an introduction to non linearity in control system

7/30/2019 An introduction to non linearity in Control System

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An Introduction to Nonlinearity in ControlSystems

Derek Atherton


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Derek Atherton

An Introduction to Nonlinearity in ControlSystems

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An Introduction to Nonlinearity in Control Systems 2011 Derek Atherton & Ventus Publishing ApS

ISBN 978-87-7681-790-9

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An Introduction to Nonlinearity in Control Systems

Contents

Preface 10

Contents

1

1.1

1.2

1.3

1.4

1.5

2

2.1

2.22.2.1

2.2.2

2.2.3

2.3

2.3.1

2.3.2

2.3.3

Introduction

What is nonlinearity?

Forms of nonlinearity

Structure and Behaviour

Overview of contents

References

The Phase Plane Method

Introduction

Basic PrinciplesThe Linear Case

The Nonlinear Case

An Example

The Phase Plane for Systems with Linear Segmented Nonlinearities

Example 1 Nonlinear Output Derivative Feedback

Example 2 Relay Position Control

Example 3 Position Control with Torque Saturation

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An Introduction to Nonlinearity in Control Systems Contents

2. 4

2.5

2.6

3

3.1

3.2

3.3

3.4

3.4.1

3.4.2

3.4.3

3.5

3.6

3.7

3.8

3.9

3.10

3.11

4

4.1

Conclusions

References

Bibliography

The Describing Function

Introduction

The Sinusoidal Describing Function

Some Properties of the DF

The Evaluation of some DFs

Cubic Nonlinearity

Saturation Nonlinearity

Relay with Dead Zone and Hysteresis

Nonlinear Models and DFs

Harmonic Outputs

Sine plus Bias DF and the IDF

Conclusions

References

Bibliography

Appendix -Tables of Describing Functions

Stability and Limit Cycles using the DF

Introduction

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4.2

4.3

4.4

4.5

4.5.14.5.2

4.5.3

4.5.4

4.6

4.7

4.7.1

4.7.2

4.8

4.9

5

5.1

5.2

5.3

5.3.1

5.3.2

5.3.3

Limit Cycle Evaluation

Stability of a Predicted Limit Cycle

DF Accuracy

Some Examples of Limit Cycle Evaluation

The Van der Pol EquationFeedback Loop Containing a Relay with Dead Zone

Feedback Loop with On Off Relay

Use of the IDF

More than one Nonlinear Element

Applications of SBDF to find Limit Cycles

Asymmetrical nonlinearity

Effect of Bias with Odd Symmetrical Nonlinearity

Conclusions

Bibliography

The SSDF and Harmonically Forced Systems

Introduction

The Cubic Nonlinearity with Two Sinusoidal Inputs

Modified Nonlinearities and the SSDF

Power Law Characteristic

Harmonic Nonlinearity

Ideal Relay

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5.4

5.5

5.6

5.7

5.7.15.7.2

5.8

5.9

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6.1

6.2

6.3

6.4

6.4.1

6.4.2

6.5

6.6

6.6.1

6.6.2

6.6.3

6.6.4

The IDF for Related Signals

More Accurate Determination of Limit Cycles

Closed Loop Frequency Response

Jump Resonance

Jump resonance region- use of the IDFSome Examples of Jump Resonance

Conclusions

References

Limit cycles in relay systems

Introduction

The Frequency Domain Approach

Properties and Evaluation of A loci

Solving for Limit Cycles

Relay with no Dead Zone

Relay with Dead Zone

Limit Cycle Stability

Some Interesting Limit Cycle Problems

Example 1 Invalid Continuity Condition

Example 2 Multipulse Limit Cycles

Example 3 Limit Cycle with a Sliding Mode

Example 4 Multiple Limit Cycle Solutions

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6.6.5

6.6.6

6.7

6.8

6.96.10

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7.1

7.2

7.2.1

7.2.2

7.3

7.3.1

7.4

7.4.1

7.5

7.6

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8.1

8.2

Example 5 Chaotic Motion

Example 6 Unstable Limit Cycles and Simulation

Forced oscillations

Conclusions

ReferencesAppendix

Controller Tuning from Relay Produced Limit Cycles

Introduction

Knowledge from the Limit Cycle

Relay Autotuning

Plant Parameter Identification

Tuning the Controller

An Example

Autotuning using the Relay in Parallel with the Controller

An Example

Conclusions

References

Absolute Stability Results

Introduction

Lyapunovs Method

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8.2.1

8.2.2

8.2.3

8.3

8.3.18.3.2

8.4

8.4.2

8.5

8.5.1

8.5.2

8.5.3

8.6

8.78.8

8.9

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9.2

9.3

9.4

9.5

9.69.7

9.7.1

9.7.2

9.8

9.9

9.10

9.10.1

9.10.2

9.10.39.10.4

9.10.5

9.11

9.12

Definitions of Stability

Positive Definite Functions

Lyapunovs Theorems

Application of Lyapunovs Method

Linear SystemNonlinear System

Definitions and Loop Transformations

Loop Transformations

Frequency Domain Criteria

Popov Theorem

The Circle Criterion

The Off-axis Circle Criterion

Examples

ConclusionsReferences

Bibliography

Design of Nonlinear Control Systems

Introduction

Linearization

Frequency Response Shaping

Nonlinear Compensation

Compensation using DF Models

Time Optimal ControlSliding Mode Control

A Second Order System

Further Comments

Fuzzy Logic

Neural Networks

Exact Linearization

Relative Degree

Input-Output Linearization

Exact State LinearizationExamples

Further Comments

General Comments

References

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An Introduction to Nonlinearity in Control Systems Preface

Preface

The book is intended to provide an introduction to the effects of nonlinear elements in feedback control systems. A

central topic is the use of the Describing Function (DF) method since in combination with simulation it provides an

excellent approach for the practicing engineer and follows on logically from a first course in classical control, such as

the compa nion volume in this series. Some of the basic material on the topic can be found in my ea rlier bo ok which is

frequently referenced throughout the text.

The first chapter provides an i ntroduction to no nlinearity from t he b asic defi nition to a discussio n of the p ossible effects

it can have o n a system and the different behavio ur that might be found, in particular whe n it occurs within a cascade

system of elements or in a fee dback loo p. T he fi nal part of the chapter gives a brief overview of the contents of the book.

Phase plane methods for second order systems are covered in the second chapter. Many systems, particularlyelectromechanical ones, can be approximated by second order models so the concept can be particularly useful in practice.

The second order linear system is first considered as surprisingly it is rarely covered in linear control system texts. The

method has the big advantage in that the effects of more than one nonlinear element may be considered. The study is

supported by several simulations done in Simulink including one where sliding motion takes place.

The third chapter is the first of three devoted to t he study of fe edback loops using DF m ethods. Although the method

is

an approxim ate technique its value, and limitations, are supported by a large num ber of e xamples co ntaining

analytical

results and simulations, including the estimation of limit cycles and loop stability. Many early papers on DFs showing

ho w

theories could be used to predict specific phenomena were supported by simulations done on analogue computers. Here

results from digital simulations using Simulink are presented and this allows much more control of initial conditions

to

show how different modes may exist dependent on the initial conditions. In particular, Chapter 5 contains some more

advanced work including som e new results on jump resonance, so some re aders may wish to omit this chapter o n a first

reading.

A relay is a unique nonlinear ele ment in that its output does not depend upon the input at all time s but is determined by

when the input passes through the relay switching levels. It is this feature which allows the exact determination of

limit

cycles and their stability in a feed back lo op. The b asic theory is presented and som e sim ple examples covere d. It is then

shown in section 6.6 how the approach can be used with computational support to analyse quite complicated periodic

mo des in relay systems. Among the topics covered for the first time i n a textboo k are the evaluation of limit cycles

with

multiple pulses per cycle, as found in a satellite attitude control system, the determination of a limit cycle with sliding

and other more advanced aspects which some readers may wish to omit.

A practical method developed in recent years for finding suitable parameters, or tuning, for a controller based on the

so called loo p cycling met hod of Ziegler and Nichols has us ed relay produced li mit cycles. Chapter 7 covers these

ideas

using both approximate analysis based o n the D F and e xact analysis based o n the relay methods of the p revious chapter.


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An Introduction to Nonlinearity in Control Systems Preface

Chapter 8 covers the topic of absolute stability namely trying to obtain necessary and sufficient conditions for the

stability

of a feed back loop with a single no nlinear elem ent. This probl ems has exercised the minds of theorists for nearly a century

but a solution seems no nearer! Several necessary but not sufficient results are presented, which dependent on ones

viewpoint may be regarded as conservative or robust. The former is usually the case when one has amathematically

defined nonlinearity and t he latter ma y be used because the res ult gives stability for any nonlinearity with certain properties,

for example, lying within a sector.

The final chapter, chapter 9, discusses quite briefly various methods which can be used for the desig n of nonlinear s ystems.

The intent has b een to provide sufficient information o n the me thods and their possible ad vantages and disadvantages.

Several of them have complete books written on the topic and more detail could not have been given without the coverage

of more specialised mathematics.

Finally my thanks to the University of Sussex for the use of an office and access to the computing facilities during my

retirement; to m y good friend Keith Godfrey and his student Roland for the computational input on jum p resonanc e and

their companionship at the races and to my wife, Constance, for her love and support.

Derek P Atherton

University of Sussex

Brighton

May 2011.

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An Introduction to Nonlinearity in Control Systems Introduction

1 Introduction1.1 What is nonlinearity?

In order to analyse the be haviour of engine ering s ystems mat hematical models are required for the vario us com ponents.

It

is commo n practice to try and obtain linear m odels as a rich mat hematical theo ry exists for linear systems. Thes e

models

will always be approximate, although possibly quite accurate for defined ranges of system variables, but inevitably nonlinear

effects will eventually be found for large excursions of system variables. Linear systems have the important property

that they satisfy the superposition principle. This leads to m any imp ortant advantages in metho ds for their analysis.

Fo r

example, in circuit theory when an RLC circuit has both d.c and a.c input voltages, the voltage or current elsewhere in

the circuit can be found b y sum ming the res ults of se parate a nalyses for the d.c. and a.c. inputs taken individually, also if

the mag nitud e of the a.c. voltag e is do ubled then the a .c. voltages and currents elsewhere i n the circuit will be doub led.

Thus, mathematically a linear system with input x(t) and output y(t) satisfies the property that the output for an

input ax1 (t) + bx2 (t) is ay1 (t) + by2 (t) , if y1 (t) and y2 (t) are the outputs in response to the inputs x1 (t) and x2 (t)

, respectively, and a and b are constants. A further point is that when the a.c. voltage is sinusoidal, which is normally

assumed in using the term a .c., then all the other voltages and currents in the circuit are sinusoidal wit h the same freq uenc y

as the input.

A nonlinear system is defined as one which do es not satisfy the superposition prop erty. The simpl est fo rm of nonline arsystem is the static nonlinearity whe re the output depend s only on the current value of input but in a nonlinear manne r,

for example

y(t) = cx(t) + dx3 (t) (1.1)

where the output is the sum mation of a linear and a nonlinear, cubic, term. This description of nonlinearity is given by a

simple m athematical expression. Practical nonli nearities which occur in control enginee ring, however, can ofte n not be

so easily described; so that appro xima ting them ma y require mo re complex ma thematical m odels, such as a high ord er

power series or a linear segmented approximation.

More commonly a nonlinear differential equation, for example

2(1.2)

will describe the behavior. From an engineering viewpoint it may be desirable to think of this equation in terms of a

block diagram, as shown in Figure 1.1, consisting of linear dynamic elements and a static nonlinearity, in this case a3

nonlinear, unlike linear, differential equations. The major point about nonlinear systems, however, is that their

response

is amplitude d ependent so that if a particular form of respo nse, or some measure of it , occurs for one input mag nitude

it may not result for some other input magnitude.


14/239

d y(t) /d(t) 2 + c (dy(t) /dt) 3 + y(t) = x((t)

cubic, which with input dy(t) /dtgives an output c (dy(t) /dt) . Unfortunately there is no general approach to sol ving

12


15/239


Figure 1.1 Block diagram in Simulink for equation 1.2

A further and very important point, is that unlike a linear operation, a nonlinear operation on a sinusoid of frequency,

f, will not produce an o utput at frequency f, alone. For exam ple, if such a sinusoid is applied to the static nonlinearity of

equatio n (1.1) it is easy to s how from substituting x(t) = a cos (~t) where ~ = 2rfthat there are outputs at

frequencies3 3

is that, as will be shown later, they exhibit several forms of unique behaviour which are not possible in linear systems.

1.2 Forms of nonlinearity

All practical systems are nonlinear and in this se ction a b rief over view is given of som e nonlinear effects that often

occur.

In initial designs it may be possible to app roximate t he nonlinear effects by linear m odels b ut invariably it will be

necessary

to finally check their effects on the system performance either in simulation or/and the real hardware. Todays digital

simulation languag es are very g ood but t o use the m efficiently fo r investigating the effe cts of nonlinearity, or to assist

in

the design of a nonlinear system, requires a knowledge of the supporting theoretical methods presented in this

book.

In typical control engineering problems nonlinearity may occur in the dynamics of the plant to be controlled or in the

com ponents used to impleme nt the control . In the latter case, for example , a valve actuator may have a de ad zone due to

friction e ffects and will certainly saturate for large inputs , so this may be referred to as an inhere nt nonlinea rity,

because

it exists although one m ight possibly pre fer this no t to be the case. Alternatively one ma y have inte ntio nal

nonlinearities

which have been p urposely designed into the system to impro ve the system specifications, either for technical or economicreasons. A good e xample of this is the on -off control used in many tem perature control systems, where the o bjective

is

to have the temperature oscillate about the required value.

Identifying the precise form of a nonlinearity may not be easy and like all modeling exercises the golden rule is to be aware

of the approximations in a nonli near model and the conditions for its validity. It might be argued that linear systems theory

is not applicable to practical control engineering problem s because they ar e always nonlinear. This is an

overstatement,

of course, but a valid reminder. All systems have actuator saturation and in some cases it might occur for relatively

low

error signals, fo r example in rotary positio n control it is not unusual for a step inp ut of say 10, or even less, to

produce

the maxim um m oto r drive torque. It simpl y is the result of goo d econom ical design as to produce linear operation to a

higher torque level would require a larger and more expensive motor. Valves used to control fluid or gas flow, apart from

having the nonlinear effects me ntioned above, can have a slightly different behavior when opening compared wit h closing


16/239

fand 3 fof magnitudes ca + 3da /a and da /4 , respectively. Perhaps the most i nteresting aspect o f nonlinear systems

due to the unidirectional pressure of the fluid.

13


17/239


Frictio n always occurs in m echanical systems a nd is very difficult to model, with ma ny quite sophisticated models having

bee n pre sented in the literature. The simplest is to assume t he three co mpo nents, illustrated in Figure 1.2, of stictio n,

an

abbreviation for static friction, Coulomb frictio n and viscous friction. As its name i mplies stiction is a ssumed to e xist

onlyat zero differential speed between the two contact surfaces. Coulomb friction with a value less than stiction is

assumed

to be constant at all speeds, and viscous friction is a linear effect being directly pr oportional to speed. In practice

there

is often a term pro portional to a higher p ower of s peed, and this is also the situatio n for many shaft loads, for exa mple

a

fan for which the drive torque typically increases as a power of speed.

Frict ion Force

V i s c o u s

Stict ion

C o u l o m b

S p e e d

Figure 1.2Three basic friction components

A mathematical expression sometimes used to approximate friction is

n (v) = (a - be- c | v| + d| v|) sgn (v) (1.3)

Here the parameters a, b, c and d are chose n to provide the desired shape ag ainst speed which initially decreases from

the value at zero speed and then starts to increase.

Many mechanical loads are dri ven through g earing rather than directly. Although geared drives, like all areas of technolog y,have improved through the years they always have some small backlash. This may be avoided by using anti-backlash

gears, which are only available for low torques. Backlash, which is a very complicated phenomenon, involving impacts

between surfaces is often modeled in a very simplistic manner. For example, the simple approach used in some

digital

simulation languages, such as Simulink, the simulation com ponent of the well -known soft ware package , Matlab, consists

of an input-output position characteristic of two p arallel straight lines with possible horizo ntal m ovem ent between

them.

This makes two major assumptions, first that the load shaft friction is hig h eno ugh for contact to be maintained withthe

drive side of the backlash when the drive slows down to rest. Secondly when the drive reverses the backlash is

crossed

and the new drive side of t he gear picks up the lo ad instantaneously with no loss of energy in the impact a nd both then

mo ve at the drive shaft spe ed. Clearly both these assump tions are never true in practice b ut no checks exist in

Simulink


18/239

to determine ho w go od they are or, indeed, when they a re co mpletely invalid. Better facilit ies for m od eling

phenomena

such as ba cklash can be found in simulation l anguages such as 20 -Sim or Dynast, which do not use the block concept of

Simulink. A good discussion of friction and backlash can be found in reference1.1.

14


19/239


Probably, the most widely used intentional nonlinearity is the relay. The on -off type, which can b e described mathematically

by the signum f unctio n, that is switches on if its inp ut exceeds zero a nd off if it g oes below zero, is widely used

normally

with some hysteresis betwe en the s witching levels. Use of this approach provides a control strategy where the

controlledvariable os cillates about the desired le vel. The switching mec hanism varies significantly according to the application

from

electro mechanical relays at low spe ed to fast electro nic switches e mploying transistors or thyristors. A co mmo n usage

of

the relay is in the tempe rature control of buildings, where typically the swit ching is pro vide d fro m a tempe rature

sensor

having a pool of me rcury on a metal ex pansion coil. As the temperat ure dro ps the coil contracts and this causes a chang e

in angle of the mercury capsule so that e ventually the mercury moves and closes a contact. When the temperature increases

the coil expands causing a change in angle for the mercury to flow and break the contact.

Electronic switching controllers are being used in many modern electric motor drive systems, for example, to regulate

phase currents in stepping motors and switched reluctance motors and to control currents in vector control drives for

induction motors. Relays with a dead zone, that is, three position relays giving positive, negative and a zero output

are

also use d. When used i n a position control system the zero output allo ws for a steady state position within the dead

zone

but this affects the resulting steady state control accuracy.

Hysteresis effects in m agnetic materials so metime s have to b e modeled . This is often done by as sumi ng a hysteresis loo p

of the form of a B-H loop for a magnetic material typically obtained for a sinusoidal input. However the shape usually

varies with the amplitude and frequency of the input and does not in fact remain constant with a random excitation.

Provided the input is sinusoidal a nd the shape does remain re asonably constant then a nonlinear functio n of the fo rm

n (x, x)o may provide a reasonable m odel as the path taken aro und the hysteresis de pends upon whether the input, x, i s

increasing or decreasing, i.e the derivative of x, xo .

its an interesting world

Get under the skin of it.

Graduate opportunitiesCheltenham | 24,9 45 + benef its

One of the UKs intelligence services, GCHQs role is two-fold:to gath er and analyse intelligence which helps shape Brit ainsresponse to global events, and, to provide technical advice for the

protection of Government communicat ion and information systems.In doing so, our specialists in IT, in ternet, engineering, languages,information assurance, mathematics and intelligence get wellbeneath the surface of global affairs. If you thought the world wasan int eresting place, you really ought to explore our world of work.

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1.3 Structure and Behaviour

From a control engineering viewpoint there are two major reasons why one needs to know about nonlinearity. Firstly

with respect to obtaining mathematical models of devices, particularly if identification techniques are being used,

and

secondly for ensuring the design meets the desired specifications when the control system is nonlinear. To

appreciate

these aspects it is appropriate to discuss very briefly in the introduction a few aspect s of b ehavio ur due to the

presence

of nonlinearity. These are dependent on the structure of the no nlinear system and the relevance of this is e xplained by

again considering a sinusoidal input.

3

If they are placed in cascade and a sinusoidal input applied the output will be a deterministic waveform containing

two frequency components one at the same frequency as the input and the other at three times that frequency. Any

cascade combi nation of linear and nonlinear ele ments will alwa ys produce a deterministic o utput for any given

discrete

input freque ncy spectrum, which in principle can be evaluated . New fre quency com po nents can only be created by the

nonlinearities and the linear elements simple alter the relative magnitudes and phases of these components.

For exam ple, if the above nonlinearity is placed before and after the linear transfer f unctio n and a sinusoid of freque ncy,

f, is ap plied at the input, then the i nput to the se cond nonlinearity will consist of the fundame ntal, f, plus third

harmonic,

3f, with m agnitudes and phases dependent on both the input sinusoidal mag nitude and frequency. These two frequenciesapplied to the se cond nonlinearity will produce an output containing the frequency com pone nts, f, 3f, 5f, 7f, and 9f. One

could define a freque ncy respo nse for s uch a cascade structure of linear and nonli near elements as the ratio of the output

at the fundamental frequency, f, to the input sinusoid at this frequency. The result, as for a linear system, would be a

mag nitud e and phase, which varies with, f, but because of the nonlinearities it would also vary with the am plitude of the

input sinusoid. Thus an approximate frequency response model for the combination could be portrayed graphically by

a set of freq uency response plots for different input a mplitudes, or gain and phase plots against amplitude for

different

frequencies.

For many problems encountered in control engineering this may prove to be a reasonably good model since many of

the linear dynami c eleme nts, like the one given, have low pas s d ynamics so that the frequency, f, will predominate at

the

output. With no linear dynamic elem ents in the com bination then these latter plots would be the same for all freq uencies

and the approximate, first harm onic, or quasi -linearized, model would b e gain and p hase cur ves as a function of the inp ut

amplitude. This r eprese ntation of a nonlinear element is kno wn as a d escribing f unction (DF), which is the topic of several

chapters. Som etimes the above mo del fo r the combination based on the fundam ental fre quency, f, only is referred to as

an amplitude and frequency dependent describing function.


22/239

Consider a nonlinearity x+ dx, and a dynamic element with transfer fu nction K/s (s + b) , where b and dare constants.

16


23/239


If alternatively we assum e the s ystem structure to consist of a feedback co mbinatio n of the linear and nonlinear

elements,3

negative gain then, a very differe nt situation is possible for the response to a sinusoid al input. Dep endent on the

valuesofb, dand the amplitude and frequency of the input, some possibilities for the output are that it is (a) approximately

sinusoidal with the same frequency as the input, similar to the aforementioned cascade connection; (b)

approximately

sinusoidal with a fre quency related to that of the input; (c) a combinatio n of primarily the input freq uency and ano ther

frequency o r (d) a waveform k nown as chaotic, which is not definable mathematically but comple tely repeatable for the

same initial conditions.

These behaviours and others are unique to nonlinear feedback systems, aspects which make such systems extremely

interesting. However, it has meant that no general analytical method is available for predicting their behavior. Several

approaches will be considered in this book all of which will be restricted in their applicability or, put alternatively, the

situations which they can address. Thus the importance of simulation studies for investigating nonlinear systems in

associatio n with analytical m ethods ca nnot be underestimated. Much of the s upport for the theoretical material

presented

in the early chapters, particularly in the 50s to 60s, was done using analogue simulation. Today simulations are done

digitally and several are included using Simulink in the following chapters to illustrate the concepts and provide solutions

for specific problems. Care has to be taken in simulating nonlinear systems particularly those with linear segmented

characteristics because of the discontinuities. Some comments are made on the simulations where appropriate.

1.4 Overview of contents

The phase plane approach discussed in chapter 2 is very useful for step response and stability studies but is basically

restricted to second order systems. However, many engineering systems, particularly in the mechatronics field, may

be

approximated by a second order differential equation so the results are still of value. It also provides a simple basis

for

understanding some of t he more advanced topics, such as optimum control and sliding mode control, covered in c hapter 9.

Stability of a feedback loop is of major imp ortance s o that it is not surprising t hat m uch early work was concerned

withthis topi c. During the 1940s enginee rs in several countries developed wha t has be come known as the describing

function

method where a no nlinearity is replaced b y an amplitude dep endent gain to a sinusoid , known as the describing function,

DF. Chapter 3 introduces the DF, s hows how its value can be calculated for vario us nonlinearities and includes a table of

results. The next two chapters, 4 and 5, deal with applications of the describing function for estimating limit cycles

and

the stability of a nonlinear fee dba ck loop. It is also shown how t he describing function can be evaluated for other than

a single sinusoidal input and applications of describing functions for a bias plus sinusoid and two sinusoids are given.

These include areas such as limit cycle stability, jump resonance and subharmonic oscillations.

The relay is a special form of nonlinear elem ent where the output is not continuously depende nt on the input. This

allows

special results to be developed for relay systems a topic covered in chapter 6. Chapter 7 deals with a design approach

that

has received m uch attentio n in recent years, namely using the inform atio n contained in relay produced limit cycles to set


24/239

with the transfer fun ction K/s (s + b) in the for ward loop and the feedback loop containing x+ dxfed back through a

the param eters of a controller, typically kno wn as relay autotuning. The topic of a bsolute stability, namel y the

development

of exact criteria for guaranteeing the stability of a feedback loop with a single nonlinearity and linear transfer

function,

is covere d in chapter 8. Depe ndent on the viewp oint these results may be re garded as robust , as they prove stability

for

variations in the nonlinearity, or conservative if one is considering stability for a specific nonlinearity.

17


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The coverage to this point, apart from chapter 7, has like many textbooks on linear control, been primarily concerned

with introd ucing analytical tools. Chapter 9, howe ver, fo cuses more on design and looks at how some of the ideas covered

and other techniques may be used for nonlinear control system design. The coverage of these topics is quite brief, some

necessarily so because of the ad ditional theoretical concepts whi ch would have to b e introduced to go into them mo re

deeply. Hopef ully, sufficient i nfo rmation, together with the references, is given for the re ader to unde rstand theconcepts

involved, their possible relevance for particular applications and how they might be applied.

1.5 References

1.1 Friedland, B, 1996 Advanced control system design, Chapter 7 Prentice Hall.

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An Introduction to Nonlinearity in Control Systems The Phase Plane Method

2 The Phase Plane Method2.1 Introduction

A significant amo unt of the research into nonlinear differential equations in the ninetee nth and early twentieth

centuries

done by m athematicians and physicists was devoted to second order differential equations. There were two m ajor

reasons

for this, namely that the dynamics of many problems of practical interest could be approximated by these equations and

secondly the phase pla ne approach allowed a graphical examination of their solutions. The s ystem s of interest in t he

late

nineteenth and early twe ntiet h centuries were fo und in fi elds such as celestial me chanics, nonlinear mechanical

systems

and electronic oscillations. This section will introduce the basic concepts of the phase plane approach and then give a

brief overview of how the method has been further developed for use in control system analysis and design.

A significant amo unt of the early develo pme nt in control theory from 1930 wa s driven by two areas, namely to achieve

better control of ind ustrial processe s and to achieve b etter pe rformance in fire control problems . The problems in the

latter

area received significantly more attention in the war years after 1939, where the requirement in many cases was

related

to the position control of radar antennas and guns in both stationary and moving situations. The dynamic equations

representing many of these position control systems could be rep resented reasonably a ccurately by second order nonlineardifferential equations. It was therefore not surprising that much of the early work on nonlinear control used the

phase

plane approach. Control engineers did make significant contributions to this field since, whereas the earlier work had

typically assume d nonlinea rities defined by continuo us mathema tical functio ns, fo r co ntrol system analysis it was often

more appropriate to approximate intrinsic nonlinearity, such as friction, or intentionally introduced nonlinearity, such as a

rela y, by linear segm ented characteristics. The approach is still useful today be cause of the fundam ental insig ht it

provides

into aspects of nonlinear system behaviour; the fact there are still many control problems which can be approximated by

second order dynamics, and also because more than one nonlinearity can be considered.

2.2 Basic Principles

The formulation used in early work on second order systems was to assume a representation in terms of the two first

order equations

xo1 = P (x1, x2)

xo2 = Q (x1, x2)

Equilibrium, or singular, points represent a stationary system for t he dynamics and occur when xo 1 = xo2 = 0

The slope of any solution curve, or trajectory, in the x1 - x2 state plane is

(2.1)

dx2dx1

= 2 =Q (x1, x2)P (x1, x2)

(2.2)

Before considering the nonlinear case further it is important to first examine the linear case, which is rarely discussed

in


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xo

xo1

detail in te xts on line ar control. Most, usually, just restrict the co ntribution to considering the step respo nse of a

second

order system.

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where A = c m.

s - (a + d) s + ad- bc = 0

m + Bm + C= 0

m1 = - B + (B - 4C) /2 and m2 = - B - (B - 4C)

An Introduction to Nonlinearity in Control Systems

2.2.1 The Linear Case

The equations for the linear situation may be written

xo1 = ax1 + bx2

xo2 = cx1 + dx2

or in matrix form

xo = Axa b

c d

The Phase Plane Method

(2.3)

(2.4)

Eliminating x2 from equations (2.3) by differentiating the first equation and substituting for xo 2 from the second equatio n

and x2 from the first yields

xp1 - (a + d) xo1 + (ad- bc) x1 = 0

which after taking Laplace transforms has the characteristic equation

2

which is the same as the eigenvalue eq uation of the matrix A, with typically replacing s when dis cussed in mathematical

texts.

The roots, or eigenvalues, of the eq uatio n depend o n the values of the ele me nts of A. If the equatio n is writte n in the form

2

then the two eigenvalues are given by

2 1/2 2 1/2/2

The solution to the differential equation is of the form

x1 (t) = K1 e

m1

t + K2 em2t

for m1 ! m2

x1 (t) = K1 em1

t+ tK2 e m2t for m1 = m2

For the linear system there is only one singular point at the origin x1 = 0, x2 = 0 of the state plane and the behaviour

of the motion near to the singular point depends on the eigenvalues.

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Trajectories cannot intersect and can only meet at a singular point. The following four cases can occur for the singular point.

(i) m1 and m2 are both real and have the same sign (either positive or negative).

(ii) m1 and m2 are both real and have opposite signs.

(iii) m1 and m2 are complex conjugates with non zero real parts.(iv) m1 and m2 are imaginary, that is complex conjugates with zero real parts.

Case (i) corresponds to a singular point known as a node, which is stable when the eigenvalues are negative and

unstable

when they are positive. The form of the trajectories near to the singular point is shown in t he simulatio n results of

Figure2

of (1,-2)T and (1,-0.5)T.

Arrows are typically placed on the trajectories showing the direction of motion which will be towards the singular point

for

a stable node, as shown in the figure, and away from it for an unstable one. Trajectories starting on an eigenvector

remain

on it as is clearly seen in the fig ure. For the stable no de all trajectories, apart from t hat starting o n the first eig envect or

tend

towards the no de along the second eige nvector, i.e the o ne with the sm allest slope. The plot showi ng trajectory

motions

from different initial conditions is known as a state or phase portrait, although the latter name is often reserved for

the

special case where xo1 = xo2 , that is a = 0 and b = 1 in equation (2.3)

Figure 2.1Phase Portrait for a Node

The singular point in case (ii) is a saddle point which is onl y theoreticall y reachable from initial conditions on the eigen v ector

of the stable solution associated with the negative eigenvalue. Any small perturbation will cause the trajectories to

diverge

as shown by the phase portrait of Figure 2.2 and move outwards as shown by the arrows.


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2.1, for the transfer function 1/ ( s + 2.5s + 1) which has eigenvalues of -0.5 and -2, with corresponding eigenvectors

21


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Figure 2.2Phase Portrait for a Saddle Point

A phas e portrait for case (iii) is s hown in Figur e 2.3 where the singular point is a focus. The focus is stable when thereal

part is negative, so that trajectories spiral towards it as marked in the figure, and unstable when the real part is

positive.

Four trajectories are shown in the figure.

Figure 2.3Phase Portrait for a Focus Showing Four Trajectories

In the final case (iv) the singular point is a centre, with the oscillatory motion producing concentric trajectories, as

shownin Figure 2.4. Their size depends on the initial conditions, which in a physical situation, such as an ideal oscillating

pendulum, is defined by the initial input energy.

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Figure 2.4Phase Portrait for a Centre

2.2.2 The Nonlinear Case

In general it m ay not be possible to obtain analytical solutio ns to even seco nd orde r nonlinear differential equations

so

the value of the phase plane a pproa ch in early work wa s to allow approximate solutions to b e obtained using

graphical

techniques for sketching phase plane trajectories. The basic approach used was to determine the slope at a sufficient

number of p oints in the state plane to allow a picture of the m otio n to be obtained s tarting from any initial conditions i n

the x1 - x2 state plane. From equation (2.2) a trajectory will have a slope rwhen it crosses the curve

rP (x1, x2) + Q (x1, x2) = 0

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By selecting a range of value s ofr, drawing the cor

an introduction to non linearity in control system

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