely tedious, as we found from the higher derivatives of tan -1 x.

There are a couple of tricks which can speed up the evaluation of awkward derivatives.

A common case is when the function is a product of two simpler functions of the independent variable.

y=u ( x ) v ( x )

y+δy=u ( x+δx ) v ( x+δx )

y+δy=(u ( x)+ δuδxδx)(v (x )+ δv

δxδx)

Expanding to first order of small quantities:

y+δy=u ( x ) v ( x )+(v ( x ) δuδx+u ( x ) δv

δx )δxThe derivative is, therefore:

dydx=v ( x ) du

dx+u ( x ) dv

dx

In particular, if v(x)=1/z(x)

dydx= 1z ( x )

dudx−u (x )z ( x )2

dzdx=z ( x ) du

dx−u (x ) dy

dxz ( x )2

This form is more suitable for finding the nth derivative of tan-1x.

The significance of this series arises from the fact that:

π4=tan−11

So that expanding tan-1x as a Taylor series, and substituting x=1, we have a means of calculating π, which converges much more quickly than our initial attempt.

Our motivation for finding the derivatives of trigonometric functions was to satisfy our curiosity as to how to go about calculating them. We shall see that they crop up regularly in problems which have nothing to do with triangles.

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6.18 Differential Equations

6.18.1 IntroductionA differential equation is one which contains derivatives and functions of derivatives of the variable of interest, as well as functions of the variable itself.

We have met the simplest form of differential equation of the form:

dydx=f ( x )

Which is solved directly by integration.

The vast majority of differential equations do not have mathematically tractable solutions, and we must revert to numerical solution methods.

Numerical methods are fine for finding the solutions in the unknown variable when the parameters defining the equations (e.g. the coefficients of the functions of derivatives) have fixed numerical values. However, the insight we seek is invariably concerned with how the parameters affect the solution. Thus hundreds, if not thousands of cases must be run before such insight can be gained into the problem by handle-cranking a computer model.

All too often huge sums are invested in producing a high fidelity computer simulation of a system, generally by managers and software engineers who are ignorant of just how useless they are at providing the insight needed to support design or take remedial action in the event of performance shortfall.

Only after the design work is over, the dust has settled, and it is time to demonstrate the system meets its requirements are such simulations required. During the process of getting there, they are more trouble than they are worth. Most of the time we are not really interested in what the answer is, but what determines the features of the solution.

There exists a class of equation, which probably doesn’t really represent any real world system exactly, but with sufficient ingenuity, may be used as a working approximation for practically any self-correcting system within clearly defined bounds of validity.

That the representations are not exact does not worry us unduly, because a self-correcting systems which use negative feedback as the correction process (like the cooling bowl of soup), are inherently robust to uncertainties in their mathematical descriptions. In any case, there exist methods which can quantify the level of uncertainty which can be tolerated, before this inherent robustness is lost.

6.18.2 First Order EquationThe differential equations of interest are called linear homogenous differential equations. We met one when considering soup going cold.

6.18.2.1 The Bowl of SoupThe differential equation took the form:

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dTdt=k (T s−T )

We converted it into a simple integral by noticing that if the temperature of the surroundings is constant, its rate of change would be zero, so it would not affect the derivative.

However, this would not be valid if the temperature of the surroundings were not constant.

More generally, this would take the form:

dTdt

+kT=k T s

The solution for temperature difference was an exponential, so that seems a reasonable guess for the actual temperature:

T=b (t ) e−at

T eat=b (t )

Differentiating:

eat dTdt

+aeatT=dbdt

Comparing coefficients:

ekt (dTdt +kT )= ddt

(T ekt )=ekt k T s

Integrating:

[T ekt ]0t=∫

0

t

k T s ektdt

In the case of constant temperature surroundings:

T ekt−T (0 )=T s ekt−T s

(T s−T )=(T s−T (0 ) )e−kt

This is expected from our earlier derivation.

6.18.2.2 Charging a CapacitorWe have seen that the discharging of a capacitor was governed by the exponential function. Changing the circuit slightly, we can consider the process of charging. In this case, the resistor R and the capacitor C are connected in series, and closing the switch connects the combination across the terminals of the battery.

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Figure 6-32 : CR Circuit Again

The objective is to find the voltage across the capacitor as a function of time. Remember that the current passing into the capacitor is given by:

i=C dVdt

And the voltage drop across the resistor is:

E−V=iR=CR dVdt

The resulting equation is:

dVdt

+ 1CR

V= 1CR

E

This is exactly the same form as the equation governing the cooling of the soup.

The left hand side is:

e−tCR ( ddt (V e

tCR ))

The solution is, therefore:

[V e tCR ]0

t

= 1CR∫0

t

EetCR dt

If E is constant:

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V etCR−V (0 )=E(e

tCR−1)

The capacitor is initially discharged, so V(0)=0, and the charging voltage is:

V=E (1−e−tCR )

6.18.2.3 Hot Air BalloonA hot air balloon flies because the weight of air it displaces is equal to its own weight. Ascent and descent is controlled by using a heat source to adjust the density of the gas in the balloon, or by discharging ballast.

The lift of the gas bag is given by:

L=( ρair−ρgas) gV

Where: ρ denotes local air density, g the acceleration of gravity and V the volume of the gas bag.

The change of atmospheric pressure with height is given by the hydrostatic equation:

dpdh=− ρg

Where p is pressure, h is height. This states that pressure falls with height by an amount equal to the weight of gas per unit area.

The pressure is related to density and absolute temperature (T) by the gas equation:

p=ρRT

Where: R is the gas constant for the gas of interest. For air it has the value287 J kg -1 K-1.

Differentiating the gas equation:

dpdh=RT dρ

dh+ ρR dT

dh

In the troposphere, the temperature drops at a rate of 6.5 degrees Centigrade per kilometre. This is called the lapse rate and is denoted α (alpha). The hydrostatic equation, the gas equation and the lapse equation are combined to yield an expression for air density as a function of altitude:

R (T (0 )−αh ) dρdh− ρRα=−ρg

Or:

R(αR−g ) ∫ρ (0 )

ρ dρρ=∫

0

h dhT (0 )−αh

The right hand side is integrated by letting

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z=T (0 )−αh

dz=−α dh

∫0

h dhT (0 )−αh

=−1α ∫

T (0 )

T (0 )−αhdzz

Integrating:

1

(1− gαR )

ln ρρ (0 )

= ln ( T (0 )T (0 )−αh )

Or:

ρ=ρ (0 )(1− αhT (0 ) )(

gαR

−1)

The sea level temperature is about 288 K, so if we are considering heights less than 1km, αh is very much less than T(0), so we are justified in treating the change in density with height as a nearly linear function:

ρ≈ ρ (0 )(1−( gαR−1) αhT (0 ) )(Apply the Binomial theorem expansion to the original formula and neglect all but the first term to get this).

A reduction in the lift gas density, will cause the balloon to rise to a new equilibrium level.

In order to avoid cluttering the equations with extraneous symbols, let:

β=( gαR−1) αT (0 )

(β is the Greek letter beta).

The change in lift gas density (assuming the balloon volume remains constant), is equal to the difference in air density between the current height and the new equilibrium height. The upwards excess buoyancy force (F) is therefore:

F=( ρ (h2 )−ρ (h1 )) gV

Expressed in terms of height difference:

F=β (h2−h1 ) ρ (0 ) gV

As the balloon is rising through the air, it will be subject to an aerodynamic drag force opposing its motion.

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At low speeds for objects the size of a typical hot air balloon we should expect the drag to be proportional to the square of the speed.

D=12ρU 2S cD

Where U is the upwards velocity magnitude, S is a reference area (typically the cross-section of the body) and cD is a drag coefficient based on the reference area chosen. We assume that the difference in density between the start and end heights is small, so that the density may be considered constant in this equation.

The balloon obeys Newton’s Second Law:

m dUdt

=F−D

Where m is the mass of the balloon and payload and t is time. This is not a particularly useful form of the equation. It is easier to solve if h is the independent variable:

dUdt

=dUdh

dhdt=U dU

dh=1

2d (U 2 )dh

The result is an equation in U2:

d (U 2 )dh

+ρS cDm

U2=2 β (h2−h ) ρ (0 )gV

m

This takes the form of a first order linear homogenous equation, like the other two examples, but illustrates the point that most of the effort is expended in formulating the problem. Unlike the mathematician, problems are not ‘given’ to the engineer. A formulation appropriate to the task in hand, making reasonable assumptions, must be found, and this is rarely a trivial exercise.

Evidently the left hand side of the equation of motion is:

e− ρS cDm

h( ddh (U 2eρScDm

h))The solution is found from:

U 2 eρS cDm

h=2 βρ (0 )gV

m ∫h1

h2

(h2−h ) eρScDm

hdh

This can be solved for height as a function of time:

dhdt=U

6.18.3 Integration by PartsOur hot air balloon example introduced an integral of the product of two elementary functions:

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∫a

b

x ex dx

We could expand this as a series and try to interpret the result in terms of familiar functions. However, when we last encountered an (at the time) intractable integral, we found that it could be evaluated by considering integrating to the left of the curve of interest, and subtracting the resulting area from the rectangle enclosing the curve.

Recall that the resulting formula was:

∫a

b

ydx=[ xy ( x ) ]ab−∫

y ( a)

y ( b)

xdy

In the case of the motion of the balloon, the function of interest is:

y=x eax

Differentiating, using the formula for the derivative of a product:

dydx=ex+ax eax

This appears to make matters worse, as it introduces an integral of the form:

∫a

b

x2 eaxdx

This situation arises quite frequently when using this method. All is not lost, however.

Consider the more general function:

y=xn eax

The derivative is:

dydx=n xn−1 eax+axn eax

Denoting, for convenience:

∫a

b

y dx=I n

Then:

I n=[ xy (x ) ]ab−n In−aI n+1

Or:

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I n+1=1a ( [ xy ( x ) ]a

b−(n+1 ) I n)

Setting n=0, we have:

y ( x )=ex

∫a

b

x eaxdx=1a ([ x eax ]ab−∫a

b

eax dx)In the example of the balloon, we need to evaluate:

∫h1

h

(h2−h )eρScDm

hdh

To reduce clutter, let:

ρS cDm

= 1hc


h2∫h

h2

ehhc dh−∫

h

h2

hehhc dh=h2hc [e

hhc ]hh2

−hc[hehhc ]hh2

+hc2 [e hhc ]h

h2

The solution for the speed squared is:

U 2=2 βρ (0 )gVρ (h1) ScD

((h2−h+hc )−(h2−h1+hc )e(h1−h)hc )

The reader is welcome to try and solve this explicitly for h as a function of time.

If h=h1, the vertical velocity is zero. The apogee is reached when this velocity is again zero.

Let:

z=(h1−hhc )The apogee height is given by:

(( h2−h1hc )+1) (1−ez )+z=0

Thinking about the problem, we notice that balloons tend to be rather large, say about 10m in diameter. Also they usually are spherical or some other bluff shape when moving vertically, so the drag coefficient based on cross section will not be much less than 1. Also the air density near sea level is expected to be about 1.23 kg m-3. A typical value for hc is about:

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hc=ρπ R2 cD≈1.23×π×102×1.0=386m

The second order term in the Taylor series expansion depends is:

12 ( h1−hhc )

2

If we are typically dealing with height changes of the order of 30m (≈100ft), ignoring this term introduces an error of about 0.3%, so we shouldn’t introduce much error if we expand the exponential term to second order.

The apogee height is given to a reasonable approximation by:

(( h2−h1hc )+1)(1−1−z− z2

2 )+z=0

The result is z=0, or:

(( h2−h1hc )+1)(1+ z2 )−1=0

z=−2( h2−h1hc )(( h2−h1

hc )+1)There will be no overshoot in height if:

z=−( h2−h1hc )This will occur if:

h2−h1=hc

Notice how the engineer brings in a wider understanding of the problem space which is not available to the mathematician. This awareness of the relative magnitudes of the quantities, and the simplifications it permits, is usually dismissed disparagingly by the pure mathematician as ‘ad hoc’. All too often it is easy to mistake one who can merely manipulate the algebra for one who actually understands the problem. Similarly, it does not follow that, because an individual can implement equations in computer code, they have any understanding of the subject matter whatsoever.

All too often the credit for the technical staff’s work goes to the software engineers who merely implemented it.

So, within the bounds of validity we have defined:

119


U 2=2 βρ (0 )gVρ (h1) ScD ((h2−h+hc )−(h2−h1+hc )(1+(h1−h

hc )+ 12 ( h1−hhc )

2

))6.18.4 More Exotic FormsThe solution for the hot air balloon vertical velocity is given by an expression of the form:

U 2=ah2+bh+c

Where a, b and c are constants, the equation for height as a function of time is found from:

dhdt=√ah2+bh+c

The closest we have come to functions of this form is with the Pythagoras relationship between trigonometric functions. The term under the square root may be written:

ah2+bh+c=(√ah+ b2√a )

2

+c− b2

4a

From which:

√ah2+bh+c=√c− b2

4 a √ x2+1

Where:

x=(√ah+ b

2√a )√c− b2

4a

Note that if the denominator took the common form : √1−x2 , we could use the substitutions:

x=sin θ

dx=cosθdθ

Also, for the form: 1+x2 we could use:

x= tanθ

dx=se c2θdθ

And for the form: 1−x2, we can factorise into one term in 1-x and another in (1+x):

∫a

b dx1−x2

=12 (∫a

b dx1−x

+∫a

b dx1+x )

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These common forms are derived directly from the properties of trigonometric functions, and elementary algebra. Many physical problems may be approximated using them. Unfortunately, the motion of the balloon is not one of them. It takes the form:

f ( x )=√x2+1If we are confident using complex numbers, we could use the substitution:

x= jsin θ

dx= j cosθdθ

Where j=√−1 .

However, I assume the reader has either not yet encountered complex numbers or has forgotten about them over the years.

We are seeking a function y(z) such that:

y2=x2+1

dxdz= y

Which implies:

y dydz=x dx

dz

dydz=x

The second derivative is:

d2 yd z2

=dxdz= y

The required function is governed by the equation:

d2 yd z2

= y

Remember, from the derivative of sine and cosine, that the equation:

d2 yd z2

=− y

Has a solution involving sine and cosine terms. The solution with a positive coefficient of y is in terms of functions, which by analogy are called hyperbolic sine (sinh) and hyperbolic cosine (cosh).

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We notice that the equation governing our desired function exhibits feedback. The feedback is positive, so we should expect the solution to diverge as the independent variable z increases.

From our discussion about e, we note that if y=ez or y=e-z:

d2 yd z2

=e z∨e− z

It follows that:

y=A ez+B e−z

dydz=x=Aez−Be− z

Where A and B are constants.

Now: y2=x2+1

A2 e2 z+2 AB+B2 e−2 z=A2e2 z−2 AB+B2 e−2 z+1

By convention A=B, so 4A2=1; A=½.

The result is the definition of the hyperbolic functions:

cosh ( z )= ez+e− z

2

sinh ( z )= ez−e−z

2

(Read this as ‘cosh z, shine z’.

So the hot air balloon height against time, and any other integral containing the √ x2+1 factor, takes the form of an inverse hyperbolic function.

6.19 The Lanchester Equation of CombatA rather grim example of a divergent solution is furnished by the equation of combat, formulated by Frederic W Lanchester, the automobile and aviation pioneer, in 1916.

Consider two opposing armies engaged in combat. We shall call them blue and red. The rate of loss of blue troops is assumed proportional to the number of red troops and vice versa.

We have:

dBdt=−kR R

dRdt=−kBB

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Where t is time, B and R are the numbers of surviving troops in the blue and red armies respectively, kB and kR are the effectiveness of the blue and red army respectively. Considering blue’s fortunes (the equation for red is similar):

d2Bd t 2

=k B k RB

From the previous section, this has solution:

B=ae√kR kB t+be−√kR kBt

When time =0, blue starts the battle with B0 troops.

a+b=B0

dBdt=−kR R=√kBk R (ae√kB kR t−be−√kBk Rt )

When time =0, red starts the battle with R0 troops.

a−b=−√ kRkB R0

Solving for a and b:

a=12 (B0−√ k Rk B R0)

b=B0−a=12 (B0+√ k Rk B R0)

We have:

B= 12 (B0−√ kRkB R0)e√kB kR t+ 1

2 (B0+√ k Rk B R0)e−√kB kRtOr, more neatly:

B=B0 cosh (√kR kB t )+√ k Rk B R0sinh (√kRk B t )

This is correct, but it doesn’t shed much light on the problem in this form. For convenience, and the avoidance of writer’s cramp, we shall make the substitutions:

√ kRkB=αe√kB kR t=x

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The equation now becomes:

B=12 (B0−α R0 ) x+

12 (B0+α R0 )

1x

Similarly:

R=12 (R0−

1αB0)x+ 1

2 (R0+1αB0) 1x

If we square both these equations and then subtract one from the other, we end up with B as a function of R:

B02−B2=α2 (R0

2−R2 )

We can derive an expression for x from either equation:

12 (B0−α R0 ) x

2−Bx+12 (B0+α R0 )=0

This is a quadratic equation having solution:

x=B±√B2−B0

2+α2 R02

(B0−αR0 )

When time=0, x=1, B=B0, so the square root must be negative.

The solution becomes:

x= (B−αR )(B0−α R0 )

√k BB−√kR B=(√kB B0−√k R R0 )e√kR kBt

The difference in the two armies increases exponentially as the battle proceeds, so that the side which is weaker will eventually be wiped out. There is a stage in the battle when it becomes evident that the situation is hopeless, and the rational commander would surrender. The history of warfare indicates that commanders feared the charge of cowardice more than catastrophic and pointless losses. It is difficult to understand why the bards wax so lyrical in celebrating military incompetence.

In the First World War, Admiral Troubridge faced a court martial for failing to expend his cruiser squadron in a futile attempt to prevent the German battle cruiser Goeben from crossing the Mediterranean to Turkey. Evidently the Navy was determined not to be beaten by the Army in the reckless stupidity stakes.

If the effectiveness of the two sides in the context of the battle, were known beforehand, the rational action would be for the weaker side to surrender without a fight, unless the purpose of the battle were to delay the enemy whilst reinforcements are being sent.

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Nearly all military tactics are geared to gaining an advantage locally, so that the enemy is outnumbered at a particular point of contact. This can only be done by introducing potential weakness elsewhere. Winning or losing depends on denying accurate knowledge on the disposition of one’s own forces to the enemy.

Napoleon relied heavily on a network of spies able to provide intelligence of potential enemies raising armies. A pre-emptive strike whilst the French Army held the advantage in numbers not only defeated the unprepared enemy, but also tacked their country on to the French Empire.

Hannibal destroyed a Roman force four times the size of his own at the Battle of Cannae, by encircling it, ensuring only the relatively small number of Roman soldiers on the perimeter could actually make contact, most of the army was confined inside the crowd and couldn’t bring weapons to bear. Similarly, tales of heroes holding bridges or narrow passes illustrate the importance of maintaining superiority only where the opposing forces met.

Modern wars rely heavily on advanced remote sensing from aircraft and satellites as well as spies on the ground, to gain accurate knowledge of enemy disposition and movements, whilst at the same time denying such intelligence to the enemy by decoys, camouflage and concealment.

To quote Sun Zi, the earliest published authority on war from the 5th Century BC; ‘where you are strong appear weak, where you are weak appear strong’.

If the two sides are evenly matched, the duration of the battle tends to infinity, and even the winning side loses nearly all its troops. The rational behaviour would be to avoid such a conflict by making sure one can match the military power of a potential aggressor. ‘ Let him who seeks peace, prepare for war’ (Flavius Vegetus Renatus).

The major problem is the build up to war is itself an unstable process, as during this stage opposing forces will try to deploy to more favourable positions for the conflict. Such action is inevitably viewed as hostile, encouraging the side whose advantage is being eroded to fire the first shot.

6.20 Impulse ResponseIf we consider the derivation of the differential equations considered so far, they take the form:

∑0

n

aid i yd t i

=f (t )

Where t is an independent variable, which the balloon example shows, is not necessarily time, although usually it is. The ai are constants.

Evidently, the left hand side is derived from the dynamic behaviour of the object of interest, whilst the right hand side is a disturbance applied to the object.

The right hand side is the description of the input, the equation itself is a mathematical model of the objects behaviour, and the dependent variable y is the output. The response to one particular input tells us very little about how the object will respond to other inputs.

We need to find an elementary input from which the output to any input may be derived.

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The idea of an impulse was mentioned with the example of rocket acceleration. My apologies for introducing a concept without first explaining it.

The impulse of a force is the product of the force and the time it is applied for, as the duration of the application of the force becomes insignificant. In essence, the force may be treated as impulsive if there is no significant motion during its application. Examples include hitting a golf ball with a club.

We may treat a continuous input function approximately as a sequence of impulses of finite duration; the approximation converges on to the function as the duration tends to zero.

If the equation is linear, the complete solution may be found by combining the outputs resulting from a sequence of impulses. The starting point is to discover the response to an impulsive input.

In general the output in response to a unit impulse applied at time t=0, may be written as h(t), so that an impulse applied at a later time (t1) will produce an output h(t-t1) . The amplitude of the input is f(t1)δt, where δt is the duration of the impulse. The output is:

y ( t )≈∑1

n

f ( iδt )h ( t−iδt ) δt→∫0

∞

f (t 1 )h ( t−t1 )d t1

If the impulse response is known, the response to any other input should be calculable from the above integral.

As an example, consider the charging capacitor:

dvdt+ 1CR

v= 1CR

E (t )

ve1CR= 1

CR∫0δt

E ( t ) etCR dt

We define δt as short compared with the characteristic time of E(t), so over the interval of integration; E(t)=E(0).

v=E (0 ) [e δtCR−1] e−tCRExpanding the exponential in the square brackets to first order:

v=E (0 ) δtCR

e−tCR

E(0)δt is a voltage applied very briefly, or a voltage ‘spike’. The effect of a spike depends on both its amplitude and duration. We are interested in the case where the product of the two has unit value (1 Volt sec), so that it may be scaled by an arbitrary input. The shorter the spike, the larger its amplitude.

Replacing the input spike strength E(0)δt, with unity, we have the system response to a unit input impulse:

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v= 1CR

e−1CR

The astute reader will notice that this equation is apparently dimensionally inconsistent. The left hand side has units of volts and the right hand side has units sec -1. This is usually glossed over, or ignored completely, since the student can get away with not fully understanding the physical meaning when manipulating the maths.

The engineer, on the other hand, must interpret the maths in terms of real world behaviour.

Ideally, the pure number 1 should be replaced with a symbol, such as I, to indicate that it has dimensions consistent with the output and the independent variable, but unity magnitude.

We usually hide this conceptual difficulty by defining a function δ(T) which is zero everywhere apart from where time=T.

∫0

∞

E (t ) etCR δ (0 )dt=E (0 )

This delta function allows us to write the answer for the impulse response immediately, provided we don’t comment on the dimensional inconsistency of the result.

Comparing the result for the impulse response with the response with no input, i.e. starting with an initial voltage across the capacitor:

V=V 0e−tCR

This should not be surprising, since the impulse response may be written:

dvdt+ 1CR

v= 1CR

E (t ) δ (0 )

Or: v0=E (0 )−CR ( dvdt )t=0

So that starting the system with an initial value of output has the same effect as applying an impulse to the input. The response following an initial displacement is called the transient response. Note that if there is no feedback, the system will remain in its initial state indefinitely.

Evidently, for any system governed by a first order homogenous differential equation of the form governing capacitor discharge, cooling soup or hot air balloons, i.e:

dydt+ay=f (t )

The transient response, and the impulse response take the form:

y= y0eλt

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Where λ (lamda) is a constant.

y0 eλt ( λ+a )=f (t ) δ (0 )

This holds for time>0. Also the exponential cannot be zero, and since we are considering transient response, neither is y0. The transient response is given immediately by:

λ+a=0

This is called the characteristic equation. We see that if it has a negative root, the feedback is negative, and the state will converge on to zero. If positive, the solution diverges indefinitely.

Most of the time we are only concerned with whether the solution is stable or unstable, we are rarely interested in solutions for specific cases expressed as functions of time. This information is immediately furnished by the characteristic equation.

6.20.1 Second, and Higher OrderThe relationship between the transient response and the impulse response may be extended to equations of any order, since if:

∑0

n

aid i yd t i

=f (t )

y0=1a0f ( t ) δ (0 )−∑

1

n

( d i yd ti )t=0

The Lanchester Equation showed that a second order equation of the form:

d2 yd t2

=ay

Has solutions in terms of the exponential function. The characteristic equation has the form:

λ2−a=0

This has one positive root, so the solution diverges.

The solution may be written in the form:

y=A cosh (√a t )+B sinh (√a t )

Recall, from the differentiation of the trigonometric functions, the equation:

d2 yd t2

=−ay

Has solution of the form:

y=A cos (√a t )+B sin (√a t )

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This is not obviously an exponential form.

We shall introduce the constant j=√−1 , mainly to give a physical meaning to it in terms of the solutions to these equations. After all, we know the form the solutions take; it is the number j which looks a bit fishy. It is more sensible to define what we don’t know in terms of what we do know, than vice versa.

The second order equation becomes:

d2 yd t2

=−ay= j2ay

This has solution:

y=A cosh ( j√a t )+B sinh ( j√a t )

When t=0, y=y(0), so A=y(0)

dydx= j √a (A sinh ( j√a t )+Bcosh ( j√at ) )

From which:

B=− j√a ( dydx )t=0

The solution in terms of trigonometric functions is:

y=acos (√a t )+b sin (√a t )

From which:

a=A ,b= jB

We can now write sine and cosine functions in terms of exponentials:

cos x=cosh jx= ejx+e− jx

2

sin x= sinh jxj

= ejx−e− jx

2 j

When the roots of the characteristic equation contain square roots of negative numbers, it means that the solution involves an oscillatory function of time. In this case y will vary sinusoidally in time

at a frequency given by √a radians per second, or 12π √a Hz.

By introducing j, we can characterise all possible solutions as exponential functions.

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6.20.1.1 Torsion Pendulum

Figure 6-33: Torsion Pendulum

A wheel of moment of inertia I is fixed rigidly to a stiff axle. A spiral spring is attached to the axle, and its free end is rigidly fixed so that it is kept stationary; it generates a restoring moment proportional to the angular deflection θ of the wheel.

The equation of motion is:

I d2θd t 2

=−kθ

Where k is the stiffness of the Torsion bar (the torque per unit deflection).

The characteristic equation is:

λ2+ kI=0

The wheel will oscillate back and forth at frequency 12π √ kI Hz.

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6.20.2 Resonant Circuit

Figure 6-34 : Inductance/Capacitor (LC) Circuit

The circuit shown consists of an inductance L, connected in parallel with a capacitor C, a short pulse is applied across the combination.

The voltage across the inductor is proportional to the rate of change of current through it. The rate of change of voltage across the capacitor is proportional to the current charging or discharging it.

We have, for the inductance:

v=−L didt

For the capacitor:

dvdt= iC

Differentiating:

d2 vd t2

=−1LC

v

In the absence of resistance, or radiation by means of an aerial, the circuit will ring indefinitely at a

frequency 1

2π √LC Hz.

6.20.3 Pin Jointed StrutConsider a problem which baffled the Ancient Greeks. How slender can a column be before it buckles and collapses?

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At any point along the strut, the bending moment is given by the end load multiplied by the deflection. From engineer’s bending theory the elastic bending the elastic restoring is proportional to the radius of curvature, the moment at any point along the strut is given by:

M=EI d2 yd x2

+Py

Where P is the end load.

Figure 6-35 : Buckling Load of a Strut

There will be a further moment arising because each particle of the beam is being accelerated laterally as the beam deflects. This manifests itself as a lateral loading on the beam. The loading is given by:

w=−ρ d2 yd t2

Where ρ is the mass per unit length of the beam.

Considering the equilibrium of an element of beam, we find the rate of change of shear force Q is equal to the loading:

w=dQdx

Also from equilibrium considerations, the shear force is equal to the rate of change of bending moment along the beam:

dMdx

=Q , d2Md x2

=w=− ρ d2 xd t 2

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The equation of motion is:

ρ d2 yd t 2

+ d2

d x2 (EI d2 yd x2

+Py)=0

For a strut which is free at the top, but prevented from lateral displacement at the base, the lateral deflection is expected to look like:

y= y tipcos( π2 xL )Where L is the length of the strut, and ytip is the lateral deflection at the tip.

d2 yd x2

=− y tip ( π2L )2

cos( π2 xL )Also:

d2 yd t2

=d2 y tipd t 2

cos ( π2 xL )The equation of motion becomes:

d2 ytipd t2

+ 1ρ ( π2 L )

2(( π2 L )2

EI−P) y tip=0


λ2+ 1ρ ( π2L )

2(( π2L )2

EI−P)=0

If the load is less than:

P= π2EI4 L2

The strut will vibrate back and forth when subjected to a disturbance.

A load greater than this will cause the tip deflection to diverge, and the strut will buckle.

This is a more interesting stability problem because it determines circumstances where a stable system can become unstable.

6.20.4 Mass – Spring – DamperThe examples considered so far containing just second and zeroeth order derivatives either diverge or exhibit a simple harmonic motion. Neither can really be described as ‘stable’ responses because there is no tendency for the dependent variable to decay to zero as in the first order case.

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The hot air balloon appeared to furnish an example in which height could be controlled without this perpetual oscillation above and below the desired height. Unfortunately, the governing equations were not linear, and do not form the basis of a general theory of control. However, they do illustrate the point that non-linear controllers can be quite satisfactory.

In the physical examples above, we notice that strain energy in a spring is converted to kinetic energy in a mass and back again, or energy is stored alternately in the magnetic field of an inductance and the electric field between the capacitor’s plates. The result is a system which oscillates indefinitely. Such systems have a repeatable period of oscillation and form the basis of a clock.

However we would not expect a real torsion pendulum to oscillate forever, and the oscillations of a resonant circuit are expected to decay rapidly in practice.

The real world contains friction and electrical resistance which convert ordered energy, such as electrical, kinetic or elastic into disordered energy (heat), which is then lost to the surroundings.

It must be emphasised that not every system governed by a second order equation represents a transformation of ordered energy. However, the presence of dissipation in these examples indicates how we may cause a more general system to converge.

If the wheel of the torsion pendulum were immersed in a viscous fluid, the oscillations would die out quickly. A reasonable model for viscous drag sets the drag force or moment directly proportional to the linear or angular velocity.

The torsion pendulum immersed in thick oil is now governed by an equation of the form:

I d2θd t 2

=−c dθdt−kθ

Where c is the viscous drag term, or the ‘damper’ term.

The characteristic equation now takes the form:

λ2+( cI ) λ+( kI )=0

From which:

λ=12 [−( cI )±√( cI )2−4( kI )]

The viscous damper provides feedback of angular velocity, whilst the stiffness provides feedback of deflection. If either introduces positive feedback, one of the roots will become positive, and the system will be unstable.

If the oil is really thick, or the stiffness weak:

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( cI )2

≫ 4( kI )The quantity under the square root is positive, so there are no complex roots. Both roots correspond to exponential decays. Unfortunately, the smaller root of the equation would tend to zero, implying an infinite settling time.

Bearing in mind the fact that the exponential decay never actual reaches zero, it might be desirable to ensure the solution retains its oscillatory behaviour.

In the absence of damping a second order system is characterised by a constant amplitude oscillation. The frequency of the oscillation is called the undamped natural frequency, denoted ω0

(omega nought). In this particular case:

ω02= kI

It is desirable to have a standard form of the equation for a second order system, so that our description of the behaviour applies to all systems, and not just this specific example.

The coefficient of the damping term has dimensions of frequency, so it is reasonable to express it as proportional to the undamped natural frequency.

cI=2 ζ ω0

Where ζ (zeta) is called the damping ratio.

The characteristic equation has the general form:

λ2+2 ζ ω0 λ+ω02=0

This is a true system representation, as it deals with behaviour independently of the actual hardware exhibiting the behaviour. Such self-corrective behaviour is to be expected at any system level. It is not, as I have heard all too often from the ignorant and innumerate, a ‘detail of implementation’.

The roots of the characteristic equation in terms of the undamped natural frequency and damping ratio become:

λ=(−ζ ± j√1−ζ 2)ω0

The explicit solution is of the form:

θ=A e−ζ ω0 t sin (√1−ζ 2ω0t )+Be−ζ ω0 t cos (√1−ζ 2ω0 t )

The sinusoidal response is multiplied by an exponential decay, so that its amplitude reduces exponentially with time.

135


Figure 6-36 : Damping of 2nd Order System - 0.7 Seems Optimum

The value of damping coefficient which appears to yield the shortest settling time is about 0.7.

Actually, when the roots of the characteristic equation are equal, the solution ceases to be an exponential function, real, complex or otherwise, but introduces terms of the form:

t n eβt

Where β is the value of the root, and n is the number of times it is repeated. However, negative real parts still ensures the solution is stable. Although the form of the solution changes, the characteristic equation remains a valid means of assessing stability.

The equation for the resonant circuit with a resistor in series with the inductor takes exactly the same form.

6.20.5 Higher Order SystemsWe can tell all we need to know about the nature of the solution of a linear homogenous equation from its characteristic equation.

If we consider a third order system, we should aim to factorise the characteristic equation it into a quadratic and a simple factor, i.e:

(λ2+aλ+b ) ( λ+c )= λ3+(a+c) λ2+(ac+b ) λ+bc

If the coefficients of the cubic are real, then a,b and c are real. So the simple root must be real. This reasoning may be extended to a polynomial of any order, so it follows that any nth order characteristic equation may be factorised into simple and quadratic factors. The complete solution is the sum of the individual exponential functions.

136


If we have the solution for a first order and a second order linear homogenous equation, we have the solution for any order, provided we can factorise the characteristic equation.

The motion associated with each simple or quadratic factor is called a mode.

We can define the behaviour required of a self correcting system in terms of the time constants of exponential decays, and the undamped natural frequency and damping ratios of second order system responses, without in any way concerning ourselves with how the behaviour is to be implemented in hardware.

6.21 ConclusionsWe have presented calculus in a restricted form, as applicable to the ideas behind the subject of automatic control.

We started off by explaining integral and differential calculus, and derived a formula which applied to every mathematical function but the one most relevant to self-correcting systems, which is the one most relevant to the problem domain.

At each stage we considered an example which introduced a feature not yet covered, and some care has been taken to ensure ideas are not used before they are explained, and illustrated with an example. This ground work is itself a useful introduction to the wider subject of calculus.

After introducing methods of solving common types of differential equations, we discovered that in most cases the effort would be nugatory, as we are not usually interested in explicit solutions for specific systems, but in the general form the solution takes for entire classes of system.

The information we sought, we discovered, can be found from an algebraic equation, with no need at all to worry about solving the governing differential equation explicitly.

137


7 More Advanced Calculus

7.1 IntroductionThis chapter does not introduce any new concepts, it is ‘advanced’ in the sense that solutions are seldom of the form of simple polynomials, but require more obscure functions. Perhaps ‘more obscure’ calculus might have been a better title. Usually we must resort to numerical solutions, to produce the ‘right’ answer, typically however with absolutely no understanding or insight into the associated problem.

7.2 Arc Length and Surface areas of RevolutionWe calculated the surface areas of a few special cases by calculating the difference in volume of two similar shapes, one enclosed inside the other. This was only possible in the special case of the sphere or the cone, where the thickness of the space between the two surfaces was uniform. In general, that is not the case.

For a finite small change in the x coordinate(δx), the curve is very nearly a straight line, so the distance along the arc (δs) may be found by applying Pythagoras’ theorem:

(δs )2= (δx )2+(δy )2

Where δy is the change in the dependent variable corresponding to the change in the independent variable (δx).

This contains second order terms, which we usually set to zero, but as long as we are dealing with finite increments we may divide through by (δx)2:

( δzδx )2

=1+( δyδx )2

Now from our initial thoughts on the derivative, we know that:

δyδx→ dydx

As δx→0.

Since 1 is a constant, it is legitimate to claim that:

δsδx→√1+( dydx )2

As δx→0.

The length of an arc may be approximated by summing finite lengths corresponding to equally spaced small intervals in the x coordinate:

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S=∑i=0

n

iw√1+((dydx )x=iw)2Where w is the width of the x interval. As the width tends to zero, the sum becomes an integral, yielding an expression for the arc length, which is now exact:

S=∫a

b

√1+(dydx )2dxSimilarly, the surface area of a body of revolution may be found from:

S=2π∫a

b

y √1+( dydx )2dxAlso, the curvature at any point is given by:

1R=dθds=dθdxdxds=

dθdx

√1+( dydx )2Where θ is the angle the tangent to the curve makes with respect to the x axis, and R is the radius of curvature. Its tangent is equal to the slope of the curve.

dydx=tan θ ; d

2 yd x2

=sec2θ dθdx=(1+(dydx )

2) dθdxThe curvature is, therefore:

1R=

d2 yd x2

(1+( dydx )2)

32

Typically, we deal with small values of slope (bending of beams, streamlines around a thin aerofoil), and the denominator is taken as unity. We can estimate the error in ignoring the denominator by calculating it for the maximum value of slope, and finding the effect on the particular problem solution.

The presence of the square root complicates the finding of analytical solutions to this type of problem.

Fortunately, we have already encountered functions which are related via square roots in this way. Recall the solution to Lanchester’s Equation, and the hot air balloon example, introduced the hyperbolic sine and cosine:

139


sinh x= ex−e− x

2

cosh x= ex+e−x

2

From which:

cosh2 x−sinh2 x=1

as can be seen by direct substitution.

Also, we have met their derivatives:

ddx

(sinh x )=cosh x

ddx

(cosh x )=sinh x

Another function which might help us is the tangent, which we met in our discussion of trigonometry:

1+ tan2 x=sec2 x

The tangent function has derivative:

ddx

(tan x )=sec2 x

Similarly, whenever expressions of the form:

√1−x2

Crop up in integrals, we might consider using the relationships between sine and cosine functions.

7.2.1 Why Bother?We may evaluate finite integrals using a computer, so why do we search for analytical solutions? The answer is that we do not seek numerical answers to specific problems, what is much more valuable is the understanding of how the solution is affected by the problem parameters.

Very few problems are ever formulated as a once-through ‘sausage machine’ producing the ‘right’ answer. Most of our analysis is concerned with finding out how things are interrelated so that we have some means of improving on each iteration. Without this insight, we are shooting in the dark, with no means of deciding in which aspect of the system the change is required.

For this reason an analytical approximation to the real world problem is worth a thousand handle-cranked computer generated numbers, and the months spent getting largely irrelevant code to work. One of the tragedies of our age is the widespread failure to grasp the difference between data and useful information.

140


We seek the inter-relationships between facts, because this is what constitutes knowledge. Facts on their own are akin to memorising the telephone directory – requiring much effort but of little use to man or beast.

7.3 The CatenaryA heavy chain supported at either end hangs in a curve in the vertical plane, and curiosity motivates us to wonder what shape the curve is (after all, we are engineers). If the question doesn’t interest you, may I suggest a career in administration or accountancy, where an under-developed imagination might prove advantageous?

A short element of the chain is in equilibrium between the change in the vertical component of tension in the chain. If the weight per unit length of the chain is ρ N m -1, the weight of a short element is ρδs, where δs is the length of the element. If the slope of the chain at this element is θ, the equilibrium of the element requires:

(T+δT ) sin (θ+δθ )−T sinθ=ρδs

Also, the element must be in equilibrium horizontally:

(T+δT ) cos (θ+δθ )−T cosθ=0

Where T is the tension in the chain.

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To first order of small quantities, the equilibrium equations become:

T cos θδθ+δT sinθ=ρδs

−T sin θδθ+δT cosθ=0

Eliminating the change in slope over the element we have an expression for the change in tension:

δT=ρ sin θδs

But: sin θ= δyδs

From which we deduce that:

dTdy

=ρ

T=ρ ( y− y0 )

Where y0 is the length of chain whose weight is the same as the minimum tension in the chain.

The shape of the curve is found from the equation for horizontal equilibrium:

ρδy cosθ=ρ ( y− y0 ) sin θ δθ

δy=( y− y0 ) tan θδθ

But:

tanθ= δyδx→ dydx

From which:

yδθ=δx

Differentiating the equation for the gradient:

ddx

(tan θ )=sec2θ dθdx=(1+ tan2θ ) dθ

dx=(1+( dydx )

2) dθdx=d2 yd x2

Substituting:

( y− y0 )dθdx= y

(1+( dydx )2)d2 yd x2

=1

Let : z=dydx; d

2 yd x2

= dzdx=dydxdzdy=z dz

dy=1

2ddy

( z2 )

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Substituting for z yields an equation for the slope in terms of the height above the lowest point:

ddy

( z2 )=2(1+z2 )( y− y0 )

This may be integrated directly:

∫0

z2 dw1+w

=2∫0

y dy( y− y0 )

The lower limits are zero on both sides because the slope is zero at the lowest point.

ln (1+z2 )=2 ln (1− yy0 )

i.e. dydx=√(1− y

y0 )2

−1

Let: p=±(1− yy0 ); dpdx=∓ 1

y0dydx

With this substitution:

− y0dpdx=√ p2−1

If: p=coshax ;√ p2−1=± sinh ax

dpdx=±a sinh ax

Hence: a= 1y0

±(1− yy0 )=± cosh ( xy0 )

For the chain to curve upwards at its centre, both the cosh and y must have the same sign. If p is taken as positive, and the right hand side negative, the shape of the suspended chain becomes:

yy0=cosh ( xy0 )−1

Resolving the sign ambiguity in this way, y=0 when x=0.

This shape is known as a catenary.

The scale factor (y0) is determined by the tension.

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dTdy = ρ;⇒

dTdx =

dydxdTdy =±ρ sinh( xy0 )

By symmetry, the tension must increase as we move away from the lowest point in the chain, so the rate of change with horizontal coordinate must be an even function, i.e:

dTdx

= ρ|sinh( xy0 )|The integration is taken over the positive x range, but we note the solution applies to negative x as well:

T=ρ y0 cosh| xy0|+T 0

Where T0 is the tension in the centre of the span.

If the semi span of the gap is L, the tension will be a maximum there. As we are typically concerned with how strong the chain must be to cross a given gap, it seems sensible to base the calculation on the strength of the chain.

T max=ρ y0cosh ( Ly0 )+T 0

T 0=Tmax−ρ y0cosh ( Ly0 )The chain can never go into compression, so the limiting value of y0 is found from the condition that the tension in the chain is zero at the centre of the span.

y0 cosh ( Ly0 )=T maxρ

This would be solved iteratively for the case of interest, and the sag of the cable found from the catenary equation.

Another possibility is for the sag in the chain to be restricted, as for a power line suspended between pylons, which requires a minimum ground clearance. The value of y0 is than found from the catenary equation, and the required cable strength is then found.

Finally, the most likely question to be asked is ‘how much cable will be needed?’ To answer that, we need the arc length.

We derived the general expression for the length of an arc in the previous section:

S=∫−L

L

√1+( dydx )2dx144


In this case:dydx=sinh( xy0 ), so that:

S=∫−L

L

√1+sinh2( xy0 )dx=∫−LL

cosh ( xy0 )dx= y0[sinh( xy0 )]−LL

Or: S=2 y0 sinh( Ly0 )7.3.1 CommentOnce we have familiarised ourselves with the exponential function in all its guises (trigonometric functions, hyperbolic functions and logarithms) we are well placed to produce approximate mathematical descriptions for a vast range of potential problems.

The difficulty lies not in the mechanical process of evaluating integrals, but in deciding how to approximate a real world problem, and estimate what we think the domain of validity must be.

In this case, some thought had to be applied to the resolution of sign ambiguities, which required an understanding of the problem in hand, rather than the abstract mathematics of a particular function.

It is for this reason that the student is usually given a range of example problems to work through, in order to develop the skill of mapping real world problems, even those considered intractable, into an approximation which is adequate for the purpose in hand.

What is missing in this approach is the overall objective which is being sought in formulating the mathematical representation in the first place. Usually one is asked to produce the ‘right’ answer, which tests the mechanical skills of manipulating formulas, but does not address the more important skill of finding a mathematical representation which is adequate for the current stage of study.

Pure mathematicians are always presented with ‘given’ problems. The engineer, applied mathematician and analyst require the ability to formulate the problem in the first place and use features of the problem to assist in finding solutions.

7.4 About Suspension BridgesThe catenary is an adequate approximation to a cable hanging between two support points such as a power cable between two pylons. It is tempting to claim that this represents the shape of the cables on a suspension bridge. However, the bridge differs from the suspended cable in that the weight of the support cables is insignificant compared with the weight of the deck. The supported weight depends on the horizontal component of position rather than the distance along the arc.

145


The equilibrium of an element of support cable is more nearly

(T+δT ) sin (θ+δθ )−T sinθ=ρδx

As before, the element must be in equilibrium horizontally:

(T+δT ) cos (θ+δθ )−T cosθ=0

This assumes that the cables are close enough together to effectively transfer the weight of the deck continuously to the cables. This is obviously an approximation, but it would be a pretty lousy design if this were not the case.

The equilibrium equations, expanded to quantities of first order, become:

δT sin θ+T cosθδθ=ρδx

δT cosθ−T sin θδθ=0

Eliminating the increment in the slope yields an equation in for the tension:

dTdx

= ρsin θ

Eliminating the change in tension:

T dθdx= ρcosθ

Eliminating the increment in horizontal position:

1TdTdθ

= tanθ

This is integrated using the substitution:

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z=cosθ ; dzdθ=−sin θ

∫0

θ

tan θdθ=∫0

θ sin θcosθ

dθ=−∫1

cosθ dzz=−ln (cosθ )

The left hand side of the equation becomes:

∫T 0

T dTT=ln ( TT 0 )

So the variation of tension with slope is given by:

T=T0

cosθ

Differentiating with respect to x:

dTdx

=T 0sin θcos2θ

dθdx

dθdx= ρT 0

cos2θ⇒s ec2θ dθdx= ρT0

We recall that:

tanθ=dydx

Differentiating with respect to x:

sec2θ dθdx=d

2 yd x2

The shape of the curve is therefore determined by:

d2 yd x2

= ρT 0

Taking the reference point as the centre of the span, where x=y=dy/dx = 0:

y= ρ2T0

x2

The tension is found from:

tanθ=dydx= ρT 0x ;sec θ=√1+ tan2θ= 1

T0√T 0

2+ρ2x2

i.e:

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T=√T 02+ ρ2 x2

The arc length is:

S=∫−L

L

√1+( ρxT 0 )2

dx

In order to evaluate this, try the substitution:

sinh z= ρxT 0


S=T 0

ρ ∫−αα

cosh2 zdz

Where: α=sinh−1( ρLT 0 )There are several ways to evaluate the integral, when all else fails it is probably worth writing it as exponential functions:

cosh2 z=( ez+e−z2 )2

=e2 z+2+e−2 z

4

Integrating:

S= ρ4T 0

[ (e2 z−e−2 z )2

+2 z ]−α

α

= ρT 0(α+ (eα+e−α )

2(eα−e−α )

2 )S= ρ

T 0 (sinh−1( ρLT 0 )+cosh α sinh α)Or: S= ρ

T 0sinh−1( ρLT 0 )+( ρT 0 )

2

L√1+( ρLT 0 )2

If your calculator doesn’t have hyperbolic functions, the inverse sinh may be written as a logarithm:

y=sinh−1 x⇒ x=sinh y= ey−e− y

2

e2 y−2 xe y−1=0

e y=x ±√1+x2

y=ln (x±√1+ x2 )

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7.5 Arc Length of an EllipseThe circumference of a circle was found by differencing the areas of concentric circles to find the area of the annulus between them, and then reducing the width of the annulus. That method cannot be applied to the ellipse because the space between similar ellipses is not of uniform width.

The equation for an ellipse is:

( xa )2

+( yb )2

=1

Where a and b are the semi-axes lengths.

Differentiating:

dydx=−( ba )

2 xy=−( ba )

2 x

b√1−( xa )2

√1+( dydx )2=√ 1−(1−(ba )2)( xa )

2

1−( xa )2

We could evaluate this using mathematical functions called (believe it or not) elliptical integrals. Now there is a whole range of somewhat specialised mathematical functions, which emerged before the days of personal computers. These were calculated numerically and presented in tables.

These functions are nothing like as ubiquitous as the exponential function and its related functions, so we would only really familiarise ourselves with them if they cropped up frequently in our current investigations. At the level the systems engineer works, the most we really need to know is the geometric shape of plots of these functions, and how the geometric features are affected by changing parameters. Our desire is to visualise the problem, and not necessarily solve explicitly for a specific case.

What we need to know is how the problem changes as we alter values of parameters; under which circumstances it is sensitive, and when it is robust to small changes.

When confronted with an apparently intractable problem, we have three courses of action:

Look it up in a standard reference, such as Abromowitz and Stegun .

Evaluate it numerically, using a fairly trivial code

Find a tractable approximation which is ‘good enough’

The first is of value if we are interested in the function in its own right. If we find ourselves repeatedly encountering it in our current piece of work, it might be worthwhile becoming familiar with the background mathematics.

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The second is fine for a particular instance, but requires numerical results for specific cases to check the operation of the code. Also, writing code brings with it its own problems which are not relevant to the task in hand, and naive solution methods frequently generate inaccurate results. To quote a crude but relevant aphorism; ‘when you are up to your arse in alligators, you tend to forget that you are here to drain the swamp’.

The third approach requires us to sit back and consider exactly what we are trying to achieve. We expect the form of the solution to be some function of the slenderness ratio of the ellipse multiplied by a representative length:

S=ak ( ba )We look for regions of the function k where we can get a reasonable approximate answer. For example, we know that when b/a = 1, the ellipse becomes a circle, and k=2π, and as b/a →0, k→4.

From the nature of the problem, we know that the arc length cannot lie outside these bounds. Two points are insufficient to sketch a curve, but we could, as a first approximation, draw a straight line between these points. However, this isn’t much use without some estimate of the error in the approximation.

Intuition would tell us that the error can’t be too great, but then intuition would soon have us burning witches and sacrificing virgins to the Moon Goddess.

We know that the arc length cannot be greater than that of the enclosing rectangle, for which:

S=4 (a+b )=4 a(1+ ba )This is a reasonable estimate for slender ellipses, but becomes ever more inaccurate as the ellipse tends to a circle. Also, the arc length is not expected to be shorter that the hypotenuse of the triangle drawn between the intersection s with the x and y axes:

S=4√a2+b2=4 a√1+(ba )2As before, the estimate error is a maximum for the quadrant of the circle.

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Using the straight line estimate, we know the estimate could be up to 27% too great or 10% too small.

We cannot say whether this is adequate or not, unless we know how accurate our analysis must be. At a first iteration, straight line approximations are desirable, as they simplify the analysis considerably.

However, without knowledge of the error implicit in the simplification, the analysis ceases to be ‘top-level’ but merely self-deception, and not worth the paper it is written on.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

Arc Length Error Bounds

Slenderness

Arc L

engt

h

The upper and lower error bounds with the box (blue) and lozenge (red) approximations to the ellipse. We know that the upper bound must be π/2 for a single quadrant, so that an approximation which exceeds this value must be invalid, regardless of how loose the tolerances on the calculation might be.

Taking the arithmetic mean f the two estimates yields an improved estimate:

S=2a(1+ ba+√1+( ba )2)For which the error will not exceed:

∆ S=2a(1+ ba−√1+( ba )2)This may be up to 17%.

7.5.1 Towards a better solutionThe estimate is at its worst s the ellipse approaches the circular quadrant, so evidently a better estimate in this region is:

S=2π

In order to reduce the clutter a bit, let:

η= xa , and β=ba

The arc length of the ellipse is given by:

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S=4∫0

a √ 1−(1−( ba )2)( xa )

2

1−x2dx

The equation becomes:

S=4 ak (β )=4a∫0

1 √ 1−(1−β2 )η2

1−η2dη

For values of β approaching unity, we the numerator square root may be approximated as a Taylor series:

√1−(1−β2 )η2≈1−12

(1−β2 )η2

∫0

1 √ 1−(1−β2 )η2

1−η2dη≈∫

0

11

√1−η2dη−1

2∫01 (1−β2)η2

√1−η2dη

The first term evaluates to the circular quadrant solution, so the second term may be considered an estimate of the error in using the circular quadrant estimate. The presence of the square root indicates a sine function substitution might be appropriate:

η=sin θ

dηdθ=cosθ

∫0

1 (1−β2 )η2

√1−η2dη=(1−β2 )∫

0

π2s ¿2θcosθ

cosθ dθ

In order to deal with the square of the sine, we recall from the formula for the cosine of the sum of two angles:

cos2θ=cos2θ−sin2θ=1−2sin2θ

It follows that:

∫0

π2

sin2θdθ=12∫0

π2

(1−cos2θ )=12

[θ−sin 2θ ]0π2=π

4

The error in using the circular arc estimate is about:

∆ S≈ π2a (1−β2 )

We can claim that the circular arc approximation is superior to that of our average of the box and lozenge approximations:

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∆ S≈2a (1+β−√1+β2 )

There exists a value of β for which the two are equal, above which it is better to use the circular arc approximation.

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

Comparison of Errors

slenderness

erro

r %

Comparing the two errors it is evident that by taking the circular arc approximation at values above β=0.62, the error is unlikely to exceed 15%.

By implication we could continue this process of taking more terms in the series expansion to get a more accurate estimate of the arc length. However, the astute reader will have noticed that we have assumed that the sum of all the terms which we have ignored may be neglected. To see that this is not always justified, consider an expansion which could be used in the vicinity of β=0:

∫0

1 √ 1−(1−β2 )η2

1−η2dη=∫

0

1

√1+ β2η2

1−η2dη=√1−β2∫

0

1

√1+ β2

(1−β2)1

1−η2dη

Expanding the square root:

∫0

1

√1+ β2

(1−β2 )1

1−η2dη≈∫

0

1

1−12

β2

(1−β2 )1

1−η2dη

Integrating the second term:

∫0

1 dη1−η2=

12∫0

1

( 11−η

+ 11+η )dη=1

2 [ ln( 1+η1−η )]01

This evaluates to infinity, which even to the sloppiest standards, cannot be considered small.

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7.5.2 About ErrorsThis is as far as we shall go with the elliptical arc problem. We can reduce the error by taking more terms, but that is just mechanical algebraic manipulation (the alligators and swamp), which adds nothing to the approach employed. It serves as an example to illustrate what the engineer does when confronting an apparently intractable problem.

The mathematician seeks an ‘exact’ solution for what is itself an approximation to physical phenomena, because mathematicians do not solve problems within any specific context. Historically, these exact solutions have proved surprisingly close fits to real world behaviour.

The medicine man uses approximations, even if they are little better than guesses, without regard to the error introduced, and can therefore ‘prove’ whatever the end user wishes to hear. This appeals to legal minds which all too often dominate corporate thinking, and which seek to ‘prove’ a particular result to be true rather than find what the actual truth is.

The engineer assumes from the outset that most of the real world problems are messy, and only approximate formulations are possible, especially at the initial stages of system analysis. It is at this stage where the errors in analysis have greatest effect, where an early misjudgement will steer the project in a catastrophically ill-conceived direction, that the uncertainties in the analysis, and their causes, must be fully understood. All too often we flow down precise system specifications from guesses and speculations disguised as analysis.

When presented with an apparently intractable problem we seek an approximation which is ‘good enough’. What is meant by ‘good enough’ depends strongly on the context in which the analysis is being conducted. If it is for part of the system which is not critical to overall performance, the tolerances can be wide. Critical elements may need to be specified to much tighter tolerances.

System engineering analysis is concerned primarily with establishing the relationships between component tolerances and overall system behaviour, without which attempts to optimise or balance the design are little better than guesses. Without this analysis, it is impossible to decide which are critical issues, and which do not matter.

It is fair to say that most of the neat analytical solutions were found years ago, so that in his/her professional life the engineer is unlikely to meet problems which have a straightforward analytical solution. However, there exists a standard solution of some form which is adequate for practically all apparently intractable problems, but it is necessary to be aware of how much additional error the approximation introduces into the analysis.

Ultimately, when the aircraft crashes or the bridge collapses, it is the engineer who must attend the enquiry. We must be satisfied that the consequences of the worst combination of tolerances are understood long before the product enters service.

7.6 ConvergenceWe know our results are wrong, that’s life, but we must know how wrong they are.

One common method of approximation is the series expansion of a function into more familiar functions. For this to be possible we must be certain that we have taken sufficient terms, and that accuracy does actually improve as we take more terms.

155


In addition, we need to be sure that the sum of all the terms that have been ignored is no greater than the final term of the approximation.

If we take a strip of paper and cut it in half, take one of the halves and cut it in half, and repeat the process, it is evident that no matter how many times we cut the remaining piece in half, the sum of all the pieces cannot exceed the original strip. In other words we know that the geometric series:

S=∑i=0

N 12i

cannot exceed unity, no matter how large N becomes.

Furthermore:

∑n

N 12i≤ 12n

The argument would still be valid if we cut the strip at 1/3rd of its length. Indeed, if we chose a different fraction at each stage, the pieces still could not exceed the original strip nor could the remainder at any stage exceed the previous partial strip.

The ratio of consecutive strips need not be constant, provided the ratio of any term to its preceding term has magnitude less than 1 for all terms, the series will be convergent, and we are justified in claiming that the error is no greater than the magnitude of the final term of our approximation.

This reasoning indicates that if the ratio of the magnitudes of consecutive terms is less than unity for all terms, the series will be convergent.

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In the elliptical arc example the expansion is:

∫0

1 √ 1−(1−β2 )η2

1−η2dη≈∫

0

11

√1−η2 (1−12

(1−β2 )η2⋯)d ηThe ith term is:

12 (−1

2 )(−32 )⋯( 12−( i−1 )) 1i !∫0

1

(1−β2) i η2i

√1−η2dη

In order to proceed we must evaluate the integral:

∫0

1η2 i

√1−η2dη=∫

0

π2

(sin 2θ )idθ=∫0

π2

sin2 iθ dθ

(this uses the sine function substitution, which we have already met).

We don’t need to evaluate the integral to prove convergence. We know that sinθ will lie in the range 0 to 1 over the limits of the integral, so the integral itself cannot evaluate to greater than π/2. Also, as i increases, the area under the curve reduces. The ratio of the integrals in successive terms is therefore less than unity.

The ratio of the magnitudes of consecutive terms is therefore less than:

157


r=|( 12−i) (1−β2 )

i |As i→∞, r →(1-β2), which is less than unity, so we are justified in treating the expansion as convergent.

7.7 Concluding CommentThe mathematically inclined may not find this intellectually satisfying. It is so easy to become distracted from our primary quest by distractions, such as spending too long trying to derive an ‘exact’ solution to an inexact problem. We must recognise the fact that such results are rare in the real world, and so the engineer uses the few analytical solutions derived in the textbooks to provide hints as to the form the best approximation may take.

By the same token it is inexcusable to employ an approximation without some understanding of the possible errors introduced by it. Intuition and guesswork are not good enough, although in recent years these appear to be preferred to proper engineering analysis.

Most of our effort is rightly spent deciding how much we can trust our results, rather than seeking results which are absolute truth. It is better to find an appropriate approximation and know the truth within defined error bounds, than it is to insist on an absolutely correct answer, and learn nothing about the truth.

It should be clear that few people are more aware of the limitations of theoretical models than scientists and engineers. Nobody expects absolute correspondence with reality, but we must know the error bounds on results produced. Without such error bounds the results are useless.

I find it amusing how much authority and kudos is so often ascribed to cost benefits analysis whose variability in the parameters swamps any marginal differences between the options. The main merit of this approach is it tells the sponsoring authority what they want to hear, whilst exposing dissenters to a largely innumerate public as irrational and ignorant. This ploy has been used by economists for years.

What is less amusing, is the tendency for important policy decisions to be based on such pseudo-scientific nonsense.

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8 The Calculus of VariationsWhen we first considered using calculus to find maxima and minima, the function whose turning points we sought was ‘given’. We could solve problems such as finding the best proportions of a given shape to enclose the maximum area, but what we couldn’t do was answer the more fundamental question of what shape encloses the greatest area with a given perimeter?

Before we consider that problem, let us address a simpler problem; what is the shortest path between two points on a plane?

In other words, find a function y(x) which minimises:

S=∫a

b

√1+(dydx )2dxThe approach is the same as we adopted in all other calculus problems, only instead of incrementing a variable by a small quantity, we modify a function by adding to it another function which is continuous and everywhere ‘small’ compared with the original function.

Let the solution be y(x), a trial solution near this is y(x)+ε(x).

Y ( x )= y (x )+ε (x )

As ε(x) is everywhere close to y(x), and is assumed smooth, we are justified in claiming its derivatives are also small quantities at all points.

The difference in arc length of the incremented function is given by

δS=∫a

b

√1+( dYdx )2dx−∫ab

√1+( dydx )2dxThe increment in the integral; δS is called the ‘variation’, hence the ‘calculus of variations’.

Expanding the first term:

∫a

b

√1+( dydx + dεdx )2dx≈∫ab

√1+( dydx )2(1+ 2 d ydx

dεdx

√1+( dydx )2 )12

dx

The variation becomes:

δS=∫a

bdydxdεdx

√1+( dydx )2dx

159


Integrating by parts:

δS=[ε dydx

√1+( dydx )2 ]ab

−∫a

b

ε ddx (

dydx

√1+( dydx )2 )dxThe function ε(x) is zero at x=a and at x=b, so the first term is identically zero. Between the end points ε(x) has a finite value, so the variation can only be zero if:

ddx (

dydx

√1+( dydx )2 )=0

This has the solution that dy/dx is a constant. In other words the path is a straight line, as we all could have guessed.

However, it is not the result itself which interests us, it is the means of deriving it which is of value, because it is evident that this may apply to more awkward problems.

More generally, we wish to find a function of x which minimises an integral of the form:

J=∫a

b

F ( y . dydx , d2 yd x2

⋯)dxIn order to avoid writer’s cramp, let us use a more compact notation:

dydx= y ' , d

2 yd x2

= y ' ', etc.

We follow the same procedure as for the shortest path problem:

δJ=∫a

b

( ∂ F∂ y ε+ ∂ F∂ y ' ε ')dxThe second term in the integrand is found by integrating by parts:

∫a

b ∂F∂ y '

ε ' dx=[ε ∂F∂ y ' ]ab

−∫a

b ddx ( ∂F∂ y ' )εdx

As before, the increment is zero at either limit (the path must pass through these points), so we have, for the variation:

δJ=∫a

b

( ∂ F∂ y− ddx ( ∂ F∂ y ' ))εdx

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The function ε(x) is finite in the interval, so the variation can only be zero if:

∂F∂ y

− ddx ( dFdy ' )=0

This expression is known as the Euler-Lagrange Equation.

8.1.1 Application to Maximum AreaThe problem we set ourselves was to determine the shape of the contour which encloses the greatest area. The area under a curve is given by:

A=∫a

b

y dx

This must be maximised subject to the constraint:

s=∫a

b

√1+( dydx )2dxis fixed.

This differs from the path between points as we have introduced a constraint into the problem. This is the more usual form of an optimisation problem.

If we were solving this problem by trial and error, we would guess a function to find A, and see if it were greater or smaller than the previous guess, then we would see the effect of our guess on the arc length. If the area were greater, but the arc length increased, we would try something else.

In effect, we would seek to maximise a combined expression which includes the constraint:

H=∫a

b

ydx+λ (s−∫a

b

√1+( dydx )2dx )Where λ is a weighting , usually called a Lagrange multiplier.

An expression of this form, with the constraints adjoined with undetermined multipliers is called a Hamiltonian.

Differentiating with respect to λ yields the constraint equation:

∂H∂λ

=s−∫a

b

√1+( dydx )2dx=0

We find the turning point in H by calculating the variation in:

J=∫a

b ( y− λ√1+( dydx )2)dxThis is given by the Euler-Lagrange Equation:

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∂∂ y

( y−λ√1+ ( y ' )2)− ddx ( ∂∂ y ' ( y−λ√1+( y ' )2))=0

This yields a set of potential solutions, which may or may not meet the perimeter constraint.

1+λ ddx (

dydx

√1+( dydx )2 )=0

Integrating:

dydx

√1+( dydx )2=c− x

λ

Where c is a constant.

( dydx )2

=(c− xλ )2(1+( dydx )

2)i.e:

dydx=

(c− xλ )√1−(c− xλ )2

y=∫(c− xλ )

√1−(c− xλ )2dx

Let (c− xλ )=cosθ

1λdxdθ=sinθ

y=∫ λcosθdθ=λ sin θ+d= λ√1−(c− xλ )2+dWhere d is a constant.

The Hamiltonian is, therefore:

162


H= λ∫√1−(c− xλ )2+ dλ dx+λ (s−∫ 1

√1−(c− xλ )2dx)

Evidently, c and d determine the position of the figure, which is irrelevant to its shape, so we may set them both to zero.

H=∫√λ2−x2dx+λs−λ2∫ dx√ λ2−x2

From the first integral it is evident that:

y=√λ2−x2

The curve is a circle of radius λ. But all we have shown is that a circle encloses either the minimum or the maximum area.

The curve must be closed, otherwise it doesn’t form a boundary at all, so the perimeter is four times the arc length of a quadrant. The arc length becomes:

s=4∫0

λ λ

√ λ2+ x2 dx=2 πλ

As expected.

In order to show this is a maximum, we only need to show it not to be the minimum, by considering the area enclosed by any shape figure we choose which has the perimeter 2πλ, and show that its area is less than that of the circle.

The obvious example is a square, which will have area:

A square=( π2 )2

λ2

Compared with a circle:

A¿=π λ2¿

Since the area of the square is less, the circle cannot be the figure enclosing the smallest area, ergo it encloses the largest.

8.1.2 CommentThe calculus of variations is seldom taught at an undergraduate level to engineering students, not because it is intrinsically difficult, but because in practice it has very few applications. The problem does not need to become very complicated before it becomes intractable. Just about all the possible analytical solutions may be found somewhere in the literature.

With the advent of high speed computers it was hoped that the analytical difficulties would go away, and methods based on calculus of variations would be used to optimise everything. This hope was

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soon dashed because the constraints introduce a combinatorial problem, which quickly exceeds the capacity of the fastest computers. There remain methods such as dynamic programming which apply variational methods to approximations of the actual constraint functions. These generally make assumptions about the nature of the function which is being maximised.

As an example, the function may contain multiple peaks, yet if all we are doing is searching for a turning point, we have no way of knowing whether or ‘maximum’ is a local maximum, or the supremum, which we are seeking.

There is something juvenile and naive about seeking ‘the best’. It pre-supposes we know what ‘the best’ actually is. Too often, the criteria which ‘obviously’ characterised the optimum solution are merely relegated to the dustbin of ‘it seemed a good idea at the time’.

In order to apply optimisation techniques we must understand the problem we are dealing with thoroughly, and must assess the appropriateness and limitations of the algorithm we are using. These are the very features of the problem modern software packages hide from the end user. In this world of spin and hype, nobody is prepared to admit that their product has limitations.

We try to look up ‘bounds of validity’ in the documentation, only to draw a complete blank. The manual waxes lyrical about the wonders one can achieve, yet the first serious problem causes it to generate garbage.

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The naive user can indeed use the tool to generate meaningless numbers, only to prove the old adage that a fool with a tool remains a fool.

We study mathematics not for the pragmatic utility of the results, but because the subtlety of thought in the more elegant proofs teaches us how to think clearly and logically. The engineer arms himself with a set of standard solutions which characterise as wide a range of physical phenomena as possible, and uses them to get a best fit on the current problem presented. Consequently the true engineer can tackle problems originating in any domain.

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9 Cybernetics and Automatic Control

9.1 IntroductionThe fundamental idea which emerges from Ross-Ashby’s work, which is presented in the early chapters of this book is the Principle of Requisite Entropy.

Implicit in much of the literature on systems engineering is the assumption that a complex system is characterised by complexity which increases with component count and diversity of components. Such is the case if the methods are not based on cybernetics.

Firstly, cybernetics doesn’t care how the behaviour of the components comes about. The same behavioural models can be made to represent a vast array of possible real world entities. In this way the diversity in the components is accommodated.

Secondly, the purpose of adding components is to reduce the complexity of the behaviour of the system, not to increase it. If complexity does increase, we are not dealing with a system at all, merely a collection of parts in close formation. There may be interactions at the secondary and tertiary function level, but not at the primary function level.

An approach which is not based on behaviours can only pre-judge the solution, since if we are restricted to organisation and structure, we must have physical entities to organise and arrange before we can commence. Indeed, the methods have difficulty identifying what constitutes primary function, and excessive emphasis on what colour the box should be appears to take precedence of whether the system will fulfil its primary function.

The cybernetic view is that a system is defined by what it does, and behaviour can be specified mathematically completely independently of the physical objects which achieve it.

Requisite entropy decrees that a system, or subsystem, is like a piece of a jig saw. It is specified completely by the shape of the hole it must fit into, and the matching up of the picture on its surface.

9.2 Implications for System SpecificationAt some relatively late stage of development we may be able to define our system by its components and sub-systems, in general that is not how we begin. Real world systems are complex in the behavioural sense, and there is no one to one mapping of behaviours on to widgets, as appears to be the underlying assumption in software-oriented methods. We cannot form any kind of hierarchical decomposition without the premise that processes map to widgets.

When dealing with World-class technology suppliers, the end user is not competent to specify anything. Rather than prescribe the system requirements directly, it is the end user’s responsibility to specify the environment with which the system interacts, and request the supplier to produce a list of costed options, which represent the best that the current state of the art can deliver, rather than prescribe actual performance.

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The Law of Requisite Entropy states that the potential design solutions, if they exist at all, will be dictated by the environment with which the system must interact.

This is called concept study. Note it is done in support of the requirements, and not in the forlorn hope of searching for a technical solution to meet the end user’s wishes, and which may not exist.

Once the customer has seen the options, and hopefully had the concept work peer reviewed preferably by a rival, or a competent consultancy, so that there is reasonable confidence in the figures, the next stage is feasibility. This costs more because it introduces mock ups, test rigs, breadboards and boiler plate prototypes, whilst concept will involve modelling and paper studies.

Finally, if there appears a worthwhile solution, the process moves on to Project Definition, at which point we have requirements which both customer and supplier are prepared to sign up to.

9.3 On Feedback The finite state machine is often taken as the paradigm for the ‘higher level’ control option in a system, as usually its function is to specify parameters or operating points or select the lower level functions required at any time.

The reason we began studying the finite state machine is it is easy to understand. Actual finite state machines have been around since the 18th Century in the form of clockwork automata. Since then, automatic gun loaders, capstan lathes, even engine valve gear emerged as examples. Implementing them in digital computers is truly a ’detail of implementation’ as far as cybernetics is concerned.

Whether mechanical cams or linkages, hydraulic or pneumatic valves, or a modern computer is used, they are conceptually identical.

The problem arises, when considering systems which are self-adapting, is the finite state machine is inherently ‘brittle’ in its operation, to the extent that it can only be used practically in a highly controlled environment.

A robot used in a factory environment works well, repeating a sequence of operations which would bore a human being silly. If the cat gets in the way, it keeps on spraying regardless.

Even a knowledge-based system, so beloved of the Artificial Intelligence community, can only select from a finite set of options. Provided the set which the environment presents is closed, that is fine. Unfortunately, the environment is not a closed system.

Also, the Turing machine presents problems when we consider adaptation. As Roger Penrose pointed out in his book ‘the Emperor’s New Mind’, changing a single bit at random does not cause the code to change its function slightly but destroys it altogether. The program in the machine must change in its entirety or not at all, which indicates that the Turing machine is a poor model for intelligence evolving by natural selection.

There may be an algorithm which can run on the Turing machine which can adapt through random mutation, but that would be an emulation of the adaptive process, not the process itself.

Ross Ashby side-stepped the issue by adding the ‘vital state’ to the set of machine states. This provided the trigger for the adaptation process. He illustrated this with his ‘homeostat’. This is

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merely a threshold below the fatal condition, which gives the machine time to adapt. Essentially the two states constitute quantised levels in what is really a single state.

The homeostat was an electronic circuit which modelled an organism interacting with the environment. It was possible to inject disturbances which could render the organism/environment system unstable. When the vital state moved out of its acceptable bounds, the ‘organism’ part of the machine selected resistors at random until the combination again became stable.

Ross Ashby’s homeostat illustrated the power of feedback. The random selection process indicated that how the feedback is done really doesn’t matter, provided it is possible to determine when adaptation has taken place. Nor is it necessary for the organism to have exact knowledge of the environment. Provided its action results in cancellation of the disturbance, the organism doesn’t care why it works.

The Principle of Requisite Entropy states that the environment throws entropy at the organism, and for adaptation to be possible, the organism must have sufficient internal entropy to destroy it.

This can come about by having a pre-defined reaction to counter every conceivable action of the environment. However, Requisite Entropy tells us that if we can detect when the entropy has been reduced to zero, no further action need be taken. This latter strategy is called feedback, and unlike the former, it is effective even if the set of environmental actions is open.

The adaptation could use knowledge based systems or fuzzy logic, genetic algorithms, neural nets, or as in the homeostats case, random selection, or any other ingenious technique which may occur to the reader. The feedback could serve to build up an internal model of the environment, in which responses could be tested, but the only way to adapt to an open environment is through feedback.

Note feedback is not just the interconnection of two processes, it is specifically a means of correcting the state of the system in response to a disturbance to drive the system entropy to zero.

Implicit in feedback is the assumption that the system is observed for long enough for the effect of the corrective action to become manifest, if we attempt to take corrective action in a shorter timescale than this, we run the gauntlet of instability.

9.4 About IntelligenceThe reader is warned that this section is philosophical in nature, and may be skipped unless such speculations are of interest. It contains the views of the author, which are not widely accepted at present, but it is left to the reader to judge their merits. My objective is to try and de-mystify ‘intelligence’, and put it on the same sound objective footing as any other phenomenon, such as time, length or temperature.

Few subjects are as important, or as ill-defined as ‘intelligence’. The position adopted here is that if a property exists, there should be an operational procedure for measuring it and by implication units by which it can be quantified, otherwise it has no place in science.

For example, if it were suggested that, say, length should be measured using different units and definitions depending on whether we are measuring human beings, animals or inanimate objects, we would have serious doubts whether ‘length’ was a real concept at all. Still worse, if we claimed

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that temperature caused by heating with a candle was a somehow inferior kind of temperature than heating with an electric element, we would quickly lose the attention of our audience. Yet, that is precisely how intelligence is described.

The world of Artificial Intelligence defines intelligence in terms of specific mental processes which must be present. Thus adaptation must take place in a specific way to be ‘intelligent’. This is akin claiming we can only use a candle flame for our definition of temperature, all other sources are not ‘real’ heat. Our definition must be eclectic, in the sense that it does not pre-suppose any particular method of implementation.

Most systems which show goal seeking behaviour, and which work, use the methods of automatic control, although the software engineer responsible for coding the algorithms usually takes the credit for them.

Requisite variety furnishes us with a clue as to how we can describe this concept without polluting our science with academic narcissism and imprecision.

In order to get an objective measure, it must apply to man, machines and animals equally well, and must not depend on any specific form of processing. Words like ‘thinking’ have no objective definition, so they are excluded from the outset as meaningless. Likewise we do not distinguish adaptive behaviour which arises from the spinal cord from that involving the brain. How adaptation is implemented has no relevance whatsoever to defining what it is. That one form of adaptation is less effective than another is understood, and that is why we must have a unit with which to measure it.

We do not dispense with numbers less than say, ten, as not ‘big enough’ to be considered ‘proper numbers’, beneath our dignity to consider. Indeed, it is usually the examples characterised by the elementary values that give us insight into the phenomenon, from which we stand a chance of progressing. Starting with the grandiose objective of simulating human intelligence is little more than academic megalomania.

Entropy is needed to destroy entropy. However, intelligence brings with it not only the destruction of entropy, but a timescale within which the entropy is reduced to zero. The intelligent subject is the one who solves the problem more quickly than the rest. This implies that we should measure intelligence in bits per second, characterising the rate at which entropy is destroyed.

This implies that we should expect intelligence to emerge in interesting times. We should not be surprised that homo sapiens emerged during a time of climatic turmoil, whilst dinosaurs reigned stupid and supreme for millions of years in steady climatic conditions.

It has been claimed that civilisation requires calm and stability to develop and survive. Actually, if the thesis of this section is correct, quiet comfortable senility is the last thing required for civilisation. War and natural disaster provides the motivation to strive to improve. Civilisations grow and expand to a point where external strife is brought to a standstill by military prowess and internal strife by law and order; the result is senility and collapse.

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Current technology has its roots in the two World Wars, The Space Race and the Cold War, we have reached the limits of performance of the technologies which emerged during these periods, so that products must be sold as fashion items, rather than for their pragmatic utility.

At a time we desperately need the truly novel, all we have is tinkering. Indeed, the idea of novelty is so alien to the modern mind, that tinkering and flogging dead horses appear to be the only recognised approaches to engineering innovation. What little innovation there is, usually falls foul of the vanity of the self-important.

As the Earth’s climate becomes more hostile to human existence, new technologies may become necessary which could be used equally well for survival in the hostile environments of other planets. Indeed, the Earth may cease to have any special merit as far as human habitation is concerned.

Without challenges to develop intelligence, the result will be regression and eventual extinction. Exploration is essential to find new questions, and hopefully lead to new discoveries. When the current technology reaches saturation, it is so tempting to believe that modern science has answered all the important questions and further discovery is impossible. I’m sure the ancient Greeks thought the same about their science, practically all of which has now been shown to be so much nonsense.

9.5 Automatic ControlAdaptation by feedback can use any method the ingenuity of the human mind can devise. Provided the principle of zeroing the entropy error is employed (i.e. the scheme is based on the principle of Requisite Entropy) it will have the ability to adapt to an unknown environment.

The greatest systematic body of knowledge which applies this principle to practical systems is called automatic control. It deals with systems which evolve in time and can be approximated by linear differential, or difference, equations. The inherent robustness of feedback ensures that quite gross approximations can be tolerated in the formation of the equations describing the interaction of the organism with its surroundings.

Most ‘systems engineering’ tends to deal with the object (usually called the ‘system’) and the environment as two separate entities. Control considers the interaction of the object with the environment as the ‘system’, the Law of Requisite Entropy explains why this is the valid approach.

A control system may find itself in any start position with equal probability but converges in time to the reference values specified at the input. In other words it begins in a high entropy state and converges on to a zero entropy state. It destroys entropy in finite time so, according to the eclectic definition of the previous section, it exhibits intelligence.

The principles which emerge from this study apply to all self correcting time evolving systems, although only the mathematically rigorous analysis permits us to find the upper bounds of potential performance. That these principles do not appear to apply to more general systems merely reflects the mediocrity of performance which is currently achieved with state space machine approaches.

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10 Stability

10.1 IntroductionAutomatic control deals with systems that may be described by sets of linear homogenous differential, or difference, equations. In reality few systems meet this criterion, but the inherent robustness of feedback alleviates the effect of disparities between the equations used to design the controller, and the actual plant behaviour. Indeed, automatic control seeks to quantify this mismatch and determine bounds on practical system performance based on this uncertainty.

The system described by the equations is usually called the ‘plant’.

In a few cases, the approximation may be accurate, and the feedback controller may be augmented with the feed forward control characteristic of typical finite state machine implementations.

One application for feedforward is in tracking a distant target with a narrow beam, high resolution sensor. Knowledge of the approximate position of the target can be used to reduce the duration of the initial search for it.

The a priori knowledge on which the feedforward controller is based normally comes from the designer’s understanding of system behaviour. In principle, it could be acquired by an outer loop observing the long term behaviour of the basic system, this could be repeated in hierarchical fashion any number of times.

The design methods are becoming ever more mechanised, to the extent that a system using currently available methods could characterise the plant using system identification processes, and design controllers, without human intervention.

The potential for hierarchical control systems in which the states of one level form the parameters of the next level, has not been investigated extensively. Existing techniques merely extend the order of basic systems until they are overwhelmed by ill-conditioning of the associated matrices. There seems to be a failure to grasp the fundamental idea that what are variables to one level are parameters to the next. This inevitably must introduce non-linear effects, which are expected to introduce chaos, as well as instability.

However, before considering potential future developments, we require a firm understanding of the basics of control theory, and the methods used.

10.2 Additional BackgroundStarting from elementary algebra, chapters 5 through 8 derive the ideas from calculus needed to understand basic ideas in automatic control. After deriving the basic ideas behind calculus, we reach the conclusion that we don’t actually need it at all to examine the stability of a system. The dynamic behaviour can be analysed by considering the roots of the characteristic equation.

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Excessive emphasis on control itself is probably misplaced, as it is the formulation of an adequate description of the plant which is intrinsically more difficult, which is why my emphasis is on examples.

The coefficients of the characteristic equation are determined by the plant parameters, so that most of the time we use them to furnish insight into why the basic system is ill-behaved. In most cases the fault may be corrected by selecting appropriate values of system parameters.

The upper bound of system performance is usually dictated by stability problems.

It is known that a slender ship can sail faster than a tub, but if too slender, it simply turns turtle. Trying to get trains to run fast by simply installing more power leads to uncontrolled swaying of the carriages, risking de-railing and damage to the track. Trying to minimise the material in a structure for reasons of economy or minimising weight introduces the risk of all manner of structural instabilities.

It is reasonable to suppose that most systems are not really describable as linear homogenous differential equations. We can usually find approximations which characterise an operating region, with some form of higher level function detecting which operating region the system is in and changing the parameters accordingly.

Before proceeding, we shall introduce notation which reduces writer’s cramp. We shall invariably be dealing with quantities which vary in time, so we will be considering terms of the form:

dxdt, d

2 xd t2

, d3 xd t3

,⋯

It is common practice to use the more compact form:

dxdt= x

(read this as ‘x dot’).

The number of dots above the variable is equal to the order of the differential.

For example, the generic second order system:

d2 xd t 2

+2ζ ω0dxdt+ω0

2 x=f (t )

Is written:

x+2 ζ ω0 x+ω02 x=f ( t )

Before the latter part of the20th Century understanding of control amounted to studying how feedback affected the solution of this type of equation.

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More typically, it is convenient to describe the plant as a set of first order simultaneous differentia equations, which may be written and manipulated in matrix form. For that reason it is necessary to revise the basics of matrix algebra.

10.3 Matrix Algebra

10.3.1 Definition and Basic Operations

Figure 10-37 : Illustration of Matrix Multiplication

Suppose we have a list of standard components, which can be used in different combinations to produce a range of possible widgets. In order to find out how much each widget costs, we could list the price of each widget in a column in a table, and on each row we could enter the numbers of each component needed for each widget.

We find the cost of parts for any particular widget by taking the terms in the row corresponding to it and multiplying each term by its corresponding cost entry in the component price column.

If we imagine a set of several suppliers, each offering an attractive loyalty discount, to the extent that it is only worthwhile sourcing from a single supplier, we can add price list columns for each, and generate the cost of widgets made using each supplier’s components, by multiplying row by column term by term.

The process amounts to multiplying one table (the component count per widget) by another table (the cost of components), to yield another table (the cost of parts for each widget).

We could write this as a single expression which just encapsulates the general idea that we multiply cost of components by numbers to get component costs for each widget:

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W=NC

Where the terms W, N and C are the tables containing the data, they are usually called matrices (the single of matrices is matrix).

If the prices change by P, we can calculate the new cost per widget by adding this increment on to the prices matrix and repeating our row by column multiplication. This is a term by term addition, so obeys the usual arithmetic laws of addition, in particular it doesn’t matter in which order the terms are taken:

C+P=P+C

However, our row/column procedure, although written the same way we would represent multiplication by ordinary numbers (scalars), is not the same. Logic would dictate that this process should have its own operator, to avoid confusion with algebraic multiplication, which like addition, would be applied element by element. However matrix multiplication is written as algebraic multiplication, by convention, and a notation is used to distinguish matrices from scalars.

Paradoxically, there are occasions when we do wish to multiply element by element, and a special operator is introduced to indicate this.

When we write two matrices next to each other we indicate this row/column operation takes place. For this to be possible the number of elements in the rows of the first matrix must match the number of elements in the columns in the second.

Note that matrices are rectangular, all rows of a matrix are the same length, all the columns are the same length. Elements containing zero are added to pad out rows and columns as necessary.

If two matrices are rectangular, as is likely in the example, because it is distinctly unlikely that the number of suppliers would equal the number of items in their catalogues, it is obvious that the operation:

C N=?

is undefined as the matrices rows and columns mismatch. I defy you to attach a meaning to this matrix product, in the cost example.

If the rows and columns are equal in length, the matrix is, quite reasonably, called ‘square’. Evidently, we can apply the matrix multiplication in either order. Call the matrices A and B.

For the product AB, any element of the result matrix is given by the sum of the product of row elements of A with corresponding column elements of B. For the product BA, the row elements of B multiply the column elements of A. The result is quite different, depending on the order of the two matrices, or:

AB≠ BA

10.3.2 Additional OperationsThere is no compelling reason why our component cost data should be organised in columns under the ‘supplier’ headings. The same data could be organised in rows. However in order to maintain

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consistent dimensions (in this case meaning compatible rows and columns, rather than fundamental units), the numbers per widget data would have to be written in columns, with ‘widget’ headings.

A matrix which has its rows and columns interchanged is called the transpose of the original matrix. We shall use the superscript T to denote this operation.

If we wish to calculate our component costs per widget matrix using these transposed matrices, we note that firstly the order of multiplication is reversed, and the result is the transpose of the product of the original matrices. i.e:

(NC )T=CTN T

So the transpose of a product is equal to the product of the transposes of the original matrices taken in reverse order. A matrix which is equal to its own transpose is called ‘symmetrical’.

10.3.3 The Identity Matrix and the Matrix InverseHaving applied the label ‘multiplication’ to an operation which is nothing of the sort, we imply an inverse ‘division’ process exists.

When we consider multiplying scalars we make use of the fact that there exists a particular number, which under multiplication does not affect the value. For scalars, that is the number one.

The equivalent question is whether there exists a matrix which has this property under matrix multiplication.

If the matrices are rectangular, the result has as many rows as the first matrix and as many columns as the second, so there is no matrix which will maintain the dimensions of either matrix, unless both matrices are square.

It follows that we can only have an equivalent to a unity quantity for square matrices.

Consider pre-multiplying this unity matrix by an arbitrary square matrix. To ensure the first column isn’t affected by this operation, the first element in each row must be multiplied by unity and the rest by zero. It follows that the first column of the unity matrix has one as its first entry, and zero for all the rest. Considering the second column of the first matrix , the second column of the unity matrix consists of zeros apart from the second element which must be unity, similarly for all the other columns.

The ‘unity’ matrix consists of zero elements everywhere but on the lead diagonal, which consists of ones. Post multiplying this by an arbitrary matrix shows that it retains its ‘unity’ property for both pre- and post- multiplication. The unity matrix is called the ‘identity’ matrix to avoid confusion with a matrix whose elements are all unity. It is usually denoted by the symbol I, or In if it is necessary to specify the size of the matrix.

Having found the matrix equivalent to the scalar unity, the next question is, given an arbitrary square matrix, does there exist a matrix such that the product of the two equals the identity matrix? If such a matrix exists we have the equivalent of an arithmetic reciprocal function, and hence a matrix equivalent of division.

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Evidently we cannot treat a matrix as a whole if we are trying to find out if an inverse exists. We are looking for relationships between the elements themselves.

A set of simultaneous equations is a classic example of a problem which may be written more compactly in matrix form:

Ax=b

Where A is n×n, and x and b are n×1, sometimes called n-vectors by analogy with 3-vectors used to describe quantities in 3D (Euclidean) space.

A solution for x can only exist if A is square. If the inverse exists, the solution is found:

A−1 Ax=A−1b

x=A−1b

Where the superscript -1 denotes a matrix inverse.

The problem reduces to finding the circumstances for which the set of simultaneous equations cannot be solved.

Evidently if any row or column consists entirely of zeros it can be removed from the matrix, as it will not affect the matrix multiplication. The result is a non-square matrix which has no inverse.

More generally, if any row consists of a linear sum of two or more other rows, it would be reduced to a row of zeros by applying the row operations used to solve sets of simultaneous equations. It follows that the inverse requires all rows of the matrix to be linearly independent. A matrix which has no inverse is called singular.

As an example, if we post multiply an n×1 matrix by its transpose, the result is an n×n matrix, but the rows cannot be linearly independent, as each row is a multiple of the first. In general, if we (post) multiply an n×m matrix (with m<n), by its transpose, the resulting matrix cannot have an inverse. A matrix of this form is called a dyadic.

Note that the inverse of the inverse is our original matrix, which post multiplies the inverse. It follows that unlike general matrices, the inverse may pre- or post-multiply the matrix.

The inverse of a product obeys the same rules as the transpose of a product.

[AB ]−1=B−1 A−1

Since:

B−1 A−1 [AB ]=B−1 [A−1 A ]B=B−1B=I

When dealing with matrix algebra, we cannot assume a square matrix will always have an inverse. If a result depends on the existence of an inverse, this must be stated explicitly.

The results presented in this section are the very basics of matrix algebra. Additional ideas will be introduced as necessary, in the context of realistic problems.

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10.3.4 Checking for Linear Independence - DeterminantsExamining a matrix by eye and deciding if one row is the sum of multiples of other rows is an unreliable process and something more methodical is needed.

Consider the pair of simultaneous equations:

a x1+b x2=α

c x1+d x2=β

In order to solve for x1, we multiply the first equation by d and the second by b and subtract one from the other:

(ad−bc ) x1=αd−βb

Similarly, solving for x2;

(ad−bc ) x2=βa−αc

Evidently, if (ad-bc) is zero, the equation set cannot be solved.

Extending this idea to a 3rd order system of equations:

a x1+b x2+c x3=α

d x1+e x2+ f x3=β

g x1+h x2+k x3=γ

Note: we skipped i, which is usually used to denote an index.

Eliminating x2 and x3, from the last two equations:

e x2+ f x3=β−d x1

h x2+k x3=γ−g x1

Using the result for the 2×2 matrix:

x2=k (β−d x1 )−f ( γ−gx1 )

(ek−hf )

x3=e ( γ−g x1 )−h (β−d x1 )

(ek−hf )

Substituting for x2 and x3 in the first equation yields the solution for x1:

(a ( ek−hf )−b (kd−fg )+c (hd−eg ) )x1=constant

The coefficient would be the same whichever pair of unknowns was eliminated.

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It is evident that repeated elimination of unknowns from any number of equations will generate a single coefficient of this form, which if zero indicates that the set of equations doesn’t have a solution.

The quantity: ad-bc is conventionally written:

|a bc d|

And is called the determinant of matrix (a bc d ).

Since the set of equations can be solved by multiplying both sides by the matrix inverse, a determinant which evaluates to zero indicates that the inverse does not exist.

The set of three equations yields a definition which may be extended to higher order matrices:

|a b cd e fg h k|=a|e f

h k|−b|d fg k|+c|d e

g h|This is the first row multiplied by the determinants of the matrices remaining after the row and column containing the corresponding row element have been removed. These reduced order matrices are called the minors.

The sign depends on the indices of the row element. In this case the entry b is in row one, column two. If the sum of the row and column indices is odd, the term is negative.

The same determinant can be evaluated from any row of the matrix.

The determinant furnishes a convenient means of deciding whether a matrix has an inverse: If the determinant is zero, the inverse does not exist.

10.4 State Space RepresentationSince linear homogenous equations are characterised by feedback, they, not surprisingly, form the basis of feedback control. They may be written in matrix form by making each derivative a state variable. Consider the second order damped oscillator:

x+2 ζ ω0 x+ω02 x=f ( t )

Let:

u= x

The equation may now be written:

u+2ζ ω0u+ω02 x=f ( t )

This is now a pair of first order equations:

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u=−2 ζ ω0u−ω02 x+ f (t )

x=u

This may be written in matrix form:

( ux )=(−2 ζ ω0 −ω02

1 0 )(ux )+(10) f ( t )

Or more compactly:

x=Ax+Bu

Where x is understood to be the two element vector of system states. u is the input to the system represented by the second order equation. A is called the ‘system’ matrix and B the control matrix.

If we consider impulse responses, u=0, so the plant equation becomes:

x=Ax

Differentiating with respect to time:

x=A x=A2 x

dn xd t n

=An x

This is the property of the exponential function.

The above relationships imply that we are justified in defining the matrix exponential as the same Taylor expansion as that for the scalar exponential.

e At=I+At+ 12!A2 t2+ 1

3!A3 t3⋯

Interestingly, although the matrix exponential is defined in this way, the individual elements of the matrix exponential are not necessarily exponential functions.

Consider the case of uniformly accelerated motion. Denoting the acceleration f, the speed u and position z, the state vector is:

x=( fuz )The system matrix is;

A=(0 0 01 0 00 1 0)

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A2=(0 0 00 0 01 0 0), A3=(0 0 0

0 0 00 0 0)

Evidently, all higher powers of A are also zero. The solution is:

( fuz )=(1 0 0t 1 0t2

2t 1)( f (0 )

u (0 )z (0 ))

This is the same as was derived by integration of the acceleration with respect to time.

If A were a scalar, stability would be determined by whether it is positive or negative. What do we mean by ‘positive’ when applied to the matrix as a whole?

10.4.1 ‘Sign’ of a Matrix We cannot say much about the ‘sign’ of a matrix without some general definition of what we mean by ‘positive’.

We can claim something is positive if it doesn’t change the sign of a positive quantity.

If x is n×1, the product:

xT x

Is the sum of the squares of the elements of x and is therefore positive. In general, the individual elements are not pure numbers, so this product would be dimensionally inconsistent; we cannot add cabbages to kings.

It is understood that we always scale our state variables to avoid this difficulty. Implicit in the above product is a scaling of the original state variables:

x=Qz

Where Q is a usually diagonal matrix whose elements are the reciprocals of the maximum desirable values of the original state variables (z).

If we multiply a system matrix A by x, the product:

Ax

still has dimensions of 1/time, or frequency. We must multiply the product by an arbitrary positive time to convert it to a vector of pure numbers. This frequency scaling is understood, and as it doesn’t affect the result, is assumed to be unity.

Quite often, the time used in the plant equation is itself scaled to some important time constant characterising the motion, so that this frequency scaling is unnecessary.

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Pre-multiplying the result by xT, the matrix becomes a weighting on the squares and products of the elements of x. By convention, the matrix is considered positive, if the product remains positive.

i.e. if:

xT AxxT x

>0

The matrix A is considered positive.

If the matrix were diagonal, some of the elements could be negative, and there would be values of the elements of x which would yield a negative result. The criterion must be strengthened, such that the expression is positive regardless of the values the elements of x may take. If this is the case, the matrix is called ‘positive definite’. If there exists values of x for which the result is zero, it is called ‘positive semi-definite’.

Reversing the criterion introduces the idea of negative definite and negative semi-definite. If the sign depends on the values of x, the matrix is ‘indefinite’.

10.4.2 Eigenvalues and Eigenvectors The only type of matrix which we can characterise in this way is a diagonal matrix.

Let:

x=Ty

Where the rows of T are linearly independent. Each element of y is a linear combination of the elements of x, and vice versa.

Furthermore, we do not wish to change the value of the denominator of the test function:

xT x= yT y=xTT T Tx

So that:

T T=T−1

As an example of a matrix of this type consider the 2×2 example:

T=( cosθ sin θ−sin θ cosθ)

Where θ is an angle.

This orthogonality requirement restricts our discussion to symmetrical matrices, which is unfortunate because the system matrix is rarely symmetrical.

We can interpret the rows as the projections of a pair of perpendicular axes which have been rotated with respect to the reference set.

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Higher order matrices of this form can be similarly considered as a set of n perpendicular axes which have been rotated with respect to the initial reference axes. Since the rows represent lines in n-space which are perpendicular to each other (whatever that means in spaces of dimension higher than 3), this type of matrix, whose inverse is equal to its transpose is called an ‘orthogonal’ matrix. It can be thought of as the rotation of an n-vector.

The numerator of the test function is:

xT Ax= yT T T ATy

Taking any pairs of rows (r) of T, the orthogonality condition implies:

ri r jT={1 i= j0 i≠ j

With a similar relationship for the columns.

We shall assume that a rotation exists such that each row becomes aligned with its corresponding axis, so that it takes on a scalar value, rather than the vector of the arbitrarily oriented original matrix row. In other words, the transformation results in a diagonal matrix.

The elements of this diagonal matrix are called the eigenvalues.

Multiplying a vector by a matrix may be considered to consist of three stages firstly a rotation into the eigenvalue axis set, followed by component-wise scaling by the eigenvalues, followed by a rotation back into the original reference frame.

This may be written as the relationship:

Λ=TT AT

Where Λ (lamda) is the diagonal matrix of eigenvalues.

The rows of T define the orientation of the eigenvalue axes, and are called eigenvectors.

As with the inverse, the process of finding the eigenvalues requires processing at the element level, so as far as matrix algebra is concerned, we assume they exist, and don’t concern ourselves with how to go about calculating them.

Using this diagonalisation we can propose the following definitions:

If all the eigenvalues are positive – the matrix is positive definite

If the eigenvalues are positive or zero – the matrix is positive semi-definite

If all the eigenvalues are negative – the matrix is negative definite

If the eigenvalues are negative or zero – the matrix is negative definite.

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Eigenvalues can occur as complex conjugate pairs. When this occurs the elements of T are also complex, so that when calculating the weighted sum squares of x, the imaginary parts cancel. The above statements apply to the real part of the eigenvalue.

The calculation of eigenvalues is covered in the following sections. For the special case of a symmetric matrix, the simplest test for positive definiteness is to expand the determinants of increasing order from the lead diagonal. If all are positive, the matrix is positive definite.

10.4.3 Calculation of EigenvaluesWe can extend the idea to non-symmetrical matrices, such as typical system matrices, by noting that the change of variable:

x=Ty

Transforms the system equation into:

T y=ATy

The new system matrix is:

Λ=T−1 AT

In the special case where A is symmetric, the transformation matrix T is orthogonal. In either case, we can set up an equation for each of the rows of A.

λ i vi=A v i

Where λi is the diagonal element corresponding to the row of interest. It follows that the eigenvalues may be calculated for a general (square) matrix by solving:

|Iλ−A|=0

If the matrix is not symmetric, the eigenvectors will no longer be orthogonal.

So we reach the conclusion that the condition for the matrix exponential eAt to be stable, is for A to be negative definite.

This is curiously similar to the stability requirement for the roots of the characteristic equation.

10.5 Stability – a Different PerspectiveConsider a system characterised by a state variable x. We know that any even function of x can be arranged such that it will have positive values regardless of the value of x.

For example:

V=x2

Will always have positive values.

Its evolution in time is:

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V=2x x

If this is negative, the system will move back to the origin if displaced.

As an example, if the speed is proportional to the sign of x:

x=−U sign ( x )

˙V=−2Ux sign ( x )=−2U|x|

This is always negative, regardless of the value of x.

We are mainly interested in the exponential case, with its feature of linear feedback.

x=−ax

V=−2ax2=−2aV

The square of the state variable is also governed by an exponential decay.

For the multi-variable case, we may define a ‘length’ of the state vector:

|x|=√( p1 x1 )2+( p2 x2 )

2⋯

Where the xi are the individual elements of x and the pi are weightings needed to ensure the calculation is dimensionally consistent (in the units sense, not the matrix size sense).

The values of the pi are frequently chosen as the reciprocals of the maximum values of the corresponding state variables.

The function:

V=|x|2

Appears a reasonable analogue of the scalar example. It may be written:

V=xT Px

In using the ‘length’ (usually called the ‘Euclidean norm’, because there are many metrics which could reasonably be described as the ‘length’). Actually, P doesn’t need to be diagonal, if it is positive definite, V will still be positive.

Differentiating:

V= xT Px+ xT P x

In order to ensure the system always drives itself to zero, we require, by analogy with the scalar case:

V=2λV

Where λ (lamda) is a constant, which is negative for a stable system.

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We could use this criterion to determine the stability of non-linear systems. For the moment, we shall restrict the discussion to the linear case:

x=Ax

V=xT ATPx+xTPAx=2 xT λPx

As x is not identically zero, we may write this as:

AT P+PA+Q=0

This equation can be solved for P. If Q is positive definite, and the equation yields a positive definite solution for P, the system is stable. Since we are dealing with symmetric matrices, this is a straightforward test.

Note that the equation must be solved for P. There exists an infinite number of choices for P, but we would be lucky indeed if they all yielded a positive definite value for Q.

The matrix equation for P is called Lyapunov’s Equation, and the method of demonstrating stability this way is called Liapunov’s direct method. It may be extended to non-linear systems.

It may be written, in terms of λ:

( AT P−λP )T+(AT P− λP )=0

i.e:

( AT−λI ) P=0

Now P is positive definite, so the equation requires the matrix (AT-λI) to be singular.

Since I is obviously symmetrical, this is the same as requiring (λI-A) to be singular.

The similarity transformation for A yields:

Tλ−ΛT is singular

If λ is equal to one of the eigenvalues of A, the rows will cease to be linearly independent, and the matrix will be singular. There exist as many solutions as there are eigenvalues, and if any of the solutions has a positive real part, the system will be unstable.

The test for linear independence is for the determinant of the matrix to be zero. Hence the eigenvalues are found from:

|Iλ−A|=0

As we have seen.

10.5.1.1 Second Order and Higher SystemsWe have seen that the damped linear oscillator equation may be written in state space form:

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(uz )=(−2 ζ ω0 −ω02

1 0 )(uz )The determinant is:

|λ+2 ζ ω0 ω02

−1 λ |=λ2+2 ζ ω0 λ+ω02=0

This is exactly the same as the characteristic equation.

More generally, for an nth order equation:

x=(−an−1 ⋯

1 0−a1 −a0

⋯ 0⋮ ⋮0 0

⋮ ⋮1 0

) xThe bottom left sub-matrix is the n-1 identity matrix, the top row consists of the coefficients of the order given by the subscript. The final column, apart from its first entry, consists of zeros. This form of the system matrix is called a companion form. More general matrices may be transformed into this form by a series of elementary row operations, amounting to a change of variable to a linear combination of the original state variables. Hence the determinant becomes:

λn+an−1 λn−1⋯ a1 λ+a0=0

This is the characteristic equation for an nth order linear homogenous equation.

We conclude that the eigenvalues are the same as the roots of the characteristic equation. However, the matrix approach shows us that each root is associated with a specific direction.

The plant equations are rarely derived in companion form, so that evaluating the determinant furnishes a more convenient method of deriving the characteristic equation than transforming first into companion form.

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11 Examples

11.1 IntroductionBefore we consider using artificial feedback to improve the behaviour of a system, our first task is to find out why it misbehaves in the first place. In the vast majority of cases adequate behaviour is obtainable by changing the values of the system parameters (e.g. adding bilge keels, changing wheel profiles, or changing the wing dihedral angle, depending on context).

In many organisations the relationship between the dynamics and the parameters of the physical components tends to fall between the mechanical, aeronautical or electronic design department and the department responsible for control system design. Identifying the fundamental reasons for the stability problems is all too often the responsibility of neither.

We have considered a few stability examples in Chapters 5 to 8. This section will introduce a range of problems from different engineering domains. This is to illustrate the point that the control engineer is not a specialist in any single domain. The control engineer must be a polymath, with sufficient knowledge of all domains to be able to communicate with specialists in each on their own terms.

11.2 Vibrating StringThe earliest eigenvalue problem was the vibration of a stretched string. Pythagoras noted that the pitch of notes was determined by the length of the string, with harmonious sounds resulting from strings whose lengths bore a simple ratio, like 2:3.

Figure 11-38 : Vibrating Stretched String

If we consider a string under constant tension T which is large compared with any change in tension due to the vibration, the equation of motion of a short length δl of the string is found from equating the force applied to it the product of its mass and acceleration. If θ is the local slope of the string, The lateral force is given by:

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F=T (sin (θ+δθ )−sin θ )

The deflection of the string is small compared with its length, so the slope will be a small angle, so:

F=T δθ

If the mass per unit length of the string is ρ, the mass of the element is ρδl, so the equation of motion for the element of string is:

ρ y=T dθdl

If the shape of the distorted string is given by the curve y=f(x),

δl δx ,θ dydx→dθdl

d2 yd x2

If the shape of the distortion is:

y= ymxsin(nπ xL )Where n is an integer and L is the length of the string, the curvature is given by:

d2 yd x2

=− ymaxn2 π2

L2 sin(nπ xL )From which the equation of motion becomes:

ymax=−n2 π2Tρ L2 ymax

Since n may have any value, there are infinite possibilities for the shape of the distorted string,

In other words there is an infinite number of possible modes, all of different frequency. The noises made by a novice violinist illustrate this point all too clearly. However, the skilful player will only excite a set of modes which gives the instrument its characteristic tone colour. Since the modes are orthogonal, it is possible to excite some and not others.

The simplest case is for the string to distort into a simple curve given by n=1. (The word ‘sine’ is derived from the Latin sinus, meaning curve). This mode of vibration is called the fundamental. The characteristic equation is:

λ2+ π2TρL2=0

The frequency of vibration is:

f= 12L √Tρ Hz

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A similar approach was used to find the buckling load of a column. A wide range of dynamic problems, involving vibration modes and critical dynamic loads, are addressed in a similar manner. As a simple extension to the above problem we could use the result from engineer’s bending theory to calculate the frequency of vibration of a beam:

ρ d2 yd t 2

+EI d4 yd x4

=0

In cases like this, where deflection depends on time and position, we should use the partial

derivative notation (∂ y∂ x ), using the Russian letter ‘da’. Since the problem is separable in x and t, it

doesn’t really make much difference which notation we use.

Although computer codes exist for this class of problem, the user should be in a position to estimate the answer roughly before using the code. A fool with a tool is still a fool, and contrary to widespread belief, the quality of the analysis depends on the quality of the analyst, not on the quality of the code used in the analysis.

11.3 Directional Stability of a Road Vehicle When cornering , the body of a road vehicle does not necessarily point in the direction of motion. Instead, the tyres distort as they rotate resulting in an angle between the plane of the wheel hubs and the direction of motion. This is known as a slip angle, although there is no actual slippage of the tyre on the road. The distortion generates forces which keep the vehicle on the road.

The tyre forces depend on the slip angle and on the load on the tyre, but for simplicity we shall ignore the dependency on load, and assume the tyre forces are proportional to the slip angle only. These are generated in an axis set which is fixed with respect to the vehicle, which is not an inertial frame of reference (we would not feel centrifugal accelerations on corners if it were).

The equations of motion must be derived in this moving axis set. If the body heading is ψ, the fixed axes components of velocity may be transformed into body axes using a 2D axis transformation matrix Ψ (psi):

uB=Ψu f

Where u is the velocity and the subscript B means body and f means fixed axes.

Differentiating:

uB=Ψ uf +Ψ u f

The force acting in fixed axes is F f=m uf , where m is the vehicle mass, so the body axes force is:

FB=ΨF f=mΨ u f

Expressed in terms of the body orientation ψ (note, the transformation matrix is orthogonal):

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Ψ=(cosψ sinψ−sinψ cosψ ) , Ψ=(−sinψ cosψ

−cosψ −sinψ )ψ=( 0 ψ−ψ 0 )Ψ

Hence, the equation of motion, expressed in body axes is:

muB=FB+m( 0 ψ−ψ 0 )uB

Expressed as a pair of scalar equations:

u=Xm+ψ v

v=Ym−ψ u

Where u,v are the body axes components of velocity and X,Y are the body axes components of force.

The first term, on the right hand side of these equations, is the one expected from Newton’s Second Law, the second means that the components of the projection of the velocity on to body axes changes as the body rotates with respect to the velocity vector.

Figure 11-39: Road Vehicle Dynamics

We assume that the vehicle is moving at constant speed (U), so that the first equation is satisfied

identically (the product ψ v is second order in small quantities, and can be neglected). The second equation becomes:

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v=Ym−U ψ

The moment equation is:

C ψ=N

Where C is the moment of inertia about a vertical axis and N is the yawing moment.

If the front axle is ‘a’ metres ahead of the centre of gravity, the slip at the front axle is (since v and ψ are small quantities).

δ f=v+a ψU

The tyre force is assumed proportional to this, so the front axle generates a force:

Y f=−k ( v+a ψU )The moment is:

N f=−ka( v+a ψU )Similarly, the rear axle forces and moments are:

Y r=−k ( v−b ψU )

N r=kb( v−b ψU )The total side force is:

Y=−k ( 2v+ (a−b )ψU )

The moment is:

N=−k ( (a−b ) v+(a2+b2) ψU )

Hence the equations of motion become:

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ψ=−k ((a2+b2)CU ) ψ−k (a−bCU )v

v=−k (2mU ) v−(k (a−b )mU

+U )ψThese are in state space form, so the characteristic equation is given by:

| λ+k ( a2+b2

UC ) k ( a−bUC )(U+ k (a−b )

mU ) λ+ 2kmU

|=0

i.e:

λ2+( k (a2+b2)

CU+ 2kmU )λ+ k2

mCU2 (a+b )2− kC

(a−b )=0

If the constant term is negative, there will be at least one positive root. This will occur if a>b and

U2> km

(a+b )2

(a−b )

So if the centre of gravity lies closer to the front axle than the rear, the vehicle will be directionally stable. If this condition is not met, instability will be encountered at some speed. The onset of instability may be delayed using a long wheel base, stiff tyres and lightweight construction.

Tyre forces also depend on the vertical load applied, so that the lateral forces arising from cornering result in different tyre forces on the left hand side compared to the right, and this must be taken into account when designing suspension systems. However, this simplified analysis does illustrate the importance of the condition of the rear tyres for directional stability.

11.4 Hunting of a Railway WheelsetA railway vehicle differs from a road vehicle in that directional stability is provided by means of a pair of rails rather than the adhesion of the tyres. Actually, railway tyres distort in the region of contact, but since they are usually made of steel, the degree of distortion is hardly discernable. The distortion of the region of contact gives rise to the forces which ultimately determine the directional stability.

The kinematic stability of a wheel set is achieved by virtue of the wheel profile. Railway wheels have flanges for safety, but these should rarely come into contact with the rail. The treads are machined with a slight taper, so that lateral displacement changes the diameter of the wheel where it contacts the rail, increasing it on the opposite side to the disturbance, reducing it on the other. The two wheels are rigidly connected together, so they have the same angular velocity. The speed of the hub of the outer wheel increases, that of the inner wheel reduces, and the wheelset steers back towards the centre.

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If pure rolling contact existed between wheel and rail, the wheel set would continue through the central position to a displacement equal to the initial disturbance on the other side, and subsequently continue to oscillate from side to side. In fact, the motion damps out below a certain speed and is amplified above the critical speed.

Figure 11-40: Coning Action of Railway Wheelset

The interaction between the wheel and rail is similar to that between a pneumatic tyre and a road surface, the amount of distortion is small, but the high Young’s modulus for steel ensures that the forces involved are high.

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Figure 11-41 : Contact Points of Yawed Axle

As can be seen from the above diagram, the position of the point of contact moves outwards on the treads as the axle yaws, but this is of second order in the yaw angle, and can be neglected.

If the speed of the wheelset is U, the angular velocity of the axle is about:

ω=Ur0

Where r0 is the radius of the wheel for straight running. If the wheelset is displaced to one side by an amount y, the peripheral speed of the wheel in contact with the rail is given by:

up=ωr0±ω ky

Where k is the wheel taper. If the angular velocity of the axle is ωz, the velocity of the point of contact is:

uc=U±d2ωz

Where d is the track gauge.

These two speeds differ slightly, resulting in elastic distortion of the wheel and rail in the region of contact. In the railway context, this is called ‘creep’, although it is practically identical to the phenomenon of ‘slip’ in road vehicle wheels. The force arising from creep is assumed proportional to this difference in speeds.

F∝(up−uc )

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The lateral force arises from the difference in inclination of the normal reactions on the rails:

y=−Uωz−Wm (2kd ) y

Where W is the wheelset load and m its effective mass.

As the wheelset displaces laterally it also rotates about an axis pointing in the direction of motion. The angle the wheelset makes with respect to the horizontal is:

θ=2kyd

If the lateral acceleration is y , the angular acceleration is:

θ=2kdy

So the inertial resistance is:

L=A θ

Where A is the moment of inertia of the wheelset about the direction of motion.

This results from a lateral force Y acting at the points of contact, so the moment is:

L=r 0Y

Or

Y= 2kr0 d

A y

From which it follows that the effective mass term is:

m=m0+2kr 0d

A

Where m0 is the actual mass of the wheelset.

The yaw equation becomes:

N=C ωz=σd2 (2Ukr0 y−dωz )

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Where σ (sigma) is the creep coefficient. The creep force is proportional to the difference in speed of

the wheel hub due to the tread contacting the rail and the axle yaw.

The characteristic equation becomes:

|λ2+W

mk2d

−U

−σ Ukdr 0C

λ+ σd2

2C

|=0

Expanding:

Cλ3+( σd2

2 ) λ2+Wm kC2d

λ+σ (Wm kd4−U

2kdr 0 )=0

For an iron tyre on a steel rail, the creep force generated per unit speed mismatch is enormous, so σ is expected to be very large, and the coefficients not containing the creep coefficient may be ignored.

The characteristic equation reduces to:

λ2+ 2kd ( W4m−U

2

r0 )=0

Evidently, the motion becomes unstable at a critical speed given by:

U2=Wr04m

The critical speed depends on the ratio of load to rolling mass. A more complete analysis includes lateral creep, which has here been treated as a rigid constraint.

Despite achieving speeds well in excess of 180mph, modern suspension design results in trains that are practically immune to hunting. This was achieved by analysing the root cause of the problem and changing the system parameters, rather than proposing any form of closed loop control.

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12 Linearisation

12.1 IntroductionIt has been mentioned several times that linear homogenous equations usually represent an approximation to the true plant behaviour. This is not an excuse to dismiss the entire edifice of mathematical systems theory, and science in general. That we use approximations is of no concern, provided our theory furnishes us with methods which quantify how wrong our plant equations can become before there is cause for concern.

The usual trick of linearization is to identify states which vary slightly over the timescales of interest, and assume all system disturbances are small quantities, so that second and higher order terms may be neglected. In essence, we start with the system in a condition defined by a design case established from earlier analysis. The issue is whether the system will remain close to the design condition if perturbed.

This is best illustrated with an example.

12.2 Stability of a Circular OrbitA particle is orbiting around a central attractor in which the law of attraction is an inverse rn relationship. For what values of n is the orbit stable?

Since the force attracts to the centre, there is no force acting tangentially, so the angular momentum is constant:

ur=h

Where u is the tangential velocity. The system is in equilibrium such that the centrifugal acceleration is equal to the force per unit mass of the attractor:

u2

r∝ 1rn

Or, in terms of the angular momentum:

h2

r3= krn

If the orbit increases by an amount Δr, force acting is

Fm= h2

(r+Δr )3− k

(r+Δr )n

Where F is the force and m the paricle mass.

Δr is small compared with r, so the increment in force is given by:

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ΔF dFdr∆r=m(−3 h

2

r 4+n krn+1 )∆r

The equilibrium condition requires:

k=h2 rn−3

The equation of motion governing Δr is:

d2 (∆r )d t 2

=(n−3 ) h2

r 4∆r


λ2−(n−3 ) h2

r4=0

Evidently, n must be less than 3 for a stable orbit.

For the case n=2, the period of the perturbation is:

T=2πr2

h=2π r2

ωr2

Where ω is the angular velocity of the radius vector from the centre of attraction to the particle. It follows that the period of small perturbations is the same as the period of the orbit. In other words, the orbit changes from circular to slightly elliptical.

12.3 Rigid Body MotionWhen predicting the motion of free-flying vehicles, such as aircraft or rockets, we have the same difficulty as with the stability of road vehicles; the forces which determine the motion are generated with respect to the vehicle body axes, but Newton’s Second Law requires the motion to be referred to an inertial (constant velocity with respect to the fixed stars) frame of reference.

It is necessary to work in the non-inertial frame of the body of the vehicle.

In the 2D case, we resolved from fixed to moving axes using a 2×2 axis transformation matrix, the rows of which were the components of lines of unit length along the body x and y axes. These lines are called unit vectors.

The 2D case is represented as the trigonometric functions of an angular rotation, and so there is a strong temptation to define the 3D axes in terms of angles. In three dimensions, the orientation of the axes of rotation aren’t fixed, whilst in the 2D case the axis is always parallel to the z-axis, and this leads to confusion.

The human eye is limited to interpretation of 2D images, and angles are strictly associated with planes rather than spaces. As the works of M C Esher illustrate, our 2D perception is easily fooled when we try to represent 3D objects on a 2D medium.

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Of course, the same self-conceit which cannot admit ignorance, and is commonly manifest as an absence of integrity, will ignore this advice. Let us admit our physiological limitations, and avoid this particular path to delusion and self-deception.

Figure 12-42 : Illustration of Ambiguity of 2D Pictures of 3D Objects

Architect’s drawings and engineering blueprints are produced using orthogonal projection for a very good reason.

We cannot extend the angle description to the 3D case because angles are inherently two-dimensional entities. We need first to define the plane in which the angle is measured.

However, defining the orientation as an orthogonal matrix whose rows are the unit vectors along the three moving axes is simpler, and a direct extension of the 2D approach.

12.3.1 NomenclatureSince aircraft and spacecraft have the least constraints on their motion, we shall derive the general equations of motion using an aeronautical convention.

The axis set chosen has the x-axis pointing from the centre of gravity towards the nose. The y-axis points to starboard (to the right, when looking along the x-axis). In order to complete a right-handed orthogonal axis set, the z-axis points downwards. This is a body axis set.

The velocity components along the x,y and z axes are denoted u,v and w respectively. The forces are X,Y and Z, the angular velocities around the x, y and z axis are p, q and r for roll, pitch and yaw respectively. The moments about the axes are L, M and N.

Consistent with the notation for the moments, the x-axis is the unit vector, (l1, l2, l3), the y-axis is the unit vector; (m1, m2, m3), and the z-axis is (n1, n2, n3). The subscript denotes the fixed axis to which the component of the moving axis is referred.

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A vector in body axes with components aB, bB, cB, becomes, in fixed axes:

(abc)=aB(l1l2l3)+bB (

m1

m2

m3)+c B(n1n2n3)

This may be written in matrix form:

(abc)=(l1 m1 n1l2 m2 n2l3 m3 n3)(

aBbBc B)

Or more compactly:

a=T aB

Where it is understood that ‘a’ is now a vector, and T is a matrix.

Resolving from fixed axes to body axes:

aB=a l1+b l2+c l3

With similar expressions for bB and cB. Or, in matrix form:

aB=TT a

In particular the fixed axes velocity is resolved into body axes:

uB=TT u

12.3.2 Translation EquationIn order to find the inertial force in body axes, the expression for the body axes velocity must be differentiated with respect to time:

uB=TT u+T T u

By Newton’s Second Law:

u= 1mX

Where m is the mass and X is the fixed axes force vector, so TT X is the body axes force vector.

There remains the derivative of T to consider. This term means that the body is rotating with respect to the velocity vector, so the components of velocity along each body axis will change in time even if the magnitude remains constant.

Consider the x axis. In a short interval of time δt, it will yaw by an amount rδt in the current y-axis direction, and pitch by an amount –qδt in the z-axis direction, so the change in the x-axis is:

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Figure 12-43 : Rates of Change of Body Axes

(δl1δl2δl3)=(m1

m2

m3)rδt−(n1n2n1)qδt

In the limit, as the time interval tends to zero, we have:

(l1l2l3)=(m1

m2

m3)r−(n1n2n3)q

Applying the same reasoning to the y and z axes:

T=T Ω

Where:

Ω=( 0 −r qr 0 −p−q p 0 )

Note: ΩT=−Ω

Taking the transpose:

T T=−ΩT T

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The translational equation of motion in body axes becomes:

uB=1mX B−ΩT

T u= 1mX−ΩuB

Re-arranging:

X B=m (uB+ΩuB )

Or, explicitly in scalar form:

X=m ( u−vr+wq )

Y=m ( v−℘+ur )

Z=m ( w−uq+vp )

12.3.3 Rotation EquationsIn order to avoid unnecessary first moment terms, we refer the rotation of the body to the centre of mass.

A general particle of the body of mass δm is located at position (x,y,z) in body axes. Since the motion is referred to the centre of gravity, during any translational acceleration, the inertial moment generated by this element of mass is exactly balanced by the other elements of mass, so there is no net moment due to acceleration of the centre of gravity.

The position of the particle in fixed axes is:

x=T x B

The velocity with respect to fixed axes is:

x=T x B=T ΩxB

The acceleration is:

x=(T Ω+T Ω ) xB

The inertial force is:

δm x=δmT (Ω2+Ω ) xB

δX B=δm (Ω2+Ω ) xB

We can derive the moment of a force from the co-ordinates of the particle in body axes and the components of force acting:

( LMN )=( 0 −z yz 0 −x− y x 0 )(XYZ )

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Or: LB=R XB

The moment equation becomes:

δLB=δmR (Ω2+Ω ) xB

Evaluating the right hand side:

(Ω2+Ω )xB=(−(q2+r2 ) x−( r−pq ) y+( q+ pr ) z( r+ pq ) x−( p2+r 2) y−( p−qr ) z−( q−pr ) x+ ( p+qr ) y−( p2+q2 ) z)

The result is the complicated expression:

δL=δm ( p ( y2+z2 )−qr (x2+z2 )+qr (x2+ y2)−( q−pr ) xy−( r− pq ) xz+(r2−q2 ) zyq ( z2+x2 )−rp ( y2+x2 )+rp ( y2+ z2 )−( r−qp ) yz−( p−qr ) yx+( p2−r2 ) xzr (x2+ y2 )− pq ( z2+ y2 )+ pq ( z2+x2 )−( p−rq ) zx−( q−rp ) zy+(q2− p2 ) yx)

Summing over the whole body, we find that the rotational motion depends on six inertia parameters rather than simply the one (mass) for translational motion, or indeed the single moment of inertia for the 2D case.

The three dimensional motion of a rigid body is potentially complex for bodies of arbitrary and irregular shape. The inertia terms are called the moments of inertia and the products of inertia.

∫ ( y2+z2 )dm=A, the moment of inertia about the roll axis

∫ ( z2+x2 )dm=B, the moment of inertia about the pitch axes

∫ (x2+ y2 )dm=C , the moment of inertia about the yaw axis

The remaining terms are called products of inertia.

In aircraft stability, it is customary to refer the x-axis to the velocity vector so that the product:

∫ xz dm=E

Is non-zero if the body of the aircraft is at a significant angle of attack in the design cases of interest (e.g. approach, landing and take-off).

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It is always possible to choose a set of body axes for which the products of inertia are zero. These are called principal axes. Generally, they are associated with the axes of symmetry, and may be found by inspection.

The moment equations with respect to principal axes are also known as Euler’s equations:

L=A p−(B−C )qr

M=B q−(C−A ) rp

N=C r−( A−B ) pq

The equations of motion contain cross product terms and are not linear.

As an aside, the equations of motion for rotation are derived from translational accelerations of the individual particles of the body. If any effects arise from the actual rotation of the particles themselves, they will not be predictable from Euler’s equations, as some who should know better seem to think.

12.4 Perturbation forms

12.4.1 Cartesian Rocket – Steady FlightA typical rocket configuration is notable for its symmetry. Often, the plan view is identical to the side view, so that the pitch and yaw moments of inertia are identical, (i.e. B=C). Also, rockets tend to be slender, so that the roll moment of inertia is much smaller than pitch and yaw (i.e. A<<B).

A fin-stabilised rocket is expected to roll slowly, so that all components of angular velocity may be considered small quantities. The moment equations become:

L=A p , M=B q=C q , N=C r=B r

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The translational equations are similar, except that the speed of the rocket is usually significant. The upper case U denotes the steady state speed, with lower case referring to small changes.

The translational equations become:

X=mu

Y=m ( v+Ur )

Z=m ( w−Uq )

The x-wise equation should be expected from Newton’s Second Law. The lateral equations introduce a term arising from the change of orientation of the body with respect to the velocity vector. If v=w=0 ,the body is at a fixed orientation with respect to the velocity (i.e. the angle of attack is constant). The velocity vector must be turning in space at the same rate as the body, which can only happen if a centripetal (if you are observing from an inertial frame outside), or centrifugal (if you are on board) acceleration, is acting.

It is important to understand that v and w are projections of the velocity vector on to body axes, they are related to the incidence (α) and sideslip (β) angles:

α=−tan−1( wU ) −wUβ=tan−1( vU ) v

U

The longitudinal and two lateral equations of motion are decoupled as far as inertial forces are concerned, so the motion is governed by three independent pairs of equations, and the3D problem reduces to three 2D problems.

12.4.2 Cartesian Rocket – High Angle of AttackHigh angle of attack is usually associated with high lift, such as would be encountered by a guided missile in a violent manoeuvre. The manoeuvre itself would be associated with high centrifugal acceleration and consequently high pitch rate. We need to know that the vehicle is stable under these conditions and not likely to increase its incidence still further, causing it to break up.

The high pitch rate couples the roll and yaw, so that the yaw equation becomes:

N=C ( r+Qp )

Where Q is now the steady state pitch rate.

Let the speed of the vehicle be VR, the steady state longitudinal and vertical components of body axes velocity become:

U=V R cos α

W=−V R sinα

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Where α is the steady state angle of attack.

We shall assume the steady state forces are accounted for in the X,Y and Z terms of the translational equations. These become:

X=m ( u+Wq+Qw )

Y=m ( v−℘+Ur )

Z=m ( w−Uq−uQ )

The inertial cross coupling with the yaw channel requires the roll rate to be kept near zero. This tends to be difficult to achieve as fin control authority tends to fall off with increasing incidence, so that most of the available control is required to trim at the high angle of attack. Very little residual control is left over for roll.

Note that the X and Z equations are now strongly coupled, so the system is governed by a higher order differential equation than was the case at low incidence. Non-linear, and potentially chaotic aerodynamic effects are also encountered at high angles of attack, rendering the design of autopilots difficult.

It is curious how ‘systems thinking’ always seems to be limited to the trivial aspects of systems organisation, with scant attention paid either to system behaviour, or the potential problems of achieving desired results. Thus the relatively mundane but easily understood problems of magazine handling appear to take precedence over the more pressing, but harder to resolve, issue of hitting the target.

12.4.3 Spinning Projectile Another case of interest to the symmetrical body is that of a spinning projectile.

As the body axes are spinning rapidly, it is difficult to interpret the results obtained from working in body axes. When observing an axi-symmetric body spinning, the eye ignores the spinning motion altogether, as any surface feature is moving so fast as to be a blur. What we see is the motion in a non-spinning axis set, and as we are seeking insight, rather than an abstract answer, we must resolve the equations of motion into this non-spinning set.

We propose a non-spinning axis set such that the body axes are rotated at an angle φ with respect to it. We resolve the body axes quantities into this axis set:

Forces:

X B=X

Y B=Y cosϕ+Z sin ϕ

ZB=−Y sin ϕ+Z cos ϕ

Velocity:

uB=u

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vB=v cosϕ+w sin ϕ

wB=−v sinϕ+w cos ϕ

Differentiating:

vB= v cosϕ+w sin ϕ−vp sin ϕ+℘cos ϕ

wB=−v sinϕ+w cos ϕ−vpcosϕ−℘sinϕ

(since: ϕ=p)

The convention for angular velocities and moments is they have direction given by their axis direction, which makes them consistent with other vector quantities.

Moments:

LB=L

MB=M cos ϕ+N sin ϕ

NB=−M sinϕ+N cosϕ

Angular velocities:

pB=p

qB=qcos ϕ+r sin ϕ

r B=−q sin ϕ+r cosϕ

Differentiating:

qB=q cos ϕ+r sin ϕ−qpsin ϕ+rpcos ϕ

r B=− q sinϕ+¿ r cosϕ−qpcosϕ−rp sinϕ¿

Substituting back into the body axes equations of motion, we have:

Y=m ( v+Ur )

Z=m ( w−Uq )

M=B q+Arp

N=B r−Apq

The first two equations remove the effect of roll from the projection term, which is not surprising. The moment equations introduce a gyroscopic term.

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12.4.4 SpacecraftSince spacecraft operate in vacuum, there is no advantage to long slender configurations, so the principal moments of inertia can be nearly equal, and Euler’s equation reduces to its simplest form:

L=A p

M=B q A q

N=C r A r

Many satellites deploy a boom to stabilise orientation using tidal forces to keep it aligned with the gravitational field, in which case the equations for a single axis of symmetry would apply.

12.4.5 AircraftA missile typically has one set of lifting surfaces for yaw, and another, usually identical set, for pitch. This is a very heavy configuration for an aircraft, which is designed for efficiency, rather than an ability to hit a target. Aircraft are noted for having a single principal lifting surface, supplemented with control surfaces.

The measurement of aerodynamic forces in a wind tunnel is most conveniently performed using a sting balance which is aligned with the airflow, so that they are referred to the velocity direction, and not the body.

The consequence of not using a principal axis set is the introduction of products of inertia, in particular, the product of inertia:

E=∫ xzdm

Is non-zero.

This does not affect the translation equations, but the moment equations become:

L=A p−(B−C )qr−E ( r−pq )

M=B q−(C−A ) rp+E ( p2−r2 )

N=C r−( A−B ) pq−E ( p−rq )

For an aircraft configuration B≠C, so the small perturbation equation becomes:

L=A p−E r

M=B q

N=C r−E p

The wind axis set introduces inertial coupling between roll and yaw. This is of no major concern because the aerodynamic rolling and yawing moments are also strongly cross-coupled.

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12.4.6 Wider ApplicationThe rigid body equations of motion apply equally well to road and rail vehicles, ships, submarines and airships, although some simplifying assumptions will usually need to be made, depending on the specifics of the design case of interest. Vehicles that rely on Archimedes principle for support must impart accelerations to the surrounding fluid when they accelerate, giving rise to ‘virtual mass’ terms which are included in the external forces and moments.

12.5 Partial DerivativesWhen considering derivatives we have so far only concerned ourselves with functions of a single variable. The increment in the dependent variable in response to an increment in the independent variable is unambiguous. However, if we are dealing with a function of two or more independent variables, the increment can only be meaningful with respect to each independent variable, whilst keeping the others constant.

For example a 2D surface may be represented as height as a function of x and y coordinates:

h=h ( x , y )

We can increment x, keeping y constant to get the slope in the x direction, finding the increment in h and taking it to the limit, results in a differential coefficient, but since its value depends on the value of y which determines the section through the curve, it is written using Russian (Cyrillic) lower case Д (‘da’) i.e:

limδx→0 ( δhδx ) y=

∂h∂x

This is called the partial derivative of h with respect to x.

The forces and moments on a flying body potentially depend on all the state variables and their derivatives, so could be functions of as many as 12 variables.

It is most unlikely that these functions will be linear, so we will always be linearising about the current design case. As an example, the drag force is expected to be a function of speed and incidence:

X=X (u .w )

The increment in drag force due to a small change in speed is:

∆ X=∂ X∂u

∆u

Having just introduced the partial derivative notation, we shall immediately discard it. In aeronautics we use the shorthand:

Xu=∂ X∂u

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The deltas are also omitted as it is understood that we are considering perturbations around a design point.

In general, we write for any force or moment F and any state x:

F x=∂ F∂ x

This is called a ‘stability derivative’.

The perturbation X force, is given by adding the contributions due to changes in all relevant states:

X=Xuu+X ww

It requires understanding of the physics of the flight vehicle to decide which of the derivatives are important. In particular, it is important to understand how the stability derivatives relate to the geometry of the vehicle.

All too often, the aerodynamicists estimate values for the stability derivatives which the control people use to design autopilots. It is usually nobody’s responsibility to see if the configuration can be tweaked to render the autopilot design simpler, or even unnecessary.

12.5.1 The Symmetrical Rocket Re-visitedThe longitudinal equation of motion is usually considered decoupled from the body motion of the rocket, so the x-wise motion is ignored. In the pitch plane, there is the normal force Z and the pitching moment M. The symmetry of the problem implies that these determined by the pitch plane quantities w (incidence) and q (pitch rate).

We have, for the pitch plane:

Z=Zww+Zqq

M=Mww+M qq

The Zw is the lift generated whenever a body or body-wing combination is rotated to an angle of attack with respect to the air flow. The lift is in the negative z direction, so Zw is expected to be large and negative. Za is the tail force arising from the increase in incidence at the tail due to the pitch rate, this is typically a very small term and is usually neglected.

The Mw term characterises the tendency for a stable vehicle to point in the direction of motion, an increase in incidence (i.e. rotating nose up) causes an increase in tail lift, causing the nose to pitch down, provided the centre of lift is aft of the centre of gravity. This is a negative incremental moment in response to s positive incidence disturbance. Mw must be negative for a stable vehicle. This is the reason why arrows and darts have flights at the rear.

The Mq term arises from the increase in tail lift due to the pitch rate causing a change in incidence at the tail. It is always negative.

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The yaw plane has analogous stability derivatives, Yv, Yr, Nv, and Nr. The same comments apply, except that with the axis convention chosen, a positive sideslip (v) must cause the nose to swing to starboard to cancel it, so that Nv must be positive for directional stability.

The significant roll stability derivatives are Lv and Lp. . A cruciform rocket tends to be stable either in the + orientation with respect to the plane of incidence, or in the X orientation. A roll disturbance will cause the plane of incidence to rotate relative to body axes, generating sideslip. For roll stability, the rolling moment resulting from this sideslip must tend to rotate the body back until the velocity vector again lies in the body plane of symmetry.

A positive sideslip causes a negative roll for stability, so Lv is negative. In rocket applications, Lv is usually ignored, as it amounts to an external excitation of the roll motion. There is no coupling of roll into yaw which would require us to consider roll and yaw in a single equation.

The Lp stability derivative is the roll damping term, it is always negative.

Including the inertia terms derived earlier, the lateral equation of motion of the rocket becomes:

B q=M qq+Mww

mw=Uq+Zww


λ2−( Zwm +M q

B ) λ−(MwUB

−ZwmM q

B )=0

Since Zw and Mq are both negative, the coefficient of λ is positive. The constant term is dominated by Mw, which must be negative. The undamped natural frequency of this mode

ω0=√ Zwm M q

B−MwUB

is called the weathercock frequency, as it characterises the tendency of a weather vane to point into wind.

12.5.2 Stability of a Spinning ShellIn order to analyse the spinning body, the equation of motion is resolved from body axes to non-spinning axes. We shall assume that the only significant stability derivatives are Zw, Yv, Mw and Nv. Furthermore, from the symmetry of the problem:

Y v=Zw

N v=−Mw

The forces and moments are:

Y=Y Bcosϕ−ZB sinϕ=Y v vBcos ϕ−Y vwB sinϕ=Y v v

Z=Y B sinϕ+ZBcosϕ=Y vw

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M=MB cos ϕ−NB sin ϕ=−N vw

N=M B sinϕ+N B cosϕ=N v v

There are no stabilising fins, so Nv is expected to be negative.

Substituting the forces into the equation of motion:

Y v v=m ( v+Ur )

Y vw=m (w−Uq )

−N vw=B q+Apr

N v v=B r−Apq

The symbol A is used to denote the entire system matrix and the roll moment of inertia, it should be obvious from the context which is meant.


|(λ−Y v

m ) 0

0 (λ−Y v

m )U 00 −U

−N v

B0

0N v

B

λ −( AB ) p( AB )p λ

|=0

This is rather tedious to evaluate. Since we are looking for relationships between parameters which determine stability boundaries, a numerical solution is a waste of everybody’s time.

The reference standard model (so beloved of non-participants, to the extent that it must be developed before anybody knows what to put in it} serves only to produce performance predictions at the end of development, for use in acceptance.

Actual understanding of the system requires the relationships between the components and subsystems, and their influence on overall system function to be determined. Such insight is provided much more effectively by approximate analytical methods of known bounds of validity, than by letting the Mongolian hordes loose on handle-cranking a time series numerical model, no matter how accurate a representation of the real would system it may be.

Knowing what the answer is (42?) does not furnish us with the knowledge needed to change it, should it be unsatisfactory.

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12.6 A Few More TricksIf we find ourselves evaluating determinants of order much higher than four, we probably haven’t employed our practical experience of the problem domain adequately.

However, if all else fails, the coefficients of the characteristic equation may be evaluated using the following procedure:

The coefficient of λn is unity. The coefficient of λn-1 is equal to –trace(A), where the trace of a matrix is the sum of the diagonal elements. The coefficient of λn-i is (-1)i multiplied by the sum of the minors of order i which can be expanded from the diagonal elements. That is all the determinants made up of the rows and columns containing the lead diagonal elements taken i at a time.

No matter how I try to explain it, it always comes out as gibberish. However, to show it is actually quite easy, we shall expand the system matrix for the spinning shell example.

Let the characteristic equation be:

λn+cn−1 λn−1+cn−2 λ

n−2⋯ c0=0

The trace is:

c3=− trace (A )=−(a11+a22+a33+a44 )=−2Y vm

The coefficient of λ2 is

c2=(|a11 a12a21 a22|+|a11 a13

a31 a33|+|a11 a14

a41 a44|+|a22 a23a32 a33|+|a22 a24

a42 a44|+|a33 a34

a43 a44|)c2=(Y v

m )2

+N vUB

+0+0+N vUB

+( AB )2

p2

Note that the number of determinants of order i is the number of different ways i objects can be selected from n; i.e:

n!(n−i ) ! i !

The first coefficient contains 4 terms, the second 6, the third 4 and the constant term is the determinant of the entire matrix. Since we know how many determinants we should have for each coefficient, it is easy to check that we haven’t missed one.

c1=−(|a11 a12 a13a21 a22 a23

a31 a32 a33|+|a11 a12 a14a21 a22 a24a41 a42 a44

|+|a11 a13 a14

a31 a33 a34

a41 a43 a44|+|a22 a23 a24a32 a33 a34a42 a43 a44

|)c1=−2(Y v

m )[( AB )2

p2+N vUB ]

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The constant term is the determinant:

c0=|A|=[(Y vm AB )

2] p2+( N vUB )

2

Since the system matrix is usually fairly sparse, a large number of the determinants implicit in this approach evaluate to zero. This method is usually quicker and less error prone compared with evaluating |Iλ-A|, and collecting up terms.

Having derived the characteristic equation, we can use our knowledge of the expected magnitudes of the system parameters to factorise it approximately. More generally, however, we need to be able to evaluate stability directly from the coefficients of the characteristic equation.

We transform the system matrix into the diagonal form of the eigenvalue matrix in order to determine the eigenvalues, so it is obvious that the two matrices have the same characteristic equation.

Applying the above algorithm to a diagonal matrix, it is evident that the trace is the sum of the eigenvalues, and the determinant is the product of the eigenvalues. So for a second order system:

c1=−( λ1+λ2 )

c0= λ1 λ2

Rather than solve explicitly for the eigenvalues, we note that if the eigenvalues are a complex conjugate pair, or both real, and have the same sign, the constant term is positive. If they are real and have different signs, one of the eigenvalues must be positive.

If c0 is positive, the real parts have the same sign, so if c1 is also positive, both eigenvalues must have negative real parts.

We can extend this reasoning to higher order systems. Taking a cubic, the coefficients, expressed in terms of the eigenvalues, become:

c2=−( λ1+λ2+λ3 )

c1=λ1 λ2+ λ2 λ3+λ3 λ1

c0=−λ1 λ2 λ3

We notice that for c0 to be positive, either all the eigenvalues are negative or two of the three are positive.

Denote the negative eigenvalue –α and the other two β and γ (gamma). The symbols denote strictly positive quantities.

c2 c1=(α−(β+γ ) ) (βγ−α (β+γ ) )=c0−(α 2+ βγ−αβ−αγ ) (β+γ )

Or: c2 c1−c0=−(α 2+c1 ) (β+γ )

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If two of the eigenvalues are positive, this will be negative, so if all the coefficients are positive, and this test function is positive, all eigenvalues will have negative real parts.

The test function may be written in determinant form:

H 2=|c2 c01 c1|

The top row consists of the even order coefficients, whilst the second row consists of the odd order coefficients.

For an nth order characteristic equation, the presence of unstable modes can be checked using a series of test functions derived from an n×n matrix of the form:

H=|cn−1 cn−3

10

cn−2

cn−1

cn−5 ⋯ 0cn−4 ⋯cn−3 ⋯

0⋯

⋮ ⋮0 0

⋮ ⋮ ⋮⋯ 1 ⋯

|The top row consists of alternate coefficients starting from that for λn-1, padded out with zeros. The second row consists of alternate coefficients starting from that for λn, i.e. unity.

The next two rows consist of the top two rows indented by one column, this is repeated until the matrix has n rows and n columns.

A matrix of this form is called a Hurwitz matrix.

The test functions are determinants of the sub-matrices formed from the top-left elements of the matrix; i.e:

H 1=cn−1

H 2=|cn−1 cn−3

1 cn−2|H 3=|cn−1 cn−3 cn−5

1 cn−2 cn−4

0 cn−1 cn−3|

⋮H n=|H|

If all the test functions are positive, all the roots of the characteristic equation have negative real parts.

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12.6.1 Spinning Shell Conditions for StabilityThe Hurwitz matrix may now be applied to find the conditions under which the spinning shell becomes stable. The Hurwitz matrix is:

H=(−2(Y vm ) −2(Y vm )[( ApB )

2

+NvUB ]

1 [(Y v

m )2

+2N vUB

+( ApB )2]

0 0

[[(Y vm AB )

2] p2+( N vUB )

2] 0

0 −2(Y vm )0 1

−2(Y vm ) [( ApB )2

+N vUB ] 0

[(Y v

m )2

+2N vUB

+( ApB )2] [[(Y vm A

B )2] p2+( N vU

B )2])

The test functions are:

H 1=−2(Y v

m )Since Yv is always negative, this will be positive.

H 2=|−2(Y v

m ) −2(Y v

m )[( ApB )2

+N vUB ]

1 [(Y vm )2

+2N vUB +( ApB )

2]|=N vUB

+(Y vm )2

Since the shell is assumed aerodynamically unstable, Nv is negative, so stability requires:

(Y vm )2

>|N vUB |

H 3=|−2(Y vm ) −2(Y v

m )[( ApB )2

+N vUB ] 0

1 [(Y vm )2

+2N vUB +( ApB )

2] [[(Y v

mAB )

2]p2+(N vUB )

2]0 −2(Y vm ) −2(Y vm )[( ApB )

2

+NvUB ]|

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i.e.

H 3=4(Y v

m )2 N vUB [(Y v

m )2

+( ApB )2]

Since Nv is negative, the shell is unstable, because all the other terms in this expression are positive.

There is no need to evaluate the fourth test function, if H3 were positive, H4 will also be positive if the constant term c0 is also positive.

The instability arises from the lift term, which for a body alone is small. The airflow does work in increasing the angle of attack, but gyroscopic action is a purely inertial effect, associated with conservative forces, and unlike tail fins cannot do work on the airflow. There does not appear to be any means of transferring energy from the shell back into the airflow, so without some means of dissipating this energy, the pitch and yaw kinetic energy must be increased throughout the flight. The yaw damping, Nr seems the only significant candidate. In any case, shells are notable for being very dense compared with aircraft, so the Yv/m term is expected to be small. If it is taken as zero, all the test functions become zero.

If we ignore the effect of aerodynamic lift, on the basis that the shell is dense, the characteristic equation retains only the even terms:

λ4+[2 N vU

B+( ApB )

2] λ2+( N vUB )

2

=0

This can be solved explicitly for λ, yielding a condition for negative roots:

p2>4|N vU|A ( BA )

Note that the analysis of the ‘simple’ system – the spinning shell which consists of only a body of revolution is more complex than that of the rocket with its two sets of lifting surfaces. This is to be expected from the Principle of Requisite Entropy. Contrary to current common misapprehension, simple systems exhibit complex behaviours. Thus, by concentrating one’s effort on the well-behaved system elements , all we are doing is deferring the evil day when the fundamental problems have to be addressed. Leaving this to the implementation stage is to court disaster.

12.6.2 The AeroplaneThe stability of an aircraft is expected to be more complex than that of a missile, but less complex than a spinning shell. Economy in weight dictates only a single set of lifting surfaces, which must provide both vertical and lateral stability. Most designs require a tail fin to provide adequate directional stability, although the ‘butterfly taii’, using steep dihedral on the tail plane to eliminate the fin, has been tried.

The measurement of stability derivatives is most easily performed in wind axes with the aircraft trimmed to the appropriate angle of attack.

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Since principal axes are not used, the inertial moments contain a product of inertia, which introduces additional terms into the roll and yaw equations.

The classical aeroplane does not exhibit the degree of symmetry of a rocket or missile, but unless the planform is asymmetric, such as in a slewed wing design, the equations may be treated as decoupled between pitch and yaw/roll.

The pitch plane moment equation is dominated by the ‘static’ stability term Mw, which characterises the tendency of the aircraft to point into wind. This is negative for a statically stable aircraft and minimum values are specified by the certifying authorities for this derivative. The pitch damping term Mq is also significant, there is also a further moment term arising from the rate of change of incidence. This arises because it takes a finite amount of time for the change in downwash from the main wing to be experienced at the tail.

Instantaneous changes in incidence angle cause lift to increase on both main plane and tail plane. The tail lift is then reduced as the increased downwash reaches it. The net effect is an increment in the steady state value of Mw, so it has the same sign as Mw. This derivative is denoted M w .

Unlike the rocket or shell examples, the longitudinal equation of motion is not decoupled from the motion, so there is an Mu term, caused by a small increase or reduction in dynamic pressure.

The pitch plane force per unit incidence Zw must be large and negative if the aircraft is to fly at all.

The longitudinal equations include Zu which is the change in lift due mainly to the change in dynamic pressure which accompanies a speed change. The Zq term arises because the tail acts a bit like a paddle, generating additional lift when the aircraft rotates in pitch.

A further consequence of including the x-wise force is the contribution of gravity. This is usually neglected from missile, shell or rocket stability, as it varies over a much longer timescale than the weathercock mode, characterising rocket stability.

The pitch plane equations of motion are:

mu=X uu+Xww−mg(θ+ wU )Xu is the change in drag force, mainly due to the change in dynamic pressure, Xw is the change in lift-dependent drag (induced drag). The angle θ is the perturbation in the pitch attitude of the body, so the incidence angle must be subtracted from it to yield the perturbation in climb or dive angle.

m (w−Uq )=Zuu+Zww+Zqq

B q=Muu+M w w+Mww+M qq

θ=q

The yaw/roll equations introduce the product of inertia E, and since aircraft usually have high aspect ratio wings (aspect ratio is: span squared divided by area) compared with rockets or missiles, the cross coupling between roll and yaw is much greater. The rolling moment due to yaw rate (L r) is

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significant. This is because the outer section of the port wing move faster than the outer section of the starboard wing with positive yaw rate (nose to starboard) causing a tendency for the starboard wing to drop. This is a positive roll, so Lr is expected to be positive.

Np, the yawing moment due to roll rate arises from increase in sideslip at the fin, which isn’t usually balanced by a ventral fin. The roll rate also increases the incidence of the starboard wing, but reduces it on the port. The difference in induced drag of the wings generates an additional yawing moment.

As with the pitch plane, gravity is present in the side force equation, proportional to the perturbation in body roll angle (φ). The lateral motion equations become:

m ( v+Ur )=Y v v+Y r r+mgϕ

A p−E r=Lv v+Lr r+Lp p

C r−E p=N v v+N r r+N p p

ϕ=p

Some manipulation is needed to transform the equations into standard state-space form.

These equations are studied extensively in specialist books on aircraft stability (e.g. Babister), and there is insufficient space here to even scratch the surface of such a complex subject.

Suffice it to say that understanding of the relative magnitudes of the various stability derivatives provides a basis for factorising the two fourth-order characteristic equations approximately, into two quadratic terms for the pitch motion and one quadratic and two simple factors in the case of the lateral motion.

The modes associated with these factors are well known to engineers and pilots and have been given names. In the pitch plane, the high frequency quadratic factor is akin to the weathercock mode of the fin-stabilised rocket, and is known as the short period pitch oscillation. The low frequency mode governs the trajectory of the aircraft in the absence of pilot input, and is called the phugoid. This is an oscillation in height and speed, with the two states in quadrature.

The lateral motion consists of three modes. The fastest mode tends to be roll subsidence which is simply a decaying roll rate. The lateral equivalent of the weathercock stability is called ‘Dutch roll’ which consists of coupled, damped oscillations in yaw and roll, such that the wingtips describe elliptical paths with respect to the centre of gravity. The low frequency mode is called the spiral mode, which like the phugoid, characterises the trajectory the aircraft would follow in the absence of pilot activity.

The effect of the stability derivatives on these modes and the relationship between the stability derivatives and the aircraft and flow field geometry, is used to design an aircraft which is naturally pleasant to fly, as opposed to the ‘white-knuckles special’ which would arise from treating them as purely abstract numbers.

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The process of approximate factorisation will be illustrated using a simpler example in the following sections.

Whereas the rocket, with its rapid response and high degree of symmetry is represented by identical second order equations in pitch and yaw, the classical aeroplane requires different quartic equations for longitudinal and lateral motion. The penalty for removing a set of lifting surfaces, is a more complex description of the dynamics.

12.6.3 The Brennan MonorailAt the turn of the Twentieth Century, Louis Brennan, the inventor of the world’s first guided weapon, turned his attention to railways. Never one to adopt the uninspired, pedestrian approaches of his contemporaries, when it came to solving the hunting oscillation which was limiting the speed of trains, he decided the best solution was to discard one of the rails, and run the vehicle on a single rail.

This approach presented the obvious problem that the vehicle would topple over, and this problem would indeed deter a lesser inventor. To overcome the problem, Brennan proposed generating stabilising moments by forcing a gyroscope to precess. As the following sections will show, this is not as crazy as it might first appear. Its bad press has more to do with the lack of integrity of experts in conventional wisdom, who were not prepared to be ousted by an upstart amateur like Brennan, than with any technical difficulties.

The gyroscope equations were derived when we considered the motion of a spinning shell. It is difficult to keep track of a particle on a rapidly spinning disc, but if we plot its path in space when the

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plane of the disc is also rotating (i.e. precessing) we see that the curvature of the path changes direction during a single spin. The resulting centripetal force appears as a moment around an axis perpendicular to both the spin axis and the precession axis. Summed over the whole disc, this becomes a continuous moment.

As with the works of M C Esher, it is our inability to perceive 3D geometry adequately that renders gyroscopic action counter-intuitive, and a source of anxiety for many who should know better.

12.6.4 Equation of MotionThe motion of the vehicle in roll, in the absence of a gyroscope is characterised by:

A ϕ=Whϕ

Where A is the moment of inertia of the vehicle about the rail, W its weight, h the height of the cg above the rail, and φ the roll angle. The characteristic equation is:

λ2−(WhA )=0

It should come as no surprise that this has one positive root.

Now we shall add a gimbal frame free to pivot about an axis perpendicular to the roll axis (it may be vertical or horizontal). We have:

J θ+H ϕ=M c

A ϕ−H θ=Whϕ

Where J is the moment of inertia of the gimbal/gyro assembly about the precession axis and H is the product of the gyro moment of inertia about its spin axis multiplied by the spin rate. This will be called the gyro angular momentum. Θ is the gimbal deflection angle and Mc is the controlling moment. With no feedback to the gimbal, the characteristic equation becomes;

(JA λ2+ (H 2−Wh )) λ2=0

This is an improvement as the modes will be neutrally stable, provided H2 > Wh.

In practice, we should expect friction in the gimbal, which we shall represent as viscous friction proportional to the gimbal angular velocity:

J θ+H ϕ=−Mω θ

Where Mω is the friction moment per unit angular velocity. The characteristic equation taking friction losses into account is not very encouraging:

JA λ4+Mω A λ3+ (H2−Wh) λ2−MωWhλ=0

The friction renders the system unstable.

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Before addressing the issue of stability, it is evident that the characteristic equation requires a constant term, this is introduced by feedback of the gimbal deflection. Without this term, there is nothing to stop the gimbal from gradually drifting on to the stops, even if the effect of friction were overcome.

J θ+H ϕ=−Mω θ+M θθ

Where Mθ is a gimbal moment due to gimbal deflection. The characteristic equation is:

JA λ4+Mω A λ3+(H2−Wh−M θ ) λ2−MωWhλ+M θWh=0

In order for the constant term to be positive, Mθ must be positive. The feedback must be positive. The mechanism implementing this must tend to push the gimbal further away from the equilibrium position, rather than pull it back. This can be implemented as a spring mechanism similar to the toggle mechanism of a bathroom light switch.

12.6.5 Approximate FactorisationIf power is lost to the gyro motor, it will start to slow down. In order to ensure there is sufficient time to bring the vehicle to a halt, and to sure it up, before instability sets in, the gyro angular momentum must be very large.

It follows, that for any practical system, the stability quartic may be split into two approximate quadratic factors:

λ2+Mω

Jλ+(H 2

JA−WhA−M θ

J )=0

λ2−Mω

HWhHλ+Mθ

HWhH

=0

The high frequency mode is called the ‘nutation’, and the low frequency mode the ‘precession’. The nutation is evidently stable, but the precession is dynamically unstable. The vehicle will tend to right itself but will oscillate from side to side with ever increasing amplitude.

The root cause of the problem is the fact that unlike a statically stable system, the equilibrium position is with the centre of gravity at its maximum height. In other words, the equilibrium position is a maximum energy, not minimum energy, state. Loss of energy due to friction will result in the gradual lowering of the cg height. It is impossible to rig up a collection of springs and dampers or other passive devices to recover this energy, so the balancing system must employ an active servo of some kind.

This differs from all previous examples in that artificial feedback is essential for stability, the system cannot be corrected by merely tweaking the values of parameters.

Positive feedback of gimbal position appears to impart static stability, implying positive feedback of gimbal rate might impart dynamic stability. At first sight, positive rate feedback would render the nutation unstable, making matters worse, because the nutation is a very rapid motion.

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12.6.6 Brennan’s SolutionHowever, we could exploit the fact that the two modes must be widely separated in frequency for robustness to loss of gyro power. If the servo used to provide the artificial feedback were too slow in its operation to influence the nutation, it could be used to provide positive rate feedback to the precession alone.

Consider a servo designed to generate an output torque in response to an input torque demand. It will tend to work by adjusting the output torque until it matches the demand. The simplest representation of a servo of this kind is a first order equation:

M s=1τ (M D−M s )

Where τ (tau) is the response time of the servo, usually called the time constant, Ms is the servo moment and MD the demanded moment.

The moment demand is a signal proportional to the gimbal rate. How this is actually implemented is irrelevant at this stage. Brennan used a pneumatic servo, August Scherl an hydraulic servo. An electric servo would be another possibility. The feedback might even be implemented manually.

The equations of motion now include the servo response:

A ϕ−H θ=Whϕ

J θ+H ϕ=−Mω θ+M θθ+M s

M s=1τ (k θ−M s )

Where k is the feedback gain.


(λ+1τ ) (AJ λ4+Mω λ3+(H 2−Wh−M θ ) λ2−MωWhλ+M θWh )− kτ λ ( A λ

2−Wh)=0

If we design the servo to have time constant:

1τ=√WhA

Then:

A λ2−Wh=A (λ−√WhA )(λ+√WhA )=A( λ−1τ )( λ+1τ )

The servo mode is present in both terms, so the characteristic equation governing nutation and precession reduces to a quartic:

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AJ λ4+Mω λ3+(H 2−Wh−M θ ) λ2+( kAτ2 −MωWh)λ+M θWh=0

Applying the approximate factorisation, it is evident that a slow servo will stabilise both the nutation and the precession.

12.6.7 CorneringThe gyroscopic moments arise from precession with respect to an inertial frame of reference, but the chassis of the monorail is an accelerating frame of reference on corners. The angular velocity of the vehicle introduces an additional gimbal angular velocity, in addition to that applied by the balancing system. With a single gyro, this additional angular velocity results in instability on corners. Brennan solved this problem by using counter-rotating gyros mounted in gimbals which precess in opposite directions. Any torques arising from the vehicle body motion would then have equal and opposite effects on the two gyros, and would cancel out.

12.6.8 Side LoadsThe stability analysis shows that the vehicle is stable. Side forces are resisted by the vehicle leaning into them until the rolling moment due to the side force is balanced by that due to the weight. It follows that the vehicle negotiates corners by leaning into them until no net side force is experienced on board.

12.7 Concluding CommentsHopefully, it should be clear that deriving the plant equations is seldom the trivial problem implied by the assumption that they are always ‘given’. It is important to analyse the dynamic behaviour of the basic plant to gain insights into the causes of instability, and the potential for selecting plant parameters which yield stable behaviour. The analysis yields stability bounds which help provide understanding of the critical design issues.

This analysis provides a sound foundation for the design of a closed loop controller to modify the plant behaviour, should it be deemed necessary.

The results take the form of analytical relationships between the plant parameters which define stability boundaries. These are infinitely more useful than numerical results which are only possible for specific values of plant parameters. Whilst the numerical results tell us exactly how stable a particular option might be, they provide very little assistance in deciding on a set of plant parameters which will result in stable, let alone optimum, behaviour.

The converse is also true, the Hurwitz matrix and approximate factorisation of the characteristic equation do not form a sensible basis for numerical approaches, they are strictly intended only for manual analysis. Also, deriving and solving the characteristic equation is a very error prone method of calculating the eigenvalues numerically. Most reliable methods employ matrix decomposition approaches such as the QR algorithm.

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13 Modifying the System Behaviour

13.1 IntroductionStability analysis forms part of the system characterisation which must precede any other activity.

It is undertaken to investigate whether the proposed system will be well-behaved, and provide the stability boundaries which usually define the upper limits of performance which may be achievable. More often than not, however, it is brought in later to explain why a badly-specified system exhibits unpredictable behaviour. Such behaviour is so often attributed to ‘emergent’ properties, but in the vast majority of cases it arises from inadequate analysis.

We need to identify the dynamic modes of the environment, rather than assume that there is a simple monotonic relationship between CO2 emissions and global warming. The rise of CO2 emissions in the 19th and 20th Centuries may well have excited an environmental mode which currently manifests itself as global warming. There is nothing to say that reducing them now will have the slightest effect. Depending on the dynamics of the atmosphere, it might even make matters worse.

We find the circumstances which result in unsatisfactory behaviour, and if we can’t solve the problem by avoiding the conditions which result in instability, or we can’t adjust the system parameters to achieve the desired behaviour, we must consider using artificial feedback.

One of the first applications of artificial feedback was probably to the Whitehead torpedo in the 19 th Century. As anybody who has ever tried to sail a toy boat will know, it is practically impossible to get it to follow a straight path without some form of remote control.

In order to solve this problem, the torpedo contained a gyroscope which served to detect the deviation of the torpedo heading from the aim direction. The gyroscope was sufficiently massive so that the controlling moments it applied to the tail fins did not upset it appreciably. The yawing motion causes displacement of the gyro in pitch, so that the pitch orientation of the gyro was expected to change but the yaw would be unaffected. Pitch motion of the torpedo was controlled by sensing depth.

Feedback was purely mechanical, with the gyro acting both as sensor and actuator, but this is irrelevant from a systems perspective.

The torpedo appeared to endow small boats with the capability to take on mighty battleships. However, the introduction of cordite in the 1890s, increasing the accurate range of heavy guns to well beyond that of torpedoes, soon dashed the hopes of the ‘Nouveau école’, who hoped to oust Britain’s naval supremacy with fleets of torpedo boats.

As is usually the case, military doctrine was dictated by the contemporary military technology. Admiral Jellicoe, at the Battle of Jutland didn’t employ Nelson’s tactics because they were incompatible with steam powered, armoured battleships having accurate gun ranges in excess of

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fifteen kilometres. The absurdity of trying to derive technological requirements from military doctrine is self-evident, although this is what actually underpins modern procurement decisions.

The torpedo only came into its own when combined with the stealth of the submarine or the speed of an aircraft. Again, these are technological developments, which drove military doctrine. They did not emerge as a result of military aspiration. Indeed, the military described them as ‘underhand and unfair’. The military attitude to novelty, even war-winning novelty, has always been obstructive.

Brennan’s monorail could not be made dynamically stable by passive elements alone, so a dashpot arrangement controlled the valves admitting compressed air into a pneumatic servo. A spring arrangement was then included in this sensor to provide positive feedback of both gimbal angle and gimbal rate.

The best known early application of artificial feed back was the automatic pilot for aircraft. We have seen that the long term modes of an aircraft consist of the phugoid and the spiral instability, which characterise the trajectory in the absence of the pilot. Applying the torpedo guidance philosophy, only this time using the gyro as a sensor, with separate servos, these modes could be suppressed and in still air at least, the aircraft could fly straight and level unattended.

The sensors which early technology could deliver tended to require very high standards of workmanship, and tended to be expensive. Consequently systems were designed to make use of as few sensors as possible. The latter part of the Twentieth Century saw a drastic reduction in the cost of sensors, making multi-sensor systems feasible at low cost. Servos, on the other hand tend to remain expensive, and unless there are strong overriding reasons, most individual controllers will be limited to a single servo.

We shall set aside the state-space description used in the basic plant stability analysis for the moment and consider a form of plant description which is more appropriate for a system composed of multiple components connected together in any possible configuration.

13.2 The Laplace TransformWe recall that the dynamics of a component are represented as the coefficients of a differential equation, the output from the component is the set of states for which the equations are solved and the input is a function of time. We have shown, that for the constant coefficient plant equation, the output is related to the input (f(t))via the impulse response:

y (t )≈∫0

∞

f (t 1 )h (t−t1 )d t1

So this integral must be evaluated for every time sample of the output function. This process is called convolution. This becomes the input to the next system component, whose output is then found from another convolution.

Evidently, this is not a very convenient way to derive the system behaviour.

Ideally, we should prefer a system representation in which the output is more simply related to the input and the component response.

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The impulse response is found by assuming the solution is of the form:

y= y0est

Although up until now we have used the symbol λ in the exponent, the use of the symbol s is for consistency with the notation convention associated with this solution method.

The non-homogenous linear differential equation becomes:

est (sn+cn−1 sn−1⋯+c1 s+c0 ) y0= f ( t )

For a linear equation, s is independent of time, so the solution can be found as:

∫0

t

y0dτ=∫0

t

y ( τ ) e−sτ dτ= 1(sn+cn−1 s

n−1⋯ )∫0

t

f (τ )e−sτ dτ

The upper time limit is arbitrary, so to avoid the inconvenience of terms in t, it will be taken as infinity. The resulting integral:

F ( s)=∫0

∞

f (τ )e−sτ dτ

Is called the Laplace Transform of f(t). The solution of the non-homogenous equation may therefore be written in terms of s, the Laplace operator, rather than time:

Y (s )=G (s )F (s )

Where G(s) is the reciprocal of the polynomial in s representing the homogenous equation of the plant in isolation. It is called the transfer function of the plant.

Now, if the output from plant G(s) is fed as input into plant H(s), there will be a new output Z(s):

Z ( s)=H ( s)Y (s )=H (s )G ( s) F ( s )

The transfer function of the combined plant is the product of the transfer functions of the components. This is precisely the formulation we require if we are to consider dynamic components connected together in a network. The components are defined in the transform domain, they are combined using the rules of elementary algebra, and if desired, the time series response to an input may be obtained from an inverse Laplace Transform.

13.3 Transforms of Common Functions

13.3.1 Derivatives and IntegralsWe justified the substitution:

dy (t)dt

→sY (s )

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on the basis of the characteristic equation. It may be shown to be a general result from the definition of the Laplace transform:

D (s )=∫0

∞ df (t )dt

e−st dt

Integrating by parts:

D (s )=[ f (t ) e−st ]0∞+s∫

0

∞

f ( t ) e−st dt

The term is square brackets is -f(0), which if the system is started from rest, is taken as zero.

It follows that:

D (s )=sF ( s )

It can be shown by repeated integration by parts, and the assumption that all higher derivatives are initially zero that:

Dn (s )=∫0

∞ dn fd tn

e−st dt=snF (s )

Quite often the D symbol is used in this sense to denote differentiation, allowing differential equations to be treated as algebraic equations. This is equivalent to using the Laplace operator, so is only mentioned because it is used elsewhere.

Equivalently:

I ( s )=∫0

∞

(∫0t

f (τ )dτ )e− st dt=−1s [(∫0

t

f (τ )dτ )e− st ]0∞

+ 1s∫0∞

f (t ) e−st dt

The integral in the first term is taken between the limits t=0 and t=∞, so is identically zero. It follows that, consistent with the use of s in place of a derivative, we may write 1/s in place of an integral.

I ( s )=1sF (s )

The Laplace operator obeys the rules of elementary algebra, with multiplication and division interpreted as the equivalents of differentiation and integration in the time domain.

13.3.2 Exponential FormsThe importance of exponential functions to the mathematical description of self-adaptive systems should be self-evident. The Laplace transform is:

F ( s )=∫0

∞

eat e−st dt=∫0

∞

e−( s−a) t dt= −1(s−a )

[e−( s−a) t ]0∞

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We can only ensure a finite result if s>a. Positive values of a (time constant) not only imply a divergent time solution, they bring into question the validity of the Laplace transform itself. Either way, an (s-a) factor in the transfer function denominator is likely to be cause of concern.

With this proviso, the Laplace transform of the exponential function becomes:

L (eat )= 1( s−a )

Second order factors are associated with sine and cosine functions. These are conveniently expressed as complex exponentials:

sinωt= ejωt−e− jωt

2 j

Where j is the square root of -1.

The Laplace transform becomes:

∫0

∞

sinωt e−st dt= 12 j∫0

∞

(e−( s− jω )t−e−( s+ jω) t )dt

Using the result for the real-valued exponential:

L (sinωt )= ω(s2+ω2 )

Similarly, for the cosine:

L (cosωt )= s( s2+ω2 )

13.3.3 PolynomialsThe input to the system may be any arbitrary time function, which could be fitted by trigonometric functions or alternatively, by polynomials.

L (tn )=∫0

∞

t ne− st dt=−1s

[ t n e−st ]0∞+ ns∫0∞

t n−1 e−st dt

This yields the reduction formula

: L (tn )=nsL (t n−1 )

If n=1: L ( t )=1s∫0∞

e−st dt= 1s2

Repeated application of the reduction formula yields:

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L (tn )= n!sn+1

13.4 Impluse Function and the Z TransformThe exponential, sine and cosine are probably the most important signals used to excite systems in order to discover their properties. The next most important is the impulse:

L (δ (0 ) )=∫0

∞

δ ( 0 ) e−st dt

The delta function is very handy for sleight of hand derivations, which leave the reader more brow-beaten than convinced. We shall consider a constant amplitude signal of duration Δt. The Laplace transform for a signal of amplitude a and duration Δt is:

∫0

∆t

ae−st dt=−1s

[e−st ]0∆t

Expanding the exponential function as a series, and assuming our error tolerances are such that a first order expansion is adequate:

∫0

∆t

ae−st dt=a∆ t

This depends on the actual value of the time interval.

How one deals with this difficulty depends on the application. In finding the impulse response of linear systems, we found that an impulse of size 1/(aΔt), is equivalent to releasing the system from rest with unit deflection.

The most important application of the impulse is in a sampled data system, such as the input to a digital processing system (not necessarily a computer; a set of shift registers or delay lines could be used, depending on application and ingenuity of the designer).

The input is sampled, with an interval between samples of T seconds. A sample of duration at least Δt is required to trigger the input circuits. So if the signal level is y(t), the impulse triggering the input is y(t)Δt. However, we are not initially interested in such low level detail as to how a particular hardware solution conducts the sampling.

At the functional level before any hardware solution is considered, we assume that the details of the sampling will be taken care of, such that it is legitimate to define the sampling as unit impulses:

∫0

∞

y (t ) δ (t1 )dt= y (t 1 )

Any dimensional inconsistency introduced by this perfect sampler is assumed compensated for in the physical system.

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We get away with apparent dimensional inconsistency because it is the information attached to processes that concern us. The physical quantities concern us only in so far as the saturations they introduce when limits are exceeded and minimum measurement intervals which are possible with affordable sensors.

Now a signal which is sampled at intervals of T seconds has the Laplace Transform:

∫0

∞

y ( t ) (δ (0 )+δ (T )+δ (2T )⋯ δ (nT ))e− stdt= y (0 )+ y (T ) e−sT+ y (2T ) e−2 sT⋯+ y (nT ) e−snT

The convention, when dealing with sampled data is to use the notation:

z=e− sT

So that:

Y ( z )= y ( t ) (1+z+z2⋯+zn )

Since z is itself a Laplace transform in disguise, it obeys all the rules of elementary algebra, and is called the Z transform.

Most modern systems are expected to implement the control processing digitally as algorithms, abdicating the credit for control system design, in the minds of managers, to software development processes and coders.

13.5 Laplace Transform Solution of Non-Homogenous EquationsA linear differential equation with input is no longer homogenous. We shall take an example which will prove important later on.

Consider a first order linear system driven by a sinusoidal input:

dydt+ay=b sinωt

Taking Laplace transforms:

( s+a )Y ( s )=b ω( s2+ω2 )

Y (s )=b ( 1(s+a )

ω( s2+ω2 ) )=b( As+B(s2+ω2)

+C

(s+a ) )This process of splitting the denominator in to simple factors whose inverse functions are known, is employed extensively in Laplace transform methods, it is known as partial fraction factorisation.

Multiplying both sides by (s+a)(s2+ω2) yields three equations in the unknown coefficients:

A+C=0

aA+B=0

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aB+ω2C=ω

Solving:

C= ω(a2+ω2 )

B= aω(a2+ω2 )

A= −ω(a2+ω2 )

Taking the inverse transforms:

y (t )=b( a sinωt−ωcosωt(a2+ω2 )+

ω(a2+ω2)

e−at)The second term is governed by the transient response, which for a stable system will decay to zero. In the longer term, the output will be:

y (t )=b( a sinωt−ωcosωt(a2+ω2 ) )

Let: sin ϕ= ω√a2+ω2 , then cos ϕ=

a√a2+ω2

y (t )=b sin (ωt−ϕ )

√a2+ω2

If the input is imagined as the angle of the shaft of an alternating current generator, the angle φ is the angle of the output relative to it. It is called the phase angle.

The phase is given by:

ϕ=−tan−1ωa

It characterises the delay caused by the system.

Also, we notice that the amplitude of the output signal is reduced by a factor; 1/√a2+ω2. This quantity is called the gain.

Applying the same approach to an nth order system:

(∑0n

ci si) y ( t )=b sinωt

Applying the Laplace transform, we have:

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Y (s )=b ( ω(s2+ω2 )

1

∑0

n

ci si )

The fraction decomposition yields:

Y (s )=b ( As+B(s2+ω2 )+∑0

n−1

Di si

∑0

n

c i si )

The second term is characterised by the transient, which will die out if the component is stable, leaving a sin(ωt) and a cos(ωt) term. The effect of a system of any order is to introduce a phase lag and a gain.

This will be considered in greater detail when we discuss frequency domain methods.

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14 Impulse Response Methods – Evan’s Root Locus

14.1 IntroductionHistorically, the design of control systems originated simultaneously in two different fields; that of aeronautics and that of electronics. The former was an extension of the stability analysis relating behaviour to aircraft configuration and flow field characteristics, whilst the latter was an extension of AC circuit theory. The equivalence of the methods used was not appreciated because of the differences in the physical hardware, a failing which is even more common nowadays, particularly amongst those who delude themselves into believing their approach to reflect ‘systems thinking’.

The aeronautical approach is largely one of assessing the behaviour in response to a sudden input, such as the pilot suddenly pulling back on the joystick. The transients were long enough to be observed directly, and pilots could describe their experiences, assuming they survived the test flight.

If we consider a system having transfer function from control input to the desired output given by G(s), we know from the Principle of Requisite Entropy that we bring the actual output close to the desired value by adjusting the input until the error between the two is zero.

In a linear system the steady state output is proportional to the input, so we fulfil the requisite entropy requirement by applying the error directly to the control input.

We make use of the fact that Laplace transforms obey the rules of elementary algebra, to derive the transfer function of the system when we close the loop. The closed loop transfer function is related to the open loop transfer function according to:

Gc (s )=k G0 (s )

1+k G0 ( s )

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Where k is a multiplier of the error signal, called the feedback gain. Note, that if the gain becomes very large, the transfer function tends to unity. In other words, the system becomes completely insensitive to the open loop plant behaviour.

Specification of components for the closed loop system cannot be based on open loop characteristics for this reason. In general, open loop systems require components to be manufactured to the tightest tolerances, requiring the highest standard of workmanship. Closing the loop obviates the requirement for extreme precision.

Practically all of control theory is concerned with finding the maximum feedback which is achievable in practice before the system goes berserk.

Usually, the feedback consists of more than a gain. An artificial dynamic system, called a compensator, is usually included to modify the plant dynamic response to permit high values of feedback gain.

14.2 Example: Missile AutopilotMy apologies to any pacifists who might be reading, but the missile autopilot is so much simpler than that of an aircraft, so it has been chosen for the sake of clarity, rather than indicative of war-mongering. The autopilot associated with the short period pitch mode of an aircraft is very similar.

The natural stability of a missile has already been considered. The equations of now need to be modified to take account of control inputs. The aeronautical convention for control deflection is ξ (ksi), η (eta) and ζ (zeta) for aileron, elevator and rudder deflections, respectively.

The control deflections generate forces and moments, characterised by control derivatives, by analogy with the stability derivatives. The control derivatives are the incremental changes in force or moment due to incremental changes in the corresponding control deflection, under the current flight condition.

The roll control derivative is Lξ, the rolling moment due to aileron deflection. Aileron deflection does not usually generate significant pitch or yaw forces or moments, but might increase drag slightly.

The pitch and yaw control moment derivatives are Mη and Nζ respectively. The convention used here is for positive control deflections to cause a positive moment about the associated axis. On a tail controlled missile the deflection is positive with the trailing edges deflected upwards, on a canard controlled missile (control surfaces ahead of centre of gravity), they are deflected downwards. In yaw the deflection is to the right for tail control, to the left for canard control.

This is not the usual definition, which is based on the direction of rotation of the control surface about the hinge line.

Since the control surfaces generate lift in order to apply moments to the body, the derivatives Zη and Yζ may also be significant. For a tail controlled missile Zη is positive, for canard control it is negative, with the signs reversed for Yζ.

Note: the trajectory modes of a missile; the equivalents to aircraft phugoid and spiral modes are dominated by the steering law used in the guidance, and are consequently irrelevant to the autopilot.

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The equations of motion in yaw (pitch is identical) are:

m v=Y v v−mUr+Y ζ ζ

C r=N v v+N r r+N ζ ζ

Taking the Laplace transforms:

((s−Y vm ) U

−N v

C (s−N r

C ))(v ( s)r ( s ))=(

Y ζ

mN ζ

C)ζ ( s )

Solving yields transfer functions relating sideslip to fin deflection and yaw rate to fin deflection:

vζ=

Y ζms−(Y ζm N r

C+N ζUC )

s2−(Y v

m +N r

C )s+( N vUC +

Y v

mN r

C )

rζ=

N ζ

Cs−(Y vm N ζ

C−Y ζ

mN v

C )s2−(Y v

m +N r

C )s+( N vUC +

Y v

mN r

C )When the transfer functions are written in terms of s, it is obvious that the states are functions of s and not time, so the explicit (s) has been omitted.

Considering the proper function of the autopilot (a concept conspicuous by its absence from most control texts), it is unlikely that the output from the system should be either sideslip or yaw rate. The guidance command takes the form of a steering command, requiring the missile to generate centripetal acceleration.

The lateral acceleration is, from the translational equation of motion:

f y ( t )= v+Ur

Taking the Laplace transform:

f y ( s )=sv ( s)+Ur (s )

After some manipulation, this yields:

f yζ=

Y ζ

ms2−

Y ζmN r

Cs−(Y vm N ζ

C−Y ζmN v

C )Us2−(Y vm +

N r

C )s+( N vUC +

Y vmN r

C )236


This is a more typical transfer function in that it consists of both a numerator polynomial and a denominator polynomial. The denominator is recognisable as the characteristic equation of the open loop plant. It is more easily derived using Laplace transforms, because the derivatives of the quantities involved do not introduce additional state variables, so the order of matrices employed is much lower.

The order of the numerator polynomial is the same as that of the denominator. This means that there is a direct link between the input control deflection and the output lateral acceleration. Physically, this is due to the direct lift of the control surface. Deflecting the control surface generates lift immediately; but it takes a finite amount of time for the body to rotate to an angle of attack. In a tail controlled missile, this lift is in the opposite sense to the wing/body lift, whilst for a canard control it acts in the same sense.

However, the canard angle of attack is greater than that of the body when trimmed to the manoeuvre angle of attack, whilst for tail control it is less, so the tail-controlled missile is expected to trim at a higher angle of attack before the control surfaces become ineffective. If the missile is statically unstable (i.e. Nv is negative) , this situation is reversed.

There is a potential benefit in making the missile statically unstable and canard controlled, but stability would have to be achieved by feedback control.

There are many other issues which must be addressed before it can be decided whether a canard design is appropriate in a particular application. The wider analysis requires full understanding of the dynamics of all options, if it is to be any more than hot air and power point.

The control deflection demand generated by the feedback cannot be realised immediately, we should expect some delay between the demand and achieved acceleration, so we can propose an exponential decay servo response:

ζζ D= 11+τs

Where the subscript D denotes a demanded quantity. The dynamics of the servo/missile combination is represented by simply multiplying the two transfer functions.

f yζ D= 11+τs

Y ζ

ms2−

Y ζmN r

Cs−(Y vm N ζ

C−Y ζmN v

C )Us2−(Y vm +

N r

C )s+( N vUC +

Y vmN r

C )

The dynamics of sensors, human beings in the loop, or any other dynamic process may be included in a similar manner.

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14.3 General Transfer Function

Figure 14-44: Open Loop Poles and Zeros

The missile example has demonstrated the point that transfer functions representing the dynamic behaviour of real world objects consist in general of a ratio of two polynomials in s with the numerator order no higher than that of the denominator. The denominator polynomial is the characteristic equation of the component in isolation.

Since it is generally found by concatenating dynamic processes, the open loop transfer function is usually factorised, so that the roots of the numerator and of the denominator may be found by inspection. The numerator roots are called the zeros, and the denominator roots are called the poles.

A general open loop transfer function will take the form:

Go (s )=Z ( s )P (s )

Closing the loop, we have:

Gc (s )=kGo (s )

1+k Go ( s )= kZ ( s )P (s )+kZ (s )

Evidently, when the feedback gain is small, the closed loop characteristic equation is nearly the same as the open loop characteristic equation, so the closed loop poles are close to the open loop poles.

As the gain tends to infinity, the closed loop poles tend to the open loop zeros, and as the number of poles usually exceeds the number of zeros, some of the closed loop poles tend to infinity as the gain increases.

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Now we could calculate the closed loop poles for a range of values of feedback gain using one of the analysis packages which are readily available, for a price. This would give us an accurate plot of the migration of poles, but without a certain degree of geometric intuition which is gained from plotting this root locus manually, it is impossible to decide how to design compensators to distort it into a more desirable shape. A fool with a tool remains a fool.

14.4 Rules for Constructing the Root LocusThe poles and zeros of the open loop system are known from the outset. These are plotted on an Argand diagram (Cartesian coordinates in which the real component is plotted in the x direction and the imaginary component in the y direction). These will either be real or appear as complex conjugate pairs (i.e. same value of real part with the imaginary parts having opposite signs).

Figure 14-45 : Root Locus Along The Real Axis

14.4.1 Real AxisStarting from the rightmost pole or zero join the poles and zeros in pairs along the real axis. The pairs may be both poles, both zeros or a pole and a zero, it doesn’t matter. Multiple poles or multiple zeros which coincide on the plot are treated as line segments of zero length for the purposes of deciding where the gaps in the real axis plot occur.

Coincident pole zero pairs are not plotted, unless they occur in the right hand half plane.

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Figure 14-46 : Asymptotes Directions and Intersections with Real Axis

14.4.2 AsymptotesAn asymptote is a straight line which the curve tends towards as s tends to infinity.

The number of asymptotes is equal to the difference between the number of open loop zeros and number of poles. The asymptotes correspond to the condition |s|→∞, for which the closed loop characteristic equation becomes:

sp− z+k=0

s is a complex number which may be written in polar form:

s=|s|(cosθ+ jsin θ )=|s|( e jθ+e− jθ

2+ j e

jθ−e− jθ

2 j )=|s|e jθThe direction, θ is called the argument. Evidently, the argument of –k is π radians (180⁰) . The directions of the asymptotes with respect to the positive real axis are therefore iφ, with i=1 to p-z, and φ given by:

ϕ= 180p−z

The intersection of the asymptotes with the real axis is given by the centre of gravity of the open loop poles and zeros taking the poles as positive and zeros as negative.

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Figure 14-47 : Angles of Departure from Poles and Arrival at Zeros

14.4.3 Exit and Arrival DirectionsThe factor (s+z) is the vector from the point s to the point –z on the Argand diagram. The argument of this number is the direction of the open loop zero at –z from the current pole. In order to satisfy the equation:

1+k Z ( s)P ( s)

The sum of the directions from the poles minus the sum of the directions from the zeros must be π radians (180⁰). The direction of the locus as it leaves a complex pole, or the angle of arrival at a complex zero, is:

θi=180−∑ θ p+∑ θ z

Where the subscript z indicates the line drawn from the node of interest to the zeros and p denotes the directions to the poles. The node of interest, whether pole or zero is not included in the sums on the right hand side. All directions are with respect to lines drawn parallel to the real axis.

14.4.4 Break Away PointsBetween pairs of zeros, and pairs of poles, the root locus branches out. At the branch out point two branches of the locus intersect the real axis, so they correspond to a condition where the closed loop characteristic equation has a repeated root. This corresponds to a turning point in Go(s), i.e:

dGo

ds=P dZds−Z dP

dsP2 =0

Or, P (s ) dZ (s )d s

−Z (s ) dP ( s)ds

=0

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This may seem a laborious exercise, but we know from the outset, that the solution must be real and lie between the two nodes, so start with the mid-point, and try a few values.

14.4.5 Value of GainIt is usually necessary to estimate the gain at positions of interest along the root locus, such as in the vicinity of the imaginary axis. If the point of interest is s1, then:

k=−P ( s1 )Z ( s1 )

Figure 14-48 : Rules Applied to Produce a Sketch of the Root Locus

14.4.6 Comment on RulesIn this age of expensive analysis packages, there is a temptation on the part of technically naive managers to imagine that high quality work is impossible without the best software money can buy.

Admittedly, it is easier to manage a software development than it is research, development or studies work, but we achieve little in pretending that all tasks are software developments, as appears to be the current wisdom. It is the quality of the analysts that determines the quality of the results, not the quality of the software employed.

No analyst worth his salt would accept the results from a model without conducting his own tests, relevant to the task in hand. Nor would he ever accept the claims of salesmen or other parties motivated by self-interest, at face value. How times have changed.

Many of the systems still in service were designed with little more than slide rules and graph paper, yet despite the advances in analysis software and revolution in sensor technology, the actual results produced are hardly commensurate.

Asking a modern analyst to demean himself with graph paper, ruler and protractor is like asking him to write with a quill pen. Unfortunately, we cannot acquire the geometric intuition which

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constitutes understanding of how the open loop nodes affect the closed loop behaviour, by handle cranking a code.

We can simulate rain showers in the computer, but they will never make us wet.

One should indeed check the manual sketch of the root locus with a mechanical calculation of the closed loop roots for a range of feedback gains. Indeed, once the design is completed, it probably is as well to reproduce it using the computer, as results from a machine, complete with spurious precision, carry more weight than equally correct manual plots on graph paper.

14.5 Autopilot RevisitedThe root locus is a numerical approach to system analysis and design, so the derivation of algebraic conditions defining constraints on the system parameters cannot be accommodated. In order to overcome this limitation, we try to minimise the reliance on magic numbers, and try to characterise the numerical quantities employed in terms which characterise the system. We note that the missile weathercock mode is characterised by an undamped natural frequency:

ω0=√|N vUC +

Y vmN r

C |We expect that nodes in the region of this frequency will have greatest effect on the corresponding mode. It would seem reasonable to scale time by this quantity so that experience gained with one missile autopilot design stands a chance of being transferable to other missiles.

We can usually derive a scaling term of this kind from the proper function of the controller which is being designed.

14.5.1 Numerical ExampleThe following values are have been derived for a typical missile configuration , although for obvious reasons, they do not represent any specific missile.

The mass is taken as 50kg, with longitudinal moment of inertia 15kg m2, and the speed is 680ms-1. The stability derivatives are:

Zw=Y v=−255.0Nm−1 s

Mw=−N v=−26.8Ns

M q=N r=−5.0 Nmrad−1 s

Zη=−Y ζ=1350Nrad−1

M η=N ζ=9500Nmrad−1

The negative Mw immediately tells us that the missile is statically stable. The positive Zη tells us it is tail-controlled. By reversing the signs of appropriate derivatives, we can investigate canard control and statically unstable missiles.

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The following analysis is artificial in the sense that it is assumed the acceleration of the centre of gravity can be measured. In practice, the centre of gravity moves during flight, and the centre of gravity is typically located within a rocket motor, which is not the easiest place to mount a sensor.

However, we shall ignore hardware related problems, until we have completed at least one iteration on the basis of perfect sensors. Sensor performance flows down from autopilot requirements.

In fact, we are ahead of ourselves in tackling auto pilot design as a first stage in the system design process. We would normally define the target set and the environment in which the targets would be encountered, and the platforms from which the weapon is to be fired. These would define requirements for the guidance loop from which missile agility, manoeuvrability and range requirements would be determined, together with launch condition constraints . Magazine handling or logistics might then dictate size and weight, but only if these constraints do not impinge on primary function.

Modern ‘systems thinking’ appears to drive system design from secondary or tertiary requirements, with no attempt at prioritising which issues need to be addressed with greatest urgency.

For the immediate purposes of illustrating root locus, we shall assume the missile design emerged from the wider system requirements.

The weathercock frequency for this missile is readily calculated as about 35 rad/sec (5.5 Hz).

The transfer function relating lateral acceleration to control deflection is:

f yη=−27 s2−9.0 s+2.16×106

s2+5.43 s+1225.0

Let σ= sω0

f yη=−27

(σ 2+0.33σ−65 )σ2+0.16σ+1

The zeros areboth real, and very nearly symmetrical about the imaginary axis at a distance of 8 characteristic frequency units.

The poles form a complex conjugate pair symmetrically placed above and below the real axis. The coordinates are about -0.1±j1.0.

As there are as many zeros as poles, there are no asymptotes. The break away point is at -0.1+j0.

As the gain increases, the poles move down towards the break away points. This reduces the autopilot natural frequency (slows down its response), but increases the damping.

Increasing the gain beyond the break away point causes one pole to move along the negative real axis to the negative zero. The other pole moves out towards the right half plane zero, introducing an unstable mode.

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Figure 14-49: Lateral Acceleration Response

As the airframe response probably required effort and ingenuity on the part of the aerodynamicists to achieve, it does not seem sensible to add complexity in order to make the missile more sluggish. However the damping should be closer to 0.7 for a satisfactory response.

In fact the situation is worse than this. The assumption has been made implicitly that the servo response is so rapid that its associated pole lies to the left of the left half plane zero. In practice, this is seldom, if ever, possible.

14.5.2 Initial Selection of Servo PoleThe placement of nodes far to the left in the Argand diagram ensures the root locus will only enter the right hand half plane for very high values of gain. There is nothing in the root locus analysis which prohibits this, so at first sight we should be able to add zeros and poles in the compensator to distort the root locus to achieve any response we desire.

In practice this is not the case. Although we must design the system to be stable, it isn’t stability which dictates where we can place the compensator or servo nodes.

Considering the servo; it is evident that speeding up its response has implications regarding power, weight, size and cost. The servo time constant must be chosen to be as long as we can get away with.

Also, the feedback signal is contaminated with noise, and generally the shorter the response time (higher the bandwidth), of the compensator, the more noise will be fed into the system. Introducing zeros, which is the usual strategy for improving stability, effectively differentiates the signal, which serves to amplify the noise.

These effects are taken into account in an initial iteration by applying a few rules of thumb:

In order to minimise the interaction between the servo mode and the airframe response, the servo bandwidth should be at least three times that of the weathercock mode.

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In order to avoid excessive noise, no compensator may have bandwidth greater than ten times the weathercock mode.

The effect of servo saturation and system noise will be considered presently. Once an initial design has been roughed out, these effects will be analysed explicitly, using methods described in later chapters.

The selection of a servo, or any other system element, independently of the remainder of the system, is a recipe for disaster. However, current ‘systems thinking’ appears to imagine that all the components can be specified on cost, maintainability and other grounds irrelevant to primary function, in the forlorn hope that they can be assembled into a working system. Dream on.

Figure 14-50 : Effect of Servo Pole

Placing the servo pole at s=-3+j0 introduces an asymptote on the negative real axis starting at the left half plane zero.

The branch on the real axis runs from the servo pole to the right half plane zero.

The two branches starting from the open loop weathercock mode poles must eventually reach a break away point beyond the left half plane zero. Before this happens,there is a significant excursion into the right hand half plane. When these two branches finally move back into the left hand half plane, we should expect the servo mode to be dangerously close to the right.

If the system without the servo was unsatisfactory, we should usually expect additional lags to make matters worse. Remember that the fundamental assumption of linear feedback is the output of the plant is a good estimate of the input, and vice versa. As we increase the feedback gain, the less tolerant the system becomes to errors arising because the open loop plant does not have time to reach equilibrium. Estimating the future output state by differentiation (adding zeros) delays the onset of instability, whilst integration (adding poles) aggravates the problem.

These principles apply for any null-seeking regulatory system, regardless of the details of its operation, and not just to linear feedback systems. Any self-adaptive process which can regulate

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itself in an environment whose disturbance set is open, must be limited by delay. There seems to be an odd perception that because the system is complex, it is somehow immune to these effects, why else are they totally ignored?

The immediate effects of the servo pole on the root locus may be found from the angle of departure from the weathercock poles. One zero is roughly at 180⁰, whilst the other is at 0⁰ the other pole of the complex pair is at -90⁰, whilst the servo pole is at 180⁰+tan-11/3. The angle of departure from the pole with the positive real part is

Θ=180-(-90+180+tan-1 1/3)+180=90-tan-1 1/3=72⁰

The servo pole repels the weathercock mode from the branch on the real axis, so that the system becomes unstable at very small values of feedback gain.

14.5.3 Adding CompensationSince the root locus starts at the open loop poles and finishes on the open loop zeros, the poles can be said to repel the locus, whilst the zeros tend to attract it. We can use this geometric intuition to decide where zeros must be placed to achieve the desired behaviour.

To begin with, we noticed that the servo mode messed things up rather badly, so placing a zero coincident with it will prevent it from interfering with the closed loop transfer function. We cannot construct a compensator having more zeros than poles, so we try a compensator of the form:

C (σ )= σ+3σ+10

This effectively shifts the servo pole well over to the left. The consequences of doing this in terms of noise and saturation will be considered in later iterations.

Placing an artificial zero over a plant pole is called pole/zero cancellation.

We notice that the right half plane zero attracts a branch of the root locus as the gain increases, and life would be simpler if it could be cancelled as well.

Unfortunately, introducing an unstable pole to cancel this zero will not work. Pole zero cancellation is more accurately called pole/zero concealment. The dynamics represented by the pole or zero which is cancelled still exists in the system, they just can’t be influenced by the control input or observed from the measurement. Introducing an unstable, cancelled pole introduces an unstable mode which we can neither see, nor influence, with catastrophic consequences.

The right half-plane zero puts an upper bound on the achievable closed loop bandwidth.

In order to pull the weathercock mode poles further over to the left, we must introduce a pair of complex conjugate zeros into the left half plane.

The obvious choice is to cancel the open loop poles and introduce new poles in a more favourable position. For example, consider the compensator:

C (σ )= σ2+0.16σ+1σ2+1.4 σ+1

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Figure 14-51 : Using a Series Compensator

This cancels the weathercock poles and effectively shifts them to a more favourable position on the Argand diagram, with damping ratio increased from the original 0.08 to a more satisfactory 0.7.

This appears to solve the airframe limitations without requiring the loop to be closed. The resulting controller involves pre-compensating, with no feedback at all.

This solution has in fact been used to avoid the expense of sensors, but the resulting system is no longer error seeking, so is inherently lacking in robustness to uncertainty in the plant and disturbances in the loop.

The pre-compensator can only deal with disturbances at the input. If we know that the disturbances, due to say, air turbulence, are likely to introduce less noise into the system than a sensor, it may not be worth closing the loop.

Without knowledge of the form of disturbances occurring within the loop, we must seek the highest loop gain we can get away with.

Zeros must be introduced further over to the left in order to attract the weathercock poles. These require two additional poles well over to the left. In order to avoid introducing excessive noise, the characteristic frequency associated with these compensator zeros cannot be much higher than 3.

If the compensator zeros are too far over to the left, the root locus originating at the weathercock poles will cease to be attracted to them, and the root locus will be attracted to the pair of zeros on the real axis. The right hand zero puts a limit on the position of the compensator zeros, and ultimately on the achievable closed loop bandwidth. In practice, it is not possible to design a satisfactory controller having closed loop bandwidth greater than half that of the right half plane zero.

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In this case, the right half plane zero is located well over to the right, so this particular upper bound will not apply. However, if the tail moment arm were reduced and, fin size increased to compensate, this zero would move closer to the origin, and would severely limit the response time which could be achieved.

Figure 14-52 : Introduction of Compensator Zeros

In the absence of the right half plane zero, we could make the gain arbitrarily large, because the only asymptotes lie in the negative real direction. In practice, all the sensor responses and the structural modes, which have been ignored, would influence the behaviour at very high values of gain.

The right half plane zero is a consequence of the reverse action of the fin. This is not experienced in canard control configurations, but these too have their own sense of humour.

14.5.4 Concluding CommentsThe example indicates that the knowledge needed to design satisfactory control systems originates from outside the control domain, and depends on a broader understanding of the factors which limit the behaviour.

Part of the problem arises because we have chosen an issue which cannot be taken in isolation from the remainder of the system. Our starting point is to specify what targets must be engaged, from what launch platforms and in what environment. The end user is over-stepping his competence in specifying details of weapon behaviour. It is up to the designers, who are the competent authorities, to flow down this information to actual system constraints.

We started at an inappropriate point, not because the methods are only applicable to ‘low level’ system issues, but because, in the missile context, we have not yet covered the material needed for the higher level analysis.

One further point is that accelerometer feedback by itself is almost never used. The impossibility of positioning the accelerometers at the centre of gravity is one reason, but in many cases the natural

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airframe weathercock response has adequate bandwidth, but is lightly damped. Feedback of angular velocity via a rate gyro is probably a more urgent requirement.

The yaw rate response is characterised by a single zero, and this lies in the left half plane, because the moment applied by the control surface does not change direction during the transient response, like the lateral acceleration.

The position of the zero is very nearly −Y vm

, and characterises the lag between the body rate and

the turn rate of the trajectory. Alternatively, it characterises the time taken for the angle of attack to reach its steady state value. For this reason, it is often referred to as the ‘incidence lag’.

14.6 Gyro MonorailAlthough more obscure, the monorail is a more self-contained problem than the autopilot. Instead of requirements flowing down from the need to hit a specified target, the need to stand upright is a pretty fundamental system requirement.

Unlike a statically stable missile, which will fly after a fashion with no active feedback, the monorail must employ feedback control to stand up. In order to apply root locus methods, we require some numerical values representative of the problem. These are derived in the following paragraphs.

14.6.1 Representative ParametersWe shall consider a 20 tonne vehicle, with centre of gravity 2.5m above the rail at equilibrium. The moment of inertia about the rail is therefore about 1.25×105kg m2.

Denoting the gyro angular momentum H, it must be large compared with the unstabilised toppling time constant:

H 2

JA100Wh

A

J is dominated by the gyro, and is equal to roughly half the polar moment of inertia about the spin axis (denoted I):

H=Iω

Where ω is the gyro spin rate in radians/sec. In order to avoid instability on corners, there must be two counter-rotating gyros, linked to precess in opposite directions. Let H1 be the angular momentum of a single gyro:

(2H1 )2

2J =2H 1

2

J =4 I2ω2

I =100Wh

Or: I ω2=25Wh

The polar moment of inertia is given by:

I=12mr 2

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Where m is the gyro mass and r is the radius. This yields an expression for the gyro mass:

m=50 Wh(ωr )2

Now ωr is the peripheral speed. Experience with gas turbine discs indicates that peripheral speeds of 400ms-1 do not present any significant azard, so we shall assume half this value. The gyro mass then works out at:

m=50× 20000×9.81×2.52002 =125 kg

Or 250kg for both. This does not seem prohibitive, as is often claimed.

Treating the gyro as a disc with thickness to diameter of 10:1, made of steel, with density 7870kg m -

3, the radius is 0.29m. The polar moment of inertia is 5.3kgm2, and the lateral moment of inertia is 2.7kgm2. The spin rate is 690 rad/sec (6600 rpm), resulting in a gyro angular momentum, for both gyros of 20700 kgm2s-1. The undamped natural frequency of the nutation mode is:

ωn=√ H 2

JA=7.4 rad/sec

The precession must be widely separated from the nutation if Brennan’s solution of applying positive gimbal rate feedback is to work.

The precession frequency should be about 1/10th of the nutation frequency; i.e: 0.75 rad/sec, to minimise the risk of interference between the modes. The gimbal deflection feedback is:

M θ=ω p2 H2

Wh

From which Mθ=491 Ns/radian.

The nutation mode will be damped so as to achieve a damping ratio of 0.7. This implies the damping moment should be:

Mω=−2 ζ nωn J

Where ζ in this context means damping ratio. This yields the result that Mω, regardless how it is implemented, must have a value -28 Ns/rad/sec.

The assumption is made that these two feedbacks are implemented mechanically. This is only because we are currently dealing with single input/single output methods.

14.6.2 Root LocusThe plant equations, with the gimbal deflection feedback and viscous damping become:

J θ+Mω θ−M θθ+H ϕ−M s=0

A ϕ−Whϕ−H θ=0

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τ M s+M s=MD

Where τ is the servo time constant and MD is the input signal to the servo.

From a systems perspective, there is something intrinsically weird about controlling the gimbal angular velocity in order to regulate the roll angle.

For the slower mode (the precession), we should expect the gyroscopic terms to dominate the roll equation of motion, which becomes:

−H θ Whϕ

So that, as far as the precession mode is concerned, the gimbal angular velocity is roughly proportional to the roll perturbation, so conceptually, the loop is regulating the roll angle.

The transfer function relating gimbal angular velocity to servo demand is:

θMD

= 1(τs+1 )

s ( As2−Wh )( AJ s4+Mω s

3+(H 2−Wh−M θ )s2−MωWhs+M θWh)

The zeros are located at ±√WhA , so that by using a servo time constant equal to this, we can cancel

the servo pole with the left half-plane zero. That is why the fifth order characteristic equation governing nutation and precession is reduced to a quartic.

There remains a right half plane zero, and a zero at the origin.

There are two more poles than zeros, so the asymptotes are perpendicular to the real axis.

The precession poles form a complex conjugate pair in the right hand half plane. If the precession frequency is low compared with the frequency of the right hand zero, the branches from these poles will be attracted to the branch on the real axis between the origin and the right half plane zero. Since all these nodes lie in the right hand half plane, the potential for a stable system does not look good.

The actual behaviour in the region of the nodes governing precession requires numerical values. The time constant on which the system parameters appear to be based is the toppling time constant of the unstabilised vehicle. Using this as a scale factor, the right half plane zero is located at 1+j0.

The nutation poles are given by:

σ 2+5.25σ+14=0

And the precession poles are:

σ 2−0.032σ+0.14=0

The servo pole is cancelled with the left half plane zero, so there is only a single open loop zero at s=+1.0.

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The nutation is stable with poles at -2.625±2.66, i.e. at 45°to the real axis. For the precession mode close to the origin these two directions are equal and opposite and cancel each other out.

Figure 14-53 : Brennan Monorail - Precession Mode, Root Locus Sketch

The precession poles are located at 0.016±j0.37. The angle of departure is given by:

θ=180+(90 )−( tan−1( 0.370.984 ))

This is -110⁰.

The break away point is given by:

P (s ) dZ (s )ds

−Z (s ) dP ( s)ds

=0

In this case, we ignore the contribution of the nutation, so the equation becomes:

(σ 2−0.032σ+0.14 )− (2σ−0.032 ) (σ−1 )=0

The break away point is either 2+j0 or -0.054+j0. The only root which lies on the real axis locus is -0.054+j0.

The same root locus has been calculated explicitly in the following diagram.

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Figure 14-54: Brennan Monorail - Precession Mode, Numerical Solution

The sketch is quicker to produce, compared with programming the calculation and plotting the result on a computer, and practice at drawing root loci yields intuitive understanding of the loop dynamics which could never be obtained from the distraction of writing code.

14.7 Comments on Root Locus ExamplesThe missile autopilot and gyro monorail were chosen deliberately to illustrate the point that practical systems can be very badly behaved. Unstable open loop behaviour, right half plane zeros and the requirement for positive feedback, are not just features contrived to illustrate arcane theoretical niceties.

Sufficient information has been presented to enable the interested reader to plot the root loci for unstable missiles and canard controlled missiles, and to consider explicit feedback of roll angle in the case of the monorail balancing system.

The remaining examples are better behaved, and consequently not so instructive, but differ from the preceding examples in that without feedback there would be no system function at all. They represent the common situation when the system has no natural feedback.

14.8 The Line of Sight TrackerIt is often necessary to be able to maintain a line of sight between a ground station and a moving object. A classical case is the tracking of a satellite so as to ensure a high-gain communications antenna is kept pointing at it. Similar techniques could be used to maintain the sight line of an astronomical telescope, (optical or radio) to a fixed point in the sky as the Earth rotates.

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All such systems have a sensor operating in the band concerned which has the capability of measuring any error between the target and the actual pointing direction. How this is achieved in a specific system is of no immediate concern. All that need be known is the sensor produces a signal proportional to the angular error.

If this angular error is fed into a motor of some kind which applies a torque to rotate the sensor in space. The equation of motion is:

I θ=T

Where T is the motor torque and I the moment of inertia of the sensor. Feedback consists of the error between the target direction and the current pointing direction (otherwise known as the bore sight error):

ϵ=θD−θ

The root locus starts at two poles at the origin which then migrate in opposite directions along the imaginary axis. Simple feedback results in an undamped oscillation which increase in frequency. In fact, the servo pole will incline the asymptotes at 60⁰ to the positive real axis, so increasing the gain actually causes the loop to exhibit a divergent oscillation.

Evidently, we need some zeros to pull the locus into the left hand half plane. A realisable compensator requires the number of zeros to be equal to, or less than, the number of poles, so at least one compensator pole must be present. This will be located well over to the left.

The servo poles would be expected to put a stability boundary on the system behaviour, but this is rather a ‘tail wag dog’ approach to system design. Servo behaviour ought to be an output from the design process not an input.

There are occasions where the design problem consists of making best use of existing components, such as servos, but even in these cases it is worthwhile finding the performance shortfall in introducing this artificial constraint. In many cases it turns out to be a false economy.

The sensor is presumably designed to track a target of known brightness at a specific range. In a military situation, the target will be doing all it can to minimise the signal which appears at the detector, failing that it will try to increase the noise, to render the angular measurement as inaccurate as possible for as long as possible. A radio telescope must see as far as it possibly can into space. For this reason, in most situations, when steering a sensor, we should expect the performance to be limited by noise in the loop.

Noise precludes the use of very high frequency poles, so will tend to reduce the practical system bandwidth. Indeed, additional high frequency poles may be introduced to prevent the system noise getting to the servos.

If targets can come close to the tracker, as can be the case with a missile tracker or gun director, there will be a further constraint imposed by the need for the servos to accommodate the maximum angular acceleration as the target flies past.

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Performance is expected to be limited by noise or maximum sightline acceleration, or both. Once these have been characterised it is possible to examine the limitations imposed by stability. Usually, stability imposes a more severe performance limitation than would be expected from a naive examination of the open loop component behaviours.

14.9 Tracking and the Final Value Theorem The sensor pointing loop differs from the other two cases in having no natural feedback, the loop behaviour is dictated entirely by the artificial feedback. The implicit, and mistaken, assumption in much finite state machine representations is that all systems are of this form.

It also differs, particularly from the monorail, in that the primary function is to follow a moving target. The input demand, rather than zero, is a function of time. Apart from stability, the quality of the system is determined by how accurately the output follows the input when the latter varies in time. It does not follow that because a system is a good regulator that it is a good tracker.

The quality of a tracker is assessed by the transfer function from the input demand to the error signal.

The transfer function is:

εy¿= 1

1+kG0 (s )=S ( s )

If the input is a polynomial in time of order n:

y¿ ( s)=n ! ysn+1

If the tracking error is to be maintained at a constant value:

ϵ (s )= εs

The steady state corresponds to s=0, so the steady state tracking error is:

ϵ= (sS (s ) y¿ (s ) )s→0

It follows that in order to track an input signal defined by a polynomial in time of order n with zero error:

( S (s )sn )

s=0

=0

The numerator of S(s) is the denominator of the open loop transfer function (i.e. P(s)). If the input is a constant signal, S(0) must equal zero for zero tracking error. This result is known as the final value theorem.

Consider the missile autopilot example. The open loop transfer function is:

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f yζ=

Y ζ

ms2−

Y ζmN r

Cs−(Y vm N ζ

C−Y ζmN v

C )Us2−(Y vm +

N r

C )s+( N vUC +

Y vmN r

C )It follows that the error transfer function is:

S (s )=(s2−(Y v

m+N r

C )s+( N vUC

+Y v

mN r

C ))(s2−(Y v

m+N r

C )s+( N vUC

+Y v

mN r

C ))+k (Y ζm s2−Y ζmN r

Cs−(Y v

mN ζ

C−Y ζmN v

C )U )From which:

S (0 )=((N vU

C+Y vmN r

C ))(( N vU

C+Y v

mN r

C ))+k (Y vm N ζ

C−Y ζ

mN v

C )UThis only tends to zero if the feedback gain tends to infinity, so the achieved lateral acceleration follows the demand with a finite error.

In order to overcome this problem the denominator of the open loop transfer function must contain an integrator. The feedback consists at least of a proportional term, plus an integral term:

k=k p+K I

s

Integral of error feedback is commonly used, not only to improve the tracking behaviour, but also to avoid the need for very high feedback gains.

The sensor pointing loop already contains two integrators, so should be expected to track a constantly accelerating sight line with constant error.

Tracking loops are characterised by the order of polynomial signal they can track with zero error. This is the ‘Type’ of the tracking loop. A loop of type n is one which can track a polynomial input of order n-1 with zero error.

14.10 The Butterworth Pole PatternWe have seen that for a second order system with no zeros (otherwise known as minimum phase), a reasonable pole pattern is given by the equation:

s2+1.4ω0 s+ω02=0

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The roots of this equation lie on the intersections of a circle of radius ω0, with two lines inclines at ±45⁰ to the negative real axis. This distribution is called a second order Butterworth pole pattern.

This would usually characterise a mode of the system. In general, we should study the dynamic behaviour to understand the system modes, and we should avoid coupling modes together by ensuring there is a separation in undamped natural frequencies.

However, if all else fails, and the problem is ‘given’ as one of pure control, with no broader knowledge of the plant, the poles may be placed in higher order Butterworth patterns given as solutions of the following equations:

s3+2ω0 s2+2ω0

2 s+ω03=0

s4+2.613ω0 s3+3.4ω0

2 s2+2.613ω03 s+ω0

4=0

The poles lie on a circle of radius ω0.

A Butterworth pole pattern is unlikely to be of much use to the Brennan monorail, which relies on the wide separation of the modes. Nor is there any benefit in coupling the short period pitch oscillation with the phugoid mode of an aircraft. In general, we would no more tune functionally different modes to the same frequency than we would tune all the strings of a guitar to the same frequency.

14.11 Status of Root Locus MethodsUnderstanding the root locus provides useful insights into how the stability of the plant is affected by the addition of compensation. It is one of many techniques which would be employed in order to gain understanding of the plant, and like all the other methods presented in this book, it is far from a panacea.

The prohibited regions on the Argand diagram are defined by considerations of, for example, noise and saturation, which lie outside the scope of an analysis of stability. The methods remain important because stability is the most important single attribute of the system.

Stability is strictly a system attribute which is not attributable to any single component, but to the behaviour of all components in concert. In that sense, it could be considered an ‘emergent’ property.

It is important also to realise that the poles and zeros of the plant can wander appreciablly as we consider the range of values the plant parameters can feasibly adopt. In principle it should be possible to ascribe probability distributions to the plant parameters and from them determine the probabilities of closed loop poles migrating into the right half plane.

Currently, such an explicit approach to characterising the system robustness has not been attempted. Instead, traditional rough rules of thumb have been extended into multi-input-multi-output system analysis with a degree of precision which is hardly consistent with the fundamental crudeness of the traditional robustness measures.

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Most plots describing control systems should probably be drawn with a marker pen or thick crayon to avoid any hint of spurious precision. Remember, if the plots were precise, we probably wouldn’t need feedback control in the first place.

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15 Representation of Compensators

15.1 IntroductionDespite the title this section does not go into the specifics of any particular implementation, but presents a more complete mathematical representation which can be used for analysis and modelling purposes.

In an era of widespread digital computing, it is unlikely that compensators will be implemented as continuous time (analogue) circuitry, although initial conceptual design may be undertaken on the assumption that signals are continuous. In essence, we start from an infinite sampling rate before we decide, after adequate analysis, what the sampling rate should be. As in all system analysis we seek a constraint which is dictated by the system proper function, even when processor speed has been specified beforehand. This is because we wish to know what the shortfall in potential performance may be as a consequence of adopting hardware which was decided on anything but system considerations.

Even if we do settle on a digital solution, the fact remains that reality is not a sampled data process, so in analysing control loops we must represent the real world as if it were sampled data, or the compensator as if it were continuous. In practice, we do both.

15.2 Modelling of Continuous SystemsAlthough the basic design work will be undertaken using manual analysis, sooner or later we are going to require a model of the aspects of the system which are of current interest. This may take the form of a time-stepping model, but as we shall see, the model which most obviously and directly represents the system is seldom appropriate for furnishing the insight needed to understand it. As the sage put it; to err is human, but to really screw it up, get a computer.

As managers appear to be unaware of any form of computer model beyond the high-fidelity reference standard model appropriate for acceptance, but useless for both analysis and design, it is advisable to have some idea as to how compensators might be represented using a high level language, such as C.

A continuous time system is typically modelled using some form of numerical integration algorithm. The Fourth Order Runge-Kutta algorithm is quite popular. As the name implies, it assumes that over the duration of the time step, the system time evolution is adequately described as a quartic polynomial in time.

It is of paramount importance that all state variables are updated simultaneously, and not piece-meal. Failure to do so introduces the risk of ‘sorting errors’ by which variables which have already been updated are used as the starting point to update other variables. The result is a spurious divergence in the solution.

The best defence against typical coding errors of this form is a sound knowledge of the expected system behaviour derived from analysis of the problem space.

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In order to ensure all states are updated simultaneously, the integration algorithm is usually designed to operate on a pair of arrays, the first containing the current values of the state variables, the second containing the derivatives. A typical implementation in the C language is presented below.

void RungeKutta4(double dt){//// Runge-Kutta 4th order integration algorithm// int i; for (i=0;i<NSTATES;i++) xn[i]=x[i]; updateAll(); for(i=0;i<NSTATES;i++) { a[i]=dt*dx[i]; x[i]=xn[i]+0.5*dt*dx[i]; } updateAll(); for(i=0;i<NSTATES;i++) { b[i]=dt*dx[i]; x[i]=xn[i]+0.5*b[i]; } updateAll(); for(i=0;i>NSTATES;i++) { c[i]=dt*dx[i]; x[i]=xn[i]+c[i]; } updateAll(); for(i=0;i<NSTATES;i++) x[i]=xn[i]+(a[i]+2.0*b[i]+2.0*c[i]+dt*dx[i])/6.0;}

The constant NSTATES is global and may be assigned using a #define compiler directive, it is the number of states which need to be integrated. The values of the state variables are stored in array x[], which in this case is a global variable. Likewise, xn[] contains the values of the states at the start of the time step. This is needed in case the time step is changed, for example in order to trap an event which happens at a specific time, perhaps within the time step interval. Arrays a,b and c hold intermediate estimates of the state, so are also of length NSTATES.

The time step is dt. The derivatives are held in array dx[].

It is imperative that the update takes place in the order indicated by the algorithm. All state variables associated with every dynamic process modelled must be updated simultaneously by calculating the derivatives in a function, which we have called ‘updateAll’. The algorithm completes four passes through this function, producing intermediate estimates of the updated state. The final

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update can be shown to be an extrapolation of the state, based on the assumption that the function is locally fitted as a fourth order polynomial of time.

Since it is reasonable to assume the integration algorithm will be of the form presented above, it is the responsibility of the modeller to write the differential equation representation in a compatible form.

The algorithm only accommodates first derivatives, so that higher order systems need to be written in state space form:

d2 ydt 2

=dvdt

Where:dydt=v

As we have seen, compensators are most frequently designed to introduce zeros in order to attract the root locus into the left half plane, so the commonest type of compensator has as many zeros as it has poles.

We need to represent dynamic systems of the form:

s+b(s+a )

This is implemented in the form:

dydt=a ( y¿− y )

yout=α y¿+βy

From which:

youty¿

=(αs+a (α+ β ) )

s+a

So that: α=1 , β=ba−1

The only other type of compensator is one with quadratic terms:

s2+cs+ds2+as+b

This would be represented as:

dvdt=−av+b ( y¿− y )

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dydt=v

yout=α y¿+βv+γy

Transforming the equations to the s–domain, and solving for α, β and γ:

α=1

β= c−ab

γ=db−1

Higher order compensators may be coded in a similar fashion, but usually we can factorise a rational polynomial into simple and quadratic factors.

The first order representations do not necessarily need to be implemented as code on a digital computer, although with current technology there seems little point considering an alternative.

In the past integration could be performed using operational amplifiers in an analogue computer, and various ingenious electro-mechanical devices were developed in the past to implement the dynamic behaviour of the compensator.

15.3 Sampled Data SystemsUsing current technology, it is most unlikely that compensators would be implemented as operational amplifiers, although such devices are readily available for a few pence. The vast majority of new systems will almost certainly exploit the flexibility afforded by programmable digital devices.

The principal characteristic of such processing of the signal is sampled in time. In what follows the sample rate is considered constant. Digital implementations quantise the signal in amplitude as well as time, but since we include analogue delay lines as possible implementations of sampled-data controllers, this quantisation need not be assumed.

It is tempting to claim that analogue signals are infinitely variable in amplitude, but in practice all signals are corrupted by noise. This sets a lower bound on the interval between analogue levels below which there is, say, an equal probability of the actual signal having the lower or the higher value. Also, all real systems are characterised by a maximum value of signal, where saturation may occur, or the system is driven into a non-linear region for which behaviour may become unpredictable.

The maximum value divided by the smallest detectable interval in the measurement is called the dynamic range of the measuring device.

Usually we select the digital least significant bit to correspond to this noise limitation, and the number of bits is chosen to match the dynamic range of the corresponding data source, so that as far as signal amplitude is concerned there is nothing to choose between the analogue and digital signals. The same quantity of information may be transmitted by either means.

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The sampled signal usually triggers some form of latch device to hold the value until the next sample. This is called a zero-order hold.

If the input signal changes during the sampling interval, the system can have no knowledge of the fact, so sampled data systems throw away information which could potentially be used in an analogue system. Again, it is worth noting that analogue systems usually have a high-frequency cut-off above which the signal fluctuates too rapidly for the system to respond. We would choose the sampling frequency such that the sampled data system cut off would be similar to that of the equivalent analogue system.

If two signals, one below a frequency given by fN=2/T and one above (T is here the sampling interval), we see that at the sample points the two signals have the same value, so they will be mistaken for the same signal. This effect is known as aliasing.

Any signal at frequency f will appear as a signal of f-fN to the sampled data system because of this effect. This is called fold-over. It is the same effect as aliasing, only expressed in the frequency, rather than the time, domain.

In order to avoid the possibility of aliasing, it is common practice to include an analogue filter in the analogue to digital conversion circuitry needed to condition the signal for use by the digital stages of the system.

15.4 Zero Order HoldThe output of the sampled data system elements is usually held at the sampled value for the duration of the sample period. In other words, the output circuits usually contain a latch. This introduces additional dynamic behaviour which is not present in the z transform of the continuous signal, and must be taken into account. The Laplace transform of a constant amplitude signal of duration T is:

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∫0

T

e−st dt=1−e−sT

s

This is the Laplace transform of the latch, otherwise known as a zero order hold (Z.O.H).

15.5 Sampled Data StabilityWe represent sampled data systems using the z-transform, which is a special case of a Laplace Transform. In particular, z-1 represents a delay of a single sample interval.

If we use the subscript n to denote the current sample, and n-i the input or output i samples before the present, we could code an algorithm of the form:

yn=−∑i=1

k

ai yn−i+∑i=0

m

bi xn−i

Where the ai and bi are constants.

The output is yn and the input for the current sample is xn. After the output is calculated, the algorithm is updated :

xm−i=xm−i+1 i=0¿m

yk−i= y k−i+1i=0¿k−1

This is called a recursive algorithm.

The algorithm’s functional behaviour is described in terms of the z-transform:

yx=∑i=0

m

bi z−i

∑i=1

k

ai z−i

The recursive algorithm translates into a transfer function in z, from which we can determine its dynamic behaviour, and its influence on the remainder of the system.

In fact, it is from its influence on the system that we determine the specification of the algorithm.

We recall that the stability criterion for the Laplace Transform representation was for all system poles to reside in the left hand half plane. The criterion for the z transform is derived from its definition:

z=e− sT

Where T is the sample period. The denominator of the z-form transfer function is a polynomial in z -1, which is equivalent to the characteristic equation, whose roots are:

z−1=esT

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For positive s, |z-1| is greater than unity, for negative s, it is less than unity. The criterion for stability in the sampled data case is for all roots of the characteristic equation to lie within the unit circle.

15.6 Illustrative ExampleTo illustrate the point that our discussion of z-transforms is not restricted to the rather limited scope of algorithms implemented on digital computers, we shall take an example from the ancient world. A computation cycle is not the only potential source of delay in the real world.

Consider a couple of Roman legionnaires firing their ballista at a moving ship, one measures the bearing, they load the ballista, train it on to the bearing, and miss. After a while, and a number of misses, it occurs to one of them that the change in the bearing between shots needs to be added to the current aim direction.

Denoting the current aim direction ψi, where the subscript i denotes the time (this is a sampled-data system with the aim/load/fire/time of flight delay between samples). This could be written as:

ψ i=ψ i−1+(ψ i−1−ψ i−2 ) Written in terms of aiming error:

ε i=(ψ i−ψ i−1)

The actual change in bearing is:

Δψ=UTR

Where U is the speed of the ship perpendicular to the line of sight, T is the time taken to prepare for the next shot, and R is the range.

If this is constant, the ballista will hit. However, operating a ballista is a tiring business, and our team is slowing down so that T is actually getting longer with each shot fired. The error in the trajectory is:

Δε=UΔTR

Where ΔT is the increase in cycle time per shot.

In order to keep the sample time fixed, we note that the slowing down of the team is identical in its effects to a speeding up of the ship between samples.

ΔU=UΔTT

The time delay of a single sample is denoted by the z-transform, z-1. In this form it is also called a backwards shift operator, for reasons which ought to be obvious.

We may represent this as a closed loop:

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RUT

Δε

1z

RUT

k

Δε

1z


In other words, the desired miss distance (zero) is differenced with the actual miss distance in the previous sample. The result is added to the aiming error due to the slowing of the reload cycle.

We have written the loop in terms of the essential variable (the error between the aim direction and the ship’s actual position).

We note that we can only observe the error when we see the splash of the ballista bolt in the water. The form of the feedback in automatic control usually takes this form, rather than implying two processes which mutually interact, we are concerned with the same process feeding back its states at a previous time. Nearly all of automatic control deals with the effect of time delay on the behaviour of the system.

Somewhat puzzled at still not hitting the target, and perhaps speculating about the presence of ‘emergent proprerties’, our legionnaires decide to increase their lead angle by a bit more or a bit less than the previous aiming error. We introduce a gain k into the feedback loop.

Introducing this gain implies the aiming error between consecutive shots is increased by a factor (1+k).

Expressed algebraically, the relationship between the desired aiming error and the achieved aiming error is:

Δε=(k ΔUTR ) (ΔεD−z−1Δε )

For convenience, set:

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K=k ( ΔUTR )The relationship between desired and actual miss distance is:

ΔεΔεD

= K1+Kz−1

We can find the behaviour of this system by calculating the response:

Δε i=−KΔεi−1

Over several time samples for a range of values of K.

Figure 15-55 : Roman Catapult - Gain < 1

For low values of gain, there is little improvement, the error remains at the level fixed by the discrepancy in time delay between shots. As the gain approaches unity, the shots initially land either side of the target, but eventually converge on a reduced aiming error. Increasing the gain appears to improve the situation.

At a gain of unity however, the situation is hardly improved, the aim oscillates ahead and behind the target indefinitely.

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Figure 15-56 : Roman Catapult - Gain = 1

Figure 15-57 : Roman Catapult - Gain > 1

Increasing the gain further results in a system which is unstable by any definition of the word. Successive bolts splash ahead and behind the ship at ever increasing distances.

By retaining the time-sampling (which in the above case was not constant) and describing the system states as sets of contiguous values we can illustrate the fundamental constraint that time delay imposes on an error-driven regulator of any description.

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With values of gain just below unity we notice an improvement in miss distance, such a tendency might encourage the class of wise fool who has an infantile obsession with superlatives to increase the gain without regard for the constraints, with catastrophic consequences.

Practically all systems contain some form of regulation which relies on feedback with delay, and performance is usually limited by divergent behaviour which even the simplest system described in this section exhibits.

15.7 Sampled Data DesignSufficient information has been presented to equip the reader to access the wisdom of any book on digital signal processing (dsp). Most dsp is concerned with fairly precise conditioning of signals, usually in an open-loop context. In comparison, the methods used in control are rather crude, because closing the loop covers a multitude of sins.

When designing controllers or regulators, we are not quite so worried about achieving precision in our signal shaping, but with deriving closed loop behaviour from the open loop system description. This is extremely difficult to achieve directly from a z transform description, so it is common practice to begin the design assuming continuous systems (Laplace Transform) methods, and transform the continuous system representation into a z-transform.

We could simply model the compensators using a numerical integration algorithm, but this is unnecessary and burdens the processor with too much work.

From the definition of the z-transform:

s=−1T

ln z

We could expand this as a Taylor series, but the logarithm expansion converges very slowly, implying a very high order polynomial in z would be needed to represent s adequately. The exponential function converges much more quickly. Consider the expansion:

1+z1−z

= 1+e−sT

1−e−sT

Expanding this to first order:

1+z1−z

=2−sTsT

From our considerations of aliasing, the sampling will be chosen such that s<2/T. So, for frequencies well below the fold over frequency, we can ignore the sT term in the numerator. This results in a bi-linear transform approximation to s:

s 2T (1−z1+z )

This is also known as Tustin’s approximation.

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15.8 Concluding CommentsWe have considered the principal issues relevant to the use of digital computers both to model continuous systems and to implement compensators as recursive algorithms. However, the ideas presented apply universally to any system containing pure delays, no matter how they arise.

The Tustin approximation is not recommended outside the domain of closed loop control which has inherent self-correcting characteristics. For z transform methods applied to digital signal processing, the reader is referred to any specialised book on the subject.

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16 Frequency Domain Description

16.1 IntroductionHistorically, interest in stability originated independently in aeronautics and in electronics. This is because these were the first areas of human endeavour which stretched the capabilities of the technology of the time to the limit. It is true that railway engineers were starting to puzzle over why trains swayed uncontrollably at high speed, but it was in these fields that it was first discovered that system performance is usually limited by stability.

As long as we are content with over-engineered mediocrity, our systems will probably be well behaved. It is in seeking superlatives that we find our naive optimisation is ill-conceived, because the spectre of instability haunts all systems. The insanity of seeking maximum performance from a system whose dynamics are not understood cannot be over-emphasised.

In aeronautics, the problem was how to make aircraft pleasant to fly, rather than white-knuckle death traps. Instability usually meant loss of the aircraft, and probably the pilot as well, so interest was dominated by transient behaviour.

In electronics, the problem was one of producing amplifiers of ever higher gain. The worst consequence of failure was an unpleasant howling in the headphones. Detection of the instability was usually long after the transient had died out. The preferred methods of analysis were a natural extension of alternating current circuit theory, which assumed a sustained sinusoidal excitation.

Of course, an experimenter who used sinusoidal excitation, varying the frequency over a wide band, for say, a ship roll stabilizer, would soon become very unpopular with the crew.

16.2 Sinusoidal ResponseWe saw from the Laplace transform solution for a system fed by a sinusoidal input, that, after the starting transient had died out, the output consists of a sinusoidal signal displaced in time by an amount characterised by the phase angle and attenuated or amplified in amplitude.

The long term output signal in response to a sinusoidal input is:

y=a sin (ωt+ϕ )

The corresponding time derivative must be:

dydt=aωcos (ωt+ϕ )

This can be simplified by noting that sin(ωt+φ) is the projection on the imaginary axis of e j(ωt+φ)

y=aℑ (e j (ωt+ϕ) )

Where Im denotes the imaginary part.

Differentiating:

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ddt

(e j (ωt+ϕ ) )= jωe j (ωt+ϕ )

The imaginary part of this is dydt . So, by treating the signal as a complex exponential, and taking the

imaginary part whenever we need to know the real time series signal, we can write:

dydt= jω y

We have already used a similar expression to derive transfer functions of system components:

L( dydt )=sY ( s )

These transfer functions may be re-written in terms of the response of the system to sinusoidal excitation, of frequency ω, by the substitution:

s→ jω

Using this substitution, the transfer function reduces to a complex number:

G (s )→ A (ω )+ jB (ω )

Where A is determined by the even order derivatives, and B by the odd order derivatives.

The inverse of the transfer function, relating output to input is:

G−1 ( jω )= A− jBA2+B2

The input is found by multiplying this by e jωt+φ, and taking the imaginary part.

The time series solution for the input becomes, with this substitution:

sinωt=a ( A sin (ωt+ϕ )−Bcos (ωt+ϕ ) )

A2+B2

Let: cos β= A√A2+B2

, sin β=¿ B√A2+B2

¿

Then: cos β sin (ωt+ϕ )−sin β cos (ωt+ϕ )=sin (ωt+ϕ−β )

The input signal is:

sinωt= a√A2+B2

sin (ωt+ϕ−β )

From which, it is evident that, the phase angle of the output relative to the input is:

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ϕ=β=tan−1 BA

And the gain (the output amplitude divided by the input amplitude) is:

a=√A2+B2

By using the s=jω substitution in the transfer function, we can calculate the effect the component has on a sinusoidal input signal as a gain and phase as a function of excitation frequency.

16.3 StabilityWhen we considered the stability of the z-transform, we actually found the region in the Argand diagram, which corresponds to the right half plane under the mapping:

z=e− sT

The astute reader would have noticed that s and z are both complex. In order to justify the unit circle criterion, we actually needed an expression for the logarithm of a complex number:

ln z=ln (x+ jy )=ln|z|(cos θ+ j sin θ )

Where: θ=tan−1 yx

We can verify this by adding two complex logarithms in the Argand diagram, and showing the result to be the logarithm of their product.

The stability of the open loop under the s→jω mapping is still found from the open loop poles. However, in order to design controllers we need to find the conditions which result in an unstable closed loop.

The closed loop characteristic equation is derived from the denominator of the closed loop transfer function:

Gc (s )=k G0 (s )

1+k G0 ( s )

Instability will evidently occur at any point where:

k G0 (s )=−1

If the frequency domain approach is to be any help to us at all, we need to find how this point relates to the contour produced by plotting G(jω). Ideally, some simple criterion like lying inside or outside it would be preferred, but unfortunately, this does not appear likely.

Consider the phase of a simple first order lag:

as+a

→ ajω+a

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At zero frequency (sometimes called DC for direct current, from its electrical origins), the gain is unity and the phase is zero. As the frequency (ω) tends to infinity, the phase tends to -90⁰. But for every simple factor in the denominator of the transfer function the phase will also tend to -90⁰. For a quadratic factor, the limiting phase angle tends to 180⁰. Any real system may have several simple and quadratic factors, so in principle the total phase lag could become much greater than 360⁰, and the contour could encircle the origin several times.

Since we cannot expect the G(jω) contour to be a simple curve such as the unit circle, or the imaginary axis, it is most unlikely that we will be able to find a simple ‘inside/outside’ criterion.

The approach adopted by Harry Nyquist back in 1932 was to find what the s-plane and the G(s) plane have in common.

We can enclose the right half plane by bounding it with the imaginary axis, which maps to G(jω), starting at the origin to a large value positive value, say +jΩ. A large area of the right half plane is then enclosed in a semi-circle of radius Ω, meeting the negative imaginary axis at –jΩ.

We expect the transfer function representing a real system to have more poles than zeros, so as |Ω| tends to infinity, G(jΩ) tends to zero. Enclosing the whole half plane maps to the locus G(jω) and the point G(s)=0. Traversing the contour around the whole right half plane s moves in a clockwise direction.

According to a theorem of complex analysis, due to Cauchy, a single clockwise encirclement of a contour in the s plane will result in N clockwise encirclements of the origin in the G(s) plane, where:

N=N Z−N P

Where NZ is the number of zeros of the transfer function, and NP the number of poles. If we apply this to the return difference:

D (s )=1+k GO ( s)

We notice that its zeros are the poles of the closed loop transfer function, and its poles are the poles of the open loop transfer function.

For closed loop stability, there must be no right hand closed loop poles. The number of open loop poles is known, so the encirclement criterion yields the number of unstable closed loop poles:

N P=N Z−N

The stability criterion becomes: The number of unstable closed loop poles is equal to the number of zeros of the return difference minus the number of clockwise encirclements of the origin by D(jω).

This is not particularly useful for design purposes, when we are trying to modify the open loop to achieve satisfactory closed loop behaviour.

We can apply the criterion to the open loop transfer function by changing the critical point from the origin to -1/k, and noting that the right half plane zeros of the return difference are the unstable open loop poles. The more useful form is: The number of unstable closed loop poles is equal to the

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number of unstable open loop poles minus the number of clockwise encirclements of the -1 point by the open loop transfer function (kGO(s)).

Stability requires the open loop harmonic plot of the open loop transfer function to encircle the -1 point as many times as there are unstable open loop poles. This is quite often quoted as a prohibition on encircling the -1 point, assuming a stable open loop. However, one reason for closing the loop could be to render an unstable system stable.

16.4 Stability MarginsImplicit in closed loop control is the idea that there is considerable uncertainty in the plant description, this is why we use feedback in the first place, and also why error-driven regulation is potentially universally applicable. It is impossible to survive within an environment which presents an open set of disturbances without the ability to learn from mistakes and adapt behaviour accordingly.

Perhaps we should insist that harmonic plots should be drawn with a thick crayon to emphasise the fact that spurious precision is neither sought nor even desired.

That our stability plot is wrong is simply to be expected from the nature of the problem, and not as so many Colonel Blimp clones seem to think, indicative of incompetence. Our system controls the system states to great precision whilst employing poor quality equipment to do so.

Where in nature do we see the precision of manufacture so characteristic of man-made artifacts? Yet creatures exist even on microscopic scales which can perform their functions to great precision. What is evolution if not error-driven regulation?

In these days of fast computers, we can in principle relate the coefficients of the transfer function to the uncertainty in the plant parameters. It is therefore possible to define a population (ensemble) of plants, each of known probability.

Generating random numbers and plotting harmonic loci are hardly numerically intensive tasks, so it should be possible to generate sample sizes in the millions. The encircling of the -1 point may then be derived as a probability, which is actually meaningful to everybody else dealing with safety, integrity or reliability.

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Figure 16-58: Gain and Phase Margins from the Nyquist Plot

Unfortunately, the world of automatic control continues to use rather arcane measures of robustness. Indeed, it is on what must be considered obsolete metrics of robustness of stability that the entire edifice of robust control has been built.

16.5 Gain and Phase MarginIt is evident that we must design our nominal controller such that the harmonic locus doesn’t get too close to the -1 point. How close is ’too close’? We need some ‘rule of thumb’ to give us our first iteration, bearing in mind the design process is itself an error driven regulator.

The effect of missing dynamics is most likely to introduce additional delays into the loop, just as the introduction of a servo lag caused a catastrophic loss of stability in the autopilot example. In addition, the missing dynamics may cause a change in gain at the frequency where the harmonic locus cuts the negative real axis.

The amount of gain needed to shift the crossing point on to the -1 point is called the gain margin. For cases where the harmonic locus crosses the negative real axis several times, this is usually based on the crossing nearest the -1 point.

The phase margin is found by drawing a unit circle centred on the origin and finding the intersection with the harmonic locus which is nearest the -1 point. The angle between a radius to this point and the negative real axis is called the phase margin.

Typical gain and phase margins used in practice consist of a factor of 2 on gain, and 30° on phase.

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16.6 Closed Loop Response from the Harmonic LocusThe purpose of the open loop harmonic locus is to predict the stability of the closed loop, and hence enable compensators to be designed so that the encirclement criterion is met.

We note that the closed loop transfer function is related to the open loop transfer function according to:

Gc (s )=kGO (s )

1+k GO ( s )

It is convention to denote the closed loop gain by the letter M (don’t ask me why), and the tangent of the closed loop phase angle as N.

Using this convention, any point (x+jy) on the open loop harmonic locus plot corresponds to a pair of values of M and N:

M e j tan−1 N= x+ jy

1+x+ jy

From this, it can be shown that the contours of constant M are a family of circles centred on (a+j0), where ‘a’ is given by:

a= −M 2

1−M 2

Having radius:

r=| M1−M 2|

Also, contours of constant N constitute another family of circles centred on (a+jb) where:

a=−0.5 , b=−0.5N

The radius is given by:

r=0.5√1+ 1N2

These are presented in Figure 16-59. The M circles are in green. These are equi-spaced on a logarithmic scale which we shall consider shortly. The vertical line though -0.5 is the unity closed loop gain case, circles to the right correspond to gains less than one, to the left they are greater than 1.

The N circles are in red. The real axis corresponds to zero degrees , and the cirecles are plotted in increments of 18° phase angle, with phase lags below the real axis and phase lead above.

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Re-arranging the closed loop transfer function expression to express the open loop transfer function in terms of the closed loop :

k GO (s )=GC ( s )

1−GC (s )

We might be tempted to try and specify a closed loop behaviour, and using this expression, deduce the open loop needed to achieve it. There are many reasons why, with single state feedback at least, this approach is doomed to failure.

Figure 16-59 : M and N Circles Defining Closed Loop Gain and Phase

Firstly, the compensator implied by this approach may not be possible; the order of the numerator may well exceed that of the denominator. The most serious problem is the nature of compensation usually required involves the introduction of zeros into the open loop. The closed loop zeros are the same as the open loop zeros, so in order to specify the closed loop, we must already know what compensation is needed, rendering the subsequent design exercise superfluous.

Some, allegedly more general, system specification methods contain the same ‘tail wag dog’ implication, although the practitioners seem blissfully ignorant of the fact.

16.7 The Monorail ExampleWe choose the Brennan monorail to illustrate the full Nyquist criterion of requiring anti-clockwise encirclements of the -1 point, since the open loop is unstable.

In more typical plant, the open loop is stable, and feedback is applied primarily to reduce sensitivity to plant uncertainties and disturbances, or to improve tracking accuracy. In such cases, we need to find the bounds on plant parameters caused by noise and saturation , before we can sensibly design

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compensators to maximise the bandwidth within these constraints. The methods needed to identify these bounds will be covered in later chapters.

Usually, experience with a particular class of system will enable the designer to place relevant, ’rule of thumb’, limitations on the possible compensator bandwidth, so that compensators designed on the basis of stability tend to furnish a good starting point which can be refined later using more advanced analysis.

On the other hand, it is obvious that the concerns of noise and saturation are mere niceties when the system is unstable and requires feedback just to stand up. For this reason, it is natural to begin the analysis of the monorail by addressing stability.

We recall from our root locus examination of the Brennan monorail, that stability requires positive feedback, whilst negative feedback is implicitly universally assumed in both root locus and Nyquist plots. The requirement shifts the critical point to +1 +j0.

It is little wonder therefore that control engineers, whose understanding is limited to a prohibition on encircling the -1 point, have difficulty with the Brennan monorail. It ‘theoretically’ can’t stand up, but in practice it can. If theory doesn’t match practice, you are using the wrong theory. However, the modern fashion is to throw the baby out with the bathwater and abandon theory altogether. It is better to find the reason for the discrepancy between theory and practice, than to proceed on the basis of blissful ignorance.

In order to avoid the complication of changing the critical point, we shall adopt an alternative control loop design, proposed by Piotr Schilovski, (the vehicle built by August Scherl used the same basic loop design as Brennan).

The Schilovski monorail used explicit feedback of roll angle. Since this is the state we actually want to regulate, this approach appears inherently more sensible.

The equations of motion are:

J θ+H ϕ=M θθ−Mω θ+M

(for the motion of the gimbal), and:

A ϕ−H θ=Whϕ

(for the rolling motion).

To recap: θ is the gimbal deflection and φ is the roll angle. J is the gimbal frame/gyro assembly moment of inertia about the precession axis. H is the gyro spin angular momentum. Mθ is the gimbal stiffness (this must tend to push the gimbal further away from the equilibrium position). Mω is a representation of the gimbal friction, and M is the gimbal controlling moment. A is the roll moment of inertia about the rail, W the vehicle weight and h the height of the centre of gravity above the rail.

Transforming to the s-domain:

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(J s2+Mω s−M θ Hs−Hs As2−Wh)(θϕ)=(10)M

From which:

ϕM= HsAJ s4+Mω A s

3+(H 2−WhJ−M θ A )s2−MωWhs+M θWh

Note: there is no left half plane zero to cancel the servo pole.

Figure 16-60 : Harmonic Locus for Schilovski Balancing System

The harmonic locus, with an arbitrary feedback gain of 100kNm per radian resulted in the above harmonic locus, which is plotted for positive values of frequency. There are two open loop poles, so the contour must encircle the -1 point twice. The plot for negative frequencies is the reflection of this contour in the real axis so that the contour from -∞ to +∞, does encircle the -1 point twice in the counter-clockwise direction.

Evidently, the chosen gain is far too high, as doubling the gain further will move the -1 point outside the contour towards the imaginary axis. Worse still, the phase margin is only about 15°.

This contour implicitly assumes an inifinitely fast servo, which is a poor approximation to reality. A fair comparison with the Brennan design would require a servo of similar response time, i.e, the characteristic time for the unstable toppling.

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Figure 16-61 : Schilovski Loop - Effect of Servo Lag

The effect of including the servo response is catastrophic, as the contour no longer encircles the -1 point.

We could try to recover the situation by cancelling the servo pole, and this option remains open until the effects of noise and saturation are taken into account. Usually, we would not dare use more phase advance than between three and ten times the servo frequency.

Doubling the gyro angular momentum has obvious effects on weight and volume, but the resulting balance loop is considerably more robust, without requiring any phase advance at all.

The nutation and precession frequencies must be kept separated as far as practical size gyros will permit. The common practice of placing modes close together in frequency, by using Butterworth pole patterns, rarely has much to recommend it. In this case a Butterworth pole pattern would prove catastrophic.

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Figure 16-62 : Stability Recovered by Increasing Gyro Angular Momentum

In this respect the monorail is far from unique. If we brought the frequencies of the trajectory modes (phugoid and spiral instability) of an aeroplane close to the handling modes (short period pitch oscillation, Dutch roll and roll subsidence), the resulting aircraft would be a death trap.

Before considering bringing in automation, there must exist a thorough understanding of the system dynamics. In particular, the relationships of the principal modes to the system functions must be appreciated.

Most of the time, we deliberately decouple functionally diverse modes by separating them in frequency. For this reason, if we find ourselves having to deal with systems of higher order than five or six, there is probably some fundamental aspect of the dynamics and/or system proper function we have overlooked.

16.8 Sampled Data SystemsPlotting harmonic loci for continuous systems is all very well, but we should expect modern systems to employ digital processing, so how are sampled data systems characterised in the frequency domain?

From the definition of the z transform:

z=e− sT→e jωT=cosωT− jsinωT

The z-transform is merely a Laplace transform in a different guise, so that all that applies to the Laplace transform applies directly to the z transform.

In order to illustrate this point, we need to consider how to represent harmonic signals in terms of the z operator. In general the z-transform of a function of time is found from the Laplace transform:

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y ( z )=∫0

∞

∑i=0

n

δ ( t−¿ ) e−st y ( t ) dt=∑i=0

n

z i y (iT )

We are actually interested in producing transfer functions in z whose output from an impulse response is the desired function of time. The input at the nth sample is zn y(nT), which we shall write x(z). Hence, we have for the transfer function:

G ( z )= y ( z )x (z )

=∑0

n

z−i y ( (n−i )T )

y (nT )

We can use this approach to produce compensators which have the same time domain impulse response as equivalent continuous time compensators, without resorting to the bilinear approximation. This method is known as impulse invariance, and is probably more relevant to digital signal processing than it is to control.

This explicit form requires summation over a sequence. For example, y(t)=1, a constant, yields:

y ( z )=1+z−1+ z−2⋯+z−n

Summing this series over n terms yields the more compact form:

G ( z )=1−z−n

1−z−1

16.9 Common z transformsIt is probably easier to start from a compact recursive form in z and find out which time series it represents by using a Taylor series expansion. Take the simplest case of calculating the current output from the current input minus the previous output multiplied by a constant gain:

G ( z )= 11+az−1

Expanding this as a Taylor series;

G ( z )=1−az−1+a2 z−2−a3 z−3⋯

Try: a=−e− βT

G ( z )=1+e−βT z−1+e−2 βT z−2⋯+e−nβT z−n⋯

Or: G ( z )=Z (eβt )

The z transform of the exponential function is, therefore:

Z (e βt )= 11−eβT z−1

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16.9.1 Sine and CosineThe functions of real interest are the sine and cosine. We shall derive these from their exponential forms:

sin θ= ejθ−e− jθ

2 j

cosθ= ejθ+e− jθ

2

From which:

Z (sinωt )= 12 j ( 1

1−e jωT z−1−1

1−e− jωT z−1 )= sinωT z−1

1−2cosωT z−1+z−2

Z (cosωt )=12 ( 1

1−e jωT z−1+1

1−e− jωT z−1 )= 1−cosωT z−1

(1−2cosωT z−1+z−2 )

A system described by transfer function G(z), excited by a sinusoidal signal is expected to have an output of the form:

G ( z )( sinωT z−1

1−2cosωT z−1+z−2 )=H ( z )+ A z−1+B(1−2cosωT z−1+ z−2 )

H(z) characterises the transient response, which is expected to die out rapidly.

The steady state time domain response is:

y (t )=α sinωt+β cosωt

Where β=B ,α=A cscωT−B cotωT

The gain is:

|G ( z )|=√α 2+ β2

The phase lag is:

tan ϕ=−βα

Consider a single delay transfer function, with a sinusoidal input:

11−a z−1

sinωT z−1

(1−2cosωT z−1+z−2 )= A z−1+B

(1−2cosωT z−1+z−2)+ C1−a z−1

This yields gain and phase:

|G (ωT )|= 1

√ (1+a2)−2acosωT

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tan ϕ= −a sinωT1−acosωT

The same result is obtained by substituting:

z=e− sT→e− jωT=cosωT− j sinωT

The resulting harmonic locus obeys the same encirclement rules as one produced by the s→jω substitution.

16.10 Frequency WarpingIn this age of computers and scientific calculators, the z→e-jωT transform does not present a problem when plotting harmonic loci. However, since the relevant Laws of Physics are described in continuous time, it is likely that the design will begin in the continuous frequency domain.

If the bilinear transform is then used to produce the compensator algorithm, we can apply a technique known as frequency warping to read the sampled data frequency response directly from the continuous system frequency response.

The bi-linear approximation is:

s≈ 2T ( z−1

z+1 )Applying the z→e-jωT to the right hand side, yields, after some manipulation:

2T ( z−1z+1 )→− 2

Tj tan(ωT2 )

The minus sign is irrelevant, as the frequency response is symmetrical about ω=0. The frequency on the original continuous signal harmonic locus, corresponding to the actual frequency ω is:

s→ jωc=2Tj tan(ωT2 )

This represents fold over explicitly, since at ω=2π(1/2T), ωc becomes infinite.

The importance of this approach lies in specification more than design, as it allows specifications defined in the continuous frequency domain to be mapped into the sampled data frequency domain. This process of shifting frequency domain specifications to account for sampling effects is called pre-warping.

16.11 DecibelsUsing the s→jω mapping, we find that the phase shift across a number of dynamic elements coupled together is the sum of the phase shifts. The gain is the product of the individual gains, and whilst it is easy to envisage a sum of quantities, a product is somewhat more difficult. To overcome this, and to accommodate systems which have a large dynamic range, it is common practice to express gain on a logarithmic scale. Using this approach, the gain of a sequence of elements becomes the sum of individual gains.

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This logarithmic unit is called the decibel. The base of the logarithm used is 10, rather than e, because in the past logarithms were used to simplify manual calculations, and tables of logarithms to base 10 were published and widely available. The base 10 is used as the number system universally employed is decimal. Despite its somewhat arcane roots, the decibel remains in widespread use.

When considering the energy conveyed by a propagation process, say sound or electromagnetic energy, it is common practice to compare the power of the actual signal with some ambient level. The commonest application is to sound. The human ear has a logarithmic response to sound level, so it is natural to take the logarithm of the ratio of the actual noise power to some ambient level, rather than the ratio itself as the measure of how loud a noise source is. The conversion from an actual ratio of power levels to decibels is:

PdB=10 log10PP0

Where P is the actual power, or power density, and P0 is the reference level. Using the decibel scale increasing the power from 6dB to 12dB makes it sound twice as loud, but corresponds to four times the actual noise power.

Wave propagation depends on the interchange of energy between two storage processes. In an electromagnetic wave they are the electric and the magnetic field, in a sound wave the elastic distortion and the inertia of the medium, in sea waves, the height and inertia. The energy stored is proportional to the square of the quantities defining the wave motion. So when comparing, for example, sound pressure levels, rather than power propagated, we must take into account the fact that the energy associated with individual quantities is proportional to their square. The appropriate decibel measurement for system states is therefore:

PdB=10 log10( v2

v02 )=20 log10

vv0

For this reason the numerical factor preceding the logarithm, when expressing the value of a gain (ratio of output state to input state) in dB is 20 and not 10.

Come to think of it, as it is merely a convention, the more usual ‘systems’ proof of ‘because I say so’, would have been adequate.

In summary, to express a gain |G(jω)| in decibels, use:

|G ( jω )|dB=20 log10|G ( jω )|

16.12 Signal Shaping in the Frequency Domain – Bode PlotsExpressing gains in decibels, we can plot frequency response explicitly as gain against frequency. Actually, if we plot against logarithm of frequency, the resulting responses become reasonable fits to straight lines.

The straight line approximation is illustrated by a first order lag transfer function:

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G (s )=ωB

s+ωB

Where ωB, the reciprocal of the time constant, is called the break frequency.

Transforming to the frequency domain yields an expression for gain:

|G ( jω )|dB=−10 log10(1+( ωωB )2

)At low values of frequency this approximates to log(1)=0, hence the red line on the frequency axis is a close approximation to the gain plot below ω=ωB.

At high values of frequency (ω>>ωB):

1+( ωωB )2

≈(1+ ωωB )2

Figure 16-63 : Bode Gain Plot - First Order Lag

This is a straight line on the gain plot of slope -20dB per decade (frequency increment of a factor of ten), or -6 dB per octave, intersecting the frequency axis at ω=ωB. This is the sloping red line of the gain plot. Apart from in the vicinity of ωB, the red straight line approximation is a good fit to the original gain plot. This part of the plot is called the cut-off.

When ω=ωB, the gain is:

|G ( j ωB )|dB=−10 log102=−3dB

For more general transfer functions, which have unity gain at low frequency, we define the bandwidth as the lowest frequency for which the gain falls below -3dB.

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It is evident that phase leads may also be represented y a pair of straight lines on the gain plot, with positive, rather than negative slopes.

The effect of cascading transfer functions is found by adding their gains at each point in the plot. For example, the gain plot where two first order cut-off regions overlap will have slope of -40dB per decade.

Similarly if the transfer function at high frequencies approaches 1/sn, the cut off will be -20n dB per decade.

The importance of a sharp high frequency cut off lies in the exclusion of the effects of noise and neglected high frequency dynamics from the desired operating frequency range.

The straight line approximation is far easier to envisage than the distortion of a Nyquist harmonic locus, and provides a straightforward approach to compensator design.

Since all polynomial transfer functions may be factored into simple and quadratic terms, we only need to show the straight line approximation works for a quadratic transfer function as well as for a simple factor.

16.12.1 Second Order Transfer Functions and ResonanceA typical second order lag transfer function is:

G (s )= 1

( sω0 )2

+2 ζ ( sω0 )+1This introduces the complication that it is characterised by two parameters; i.e. there is the damping ratio as well as the undamped natural frequency.

289


Figure 16-64: Bode Gain Plot - Second Order System

The quadratic factor gain plot exhibits the expected 40 dB per decade roll off above the undamped natural frequency, and appears to be a good fit to the 0dB curve at lower frequencies.

The response around ω=ω0 depends on the damping factor. The lower the damping factor the higher the gain. As far as compensator design is concerned, such gain peaks would be avoided by using a damping factor of about 0.7.

However, such peaks may well occur within the plant. Structural modes in particular have tiny amounts of damping and the gain peak is very high indeed, so that if the structure is excited at its undamped natural frequency very large amplitude oscillations will be observed. The classic example is of the opera singer hitting the right note to shatter a wine glass.

This effect is known as resonance, and does have application for example in radio, where it is used as a means of tuning to a particular frequency whilst excluding all others. In the control context it is usually a nuisance.

Resonance can occur in a rocket rolling at a rate approaching the weathercock frequency. From a body axes point of view, the plane of incidence is rotating around the body at the roll rate. In the pitch or yaw plane this appears as a sinusoidally varying incidence angle. As the roll rate approaches the weathercock frequency, resonance will occur, causing the rocket to reach high angles of attack, which may cause it to break up.

Figure 16-65 : Bode Phase Plot - Second Order System

The lower the damping, the narrower the resonance peak becomes, so that by plotting data obtained from measurements on the plant it is easy to miss the most dangerous peaks from the gain plot, as it could easily fall between samples.

In order to detect resonance, it is better to examine the plot of phase against frequency.

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For the second order lag, the phase φ is given by:

tan ϕ=−2ζ ω0ω

(ω02−ω2 )

Differentiating:

sec2ϕ dϕdω=−2ζ ω0 (ω0

2+ω2 )(ω0

2−ω2)2

Or:

( (ω02−ω2 )2+4 ζ 2ω0

2ω2

(ω02−ω2)2 ) dϕdω=¿−2ζ ω0 (ω0

2+ω2 )(ω0

2−ω2 )2

The slope (for a linear frequency scale) is:

dϕdω=

−2 ζ ω0 (ω02+ω2 )

(ω02−ω2 )2+4 ζ 2ω0

2ω2

At resonance (ω=ω0), the plot is at its steepest. It becomes steeper as the damping reduces. Using a logarithmic scale for frequency the effect of damping on slope is expected to be even more profound. A narrow resonance peak is therefore easily detected on the phase plot as a sudden change in phase between frequency samples.

This effect is illustrated by the phase plot of the example second order lag. The lower the damping, the steeper the curve near resonance

16.13 Gain and Phase Margins from Bode PlotsThe determination of gain and phase margins requires the gain an phase of the open loop to be plotted to a common frequency scale. The two plots are positioned one above the other, and the pair are referred to as a Bode plot.

291


Figure 16-66 : Stability Margins From Bode Plots

The phase margin is found from the frequency at which the gain becomes 0dB, called the gain cross-over point. The difference between the actual phase and 180° at this frequency is the phase margin.

Similarly, the value of gain at the frequency where the phase is 180°+360n°, where n is the number of encirclements of the origin, yields the gain margin.

16.14 Phase UnwrappingMost automatic plotting procedures tend to plot phase between 0 and 360°, skipping back to zero at each encirclement. Since we associate sudden changes in phase with resonance peaks, this spurious discontinuity in the phase plot can be misleading. It also looks awful.

In order to overcome this problem it is advisable to use phase unwrapping to get a continuous plot of phase against frequency.

Itoh’s algorithm furnishes a means of phase unwrapping which is both simple and robust.

Suppose we have a pair of adjacent phase samples, one less than 360°, and one greater:

ϕi=360−δ

ϕi+1=ϕ i+1

The difference between these is:

292


Δi=(ϕi+1+δ )−360

We notice, however, that the true phase difference between these consecutive samples is:

d i=ϕi+1+δ

Now:

tan Δi=tan d i−tan 2π1+tan d i tan 2π

=tan d i

So:

d i=tan−1 tan Δi

The arctan would need to use the atan2() function, covering the full 360° range, so that the algorithm would be coded using:

d i=atan2 (sin Δi ,cos Δi )

The procedure would calculate all the Δs between adjacent samples, then would use the arctangent to calculate the true differences. Starting from the first sample the true increments would be added in turn, creating an unwrapped phase sequence.

16.15 The Nichol’s ChartThe Nichol’s chart is another popular means of presenting the harmonic locus by plotting gain in dB against phase angle. The charts are usually printed with M and N contours, indicating the closed loop characteristics.

Gain margin is read from the intersection of the open loop harmonic locus and the 180° phase ordinate, and the gain margin is read from the 0dB gain axis. The intersection of the locus with the -3dB gain yields the bandwidth of the open loop transfer function.

16.16 A Word of WarningThe frequency domain approach implicitly assumes the transient behaviour is well behaved. Typically, however, we are seeking to improve stability by introducing phase advance to try and offset the effects of system lags. These can generate very large starting transients, which can displace the system away from its equilibrium state. It is common practice to disconnect the output from phase advance networks until the starting transient has died out. This implies that the loop may be briefly unstable, but the starting transient is expected to be more harmful than a brief period of instability.

It is tempting to remove the effects of a resonance peak by including a notch filter at the resonant frequency; i.e:

G (s )=s2+2ζ 1ω0 s+ω0

2

s2+2ζ ω0 s+ω02

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Where ζ1<<ζ. This tries to cancel a pair of lightly damped poles with a double phase advance. The transient response of this type of transfer function is presented in Figure 16-67.

Notch Filter Step Response (6.28sec Period)

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60 70

Time (sec)

Out

put

Figure 16-67 : Transient Response of a Notch Filter - It Rings at the very Frequency it is Intended to Suppress

The notch filter actually rings at precisely the frequency we are trying to suppress, so the transient behaviour will actually excite the unwanted behaviour. The filter would be by-passed until the transient dies out.

16.17 Concluding CommentBoth Root Locus and frequency domain methods afford insights into the causes of instability and suggest potential solutions. To claim one is superior to the other is rather like claiming a screwdriver is a ‘better’ tool than a hammer, the comparison is meaningless.

Much of the literature appears to concentrate on panacea methods which can be applied by handle-cranking a standard design package, with minimal understanding of the actual plant behaviour.

The approach adopted here is to emphasise the importance of understanding the dynamics of the system of interest, from which the appropriate choice of analysis and design methods becomes apparent. None of the techniques is a panacea, but each potentially offers a different insight into the system behaviour.

Hopefully, this chapter will help the reader to at least partially understand the gibberish which practitioners of control insist on using in preference to plain English.

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17 Time Varying Systems

17.1 IntroductionThe discussion so far has been restricted to systems describable by differential or difference equations having constant coefficients, or coefficients which vary gradually with operating regime. There also exists an important class of system which is required to be close to a specified state at a specific time; the state of the system at any other time is usually of little interest. Examples are aircraft auto land systems, spacecraft docking and missile guidance. These are known collectively as terminal controllers.

The problems are formulated in terms of the time of flight from the current state to the end state. It is usual to define the start time as the initial time of flight to the end state (denoted T), so that the problem is naturally defined in terms of time to go (T-t), where t is elapsed time.

Problems of this type are characterised by solutions of the form:

y= y0(T−tT )n

Where n must be positive and real if the state y is to reduce to zero at the end time. We are not concerned with stability as in the time invariant case. Also the entire flight constitutes a transient, so a steady state harmonic locus is also meaningless.

The higher order derivatives are given by:

dm yd tm

= y0(−1T )

m n !(n−m )! (T−tT )

(n−m )

With this substitution, the governing differential equation reduces to an algebraic equation in n, resembling the characteristic equation of time-invariant systems.

Spacecraft docking and auto land require both position and velocity relative to the target to be small, whilst the homing missile only needs to control terminal miss distance, so represents a simpler example to illustrate the methods used in the analysis of this type of system.

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17.2 Missile Homing

Figure 17-68 : Homing Missile KInematics

We assume the missile is aimed at a distant target, but because the aim is invariably incorrect, and the target moves away from the initial aim point during the engagement, the missile must correct its course in order to hit it.

The deviations of missile and target from the initial aim direction are considered small. It is also assumed that the missile speed is constant and the target velocity is constant (i.e. the missile heading γ may change, but the target maintains its flight direction).

A primitive homing law, which is no longer used, steers the missile towards the current target position. This is called pursuit guidance.

The range from missile to target is denoted R. The direction of the sightline with respect to the aim direction is φ. The look angle relative to the missile flight path is:

α=ϕ−γ

Where γ is the missile heading.

It is usually assumed that the missile body is approximately aligned with its velocity direction, so that the look angle can be measured with a body-mounted sensor. Sometimes the sensor is mounted in an aerodynamically stable forebody to ensure that it aligns with the velocity vector. Such structures are evident on the noses of guided bombs.

The engagement is determined entirely by the motion of the line of sight between the missile and target, so we need to derive the relationship between the sight line and the missile and target behaviours.

The sight line, referred to the aim direction, and assuming angles are small, is given by:

296


ϕ=yT− ymR

Or:

Rϕ= yT− ym

Differentiating:

R ϕ+Rϕ= yT− ym

And again:

( R ϕ+ R ϕ )+( R ϕ+R ϕ )= yT− ym

Now the constant speed assumption implies R=0, and the constant target velocity implies yT=0, so that the sight line direction is related to the missile lateral acceleration via:

2 R ϕ+R ϕ=− f z

Where fZ is the missile lateral acceleration (latax).

The pursuit homing law sets the lateral acceleration proportional to the look angle.

f z=k p (ϕ−γ )

Also, the lateral acceleration is equal to the centripetal acceleration:

f z=Um γ

The constant speed assumption implies:

R=−U c

Where UC is the closing speed.

The sight line equation becomes:

−2U c ϕ+U c (T−t ) ϕ=−k P (ϕ−γ )

In order to make the problem tractable, we shall vary the pursuit gain with time to go:

k p=kU c

2

R=k

U c

(T−t )

Evidently, if the look angle does not reduce to zero, the gain, and hence the lateral acceleration will increase without bounds. In reality, the controls will saturate.

Using this substitution:

kγ=(T−t )2 ϕ−2 (T−t ) ϕ+kϕ

297


Or: k γ= (T−t )2 ϕ−4 (T−t ) ϕ+( k+2 ) ϕ=−k U c

U m( (T−t ) ϕ−2 ϕ )

This equation is in terms of ϕ ,which is usually called the ‘sight line spin rate’, despite the fact that spin is usually associated with rotation about the specified line.

Using the substitution:

ωs= ϕ

The equation for sight line spin becomes:

(T−t )2ωs−(4−k U c

U m )(T−t ) ωs+(k (1−2U c

U m )+2)ωs=0

17.3 Significance of Sight Line SpinBefore proceeding with a solution to the pursuit equation it is worth taking a moment to consider why sight line spin rate should be important when considering the problem of hitting the target.

If the missile takes no action, the miss distance which will result from continuing on its current course is:

xmiss=( yT− ym )+(T−t ) ( yT− ym )

Substituting quantities from the sight line kinematic equations:

xmiss=Rϕ+(T−t ) ( R ϕ+R ϕ )=Rϕ− RR

( R ϕ+R ϕ )

Or: xmiss=U c (T−t )2 ϕ

The sight line spin rate is therefore proportional to the predicted miss distance. This miss distance, based on missile inaction, is frequently called the ‘zero effort’ miss distance.

We should therefore expect any self correcting homing loop to feed back an estimate of the miss distance, so we should expect the governing equation to have sight line spin as the dependent variable, as is the case with pursuit guidance.

Since the loop is closed we should expect it to be robust to the approximations implicit in the derivation of the governing equations.

17.4 Pursuit SolutionThe pursuit equation has been somewhat artificially written in a form which has solution of the form:

ωs=ωs0(T−tT )n

Differentiating:

298


ωs=−nωs0

T (T−tT )n−1

ωs=n (n−1 )ωs0

T 2 (T−tT )n−2

Substituting back in the pursuit equation yields an algebraic equation for n:

n (n−1 )+n(4−k U c

U m)+(k (1−2

U c

Um)+2)=0

For pursuit to be at all viable, this must have real positive roots.

Actually, it might be possible to choose n such that the miss distance tends to zero, but there is nothing in the equation to prevent the latax from saturating. For this reason, it is advisable to write the equation in terms of lateral acceleration.

f z=−2U cωs+U c (T−t ) ωs=− (n+2 )U cωs0(T−tT )n

In this case, if the sight line spin tends to zero, so does the lateral acceleration.

Solving the auxiliary equation for n:

n=12 ((k U c

U m−3)±√(3−k U c

Um )2

−4(k (1−2U c

Um )+2))Let k→∞; the roots of the equation become:

n→kU c

Um

And:

n→ 12 (k U c

Um−k

U c

U m (1−4 (1−2U c

U m )k ( U c

U m )2 )

12)

Expanding the square root to the first term of a Taylor expansion:

n→(Um

U c−2)

It follows that, in order to avoid saturation:

299


Um>2U c

From the diagram it is evident that a successful engagement is only possible from a rear aspect, very close to a tail chase.

In an actual tail chase:

U c=U m−UT

A successful engagement is then only possible if UT < Um <2UT.

It appears that pursuit is aptly named because the missile must chase a moving target. Also, it should not really be used against stationary targets, as it is in guided bombs.

The earliest missile seekers were not sensitive enough to detect anything cooler than the jet pipe of the target, so this restriction to rear hemisphere engagement was not such a severe handicap. In fact combat pilots had trained for years to get into the enemy’s rear hemisphere in order to engage with guns.

However, this limitation renders homing using pursuit guidance useless for surface to air weapons, as the target can only be engaged after it has performed its mischief.

Figure 17-69 : Pursuit Guidance - Only Effective in a Tail Chase

17.5 Seeking a More Satisfactory Navigation LawThe absence of forward hemisphere coverage renders pursuit guidance useless for a surface to air missile (SAM), where the objective is to keep the enemy away from the defended area.

In this section we derive a guidance law based on the actual requirements, expressed mathematically.

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In order to hit the target, the predicted miss distance should be reduced to zero within the time to closest approach. Expressed algebraically, we want the miss distance to reduce according to an expression of the form:

x=x0(T−tT )N

Where x0 is the initial predicted miss and N is a positive constant.

For error-driven regulation, the rate of change of miss distance should be proportional to the current miss distance.

The guidance law is derived by differentiating the above expression:

x=−x0TN (T−tT )

N−1=−N x

(T−t )

This takes the form of an error-driven regulator, as required by the Law of Requisite Entropy. It defines the proper function of the missile. In order to determine how to implement it, it must be expressed in terms of the quantities which can be measured and are available as controls.

Differentiating the expression for predicted miss distance :

x=( yT− yM ) (T−t )−( yT− yM )+( yT− yM )

By hypothesis, the target velocity is constant (target lateral acceleration is zero), so the rate of change of miss distance is directly proportional to the lateral acceleration:

x=−f y (T−t )=−Nx

(T−t )=−NU c

(T−t )2

(T−t )φ

Simplifying yields the guidance law which will achieve the desired behaviour:

f y=NU c φ

Thus by setting the lateral acceleration proportional to the product of the closing speed and sightline spin rate the missile will achieve the desired behaviour. In fact, as the guidance law is a closed loop, it will automatically correct for the errors arising from the approximations made in its derivation. This guidance law is by far the commonest in use and is called proportional navigation.

The equation of proper function implies that the number N, usually called the navigation constant, can have any positive value. It clearly cannot be negative because the result would then diverge with time to go. In practice, further constraints apply which restrict the values that can be used.

17.5.1 Lateral Acceleration HistoryConsider how lateral acceleration varies with time to go. It is proportional to sight line spin rate, so the homing guidance equations should be expressed in terms of sightline spin.

Differentiating the relative lateral velocity :

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yT− yM=R φ+2 R φ+R φ

By the constant speed hypothesis:

R=0 , yT=0

In order to simplify the notation, we will use:

ω=φ

Equation (12) becomes:

−f y=2 R ω+Rω

Substituting for closing speed and lateral acceleration:

−NU cω=−2U cω+U c (T−t ) ω

Which, re-arranged yields:

ω=− (N−2 )(T−t )

ω

This is directly soluble by separation of the variables:

∫ω0

ωs dωω

=−(N−2 )∫0

tdτ

(T−τ )

Where 0 is the initial value of sight line spin rate.

Integrating:

[ ln (ω ) ]ω0

ωs =(N−2 ) [ ln (T−τ ) ]0t

Hence:

ωs=ω0(T−tT )(N−2 )

This resembles the miss distance equation, except that the index on time to go has been reduced by 2.

This factor of 2 arises because, in the absence of missile lateral acceleration, the combination of the rotating sightline and velocity component along the sightline results in an apparent Coriolis acceleration of the target with respect to the missile. The actual lateral acceleration must exceed this Coriolis threshold before the missile can actually make progress towards the target.

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The navigation constant must therefore exceed the value 2, otherwise the sightline spin rate, and hence lateral acceleration, will increase without bound. In practice, the control will saturate some time before impact, and the miss distance will increase substantially.

17.6 Blind RangeIn practice we should not expect the sensor to work up until impact, there will usually be a blind range within which the sensor information is useless. For the constant speed case, this is expressed as a time to go. The sightline spin is given by:

ω=ω0(T−tT )(N−2 )

Hence guidance takes place until:

ωB=ω0 ( tBT )(N−2 )

The guidance command is assumed set to zero within the blind range so that from

T-t=tB, the sight line spin rate becomes:

ω=ωB( (T−t )tB )−2

Hence the sightline spin increases without bounds, but since the guidance loop is not closed, there is no corresponding control saturation.

The predicted miss distance is given by:

x=U c (T−t )2ω

Hence: ω0=

x0U cT

2, so that:

x=x0(T−tT )N

This reduces until:

xB=x0 ( tBT )N

For time to go less than the blind range:

x=xB

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Because N=0 within the blind range. It follows that zero latax against a constant speed target results in the miss distance at the blind range being maintained until the point of closest approach.

Introducing the fixed blind time into the basic equation yields a result which is more typical of general homing problems. Real missiles are characterised by fixed time constants whilst the sightline kinematics introduce functions of time to go. The miss distance performance then becomes a function of the time of flight.

More complex systems, which for example, include the effects of missile lags, rarely have straightforward analytical solutions like equation (34), from which the relationship between time of flight and miss distance can be seen immediately. The equations of motion must be integrated forwards in time for a number of flight times, and the terminal miss distances found for each case. This results in a set of miss distances as a function of time of flight (T).

Differentiating the blind range miss distance with respect to flight time (T):

dxBdT

=−Nx0 tB( tBT )N−1

=−NxBT

If the analytical solution were not available, we could find it by integrating this equation, but how do we derive this equation without knowing the analytical solution?

Equation bears a striking resemblance to the original equation for miss distance:

dxdt+ N

(T−t )x=0

The principal difference is the blind range equation generates the miss distances for a range of flight time cases, but the original equation models a single time of flight.

We shall see that this similarity is not mere coincidence.

17.7 SensitivityIn general, the miss distance may be found by integrating the equations governing the engagement with respect to time. One calculation yields a single value for one set of conditions, which is not likely to furnish much insight.

The object of study is to gain understanding, not to write computer code or merely generate data. What we want of our code is to be able to see how miss distance is influenced by all the disturbances and system parameters, so that we can develop the insight necessary to identify problems and propose solutions.

In fact, we only calculate miss distance as an intermediate result. If running a conventional model forwards in time, we can only apply disturbances one at a time and hence assess how each disturbance feeds through to miss distance.

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Blind Range at 5 Seconds to go

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

Time (sec)

Mis

s D

ista

nce

Impu

lse

Res

pons

e

T-t1=0T-t1=2T-t1=3

Figure 17-70 : : Blind Range Solution; All Responses are Just Scaled Copies of the Corresponding Section of the Lowest Plot

Consider the numerical solution of the single state system for predicted miss over a flight time T having a fixed blind range. Evidently, the solution at time t1 between 0 and T is the start condition for a time of flight (T-t1). If this is scaled to unit initial miss, the final miss is scaled by the same amount.

However, if the system were described by more than one state, the solution from (T-t1) to T will not be a scaled copy of the final section of the impulse response but a linear combination of the impulse responses for each state.

The miss distance will be, in general, a linear combination of the inputs at (T-t1), so a solution must be sought for each input. This set of forward time solutions would then yield a set of simultaneous equations from which the sensitivity of the miss distance to each input could be determined.

More interestingly, if we had a set of equations which produced the same solution as our original model, but in terms of total flight time, the result would be the initial state corresponding to a particular miss distance, for all times of flight.

In short, every time step of this ‘total’ time solution would yield the sensitivity of the final miss to the inputs at the start of the timestep, whilst as it stands, the intermediate time steps of the forward model would usually be discarded.

From the above discussion it is evident that the simplest approach requires a new equation set, which is solved with respect to (T-t1), where T is the time of flight ambit of interest, and t1 is the start

305


time of a particular run ending at time T. (T-t1) is the time of flight for this case. Also, the state variables which are sought are not the forward time states, because we are not interested in inputs such as sensor measurement error or target acceleration, but in the sensitivity of the final state to these inputs.

Let the sensitivities be y, these are related to the forward states x by:

x (t ) yT (t )=I

N.B. In this context the superscript T denotes a transpose, not the ambit of flight times.

In other words, the sensitivity of any state to itself at the same moment in time must be unity.

This holds true regardless of the value of t, so:

x (T−t1 ) yT (T−t1 )=I

Now the forward equation set is:

x=A (t ) x

We can calculate x(T-t1) by changing the independent variable:

dxd (T−t1 )

=−dxdt=−A (T−t 1) x

Differentiating the forward equation set with respect to T-t1:

yT dxd (T−t1 )

+( dyd (T−t1 ) )

Tx=0

Substituting the equation for x, yields the required equation:

xT (−AT (T−t1 ) y+ dyd (T−t1) )=0

The required equation set becomes:

dyd (T−t1 )

=AT (T−t1) y

The sensitivities (adjoint states) are calculated as a function of time of flight for the ambit of times of flight by solving this equation. In state space form, the system matrix is the transpose of the forward system matrix, but with functions of forward time expressed as functions of time to go.

If A is a scalar, as for the solution with the blind range, the adjoint equation is the same as the original forward equation. In general it is not.

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17.7.1 Block Diagram Adjoint

Figure 17-71: Forward System Block Diagram

Consider the forward system 2 state system characterised by:

( x1x2 )=( A11 ( t ) A12 (t )A21 ( t ) A22 (t ) )( x1x2 )

This would be represented in a block diagram like Figure 17-71.

The adjoint system is:

( y1'

y2' )=(A11 (T−t1 ) A21 (T−t 1)

A12 (T−t1 ) A22 (T−t ) )( y1y2)

Where the prime denotes differentiation with respect to (T-t1).

Figure 17-72: Adjoint System Block Diagram

The adjoint system, in block diagram form, is presented in Figure Figure 17-72.

The signal flow has been reversed compared with Figure Figure 17-71, and the summing junctions have been replaced by connections and vice versa.

307


These considerations reduce the construction of the adjoint system from the original system to a simple recipe:

Arrange the original system so that all inputs are unit impulses. This may involve introducing further integrators to convert step inputs to impulse inputs.

Write all functions of forward time as functions of adjoint (backwards) time. Reverse the signal flow such that the miss distance is a unit impulse input to the new

system. Replace summing junctions with connections and connections with summing junctions. Where the forward system differences signals, introduce gains of -1 to convert comparators

to summing junctions. Take outputs from the nodes in the block diagram which correspond to the variables whose

effect on miss distance is sought. The adjoint variable is the sensitivity of the miss distance to this input in the forward model.

The adjoint model may be run once to generate the sensitivity of the miss distance to all input disturbances for all flight times of interest in a single run.

17.8 Example: Homing Loop with First Order LagAny real missile will take a finite amount of time to respond to changes in the sight line spin. The sensor will have filters associated with it to discriminate targets from noise, which usually rely on the persistence of target signals compared with noise and hence require the scene to be observed over a finite length of time. Even if the signal processing were instantaneous, the mechanical steering of the seeker introduces a finite lag into the system. In addition, there will always be a finite lag between the generation of the guidance command by the seeker and the lateral acceleration being reached, as the airframe must rotate to the desired angle of attack to achieve the desired acceleration.

In the following discussion, we shall consider the missile response dominated by a single first order lag. In effect we assume the sight line estimation involves noisy measurements which require a long time constant to achieve an acceptable signal to noise, so that the remaining lags, which are expected to be of higher order, have an insignificant effect on the response.

The guidance command is:

f y=NU cωe

It is assumed the closing speed Uc is known. ωe is the estimate of sight line spin rate found from the guidance filter. It lags behind the true sightline spin rate:

ωe=1τ (ωs−ωe )

Where τ is the guidance filter time constant.

The sightline kinematics are given by equation (12):

f T− f y=U c (T−t ) ωs−2U cωs

308


Equations (43), (44) and (45) define the system behaviour. In this form it is not particularly useful for adjoint simulation as the miss distance is given in terms of sight line spin:

x=U c (T−t )2ωs

Now x in the above equation is a predicted miss distance, but in the adjoint model, this is only assigned a value when (T-t)=0, where it is simply the difference in lateral position between the missile and the target.

A more suitable set of equations replaces sightline spin equations by an expression for the difference in lateral position, y:

y=f T−f y

A principal source of noise is in the measurement of the sightline direction, as real sensors have a finite angular resolution, so that it is necessary to calculate the sight line direction ψ and differentiate it to find ωs.

ψm=y

Uc (T−t )+nψ

Where the subscript m indicates a measured quantity, and nψ is the measurement noise.

The relationship between the sight line direction with respect to the aim direction, and the estimated sightline spin rate is, from differentiating sight line direction and including the filter lag:

ωeψm

= s1+τs

The equations are are now in a suitable form for constructing the adjoint model.

The block diagram of the forward model is presented in Figure 17-73.

Figure 17-73 : Forward System Block Diagram

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Before considering the adjoint model, consider what can be done by running the forward model. There are three states, so we consider the solutions obtained from applying impulses into each of the states at the start time.

Impulse Applied at Filter State

-600

-400

-200

0

200

400

600

800

1000

1200

1400

0 1 2 3 4 5 6 7 8 9 10

Time (sec)

Am

plitu

de slSpinVelocityPosition

Figure 17-74 : Forward System, Impulse Applied at Sight line Spin State at Time=0

Impulse at Lateral Velocity

-30

-20

-10

0

10

20

30

40

50

0 1 2 3 4 5 6 7 8 9 10

Time (sec)

Ampl

itude slSpin

VelocityPosition

Figure 17-75: Impulse Applied at Relative Velocity at Time=0

Impulse at Relative Position

-30

-20

-10

0

10

20

30

40

50

0 1 2 3 4 5 6 7 8 9 10

Time (sec)

Ampl

itude slSpin

VelocityPosition

310


Figure 17-76: Impulse Applied at Relative Position at Time=0

The three state variables are denoted x1, x2 and x3 respectively, with x3 as the miss distance. We are seeking a function y such that:

yT (T−t ) x (t )=x3 (T )

There are three forward time solutions for x(t) found by applying an impulse at each state variable in turn at time t=0. The solutions will be denoted ha, hb and hc (corresponding to the above three numerical solutions) respectively.

This yields a set of three simultaneous equations to be solved for y:

yT (T−t ) [ha (t ) hb (t ) hc (t ) ] x (0 )=[h3 a (T ) h3b (T ) h3c (T ) ] x (0 )

The matrix:

Φ (0 , t )=[ha (t ) hb ( t ) hc (t ) ]

Whose columns are the responses at time t to a unit impulse applied to each state variable at time=0, is usually called the transition matrix. The right hand side of the sensitivity equation (the equation for y) is the third row of:

Φ (0 , T ) x (0 )

Also:

x (T )=Φ (0 ,T ) x (0 )x ( t1 )=Φ (0 , t1 ) x (0 )

Hence: x ( t1 )=Φ (0 , t1 )Φ−1 (0 , T ) x (T )

Solving, by taking the inverse of the matrix of coefficients:

yT (T−t1)=Φ3 (0 ,T )Φ−1 (0 ,t1)

Now, by definition:

yT (0 )= (0 0 1 ) ;

so the above equation may be written:

yT (T−t )= yT (0 )Φ (0 , T )Φ−1 (0 , t1 )

Or: yT (T−t1) x ( t1 )= yT (0 ) x (0 )

y(0) is a constant, and so is the input to the forward system x(0).

311


In summary, solving the adjoint set of equations is exactly the same as obtaining the required sensitivities from the forward time transition matrices by solving three simultaneous equations for each time step.

The extra overhead of solving simultaneous equations of order equal to the system is redundant, as the same answer is produced by solving the adjoint set directly.

The adjoint loop is:

Figure 17-77: Adjoint Model of PN Homing Loop with First Order Lag

The individual blocks have retained their positions and names, so that the correspondence between the adjoint of Figure 17-77 and the original system ( Figure 17-73) is reasonably apparent.

At the top of the diagram, the original calculation of time to go has been replaced by adjoint time, which is time of flight. Since the reciprocal of this is what is actually used, a small increment (delta) is added to the adjoint time to avoid numerical overflow problems.

All the blocks, and the associated signals, have been reversed in direction.

The miss distance input feeds in via summing junction into what would be a connection to the model to observe miss distance in forward time.

Similarly, the summing junction in Figure 17-73 calculating the difference between target acceleration and missile acceleration has become a connection, with the negative sign absorbed into the gain Uc.

The major difference between the two loops is the introduction of an integrator before the output. This is necessary because the original input is a step function, whilst the adjoint model requires impulse inputs. The integrator in the forward model would serve to convert an impulse into a step, but it is more convenient to code a step input than it is an impulse, so this integrator is not present in the forward model.

312


Alternatively, the state-space representation of the system, with target included is 4 th order and not third order, and Figure 17-77 represents the explicit representation of the fourth order state-space adjoint equation.

Finally, the constant coefficient transfer function representing the guidance filter is self-adjoint, so is simply reversed in direction with no further modification.

Adjoint Run - Miss Distance Sensitivity

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Flight Time (sec)

Mis

s di

stan

ce p

er U

nit T

arge

t Acc

eler

atio

n (s

ec^-

2)

Figure 17-78: Adjoint Run for Miss Distance

The result obtained from the adjoint model is presented in Figure 17-78. The time step chosen was 1mS, and run times up to 2.0 seconds have been considered, so the single run is equivalent to about 1000 forward simulations.

However, we can do better than this.

It is unlikely that consideration will be limited to the effect of a single input, there are many more effects of interest. Two more effects are included in this simple model, these are included in Figure 17-79.

The first consideration is probably more fundamental than target manoeuvre; it is the ability of the loop to remove the initial aiming error, this is the initial component of velocity perpendicular to the aim direction. Note that the aim direction is defined as a line passing through the initial position of the missile, so there is no position contribution to the aiming error.

313


Figure 17-79 : Forward Model with Measurement Error and Initial Aiming Error

The second additional input is the error in measuring the direction of the target relative to the missile. Some error is expected regardless of the nature of the sensor, and understanding the effect of sensor error on miss distance has obvious relevance to the specification of the sensor. High accuracy is usually associated with high price, so it is of great importance to find a sensor which achieves the desired miss distance performance, without breaking the bank.

Unlike the forward system, which must be run for each disturbance and each time of flight, these disturbances may be introduced into the adjoint model, and their effects found in a single run.

Figure 17-80 : Adjoint with Additional Error Sources

The adjoint system is as presented in Figure 17-80. The aiming error is a step function in the forward model; an impulse would have been fed in on the other side of the integrator marked ‘relative velocity’, which is why the output is taken from this point in the adjoint model.

The angle error is simply the output from the adjoint guidance filter, this could also be scaled against range to account for any range dependency of the angular noise.

314


The three sensitivities are obtained for all times of flight and for all three disturbances from a single run. The results are presented in Figure 17-81 and Figure 17-82.

Sensitivity of Miss to Aiming Error

-100

-50

0

50

100

150

200

250

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Time of Flight (sec)

Mis

s D

ista

nce

per r

adia

n Ai

min

g Er

ror

Figure 17-81: Adjoint Model Calculation of Miss per Unit Aiming Error

Miss Distance Sensitivity to Sensor Angle Error

-120

-100

-80

-60

-40

-20

0

20

40

60

80

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Flight Time (sec)

Mis

s D

ista

nce

per u

nit M

easu

rem

ent E

rror

(met

res

per r

adia

n)

Figure 17-82 : Adjoint Model Calculation of Miss per Unit sight line measurement error.

Note that aiming error is found from an initial lateral velocity impulse.

There may be a temptation to put an initial position impulse in the forward model to ostensibly find the effect of lateral position offset in the adjoint model. However, if we consider how the linearised

315


equations were formulated in the first place, this would be nonsense. The reference axis, by definition, passes through the launch point, so a ‘position offset’ is meaningless.

17.9 Concluding CommentIf the system is time varying, there is usually a specific time at which the state must be known, with the state of the system at any other time being of little interest. The objective of the excercise is to find the effect on this end state of disturbances which occur at earlier times. By far the most efficient way of achieving this is by constructing the adjoint system from the forward time equations, and solving for all disturbances and all times to go in a single run.

The homing missile is presented as an example, but the same method applies to all time varying systems, such as spacecraft docking and aircraft autoland systems.

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18 Rapid Non-Linearities

18.1 IntroductionThe Law of Requisite Entropy implies that system analysis consists essentially of identifying the set of constraints under which the system must work, from which the design will flow down. We must identify the hole in the jigsaw into which our piece must fit.

The system we develop must in some way be the best that can be produced within the constraints. However, in seeking superlatives we sooner or later run into stability issues. ‘Optimum’ structures flap about in the wind like flags, the slender ship turns turtle, the overpowered motor car cannot negotiate bends, putting more power into the locomotive merely makes the train sway violently, and optimising our returns results in the inevitable market crash.

The past few chapters indicate that the spectre of instability can be exorcised by understanding and modifying the system dynamics. Initially, a passive solution should be sought, by adapting the system parameters. This may result in loss of primary system performance, so that the system almost certainly would not be operating near the naive optimum.

If we cannot access the plant to improve its inherent behaviour, we can modify the dynamics of how we interact with the plant to achieve the same end. This is how the wise royal adviser interacts with the foolish king to ensure the worst consequences of stupidity are avoided. Wives have manipulated their husbands in this way for millennia.

There are limits, however, to how far we can tame the beast.

The fundamental cause of instability is the implicit assumption that the output of the plant can be used as an estimator of the input, so that the control is driven by the error in the output as in the ideal null-seeking regulator. Plant delays ensure this is never the case, but feedback renders the plant insensitive to this mismatch. If we try to drive the closed loop plant significantly faster than the response time of the open loop plant, we are pushing our luck.

It is because delay is the fundamental reason for instability that we resort to phase advance in the compensator in order to recover stability. The phase advance is merely a predictor of the future output. Ideally, we should build a compensator which was the reciprocal of the plant transfer function. This would be impossible because the plant usually has more poles than zeroes, so the resulting compensator is not realisable, also right half plane zeros become right half plane poles of the compensator.

The first concern is that prediction amplifies errors in the current estimate, so if the measurement of the output is wrong, which it will be if a real sensors is used, the error will be amplified. The second concern may be understood if we dissect the implementation of a phase advance transfer function:

G (s )=1+ατs1+τs

317


Where τ (tau) is the time constant and α the advance ratio. In order to implement this either as an algorithm or as a circuit or mechanism, the order of the numerator must be less than that of the denominator.

G (s )=α+ 1−α1+ τs

The advance ratio is greater than unity, usually between 3 and 10. We see that introducing the phase advance inevitably introduces a large amplification factor. The servo demand and control deflection will be amplified by a similar amount, so that either a more capable servo is required, or the servo will be forced on to its stops.

Servos are expensive, and if rapid response and large servo effect is required from them, they can be heavy, bulky and power hungry as well. Rather than over-specify and require the servo never to saturate, it is advisable to investigate the effect of saturation, in order to see what we can get away with. In so doing we will consider how to deal with all other non-linear effects which act instantaneously as far as the characteristic time constants of the plant are concerned.

18.2 Describing the Distorted SignalWe have seen in the previous chapter how sinusoidal excitation of the plant yields information about its stability. If we feed a sine wave through a saturation, the output is unaffected if the input amplitude lies below the saturation level. Above this level, the signal is clipped. We can still represent this distorted signal as a series of sine waves, rather like the Taylor expansion used for polynomial fits to functions. The output may be represented as:

y (t )=∑i=1

∞

ai sin iωt

This is called a Fourier sine series. More generally, the non-linearity will introduce a phase shift, so that the output will take the form:

y (t )=∑i=1

∞

ai eijωt

This is the more general Fourier series.

18.3 Fourier Series Curve FitIn order to evaluate the coefficients we need to address the more general issue of fitting a curve with a set of simple functions.

We can write a single sample of the output in the form:

y (t 0 )=(e jωt 0 e2 jωt 0 ⋯ )(a1a2⋮ ) For a set of samples taken at consecutive times:

318


( y (t 0 )y (t 0+δt )

⋮ )=( ejωt 0 e2 jωt0 ⋯

e jω (t 0+δt ) e2 jω( t0+δt ) ⋯⋮ ⋮ ⋱)(a1a2⋮ )

This is written more compactly in matrix form:

[ y ]=[E ] [a ]

The error between the actual time series and the curve fit is:

[e ]=[E ] [a ]−[ y ]

The sum squared error is:

ε=[e ]H [e ]=( [a ]H [E ]H− [ y ]H ) ( [E ] [a ]−[ y ])

Where H is the transpose of the complex conjugate of the original matrix.

Differentiating with respect to [a] yields the curve fit with the least sum squares error:

(∂ ε∂a1∂ ε∂a2⋮)=[E ]H [E ] [a ]− [E ]H [ y ]=0

The least squares curve fit coefficients are:

[a ]= [ [E ]H [E ] ]−1[E ]H [ y ]

The quantity:

[ [E ]H [E ] ]−1[E ]H= [E ]¿

Is called the pseudo-inverse of non-square matrix [E]. Note that the pseudo-inverse only exists if there are more samples than coefficients.

This is a general result for fitting a curve to any set of more tractable functions.

For the specific case of sine functions, we need to investigate the inverse of [E}H[E].

The terms on the lead diagonal are all equal to:

f ii=∑k=0

n

e ijω (t 0+kδt ) e−ijω (t 0+kδt )=n

The off-diagonal terms become:

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f ℑ=∑k=0

n

e( i−m) jω (t 0+kδt )

If the intervals between samples tends to zero, this sum becomes an integral:

f ℑ→1T ∫0

T

e (i−m ) jωt dt

If we choose T=2π/ω, this integral becomes identically zero for all i ≠ m

The inverse is:

[ [E ]H [E ] ]−1=1nIn

The ith coefficient is found from the ith row of [E]H:

a i=1n∑k=0

n

e−ijω ( t0+kδt ) y (t 0+kδt )

Or:

a i=1nδt∑k=0

n

e−ijω ( t0+kδt ) y (t0+kδt )δt

With sufficient finely spaced samples, the sum becomes an integral over the interval: nδt=T=2π/ω.

The coefficient is found from:

a i=ω2 π∫0

2πω

y ( t ) e−ijωtdt

18.4 The Describing FunctionWe could calculate the output of the non-linearity for a set of harmonics fitting the shape of the distorted signal. However, this is excessive, and not really necessary, because most real systems exhibit a low-pass frequency response, so the higher harmonics will be attenuated.

We can get an adequate representation of the effect of the non-linearity by considering only the fundamental harmonic of the distorted signal.

To illustrate this point, consider the effect of saturation at ±vs on a sinusoidal signal of amplitude X.

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Figure 18-83 : Effect of Saturation

The output from the saturation is the clipped sine wave presented in figure Figure 18-83

This is described, over the interval 0 to 2π/ω, as:

y=X sinωt t< θω

y=v θω<t< π

ω− θω

y=X sinωt (π−θ )ω

<t< π+θω

y=−v (π+θ )ω

<t< (2 π−θ )ω

y=X sinωt (2π−θ )ω

< t< 2πω

Where, from the figure it can be seen:

sin θ= vX

The coefficient of the fundamental is:

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a1=ω2 π∫0

2πω

y e− jωt dt

The input is a sine wave, which when expressed as a complex exponential, is :

x=X sinωt=X ejωt−e− jωt

2 j

During the unsaturated times;

∫t1

t2 (1−e−2 jωt )2 j

dt=¿(t 2−t1 )2 j

−(e−2 jωt1−e−2 jωt 2 )

4ω¿

The contributions of the non-saturated regions to the integral are:

( θω−0)+( π+θω −π−θω )+(2 πω −2π−θ

ω )2 j

=−2 j θω

And:

(e−2 jθ−1 )+(e−2 j (π+θ)−e−2 j( π−θ) )+(e−4πj−e−2 j (2π−θ) )4ω

=−e2 jθ−e−2 jθ

2ω= j sin2θ

ω=2 jωvX √1−( vX )2

(since: e± j2πn=1 )

The integral over the saturated regions becomes:

1X∫t 1

t 2

±v e− jωt dt=±v 1jωX

[e− jω t1−e− jωt 2 ]

Or: vjωX [ (e− jθ−e− j(π−θ) )−(e− j (π+θ )−e− j (2π−θ ) ) ]=4 v

jωcosθ= 4 v

jωX √1−( vX )2The Fourier coefficient for the fundamental becomes:

a0=− j1π (θ− v

X √1−( vX )2)This is the coefficient of the complex exponential signal, so the output is of the form:

y=e jωt

This is not in a form which can be compared with the input in order to determine the effect of the non-linearity on gain and phase. The output must be in the form of a sine wave:

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y= ejωt−e− jωt

2 j

Evidently we must apply a gain of 2j to the coefficient if it is to characterise the effect of the non-linearity on a sine wave. The coefficient becomes:

N (X )= 2π (sin−1( vX )−( vX )√1−( vX )2)

We see that when v=X, N(X)=1.0, as should be expected, because the sine wave is no longer clipped.

The function N(X) is called a describing function.

Other non-linearities may be characterised in exactly the same manner.

18.5 Application of Describing Function to the Harmonic LocusThe open loop transfer function with a describing function representing the non-linearities (usually in the servo) is of the form:

GO ( jω )=N ( X ) kG ( jω )

The effect on the harmonic locus is to change the critical point from -1 to the region defined by -1/N.

If the harmonic locus and describing function reciprocal intersect, the system will oscillate about the equilibrium position at a frequency given by the harmonic locus at the point of intersection, and with amplitude given by the describing function. This steady state oscillation is quite common in systems with non-linear elements, and is called a ‘limit cycle’.

18.6 Bang-Bang ControlServos are expensive, so it is likely that at the beginning of the analysis at least, the designer will be required to assess the suitability of existing servo designs for the system under consideration. Such work is needed to decide whether the effort needed to develop a new servo will be commensurate with the benefits in overall system performance improvement. Since servo lags have a profound effect on stability, much of this assessment can be undertaken using the methods already covered.

Upper bounds on system bandwidth imposed by limitations of the servo are readily derived by applying the methods of the previous chapters. Naturally, of course the ‘generalist’ systems engineer would never demean himself with such ‘theoretical niceties’ as basic stability.

Much of the expense of the servo arises from the apparent need to make its behaviour linear. This is difficult because most actuation methods involve electric or hydraulic motors and are characterised by rate limits as well as position limits. The insistence on linearity, just for the convenience of the designer, would hardly justify incurring the extra expense, if the system can be made to work with a simpler design.

One option is to employ a servo which applies its full effect or none at all. In practice such a servo would ‘chatter’, i.e. switch back and forth as fast as it can in response to system noise. Rather than the pure all or nothing response, it is normal practice to include a ‘dead band’ region so that the

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input must exceed a certain threshold before the servo will respond. The width of the dead band is chosen such that the probability that the servo is triggered by noise is small.

We derive the describing function by following the procedure of the previous section of considering the effect the servo has on a sinusoidal excitation of amplitude X.

If the dead band is of width 2d, nothing happens until the input exceeds d, which occurs at time θ/ω

Figure 18-84 : Relay with Dead Space Response

Where:

θ=sin−1( dX )The describing function is found to be:

N (X )=2 Fπ √1−( dX )2

18.7 Schilovski MonorailThe use of this type of actuation (known colloquially as ‘bang-bang’ control) on a Schilovski monorail balancing loop is illustrated by the harmonic locus of Figure 18-85.

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Figure 18-85 : Schilovski Loop with Bang-Bang Actuation

The describing function is the red plot. It starts at -∞, at zero amplitude, then the amplitude increases to infinity at point B, which is -1/F +j0.

If the limit cycle amplitude is below the value corresponding to the intersection with the harmonic locus, the system is unstable, so the amplitude will increase until the intersection is reached. Also, if the amplitude is greater than the value at the intersection, the critical point will lie inside the contour and the system will be stable. The amplitude will then reduce until the intersection is again reached. This intersection therefore corresponds to a stable limit cycle. The further over to the right point B is located, the lower the amplitude of the limit cycle, unless the describing function crosses the harmonic locus again at A.

If the servo force F is too low, the describing function will lie entirely to the left of the harmonic locus and stability will be impossible. If it is too high, point B will lie between point A and the origin. The system will then be unstable, causing the response to move to point B, and infinite amplitude.

Schilovski actually implemented control in this manner using a clutch mechanism to extract the actuation torque from the gyro itself, obviating the need for a separate servo.

18.8 More General Servo ConsiderationsMost means of applying actuator displacements are rate, as well as position, limited. This introduces a phase lag on the input, which is likely to have a de-stabilising effect on the closed loop. Since such constraints are inevitable, it may be more sensible to operate the servo either at its maximum rate

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or not at all, rather than reducing system bandwidth just to keep the servo within its linear operating range.

The describing function for a servo of this type is found from the output in response to a sine wave input signal.

Figure 18-86 : Servo with Rate and Position Limits

We see immediately that there is a 90° phase shift between input and output, which should be expected from a component containing an integrator.

If the input amplitude is greater than the dead band (denoted ±v/2), the velocity output is zero, which implies the servo stays where it is. There is no reason to believe that this will be the zero position. The description of a servo with both rate limits and position saturation as two consecutive non-linearities is incomplete, as it is evident that there is no ‘open loop’ control over the servo position.

In assuming the equilibrium position is about the zero deflection, we fail to take account of the fact that we are usually dealing with linearisations about an operating point. If the servo represents, for example, the elevator actuation of an aircraft, a finite deflection will normally be required to trim the aircraft at its steady state angle of attack. Similarly, the presence of lateral wind forces, or accelerometer bias, in the Schilovski loop become manifest as non-zero gimbal deflections at equilibrium (inertial side forces do not introduce any bias).

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In order to apply describing functions, we implicitly assume the zero level is an operating point which will change slowly compared with the response time of the control loop.

In the aircraft example, the steady load is taken out with a trim tab, in the monorail the centre of gravity may be shifted laterally.

By implication, the controller containing the non-linearities is part of a hierarchy of control systems.

The presence of the integrator implies the servo must itself contain a feedback loop comparing the actual deflection with the desired deflection.

Figure 18-87: Non-Linear Servo Schematic

These points are illustrated by the schematic loop of Fig Figure 18-87. The presence of the integrator would render any attempt at open loop control precarious, to say the least, so the actuator is controlled by the difference between the desired deflection and the current deflection.

The velocity limit usually arises from a flow rate or current limit in the actuator motor, which is represented here as a limit on the input.

The continuous dynamics are second order, so the harmonic locus will not intersect the negative real axis. The gain margin is therefore infinite.

Since we are assumed to operate well away from the deflection limits, stability is determined by the relay with dead space non-linearity. The phase margin is found from the intersection with the harmonic locus at a distance given by point B on the describing function plot. If this is less than about 30°, there is a risk of a limit cycle. A system which oscillates about the equilibrium state is described as ‘hunting’.

The phase margin is dictated by the maximum servo rate and the time constant of the motor.

In order to use the servo within a loop it is necessary to represent it as a linear system, for inclusion in the parent subsystem.

We note from the describing function that for signals significantly greater than the dead band, the relay acts as a gain of 2F/π.

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Using this approximation, the closed loop transfer function is:

G (s )=

2Fπ

1τ

s2+1τs+ 2 Fπτ

The undamped natural frequency is:

ω02=2Fπτ

Note that the damping depends entirely on the motor response, which therefore dictates the maximum servo rate and hence bandwidth. These are hardware limitations, which introduce penalties in terms of cost, size, weight and power consumption. These latter considerations impose restrictions on the servo bandwidth, but at the same time, the servo bandwidth imposes significant limitations on overall system bandwidth and proper function.

The servo design is therefore a compromise of maintaining adequate system response time, whilst avoiding excessive weight, etc.. Note these latter constraints are secondary to achieving adequate response time, but the modern tendency is to employ a tail-wag-dog approach of dictating subsystem performance on the basis of secondary system requirements, thus rendering primary function difficult, if not impossible, to achieve.

18.9 The Phase Plane

18.9.1 Dominant ModesThere is no shortage of authorities which recommend that the designer should aim to ensure the system modes are closely spaced by, for example, employing a Butterworth pole pattern. This author has consistently insisted that the mode frequencies should be determined from their proper function. This understanding comes from experience with the specific plant to be controlled. In general, we would no more set the modes to the same frequency than we would tune the strings of a violin to the same note. If you lack this understanding of the original open loop plant, you are in no position to consider modifying it.

We use the differences in frequency to decouple different behaviours. This enables us, for example, to control the attitude of an aircraft, as well as its trajectory, from a single set of controls. Imagine trying to ride a bicycle in which the balancing modes and steering modes were strongly coupled together.

If we cannot find an approximate factorisation of the characteristic equation, the implication is that modes are closely coupled, which indicates there is something wrong with the plant itself. With this decoupling, we can study modes in isolation.

When we try to speed up our system response, or improve its behaviour in some other way, the result is that coupling between modes and cross-coupling between functionally independent channels increases to such an extent that this assumption breaks down. Our fundamental concern is with how much feedback we can have before the effects of all our approximations and uncertainty

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in the plant dynamics undermines the assumptions we have made. In short, the most important single concern when introducing active feedback control is with the robustness of our design.

At present we use the idea of gain and phase margins to hopefully account for such uncertainty, together with separations in characteristic frequencies of modes of at least a factor of 3, (preferably 10).

With the proviso that we shall investigate robustness explicitly, it is often the case that we can deal with modes in isolation. A mode is represented as either a simple or a quadratic factor, and familiarity with the specific plant behaviour will usually indicate which modes are of immediate interest.

We have already done this in the case of the rocket weathercock mode, in assuming it to be separable from the trajectory mode (the phugoid).

As a further example, consider the gyro monorail; our concern is with the stability of the precession mode, which is dynamically unstable. We can write the equations of motion, ignoring the terms which are significant at high frequency, as:

Gimbal:

−H ϕ=M θθ+M c

Roll motion:

H θ=Whϕ

Experience with a toy gyro will soon show that the most obvious effect is the tendency of the spin axis to align itself with the precession axis. The effects of the expected inertia resistance (proportional to angular acceleration) are transient, and not noticeable. This counter-intuitive behaviour unfortunately makes the gyroscope an ideal candidate for the active element in the crackpot inventor’s proposals.

This is now a second order system.

18.9.2 Representation of a Second Order PlantThe importance of a second order system is its motion can be represented completely on a two-dimensional diagram. We have seen from the works of M C Esher, that pictorial representation of higher dimensions, although possibly aesthetically pleasing, is ambiguous , and the reader is warned against such self-deception. Beyond two dimensions; algebra reigns supreme.

Consider the general linear, constant coefficient second order system:

v=−2 ζ ω0 v−ω02 y+ f

y=v

We eliminate time by expressing it in terms of y:

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v=dvdt=dydtdvdy=v dv

dy=1

2d v2

dy

The second order plant is described by the single equation:

12d v2

dy=−2ζ ω0 v−ω0

2 y+f ( v , y )

18.9.3 Unforced PlantA diagram is only of any value if we know how to interpret it. The most important features become apparent if we consider the simplest case of a system with no input and zero damping.

Figure 18-88 : Phase Plane - Linear Oscillator - Locus Direction is Clockwise

The equation is usually plotted with displacement as ordinate (horizontal axis) and its rate of change as abscissa.

With no damping, the system conducts a simple harmonic motion so that the displacement and speed are in quadrature (displacement lagging speed by 90° phase lag). This results in the elliptical contours shown, produced by starting the system at y=1, v=0 and y=2,v=0.

The most important feature of the phase plane is that all possible trajectories form clockwise paths around the origin as time increases.

In quadrant A, the velocity is positive, so it is impossible for the displacement to reduce in this region, the slope can never become negative here.

In quadrant B the speed is negative, so the slopes must be negative here. By similar reasoning, the slope must be positive in quadrant C and negative in quadrant D.

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We know that adding damping will cause the amplitude to decrease with time. It could be argued that in many circumstances, drag is more nearly proportional to the square of the speed, so the plant may be better represented with an equation of the form:

d v2

dy=−c|v2|−ω0

2 y+f ( v , y )

Figure 18-89 : Non-Linear Damping

As expected, adding the damping causes the plot to spiral in towards the origin. Using the square law, the damping at low amplitudes is poor, so it takes a large number of cycles for the disturbance to decay. This is worth bearing in mind when we approximate a square law damping term with a linear (viscous) damping representation.

Introducing negative damping, as expected, causes the plot to spiral away from the origin.

Phase plane representations are used extensively in the analysis of non-linear systems, which have several unstable and stable equilibrium positions, rather than just the origin. These equilibrium points are called ‘attractors’ because of the tendency to spiral in towards them, or spiral out from them. With a second order system, such as a linear mode studied in isolation, it is only possible to have the three types of attractor presented here.

With a highly non-linear model, the perturbation equations become problematical, as it is extremely difficult to separate ‘states’ from ‘coefficients’. In effect the dominant mode changes from one centred on one attractor to one centred on a different attractor, but is not stable about either. The result is a trajectory which orbits both with an apparently random switching between the two

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attractors. Although the trajectories are practically unpredictable, they stay in the vicinity of the attractors, exhibiting non-periodic orbits about both. This ‘multiple attractor’ which can affect continuous time systems of order three or higher, has been given the rather pretentious name of ‘strange’ attractor, and is fundamental to characterising chaotic motion.

Figure 18-90: Negative Damping - Locus Spirals Outwards

As engineers, we are not concerned with chaos as such, but on determining the circumstances under which it occurs, and making sure our systems remain in an ordered operating regime. Chaos theory comes into its own when studying natural phenomena beyond human control. Our job is to bring nature to heel, as far as is possible.

It seems a reasonable speculation that left to themselves, uncontrolled natural selection processes will lead to chaos, because that is the limit of adaptation before actual instability sets in. The reason evolution cannot tell us which species will dominate in a thousand years’ time is the population equations are chaotic in nature. However, to operate an entire economy, on which the survival of mankind depends, in the chaotic region, must be considered the most extreme act of folly of the human race.

18.10 Introducing ControlIt is evident from the basic stability analysis of the equations of motion that the precession mode of the gyro stabilised monorail is unstable. The phase plane plot for the basic plant resembles the outward spiral of the unstable attractor.

Using the parameters derived earlier, with the level of friction we would associate with well-lubricated gimbal bearings, results in the above open loop phase plane. The system is clearly unstable, but what is not presented is the time taken for the oscillation to build up from about half a degree to about 1.5 degrees. This is about 50 seconds. The result is important because we need to

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know how the system will behave in the event of servo failure. Provided the servo does not actually cause the gimbal to lock, there should be ample time to bring the vehicle to a stop.

Figure 18-91 : Gyro Monorail with no Feedback Control

If we consider the Fourier series fit to the output from a bang-bang actuator, it is evident that there will be significant energy in the higher harmonics, so there is a risk of de-stabilising the nutation, if gimbal rate feedback is employed. In order to avoid this risk, we restrict our consideration to roll angle feedback.

The phase plane is just another tool which provides insight into the system behaviour, it is not a panacea solution, nor is it a numerical ‘sausage machine’ to be used by unskilled labour. The designer must be aware of the potential dangers of taking a single mode in isolation.

We shall introduce the bang-zero bang control used to illustrate the describing function approach.

This is represented on the phase planes as switching curves which indicate the conditions when the servo force is applied. In this case, if the roll angle lies within the deadspace, the gimbal control moment is zero, otherwise it provides a constant control torque in a sense which stabilises the system.

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Figure 18-92: Effect of Relay with Dead Space Control –Roll Angle

Figure 18-93: Effect on Gimbal Angle

The roll angle plot indicates that the trajectory will spiral in until it is approximately bounded by the switching curves. Superficially, this appears reasonable. However, a glance at the phase plane plot

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for the gimbal angle shows that the situation is far from satisfactory. The maximum gimbal deflection approaches 1 radian, which is well beyond the small angle assumptions implicit in the equation of motion. In practice the gimbal would hit the stops and the vehicle would topple.

Figure 18-94: Excessive Gimbal Deflection Indicates a Requirement for Larger Gyros

The gimbal limits may be respected by increasing the gimbal deflection feedback. However, this requires an increase in gyro angular momentum in order to maintain separation between nutation and precession modes.

A system which appears stable according to our basic analysis is easily rendered unstable by system non-linearities. Typically, the high gains and phase advance which appear to ensure rapid system response are precisely the measures which aggravate the effects of non-linearities.

Any method of analysis of the robustness of the system must take into account the need to avoid saturation of the system states.

18.11 Hierarchical System ConsiderationsAutomatic control is frequently introduced with words to the effect; ‘given a system described by the linear time invariant equation:

x=Ax+Bu...’

The view of the author is that much ingenuity is involved in deciding what aspects of system dynamics are relevant to the problem in hand, so the equations are rarely ‘given’. Indeed if our sole knowledge consists of the numerical values of the elements of the matrices, it is difficult to see how we could offer advice as to how the plant could be modified to render it more readily controllable.

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Furthermore, the implicit assumption is that the states are similar in nature at all levels in the system. This is never the case. The higher level functions are different in nature from the lower level functions, but are equally describable as linear differential equations.

The skills of the soldier on the field of battle are different in nature to those of the general. What are constants to one are variables to another. However, at the end of the day the evolution of any dynamic process can be modelled as a set of differential or difference equations. We do not throw the baby out with the bathwater, just because snobbery and self-importance does not admit that ‘high level’ system function is just as amenable to dynamic analysis as any other level.

To illustrate this point, consider a practical example of a simple hierarchical system.

18.12 Scanning SystemA scanning system for a certain type of sensor consists of a mirror driven by a pair of servomotors. In order to avoid backlash the mirror is supported on flexible mounts, and the direction of pointing in azimuth and elevation are measured using shaft encoders.

The mirror is required to follow a scan pattern consisting of sinusoidal motions in azimuth and elevation. The frequencies of the azimuth and elevation scans are to be controllable, as is the phase of each with respect to a specified time datum, and the amplitude of each.

At first sight, this appears a simple matter of feeding in a demand which varies at the correct frequency, amplitude and phase into a pair of tracking loops. However, we know that the output will be corrupted by higher harmonics arising from non-linearities, and even if the tracker were linear, there would inevitably be a difference in amplitude and phase, between the input and output.

If the scan pattern were fixed, a simple solution would drive a resonant system open loop. The disadvantage of this approach is the problem inherent in all open loop approaches; it is extremely sensitive to the system parameters. So a relatively inflexible system could be built at great expense, as the scan frequency requirement places severe constraints on the mass of the mirror employed.

In order to employ lower quality hardware, and to obtain control over the scan pattern, the loop must be closed, and the inherent robustness of closing the loop would obviate the need for high quality components.

The system therefore consists of the plant, consisting of the mirror, its mounts, the motors and the encoders, and two controllers, one to point the mirror, and the other to control the amplitude and phase of the scan pattern.

Now amplitude and phase are not describable as linear functions of the bore sight angle, so the controller is non-linear. It introduces the type of non-linearity we would expect in any hierarchical system. The states of the parent system are not linearly related to the states of the subordinate system. Evidently, this is always the case in a hierarchical system.

Evidently if the loop which adjusts amplitude and phase has similar response time to the inner pointing loop, the system will be non-linear, possibly even chaotic. We need to know how close in frequency the two may be brought before the system starts to misbehave.

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An open loop system would be expected to operate near resonance in order to minimise the motor power required. This would also be desirable for a closed loop system. However, we have seen that at resonance the slope of the phase curve against frequency becomes infinite, so we should anticipate that controlling phase is likely to become problematical.

Figure 18-95 : Amplitude and Phase Response of Basic Plant

A frequency error is the derivative of a phase error, so if we are to control phase we need to operate in a reasonably linear part of the response curve. This corresponds to 90 degrees phase lag, but unfortunately, the amplitude plot is at its peak at this point. It is concluded that control of both amplitude and phase near resonance is unlikely to be possible.

337

+ 200

21

2 ssk

ssk1.01

12

Compensator

Plant

DemandAchieved


The inner control loop is therefore designed to ensure a near linear response in both amplitude and phase. One option is to increase both frequency and damping by means of a phase advance compensator, to ensure the response is at least monotonic, if not reasonably linear.

Figure 18-96: Inner Control Loop

With suitably chosen parameters, the worst effects of operating near resonance can be avoided.

The final value theorem states that in order to track an input expressed as a polynomial in time, the loop must contain as many integrators as the order of the input polynomial plus one. This is unfortunate if the task is a sine wave, which is represented as a Taylor series with an infinite number of terms.

338


Figure 18-97 : Gain and Phase Responses with the Inner Loop Controller

One approach is to control amplitude and phase explicitly using an outer loop, which generates a sinusoidal input corrected for amplitude and phase lag.

The relationship between the states of the outer loop (amplitude and phase) are not linearly related to the inner loop states, so the system cannot be represented as:

x=Ax+Bu

A moment’s reflection will reveal that this is the usual situation for a hierarchical system. The different levels have different functions. Higher level system functions determine operating points and parameters for subordinate systems. The overall system is far from linear, but that should not worry us if we know the limitations of our linear theory. Indeed, we are far better off than the buffoons who imagine the dynamics of complex systems are adequately represented as finite state machines.

18.12.1 Outer LoopA possible linear model of the outer loop is presented in figure Figure 18-98. The phase and gain responses are their steady state values, so it follows that the characteristic time constants of the outer loop must be chosen to allow time for the transients of the inner loop to die out, this is fundamentally where we get the 3 to 10 times rule of thumb from.

Subject to this constraint, we may deduce values for the various feedback gains.

However, this conceptual representation of the outer loop bears very little resemblance to what is actually implemented, its purpose is to enable the control loop function to be understood and analysed.

The actual implementation requires the amplitude and phase of the output signal to be estimated.

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Figure 18-98: Outer Loop Controller - Controls Gain and Phase, not the Position and Velocity

Given that the output is of the form:

y I=a sin (ωt+ϕ )

We can find the amplitude and phase by constructing a quadrature signal:

yQ=1ωd yIdt

This is produced by passing the output through a differentiator of the form:

G (s )= s1+τs

The amplitude of the output is:

a=√ yI2+ yQ2

The phase lag due to the inner loop lags is added as a phase lead to the input signal. The phase lag is given by:

sin ϕ=sin (ωt+ϕ )cosωt−cos (ωt+ϕ )sinωt

cos ϕ=cos (ωt+ϕ )cosωt+sin (ωt+ϕ )cosωt

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Or:

sin ϕ=1a ( y Icosωt− yQ sinωt )

cos ϕ=1a ( yQ cosωt+ yI sinωt )

Where sinωt is the input to the inner loop.

It should now be obvious why the closed loop avoids the resonant region, with its potentially infinite gain.

18.13 Concluding CommentsIf control theory were restricted to linear systems, its value would be severely limited, because of all dynamic systems, very few are actually linear. The issue is not one of controlling non-linear systems but of knowing under what circumstances the linear approaches break down. They are considerably more robust and universally applicable than they are given credit for.

By maintaining a high level of abstraction, which only a mathematical approach permits, we can study system behaviour independently of any specific implementation. In stark contrast, most current, non-mathematical approaches, are (in practice) strongly influenced by specific hardware characteristics, and are anything but abstract.

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19 The Representation of Noise

19.1 IntroductionNoise has been mentioned in earlier chapters as one of the reasons for avoiding very short time constants in compensators. It is hoped the reader has some intuitive idea of what is meant – consider for example the sound of static from a radio receiver which is not tuned to any station.

A noise signal is one in which contains no useful information. What constitutes noise depends on the context, but it is evident that we are concerned with values of system states, which are corrupted by random disturbances and instrument errors, these limit the accuracy to which the states may be known. This chapter considers how we represent such noise sources.

Whilst saturation yields the upper bound on the magnitude of the signals passing through the system, the separation in signal levels which can be distinguished is determined by the level of noise. Hence the maximum and minimum intervals, which ultimately determine the amount of information which can pass through the system are dictated by saturation and noise. The ratio of the saturation value to the measurement interval is called the system dynamic range.

In principle, the noise level would define the quantisation interval to convert analogue signals to digital. It is therefore reasonable to characterise even the original analogue signal as so many bits, corresponding to the logarithm to base 2 of the dynamic range. A moment’s reflection will reveal that this is the same as the information capacity of the process conveying the information.

The reason our earlier ruminations, based on response time, are incomplete, is we have only addressed the time response aspect of the system ‘intelligence’, in order to define it completely, we need to know the quantities of information flowing.

In the real world this information is represented as positions of controls, angles of attack, gimbal deflection angles, etc., but cybernetics views these as no more than writing of the same information on a different medium with a different type of ink.

19.2 CorrelationWhat is fundamental to any broadening of the application of a mathematical theory of systems is the ability to compare two signals and determine whether they are the same. Ideally, we should also seek some metric of how dissimilar the two signals are. The latter requirement is really a luxury, because as Ross Ashby’s homeostat, and indeed, evolution by natural selection illustrate, the adaptation can be random, but the ability to identify the null state is essential for error driven regulation. To recap fundamental results, error driven regulation is the only way of dealing with an open set of disturbances.

How, in general, do we compare two signals?

19.2.1 The Same, Or Different?As an example, an estate agent might list all the attributes of the properties on his books. In order to avoid ambiguity of natural language, each field describing the property would be selected from an

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appropriate keyword list, which would appear as a pull down menu when the data is entered, thus ensuring exact matches between enquiries and data fields. Data specific to particular properties, such as a photograph, or any additional comments on the property would probably be entered explicitly.

We might expect price, location, type of dwelling, number of bedrooms, number of reception rooms, etc. to be specified. In an ideal world, the estate agent would enter the customer’s requirements using the standard keyword list and out would come a list of properties matching the customer’s requirements. In practice, only rarely would anything at all come out.

Depending on how many properties are on the books, the estate agent would play with the inputs until about five or six options emerged. The simplest option would be to score a one for each attribute which matched , and zero for each which did not, and summing the result, the five or six with the highest score would be selected. This has the difficulty of weighting less negotiable attributes, such as price, number of bedrooms, and perhaps location equally with cosmetic or irrelevant attributes.

Most databases require a certain ingenuity both in entering data and on retrieving it. Does a two bedroom house with planning permission for a loft conversion constitute a 2 and a bit bedroom house? What if a separate dining room could be used as a bedroom? The funny little room currently filled with junk, could that be treated as a potential bedroom? Architect’s aberrations, and weird extensions and alterations quickly defy the naïve options of the input lists.

We could extend the database by making the options themselves database fields into which we can enter a number reflecting how accurately the term describes the actual property. In essence, we deal with the vagueness of the real world by replacing the crisp sets of the original database with fuzzy sets. This would yield a, somewhat arbitrary, metric indicating how well each attribute meets the customer’s requirements.

This could be extended further by deciding the relative importance of each attribute, by applying a large weighting to the important attributes, and low weighting to unimportant. This results in an overall figure of merit, from which the ‘best’ options may be selected. The problem arises, as in all ‘soft’ analysis methods, of how we attach meaningful weightings to the attributes.

Staying with single bit data, consider the representation of letters of the Latin alphabet represented as arrays of dots, as was once actually done on low cost printers. For simplicity, each character will be defined as a 59 dot array. The characters A and B are presented. The object of the exercise is to be able to distinguish between these symbols using an algorithm. We will use the image of one letter, say A, as a template to compare with a candidate, to see if they match.

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Figure 19-99: Pattern Matching Requires the Noise to be Characterised

Following the database matching idea, we increment our metric by 1 when the candidate and template pixels are both high and 0 when they do not. Comparing the A and B,

For the top row, 3pixels match, two don’t = 3For the 2nd, two pixels are high on both=2For the 3rd, 2 pixels match=2For the 4th, 4 match, 1 doesn’t=4For the 5th, 2 match=2For the 6th, 2 match=2

For the 7th, one matches=1For the 8th, none are set=0For the 9th, none are set=0

The total value is 16, whilst for a correct match it would be 18 so the ratio=0.89, could be interpreted as a measure of how much the letter B resembles the letter A in this script.

19.2.2 Pattern MatchingAs it stands, this is not particularly useful. It could be argued that the legibility of the script could be assessed pseudo-objectively by comparing every character with every other and summing the ambiguity values of all pairs.

Meaning can only be ascribed to these numbers, if due to imperfections in the print process pixels are omitted and additional ones included at random. These spurious pixels constitute noise. Knowledge of the process would enable us to assign a probability of spurious pixel mismatch from which the probability of confusion of characters may be determined. Without this knowledge, the metric is meaningless.

In order to make the results more general, we treat the pattern we are seeking as a (not necessarily continuous) function of two variables:

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f=f (x , y )

The region being checked for the presence of this pattern is also a function of the position coordinates, but these are referred to the region itself, and not the template, so the coordinates will be denoted h,k rather than x,y.

g=g (h , k )

We shall assume that the region is to the same scale and the same orientation as the template, so that an exact match of the template with the region is possible. We test an area equal to the template at the coordinate (h,k) by comparing each point

Dealing with functions of two variables adds unnecessary complexity to the problem. Imagine the page is scanned as a raster pattern, in much the same way it is read. This is always possible in practice because the scan line width will be equal to the resolution, which will always be finite. In order to convert from a 2D matrix to a 1D signal we need to know the raster width, W.

The area of interest is located between the k th scan and the (k+b) scan, where k is the vertical coordinate in the raster, and b is the height of the area of interest. Expressed as a time series, the start of the area of interest is offset from the start of the scan by kW samples, and has duration bW samples.

Within this interval, the area of interest is offset by h samples (where h is the x coordinate), and is ‘a’ samples in duration, where a is the width of the symbol, followed by a delay of W samples and another block of samples of length ‘a’, this repeats across the entire range of scans. The mapping from two dimensions to one involves a fairly simple algorithm, and indeed corresponds to how data is usually stored in digital computers. The algorithm may be extended to higher dimensions enabling searches in scale, orientation and other common distortions, to be mapped into a single dimension.

The total number of samples, and the magnitudes of individual samples is not affected by this process, so the entropy is unchanged. The same data could be scanned vertically, and there would be no entropy change, so the mapping to a single dimension does not alter the information content, nor should it matter how the data is scanned, provided each sample is covered only once per scan.

Using the algorithm, the area of interest is transformed to a 1D sequence covering just the area of interest. So it is now a function of the sample number, which may correspond to time, or some more abstract independent variable.

The problem is now in a form amenable to mathematical analysis. The problem of pattern matching, whether printed characters, images of objects or spoken words, reduces to ‘how well does function f(t) fit function g(t), over the range (0,T), where t is an arbitrary independent variable?’.

The difference between the two signals is the error signal:

ε ( t )= f ( t )−g ( t )

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We have already solved this problem when we considered a least squares fit of a set of elementary functions to a given time series, when seeking to characterize the sine wave signal distorted by rapid non-linearities. In that case the ‘template’ function was the first harmonic of the output signal.

If we fit f(t) to a set of elementary functions so that it takes the form:

f ( t )=∑1=1

n

c i f i (t )

For m samples, these quantities become matrices:

ε=Fc−g

Where is now an m-vector, F is an mn matrix, c is an n-vector, and g is an m-vector of samples.

We found in Chapter 18 that the coefficients are given by:

c= [FT F ]−1FT g

In the special case of a single function (template) fit to the data g(t), there is only one function, and hence one coefficient, which we may interpret as the measure of the fit in a least squares sense. In this case:

FT F=∑1

m

f (t j )2

FT g=∑1

m

f ( t j ) g (t j )

Hence the metric becomes:

c=∑1

m

f ( t j ) g (t j )

∑1

m

f (t j )2

In other words, we compare every sample with the template and sum the result, we then scale the result by the same process applied to the template itself.

This comparison process is called correlation.

We can map n-dimensional data to a single dimension using a raster-scanning approach, because we are always dealing with finite dynamic ranges, and in fact the quantities f and g are not limited to scalar numbers, the approach applies to single bit (text string) data as well as continuous data, multi-dimensional data and scalars.

This metric therefore appears fairly universal, it is used explicitly in experimental curve fitting and implicitly in practically every form of pattern analysis. It is usually given the name correlation coefficient.

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In analysing correlation problems, it is usually more convenient to deal with continuous expressions, assuming the sampling interval is infinitesimal. The sums reduce to integrals. Furthermore it is usual to define the template such that its self-correlation is unity. The correlation becomes:

c (T )= 1T ∫0

T

f (t )g (t )dt

Returning to the original problem, the objective is not just to see whether a pattern is present in the data set, but whereabouts it is located.

However, the setting of thresholds to decide whether the correlation function is sufficiently high, requires knowledge of processes which degrade the pattern, i.e. the noise which may be present. We need to know how likely it is that the threshold crossing is spurious.

19.3 Auto CorrelationIf we select symbols (e.g. letters) at random and put them in sequence, it is unlikely that they will spell out English words. We expect a constraint to be present which restricts the options which may be selected following a particular letter. In fact we need to take account conditional probabilities of up to about four consecutive characters before the resulting sequence starts to generate credible English words.

Conveying meaning appears to require a restriction on the entropy of the message to below the potential capacity of the channel. Conversely, if we compare successive samples of a message, we expect to see significant correlation, if the message conveys meaning.

We can use this idea to detect whether a time series contains a useful signal is to investigate it for trends. We have seen from our deliberations with the z transform that functions of time may be represented as recursive algorithms, so that the current output depends in some way on previous outputs.

We can detect whether such a relationship exists by comparing the output with itself, shifted in time:

ck=∑i=0

∞

y i y i+k

The resulting sequence is called an autocorrelation. For a continuous signal, the sequence becomes a function of a time offset:

c (τ )=∫0

T

y (t ) y (t+ τ )dt

Where it is understood that T is long enough to cover the maximum system time constant of interest.

If we consider the impulse response from a first order lag:

y (t )=e−at

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The autocorrelation function becomes:

c (τ )=∫0

T

e−at e−a (t+τ ) dt=−e−aτ

2a[e−2at ]0

T→e−aτ

2a

There is an observable pattern in the signal.

Fig Figure 19-100 is a plot of the outcome of throwing two dice 100 times. The average (7) has been subtracted from each throw in order to produce a nominally zero-mean sequence.

0 20 40 60 80 100 120

-6

-4

-2

0

2

4

6

Figure 19-100: Sequence of Dice Throws

0 20 40 60 80 100 120

-200

-100

0

100

200

300

400

500

600

700

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Figure 19-101 : Autocorrelation of Random Sequence

The correlation plot for this sequence indicates that the correlation is high only if there is no offset in time. For all finite time offsets the correlation is much lower than the peak.

One might expect that the longer the sequence, the lower the correlation will be, away from the zero time offset. With an infinite series with samples separated by infinitesimal intervals, it looks like the autocorrelation for a random sequence will become a delta (impulse) function.

We can then define a random process as one whose autocorrelation function reduces to an impulse.

19.4 Characterising Random ProcessesA random process is characterized as an infinite series of joint probability density functions (pdf), each one of which characterizes the probability of finding the amplitude x of the process at some time. For example, for three discrete time samples we may define a pdf:

pdf ( x1 , x2 , x3 : t1 ,t 2 , t3 )Δx1 Δx2Δx3=probability [x1<x (t1 )<x1+Δx 1 , x2< x (t2)< x2+Δx2 , x3<x (t 3)< x3+Δx3 ]

As the population of individual samples is infinite, we must revert to a statistical description of the process, and leave the lies and damn’ lies to the software developers.

The process is characterized typically by the first and second moments:

Mean:x (t )=∫

−∞

∞

x pdf ( x , t )dx

Mean Square:x2 (t )=∫

−∞

∞

x2 pdf ( x , t )dx

Variance:σ 2 (t )=∫

−∞

∞

(x− x (t ) )2 pdf ( x ,t )dx

Correlation function:ρ (t , τ )=∫

−∞

∞

∫−∞

∞

x1 x2 pdf (x1 , x2 : t , τ )dx1dx2

These may be defined for vector functions, in which case the correlation function becomes a correlation matrix.

Note: ρ (t , t )=x2 (t )

Because x can have only one value at any instant in time, so if τ=t , then x1=x2 . (we have seen this with the dice example)

In the following, and indeed implicitly in Monte Carlo simulations, it is assumed that the statistical processes are stationary (the statistics are invariant with respect to translation in time), and ergodic

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(the statistics taken from a single run over time are the same as the statistics of an ensemble of processes).

It is practically impossible to prove that these conditions ever apply, but any error will be present in a Monte Carlo just as in an analytical approach.

These assumptions simplify the expressions for the process statistics:

x (t )= limT→∞

1T ∫

−T2

T2

x ( t )dt

σ 2 ( t )=limT→∞

1T ∫

−T2

T2

(x (t )− x ( t ) )2dt

ρ (t 1)=limT→∞

1T ∫

−T2

T2

x (t ) x ( t+t1)dt

19.4.1 White NoiseTaking a simple example, if there is no correlation between one instant and the next:

x (t ) x (t+ t1)=0 , |t1|>0

=n2 , t1=0

Where n is taken from the input random sequence (it is assumed x(t) has zero mean).

The individual samples must have finite duration Δt , so that the time sequence of is in fact a set of samples of length:

N= TΔt

The variance of the sequence is given by:

σ 2= 1N∑1

N

n2

But the correlation is given by:

ρ (t1)=σ2 Δt=W

A random sequence for which this equation applies is called white noise. In practice white noise is a process which has correlation time much shorter than any of the time constants relevant to the current problem.

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It is evident that the correlation function of white noise does not have units of the input quantity squared; it is multiplied by a time constant, in this case, Δt . A white noise source is one for which Δt→0, implying the amplitude of individual samples increases without bound.

Any practical attempt to measure white noise intensity will be limited by the sample interval that can be used, so a given noise source might be expected to yield different answers for different intervals.

However, we ascribe to white noise the property that the variance of the noise reduces proportionately as the sample interval increases, so that the estimate of white noise intensity calculated by correlation ought to be independent of the actual sample length used. The justification for this is the noise ‘energy’ (proportional to the square of the amplitude, and thus the variance), is proportional to the sample length.

Provided the sample size is small enough to detect any potential high frequency cut-off within the range of frequencies of interest, the sample time should not matter.

For this reason, the arbitrary sample length is absorbed into the term W, which is what the correlation process actually measures, and is independent of the sample length for a white noise process.

Like geometric points, particles, weightless pulleys, inextensible strings and rigid bodies, white noise is another useful fiction which is never absolutely true, but yields deep insights into the problem. There is no shame in approximation; it is ignorance of the error bounds which renders results useless.

19.4.2 System Response to White NoiseWhite noise intensity is therefore always expressed as a signal squared over a bandwidth, e.g. volts2 per Herz. It may be thought of as the density the noise spread across the frequency domain, hence the more usual term; power spectral density.

The random signal at the output of a first order lag is the limit of the following sum:

x (t )=∑ n (iΔt ) exp(− (t−iΔt )τ )

Where τ is the time constant of the lag. n(iΔt) is the value of the random input at time iΔt, and i is an integer. This is assumed uncorrelated from one instant to the next, like successive throws of a dice. Also if (t-iΔt) is negative, it will be taken as zero in this equation. It follows that:

x ( t+t 1)=∑ n (iΔt )exp(− ( t+t 1−iΔt )τ )

If the random sequence has zero mean, the only significant contributions over a population of samples will be from terms resulting from the same inputs:

x (t ) x (t+t 1)=∑ n2 ( iΔt )exp (− 2 ( t−iΔt )τ )exp(− t1τ )

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In this form, the correlation depends on the time of application of each random impulse, as well as the current time, so that it does not take the simple form of the original white noise sequence. The time of application of the impulse relative to current time is important, whereas in the white noise case, the time samples could be taken in any order.

We are dealing with the forward solution of a linear homogenous differential equation of constant coefficients, whilst the right hand side of our response equation is expressed in adjoint time. The input is taken from a random sequence, so is not an explicit function of time. The statistics are not changed by taking adjoint time.

If we let ta=t-iΔt the impulse response of the first order lag when expressed as an adjoint, is common to all inputs, so we are justified in summing them over adjoint time, because a transfer function with constant coefficients is self-adjoint.

The impulse response of the first order lag does not take account the finite width of any real excitation. The input is assumed to be a rectangular pulse of finite width. So in order to evaluate the correlation, it is necessary to calculate the output from a first order lag to a pulse of width Δt: The response during the pulse is:

x (t )=n(1−exp(− tτ ))So the amplitude at the end of the pulse is:

x (Δt )=n(1−exp(− Δtτ ))≈n ΔtτSince Δt will always be chosen such that it is much less than τ. The finite width pulse does not affect the correlation beyond modifying the input. The correlation therefore becomes:

1T∫0T

exp(−2( taτ ))exp(− t1τ )dt a= τ2T (1−exp(−2 T

τ ))exp (− t1τ )The left hand side of equation of this is common to all samples, of which there are N., so the correlation coefficient is:

ρ (t 1)=TΔt∑1

TΔt

n2( Δtτ )2 τ2T

exp(−t 1τ )=W2 τ exp(−t 1τ )As t1→0 this tends to the variance:

E (x2)=ρ (0 )=W2 τ

Where E() is the expected value.

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What we have calculated is the output variance of a system fed by white noise. This time domain calculation is seriously tedious. An alternative approach is considered later in this chapter.

19.4.3 Implications for Time-Varying SystemsA practical time-domain model of white noise uses a sample period much shorter than any time constant of the system (which, by implication, assumes the response times of the components have been found long before we think about computer modelling). If this sample period is Δt, a reasonable approximation to the white noise process is a sequence of pulses of rms amplitude:

y=√WΔtWhere W is the white noise intensity. This is not very precise, but since the only use for explicit time series white noise generators is for inherently poor Monte Carlo simulations, the small error introduced by this approximation is unlikely to be noticed.

The time domain solution in effect replaces the time domain signal fed into a single system with an ensemble of systems, each fed with an impulse taken from the random distribution. The random process must be ergodic for this to be valid. The correlation integral is evaluated using the adjoint solution.

Now, the above reasoning could be applied to a system with time varying coefficients. In order to reduce the problem to a single time variable, the equations must be evaluated in adjoint time. Hence the result is directly applicable to adjoint models, but not to their equivalent forward time models. The output yi

2 integrated over adjoint time is the rms value of the corresponding (forward model) input taken over the ensemble of run times. The rms miss caused by this input is:

x2 (T )=W i∫0

T

yi2dt a

Where Wi is the white noise description of the input noise signal.

Thus, by squaring and integrating the adjoint outputs with respect to adjoint time, we obtain the miss distance statistics without resorting to Monte Carlo techniques. Also, unlike a typical Monte Carlo, the partitioning of miss distance between the noise sources is found from a single run, so the relative significance of each of the noise sources is readily apparent.

Not only have we saved a thousand runs to cover the time to go ambit, this figure is multiplied by the number of disturbances and the number of Monte Carlo replications. In a single afternoon we can complete an analysis equivalent to several million individual runs.

Target manoeuvre may be described as a white noise process, but more typically a target would be expected to conduct a deterministic manoeuvre, but at random time. The adjoint for this is similar to equation, but divides the integral by the time of flight.

x2 (T )=fT2

T ∫0T

yfT2dta

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Also, for aiming error, which is a discrete distribution, rather than a white noise process, the interpretation of the input miss distance as an rms value taken over the ensemble implies the deterministic aiming error should also be interpreted as an rms value, there is no need to modify the existing deterministic description.

The signal can usually be interpreted as a deterministic time varying signal with a zero mean random signal superimposed on it, so restricting our consideration to zero mean random processes does not lose generality.

19.5 Frequency DomainCalculating the time domain system response to white noise is very tedious, as is calculating a linear system response to any arbitrary input in the time domain. It was only presented here to illustrate the point that it applies directly to the adjoint model of a time-varying system.

We found that the Laplace transform furnished a means of overcoming the difficulty of obtaining time domain solutions by mapping the time series functions into the s operator domain, where convolutions were converted to straightforward multiplications. We need a similar process to deal with noise signals.

19.5.1 The Fourier TransformWhen considering the distortion of a signal passed through a non-linear element we fitted the distorted signal with a series of complex exponentials, representing sine and cosine series. We found that the coefficient of the ith harmonic is given by:

a i=ω2 π∫0

2πω

y ( t ) e−ijωtdt

Now, consider the case where the fundamental frequency is so low, that the resulting time series signal is indistinguishable from d.c over the time scale of interest:

a (iδω )= δω2 π∫0

2πδω

y (t ) e− jiδωt dt

This can be thought of as a finely sampled continuous function of frequency:

a (iδω )=δ (iδω )g (ω )δω

i.e. the a(iδω) is the area of the infinitesmal strip under the function of frequency. When δω→0, the continous function in the frequency domain becomes:

g (ω )→ 12π∫0

∞

y ( t ) e− jωt dt

This is called a Fourier Transform, it resembles a Laplace Transform, with s=jω. Hence the frequency domain description of transfer functions is precisely the same as the Fourier transforms of their impulse responses.

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This has immediate practical application in finding the frequency response of the system. Subjecting the system to an impulse and then applying a Fourier transform to the resulting time series response, yields the frequency response. The experimenter testing a ship’s roll stabilizer need not become unpopular with the crew, after all.

19.5.2 Application to NoiseIn the time domain, the use of white noise implies infinitesmal time intervals and infinite amplitudes, so that in practice rather clumsy approximations need to be employed. When considering the Fourier series decomposition of the random signal, these difficulties disappear, and white noise becomes a very handy signal description.

This also reflects the way in which noise was originally measured. Rather than deriving the power spectral density by explicitly calculating the correlation coefficient over a set of samples, a narrow notch filter was scanned over the frequency range of interest. As with the time domain calculation, the uniform distribution of the noise energy implied that the intensity (power spectral density) obtained by this method should be insensitive to the notch width.

The frequency domain description of white noise is obtained from the Fourier transform of its autocorrelation function:

Q (ω )= 1T ∫−∞

∞

exp (− jωt 1)∫−T

2

T2

x ( t ) x ( t+t1) dtdt1

The time domain solution transforms this to adjoint time, producing a time dependent term common to all members of the ensemble. We represent the sequence as finite width pulses, rather than impulses:

x (t ) x (t+t1)=n , 0<t1<Δt=0 , otherwise

Q (ω)= 1T ∫−∞

∞

exp (− jωt 1)n Δtdt 1

Now t1 only adopts positive values. So taken over the ensemble:

Q (ω )=σ2∫0

∞

exp (− jωt1 )dt1

Q (ω)=σ2 (1−exp ( jωΔt ) )jω

≈σ2 Δt

This is the same as the time domain definition obtained by correlation, so white noise may be described by the same constant whether in the time or frequency domains.

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The frequency response of a constant coefficient system is obtained from the transfer function expressed as a Laplace transform by the substitution:

s→ jω

If the system is described by transfer function G(s), the amplitude of the response to a harmonic input at frequency ω is given by:

|G ( jω1 )|2=G ( jω )G (− jω )

The rms value taken over all frequencies is:

S= Δt2π ∫

−2 πΔt

2πΔt

G ( jω )WG (− jω )dω

As Δt→0.

The maximum frequency corresponds to the minimum sample period, so that the frequency domain and time domain descriptions are consistent, and the same signal energy is present in both. In the time domain we consider components which change in time in the range:

Δt<t s<T

With white noise we assume the energy is equally partitioned between time ranges, hence the variation of variance with sample interval.

The equivalent frequency domain description partitions the same energy equally over the corresponding range of frequencies:

2πT<ω< 2π

Δt

In both calculations we ignore the lower limit altogether, but there is an infinitesimal region about the origin which is usually ignored. What this means in practice is the sample period and observation duration must be such that this region contains insignificant energy in both integrals.

Integrating over the frequency range yields the same answer as calculating ρ(0). Too many texts appear to have forgotten why the 2π factor is included in the integral over the frequency domain.

So we expect from energy considerations:

S= ρ (0 )=E (x2)

Equation (73) uses the power spectral density of the white noise directly, so we do not need to mess about with finite width pulses which can only be interpreted as white noise inputs at the final stage of the calculation. Also, this is an inherently easier calculation than a correlation, but is restricted to time-invariant systems.

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As an example, consider the first order lag:

G (s )= 11+ τs

, G ( jω )= 11+ jωτ

Hence:

S= W2π ∫−∞

∞ dω1+ω2 τ2

= W2πτ

[ tan−1(∞ )− tan−1

(−∞ ) ]=W2 τ

=ρ (0 )

The same answer as was derived in the time domain is obtained much more readily.

19.6 Application to a Tracking SystemA sensor detects a distant target, and contains all the processing needed to follow the target within its field of view. If the target is distant, the amount of energy arriving at the sensor is expected to be small, so some magnification is necessary.

Typically, the higher the magnification, the narrow is the possible field of view, so if the target is moving it may be lost from the field of view.

The detector, and its associated signal processing, will usually generate noise as well as the desired signal.

If the target is a distant astronomical object, the tracking system may use knowledge of the rotation of the Earth to maintain the approximate sight line direction, but usually feedback control will be needed as well. In the military context, where no such cooperation is expected of the target, feedback is all that can be employed.

There are two basic design options for the feedback. The boresight error may be measured and fed back, using phase advance compensation to stabilise the loop, or sight line rate as well as boresight error may be estimated and fed back.

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Figure 19-102: Typical Tracking Loop

We shall ignore any a priori knowledge we may have, so that the tracker must work completely by feedback. Also we shall stick to the single input/single output loops that we have studied so far. A possible control loop is presented in fig Figure 19-102.

The detection, recognition and target loyalty require processing in the sensor which is expected to take a finite amount of time. The presence of the double integrator arising from the equation of motion implies that some phase advance compensation must be necessary. In addition, we assume the servo force takes a finite amount of time to build up.

We require the boresight error to be kept within the field of view when following a rotating sight line.

The open loop transfer function is:

G (s )= K1+T p s

1+ατs1+τs

11+T a s

Is2

The boresight error is:

ε ( s )=θD−θ=θD−( ε (s )+n )G (s )

Where θ is the sight line direction and the subscript D denotes a demanded quantity, n is the noise sequence. It follows that;

εn= G (s )1+G ( s )

From which the rms error may be calculated:

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ε 2= 12 π∫0

∞

( G ( jω )1+G ( jω ) )W ( G (− jω )

1+G (− jω ) )dωWhere W is the power spectral density of the measurement noise. Note, that in order to attenuate the noise G(s)→0.

The deterministic tracking error is:

εθD= 11+G (s )

So accurate tracking of the input requires G(s)→∞. There is clear a conflict between attenuating noise and at the same time accurately following a moving sight line.

The transfer function:

S= 11+G (s )

is known as the sensitivity, and:

T= G (s )1+G ( s)

is the transmission, or complementary sensitivity. These become important when we consider design methods based on closed loop behaviour.

20 Line of Sight Missile Guidance

20.1 IntroductionThe earliest designs of anti-air target missiles used some form of line of sight guidance. This was a natural development of anti-aircraft artillery, which unlike the homing guidance, covered earlier, does not require an expensive sensor to be expended with the missile. Unlike the homing system, the coefficients of the transfer functions are usually constant, so we may use this example to illustrate the ideas of the previous chapter.

We should expect a deterministic miss distance to arise because the line of sight is unlikely to be fixed, or even rotate at constant angular velocity if a target is flying a straight line trajectory at constant speed, offset from the launch point. However, for the moment, we shall restrict ourselves to the statistical miss distance which arises from the noise sources within the loop.

20.2 Line of Sight VariantsFrom the point of view of fundamental kinematics, line of sight guided missiles may be implemented in one of two basic forms, the oldest is called a beam rider, but later systems tended to use a form known as command to line of sight (CLOS). The reduced cost of modern seekers has rendered this technology in either form, more or less obsolete. The pacifist may therefore read this section with a

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clear conscience knowing that this example is chosen for its efficacy in explaining an academic point, rather than any practical value it might have in the construction of weapons of war.

20-103 Line of Sight Guidance Basic Options

Some authorities treat beam riders and CLOS as equivalent but there is an important difference, which ensures that CLOS usually has superior performance to beam riders. In both systems a tracker is located on the launch platform, which usually is either stationary or moving slowly compared with the speed of the target.

In a beam rider system, the tracker maintains its bore sight pointing at the target, and a guidance beam is transmitted, having a common bore sight with the tracker. The missile has rearwards facing sensors which serve to detect its position from the bore sight. How this is achieved is the subject of subsequent iterations of the system analysis.

In a command to line of sight system the target is again tracked with a sensor on the platform, but the same sensor also tracks the missile. The error between the target and missile positions is used to derive a guidance signal which is transmitted to the missile via a command link.

Superficially, these appear identical guidance philosophies, but a moment’s reflection will reveal that the CLOS system does not require the tracker bore sight to be exactly aligned with the line of sight to the target as the sensor can measure the difference in direction from the true sight line to the target and the missile sight line. In short, the tracking loop in the beam rider is connected in series with the guidance loop, whilst with CLOS the two are in parallel.

Another advantage of the CLOS loop is the ease with which the acceleration arising from the rotation of the beam may be added into the guidance command as a feed forward (open loop) signal, to improve the tracking behaviour of the guidance loop itself.

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20.3 AnalysisThe two guidance loops are presented in Figure 20-104. In both cases the system has been drawn in a form which represents its proper function, the input is the target behaviour and the output is the miss distance.

20-104 Comparison of Beam Rider and CLOS Loops

The principal difference is evident from these diagrams; the target position is fed directly into the guidance stage in the case of CLOS, whilst it is fed through the tracker response in the case of the beam rider. There are several noise sources in the loop, but for simplicity we have restricted our consideration to that originating from the measurement errors. For reasons of clarity, generality and security, the exact forms of the transfer functions G(s) and H(s) are not given.

In the form presented in Figure 20-104 the response of the miss distance to target behaviour can be found, but for the current analysis, we shall restrict ourselves to the effect of sensor noise on miss distance. Since the system is linear over the range of state variable magnitudes considered we may repeat the analysis for all the noise sources and find the total effect on miss distance by summing the variances due to each source and taking the square root.

For the moment, we shall just consider the noise in measuring the target position. This is associated with two-way path loss which is expected to be much greater than the one-way path loss of the beam rider, and the CLOS missile would usually carry some form of beacon, so that its signal strength is expected to be high.

In order to determine the effect on miss distance, we require the transfer functions from the noise source to the miss distance. In the CLOS case, this is fairly simple because the response of the tracker is irrelevant to miss distance, provided the target remains at all times within the field of view

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of the sensor. The relationship between the target tracking noise and miss distance is determined entirely by the guidance.

Denoting the miss distance ‘d’ and the tracker noise ‘n’, it is evident from Fig ??? that:

d=−H ( s ) (n+d )

Or:dn=−H (s )1+H (s )

The variance of the miss due to the tracker noise is found from the frequency domain noise integral:

σ d2= 1

2π ∫−∞∞

( H ( jω )1+H ( jω ) )( H (− jω )

1+H (− jω ) )N dωWhere N is the power spectral density of the tracker noise, and the symbol σ is used to denote the standard deviation in this context..

For the beam rider the transfer function includes both tracker and guidance response:

dn=( G ( s)

1+G (s ) )( H (s )1+H ( s ) )

For the same bandwidth guidance loop the beam rider response will be slower.

20.4 CommentsWe note that in order to respond to deterministic target behaviour, such as evasive manoeuvres, the system bandwidth must be high. However, increasing the bandwidth admits more sensor noise into the loop causing an increase in RMS miss distance. A compromise must be found which ultimately depends on the expected target behaviour and its signature at the wavelength at which the sensor operates.

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Needless to say, military targets are highly manoeuvrable and have the smallest signature possible at the wavelength the enemy sensor operates. If the target is un-manoeuvrable, slow and easy to see, the chances are it is a civilian vehicle, and not a legitimate military target at all.

Note that we do not describe missile systems according to the sensor type or any other constructional details (e.g. ‘wire-guided’, ‘heat seeking’), instead we start from the definition of the target characteristics, the platform characteristics, environment, available budget and state of the art constraints. Once these are defined it is possible to flow down to an appropriate set of potential design solutions. There is a worrying modern tendency to reverse this process, and try to build a system from pre-defined parts, rather like Lego, in the forlorn hope that the resulting collection of parts will form a system.

The astute reader will doubtless notice that the target noise is expected to be a function of the range because the energy received from the target is expected to diminish with increasing range. For this reason the assumption of constant power spectral density throughout an engagement is a bit fishy. The reality is, our noise integral is not an estimate of the variation of RMS miss over a particular fly out. Instead, it is the miss distance of an ensemble of fly outs with intercept occurring at a particular range. In fact, the result is equivalent to an adjoint run, but with a system which is self-adjoint. The analysis would be repeated for a range of nominal intercept ranges.

The limitation to small perturbations is not really a limitation, if it is understood that a system in which the miss distance were a large quantity would not merit extensive further analysis, but would be immediately dismissed as useless.

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21 Multiple Outputs

21.1 IntroductionThe compensator design methods presented so far have been for the classical case where the system proper function is to control a particular state, so we design the system to drive the error in the state of interest to zero. This is a single-input/single output (SISO) system. We should expect the plant to be designed such that functionally independent channels will be decoupled from each other, so more complex systems are expected to consist of a collection of SISO systems in close formation.

For the above reasons, SISO systems are by far the commonest, and since sensors are expensive compared with processing, it is not immediately obvious why we should spend money on additional sensors, unless there were clear benefits in doing so.

Adding sensors is expected to reduce the amount of phase advance which is typically required to stabilise a SISO system, and usually provides a means of increasing the closed loop bandwidth without running into noise and saturation problems.

Actually, most of the examples considered in earlier chapters do contain implicit multiple feedback loops, but they are inherent in the dynamics of the plant. The unstable gimbal mount of the gyro monorail is a gimbal feedback implemented mechanically. The weathercock stiffness and damping of a rocket arises from its configuration and its interaction with the airflow.

The way we treat a system with a single input, but multiple outputs is much the same as we accommodate the inherent feedback of the plant. As there is a single control input, there is a corresponding reference input consisting of the required value for the output. This means that once we have designed a compensator for the multi-output case, we can assess its robustness by generating Bode plots, harmonic loci or root loci for all loops.

Note that this is only possible because we have a phase reference in the form of a single reference input. In the multi-input multi-output (MIMO) case (i.e. a collection of cross-coupled SISO systems) no such phase reference exists, and the idea of a ‘phase margin’ becomes meaningless.

The problem arises that the dynamics of these additional artificial loops are not ‘given’, but a matter of design. The following sections suggest possible approaches.

21.2 Nested LoopsThe simplest (and quite common) case to consider is when the additional measurements control a loop which is nested completely within the main loop. This is illustrated by the sensor tracking loop which has some means of measuring the slew rate of the sight line (e.g. a rate gyro mounted on the antenna).

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Achieved acceleration

Rate feedback

Acceleration demand

22 2 nnn ssbas

22

2

2 nnn ssss

G1 kr


Figure 21-105: System with Nested Loops

Evidently, the innermost loop is designed first, and then the outer loop. This works because there is a direct causal relationship between velocity and position.

Figure 21-106 : More Typically, the Loops are not Nested – the Outputs are Caused by a Common Input

The idea can be extended to more general systems.

Taking the rocket autopilot example, we saw that the natural airframe response is usually lightly damped, so the first vice we will want to eliminate is the sustained weathercock oscillation. We can do this by applying body rate feedback.

Now it is evident that the body rate loop is not nested within the lateral acceleration loop, both are caused by the control deflection, and consequently the two are in parallel.

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2111

2 2 ssbas

Closed rate loop

basss

2

‘constraint’ transfer function

ka


By re-arranging the transfer functions we can get an apparent relationship between the lateral acceleration and the body rate. The transfer functions relating lateral acceleration to control defection, and yaw rate to control deflection are (from chapter ???)

f yζ=

Y ζ

ms2−

Y ζmN r

Cs−(Y vm N ζ

C−Y ζmN v

C )Us2−(Y vm +

N r

C )s+( N vUC +

Y vmN r

C )

rζ=

N ζ

Cs−(Y vm N ζ

C−Y ζ

mN v

C )s2−(Y v

m +N r

C )s+( N vUC +

Y v

mN r

C )These characterise the physics of the problem. We can contrive a relationship between the body rate and the lateral acceleration by dividing one transfer function by the other.

f yr=

Y ζ

ms2−

Y ζmN r

Cs−(Y vm N ζ

C−Y ζmN v

C )UN ζ

C s−(Y v

mN ζ

C −Y ζmN v

C )Using this relationship, we can draw the loop as if the body rate loop were nested within the lateral acceleration control loop:

Figure 21-107 : They may be Treated as Nested by Introducing a the Ratio of the Numerators as an Additional Transfer Function

Although the transfer function looks odd (the order of the numerator is greater than that of the denominator), it does yield useful insights into the plant dynamics. The constraint transfer function consists entirely of plant zeros, which are inherent in the plant dynamics. Unlike the plant poles,

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these cannot be moved about by feedback. Left half plane zeros may be cancelled by compensation, but why bother, when it is the right half plane zeros that are the source of so many woes?

If we want high manoeuvrability from the missile, it will be designed to be statically neutrally stable (i.e. Nv≈0), also the direct lift due to the fin deflection (Yζ) is expected to be small, as is the yaw damping (Nr). The constraint equation becomes:

f yr=

−(Y v

mN ζ

C )UN ζ

C s−(Y vm N ζ

C )= U

−( mY v )s+1The steady state centrifugal acceleration is Ur. What this equation means is that the achieved lateral acceleration lags behind the steady state value implicit in the body rate. The time constant of the lag is –(m/Yv), which is consequently known as the ‘incidence lag’.

In fact the numerator for the body rate transfer function cancels with the denominator of the constraint equation, so we design the lateral acceleration loop with the plant formed by closing the loop around the body rate.

Increasing the number of feedback states beyond two leads to issues of how the loops are to be nested.

21.3 Pole PlacementThe state space representation of a system with input is:

x=Ax+Bu

Where x is the state variable and u is the control input. For a single input system B will consist of a single column. The control will have least effect on the closed loop system if it affects only a single state directly, i.e. if it has only a single non-zero element.

With a control matrix (B) in this form, any feedback can only affect a single row of the system matrix.

If we modify a single row of the system matrix, then, in order to change all n coefficients of the characteristic equation, we must modify all n row elements. As each column corresponds to a state variable, it follows that if we want to place the closed loop poles anywhere in the Argand diagram, we must feedback all n state variables.

However, whilst it is obvious that we need the n feedback gains to modify n coefficients, it isn’t clear under what circumstances full state feedback will actually present a solution.

To illustrate this point, consider a system governed by the equations:

x1=a11 x1

x2=a21 x1+a22 x2+u

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Evidently, no matter what we do with the control u, we will not affect x1. If the system described by the first of the above equation is stable, this may not matter; we simply do not have complete control over the closed loop poles.

The example is easy to identify as a pair of disparate systems, with one exciting the other. More often than not the coupling of the control into the system is not so obvious, and the presence of states which are not modifiable by the available control cannot be determined by inspection.

This ability for the control to influence every state is, somewhat mischievously, called ‘controllability’, even though plant which cannot meet this criterion may indeed be controllable in the more usual sense.

We shall call this type of controllability, ‘state controllability’, to indicate that it is no more than an essential pre-requisite for pole-placement methods.

The following section presents the constraints on the system and control matrices required for this type of controllability.

21.4 Algebraic ControllabilityIf we consider the special case of a system represented as an nth order linear differential equation, it is represented in state space with a system matrix of the form:

A=(−cn−1 −cn−2 ⋯ −c1 −c0

1 0 ⋯ 0 0⋮0

⋮ ⋱0 ⋯

⋮ ⋮1 0

)Where the top row consists of the coefficients of the characteristic equation. The remaining rows consist of the (n-1)×(n-1) identity matrix, followed by an (n-1) element column vector of zeros. This is known as a companion form, for doubtless a good reason, which isn’t too clear from this context.

If the control matrix takes the form:

B=(b0⋮0)

It is evident that full state feedback of the form:

u=∑i=1

n

k i xi=KT x

Where the ki are gains, will modify each of the elements of the first row:

−cn−i'=−cn−i+bk i

Where the prime denotes the coefficient of the closed loop characteristic equation.

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In this special case, it is evident that each coefficient may be assigned arbitrary values, so that the poles may be placed anywhere we choose.

More generally, we need to find out whether a control exists which can move the system states from any value to any other value.

The governing equation is:

x=Ax+Bu

Differentiating:

x=A x+B u=A (Ax+Bu )+B u=A2x+ [AB B ] (uu)Repeating this process n times:

dn xd t n

=An x+ [An−1B ⋯ B ]( dn−1ud tn−1

⋮u

)In order to proceed we need to use a result known as the Cayley-Hamilton theorem.

We know that the eigenvalue matrix must satisfy the characteristic equation:

Λ n+cn−1 Λn−1⋯ I c0=0

The eigenvalue matrix is obtained from the system matrix by means of a similarity transform:

Λ=T−1 AT

Multiplying both sides by the eigenvalue matrix:

Λ2=(T−1 AT )T−1 AT

(Note: this assumes there are no repeated eigenvalues)

It follows that:

Λ2=T−1 A2T ,⇒ Λn=T−1 AnT


T−1 (An+cn−1 An−1⋯ c0 I )T=0

Since T is non-singular, it follows that the matrix satisfies its own characteristic equation.

It also follows from this result, that:

An=−cn−1 An−1−cn−2 A

n−2⋯ c0 I

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Differentiating the plant equation for the n+1 th time:

dn+1 xdtn+1

=(−cn−1 An−cn−2 A

n−1⋯ c0 A ) x+ [An−1B ⋯ B ]( dn−1udt n−1 −cn−1u

⋮u−c0u

)We can repeat this process an infinite number of times, substituting for An as appropriate. We can represent x as an infinite time series by repeatedly integrating the highest order derivative. However, it is impossible to find the control input corresponding to an arbitrary state history unless the matrix in square braces has an inverse.

The state controllability criterion reduces to the requirement that [An-1B An-2 B…B] has an inverse.

In our simple example:

A=(a11 0a12 a22), B=(01)

[AB B ]=( 0 0a22 1)

This has a zero row, so the inverse does not exist.

21.4.1 MonorailThe state controllability requirement does not necessarily determine whether or not a viable controller can be designed, but can provide insights into the system dynamics. For example, consider the gyro monorail. We wish to know whether it is state controllable, and also why.

The plant equation with no feedback, either passive or active, is:

J θ+H ϕ=M

Aϕ−H θ=Whϕ

The system state vector is ( θ θ ϕ ϕ )T , for which the system matrix is:

A=( 0 0 −HJ

0

1 0 0 0HA0

00

0 WhA

1 0) ,B=( 1J000 )

The controllability test matrix is:

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Γ=(0 −H2

A J2

−H2

AJ 2 0

0 1J

1J

0

−H3

A2 J 2+HWhA2 J

0

0 HAJ

HAJ 0

0 0)

The determinant evaluates to:

|Γ|=H2WhA3J2

It is actually the toppling moment which renders the system controllable. Physically, this means that the equilibrium position is determined by the balancing of the moment due to weight against the disturbance, with the inertial (gyroscopic) torques providing a zero mean correction to perturbations around the equilibrium.

21.4.2 Road VehicleA fairly crude representation of the directional stability of a road vehicle was derived in Chapter ???:

ψ=−k ((a2+b2)CU )ψ−k (a−bCU )v


+U )ψWhere ψ is the body axis direction and v is the slip velocity. U is the vehicle speed, a the distance of the front axle ahead of the centre of gravity and b the distance of the rear axle behind the cg.

C is the moment of inertia in the yaw plane, m the mass of the vehicle and k is the slip force coefficient.

The vehicle is steered by deflecting the front wheels with respect to the body. This generates a yawing moment; kaζ, where ζ is the steering deflection, in addition there will be a side force; kζ.

The equation of motion, with the steering control input becomes:

ψ=−k ((a2+b2)CU )ψ−k (a−bCU )v+kaC ζ


+U )ψ+km ζ

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If the steady state slip at the centre of gravity where zero, the body would point in the direction of motion.

Re-arranging the plant equations with v=ψ= v=0:

( a2+b2

CU )ψ+ aC ζ=0

( k (a−b )mU

+U )ψ+ km ζ=0

Since neither the steering deflection, nor the yaw rate, will in general be zero, so this is only possible if the matrix of coefficients is singular:

| a2+b2

CUaC

( k (a−b )mU

+U ) km|=0

This corresponds to a speed:

U 2= (a+b )( ba )( km )At this speed the vehicle neither oversteers nor understeers, but from the point of view of handling the steering is perfect. It is usually preferable for the car to oversteer than understeer, so this speed ought to be low. This implies the centre of gravity should be nearer the rear axle than the front, which conflicts with the basic directional stability requirement.

The controllability test matrix determinant is:

|Γ|=U 2−( km ) (a+b )(ba− Cma2 )

At this speed using feedback gives us no control over the frequency of the associated mode, but we can adjust the damping. This is hardly a cause of concern. Cars have been built for years oblivious of the fact that there is a speed at which they are apparently ‘uncontrollable’.

We obtain a further interesting result if we replace the slip velocity with the actual velocity direction:

γ=ψ+ vU

γ=ψ−k ( 2mU 2 ) (γ−ψ )U−(k (a−b )

mU2 +1)ψ+ kmU ζ

Or: γ=−( 2kmU )γ−( k (a−b )mU 2 )ψ+( 2kmU )ψ+ k

mU ζ

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ψ=−k ( a−bC ) γ−k ( a2+b2

CU )ψ+k ( a−bC )ψ+ kaC ζThe system matrix then becomes:

A=(−2kmU

k (a−b )mU 2

2kmU

−k ( a−bC ) −k ( a2+b2

CU ) k (a−bC )0 1 0

)The determinant of this is zero, which implies that the equilibrium yaw orientation and heading are not calculable from the control input. This is also the constant term in the characteristic equation, so the system has a pole at the origin. This is consistent with experience; we turn the wheel to change direction, and centre it when the new heading is reached; this neutral mode is not a cause for concern.

When we find neutrally stable modes, we need to interpret them in terms of physical reality before we can decide whether we should worry about them.

21.5 Comment on Pole-placementFor low order systems, we can evaluate the closed loop characteristic equation directly. For higher order systems there exist standard numerical algorithms (e.g. the Bass-Gura algorithm) to calculate the gains needed to place the closed loop poles in their specified positions.

This raises the question of where the closed loop poles should be placed. We cannot answer this question without knowledge of the proper function of the plant. Usually we seek to separate functionally independent modes. Imagine trying to use a mouse whose interval between the key presses of a double-click was comparable to the time taken to move it on to an icon.

If, for some reason, such understanding of the plant dynamics is unavailable, a Butterworth pole pattern might be considered.

Since we expect our states to be limited by saturation in practice, we might determine our gains using a method which tries to keep the states within their amplitude bounds. This is the idea behind linear/quadratic, so-called ‘optimal’ control.

Full state feedback appears to allow us to place the closed loop poles anywhere we like on the Argand diagram. This seems too good to be true, and of course, it is. We have the same constraints on pole locations as we did with the root locus method.

There is nothing in the method which reveals any requirement for integral feedback needed to track time varying inputs. Integrators must be anticipated as necessary and included in the plant equations before pole placement can be applied.

We can put the poles where we like, but what about the zeros?

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In general, we are interested in a outputs from the plant which are linear combinations of the states. For this reason we distinguish the states from the outputs:

y=Cx+Du

Where y is referred to as the plant output, it consists of the vector of quantities which are available for measurement. In the previous section we have assumed C to be the identity matrix.

The matrix D is included to cover cases such as the missile autopilot, which has a direct link from fin deflection to lateral acceleration.

The structure of C and D determine the system zeros. Consider a system in companion form:

A=(−cn−1 −cn−2 ⋯ −c1 −c0

1 0 ⋯ 0 0⋮0

⋮ ⋱0 ⋯

⋮ ⋮1 0

)(any system matrix can be transformed into this form). The output tracking the reference input is expected to include the nth state variable.

If the full state feedback, as is needed for pole placement is applied, C becomes the identity matrix and D=0. The transfer function relating input to output will feed back all derivatives of the nth variable, so that the order of the numerator will be one less than that of the denominator in the closed loop transfer function:

GC (s )=bn−1 s

n−1+⋯sn+cn−1 s

n−1+⋯

At high frequency, this approximates an integrator:

GC ( s )s→∞

→bn−1

s

The high frequency roll-off is 20dB per decade, whilst for an nth order system we should expect it to be n times steeper. From a noise perspective, this is not a good feature, because we wish to exclude all the noise above our design bandwidth. This can only be achieved with a sharp cut off.

Quite apart from the poor high frequency roll-off, it is usually impossible to measure all the states of a system. The quantities which are available for measurement are the outputs y. These are related to the actual state variables via matrix C, which is determined by the plant dynamics. The elements of C and D determine the system zeros, which are inherent in the plant. Unlike the poles, we cannot place the zeros anywhere we like by applying feedback.

Usually, not all the states are available for feedback. However, we found that systems with many states but a single measured output could be made to work, so presumably having more than one output ought to improve matters. We can still apply full state feedback if we can estimate all the states from the outputs.

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21.6 The Luenberger ObserverSince, in order to design the feedback, we must have a set of differential equations describing the system, it is reasonable to use the same equations to estimate the state of the plant. The astute reader will see the flaw in this reasoning, if we knew the dynamics of the plant with sufficient precision that we can estimate its states from a model of the plant, we would not need closed loop control in the first place.

We correct the model estimates by feeding the difference between the measurements and the estimated values of the measurements. Let the observer take the form:

˙x=E x+Fu+K ( y−C x )

The circumflex (hat) is used to indicate that it is an estimate of the corresponding state variable (read it as ‘x-hat’). E is an n×n matrix F is n×1 and K is n×m, where m is the number of measurements.

The plant equation is:

x=Ax+Bu

Evidently, E=A and F=B.

A system model, fed with the error between measurements and measurement estimates, is called an observer of the system.

The closed loop system equation for the observer is:

˙x=( A−KC ) x+Bu+Ky

Comparing this with the full state feedback controller:

x=( A+BG ) x

Where G is the gain matrix.

Consider the transpose of the homogenous part of the observer:

˙xT= xT (AT−CT K )

This is similar in form to that of the plant with full state feedback. In the latter case we derived a controllability condition on the A and B matrices, this problem is analogous. Following a similar reasoning to the derivation of the controllability criterion, we can obtain an ‘observability’ condition:

ΓO=[CT ATCT ⋯ ( An−1 )TCT ]In the case of a single input system this is n×n and essentially determines whether the full state can be estimated from the one input. More generally, it will be n×(nm), so the criterion reduces to the requirement for ΓO

T to have at least n independent rows, or in matrix jargon; the observability matrix must have full rank.

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As with the controllability criterion, failure to meet this criterion doesn’t mean the system is necessarily unsatisfactory – it merely means we can’t construct a full state observer. If the associated mode is well behaved, who cares?

Consider the simple case:

u=0

x=u

y=u

Or: A=(0 01 0) ,C= (1 0 )

Hence: ΓO=(1 00 0)

This is singular. It is impossible to estimate position from a measurement of velocity, as should be intuitively obvious.

The observer is a feedback system in its own right, using measurements to correct the state estimates rather than to control the states themselves. For this reason the observer and the controller are often called ‘dual’ problems.

The gain matrix K, for the observer may be found by pole placement methods, as in the case of the controller. However, we noticed that compensator design required poles well over to the left of the Argand diagram, implying that performance would be limited by excessive noise bandwidth. It is possible to design the observer such that the variance in our state estimate is a minimum, and this approach leads to the Kalman filter.

If the observer is designed using pole placement, it is usually called a Luneberger Observer.

21.7 The Separation TheoremA system which uses full state feedback together with an observer to estimate the states is governed by the following equations:

x=Ax+BG x

˙x=( A−KC+BG ) x+KCx

The error in the estimates is:

ε=x− x

The equations become:

x=( A+BG ) x−BGε

ε=( A−KC ) ε

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The poles of the observer equation are not affected by the controller poles, neither are the controller poles influenced by the observer poles. The observer gains may be calculated independently of the controller gains. This result is known as the separation theorem. It permits the observer to be designed independently of the controller.

The separation theorem relies on the plant equations used in the observer matching the actual plant equations. Usually, however, the plant equations are not known, or they at least constitute a gross approximation to physical reality.

Assume the system, control and output matrices are in error, the observer becomes:

˙x=( [A+ΔA ]+ [B+ΔB ]G−K [C+ ΔC ]) x+KCx

The error equation becomes:

ε=( [ A+Δ A ]+ ΔBG−K [C+ΔC ] ) ε−( ΔA+ΔBG−ΔC ) x

Where the Δ denotes an uncertainty. We can apply the separation principle in the initial design, but it is evident that some adjustment may be necessary when plant uncertainty is taken into account.

21.8 Reduced Order ObserversIt seems a bit odd to design an estimator for those states which are actually measured. In cases where the measurement noise is high, this is actually a reasonable approach. However, more typically we are better off using the estimator only for states which are not available for measurement.

We re-order the state vector, and by implication, the system and control matrices, so that it is partitioned between the states that are measured and the states that are estimated. There will be a corresponding partitioning of the measurement matrix.

C=⌈C1∨0⌉

Where C1 has an inverse.

The system and control matrices will be similarly partitioned:

A=[ A11

A21

¿¿A12

A22 ]B=[ B1

B2 ]The full state observer is:

˙x=A x+Bu+K ( y−C x )

˙x2=A22 x2+A21C1−1 y+B2u

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The poles of this observer are the same as the eigenvalues of matrix A22. In general, these will not be favourably located. However, we should have realised by now, that in order to move poles to more favourable positions, we must include zeros. These are implemented by feeding the input both through a dynamic process and directly to the output, and combining the two. In other words we estimate the unmeasured states using a compensator of the form:

x2=Ly+z=LC1 x1+z

z=Fz+My+Nu

Differentiating the first of this pair:

˙x2=LC1 ( A11C1−1 y+A12 x2+B1u )+F ( x2−Ly )+My+Nu

Or: ˙x2=F x2+LC1 A12 x2+(LC1 A11C 1−1−FL+M ) y+ (LC1B1+N )u

The error in the observer estimates is:

ε= x2− ˙x2=Fε+ (A22−F−LC 1 A12) x2+(( A21−LC1 A11 )C1−1−FL+M ) y+(B2−LC1B1−N )u

The observer poles are the eigenvalues of F. In order to ensure these are distinct from the controller poles, the remaining matrices must be chosen such that:

A22−F−LC1 A12=0

M=FL−( A21−LC1 A11 )C1−1

N=B2−LC1B1

With these values, the separation theorem applies, with the usual proviso that the separation theorem never really applies, but this approach is not a bad method of deriving a reasonable compensator quickly.

The limitations of the separation theorem only represent a problem if we delude ourselves that control system design is a once through ‘sausage machine’ process requiring no broader understanding of the plant behaviour, than is needed to handle-crank algorithms devised by wiser minds. Such an approach betrays spectacular ignorance of the engineering enterprise.

21.9 Concluding CommentsOn re-introducing the state space representation, with its matrix algebra rather than complex numbers and operator notation, we have also subtly changed the approach from predicting how modifications to the open loop affect the closed loop, to designing the closed loop directly.

When we come to consider multi-input/multi-output systems, it becomes evident that extrapolating from SISO methods is not particularly helpful, and the most successful methods tend to work directly with the design requirements of the closed loop plant.

If good reason is required to justify the expense of additional sensors for multi-state feedback, a positively watertight case must be presented for introducing the expense, weight, volume and

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power requirements of additional servos. For this reason MIMO systems tend to be rare; found more frequently within the hallowed cloisters of academia, rather than in the world of practical technology.

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22 Respecting System Limits – Optimal Control

22.1 IntroductionIntroducing artificial feedback, if adjusting system parameters does not yield an adequate solution, has introduced the problem that constraints on compensators beyond ensuring stability, do not emerge directly from the mathematics. In practice, the designer is limited to off-the shelf sensors and servos and is restricted to seeing the best he/she can do with what is available. Once these hardware constraints are introduced, this problem largely disappears.

In order to decide which bought-in items are most critical to achieving system performance the loop is designed with incremental hypothetical improvements in component noise, saturation and bandwidth to understand the sensitivity of overall system performance to component characteristics.

However, we are not always in the position of seeking the best that can be achieved with ‘given’ components. Sometimes the bandwidth requirement flows down from overall system requirement, in which case, such things as servo requirements are the outputs from the study. As an example, the miss distance of a guided weapon against a specified target depends critically on the guidance loop bandwidth, which flows down from the target signature and behaviour. There is little point considering noise or saturation constraints if the basic kinematics are inadequate. In such circumstances, where bandwidth is driven from fundamental system function, the output of the design process consists of servo and sensor requirements specifications.

Now, contrary to a few user requirements I have actually come across, we cannot specify both system performance and subsystem constraints. We can either flow down component requirements from system requirements, or we can see what we can do with what is available. Sometimes one component shortfall can be compensated by another’s exceeding expectations. However, one must apply the methods presented here to decide if such trade-offs are possible.

When flowing down requirements, it is quite easy to characterise system components as bandwidths, but as we have seen, saturation and noise prevent high bandwidth from being achieved. In order to flow down to noise levels and saturations, our frequency domain and impulse response methods need to be supplemented with methods based on dynamic range constraints. If both methods yield similar results, we can have reasonable confidence in our design.

The methods presented in this chapter complement classical methods, and were once briefly considered the ultimate approach to control system design.

If we dispense with the notion that any single method will meet all our needs and make use of all the tools at our disposal, at least we will stand a chance of understanding why our controller doesn’t work, which will enable us to do something about it.

22.2 Optimal Control In every walk in life, there is always some naive zealot who insists on ‘the best’, blissfully unaware of the fact that what constitutes ‘the best’ of anything is very much a matter of debate, it all boils down

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to what our criteria for quality are. Most of the time this juvenile pre-occupation with superlatives tends to be the domain of the snob, ascribing fantastic value to artefacts of minimal, if not zero utility, beyond appealing to even bigger fools.

The ultimate performance of time varying systems is expected to be stability limited, hence the emphasis on stability throughout this book. Optimisation algorithms tend to be notable for their extremely poor numerical stability, and usually need to be used with extreme caution by an individual familiar with both the problem domain, and the details of the optimisation algorithm. All too often they are presented as turnkey applications with an attractive user interface, which allows the naive user to venture freely where angels fear to tread. Some codes are merely stupidity amplifiers.

The definition of the ‘best’ regulator in the context of what is known as ‘optimal’ control is one which minimises a quadratic performance index:

J=x (t f )T Sx ( x f )+∫

0

t f

xTQx+uT Rudt

Where Q is a weighting matrix which can be used to scale the states, or combinations of states, by their saturation values, and R is similarly a weighting matrix for the controls. This does not mean that the method ensures that saturation limits will not be reached, but renders saturation equally likely on all states and controls. Q, R and S are n×n symmetric matrices, where n is the number of states.

The objective is to choose a control (u) which minimises the least squares error in the state. This is implicitly a regulator, rather than a tracking problem.

The performance index is open to a range of criticisms, which we shall not go into here, except to mention a result which has crept into the folklore of proportional navigation missile guidance. Applying the quadratic performance index to the terminal homing problem yields a navigation constant of 3 as the ‘optimum’.

However, the only reason for choosing a quadratic performance index is to make the problem mathematically tractable. If we based the performance index on n rather than 2, the optimal control solution would yield a navigation constant of:

N=2n−1n−1

When n=2, this reduces to the much-quoted value of 3, but as n→∞, this reduces to the value 2, which as we have seen, is the limiting value of navigation constant.

Using n→∞, is more appropriate for the frequency domain, where we wish to suppress resonance peaks, rather than to average out errors over the entire frequency range. This is the basic idea behind H-infinity optimisation (a name strongly indicative of academic in-breeding), which we shall consider presently.

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22.3 Optimisation Background

22.3.1 Principle of OptimalityPractically all optimisation problems consist of finding a path in the solution space which minimises or maximises some performance index. Usually we assume only one such path exists, but particularly if there is uncertainty in the parameters, this may not be the case. The principle of optimality states that if an optimal path exists between a certain intermediate point and the end point, the optimal path from any other point must pass through this same intermediate point.

This is illustrated with the simple network consisting of nodes which might represent events on a PERT chart, or way points on a routing problem, and connecting arcs, which could represent distances, activity durations, survival probabilities, or the myriad of other data relevant to problems which can be represented as networks.

The object of the exercise is to find a path through the network, which maximises or minimises some cost function.

Figure 22-108: Illustration of Principle of Optimality

A typical minimising problem might be to find the critical path through a complex plan. In our example we need to find the path from A to B which minimises the total cost. The cost of traversing each arc is marked on the network. We could perform an exhaustive search of all possible routes, but the number of nodes does not need become very large before the number of potential routes becomes prohibitive.

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In order to reduce our search space we apply the principle of optimality by considering first the potential paths which lead to B.

If we zoom in on the right hand edge, we can examine the paths from the nodes closest to the target node B, e.g. node x. There are two potential paths; the red one with a cost of 16, and the blue with a cost of 18. To advance from x we follow the red path. Evidently, when trying out paths up to node x, we need advance no further, we just use this result. Similarly we can derive costs of (18, 29) and (20,32) for the direct and indirect (via node x) paths from the nodes above and below x. Similarly, we no longer need to consider the paths in the network to the right of these nodes as we have the minima 18,16 and 20 as costs to completion for the three start nodes of all paths to the right.

Figure 22-109 : Final Stage of the Optimal Path

This process may be repeated, only considering the path as far as the set of nodes for which the minimum cost functions are known.

In this example, applying the optimality principle does not appear to save much effort, but the benefit increases with the complexity of the network. For obvious reasons, an illustrative example must be simple in order to explain anything.

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22.3.2 Application to the Control ProblemThe principle of optimality implies that we must work backwards from the end state in order to derive candidate control schemes which minimise the weighted sum square of control and state from an arbitrary start condition.

The idea may be illustrated using a scalar example, as the usual matrix presentation is probably a distraction to most readers, it certainly causes writer’s cramp.

The system to be controlled has a single state and a single control input:

x=ax+bu

Where a and b are constants (actually, they can be time-varying). The performance index is:

J=s x (t f )2+∫

0

t f

q x2+r u2dt

We shall assume that control is applied at discrete intervals of duration (Δt) and is maintained at a constant level throughout the interval. The performance index J is calculated over a fixed time interval (tf=NΔt).

Using the principle of optimality, we work backwards from the end state. We need to define cost to completion associated with the control strategy:

J¿ (t f−t )=s x (t f )2+∫

t

t f

q x2+ru2dt

We have:

J¿ (0 )=s x (t f )2

Stepping back a single control interval:

J¿ (∆t )=s x (t f )2+ ∫

t f−∆t

t f

q x2+r uN−12dt

From the plant equation:

x=ax+bu

Solving:

xN=xN−1ea∆ t+ b

a(ea∆t−1 )uN−1

If the time interval is small compared with 1/a, this becomes:

xN=(1+a ∆t ) xN−1+b ∆t uN−1

The cost to completion at time tf – Δt is:

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J¿ (∆t )=s (x N )2+∫

0

∆t

q (1+at )2 x N−12 +2qbt (1−at ) xN−1uN−1+ (b2 t2+r )uN−1

2 dt

The integral to first order in Δt becomes:

∫0

∆t

q (1+at )2 xN−12 +2qbt (1−at ) xN−1uN−1+(b2 t 2+r )uN−1

2 dt ≈ (q x N−12 +ruN−1

2 )∆ t

The cost to completion is:

J¿ (∆t )=s ( ( (1+a∆ t ) x N−1+b∆ t uN−1) )2+q xN−1

2 ∆ t+r uN−12 ∆ t

This is a minimum if the control is chosen such that:

∂J ¿ (∆ t )∂uN−1

=0

Or:

2 sb ∆ t ( (1+a ∆ t ) xN−1+b ∆ t uN−1 )+2 ruN−1∆ t=0

This yields the result: uN−1=brs xN−1

The ‘optimal’ control is linear state feedback. Also, the second differential is positive, so this yields a minimum cost function. Both these results are a consequence of the quadratic form of the performance index, which was chosen for convenience, not because of its relevance to the problem.

The problem reduces to finding a gain k such that the performance index:

J (t f )=s x (t f )2+∫

0

tf

q x2+ rb2

( x−ax )2dt

Is a minimum.

The time series solution for x is:

x=x (0 ) e( (a+bk ) t )

If the system is stable, the final value of the state is expected to be small if the final time is taken as a multiple of the settling time. The performance index is then expected to be dominated by the integral.


x (0 )2∫0

tf

(q+r k2 )e (2 (a+bk ) t )dt=x (0 )2(q+r k2 )2 (a+bk )

[e2 (a+bk ) t−1 ]

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It is difficult to see how the performance index could be minimised if a+bk were positive, so it is evident that the exponent must be negative. It follows that as the final time tends to infinity, the performance index tends to:

J (∞ )→−x (0 )2(q+r k2 )2 (a+bk )

Differentiating with respect to k to find the gain to minimise J(∞):

dJdk=0=x (0 )2 4 rk

(a+bk )−2b (q+r k2 )4 (a+bk )2

=x (0 )2 2rka−bq+br k2

2 (a+bk )2

The optimal gain is given by:

2 rka−bq+br k 2=0

For consistency with the rest of the optimal control literature, and the form of the feedback control derived in the previous section, we shall introduce a variable p defined as:

k=−pbr

The optimal gain is given, in terms of this parameter, by:

2 pa+q− p2

r=0

The derivation for an nth order system, is more cumbersome on account of the constraints of matrix algebra. It may teach us a lot about matrix algebra and constrained optimisation, but very little about control. The matrix equation is:

PA+AT P+Q−PBR−1BT P=0

This is sometimes called an algebraic Riccati equation. The first three terms are recognisable as Liapunov’s Equation, which we have already met. The fourth term is quadratic, so is expected to be at least positive semi-definite. We require a positive definite solution for P for the system to be stable.

The gain matrix is given by:

K=−R−1BT P

The closed loop system equation becomes, with this feedback:

x=( A+BK ) x

This form of ‘optimal’ control is called ‘LQ’ for Linear/quadratic, indicating that the plant equations are linear, whilst the performance index is quadratic.

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22.4 Gyro MonorailUsually the Riccati equation must be solved numerically for specific cases, so several solutions are needed to provide the insight into how the system parameters relate to the plant behaviour. With modern computers, hundreds of solutions may be generated quickly, and this to some extent offsets the disadvantage of a method which has all the hallmarks of a sausage machine to be cranked by unskilled labour. Used sensibly, however, this method has much to recommend it.

One of the few cases for which the Riccati equation has an analytical solution is the gyroscopic monorail. This, and the fact that it exhibits both right half plane poles and zeros, is the principal reason why this engineering curiosity features so prominently in this book.

We shall make the assumption that the gyro momentum is large, so that the nutation and precession modes are widely separated. We shall use the LQ method to obtain values for the gimbal feedback and roll angle feedback gains.

The plant equations governing the precession are:

H ϕ=M

−H θ=Whϕ

To recap; H is the gyro angular momentum, W the monorail weight, h the height of the centre of gravity above the rail and M is the controlling moment applied to the gimbal. The roll angle is φ, and the gimbal deflection is θ.

The system matrix is, therefore:

A=( 0 0−WhH

0) The control matrix is:

B=(10)We characterise the servo moment by a maximum value Mmax, so that the control weighting matrix becomes:

R= 1Mmax

2

We scale the roll angle and gimbal deflection by φm and θm respectively:

Q=12 (

1ϕm

2 0

0 1θm

2 )387


The factor 2 is included so that:

|xTQx| 1

When the states are near their scaled values. The control weighting is similarly scaled if there is more than one control input.

The algebraic Riccati equation is:

( p11 p12p12 p22)( 0 0

−WhH

0)+(0 −WhH

0 0 )( p11 p12

p12 p22)+12 (

1ϕm

2 0

0 1θm2 )+P(10)(Mmax

2 ) (1 0 )P=0

This reduces to three simultaneous quadratic equations.

2WhHp12−

12 ϕm

2 +Mmax2 p11

2 =0

WhHp22+Mmax

2 p11 p12=0

Mmax2 p12

2 − 12θm

2 =0

Which may be solved for the elements of P:

p12=−1

√2θmMmax

p11=1

√2ϕmMmax √1+ 2√2ϕm2

θmMmax

WhH

p22=HWh

12ϕmθm √1+ 2√2ϕm2

θmMmax

WhH

Actually, p22 is redundant. The resulting feedback gains are:

M ϕ=−Mmax

√2ϕm √1+ 2√2ϕm2

θmMmax

WhH

For the roll angle feedback.

M θ=Mmax

√2θm

For the gimbal angle feedback.

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This isn’t a bad approach to the monorail problem as it is evident that gimbal deflection must be kept small. In fact, there will always be an amplitude limit on the states. If not set by physical saturations, they will be determined by the effects of the small perturbation assumptions implicit in the plant equations.

It should be noted that this example is very much the exception, and most real systems require a numerical solution. Indeed, the emergence of commercial analysis packages has been a mixed blessing, as experience in using bespoke software has become valued over the ability to do anything useful with it.

Given that neither managers nor the HR departments know the first thing about the subject, this placebo approach of valuing software packages over skill has become almost universal. Why technically naive managers are given responsibility over cutting edge technology remains a mystery.

22.5 The Optimal ObserverThe linear-quadratic approach is merely pole placement with a particularly fancy algorithm, and suffers from the same 20dB per decade roll-off at high frequency. Like the pole placement method, it relies on full state feedback, so the separation theorem must be invoked to permit an observer to be constructed to estimate those states which are not measured.

As there is an algorithm based on signal amplitude which appears to generate reasonable first-cut designs, the question arises as to whether a similar approach is possible with respect to the observer design.

The controller appeared to be limited by saturation, so the LQ approach sought to keep states and controls within bounds. The corresponding observer limitation, which prevents us from placing observer poles too far over to the left, appears to be noise.

If we just introduced noise originating from sensors, the problem cannot be formulated as an observer; we would need to trace the noise through to, for example, the servo, and determine levels which give rise to excessive power consumption, vibration or some other system limitation.

What follows is a derivation of a type of Kalman filter, limited in scope to the problems that interest us, i.e. linear time-invariant systems. If our interest were in signal processing, we would adopt a more general approach.

We assume the plant equations contain uncertainty, not in their parameters or missing dynamics, but in the form of random disturbances. Examples might be wind gusts or vibration. The plant equations are modified to take this random excitation into account:

x=Ax+Bu+Fv

Where v is a vector of zero mean white noise processes.

Also, and less controversially, the measurements are assumed corrupted by noise:

y=Cx+w

Where w is also a vector of zero mean white noise processes.

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Our objective is to use our knowledge of the disturbance and noise statistics to design an observer which produces an estimate with the least variance.

22.5.1 Covariance EquationIn order to proceed, we must extend our scalar expression for the response of a system to white noise to the vector case. To simplify matters we shall ignore the control input and consider the response of the simpler system:

x=Ax+Fv

The white noise input is tricky to define as an explicit function of time, so we shall treat it as a discrete random sequence which is updated at a sampling interval of Δt, which is short.

compared with the time constants of the system we are dealing with. A single sample has a vector of values v(i) at time iΔt. The time series solution is:

x=e At x (0 )+∆ t [e At Fv (0 )+e A ( t−∆t ) Fv (1 )+⋯+e A (t−k ∆ t )Fv (k ) ]

In the vector case, the scalar variance is replaced with a covariance matrix, the diagonal terms of which are the component variances, whilst the off-diagonal terms account for correlation between the components.

Figure 22-110 : White Noise Sequence

It is given by:

X=E (x xT )

Where E is the expected value, i.e. the value found by averaging over a large ensemble of cases.

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The covariance is:

X=e At x (0 ) x (0 )T [eAt ]T+(∆ t )2 [eAt Fv (0 ) v (0 )TFT [e At ]T+eA (t−∆t )Fv (1 ) v (1)T FT [ eA (t−∆t ) ]T⋯ ]All the terms in x(0)v(i)T are zero, when taken over the sample, because v has zero mean.

Since the noise is uncorrelated between samples v(i)v(j)T=0, unless i=j. Also, the disturbance is a stationary process, so that:

E (v (i ) v (i )T )=V ∀ i

X=e At X (0 ) [eAt ]T+ (∆ t )2∑0

k

e A ( t−i ∆ t )FV FT [∑0k

e A (t−i ∆ t )]T

As Δt→0

X→eAt X (0 ) [e At ]T+∫0

t

eA (t− τ ) dτ FV FT [∫0t

eA (t−τ )dτ ]T

X=e At X (0 ) [eAt ]T+A−1 eAt FV FT [ A−1e At ]T

Differentiating yields an expression for the time evolution of the state covariance:

X=AX+X AT+FV FT

Assuming A is stable, this converges on to a steady state solution:

AX+X AT+FV FT=0

This correlation equation bears a strong resemblance to the Liapunov equation used to test stability.

22.5.2 The ObserverThe observer structure is the same as for a Luneberger observer:

˙x=A x+Bu+K ( y−C x )

The error in the observer estimate is:

e= x− ˙x=Ae+Fv+KCe−Kw

Or: e=( A+KC ) e+(Fv−Kw )

This takes the form of a homogenous equation fed with white noise. The white noise source is a combination of disturbance and measurement noise.

q=Fv−Kw

From which: q qT=FvvT FT−Fv wT K T−Kw vT FT+KwwT KT

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We would not normally expect the disturbance noise to be correlated with the measurement noise, so the expected value of the observer noise input is:

E (q qT )=FV FT+KW KT

Where W is the measurement noise covariance matrix.

The steady state covariance equation for the observer becomes:

( A+KC )X+X (A+KC )T+FV FT+KW KT=0

In order to get some idea of the form of the solution, let us consider the single state case, in which the matrices become scalars.

2 (a+kc ) x+ f 2 v+k2w=0

Or x=−( f 2 v+k2w )2 (a+kc )

, dxdk=

4kw (a+kc )−2c ( f 2 v+k 2w )4 (a+kc )2

=0

This has a turning point where:

2kwa+k2 cw−c f 2 v=0

For consistency with the Kalman filter literature, let;

k=−cwp

The minimum variance gain is found from:

2 pa+ f 2 v− p2 c2

w=0

The matrix equivalent of this expression is an algebraic Riccati equation, similar to that derived for LQ control:

AP+P AT+FV FT−PCTW−1CP=0

Since the gains are calculated from a similar equation, the observer problem is often called a ‘dual’ to the control problem.

The gain matrix becomes:

K=−PCTW−1

We call this a minimum variance filter, rather than a Kalman filter, to indicate that Kalman filtering is a major area of study in its own right, which we have only touched on. All we have done is presented the fundamental ideas behind the Kalman filter, without, I hope, obscuring the principles with gobbledegook.

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22.5.3 Tracker ExampleAnalytical solutions of the algebraic Riccati equation for the observer are as rare as they are for the controller. We are restricted to specific numerical solutions, waving the Kalman magic wand, and abracadabra - we have ‘the best’ observer.

We shall consider the problem of tracking a faint object by means of a steerable antenna. The antenna is subjected to wind gusts, which usually have a defined spectrum, but we shall assume a Gaussian white noise process. The tracker specification is expected to be based on a requirement to track a target of known brightness to a specified accuracy at a particular range, whilst minimising the power and antenna size and weight. In such circumstances, we might expect performance to be noise limited.

We shall assume only the boresight error is available as a measurement.

The system equation is:

A=(0 01 0)

V=σG2

F=( 1J0 )C=(0 1 )

W=σm2

Where σ denotes standard deviation, subscript G refers to the gust torque, J is the moment of inertia of the antenna, and subscript m refers to the measurement noise.

Evaluating the terms in the algebraic Riccati equation.

AP=( 0 0p11 p12) , P AT=(AP )T=(0 p11

0 p12) , FV FT=( σG2

J20

0 0)And:

PCTW−1CP= 1σ m

2 ( p122 p12 p22

p12 p22 p222 )

The Riccati equation becomes:

393


(σ G

2

J 2−1σm

2 p122 p11−

1σm

2 p12 p22

p11−1σm

2 p12 p22 2 p12−1σm

2 p222 )=0

Solving:

p12=√σG2 σm2J

, p22=√2σm2 p12 , p11=p12 p22

σm2

The gain matrix is:

K=( 1J √ σG2σm2

√ 2J ( σG

2

σm2 )

14 )

And p11 is redundant.

The gain matrix contains the ratio of the variances. This is a general feature of the solution, so white noise intensities could be used to characterise the noise sources, instead of variances, without affecting the result.

22.5.4 Alpha beta Filter‘Alpha beta’ and ‘alpha beta gamma’ filters are observers based on the assumption that target motion is adequately represented as constant velocity, or constant acceleration. It is assumed that position is measured, differenced with the filter estimate of position and fed back via gains, usually denoted α, to the position estimator , β/Δt to the velocity estimator, and 2γ/Δt2 to the acceleration estimator. Here Δt is the sampling period. Values of α, β and γ are usually found by trial and error using a system model.

22.6 CommentThe gain matrix indicates that as the measurement noise increases, the observer gains reduce, and the observer pays more attention to its internal model of the plant, and less attention to the measurements. This effectively moves the observer poles to the right, into the region on the Argand diagram where we should expect to find the controller poles.

We know that the plant equations in the observer are approximations, so the separation theorem doesn’t strictly apply, but we encounter the most serious interactions between observer and controller when their poles are adjacent.

It is this ‘open-loop’ behaviour, which contravenes the fundamental principles of the error-driven regulator. This renders the Kalman filter approach questionable when applied to control systems. Kalman filters provide the best observers when plant equations are known accurately, and all uncertainty is indeed attributable to disturbances and measurement errors. A classical example is in

394


inertial navigation, where the sensors are characterised and calibrated to the limits of available precision, and the statistics of the residual error bounds obtained. Elsewhere, the claim to optimality is a bit mischievous.

It is reasonable to claim that if the system designed using a Kalman filter based observer becomes unsatisfactory due to the interaction with the controller poles, we need to consider whether the candidate sensors are really up to the job, and higher performance sensors might be needed. The Gaussian disturbance noise may be interpreted as a characterisation of the variation in system dynamics over an ensemble of systems, rather than an actual physical disturbance of an individual system.

All observers based on the separation theorem remain blissfully ignorant of the conflict between accurate kinematic tracking of a moving target and minimisation of statistical tracking error.

As with all the other ideas presented in this book, the reader is encouraged to grasp the underlying principles, and to understand where each has its application. We are not looking for the Philosopher’s Stone, merely a set of tools, which if used skilfully, will yield insights and effective designs.

For a while it appeared that the control problem was ‘cracked’; with controllers designed using LQ ‘optimal’ methods and observers with Kalman filter methods (the resulting systems are called LQG controllers), each designed in isolation by invoking the separation theorem. Indeed, before 1980, very little was accepted for publication unless it was based on the virtues of these methods. When practitioners started to use them in anger, however, their limitations rapidly became manifest.

Designing our observer on the one-dimensional criterion of minimum variance is only ever going to be a partial solution.

The absence of the noise spectrum from the Kalman filter may not be a cause of concern if it is used in a more or less open loop manner, as is frequently the case. If the output is to be fed directly into other elements in a closed loop system, we really need to know something about its frequency content so that we know which modes it might excite. Typically, we would desire a much sharper high frequency cut off than 20dB per decade. Such a gradual roll off is in danger of exciting modes which have been neglected from the analysis, in particular, lightly damped structural modes could be excited, with catastrophic consequences.

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23 Multiple Inputs

23.1 IntroductionWe have seen that additional outputs give us considerable control over the closed loop pole locations, which may justify the inclusion of additional sensors within the loop. Multiple inputs imply multiple servos, which remain potentially expensive system elements, and will only be considered if there is good reason. The ability to manipulate matrices does not qualify as a good reason.

It is difficult to see how the proper function of any plant could be specified as requiring cross-coupling between functionally independent channels. MIMO systems emerge from processes which are inherently strongly cross coupled, or become cross coupled for over-riding cost, weight or power consumption reasons.

An example of the former is the environmental control of a greenhouse where the temperature, humidity and carbon dioxide concentration interact, but each must somehow be controlled independently of the other two.

The most ubiquitous example of the latter is the classical aeroplane. Unlike a missile, which is designed for manoeuvrability and rapid response, it is not feasible in an aeroplane to have separate lifting surfaces for yaw and pitch. Only a single lifting surface is available, and this introduces inevitable-cross-coupling between roll and yaw, which is absent from the more complex configuration of the missile. As should be expected from the principle of requisite entropy, simplification of the configuration leads to more complex (NOT simpler) behaviour.

It cannot be emphasised strongly enough that we do not begin control system design until the dynamics of the open loop are fully understood, and we know the reasons why cross-coupling exists and cannot be avoided. If we find that the plant is heavily cross-coupled with no frequency separation between its modes, the wisest course of action is probably to scrap it, and not waste time trying to make the proverbial silk purse from the sow’s ear.

We have seen that instability is generally encountered as we try to speed up the response time in a single-input/single output system. In the MIMO case this is aggravated by parasitic cross coupling between functionally independent channels, and this must be taken into account when we try to achieve high system bandwidth, so there is usually practical merit in studying MIMO systems.

For a while it appeared that the LQG approached furnished a means of designing systems on the basis of state limits and noise levels. Indeed, used sensibly, these methods can be very effective. However, the absence of explicit stability margins and the questionable assumptions behind the separation theorem has led to some precarious designs.

This is an on-going area of research, and the literature is very much in a state of flux. With few practical applications and even fewer opportunities for practitioners to publish, the available text books generally remain opaque to the average engineer.

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23.2 Eigenstructure AssignmentWe found in the monorail example that decoupling between modes becomes possible by separating the modes in frequency. This may be thought of as analogous to having different effects arising from double-clicking and single clicking a computer mouse.

The idea behind modal separation is not dissimilar to the idea behind the derivation of the linear small perturbation equations from inherently non-linear equations. We are really only concerned with the stability of set points, which are defined outside the current loop. The value of a particular state variable, controlled by a low frequency mode, is effectively an equilibrium value as far as a high frequency mode is concerned. Similarly, the rapid fluctuations in the state associated with the faster mode represent a zero mean noise source, above the cut off frequency of the slower mode.

This is the reason why, if we nest loops, we cannot expect the outer loop to respond more quickly than the inner loop. It is why the ‘low level’ dynamics dictate the behaviour of the overall system. So much for the nonsensical dogma of ‘top down’ analysis.

Thus, we can have independent control over the pitch transients of an aircraft as well as the long term control of the flight path, through a single elevator control.

We found that full state feedback gives us the option to place the closed loop poles where we wish, although we included the health warning that they should reflect the modal behaviour of the plant.

A similar result exists for the multi-input case. Not only can the mode frequencies (eigenvalues) be influenced by feedback, but the extra control can be used to suppress or encourage the participation of different states in each mode (i.e. the eigenvectors may be altered).

We would not begin to think of using this method without first acquiring a fundamental understanding of the open loop plant behaviour.

As with all methods in this and later chapters, only the bare bones will be presented here.

23.3 Using Full State FeedbackWe shall assume that estimates of all the state variables are available via some form of observer.

With full state feedback, and a state-controllable system, we can place the closed loop poles anywhere we choose. We also have (potentially) control over as many elements of the eigenvectors as we have control inputs. Thus with two inputs we can set to zero elements of the eigenvector corresponding to states which we do not wish to take part in the corresponding mode.

This manipulation of the eigenvector must be undertaken with care. It is evident that any attempt, for example, to remove the pitch rate from the short period pitch oscillation mode, is likely to lead to some bizarre results.

23.3.1 Basic MethodWe start by specifying a desired closed loop eigenvalue and the elements associated with the eigenvector (mode shape). The number of eigenvectors which may then be specified is equal to the maximum of the rows of the measurement matrix and the columns of the control matrix (assuming

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both have full rank). The number of elements of each eigenvector which may be assigned is equal to the minimum of these two.

Taking lateral autopilot of a monoplane, the modes of interest are Dutch roll and roll subsidence. The spiral mode is taken care of by the trajectory guidance. The state vector is:

[ vrpφ ]Where v is sideslip, r is yaw rate and p is roll rate, φ is the perturbation roll angle with respect to the plane of incidence in a steady turn. The plant equations contain cross-coupling derivatives e.g. Lv (roll due to sideslip, or dihedral effect) and the obvious manoeuvre coupling via the roll perturbation.

Any particular mode consists of an exponential (simple or complex) function of time multiplied by a vector defining the amount by which each state is affected by the particular mode.

The plant equations reflect the fact that the open loop plant is usually designed such that the each control input is associated with a mode which excites specific states, whilst avoiding excitation of others. It is this inherent decoupling which permits separate autopilots to be designed for the yaw, pitch and roll channels of skid to turn missiles. Even here there is weak cross-coupling, but adequate stability margins generally ensure these have minimal effect as far as the closed loop control is concerned.

The monoplane configuration arises largely because of the economy in terms of weight and profile drag of having a single principal pair of lifting surfaces. The penalty is an inability to apply lateral forces rapidly, as the limited control authority is required for simultaneous roll and pitch manoeuvres, whilst regulating the sideslip near zero. This is particularly serious during a reversal of (fixed axes) yaw command direction, when saturation is almost inevitable. This is inherent in monoplane configured vehicles, and no amount of clever autopilot design will compensate for this intrinsic limitation of the plant.

What we can do with eigenstructure assignment is ensure that the Dutch roll and roll subsidence modes are adequately decoupled. However, we can do more than that.

Whether bank to turn or skid to turn, an explicit method for decoupling roll and yaw is necessary, and eigenstructure assignment is a potential option to consider for this purpose.

23.3.2 Fitting the EigenvectorsIn the monoplane case we have two modes of interest, and what we wish to do in the first instance, is use the control to decouple Dutch roll from roll subsidence, although the two are strongly coupled in the plant equations.

We have two control inputs, so we can specify two elements of the eigenvector, e,g. consider Dutch roll, with the rows of the eigenvector corresponding to the rows in the state vector:

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Sideslip:[1¿¿0 ]

In other words, the Dutch roll pole is associated with sideslip but not roll perturbation. The asterisk denotes that these other elements do not concern us. Similarly for yaw rate we expect.

Yaw rate:[¿1¿0 ]

The eigenvalues will form a complex conjugate pair for the Dutch roll mode.

Similarly, we have for the roll subsidence mode:

Roll rate:[0¿1¿]

Roll perturbation:[0¿¿1 ]

The roll perturbation, φ influences sideslip and yaw rate via the lateral force equation, but the only feedback to the roll perturbation is via the roll rate.

We specify two pairs of complex conjugate eigenvalues corresponding to the time responses of these two modes.

For each eigenvalue, we calculate the values of the transfer matrix from output state to control input:

T i=( Isi−A )−1B

Where the plant equation can be written in the form:

x=Ax+Bu

Where x is the state, u the control input, A the system matrix and B the control matrix.

The control eigenvector νi corresponding to the specified state eigenvector is given by:

ηi=T iν i

Where ηi is the specified state eigenvector. The transfer matrix must be re-ordered so that the rows and columns associated with the * entries are distinct from those containing numerical values and can consequently be ignored.

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The resulting re-ordered transfer matrix has nu rows and nx columns and will be denoted Li (note: this is not to be confused with a roll derivative).

The control vector corresponding to the ith specified eigenvector is obtained from the pseudo-inverse of Li :

νi=[LiT Li ]−1 L

iTηi

The eigenvectors which are actually achievable are:

μi=T i ν i

It is usual to display this intermediate result, to allow the designer to adjust the eigenvalue to more nearly fit the desired eigenvector.

Absurd combinations of eigenvector elements, e.g. requiring zero values for the derivative of a state which is desired, are not expected to fit the actual plant very well. Some understanding of the dynamics of the plant is required before the technique can be used effectively.

23.3.3 Calculation of GainsOnce a set of eigenvectors have been decided on and appear feasible, it is necessary to calculate the feedback gain matrix F, where:

u=Fx

We shall assume full state feedback for the sake of clarity. Methods are described in the literature for output feedback, but for the moment it will be noted that this is really a pole placement method for which the separation theorem applies, so a full state observer may be constructed to estimate the state from the available output. The robustness of this process must be tested afterwards.

From our achievable eigenvector we have:

μi si−Aμi=BF μi

For all nx we have:

ΝΛ−AΝ=BF Ν

Or:

Z=BFN

Where N is the matrix whose columns are the achievable eigenvectors, Λ is the diagonal matrix of corresponding eigenvalues.

If the eigenvalues are distinct, the inverse of Ν exists, so the gain matrix is given by:

F=[BT B ]=1BT ZN =1

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This is a least squares fit, which may result in the closed loop poles being displaced from their nominal positions, further analysis is required to ensure the resulting system meets stability and robustness criteria.

Quite often the control matrix is of the form:

B=(I u0 )In which case F is calculated by ignoring the lower nx-nu rows of ZN-1.

More generally, the control matrix is transformed into this form by the following method:

The control matrix is of the form:

B=(XY )Where X is nuхnu and Y is (nx -nu )хnu.

Pre-multiply this by matrix E which is partitioned to match that of B:

E=( L MN P )

Such that:

EB=(I0)The sub-matrices are found from:

LX+MY=INX+PY=0N=−PYX−1

L=X−1−MYX−1

Sub-matrices M and P may be chosen at random. The matrix ZN-1 is pre-multiplied by E, and the final nx-nu rows ignored.

23.3.4 Comment on StabilityThe process of assigning eigenvectors amounts to a least squares squares fit, which can go badly wrong, so this is not a once-through handle-turning exercise. The resulting loops will need to be analysed, and additional compensation introduced as required.

However, the method has the advantage of relating design closely to the actual system proper function requirements, rather than requiring a working solution to spontaneously appear through esoteric magic.

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23.4 Loop MethodsThe single-input/single-output design methods are all based on understanding how the open loop dynamics influences the closed loop plant. This information is used to design compensators to distort the appropriate plot into one which has more favourable characteristics, as measured by gain and phase margins.

We have deviated from this approach when we introduced the state-space methods, in the form of pole placement and its variations, and have worked directly with the closed loop. The problem with this approach is the impossibility of defining the stability margins, so the robustness of the design solutions remains little better than a matter of faith.

At about the same time as most people were adopting LQG exclusively as the way ahead, extensive work was undertaken to extend the ideas of SISO design, to MIMO system. Whilst these methods have practical merit, the whole loop-based approach is inherently unsuitable for MIMO systems, which is why LQG was heralded as the only general method which could tackle this type of system.

23.4.1 General ConsiderationsTransfer functions are derived either by combining the transfer functions of the individual components making up the plant or from the differential, or difference, equations defining the plant behaviour. In a multi-input multi-output system we are evidently dealing with matrices of transfer functions. As servos tend to be considerably more expensive than sensors, we would typically expect the number of outputs to exceed the number of inputs, so our transfer matrices are rarely expected to be square.

We note that the plant poles remain the eigenvalues of the system matrix, regardless of the numbers of inputs . When deriving the transfer matrices from component transfer functions we need to be careful with pole-zero cancellation, especially if the loop contains improper transfer functions, such as was present in the missile two-loop autopilot example.

Most controllers are designed to maintain performance about an ensemble of operating points, so we should expect the commonest reference input to be a fixed set point. To begin with, therefore, we would seek combinations of inputs and outputs are most strongly related at equilibrium. In order to perform this analysis of the inherent controllability of the plant it is necessary to scale the inputs and outputs by dividing them by their limiting values.

A set of n feedbacks (where n is the order of the system) are chosen using this information together with the state controllability criterion, such that the poles are placed appropriately.

In practically all MIMO analysis and design it is necessary to scale all quantities, and modify the plant equations to reflect the scaled values of the variables.

To date, we have used ‘input’ to mean the inputs to the open loop plant, and ‘outputs’ for the sensor measurements. We shall introduce the term ‘reference’ to mean the input to the closed, and ‘outputs’ to refer to the measurements of the quantities the system is required to control.

With this proviso, that we are dealing with a system that has already passed through at least one design iteration, we are justified in treating the modified plant transfer matrix as ‘square’.

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23.4.2 Nyquist CriterionWhen we consider extending frequency domain analysis to multi-input systems, we really need to consider whether what we are doing makes sense. Any method which considers phase information must be clear regarding what the phase reference is. The single input case has the phase of the input as a reference. If we have more than one input, is it reasonable to imagine that both inputs will necessarily be synchronised in phase? Surely the phase of one input is independent of the other and any combination of phase is possible.

We can derive a stability criterion analogous to the SISO Nyquist criterion, using the determinants and eigenvalues of the transfer function matrix, and an identical encirclement criterion arises. However, as an extrapolation of the SISO case, it implicitly assumes phase synchronicity between the inputs, which is never the case.

Figure 23-111: Gershgorin Circles

In fact, in the MIMO case, we have great difficulty in defining what we mean by ‘phase’. This is unfortunate because we have identified phase lag as the principal cause of loss of stability in the single input case. There is therefore a culture shock awaiting the traditional control engineer, in discovering that the familiar approaches are of questionable value, and we are better off dealing with the closed loop directly, rather than using the more familiar loop-shaping approach.

However, if the plant is not strongly cross-coupled we can still deal with it as if it were decoupled.

Consider a plant having transfer function:

G (ω)=(G11 G12

G21 G22)

The matrix elements are responses to sinusoidal inputs of unit magnitude.

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The first output is given by:

y1=G11u1+G12u2

Now u1 and u2 have unit magnitude but the phase of u2 with respect to u1 is random, so any point on the actual harmonic locus lies within the circle centred on G11, of radius |G12|. More generally, the harmonic locus lies within a band defined by circles of radius equal to the sums of the magnitudes of the off diagonal elements, centred on the diagonal elements. This is known as Gershgorin’s theorem, and furnishes a means of applying Nyquist’s criterion to a system which isn’t too far from diagonal. In fact, such systems are quite common.

The circles superimposed on the harmonic locus are called Gershgorin circles, and likewise the regions encompassing all the Gershgorin circles are called Gershgorin bands.

If the Gershgorin bands are narrow near the -1 point, the system is said to be diagonally dominant.

If the Gershgorin band includes the -1 point, the system is not necessarily unstable, so large diameter Gershgorin circles near the -1 point will usually lead to excessively conservative designs.

23.5 Sequential Loop ClosingIt is often the case that additional controls serve specifically to control modes which are widely separated from each other. The throttle control of an aircraft usually has a very slow response, as it is dictated by the dynamic behaviour of the engines, but it usually influences the trajectory modes, having negligible effect on the short period handling modes. The same can be said about the trim tab. The pitch plane control appears to be a three input system, but it is evident that the associated modes are decoupled by virtue of their frequency separation.

Similarly, we may refine the gyro monorail by including an additional input causing the centre of gravity to shift laterally, so that the vehicle will not be affected by steady state contact forces such as offset loads or side winds (inertial forces are automatically dealt with by the basic system, which will cause the vehicle to bank on bends). Since the objective is to balance almost steady state disturbances, the associated mode is expected to be very slow compared with those of the balancing system.

A submarine has ballast tanks to control buoyancy, but they are useless for controlling depth. That requires the dynamic lift, and rapid response, of hydroplanes.

Many systems are of this form, where it is impossible to force the actuation of slow modes to influence higher frequency modes without excessive power or saturation, by virtue of the fundamental dynamics of the system.

In such cases, we should expect to be able to close the loops one at a time, starting from the highest frequency mode.

Designing the loops to accepted values of gain and phase margin should be sufficient to ensure the interaction between the modes should be minimal.

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23.6 Diagonalisation Methods

23.6.1 ‘Obvious’ ApproachA naive approach might imagine designing the compensator in two stages; the first to render the plant/compensator system diagonal, and the second to adjust the dynamic behaviour. This will not work for a number of reasons, the first of which is that the inverse of a general plant is expected to introduce improper transfer functions (numerator order exceeding denominator), which are impossible to realise. Secondly, in the subsequent decoupled single loop analyses, the effect on robustness of the diagonalisation compensator is effectively ignored. Since in many cases this may be where most of the phase loss occurs, the resulting loop design becomes meaningless.

We may design the system such that for quasi-static signals (such as we should expect from the reference input), the system is decoupled. This amounts to making the actual error signals linear combinations of the errors between reference inputs and plant outputs, to offset the static cross-coupling in the plant.

The simple diagonalisation of the plant by calculating its inverse also raises issues of integrity, in that each output is fed to every input, so that failure of a single sensor could cause simultaneous failure of all channels rather than just one.

23.6.2 Commutative CompensatorsThe idea behind commutative compensation is to produce a transformation which diagonalises the plant transfer matrix using a similarity transform, rather like the eigenvalue/eigenvector decomposition of a system matrix. We seek a matrix W such that:

G (s )=W ( s ) Λ (s )W (s )−1

Where Λ(s) is diagonal. The similarity transform itself does not affect the actual plant dynamics, so that a compensator may be designed using single loop methods. The diagonal compensator may then be transformed back into the same space as the original plant, where it would actually be implemented.

Obvious candidates for W are the eigenvectors of G(s). Unfortunately, these invariably contain improper transfer functions, which cannot be realised in practice. However, there is extensive literature, and even design software, available which enable close approximations to the ideal commutative compensators to be designed.

The details of such methods lie outside the scope of a presentation which aims at breadth, rather than depth, of cover.

23.7 CommentLoop based approaches, in which the effect of changes to the open loop plant on the closed loop is used to design compensators, are extremely effective for single-input/single output systems. Their extension to MIMO systems requires a certain ingenuity and creativity together with comprehensive understanding of the plant dynamics. The former is usually effectively suppressed by our academic institutions, while the need for the latter is seldom recognised by a technically naive management.

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The absence of a single phase reference, which is so fundamental to single input systems, would indicate that loop methods are not really transferrable to the MIMO case. This is perhaps the reason why methods which worked directly with the closed loop, such as pole placement and LQG, quickly overshadowed loop based methods, until their potential lack of robustness became apparent.

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24 Closed Loop Methods

24.1 IntroductionThe control systems which have been designed in the past tended to be mechanisations of tasks originally performed by human beings. Contrary to what appears to be the current wisdom, human beings have severe limitations, which require tasks to be largely decoupled, either in separate physical channels or in time, with different modes requiring attention over different time periods. Just channelling all the information to a human being without filtering it down to the relatively few bits per second he/she can handle, is a recipe for disaster. Admittedly, there is always the cop-out of ‘pilot error’ as a fig leaf for all manner of poor system design.

The control engineer views the human being as merely another dynamic component, whose behaviour when performing a specific task is not as difficult to characterise as human vanity believes.

These limitations are reflected in the plant. So when we come to automate existing tasks, they are usually nicely decoupled into functionally independent channels or separated in frequency for which intervention is required. SISO methods are adequate, so that multivariable control techniques seem to lack relevance in the real world.

For this reason, and the fact that most presentations are well beyond the event horizon as far as clarity is concerned, there has been a reluctance on the part of practitioners to study the methods which have emerged over the past 30 years. If control system specialists see no merit in investing time in the newer methods, how much less likely is the world of systems engineering to be impressed.

The more we ask of the machinery, the greater the effect all the cross-coupled neglected dynamics will have. The chemical industry furnishes us with many processes which are strongly cross-coupled, and the consequence of merely increasing flow rates to speed up production is unlikely to be a pretty sight.

Systems of the future will be muti-variable, and as we have seen, SISO methods really are not up to the job. At best we can pretend that ‘square’ systems can be treated more or less like multiple SISO systems, but generally systems only become square AFTER we have designed the controller. Square plant is a rarity. In general, we cannot use loop methods for MIMO systems, so we are forced to adopt methods which apply directly to the closed loop.

Also, our characterisation of robustness, in terms of gain and phase margin, has served well as ‘rules of thumb’, but outside the hallowed cloisters of control engineering, they are meaningless. We are better off describing our stability margins in terms of the variability in the values of the plant parameters which can reasonably be expected. These are meaningful to other specialists. They provide a means by which the control engineer can specify tolerances on components. In principle, the information could be used to define the probability of a design becoming unstable in service. This is what the safety people actually want to know.

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The following begins with SISO systems, to see how we characterise the closed loop in order to address the actual design issues which confront us. We use the idea hinted at when we considered the effect of noise on a tracking sensor, that there is a conflict between suppressing noise originating within the loop and at the same time ensuring accurate kinematic tracking of the reference signal. Unlike the traditional SISO approaches, these methods may be extended readily to the MIMO case.

24.2 The H-infinity NormMost practitioners would have come across the term H∞, and wondered, like the author, about the publishing standards which permit mathematical symbols to enter the abstract, let alone the title, of a paper. Perhaps it indicates a limited command of the language, which cannot express the idea in plain English. Perhaps the concept is not fully understood by the originator, to the extent that it can be expressed in natural language. Perhaps that once explained, it loses the mystic air of expertise which seems to be associated with gobbledegook. Well, the term is now established and must be added to the list of control engineer’s jargon (together with M and N circles, phase margin, etc.)

Although the idea is not needed immediately, we shall dispel the mysticism before proceeding.

We mentioned, when optimising a signal in the time domain, that minimising least squares error seemed a reasonable basis for finding the ‘best’ control. It was pointed out that this was just a matter of convenience; we could have raised the error to the power p. As p tends to infinity, large deviations from the general trend of the data become penalised more heavily.

In the time domain the rms value of a quantity is often associated with the energy conveyed, so does have some physical significance. In the frequency domain the energy of the signal distributed over the spectrum does not have the same significance. In the frequency domain our desire is to avoid resonance peaks, so that our characterisation of the mean deviation from the desired frequency response must avoid smoothing out such features.

We can define a norm of a set of data values as:

x p=( 1N∑i=1

N

|x i|p)

1p

The norm of the random data set for several values of p is presented in the figure. We see that as p→∞, the norm tends to the maximum peak, or the supremum, of the set. This is the vector infinity norm. This illustrates the idea behind H∞ methods, but is not in itself a definition of the H∞ norm. Perhaps calling it the ‘resonance peak alleviation’ method would have made up in accessibility what it lacks in accuracy. H∞ seems a spectacularly pretentious way of saying ‘the biggest’.

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Figure 24-112 : Illustration of the Effect of Index on Sample Norm

24.3 MIMO SignalsIt has been suggested that in the absence of a common input to act as a phase reference, phase information is meaningless in MIMO systems. By implication, the sinusoidal excitation of the system is of little use without exploring all possible phase relationships between the inputs. Nyquist, Bode and Nichols charts become misleading. We must deal with signal magnitudes.

Consider the plant:

y=G ( s)u

The magnitude of the output is:

yH y=uHG ( s )HG ( s)u

Where H denotes a complex conjugate transpose. This deals strictly with signal amplitudes, and is only meaningful if outputs and inputs are scaled by their limiting values, such that the resulting amplitude is a pure number.

The matrix G(s)HG(s) is symmetric, so its eigenvectors are orthogonal. Furthermore, its eigenvalues are all either positive or zero.

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The eigenvalue decomposition becomes:

G (s )HG ( s )=U H Σ2U

Where Σ2 is a diagonal matrix of eigenvalues.

If v=Uu

vH v=uHUHUu=uHu

Since U is orthogonal. This means the relationship between the magnitudes of input and output vectors depends only on the eigenvalues of GHG.

This is all very well, but controlling the vector magnitude of the output is going a bit too far in our quest to avoid using signal phase. We desire to control individual outputs.

The elements of Σ are gains relating the inputs to the outputs. The degree to which each input is amplified by each gain depends on the rows of U. These may be thought of as input directions corresponding to the gains. We can find the directions of the output:

GH y=GHGu=UH Σ2Uu

yHGGH y=uHU H Σ 4Uu

The magnitude of the output can only be invariant if:

GGH=Y Γ2Y H

Where Y is orthogonal.

For consistency with the expression for GHG:

Γ=Σ

And G may be written in the form:

G=Y ΣU

The ith column of Y characterises the effect of the ith gain on the output, whilst the ith row of U determines the input which affects the ith output (all other inputs having no effect).

The elements of Σ are called the principal gains, or the singular values, and the partitioning of a rectangular matrix into two orthogonal matrices and a diagonal matrix of singular values is called a singular value decomposition.

The singular value decomposition of the system matrix is of interest in its own right, as it reveals how inputs affect outputs over the frequency range, revealing clues as to how the system can be partitioned into functionally independent modes.

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Note that even if G is square, the system response is determined by the principal gains (singular values) and not the eigenvalues. The eigenvectors corresponding to eigenvalue decomposition are not orthogonal, as is needed to associate specific input and output directions with each gain.

Note that the singular value decomposition is not a similarity transform. The left hand matrix multiplying the diagonal matrix of singular values is not the inverse of the right hand matrix.

For a matrix which is a function of frequency (i.e. a transfer function), the H∞ norm is the supremum of the maximum principal gain, taken over the frequency range, i.e. the highest resonance peak of all the possible system responses.

24.4 Closed Loop Characterisation We have introduced the idea of input and output directions for MIMO systems, which impose limitations on achievable performance and robustness. These are additional considerations which will be addressed once the issues of working just with signal magnitude have been considered in the SISO case.

We saw in the construction of Nichols charts, that simply specifying the desired closed loop response was insufficient to enable compensators to be designed, because the closed loop zeros are the same as those for the open loop. Typically, we would expect the compensator to introduce additional zeros, so apparently this approach would require the compensation to be known, before it could be applied. ‘Tail wag dog’ methods often arise when approaches are not properly thought through.

However, we recall from considering the tracking properties of a steerable sensor (e.g. for communicating with a distant space vehicle), that there exists a conflict between minimising the response to sensor noise whilst accurately tracking a moving target. In fact, this is an extremely common problem, and enables the closed loop to be described by a pair of transfer functions, one characterising response to the input, the other the response to noise.

We have:

T= KG1+KG

Where K is the compensator transfer function, and G is the open loop transfer function. T is the usual closed loop transfer function, or loop transmission.

The response to disturbances (and the error response to the input) is characterised by:

S= 11+KG

This is called the sensitivity.

We have two transfer functions characterising the closed loop, one of which has the open loop zeros, and one which doesn’t. By considering the conflict between accurate tracking of the input, and minimising the response to noise, we have a potential technique for designing the compensation.

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24.4.1 Tracker Example

Figure 24-113: Tracking Loop with Measurement Noise and Gust Disturbance

The sensor loop is as shown. The size and weight will presumably have been determined from initial detection capability, volume and cost considerations. Presumably the operating environment would be specified, so that local gust loading on the structure would be known.

The loop consists of the desired sight line direction θD, which is not measured explicitly. The sensor measures the boresight error between the desired direction and the actual sensor pointing direction. The system could be driven by an explicit boresight demand during initial acquisition, involving an outer loop, optimised to minimise the acquisition time.

The signal from a distant target is expected to be weak, so the boresight error will be contaminated with noise.

We expect there to be some filtering of the boresight error combined with compensation for the loop dynamics (Kf). This feeds into some form of servo (Gs) to steer the sensor. The servo applies torque to the sensor, as does the wind. The total torque is the servo plus wind gusts.

The wind gust is represented by a white noise source filtered so as to represent its expected spectral content (Gd) .

24.4.2 CommentThis is actually a very high level view of the tracker. We are only considering a single channel, and we may choose to feed back the angular velocity of the sensor, rather than rely on position alone. This is expected to introduce additional biases and noise, the consideration of which would be deferred to a later iteration.

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This represents the bare bones of what is needed to understand the trade-offs between servo requirements and sensor detectivity, and to determine realistic power, weight and volume requirements for the tracker.

If we are tracking a distant space craft, the sight line direction is expected to move slowly but the signal will be weak, so we expect the compensator time constant to be large and the signal energy may be integrated over a significant period of time whilst the boresight is held nearly constant. If we are tracking an incoming missile, loop response may be more important. We can frame the requirements in terms of the signal to noise, or some other means of characterising the noise input, and an overall loop response time.

More often than not, we need to find the best response time which can be achieved using candidate sensors, and analysis of the higher level process will determine whether the available performance is adequate.

Reducing response to measurement noise also slows down the response to target motion. Since the sensor measures bore sight error, the reference signal (the true sight line direction) is not available as an independent measurement. It is therefore impossible to apply a pre-filter to modify the demand, as is frequently suggested.

The problem is not resolvable using control methods alone. We must proceed to a finer level of detail. Typically, we expect the more sensitive sensor to be larger, e.g. a larger dish is needed for greater gain, which is more vulnerable to gusts. There may be a trade off between sensor noise and disturbance noise at a secondary level of analysis. By ensuring that disturbance noise is dominant by virtue of the sensor design, we can optimise the control to maximise disturbance rejection.

However, there are cost, weight and power implications in increasing the size of the plant, and it is evident that a compromise is needed. Any ‘optimisation’ which does not take the dynamic behaviour of the entire loop into account isn’t worth the paper it is written on.

24.4.3 Input Output RelationshipsFrom our tracking loop we can determine the output in terms of the inputs and disturbances.

θ=(θD−θ )K G sG p+mK GsG p+dGdGp

We are also interested in the input to the servo, as this characterises saturation.

u=(θD−θ )K+mK

Re-arranging:

θ=K GsG p

1+KGsG pθD+

KGsG p

1+KG sG pm+

GdG p

1+K GsG pd

And:

u= K1+K GsG p

θD+K

1+KG sG pm−

K GdG p

1+K GsG pd

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Or in terms of the sensitivity, S and transmission, T:

θ=T (m+θD )+SGdG pd

u=KS (m+θD )−K SG dG pd

The tracking error is:

e=θD−θ=SθD−Tm−SGdG pd

In this case the measurement noise was added into the boresight error measurement, because this is what the sensor actually measures. More typically, the measurement error is added to the output θ. The sole effect of which is a reversal of sign, which is irrelevant to a zero mean white noise process.

Evidently, accurate tracking and complete disturbance rejection implies S=0 and T=1, or K→∞. But suppression of measurement noise requires T→0, in direct conflict to the requirement for disturbance rejection. We must seek a compromise. More often than not, we design the plant such that one or other noise source is dominant. Hence radar dishes are invariably housed inside radomes, to ensure that performance is limited by detector sensitivity.

The processing may involve long observation times which dominate the compensator response, such that the details of the motion of the antenna become secondary. Some sensors may be electronically steered, in which case the system delays are determined almost entirely by the processing delays.

This compromise between measurement and disturbance noise was resolved in the Kalman filter by producing a minimum variance state estimate.

We have considered explicitly examining the effect of feedback on loop stability using ‘classical’ methods. It was also found that stability could be addressed implicitly by minimising tracking error over time using the linear quadratic method to calculate the gain matrix. We may also address the stability of the closed loop in the frequency domain by using a measure of mismatch between the desired and actual frequency responses which is sensitive to resonance peaks, which occur as instability is approached. The appropriate measure is the H∞ norm.

For the reasons discussed, the transmission T does not form a good basis for compensator design, but the sensitivity S does not share its zeros with the open loop transfer function, so could be used. We see that it determines the response to disturbances originating within the loop. These may arise not only from physical disturbances such as wind gusts or vibration, but when considering an ensemble of control loops, they may represent uncertainty in the plant parameters.

In considering the sensitivity, we can not only synthesise controllers, we can also design them to achieve robustness to specified uncertainties in plant parameter values.

24.4.4 Weighted SensitivityWe specify the closed loop behaviour as a weighting function (wp), which is the reciprocal of the desired sensitivity, expressed as a function of frequency. (The subscript p denotes performance, as opposed to robustness). We would choose a function which reflects the requirements for

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sensitivity. The gain should be low at low frequency, which implies large feedback gain. However, this cannot be maintained in the region of gain crossover (the frequency which characterises the open loop bandwidth), without risking instability or control saturation. A typical weighting might be:

wP=

sM+ωB

s+AωB

Where ωB is the desired bandwidth, M is the high frequency gain, A the low frequency gain and the integrator is included in the denominator because we suspect integral of error is probably needed for accurate tracking. The need for integral control cannot be derivable from methods based on zero mean noise processes.

If we are concerned with high frequency cut-off, we could try:

wP=(s

M1N

+ωB

s+A1N ωB

)N

To increase the slope below the specified bandwidth to 20N dB per decade.

The problem reduces to finding the compensator K, for which the H∞ norm of the expression:

|w pS|

Is less than 1.

The method is, more or less, directly applicable to the MIMO case, only dealing with a sensitivity matrix:

S= (I+GK )−1

Rather than the magnitude, the maximum singular value is used.

The sensitivity, at large values of gain is determined by the smallest singular value of the plant. If this is small, the control input will have little effect on the output, and there will be a risk of saturation, because high feedback gains will be needed. With scaled inputs and outputs, we need the lowest singular value of the plant to be greater than unity, otherwise it is unlikely to be controllable.

24.5 Stacked RequirementsThe sensitivity may yield controllers with phase advance, but there is no control over response above the specified bandwidth. In particular, there is no control over the high frequency cut-off needed to suppress measurement noise, nor is there any attempt to limit the control input.

A more satisfactory approach would seek to bound the sensitivity, the control effort and the tracking error by using a matrix of requirements:

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N=[ ωPSωTTωuKS ]

The problem then reduces to finding a transfer matrix K which minimises the H∞ norm (the highest peak of the largest singular value). The functions ωT and ωu characterise the desired closed loop response and control input respectively.

24.5.1 RobustnessThe use of H∞ as a means of compensator synthesis, on the face of it, does not appear to offer any advantage over linear-quadratic methods. It’s advantage lies in the fact that it can be adapted to account for plant uncertainty, which may be included in the compensator design. The uncertainties impose constraints on the singular values of both sensitivity and loop transmission, in many respects reminiscent of the Gershgorin bands encountered in diagonally dominant systems.

A simple introduction to this subject would probably be misleading, and in any case, the author does not consider himself competent to explain it in detail. However sufficient has been presented to indicate where and when the methods are appropriate, and that is all that is required of an introductory text.

As we have introduced the idea that it is the effect of plant delays (phase) which is the principal reason for the error signal failing to reflect the true state error, the idea of designing systems without reference to phase, represents a major departure from fundamental principles. It is generally the additional phase lag associated with neglected high frequency dynamics, which is of principal concern. Gain errors arise mainly from uncertainties in the plant parameters which are actually present in the system model.

The H∞ approach effectively assumes the worst possible phase is associated with each gain uncertainty, so is truly robust, if probably excessively conservative.

24.5.2 Phase in MIMO SystemsThe phase reference is the input, so that a multi-input system has multiple phase references. There are as many harmonic loci as there are potential loops. We might expect a system to have up to, say 3 inputs and 15 outputs, yielding 45 possible harmonic loci. All 15 outputs could potentially be associated with each input, so we should expect three plots of 15 loci, from which the worst gain and phase margin associated with each input may be determined.

The computation needed to generate the full population of potential loops is trivial by modern standards.

However, the harmonic loci can only be produced once all the loops have been closed, and each loop for each input cut one at a time to see the consequences of additional phase loss (delays) in each feedback path. The phase margins may be found, but some other means is needed to generate the candidate design in the first place.

Ultimately, one wonders whether it is a good idea to agglomerate the system description into a single matrix at all. It is the information flows between components which is of greatest interest. We are merely hiding the connectivity information within a single mathematical entity. This was

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highly successful in the time domain with state space methods, where the entire information content of the matrix is contained in its eigenvalue decomposition.

The frequency domain description, although mathematically valid, does not appear to benefit as much from the matrix description. If anything it obscures, rather than clarifies the problem. We are probably better off sticking to scalar representations of the individual components interconnected in a network, so that it becomes possible to decide the phase lag associated with each path through the network, from which the margins may be deduced.

Indeed, the classical control engineer (and the general systems engineer) is much happier dealing with networks of various kinds to characterise their systems, than with a matrix algebra description which completely omits the most important signal characteristic.

We actually need to know how each individual input affects each output, so it is unwise to try and deal with inputs and outputs as vector quantities. This is the background understanding which underpins any attempt to automate the system. Singular values appear just the best way of describing an inherently poor system representation.

24.6 Concluding CommentAutomatic control is an engineering discipline, not a pure science, and research budgets need to be predicated on the potential of the methods produced, not on awards which academics may give each other. For many years now, academia has seriously neglected its paymaster, yet continues to expect support for producing little which is presented in a form which line engineers can use.

Despite the fact that this approach has been known for at least 30 years, the literature on the subject of MIMO control remains in a state of flux. This, in the author’s opinion, arises from the absence of practitioners in the loop to determine what is actually useful.

The best textbooks are written by those academics whose expertise has been required in the unforgiving world of commercial engineering, although even here the target audience appears to be graduate mathematicians, rather than engineers. The commonest objective appears to be to make graduates aware of engineering, rather than helping engineers to understand the method.

For these reasons, only a very sketchy outline of the method is presented, which it is hoped is sufficient to enable the reader to form an informed opinion as to whether the methods are appropriate to his/her task. Its application requires a deeper understanding than is possible to present in a single chapter. However, there are standard MATLAB algorithms available, which can be handle-cranked by the ignorant, and it must be admitted, they generally produce very good controllers, with nobody knowing exactly why.

From the perspective of requisite entropy, MIMO systems introduce the additional problem that the inputs and outputs do not act in the same direction, so that one ceases to be an estimate of the other. This is in addition to the effect of delay. Many systems may appear decoupled at low frequency, but start to encounter cross-coupling as we try to increase the bandwidth. The input and output directions obtained from singular value decomposition help us to detect the onset of this particular limitation. When we ask too much of a SISO system we encounter instability. MIMO systems are far less forgiving, and we have the potential to screw up big time.

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On the positive side, these are the only methods which truly apply to MIMO systems, and address the issue of robustness which was overlooked in ‘optimal’ control. Also, the approach to robustness is inherently superior, and actually more in keeping with traditional engineering thinking.

Nobody outside the control community has any idea what a gain margin or phase margin is, and indeed these terms are all but meaningless in the MIMO case. It must be better to base our stability margins on explicit awareness of the uncertainty in the plant. Admittedly, plots of singular values are probably even more obscure, but it should be possible to characterise parameter errors as statistical distributions and quote the stability and performance margins as probabilities that the uncertainty in the plant will lead to instability.

The H∞ approach originated in the 1980s, when computer processing power was severely limited. Perhaps, with modern computing power, explicit Monte-Carlo methods could be applied to the characteristic equation, and appropriate distributions fitted to the resulting closed loop pole positions on the Argand diagram.

It makes sense to specify a probability that the plant will misbehave, because this is what the wider community wishes to know.

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25 Catastrophe and Chaos

25.1 IntroductionThis, and the following chapter, present material which is peripheral to the control engineer’s stock in trade, and will only be of interest to readers having a more philosophical interest in system science.

It has already been mentioned that linear systems are actually quite rare. In cybernetics, we are fully aware that our equations are wrong. That is not the issue. The methods are concerned with finding how wrong we can be before our linearised analysis fails us. This boils down to how much performance we can get from our system before its behaviour becomes unpredictable. Feedback introduces inherent robustness, such that any interaction with an open set of disturbances must use feedback in some form.

Compared with open loop control, feedback substantially reduces the accuracy requirements and cost of system components, and endows the system with substantial disturbance rejection, where the open loop has none.

As part of this smokescreen, a considerable amount of pretentious nonsense has been written on the subjects of catastrophe theory and chaos theory, as alternatives to linear system theory. In fact both these areas of study actually complement linear theory. As in linear system theory, we seek to characterise and explain extremely complex behaviours by means of simple expressions.

To those not versed in basic cybernetics, the idea that simple systems can exhibit complex behaviours appears anomalous. However, if we are familiar with the Principle of Requisite Entropy, it comes as no great surprise. Indeed, it is expected.

25.2 CatastropheBoth catastrophe theory and chaos theory deal with non-linear systems. Taking small perturbations about any single condition we see that the characteristic equation can have positive roots at some values of the state variables. In catastrophe theory, the system moves quickly to values of the state variable which result in negative roots for small perturbations around the new position. The system starts at one stable equilibrium point and a coefficient is varied until this equilibrium ceases to be stable, so that the system moves to a new stable equilibrium condition. This sudden transition between two stable equilibria is known as a ‘catastrophe’ because the interval the transition takes place in is very short, compared with the rate the controlling coefficient is varied.

Catastrophe theory deals only in static equilibria, without reference to the nature of the transition between them, or whether the system will actually settle down at all. When the dynamics of the transitions are of interest, we find ourselves in the field of chaos.

The idea is to characterise the state of the system as a potential energy which is a function of position coordinate (x). If a perturbation in x increases the potential energy, it represents a stable equilibrium, otherwise it is neutral or unstable. This was actually the idea behind Liapunov’s direct method, but in general, the ‘potential energy’ analogy is not very helpful.

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25.3 The ‘Fold’The simplest illustration of this is called the ‘fold’ catastrophe. It serves to illustrate the idea. But has little practical value.

It has the ‘potential’ function:

V=x3+ax

Where ‘a’ is the controlling parameter. This may be thought of as a surface in 3D space, each vertical cross section of which is a cubic curve.

The roots of a cubic equation may be found by means of the following algorithm.

Given:

x3+a2 x2+a1 x+a0=0

The nature of the roots can be determined by first evaluating:

q=a13−a22

9

r=16 (a1a2−3a0 )−

a23

27

If q3+r2<0

All roots are real, otherwise the curve has one real root and a pair of complex conjugate roots. The second case has no turning points, so there can be no equilibia. The first case has one peak (unstable equilibrium) and one trough (stable equilibrium).

In this case a0=-V, a1=a and a2=0.

q=a3,r=V

2

The condition for three real roots is:

a3← 274V 2

This requires a to be negative.

As a is varied from an arbitrary large negative value, the system remains in the equilibrium position corresponding to the trough, until as a approaches 0, the equilibrium disappears. It has been argued that this could represent irreversible catastrophic failure. The point where the control variable a approaches zero is called the ‘tipping point’.

The stable equilibrium line is:

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x=−√ a325.4 The ‘Cusp’The most commonly used catastrophe model is the cusp.

The author is dubious about representing 3D objects pictorially on 2D paper, because this introduces ambiguity. The idea of trying to represent higher dimensional objects in 2D really does cross the threshold of self-deception.

Instead of representing the potential hypersurface ( i.e. 4D plot of V), we draw a 3D contour representing the set of potential equilibrium positions. The pictorial representation of the cusp catastrophe is usually the derivative of V with respect to the system variable x, plotted against two control variables (a and b). The potential function is:

V=x 4+a x2+bx

The equilibrium surface is given by:

dVdx

=0=4 x3+2ax+b

This is similar in shape to the potential surface of the fold catastrophe.

Figure 25-114 : Equilibrium Surface for the Cusp Catastrophe

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The fold in the equilibrium surface is called a cusp. Nothing interesting happens if ‘a’ is positive, as this corresponds to an equilibrium position beyond the apex of the fold.

With negative ‘a’, increasing b from a negative value as far as the cusp doesn’t raise any problem. However, the only equilibrium position beyond this point is on the other side of the cusp. The state variable changes suddenly to accommodate this transition.

Reducing the value of b does not cause an immediate transition back, but the system moves to the edge of the cusp before transitioning back. This effect is called hysteresis.

Rather than characterising a truly catastrophic event, the cusp models the sudden transition between two levels in the state variable. It behaves as a binary switch with hysteresis.

25.5 CommentThere are several catastrophe models, involving more control variables and states, but these two illustrate the principle.

Since it was first formulated, catastrophe theory has acquired some strange bedfellows, and is often central to some pieces of pseudo-scientific nonsense. However, there are circumstances where it can be useful. The author has used it personally to explain the behaviour of a command to line of sight missile in multipath. In that particular case, the missile trajectory appeared to transition spontaneously to different nulls in the tracking radar polar diagram.

The attraction of catastrophe theory is its ability to combine continuous signals with the transitions of a finite state machine, without using explicit logic.

It has been mentioned in earlier chapters that finite state machines cannot deal with an open set of disturbances, and the Turing machine cannot adapt by random mutation. It appears that catastrophe theory might furnish a means of circumventing these limitations. In principle, the control variables of a catastrophic system could be adapted by feedback, and an organism which employs catastrophe, as opposed to Boolean logic, appears to have the potential to evolve by natural selection.

The weakness of the Turing machine is its inherent sequential processing, which is like a chain; if one link breaks, so does the chain. If processing is done in parallel, with relatively short sequences, it may be possible to devise a processing machine which can adapt as an error-driven regulator.

This seems a reasonable speculation in view of the incredible feats of processing achieved by nervous systems whose elements trigger in milliseconds, rather than the nanoseconds of electronic devices. It is difficult to see how this could be achieved with such slow elements without massive parallelism.

We can imagine the processing power of a system which has both electronic switching times and biological parallelism.

If catastrophe is used as the process of adapting a finite state machine process, it must follow that the finite state machine is subordinate to one employing continuous signals. This is at variance with current wisdom which appears to believe that information should become more quantised, the higher up the system hierarchy it is used.

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The fact that we don’t see finite state machines in nature, should raise concerns regarding the modern perception that they furnish a universal representation of system dynamics.

25.6 Chaos

25.6.1 Adding Some DynamicsWe can treat the cusp catastrophe as a dynamic system by adding a kinetic energy term to the potential energy:

V=12

( x )2+x4+a x2+bx

Differentiating:

x x=−(4 x3+2ax+b ) x

i.e: x=−(4 x3+2ax+b )

Everywhere on the equilibrium surface x=0. We shall denote the equilibrium value of the state on this surface as X, and use the lower case x to denote a perturbation with respect to equilibrium.

x=−4 (X3+3 X2x⋯ )−2a (X+x )−b

Equilibrium requires: 4 X3+2aX+b=0

The small perturbation equation is:

x=−(12 X2+2a ) x

We can write this in terms of the control variable b, by solving the equilibrium equation for X. An analytical solution only exists for the case of a single real root, but we are interested in the region of the cusp, where there are three real roots. Instead, we shall calculate b for a range of values of X, resulting in the following plot of ‘stiffness’ as a function of b:

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Figure 25-115 : Stable and Unstable Regions for the Cusp Catastrophe

The small perturbation equation is of the form:

x=−cx

If c is positive, the system is neutrally stable, and the state will oscillate about the equilibrium condition indefinitely. If negative it is unstable, and the system will diverge from the nominal equilibrium point. The unstable region corresponds to the region between the two turning points in the equilibrium surface.

The resulting system will not actually settle down as predicted by catastrophe theory unless we introduce a damping term.

x=−d x−cx

Where the coefficient d characterises the damping. With this modification, the system should behave dynamically as predicted by catastrophe theory. For optimal damping, we shall choose:

d=1.4√c

i.e. a damping ratio of 0.707.

We shall illustrate this by starting the system on the cusp with X=0.4, and plotting the transition to the stable equilibrium position at X=-0.8.

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Figure 25-116 : Transition from Unstable to Stable Equilibria

The transition is illustrated by the above phase plane plot. Note that the system starts at the unstable equilibrium point on the right hand side of the plot.

The motion is determined by the nature of the stability at the start and end points, and most descriptions of non-linear system call these points ‘attractors’.

25.6.2 AttractorsStability of non-linear systems cannot be quantified by a simple criterion. There is no simple extrapolation of the right half plane or unit circle criteria to non-linear systems. We can only talk about stability of perturbations about nominal equilibrium positions. The potential equilibrium positions are calculable from the steady state solutions of the non-linear equations, so our restriction to examining the perturbation of states around the various equilibria appears to cover all eventualities.

In system theory, we deal with potential equilibria, i.e. the equilibrium positions which are determined by setting all time derivatives to zero. Catastrophe theory, and a lot of quasi-scientific theories, appear to assume that these equilibria are actually reached, and indeed characterise the problem. We have seen, however, that equilibrium is only reached and maintained for a particular set of system parameters.

These potential equilibrium points are called ‘attractors’, to indicate that although there might be a tendency for the system to move towards them, usually they never get there.

Linear theory yields three types of attractor; stable, neutral and unstable, as shown in the following figures.

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25.7 The Lorenz EquationLinear theory indicates that there are three types of attractor, the system may converge on a stable equilibrium point and stay there, the system may cycle around the nominal equilibrium indefinitely, or the system may diverge away from the nominal equilibrium position. The reason for this is the fact that a characteristic equation can be factored only into simple and quadratic terms, and the only behaviours the corresponding modes can exhibit correspond to the three basic types of attractor.

Consider now a system governed by the non-linear equations:

x=σ ( y−x )

y=ρx− y−xz

z=−βz+xy

Our analysis starts by linearising about a reference point X,Y,Z:

x=σ ( y−x )

y= (ρ−Z ) x− y−Xz

z=Yx+Xy−βz

The system matrix is:

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A=( −σ σ 0( ρ−Z ) −1 −XY X −β )

From which, the characteristic equation is:

s3+(1+σ+ β ) s2+[σ (1+Z−ρ+β )+β+X2 ] s+σ [ β+X2+Zβ−ρβ+XY ]=0

The Hurwitz test matrix is:

H=((1+σ+ β ) σ [ β+X2+Zβ− ρβ+XY ] 01 [σ (1+Z−ρ+ β )+β+X 2 ] 00 (1+σ+β ) σ [ β+X2+Zβ−ρβ+XY ])

The first test function is:

T 1=1+σ+β

This depends only on the system parameters.

The second and third test functions are polynomials in X, Y and Z. Since these can adopt arbitrary values, it is evident that there will be regions of state space where the system will be unstable.

From a control perspective, that is the end of the matter. We detect the unstable regions and develop controllers to stabilise them. Furthermore, we consider the constraints on the size of perturbations and system bandwidth which arise from the non-linearities, in order to ensure robustness.

However, the Lorenz equation was not formulated for a system over which control could be feasibly exercised. It was a simple model of the weather. Linear analysis indicates that the solution would just diverge. However, usually we want to know a bit more about the weather.

We know that weather is not a divergent phenomenon, although the long term effects of global warming may force us to revise this perception. If anything it appears cyclic, but not predictable by simple harmonic models. The cycles appear aperiodic.

25.7.1 Explicit SolutionSolving the Lorenz equation for σ=10, ρ=27 and β=2.666 results in the somewhat surprising plot of figures Figure 25-117, Figure 25-118 and Figure 25-119. Rather than a divergence, the system appears to orbit around a pair of attractors, but doesn’t conduct periodic orbits about either.

Evidently the unstable mode near one attractor is a different mode to the one which is unstable about the other. The system appears both unstable and stable at the same time. The result from linear theory is therefore pessimistic, because the system does not simply diverge to infinity, as predicted.

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Figure 25-117 : Lorenz Equation Solution: x-y Plane

Figure 25-118: Lorenz Equation Solution : x-z Plane

Figure 25-119 : Lorenz Equation Solution : y-z Plane

The neither stable nor unstable attractor has become known as a ‘strange’ attractor. It indicates that non linear systems are not necessarily unstable in certain regions, but many exhibit this

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‘strange’ behaviour. Unlike artificial systems which must be designed for stability and robustness of stability, naturally occurring systems can and do evolve well into this region of aperiodic, unpredictable behaviour.

In fact, linear dynamic systems are such a rarity outside the field of artificial systems, that linear theory is all but useless for studying naturally occurring dynamic phenomena. Actually, it is difficult to see how life could have evolved without chaos to stir the soup.

However, if it is within our power to control a system, it would be the height of folly to allow it to operate in the unpredictable chaotic regime. Yet, this is precisely how human economies are managed.

It is a case of accepting what we cannot control, but controlling everything we can. Cybernetics furnishes us with the wisdom to tell the difference.

25.8 The Logistics EquationThe Lorenz equation is most frequently cited because it was allegedly the first to draw our attention to the phenomenon of chaos. Doubtless it had been encountered earlier, but rather than gaining the laurels of a new discovery, the programmer responsible was probably dismissed for incompetence.

It really isn’t a surprise to discover the weather is difficult to predict. That was established many centuries ago. The surprise is that the unpredictability can be demonstrated with a deterministic model.

The Lorenz equation requires three parameters, and is probably not the best option for studying the basics of chaos. A simpler model is one used to represent predator/prey interactions or population growth. We sample the population at equal intervals. If the population last year was x, we expect it to increase this year by an amount kx:

xn+1=xn+k xn

But this is a formula for exponential growth. We should expect a regulatory influence to exist, such as predation, overcrowding or starvation. A certain proportion of last years’ population will not survive to breed. We assume the proportion reduces the higher the population, so we have:

xn+1=α (xn+k xn ) ( β−xn )

Where β is the proportion which can breed when the population is very small (αβ=1). Defining a new variable:

y= xβ

The equation for population growth becomes:

yn+1=γ y n (1− yn )

Where γ is a constant.

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The steady state is found by setting yn+1=yn.:

y∞=1−1γ

The linearised equation about a population Y becomes:

yn+1=γ y n−2 γY yn

Taking the z transform:

z−(1−2Y ) γ=0

Stability requires z to lie within the unit circle:

|γ−2Yγ|<1

Or:

γ2 (1−2Y )2<1

Substituting Y=y∞:

γ<3

If the population were to breed faster, the equilibrium would not be stable.

In practice, real populations move well into the unstable region, and oscillate between short-lived equilibria , or collapse altogether for no apparent reason.

A naive analysis would delude us into thinking that the equilibrium population is given by y∞=1, whilst it is evident that we do not dare exceed y∞=2/3.

25.9 Bluffer’s GuideThe original plan had been to expand on the logistics equation, to cover the basic methods used in chaos theory. I have decided against this, partly because I am not best qualified to cover the subject with any authority, but primarily because it is not really relevant to the subject of cybernetics.

Chaos, like signal processing, is a close relative of cybernetics, and despite appearances (they all use mathematical formulation), the subjects are different. They have different objectives. However, a lot of chaos ‘buzz words’ have crept into the systems science vocabulary, so I shall limit the discussion to definitions of these terms.

25.9.1 Basin of AttractionIf we move beyond the linear stability region, the system behaviour may become characterised by two or more attractors. A basin of attraction is associated with each of these attractors, and consists of the set of all starting points which end up converging towards that particular attractor. If we are trying to control a system, the basin of attraction should be the whole of state space.

Frequently, the basins of attraction are fractals, and this is one reason for the sensitivity of the system to small changes in initial conditions. They also make cool T-shirts.

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25.9.2 BifurcationIf we solve the equation of motion for different values of the control parameter, we expect the long term solution to converge on a single value, so a plot of final value against parameter is expected to be a monotonic curve. Beyond the stability region, for a fixed end time, the behaviour is expected to be oscillatory, and the plot splits into two branches. Increasing the control parameter still further, causes a doubling of the frequency of oscillation and each branch of the plot again splits into two branches.

25.9.3 Cobweb PlotFor a sampled data system, such as the logistics equation, we plot the current value of the state against its previous value. The equation is of the form:

xn+1= f (xn )

Starting at point x0, a line is drawn vertically to intersect f(x). This is projected horizontally on to the line y=x, and vertically again on to f(x). This is repeated until the solution converges or the oscillation can be characterised.

25.9.4 Feigenbaum NumberThe bifurcation of the plot of final value against control parameter takes place at a definite ratio of the changes in the control parameter between successive bifurcations. As the number of bifurcations approaches infinity, this ratio approaches the value 4.669. This is independent of the actual form of the non-linear function.

25.9.5 Liapunov ExponentA Liapunov exponent characterises the divergence of solutions of a chaotic system. It is effectively the time constant of an exponential fit to the difference between the two solutions as time evolves. It characterises the maximum time span the model may be used over to predict future events.

25.9.6 Poincaré SectionA system characterised by continuous rather than sampled-data dynamics may be investigated for features such as bifurcation and symmetry by converting it into a sampled data system using a sampling frequency which in some sense characterises the motion. (e,g, the period of stable oscillations before the system is driven into the unstable regime).

25.9.7 Symmetry BreakingIn addition to bifurcation, operating the system beyond the stable region frequently introduces an asymmetric motion, even though the equations exhibit symmetry.

25.10 Concluding CommentWhatever else can be said about chaotic systems, from a cybernetics perspective they are bad news. It appears however, that they are detectable from the stability characteristics of the linearised systems. To attempt to control a non-linear system explicitly is to miss the point. We have a sound body of knowledge on which to base our control systems , central to which is a means of deciding whether such control is possible, and what its limitations are.

The word ‘cybernetics’ comes from the Greek word for ‘helmsman’. It is about taking control of systems, rather than characterising the unfolding disaster of the careering train. As technology

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advances, the ability of mankind to acquire mastery over the environment ought to improve. The current ethos of cowering in the rear of the cave, too afraid to intervene, and allowing natural phenomena regularly to slaughter entire populations, simply isn’t acceptable.

Managing fish stocks by simply stopping fishing may not address the problem of overfishing. Perhaps the elimination of a natural predator could have pushed the dynamics into the chaotic region. As long as we base our ideas on the naive direct cause-and-effect, we are going to be surprised at the results.

In many ways the setting of prices in a ‘free’ market resembles the predator/prey system, and we should expect similar dynamic behaviour. The belief that equilibrium prices set using static ideas are stable, seems a very dangerous assumption. Much of human history indicates that they are nothing of the sort.

Systems are not like cars, which we simply drive, they are like donkeys; if we push them too hard, they simply turn around and kick.

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26 Artificial Intelligence and Cybernetics

26.1 IntroductionThis chapter is peripheral to the mathematically rigorous control theory covered in earlier chapters, and is presented for the sake of interest.

The present author has had very little practical experience of artificial intelligence methods, so is not well placed to offer anything but a rudimentary summary of the established techniques.

There is a conceptual problem, in that AI defines intelligence in terms of specific mental skills which humans exhibit, and which the discipline of AI seeks to emulate mechanically. I have already explained my objection to defining a quantity by specifying the manner in which it is implemented, rather than by its actual measurable effect.

Indeed, the author tends to view the relationship between AI and automatic control as akin to that between astrology and astronomy. The former is needed to attract funds from a technically naïve establishment, whilst the latter undertakes real science. But that is a particularly harsh personal view. The reader is encouraged to acquire the facts and form his or her own opinion.

It has been mentioned that the specific mental skills of human beings may be irrelevant to, for example a Blue Whale, which according to the principle of requisite entropy, would almost certainly require a different set of mental skills to survive in its environment. There is something chauvinistic and parochial about seeking to emulate human intelligence, as opposed to finding out what intelligence actually is.

The following describes techniques which are established, or have been tried, with varying degrees of success.

26.2 Intelligent Knowledge Based SystemsIntelligent knowledge-based systems or ‘expert systems’ (note the tendency to hyperbole of the AI world), are essentially databases with an automated search process attached.

Knowledge differs from data in that it consists of inter-relationships between facts, rather than just the facts themselves. Recall the estate agent’s database of chapter 19, the estate agent looks for values for each of the attributes describing the properties which match the customer’s requirements. Much the same effect could be achieved by presenting the customer with a sequence of questions, the answers to which would trigger ‘if…then’ rules to present the next question.

However, an expert system can use the relationships between its rules to find the questions which give rise to particular results. This gives it the ability to backtrack, if a sequence fails to reach the desired result. This feature is essential if the expert system is to be used within a closed loop.

We can imagine that the attributes are descriptors of the state of the system, and a particular combination of attributes triggers an activity which the system performs on the environment, which

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hopefully will change the state to a desirable one. If the actual state is not as desired, the IKBS must be able to select an alternative.

This conceptually yields a measure of how near different states are in terms of the number of rules that must be re-tried before the correct course of action is chosen.

We might be able to organise the IKBS in hierarchical form, with highly quantised states at the highest levels corresponding to initial stages in a sequential set of operations.

From a cybernetic perspective, we are not concerned with how to implement an IKBS. Our concern is with characterizing its dynamic range and time response, which are essential to understand its inclusion within a broader system. Needless to say, expert systems are never characterized in this way.

The process is intelligent, in that it reduces entropy, it is knowledge-based in the sense that it deals with relationships between data items, but a ‘system’ it is not.

Computer Science tends to call data ‘knowledge’, and knowledge (i.e. the organisation of data) as meta-knowledge. Since it is impossible to tell from a single sample whether it constitutes signal or noise, this appears an unwarranted promotion for what could be, and frequently is, meaningless nonsense.

26.3 Fuzzy LogicExpert systems have found widespread applications in diagnostics tools, where the user is not only presented with possible solutions, but also the line of reasoning which led to the result. They deal with highly-quantised, multi-dimensional information typically expressed as data fields, each of which contains a limited number of options.

An expert system is conceptually a finite state machine, and as such is extremely difficult to design to adapt, as is required by requisite entropy.

Although feedback might be applied as a random selection, if there are a large number of stages in a sequence of operations, the adaptation time is likely to be astronomic. A genetic algorithm, for example could be used to devise sequences of operations, by combining those which achieve a result closest to the objective. This raises the question as to what we mean by ‘close’. If the problem is one of guessing the winning lottery number, every possible choice is equally ‘close’ to the winner as all are potential winners.

As Ross Ashby pointed out, Nature is not that perverse as to present us with a lottery each time we need to adapt. However, it is evident that if we can somehow characterise the difference between the desired outcome and the actual outcome by some metric, we should be in a better position to decide on a change in behaviour which will correct the error.

In order to overcome the inherent brittleness of sequential processing and rigidly defined sets of data items, the idea of a fuzzy set was introduced in the 1970s.

With fuzzy sets we try to accommodate the frequently occurring problem that real world entities simply don’t fit neatly into the artificial categories specified by human beings. We considered a few examples in the estate agents case. Rather than define an entity as a member of one set, fuzzy

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reasoning allows us to assign it a degree of membership in more than one, possibly all of the categories we choose to define.

What we are trying to represent is the vagueness associated with our artificial categories. It must be emphasised that we are not trying to define the probability that a specific entity will find itself in one set or another. Probability could be used, but we would need to consider conditional probabilities and apply Bayes’ Theorem. This confusion of degree of set membership with probability has given fuzzy logic a bad name amongst statisticians.

We define a set membership value by first defining the attributes an entity must have to be the quintessential member of a particular category. The membership value is the degree to which the current entity resembles the definitive member of each category. In practice, this is highly subjective, but is a way of capturing human perception of similarities and differences between objects.

Indeed it is in mechanising tasks undertaken by human beings where fuzzy sets are claimed to have some merit over mathematical analysis of the system dynamics.

Suppose we are heating a room with a heater whose output is variable from zero to a defined maximum power. We can define the heater response as high or low according to the degree of membership of the ‘low’ or high fuzzy sets of heater settings.

Similarly we can describe the room temperature in terms of the degree of membership of the sets ‘cold’, ‘moderate’ and hot.

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The control of the room temperature takes the form of a set of fuzzy rules:

‘if the room is hot set the heating low’

‘if the room is cold, set the heating high’

‘if the temperature is moderate set the heating low’

This differs from conventional logic rules in that they are applied simultaneously, rather than sequentially, and the output of each is the set membership value of the associated state of the room.

There are many ways of converting the result obtained from the rules into a heater setting. The simplest is to apply the rule which has the highest output and apply the value to the appropriate heater set. The heater setting is then the value corresponding to the value of membership function of the associated set.

Whilst this overcomes the limitations of crisp sets, the plant behaviour is ignored, so that stability or robustness are matters of faith, rather than demonstrable. However, in mechanising activities undertaken by human beings we may be reasonably confident that the processes are likely to be stable. If nothing else, the fuzzy sets approach illustrates the liberties which can be taken with feedback control.

26.4 Neural NetsFuzzy sets and expert systems are essential ly techniques for capturing human knowledge and skills and implementing them mechanically. Neural nets were introduced as an attempt to extend the emulation to the level of nerve cells.

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Each ‘neuron’ applies weightings to a set of inputs which are then combined and fed through a non-linear function. Usually large numbers of these elements are arranged, usually in three layers denoted input, hidden and output layers. The weightings are determined by training the neural net with a number of examples representative of the task which the neural net is intended to perform.

Whilst considerable success has been claimed, and some actually achieved, there is nothing to prove that a neural net will work for anything outside its training set. It is impossible to prove that it will without some indication as to how the neural net is actually performing its task.

26.5 Concluding CommentThe AI community, like the Computer Science community, seem to set their system boundaries around their particular application, which they then proceed to call a ‘system’. This is a somewhat parochial view, yet it appears to dominate modern systems thinking.

In order to use these elements within a cybernetic system they need to be characterised in some way in terms of their time evolution and errors, otherwise they can only be employed as stand-alone applications. Rather than produce homomorphs of behaviour, which are actually useful for proper systems analysis, we have no option but to implement the actual software explicitly.

In order to use a component which employs AI techniques within a system, it is necessary to characterise it as a system element, in terms of its time evolution, input/output relationships and dynamic range. This requires insight and understanding beyond the current ‘suck it and see’, approaches which seem prevalent. The tendency to design controllers without reference to plant delay is a distinctly worrying development.

The introduction of non-linearity into the system is not a virtue, but a vice, for the reasons which have been presented repeatedly in the earlier chapters of this book. It is bad enough for the plant to be non-linear, without compounding the problem by making the controller non-linear as well, particularly as there is no attempt to characterise the bounds of validity of the method employed.

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27 Conclusions

27.1 IntroductionI have tried to introduce ideas in order of importance, starting with the fundamental principle of cybernetics : the Principle of Requisite Entropy, from which the necessity for feedback is derived.

Even in an apparently open loop controller, the parameters are determined by an outer loop involving the acquisition of the knowledge of system behaviour, which itself followed an iterative process. This process might be the general conjecture/experiment cycle of science, culminating in knowledge in the designer’s brain, or it may take the form of a higher level automatic process whose function is to acquire information on the system behaviour.

For example, economic policy may be tested in a model, and the actions which appear to give the desired results are proposed for implementation. The loop is then closed by the actual response of the economy. Invariably, the predictions are wrong, so the model must be modified to take this into account. This approach indicates a belief that the system is in fact controllable, observable and stable, when all the indicators point to chaos. We are better off with a crude model of the dynamics of the system from which the systemic issue of stability may be addressed rather than concentrating on calculating the wrong answers to great precision.

In general, we do not care how the error null is achieved, it may be implemented in ways which are limited only by ingenuity and imagination. Ross Ashby, and Nature use random selection.

We can say nothing about the system behaviour until we have characterised its time evolution. When the system elements are represented as difference, or differential, equations, this is a direct mapping. More generally, a level of abstraction is required to identify the dynamic processes, and characterise them into differential (or difference) equations which can be linearised around the set of circumstances which interest us.

At a minimum, we can represent the time evolution of a process as a pure delay. Typically, we should characterise the input and output in such a way that the duration of the delay may be related to the attributes of the signal. For example, we might reasonably assume that the greater the entropy change between input and output, the longer the processing will take.

27.2 HierarchyThe important point, which has been hinted at from time to time is that the state variables do not have the same meaning throughout the system . What are parameters to a lower level system may well be states of a higher level system. The functions are different in nature, to the extent that they cannot be lumped together in a single system matrix.

We saw from the sensor scanning example that this does not matter, so long as we keep the processes response times widely separated. Indeed, our concerns for the effects of non linearity are ill-founded; we should never operate the higher level system any faster than between 1/3 rd and 1/10th the speed of its slowest subordinate system.

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Also ,we have seen that our linear analysis will indicate whether we risk not only instability but chaos, and we can determine the bounds on the signals which will push the system into a chaotic regime.

The dynamic model at any level in the system would typically treat lower level elements as infinitely stiff and higher level signals as constants. In view of the ratio of response times, a hierarchy consisting of elements of equal intelligence, must contain between a third and a tenth as many elements in each hierarchical layer as there are in the layer beneath it. A hierarchical pyramid emerges from purely cybernetic principles, based on an egalitarian assumption.

In human hierarchies some myth of intrinsic superiority is usually invoked to justify the right to abuses of power characteristic of leaders. The fact that coercion and oppression tend to figure highly in establishing and maintaining this hierarchy, leads to doubts that our masters really are our ‘betters’.

27.3 IntelligenceDespite appearances, we human beings aren’t really intelligent. Our brains are specialised to a particular set of intellectual activities which presumably were of benefit for survival, and most ‘intelligence’ tests are geared to measuring performance at those particular tasks. Those intellectual skills which lead to financial success figure highly in validation of the tests. In Western society, being white and middle class tends to lead to a higher probability of financial success in later life, ergo ‘intelligence’ is strongly genetic in origin.

In order to handle life as we find it, we put together a set of myths, or heuristics, which we know are not true but assist in helping us understand the world in terms which are familiar. As long as we know we are dealing with convenient fictions, there is no problem. The difficulties begin when we treat them as truth.

Science has sought , through the acid test of experimental verification, to distinguish speculation from fact. When formulating heuristics to describe the world, however, we can tolerate the odd counter example. This is legitimate, provided we don’t misrepresent the area of study as a ‘science’.

Cybernetics lies above the level of this debate. We do not care how the models originate. Our definition of ‘intelligence’ is therefore independent of the specific intellectual feats of the human mind, but is in principle extendable to non-human, and mechanical minds.

The more capable organism may well act out its options in an internal model before committing itself to action. We could ascribe some meaningless term, like ‘thinking’, or ‘imagination’ to this process, but to do so would miss the point. In defining a metric, we need to observe the phenomenon and quantify it, not dictate how it comes about.

27.4 Seeking NullIn order to extend the linear theory analysis to higher levels in the hierarchy it is necessary to find a way of describing the data items in such a manner that it is possible to decide whether the achieved result is equal to the desired result, within the bounds of uncertainty characterising the process. If

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all the possible results can be defined as an enumerated type, all we have to do is search through the set of possible outcomes and compare them in turn with the actual outcome.

For trivial systems with few alternatives, an exhaustive search is probably adequate.

More generally, we need a faster search process. As an example, suppose our objective is to identify a target from a high resolution image. The target may be at any orientation with respect to the observer, and it may come from a large set of potential targets. Furthermore, it is not good enough simply to match the best fit to the known target set, sufficient knowledge of the noise should be available to return an ‘unidentified’ classification.

A search matching all possible targets at all resolutions is probably prohibitive in the time available. We need some metric of similarity before we can determine how well a target resembles a template. The correlation integral introduced in Chapter 19 fulfils this requirement, but doesn’t help as far as run time is concerned.

Supposing a rectangle is fitted to cover the extremities of the target in the image. The aspect ratio of this rectangle should immediately yield information about the target, but this will be ignored. We fit the image with a raster initially of 2×2 pixels, producing a sequence of four samples characterising the image.

Only a proportion of the targets will match this low resolution image. We then repeat the process with a 4×4 pixel image. This is repeated until the full image resolution is achieved, rejecting all but the set of targets which ‘look like’ the actual target at each resolution level.

In practice, the correlation would be done with a Fourier transform, with upper frequency bound set to the reciprocal of the pixel width. In essence, the image would be sampled at ever increasing spatial bandwidths. The algorithm could be arranged, such that the lower bandwidth results could be used to shorten the calculation of the higher bandwidth results.

In characterising the targets in this way, we have a means of quantifying how similar two targets are. This information might itself be useful in generating an error signal to drive the system to its result.

Producing metrics which characterise the data such that their values indicate how close together the corresponding data items are requires considerable ingenuity, but is central to establishing a cybernetic representation of the process from which performance and stability may be assessed. The alternative is the extremely expensive and unreliable process of explicitly constructing the system and testing it.

27.5 And Finally...There is a common misapprehension, particularly amongst the class of incompetent manager who view mathematics as a ‘specialisation’, that the methods of cybernetics apply only to the lowest levels of system operation. Hopefully, the material presented here will go some way to dispelling this myth and exposing it as the fig leaf for ignorance which it is.

To quote Lord Kelvin:

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‘When you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginnings of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science’

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