singular perturbation methods in control analysis and design classics in applied mathematics

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Singular PerturbationMethods in Control

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Petar KokotoviC, Hassan K. Khalil, and John O'Reilly, Singular Perturbation Methods inControl: Analysis and Design

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Singular PerturbationMethods in Control

Analysis and Design

Petar KokotovicUniversity of California

Santa Barbara, California

Hassan K. KhalilMichigan State University

East Lansing, Michigan

John O'ReillyUniversity of Glasgow

Glasgow, United Kingdom

513J1LSociety for Industrial and Applied Mathematics

Philadelphia

Copyright ©1999 by the Society for Industrial and Applied Mathematics.

This SIAM edition is an unabridged, corrected republication of the work first published byAcademic Press, London, 1986.

10 9 8 7 6 5 4 3 2 1

All rights reserved. Printed in the United States of America. No part of this book may bereproduced, stored, or transmitted in any manner without the written permission of thepublisher. For information, write to the Society for Industrial and Applied Mathematics,3600 University City Science Center, Philadelphia, PA 19104-2688.

Library of Congress Cataloging-in-Publication DataKokotoviC PetarV.,

Singular perturbation methods in control: analysis and design /Petar KokotoviC Hassan K. Khalil, John O'Reilly.

p. cm. • • (Classics in applied mathematics ; 25)Originally published : London: Academic Press, 1986.Includes bibliographical references and index.ISBN 0-89871444-3 (pbk.)1. Automatic control 2. Control theory 3. Perturbation

(mathematics) I. Khalil, Hassan K., 1950- . II. O'Reilly, J. (John) III. SeriesTJ213.K3924 1999629.8--dc21 99-37307

SlxUTL is a registered trademark.

CONTENTS

Preface to the Classics Edition

Preface

1 TIME-SCALE MODELING1.1 Introduction 11.2 The Standard Singular Perturbation Model 21.3 Time-Scale Properties of the Standard Model 9

Case Study 3.1: Two-Time-Scale PID Control 151.4 Slow and Fast Manifolds 171.5 Construction of Approximate Models 221.6 From Nonstandard to Standard Forms 281.7 Case Studies in Scaling 35

Case Study 7.1: Dimensionless e in the DC-Motor Model 36Case Study 7.2: Parameter Scaling in an Airplane Model 37Case Study 7.3: State Scaling in a Voltage Regulator 40

1.8 Exercises 431.9 Notes and References 45

2 LINEAR TIME-INVARIANT SYSTEMS2.1 Introduction 472.2 The Block-Triangular Forms 492.3 Eigenvalue Properties 562.4 The Block-Diagonal Form: Eigenspace Properties 602.5 Validation of Approximate Models 672.6 Controllability and Observability 752.7 Frequency-Domain Models 842.8 Exercises 882.9 Notes and References 90

xi

xiii

90

viii CONTENTS

3 LINEAR FEEDBACK CONTROL3.1 Introduction 933.2 Composite State-Feedback Control 943.3 Eigenvalue Assignment 1023.4 Near-Optimal Regulators 1103.5 A Corrected Linear-Quadratic Design 1283.6 High-Gain Feedback 1363.7 Robust Output-Feedback Design 1433.8 Exercises 1513.9 Notes and References 155

4 STOCHASTIC LINEAR FILTERING AND CONTROL4.1 Introduction 1574.2 Slow-Fast Decomposition in the Presence of White-Noise Inputs 1584.3 The Steady-State Kalman-Bucy Filter 1664.4 The Steady-State LOG Controller 1744.5 An Aircraft Autopilot Case Study 1824.6 Corrected LOG Design and the Choice of the Decoupling Transformation 1864.7 Scaled White-Noise Inputs 1914.8 Exercises 1944.9 Notes and References 198

5 LINEAR TIME-VARYING SYSTEMS5.1 Introduction 2015.2 Slowly Varying Systems 2025.3 Decoupling Transformation 2095.4 Uniform Asymptotic Stability 2165.5 Stability of a Linear Adaptive System 2215.6 State Approximations 2265.7 Controllability 2295.8 Observability 2385.9 Exercises 2435.10 Notes and References 247

6 OPTIMAL CONTROL6.1 Introduction 2496.2 Boundary Layers in Optimal Control 2496.3 The Reduced Problem 2606.4 Near-Optimal Linear Control 2686.5 Nonlinear and Constrained Control 2746.6 Cheap Control and Singular Arcs 2806.7 Exercises 2846.8 Notes and References 286

CONTENTS ix

7 NONLINEAR SYSTEMS7.1 Introduction 2897.2 Stability Analysis: Autonomous Systems 2907.3 Case Study: Stability of a Synchronous Machine 3017.4 Case Study: Robustness of an Adaptive System 3087.5 Stability Analysis: Nonautonomous Systems 3127.6 Composite Feedback Control 3157.7 Near-Optimal Feedback Design 3217.8 Exercises 3337.9 Notes and References 337

References 339

References added in proof 356

Appendix A Approximation of singularly perturbed systems driven by white noise 357

Appendix B 365

Index 367

PREFACE TO THE CLASSICS EDITION

Singular perturbations and time-scale techniques were introduced to controlengineering in the late 1960s and have since become common tools for the modeling,analysis and design of control systems. The 1986 edition of this book, reprinted herein its original form, provides the theoretical foundation for representative controlapplications. Its bibliography includes more than 400 references. Their number hasbeen steadily growing since the 1986 publication of the book.

Recent publications show a significant diversification in control applications ofsingular perturbation techniques. Most current developments can be divided in threegroups. The first group comprises the use of singular perturbations in new controlproblems. One of these stresses the geometric ("slow manifold") aspect of two-time-scalesystems and provides, via high-gain feedback, an interpretation of the important conceptof zero dynamics. Multi-time-scale behavior is also analyzed in high-gain observers,which are employed for semiglobal stabilization of nonlinear systems. A singularperturbation analysis reveals the dangers of large magnitude transients ("peaking") inhigh-gain feedback systems and serves as a basis for low-gain/high-gain designs in whichsuch undesirable transients are avoided. Singularly perturbed H^ control systems havealso been studied, including Hw estimators with small noise.

The second group of new results encompasses extensions and refinements of earliertheoretical concepts. More general singularly perturbed optimal control problems havebeen solved with a broader definition of the reduced (slow) problem based on averaging.New results have been obtained on stability of singularly perturbed systems with the helpof the recently introduced concept of input-to-state stability. Analytical tools for multi-time-scale analysis of Markovian systems have been advanced.

The third group of new results includes diverse problem-specific applications. Forrobotic manipulators, the slow manifold approach has been employed to separatelydesign the slow (rigid system) dynamics and the fast (flexible) transients. Electricmachines and power systems have been an area of major applications of multi-timemethods for aggregate (reduced order) modeling and transient stability studies. Singularperturbations are continuing to be among the frequently used tools in chemical kinetics

xii Preface to the Classics Edition

and flight dynamics. A new application area for multi-time-scale methods are modelsof manufacturing systems.

While these new developments go beyond the topics covered in this book, they arestill based on the methodology described here, which continues to be their commonstarting point.

Petar Kokotovic

Hassan K. Khalil

John O'Reilly

PREFACE TO THE FIRST EDITION

Ever since Prandtl's work at the beginning of this century, singular perturbationtechniques have been a traditional tool of fluid dynamics. Their use spread to otherareas of mathematical physics and engineering, where the same terminology of"boundary layers" and "inner" and "outer" matched asymptotic expansions continuedto be used. In control systems, boundary layers are a characteristic of system two-time-scale behavior. They appear as initial and terminal "fast transients" of state trajectoriesand represent the "high-frequency" parts of the system response. High-frequency andlow-frequency models of electrical circuits, which have had a long history of their own,are naturally incorporated in the new two-time-scale methodology.

In the control literature, the singular perturbation approach of Tikhonov (1952)and Vasil'eva (1963) was first applied to optimal control and regulator design byKokotovic and Sannuti (1968) and Sannuti and Kokotovic (1969a, b) and, morespecifically, to flight-path optimization by Kelley and EdeBbaum (1970) and Kelley(1970a, b, 1971a, b). Applications to broader classes of control problems followed atan increasing rate, as evidenced by more than 500 references* surveyed by Kokotovic(1984) and Saksena, O'Reilly and Kokotovic (1984). For control engineers, singularperturbations are a means of taking into account neglected high-frequency phenomenaand considering them in a separate fast time-scale. This is achieved by treating a changein the dynamic order of a system of differential equations as a parameter perturbation,which, being more abrupt than a regular perturbation, is called a singular perturbation.The practical advantages of such a "parameterization" of changes in model order aresignificant, because the order of every real dynamic system is higher than that of themodel used to represent the system.

Chapter 1 shows that the parameterization of the model order using a perturbationparameter e to multiply the derivatives of some of the state variablesis common to most physical systems with small time constants, inertias and otherphysical quantities which can be expressed as multiples of £. Neglecting these quantitiesmeans setting £ = 0, thus eliminating some of the derivatives from the model and hencereducing its order. It is further shown how a broader class of dynamic systems can bebrought to this "standard" singular perturbation form.

xiv Preface to the First Edition

A geometric property of this form is that it possess a slow manifold which isan equilibrium manifold for the fast phenomena. Examples in Chapter 1 includemodels of DC motors, PID controllers, aircraft, voltage regulators and adaptive systems.Their study is pursued in subsequent chapters.

Chapter 2 is dedicated to properties of linear time-invariant systems that exhibittwo-time-scale behavior because of the presence of both a group of small eigenvalues anda group of large eigenvalues. These systems can be transformed into a slow and a fastpart, each of which can be separately analyzed for stability, controllability and othersystem properties.

Chapter 3 deals with linear feedback control design for linear time-invariantsystems. For state-feedback control, design proceeds by way of an exact decompositionof the state-feedback control problem, be it an eigenvalue assignment problem or anoptimal linear regulator problem, into separate slow and fast problems. Advantage isthereby taken of the singularly perturbed nature of the problem to design a well-conditioned composite feedback control, the sum of slow and fast controls, which solvesthe original ill-conditioned control problem to within a specified order-of-E accuracy.Conditions are also provided for overcoming the fact that, unlike state feedback, staticoutput-feedback design based upon a slow model is nonrobust in the sense that it maynot stabilize the original system.

Singularly perturbed systems with white-noise inputs are more complex because inthe limit their fast variables also behave as white-noise processes. In optimizationproblems this leads to unbounded functionals. Chapter 4 shows that a two-time-scalenear-optimal design of filters and regulators is possible with major savings in on-line aswell as off-line computations.

The remaining three chapters deal with time-varying and nonlinear problems,where the benefits of a separation of time scales are even more pronounced. Aperturbation approach is often the most effective, and sometimes the only way to solvesuch problems.

Time-varying systems in Chapter 5 differ from their time-invariant counterpartsin Chapter 2 owing to time variations of the coefficients as a potential new cause ofmultiple-time-scale behavior. It is shown, however, that if the parameter variations areslow compared with the fast dynamics, then the time-scale phenomena remainqualitatively the same as in time-invariant systems. The methods of Chapter 5 takeadvantage of this structure by introducing a "frozen" parameter treatment of the fastsystem and showing when this conceptually appealing approximation is valid.

Control problems in Chapter 6 are of the trajectory optimization type. They arerestricted to finite time intervals, and hence must take into account the fast phenomenaat both ends of the trajectory. Initial and end conditions for these "boundary layer"phenomena must be properly "matched." Chapter 6 circumvents the matchingdifficulty by an explicit use of the Hamiltonian property of optimization problems.

Preface to the First Edition xv

Thanks to this property the initial and end layers can be separated by an exacttransformation.

Nonlinear stability and stabilization problems in Chapter 7 are greatly simplifiedby exploitation of the two-time-scale system property. A Lyapunov function for anonlinear singularly perturbed system is constructed via a two-time-scale stabilityanalysis, which is in many respects analogous to the linear analysis of Chapter 5 becausethe slow states appear as slowly varying parameters in the fast system. A two-stagefeedback design, the so-called composite control design, is used to obtain stabilizing andnear-optimal feedback controllers. Unlike the finite-time near-optimal control ofChapter 6, the feedback nature of the near-optimal composite control is required forstabilization over an infinite time interval.

The chapters of the book may be studied sequentially or in a number of other ways.For example, readers interested in stochastic control would concentrate on Chapter 4after familiarizing themselves with the contents of Chapter 3 on linear feedback control.Chapter 5 on time-varying systems could be read immediately after Chapter 2. Otherpossibilities are that Chapter 6 on optimal control and Chapter 7 on nonlinear systemscould be proceeded to immediately after Chapter 1 and Chapter 5.

Ac knowledgmentsFrom its initiation almost two decades ago, the research in singular perturbation

methods for control system analysis and design described in these pages has beenencouraged and fostered by M. E. Van Valkenburg, J. B. Cruz, Jr., W. R. Perkins andother friends at the Coordinated Science Laboratory and the Electrical and ComputerEngineering Department, University of Illinois at Urbana. The Department of ElectricalEngineering and Systems Science, Michigan State University, East Lansing and theDepartment of Electronic and Electrical Engineering, University of Strathclyde,Glasgow have also been highly supportive of this effort.

While responsibility for any errors or shortcomings in this book rests with theauthors, its inspiration and content owe much to many colleagues, friends, co-workersand supporters. We are particularly thankful for the contributions of K. W. Chang, J.H. Chow, Z. Gajic, A. H. Haddad, F. Hoppensteadt, P. A. loannou, S. H. Javid, R. EO'Malley, Jr., G. Peponides, A. Saberi, P. Sannuti, V. I. Utkin, A. B. Vasil'eva, R. R.Wilde, R. A. Yackel and K. K. D. Young.

Finally, thanks are due to Rose Harris and Dixie Murphy for their unceasing effortsin typing the many drafts that went to make the major portion of the book, and alsoto Bob MacFarlane for the artwork, all of whom are with the Coordinated ScienceLaboratory of the University of Illinois.

P. V. Kokotovic

H. K. Khalil

J. O'Reilly

* A selection from these references is reprinted in a volume published by IEEEPress, edited by Kokotovic and Khalil (1986).

1 TIME-SCALE MODELING

1.1 Introduction

When a control engineer uses singular perturbation methods to solveproblems in his field, the first problem he faces is one of modeling, that is,how to mathematically describe the system to be controlled. Modeling forcontrol is parsimonious and implicit. It is parsimonious, because the modelshould not be more detailed than that required by the specific control task.It is implicit, because the extent of the necessary detail is not knownbefore the control task is accomplished. Typical control tasks are optimalregulation, tracking and guidance. Since these tasks are to be accomplishedin the inevitable presence of unknown disturbances, parameter variationsand other uncertainties, the control system must possess a sufficient degreeof robustness or insensitivity to the extraneous effects.

How do singular perturbation techniques respond to this challenge?Their key contribution, from which all other benefits follow, is at the levelof modeling. Control engineers have been simplifying their models longbefore they were told that what they were doing was a singular perturbation.As our bibliography shows, they became aware of the new toolabout fifteen years ago and have been increasingly interested in it eversince.

For the control engineer, singular perturbations legitimize ad hoc sim-plifications of dynamic models. One of them is the neglect of "small" timeconstants, masses, capacitances, and similar "parasitic" parameters whichincrease the dynamic order of the model. However, a design based on asimplified model may result in a system far from its desired performance,or even in an unstable system. If this happens, the control engineer needsa tool that will help him to improve his oversimplified design. What isrequired is to treat the simplified design as a first step, which captures the

1

2 1 TIME-SCALE MODELING

dominant phenomena; then the disregarded phenomena, if important, aretreated in the second step.

It turns out that asymptotic expansions into reduced ("outer") andboundary-layer ("inner") series, which are the main characteristics ofsingular perturbation techniques, coincide with the outlined design stages.Because most control systems are dynamic, the decomposition into stagesis dictated by a separation of time scales. Typically, the reduced modelrepresents the slowest (average) phenomena, which in most applicationsare dominant. Boundary layer (and sublayer) models evolve in faster timescales and represent deviations from the predicted slow behavior. The goalof the second, third, and later, design stages is to make the boundary layersand sublayers asymptotically stable, so that the deviations rapidly decay.The separation of time scales also eliminates stiffness difficulties and pre-pares for a more efficient hardware and software implementation of thecontroller.

In this chapter, some of the basic concepts of singular perturbationasymptotics and time-scale modeling are introduced by way of illustrativeexamples. So as to develop a more intuitive feel for time-scale phenomena,technical details and proofs are postponed to subsequent chapters. Sincethe rest of the book is design-oriented, some of the modeling examples,developed in this chapter, will later be used to illustrate various analysisand design results.

1.2 The Standard Singular Perturbation Model

The singular perturbation model of finite-dimensional dynamic systems,extensively studied in the mathematical literature by Tikhonov (1948,1952), Levinson (1950), Vasil'eva (1963), Wasow (1965), Hoppensteadt(1967, 1971), O'Malley (1971), etc., was also the first model to be used incontrol and systems theory. This model is in the explicit state-variable formin which the derivatives of some of the states are multiplied by a smallpositive scalar e, that is,

where a dot denotes a derivative with respect to time t, and / and g areassumed to be sufficiently many times continuously differentiable functionsof their arguments x, z, E, t. The scalar e represents all the small parametersto be neglected. In most applications, having a single parameter is not arestriction. For example, if Tl and T2 are small time constants of the same

1.2 STANDARD SINGULAR PERTURBATION MODEL 3

order of magnitude, O(7\) = O(T2), then one of them can be taken as eand the other expressed as its multiple, say Tl = e, T2 - oce, where a =T2/Tl is a known constant. We postpone the discussion of important scalingissues and the meaning of the O-symbol until Section 1.7. In this section,our goal is to introduce e as a modeling tool.

In control and systems theory, the model (2.1), (2.2) is a step towards"reduced-order modeling", a common engineering task. The orderreduction is converted into a parameter perturbation, called "singular".When we set e = 0, the dimension of the state space of (2.1), (2.2) reducesfrom n + m to n because the differential equation (2.2) degenerates intothe algebraic or transcendental equation

where the bar is used to indicate that the variables belong to a system withe = 0. We will say that the model (2.1), (2.2) is in standard form if andonly if the following crucial assumption concerning (2.3) is satisfied.

Assumption 2.1

In a domain of interest (2.3) has k ^ 1 distinct ("isolated") real roots

This assumption ensures that a well-defined n-dimensional reduced modelwill correspond to each root (2.4). To obtain the rth reduced model, wesubstitute (2.4) into (2.1),

and keep the same initial condition for the state variable x(t) as for x(f). Inthe sequel, we shall drop the subscript i and rewrite (2.5) more compactlyas

This model is sometimes called a quasi-steady-state model, because z, whosevelocity z = g/e can be large when e is small, may rapidly converge to aroot of (2.3), which is the quasi-steady-state form of (2.2). We will discussthis two-time-scale property of (2.1), (2.2) in the next section.

The convenience of using a parameter to achieve order reduction alsohas a drawback: it is not always clear how to pick the parameters to beconsidered as small. Fortunately, in many applications our knowledge ofphysical processes and components of the system sets us on the right track.Let us illustrate this by examples. In all four examples the control input uis a given differentiable function of time, u = u(f).


Example 2.1

A common model for most DC motors, shown in Fig. 1.1, consists of amechanical torque equation and an equation for the electrical transient inthe armature circuit, namely,

where /, u, R and L are the armature current, voltage, resistance andinductance respectively, / is the moment of inertia, co is the angular speed,

Fig. 1.1. Armature controlled DC motor.

and ki and kco are respectively the torque and the back e.m.f. developedwith constant excitation flux 0. In practically all well designed motors, Lis small and can play the role of our parameter e. This means that co = xi = z and the model (2.7), (2.8) is in the standard form (2.1), (2.2) wheneverRJ=Q. Neglecting L, we solve

to obtain

which is the only root (2.4), and substitute it in (2.7). The resulting model

is the commonly used first-order model of the DC motor in the form of(2.6). We shall return to (2.7), (2.8) for a discussion on scaling in Section1.7.

4

1.2 STANDARD SINGULAR PERTURBATION MODEL

Example 2.2

In the feedback system of Fig. 1.2(a) with a high-gain amplifier K and anonlinear block N(z), the choice of e is not as obvious. However, any

Fig. 1.2. High-gain loop (a) approximated by the inverse of the feedback path (b).

student of feedback systems would pick e = l/K, where K is the amplifiergain, and obtain

To be specific, let N(z) = ez - 1 and obtain from (2.3) and (2.4) applied to(2.12),

5

or


Hence Assumption 2.1 is satisfied whenever u - x > -1. Then, the reducedmodel is

This model is represented by the block diagram in Fig. 1.2(b) in which theloop with infinite gain e = 0 is replaced by the inverse of the operator inthe feedback path.

Example 2.3

Using capacitor voltages vi and v2 as the state variables, the RC circuit inFig. 1.3(a) is described by the equations

Fig. 1.3. Low-frequency "equivalents" of the circuit (a) when C2 is small (b) and when ft, issmall (c).

6

1.2 STANDARD SINGULAR PERTURBATION MODEL 7

Suppose first that the capacitor C2 is small; that is, let C2 = e. ThenVi= x, v2 = z, and the root (2.4) obtained after letting e = 0 in (2.16) is

which represents an ideal voltage divider formed by the series connectionof R\ and R2. When (2.17) is substituted into (2.15), the reduced model ofthe RC circuit is

and represents the circuit with C2 removed, as in Fig. 1.3(b), a common"low-frequency equivalent" of the original circuit in Fig. 1.3(a).

In all the examples considered thus far, the original models are in thestandard form (2.1), (2.2) because Assumption 2.1 is obviously satisfied.To avoid a misleading conclusion that this is always the case, let usreconsider the RC circuit in Fig. 1.3, but this time assuming that the smallparameter is the resistance R^.

Example 2.4

Multiplying (2.16) by RjR2 and letting Rl = s, (2.15) and (2.16) become

If this model were in the form (2.1), (2.2), both Vi and v2 would beconsidered as z-variables and (2.3) would be

However, Assumption 2.1 would then be violated because the roots of(2.3), in this case v\ = v2, are not distinct. The question remains as towhether the model of this PC network can be simplified by setting e = 0,that is, by neglecting the small parasitic resistance R^. For a simple answerwe multiply (2.19) by C\, (2.20) by C2, add them together and obtain anequation without e, that is,


This suggests that instead of I?! we can use

(2.23)

as a new voltage variable, which, along with v2 = z, transforms (2.19),(2.20) into

Now (2.3) becomes

and it satisfies Assumption 2.1. The substitution of z = ^ into (2.24) resultsin the reduced model

describing the circuit in Fig. 1.3(c). Every electrical engineer would proposethis circuit as a "low-frequency equivalent" of the circuit in Fig. 1.3(a)when R! = e is small.

Most of the singular perturbation literature assumes that the model (2.1),(2.2) is in the standard form; that is, it satisfies Assumption 2.1. Theimportance of Example 2.4 is that it points out the dependence of Assump-tion 2.1 on the choice of state variables. In most applications a goal ofmodeling is to remain close to the original "physical" variables. This waspossible in our Examples 2.1, 2.2, 2.3, but not in Example 2.4, where anew voltage variable (2.23) had to be introduced. However, few engineers,accustomed to the simplified "equivalent" circuit in Fig. 1.3(c), wouldquestion the "physicalness" of this new variable. On the contrary, physicalproperties of the circuit in Fig. 1.3 (a) are more clearly displayed by thestandard form (2.24), (2.25).

For a simple RC circuit we have been able to find a way to convert anonstandard model into the standard form. A systematic approach tononstandard models will be presented in Section 1.6.

1.3 TIME-SCALE PROPERTIES OF STANDARD MODEL

1.3 Time-Scale Properties of the Standard Model

Singular perturbations cause a multi-time-scale behavior of dynamic sys-tems characterized by the presence of both slow and fast transients in thesystem response to external stimuli. Loosely speaking, the slow response,or the "quasi-steady-state", is approximated by the reduced model (2.6),while the discrepancy between the response of the reduced model (2.6)and that of the full model (2.1), (2.2) is the fast transient. To see this, letus return to (2.1)-(2.6) and examine the variable z which has been excludedfrom the reduced model (2.6) and substituted by its "quasi-steady-state" z.By contrast with the original variable z, starting at f0 from a prescribed z°,the quasi-steady-state z is not free to start from z°, and there may be alarge discrepancy between its initial value

and the prescribed initial condition z°. Thus, z cannot be a uniformapproximation of z. The best we can expect is that the approximation

will hold on an interval excluding f0> that is, for t E [rl5 T\ where tl > t0.However, we can constrain the quasi-steady-state x to start from theprescribed initial condition x°, and hence the approximation of x by x maybe uniform. In other words,

may hold on an interval including ?0, that is, for all t in the interval [t0, T\on which x(i) exists.

The approximation (3.2) establishes that during an initial ("boundarylayer") interval [t0, ti] the original variable z approaches z and then, during[ti, T], remains close to z. Let us remember that the speed of z can belarge, z = g/e. In fact, having set e equal to zero in (2.2), we have madethe transient of z instantaneous whenever g =£ 0. Will z escape to infinityduring this transient or converge to its quasi-steady-state z?

To answer this question let us analyze ez, which may remain finite, evenwhen e tends to zero and z tends to infinity. We set

9

1.3 TIME-SCALE PROPERTIES OF STANDARD MODEL 11

Assumption 3.2

The eigenvalues of dg/dz evaluated, for e = 0, along x(t), z(r), have realparts smaller than a fixed negative number, i.e.

Both assumptions describe a strong stability property of the boundarylayer system (3.6). If z° is assumed to be sufficiently close to z(f0), thenAssumption 3.2 encompasses Assumption 3.1. We also note from (3.9)that the nonsingularity of dg/dz implies that the root z(t) is distinct asrequired by Assumption 2.1. These assumptions are common in much ofthe singular perturbation literature (e.g. Tikhonov, 1948, 1952; Levins1950; Vasil'eva, 1963; Hoppensteadt, 1971). These references contain thproof and refinements of the following fundamental theorem.

Theorem 3.1

If Assumptions 3.1 and 3.2 are satisfied, then the approximation (3.3),(3.7) is valid for all t E [f0, 7], and there exists f j 2= r0 such that (3.2) is validfor all re [ r l 5 T}.

We will refer to this result as "Tikhonov's theorem", although the formin which it is stated here is due to Vasil'eva and other authors. We do notgive the proof of this theorem. Instead, we provide independent proofs ofits special cases as they are needed in subsequent chapters of this book.This approach is more in tune with control applications, in which require-ments (3.8) and (3.9) are to be satisfied by controller design. Let us onlypoint out that the proof of Theorem 3.1 makes use of two-time-scaleasymptotic expansions of x, z, having some terms defined in r-scale andothers in r-scale. We illustrate this technique by substituting only the firstterms

into (2.1), (2.2). Then, expressing the derivatives of x and z with respectto both t and r, using (3.4), the 'system (2.1), (2.2) is rewritten as


Now, requiring that x(f) satisfy the reduced model (2.5), we obtain from(3.H)

and hence dje(T)/dr-> 0 as £-> 0. Therefore, X(T) = const for e = 0, and,since x(t0) = x°, this constant is zero; that is, x(i) = 0 for e = 0. This showsthat if the bracketed quantity on the right-hand side of (3.13) remainsbounded, then x(t, e) —> x(t) as e-> 0. That the error is not larger than O(e)would follow from a more detailed analysis of (3.13). Using the fact thatX(T) = 0, and requiring that z(t) satisfy (2.3), we obtain from (3.12)

where z is determined by differentiating (2.4) with respect to t. We notethat (3.14) is a more general form of the boundary layer system (3.6). If in(3.14) we substitute t = t0 + er, using (3.5), and then let e-» 0, we obtain(3.6) as the limiting case of (3.14). For this reasoning to make sense, theequilibrium z(r) = 0 of (3.6) is required to possess the stability propertiesspelled out in Assumptions 3.1 and 3.2. General asymptotic expansionprocedures use two power series in e to represent x(t, e). The coefficientsof the first series are functions of t, and those of the second are functionsof T. Two analogous series are used to represent z(t, e). After these seriesare substituted into (2.1), (2.2), equations similar to (3.11), (3.12) areobtained and matched, term by term, to calculate the coefficients of theseries. We will comment on such higher-order approximations in sub-sequent chapters in connection with specific applications. Here, variousaspects of Theorem 3.1 are illustrated by examples.

Example 3.1

To apply Theorem 3.1 to the high-gain feedback system in Fig. 1.2(a) weexamine the boundary layer system (3.6) corresponding to (2.12), whichdescribes the fast transient in the inner feedback loop of Fig. 1.2(a); namely,using (2.13) and «(f0) = u°,

Since -x° + 1 + w° > 0 by Assumption 2.1, as in Example 2.2, the equi-librium z = 0 of (3.15) is uniformly asymptotically stable for all z(0) and


Assumption 3.1 is satisfied. Also, for e = 0, z = z, we have

and hence Assumption 3.2 is satisfied for all z. We conclude that Theorem3.1 holds whenever u - x > -1, which is also a necessary and sufficientcondition for the existence of the reduced model (2.15). While this restric-tion is imposed on the two-time-scale approximation, the solution of theexact system (2.11), (2.12) for e > 0 is not restricted and it exists even whenu-x< -1.

Example 3.2

In this example there are three possible but only two valid reduced models,determined by the initial condition for z. Consider the system

In this case (2.3) is

and has three distinct roots

that is, there can be three reduced models. Analyzing the boundary layersystem

we can readily check that Assumptions 3.1 and 3.2 hold for z = —xt ifz° < 2, and for z = 4 if z° > 2. Both assumptions are violated by z = 2,which is an unstable equilibrium of (3.21). Hence, there are only two validreduced models

Note that the solution x = (1 - g?2)"1 of x = \x2t escapes to infinity att = 2V2. However, Theorem 3.1 still holds for t £ [t0, T\ with T< 2V2.


Fig. 1.4. Exact (solid) and approximate (dotted) solutions of (3.17), (3.18).

This is illustrated by simulation results in Fig. 1.4 for four different valuesof z°, two for each reduced model. The approximate (dotted) and theexact (solid) trajectories for z are virtually indistinguishable, although e isrelatively large at e = 0.3. The trajectories in Fig. 1.4 clearly exhibit a two-time-scale behavior. They start with a fast transient of z(t, e) from z° toz(r), which is the so-called boundary layer, or the "inner" part of thesolution z(t, e). After the decay of this transient, they remain close to z(f),which is the slow ("outer") part of the solution, also called the "quasi-steady-state" of z(r, e). In general, the "quasi-steady-state" varies withtime, as does z(t) = -te~l. Exceptionally, as in the case z = 4, the "quasi-steady-state" can be a true steady state of z. For a fuller understanding ofthe general boundary layer system (3.14) it is useful to see the meaning ofthe term ez(t) for z(t) - -re"' in this example.

While the preceding two examples treat boundary layers as correctiveterms to the reduced slow solution, which is the more relevant part of theexact solution, in many applications both fast and slow parts are important.In such applications, the two-time-scale approximation (3.3), (3.7), vali-dated by Theorem 3.1, is a tool for separate analysis and design of the slowand fast subsystems, the main topic of this book. The following case study


formulates one of the classical control designs, the PID regulator, as a two-time-scale problem. This study is a precursor of methods to be developedin Chapters 2 and 3, and will be reconsidered there in more detail.

Case Study 3.1 Two-Time-Scale PID Control

A classical feedback loop appearing in most single-variable process controlproblems is shown in Fig. 1.5, with the transfer function of the PID blockdefined by

where kp, kD and ki are the customary tuning parameters, d is an unknowndisturbance, y is the output to be regulated to the desired set-point r, and

Fig. 1.5. PID control system.

s is the Laplace transform complex variable. In this example we shall treatr and d as step inputs, that is, as constants for t > 0. The presence of theintegral term l/s in (3.23) will then guarantee that the regulation errore(t) —> 0 as r-» so, provided that the feedback loop is asymptotically stable.The property that for constant disturbances the integral control actionreduces the regulation error to zero is true for all plants, linear or nonlinear,for which the loop remains asymptotically stable. The task of the pro-portional gain kp is to speed up the process, and that of the differentialterm kD is to stabilize it. A classical design approach is to first choose kp

and kD for the desired transient performance and then to insert a smallintegral gain k{ to slowly regulate the error to zero. In this way, the integralterm contributes to the steady-state accuracy of the regulation, withoutmuch interference with the loop stability properties. For the sake ofargument, assume that the plant transfer function is


where k0 is the gain of the plant and is assumed to be one, kQ = 1, withoutloss of generality. This plant can be thought of as a motor driving a heavyinertial object, or as a two-reservoir system. In the first case the disturbanced can be a load torque, while in the second case it can be a constant outflowor inflow.

To model this feedback loop as a singularly perturbed system, we needto mathematically express the fact that the integral action is much slowerthan the other two actions. This will be so if its gain ki is O(e)\ that is, ifki = ek}, where £r is 0(1). So as to simplify notation, we also introducek1 = kp, k2 = kPkD, k3 = kpki and rewrite the controller transfer function(3.23) as

Choosing the state variables in the fast time scale T = t/e,

we obtain the following state representation of (3.24), (3.25):

Equation (3.29) shows the reason for denoting the time variable as the fasttime scale r. Namely, when e = 0 the integral action e3 is constant, whichis the characteristic of the slow variable's behavior in the fast time scale;see (3.13) and the discussion following it. To observe the slow transientcaused by the integral action, we rewrite the system (3.27)-(3.29) in theslow time scale t = er. In doing so, we also identify e\ = z\, e2 = z2 as thefast variables and e3 = x as the slow variable, and remind the reader thata dot denotes the derivative with respect to the slow time variable t. Thesystem (3.27)-(3.29) is thus brought to the standard form (2.1), (2.2),namely,

1.4 SLOW AND FAST MANIFOLDS 17

To satisfy Assumption 2.1 we need &i =£ 0, while to satisfy Assumptions3.1 and 3.2 we must choose kl and k2 in the PD part of the design suchthat the matrix in (3.31), (3.32) be Hurwitz, that is,

For the I-part of the design the reduced model of (3.30)-(3.32) is obtainedwith z2 = 0, zl = -(kijc + d)/kl; that is,

Equation (3.34) shows that the slow transient due to integral control isadjusted by the choice of the tuning parameter £I5 while its extent isproportional to d and inversely proportional to kpki. In this case, theboundary layer system (3.6) is

that is, it is a homogeneous system with the matrix (3.33), used for the PDpart of the design. The fact that the disturbance term does not appear in(3.35), but appears in (3.34), clearly indicates the roles of the PD and Iactions of the controller (3.25).

1.4 Slow and Fast Manifolds

It is recalled that, as functions of time, the solutions x(t, e), z(t, e) of thesingularly perturbed system (2.1), (2.2) consist of a fast boundary layer anda slow quasi-steady-state. Of the two components x(t, e) and z(t, e), thelayer is significant only in z(f, e), while x(t, e) is predominantly slow sinceits layer is not larger than O(e). Our aim in this section is to give a geometricview of the two-time-scale behavior of x(t, e), z(t, e) as trajectories inRn + m. We consider a simpler form of the system (2.1), (2.2), namely,

where the dependence of / and g on e and t is suppressed. Insofar as e isconcerned, this is done only to simplify notation, while the elimination of


the dependence of / and g on t is made to allow us to use the concept ofinvariant manifolds.

In the (n + ra)-dimensional state space of x and z, an ^-dimensionalmanifold Mf, depending on the scalar parameter e, can be defined by theexpression

where it is assumed that 0 is a sufficiently many times continuously dif-ferentiable function of x and £. This expression reduces the dimension ofthe state space from n + m to n, by restricting the state to remain on M£.For example, if in R3 we have n = 2 and m = 1, then M£ will be a surfacedefined by one scalar equation (4.3), while for n = 1 and m = 2 the twoscalar equations (4.3) will define a curve. For Me to be an invariant manifoldof (4.1), (4.2) the expression (4.3) must hold for all t > t* if it holds for t =f*; that is,

Differentiating this expression with respect to t, we obtain

Multiplying (4.5) by e, and substituting x and z from (4.1), (4.2) and z from(4.3), we obtain a manifold condition

which 0(*, e) must satisfy for all x of interest and all e E [0, e*], where e*is a positive constant. To begin with, let us analyze Me for e = 0, that is,M0 defined by (4.6) at E = 0,

In (4.7) we recognize the expressions (2.3), (2.4) obtained by formallyneglecting e in (2.1), (2.2). There can be several functions cf>(x, 0) satisfying(4.7), and hence several manifolds M0. By Assumption 2.1, they have npoints in common, that is, they do not intersect each other and are non-tangential in the domain of interest. For a further interpretation of M0 werewrite (4.1), (4.2) in the fast time scale r, defined by (3.4), (3.5). From

we conclude that at e = 0 both cbt/dr and dz/dr are zero for z = <£(#, 0).


Hence A/0 is an equilibrium manifold of (4.8) at E = 0. If Assumption 3.2is satisfied for all x, z G M0 then A/0 is a stable (attractive) manifold.

Remark 4.1 For the remaining discussion in this section, it is sufficientthat A/o be "conditionally stable"; that is, we can relax the requirement<-c in (3.9) to =£0. This relaxed form of Assumption 3.2 will be used inChapter 6 when dealing with optimal trajectories having boundary layersat both ends.

A consequence of Theorem 3.1 is that the existence of a conditionallystable equilibrium manifold M0 of (4.8) for E = 0 implies the existence ofan invariant manifold ME of (4.1), (4.2) satisfying the manifold condition(4.6) for all e G [0, £*] and converging to A/0 as E—> 0; that is,

Further insight is gained if the deviation of z from ME is represented by anew variable

In terms of x and r] the system (4.1), (4.2) becomes

The invariant manifold Af£ is now characterized by the fact that r] = 0implies r) = 0 for all x for which (4.6) is true. Thus, if r](f0) = 0, all we needto solve is

-which is the exact slow model of (4.1), (4.2) valid for jc, z G M£. The reducedmodel (2.5) is its approximation in which (f>(x, E) is replaced by (f>(x, 0).We call Af£ a slow manifold of (4.1), (4.2), stressing that to each slowmanifold there corresponds a slow model (4.13). To introduce the notionof a fast manifold we examine the behavior of 77 in the fast time scale Substituting (4.10) into (4.8) and letting e-» 0, we see that rj at E = 0 isdefined by the boundary layer system

which we have already encountered in (3.6). This system describes the


trajectories x, 77, which, for every given x°, lie in a fast manifold Fx definedby x = x° = const, and rapidly descend to the equilibrium manifold A/0.For e larger than zero but small, the fast manifolds are "foliations" ofsolutions rapidly approaching the slow manifold ME. Let us illustrate thispicture by two examples.

Example 4.1

For the system

the manifold condition (4.6) is

Fig. 1.6. Trajectories of the system (4.15).

and it is easily checked that it is satisfied by

In this very special case Me not only has a simple analytical expression, butit even coincides with M0, because (4.17) does not depend on e. Sincedg/dz = -1, the slow manifold is stable, see the plots in Fig. 1.6. The slowmodel (4.13) for z(f0) = -[*('o)]4/3 and all x(t0) is x = -x5.


Example 4.2

To illustrate the possibility of several slow manifolds we consider the system

which is a variation on the theme of Example 3.2. The manifold condition(4.6) is

and is not easy to solve. However, at e = 0 we have three equilibriummanifolds defined by

Since they intersect for some negative values of Jt, let us assume that onlyx 5: 0 is of interest. From

it is observed that M\ and MQ are stable, while MQ is unstable.

We close this section with a brief look at the linear system

which will be studied in detail in Chapter 2. The manifold condition (4.6),in this case

can be satisfied by a 0 linear in x,

which, when substituted in (4.24), results in

a

It is shown in Section 2.2 that a solution L(e) of this equation exists and isclose to L(0) = A & A 21- The substitution

into the system (4.22), (4.23) reduces it to the block-triangular form

and separates (4.29) as the exact fast subsystem. The slow manifold Me

defined by z = L(e)x is the eigenspace corresponding to the slow eigen-values of (4.22), (4.23), that is, the eigenvalues that remain finite as e-» 0.

1.5 Construction of Approximate Models

It is in the spirit of engineering, control engineering in particular, to simplifythe models used for design. Apart from obvious practical reasons, thereare important conceptual advantages in this seemingly pragmatic approach.Phenomena that may be hidden in an exact but complex model sometimesappear in transparent clarity in a simplified model. Simplified two-time-scalemodels used throughout this book are based on explicit approximations ofthe manifold condition (4.6). To each approximation of $(x, e) therecorresponds an exact transformed model analogous to (4.11), (4.12).Approximate slow and fast models, analogous to (4.13) and (4.14), arethen constructed by neglecting various e-terms in the exact transformedmodel. In this section the main steps of such a model building procedureare outlined. The neglect of e-terms, which at this stage is ad hoc, will besubsequently validated in Chapter 2 for linear time-invariant systems, inChapter 5 for linear time-varying systems, and in Chapter 7 for nonlinearsystems.

The approximation procedure starts by substituting into the manifoldcondition (4.6) a power series for 0(jc, e),

and calculating cpo(x), (p\(x) etc. by equating terms in like powers of e.This requires that the functions / and g also be expanded as power seriesof e. The more important of the two is the series for g:

where dg/dz and all higher derivatives are evaluated at x, z = q>0(x). An


1.5 CONSTRUCTION OF APPROXIMATE MODELS 23

immediate consequence of (4.6), (4.7), (5.1) and (5.2) is

With (po(x) known, we proceed to equate the e-terms of power one, e1,and obtain

Although with (po(x] and <p\(x) known, one can equate e-terms of powertwo, £2, and obtain q> 2(x), etc., the approximation (f)(x, e) = cp0(x) + £(p\(x)will suffice for most applications. The substitution of

into (4.1), (4.2) results in an exact transformed model analogous to (4.11),(4.12), namely,

The essential difference between (4.11), (4.12) and (5.6), (5.7) is that 77 =0 is not an invariant manifold of (5.6), (5.7). However, from (5.5) and adetailed examination of (5.7), we see that 77 = 0 describes a manifold 0(e2)close to the invariant manifold of (5.6), (5.7). Furthermore, it will be shownlater that when rj is 0(1) its effect on x in (5.6) is only O(e). Thus neglectinY] in the slow model (5.6) will cause an O(e) error in x and, if r] itself isO(e), then this error will be O(e2). This suggests that (5.6) without r\ canbe used as an approximate slow model. We note that (5.6) improvesthe reduced model (2.5) by adding the correction term ecp^x), which isconceptually important, because it allows us to analyze the dependence ofthe properties of the slow model on e. This point will later be illustratedby an adaptive control example. As for the construction of an approximatefast model, (5.7) offers several possibilities, one of which is to neglect thee-terms of power two, e2, and higher, to obtain

Specializing (5.3) and (5.4) for the linear system (4.22), (4.23), we obtain


and hence the exact transformed model (5.6), (5.7) is

where

Neglecting e2a in (5.11), this model is written compactly as

where ^4S and >lf denote the corresponding matrices in (5.10), (5.11), and77° is expressed in terms of the original initial conditions *° and z°, accordingto (5.5). Neglecting 77 in (5.12), one can use x = A<x as a slow approximatmodel, and ET) = A{r] as a fast approximate model. However, this is aconvenient place to show, first, that the effect of 77 on x is O(e) and, secondthat it can be further reduced. Expressing 77 in terms of £77, using (5.13)and substituting it into (5.12), one obtains

which suggests that a new slow variable should be

Using this variable, (5.14) becomes

and the Tj-term is now O(e). This process can be continued and the presenceof 77 in the slow model can be progressively reduced to any desired poweof £. Further details are presented in Chapter 2.

With almost the same ease we can develop slow and fast approximatemodels for nonlinear systems that are linear in z, that is, for

where A12 andA22 can also be functions of*, but, to simplify the derivations,

1.5 CONSTRUCTION OF APPROXIMATE MODELS

we treat them as constant matrices. In this case

and the substitution of (5.5) and (5.19) into (5.17), (5.18) gives

where

25

The remarkable analogy of (5.20), (5.21) and the linear system (5.10),(5.11) shows how, in systems with slow nonlinearities , the linearity of thefast phenomena is preserved. Adaptive systems with fast linear parasiticsbelong to this class, and are now illustrated by an example.

Example 5.1

The equations of the adaptive system in Fig. 1.7 are

Fig. 1.7. Adaptive system with parasitics.


where the plant parameter a and the small parasitic time constant e > 0 areunknown. A result of adaptive control theory (Narendra, Lin and Valavani,1980) is that the update law (5.24) for the adjustable gain k stabilizes thesystem for all a = constant in the parasitic-free case, that is, when e = 0.A task of the so-called robustness analysis is to determine how the presenceof the parasitic time constant e affects-the stability of the system (5.23)-(5.25). To express (5.23)-(5.25) in the form (5.17), (5.18), we let

and evaluate the quantities needed for the model (5.20), (5.21), that is,

Finally, we obtain the model (5.20), (5.21) as

For £ = 0 and r] = 0, (5.28), (5.29) represent a globally stable adaptiveregulation. The presence of e-terms in (5.28) and (5.30) shows that theparasitic time constant may lead to a loss of stability, which illustrates howa transformed model directs our attention to possible causes of non-robustness. The robustness analysis of adaptive control problems will bepursued in Chapters 5 and 7.

In this and the preceding section, the discussion has been restricted tothe case of time-invariant models without inputs; see (4.1), (4.2). Let usnow show how to remove these restrictions by introducing appropriatemodifications in the modeling procedure. First, if the functions / and gdepend explicitly on time t, the model (4.1), (4.2) can still be used byextending the *-part of the state space to include xn+l = t as an additionalstate variable. The additional state equation is xn+l = 1 and the cor-responding component of the function / is fn + \ = 1. The most importannew property of the model (5.6), (5.7), and the approximate models derivedfrom it, is the appearance of the time derivatives

1.5 CONSTRUCTION OF APPROXIMATE MODELS 27

which are required to be 0(1) or less, so that the term ey\(x) in the slowmodel (5.6) and the term edcpQ/dx in the fast model (5.8) remain O(e). Inother words, for a "quasi-steady-state" to be "slow", it should not containfrequencies higher than 0(1) and it should certainly not contain terms suchas sin(r/e).

Our next two remarks concern the effects of an inclusion of time-dependent inputs in the model (4.1), (4.2). The effects of slow inputs onslow models and fast inputs on fast models are treated without furthersimplification. However, the effects of slow inputs on fast models and fastinputs on slow models can be approximated using the development in thisand the preceding section. Let us consider the case of linear time-invariantsystems for which the procedure is straightforward.

The presence of a slow input us(t) into a fast model, namely,

implies that the "quasi-steady-state" z(t) of z is z(t) = -AflBfUs(i). Thediscrepancy between z and z is

which, when substituted into (5.32), yields

Requiring that «s(f) be 0(1) or less, that is, bounded by a constant inde-pendent of e, we assure that the "quasi-steady-state" of 77 j is only 0(e) orless. If this accuracy is satisfactory, we can treat r]^ as our fast variable and(5.34) as the fast model. For higher accuracy, the step (5.33) can berepeated by introducing 772 and transforming (5.34) as follows:

with a further requirement that «s be 0(1) or less. This procedure can becontinued, each step generating a term in a series expansion for z. To relatethe model (5.35) to (5.11), we can assume that ws is generated by a slowmodel «s = Asus. Then the substitution of ws = Alus into (5.35) will make(5.35) appear in the form (5.11).

The effect of a fast input «f on a slow model, that is,

can be reduced to our previous discussion of (5.14)-(5.16) by consideringthat wf is the output of a stable fast model;


Letting BsCf = A12, and using (5.15), we obtain

This procedure can be continued by defining £2> which would lead to anapproximation of x by the solution of |fc = /ls^ and a power series in ebeginning with the term eB^C^A^r] which is O(e). This proves that if wesimply disregard «f in (5.36) and obtain x from x = A^x, the error will notbe larger than O(E). It is important to note that, by construction (5.37),the signal uf does not contain frequencies lower than O(l/e).

A more general stable fast model (5.37) can have purely oscillatorymodes. Then wf remains bounded, but, in addition to the exponentialldecaying terms, it also contains the terms of the form sm(t/e). For suchinputs the system (5.36) acts as a low-pass filter, and hence their effect onx is only O(e). This conclusion will be helpful in Chapter 4, where we dealwith stochastic problems. If instead of (5.36) we consider a nonlinear slowmodel, then/lf in (5.37) should be Hurwitz, that is, high-frequency periodicinputs such as sin(f/e) should be excluded. This is an important caveat,because a persistent high-frequency signal may produce an effect 0(1) dueto nonlinear averaging in the slow model. Problems of this sort are handledby averaging techniques such as those of Krylov-Bogolyubov (Bogolyubovand Mitropolsky, 1961), further extended by Hale (1980). They are notconsidered in this book, and the appearance of high-frequency oscillatorymodes is excluded by Assumption 3.2.

1.6 From Nonstandard to Standard Forms

The two-time-scale properties of the standard model (4.1), (4.2) have beencharacterized in terms of the geometric properties of the same model (4.8)considered in the fast time scale r at e = 0. The latter model was shown topossess at least one equilibrium manifold M0, defined by z = (j>(x, 0) =<po(x), and a family of fast manifolds Fx on which x as a function of i isconstant, while z converges to A/0. A further characteristic of the standardform is that the fast manifolds Fx are "parallel to the z-axis"; that is, theyare translates of the subspace x = 0. If we now perform a nonsingularchange of coordinates, the transformed model may no longer be in thestandard form, but will still possess the same two-time-scale properties.What may be lost is the property that some variables are predominantlyslow (the former variables x), while the others exhibit a boundary layerbehavior (the former variables z). Instead, all the new variables may possessboundary layers and converge to quasi-steady-states. However, this more

1.6 FROM NONSTANDARD TO STANDARD FORMS 29

general situation is still characterized by the existence of the equilibriummanifolds M0 and the fast manifolds Fx, although their positions and shapesin the new coordinate system are different. In particular, Fx need no longerbe parallel to any of the coordinate axes.

What has just been described is that a change of coordinates may "hide"the time-scale properties already discovered from a model in standardform. Of course, we are interested in a reverse step that would convert anonstandard model into the standard form and thus reveal the hidden timescales.

To begin with, let us consider a fairly general linear model written, forconvenience, in both slow and fast time scales, namely,

Clearly, if detF(O) =£ 0, the crucial Assumption 2.1 would be satisfied andall the components of v would be fast. Thus, for two or more time scalesto appear we need detF(0) = 0.

To simplify notation, let F(e) = F0 + eF^e) and assume that the dimen-sion of the null-space N of F0 is n. Our next assumption is crucial. It statesthat the range-space *3l and the null-space JV of F0 span Rn+m, that is,

This means that F0 has n zero eigenvalues with the eigenspace N(Fo) andm nonzero eigenvalues with the eigenspace Sft(F0). Therefore J^(F0) is theequilibrium manifold M0.

To select a fast variable r\ we choose m linearly independent vectors inRn + m orthogonal to JV(F0) and arrange them as the rows of the m x (n + m)matrix Q such that

In particular, Q can be formed of any m independent rows of F0. Therequirement that 77 be zero on M0 is satisfied by the m-vector Qv. Thus wetake Qv as a candidate for the fast variable r\.

To select a slow variable x we recall that, as a function of r, the variablex was constant at e = 0. Hence we look for a constant quantity in (6.1). Itis found with the help of an n x (n + m) matrix P whose rows span the leftnull-space of F0, that is, FF0 = 0. Then at e = 0 we have

Therefore the n-vector Pv, which remains constant for all rs:0, is ourcandidate for the slow variable x.


In the proposition below we show that this choice of x and r\ indeedtransforms (6.1) into the standard form (2.1), (2.2).

Proposition 6.1

If F(0) = F0 satisfies (6.2) then the change of coordinates

transforms the system (6.1) into

which is in standard form since ^22(0) is nonsingular.

Proof The inverse of T is T 1 = [V W], where the columns of V andW are bases for N(FQ) and 9t(A0) respectively. Since PF0 = 0 and F0V = 0we have

and hence An(e) = PF,(e)V, An(e) = PF^W, A2l(e) = QF^V andA22(£) = QF0W + £QF{(e)W. The m x m matrix A22(0) = QFQW is non-singular by (6.2). It is the only nonzero block of TF0T~l, and as such itcontains the m nonzero eigenvalues of F0. D

It should be noted that in (6.7) the variable x appears multiplied by £.For this reason we have chosen the notation r], rather than z. Althoughr] - 0 is not the slow manifold M£ of (6.6), (6.7), it is O(e2) close to it,because the quasi-steady-state of rj is O(e).

Example 6.1

Returning to Example 2.4, a more systematic solution proceeds as follows.The F-matrix in (2.19), (2.20) is


and hence Q = [-1 1]. The matrix P = \pv p2] is determined bypl/Cl =p2/C2 and we can take either pl = C\, p2 = C2, or

as we did in (2.23). The choice (6.10) is preferable if we want the variablex to represent a voltage, while the choice oipl = Cl, p2 = C2 would makex a sum of the charges in Ct and C2.

A nonlinear analog of (6.1) is the system

and its equilibrium manifold M0 in the T-scale is the set of all v such thath(v, 0) = 0. Assuming that M0 is n-dimensional, we can reduce the n + mequations h(v, 0) = 0 to m independent equations 9(v) = 0 such that

Again the quantity 0(v), which is zero when v is in the equilibriummanifold, will be used as the fast variable. To find a candidate for our slowvariable we look for a quantity that, in the fast time scale r, is constantwhen e = 0. Suppose that the n-vector o(v) is such a quantity, that is,

where 5 is a compact set in Rn+m relevant to the application at hand. Acrucial assumption, analogous to (6.2), is

Under this assumption, the change of coordinates

converts the system (6.11) to the standard form (4.1), (4.2). This is illus-trated by two control problems of practical significance.


Example 6.2

Consider a nonlinear system with linear high-gain feedback

In this case

and hence Cv = 0 implies h(y, 0) = 0. So we will take 77 = d(v) = Cv. Nextwe observe from

that if the n x (« + m) matrix P satisfies P5 = 0 then the constant quantityo(v) is Pv, which we take as x. In this way, our proposed "standardizing"transformation is

and the condition (6.14) requires that it be nonsingular. This will be thecase if

which also ensures that (CB)~l exists. Our standard form of (6.16), (6.17)is therefore

where v is to be expressed in terms of x and 77 using the inverse tranformation of (6.20). The discussion of this problem will be continued inChapter 3, where it will be shown that for the stability of the feedback loopwe need that ReA(C5) be negative, as also required by Assumption 3.2.

Example 6.3

Energy modeling of aircraft flight has been behind many applications ofsingular perturbation techniques to guidance and control (see e.g. Kelley,


1973; Calise, 1981; Ardema 1979). The point-mass model of symmetricflight in the fast time-scale T is

where H = geometric altitude, V = velocity and y = flight path angle areconventional state variables, while g is the acceleration due to gravity, / =L/w is the lift per unit weight w and eA is the difference of thrust minusdrag per unit weight; both / and A are functions of H and V. More elaboratemodels include an equation for the change of weight, but for our purposesthe assumption that the weight is constant is acceptable.

The system (6.24)-(6.26) is in the nonstandard form (6.11), and thecondition /i(v,0) = 0 defines the equilibrium manifold M0 as siny = 0,/ = 1, which is a level flight with lift balancing the weight. A constantquantity at e = 0 in r-scale is provided by the fact that without thrust anddrag the energy is conserved. Thus, multiplying (6.24) by g and (6.25) byV and adding them together, we get at e = 0, for all T s= 0 and all H, Vand y,

Hence as our slow variable we take E = o(v] = gH + W2, and obtain, inr-scale

Although we have not defined the new fast variables yet, this model is instandard form (4.1), (4.2). Our development suggests that it]l = siny and7/2 = I ~ 1 be the fast variables. For rj^ = y, noting that YII = sin y = y forsmall y, an interesting change, according to (6.15), would be to eliminate


H using r\2 = I - cos y. However, we can retain the classical model (6.28)-(6.30), which in the slow time scale shows, for example, how to choose A(i.e. thrust) and altitude H to minimize fuel. The boundary layer part ofthe solution corresponds to climb, dive or landing.

Our presentation thus far has concentrated on two-time-scale pheno-mena. More complex models may involve three or more time scales. Forexample, a standard three-time-scale model is

where z is faster than x, and v is faster than z. The corresponding timescales are

In the fastest time-scale 6 the system (6.31)-(6.33) is

and, as e—> 0, both ;c and z are constant, while v(Q) is determined fromthe sublayer system (6.37). In the intermediate scale T the same system is

and, as e—>Q, only JC(T) is constant, while Z(T) and V(T) are determinedfrom the standard singularly perturbed system (6.39), (6.40). This system is

1.7 CASE STUDIES IN SCALING 35

the boundary layer subsystem of the original system (6.31)-(6.33), and has(6.40) as its own boundary layer subsystem.

1.7 Case Studies in Scaling

It has been assumed thus far that the time-scale behavior is dictated by thesmallness of a scalar parameter £ > 0. Although in the limit as e — » 0 fastsolutions exhibit boundary layer "jumps", that is, their limits are notuniform, a uniform approximation (3.7) is achieved by an additive composition of the solutions of the fast and slow models. The fact that, say,x(t) is a uniform approximation of x(t, e) over an interval of t is expressedby the statement that over this interval the difference x(t, e) — x(t) is O(e).The "big 0" symbol is defined as follows.

Definition of O(e)

A vector function f ( t , e) G R" is said to be O(e) over an interval [r l5 12] ifthere exist positive constants k and e* such that

where | - | | is the Euclidean norm.

In the asymptotic analysis the "small o" symbol is also common andmeans that /(?, e) is o(e) if

For simplicity, we will use only the "big 0" symbol. In some cases we willbe able to give estimates of constants k and e* in (7.1) and thus quantifythe corresponding O(e) approximations. Otherwise, we will be satisfied byO(s) being an "order of magnitude relation", valid for "e sufficientlysmall." As an example, let us compare 2e and 8e2, considering 2e asO(e) and 8e2 as O(e2). However, 8e2 < 2e only if e G [0, i], that is, e is"sufficiently small" only if it is less than \. Thus, applying the results of anasymptotic analysis to a problem with a given value of £, requires a prioriknowledge that this value is indeed "sufficiently small".

But what is the meaning of e and when is it small? This raises theperennial issue of scaling, which we now address in three short case studies.


Case Study 7.1 Dimensionless e in the DC-Motor Model

In some models the time and state variables can be scaled to exhibit E asa dimensionless ratio of two physical parameters such as time constants.The simplest illustration is again the armature controlled DC-motor model(2.7), (2.8), rewritten here as

where

The relative unit for current is chosen such that in the steady state whenur = 1 and cor - 0, then from (7.4) ir= 1. This gives / = kQ/R, so that(7.3), (7.4) become

where

are the well known electrical and mechanical time constants respectively.Since Tm > Te, we let Tm be the time unit, that is, t r = t/Tm, and rewrite(7.6), (7.7) as

This scaling has brought us to a physically meaningful dimensionlessparameter


Model (7.9), (7.10) will be used for illustrative purposes throughout theremaining text.

Case Study 7.2 Parameter Scaling in an Airplane Model

A slow "phugoid mode" and a fast "short-period mode" are well knowntime-scale characteristics of the longitudinal motion of an airplane. Wenow show how these characteristics are exhibited by first scaling and thenapplying the modeling methodology of Section 1.6.

Assuming, as in Fig. 1.8, that the longitudinal motion consists of pitchingand displacement in the plane of symmetry (Etkin 1972), we apply Newton'slaw and obtain the following nonlinear model:

where T, L, D, mg and M are the thrust, lift, drag, weight and moment,while a, y and 9 are the angle of attack, flight-path angle and pitch angle,respectively; / is the moment of inertia, T is the time corresponding to the

fig. 1.8, parameter scaling in an airplane model.


fast motion and 6 = a + y. The linearization of this model for the straightsteady flight with velocity V0 yields, after neglecting some small terms,

where 6, a, q and 6 are respectively incremental pitch angle, angle ofattack, pitch rate and elevator position, while v = (V - V0)/V0 is thenormalized incremental velocity and the subscripts denote the partialderivatives.

There is no obvious candidate for e in (7.15). However, for typicalnumerical values (Elliott, 1977) the small quantities are

while the quantities

This suggests that e be introduced as the ratio of the largest of the smallquantities (7.16) versus the smallest of the large quantities (7.17). Toachieve this we rewrite the 4 x 4 system matrix of (7.15) in the form F(e) =F0 + eFl and partition it into 2 x 2 matrices as folows:

In this way the nonzero entries of FU1, namely Xv/e, -g/eV0, and thoseof F211, namely ZvV0/e, MvVo/e are all scaled to be 0(1). It is also importantto note that the entry Xa/VQ of F120 is O(e).

With the scaling (7.18) the linearized model (7.15) has become a specialcase of the nonstandard model (6.1). Thus we apply Proposition 6.1 to(7.15) scaled by (7.18), which, for generality, we first rewrite as follows:


where F^o exists. The matrix P = [P^ P2] of Proposition 6.1 must satisfy

If we chose P l = InXn then P2 = -F120F220. Thus our slow variable is

Since F220 is nonsingular the expression Qv = 0 reduces to f2 = 0, so wecan take Q = [0 Imxm]> and hence our fast variable rj = v2. The trans-formed model is (6.6), (6.7), where

Specializing the steps (7.19)-(7.22) for the airplane model (7.15) scaledas in (7.18), we have n = 2, m = 2 and

Thus r\ = v2 implies that a and q are our fast variables, while the slowvariables can be evaluated from (7.21).

In the traditional analysis (Etkin 1972) of the longitudinal model (7.15)the slow "phugoid mode" and the fast "short-period mode" are approxi-mated to reveal the factors influencing natural frequencies con, and dampingratios £,-, which provide important stability information. It is of interest tocompare one of these approximations with the approximations offered byour model matrices (7.22). From the standard model (6.6), (6.7) we knothat the eigenvalues of the matrix

approximate the slow mode. (Note that An is multiplied by e in order toreturn to the fast time scale i in which the original model (7.15) wasdefined.) Matrix (7.24) shows that the slow mode's wl is


and its damping ratio £ can be evaluated from

This approximation is more accurate than a simpler approximation

found in standard texts such as Etkin (1972). It is of conceptual importance,however, that, while Etkin's approximation (7.27) is obtained via severalheuristic steps, the approximation (7.25), (7.26) follows from the analyticprocedure summarized in Proposition 6.1. Moreover, in the common situa-tion when

the approximation (7.27) follows from the singular perturbation approxi-mation (7.25), (7.26). For the NASA F-8 numerical values given by Elliot(1977), the error of (7.25), (7.26) is 0.8% and that of (7.27) is 1.9%.

The fast-mode matrix A22 defined by (7.15), (7.18) and (7.22) is

which, using (7.16) and (7.17), may be approximated according to

and then the fast mode approximation

is the same as in Etkin (1972).

Case Study 7.3 State Scaling in a Voltage Regulator

While in Case Study 7.2 only system parameters are scaled by e, we nowconsider a model in which a state variable is scaled by e. This scaling isfundamentally different from the scaling by relative units used in Case


Study 7.1. The IEEE Type 2 voltage regulator is shown in Fig. 1.9, and,neglecting saturation SE, is described by

where A = Vref - Vbus. To represent this model in a singular perturbationform we examine typical numerical values of its parameters:

attempt: We assume that the fast phenomena are caused, as inExample 2.2, solely by high gain £A, and exhibit this assumption by

First


rewriting the system matrix of (7.32) as

where the coefficients a l 5 a2, • . . , are known from (7.32) and (7.33). It iseasy to verify that with the scaling e = l/KA this matrix cannot be broughtinto a standard form via the procedure of Proposition 6.1. However, if wedefine e and the states as

that is, if we scale one of the states by e then the system (7.32) becomes

Although it would seem that we have reached a standard form, this is notso because our crucial Assumption 3.2 is not met since

violates the requirement (3.9).

Second attempt: Recognizing that, in addition to high gain KA, the smalltime constant TA also contributes to the appearance of the fast phenomena,we scale TA by e and define

Although z2 = eV as in (7.35), the scaling (7.38) is different, because e isdifferent. With the parameter and state scaling (7.38), the system (7.32)becomes

which is in the standard form satisfying Assumption 3.2, because using the

1.8 EXERCISES 43

parameter values of (7.33)

The two models, (7.36) and (7.39), behave differently insofar as theirfast modes (7.37) and (7.40) are concerned. As e—» 0 the fast eigenvaluesof (7.36) become purely oscillatory, while those of (7.39) remain welldamped.

An important consequence of the state scaling z2 = eV is that it allowsthe original variable V = z2/£ to be O(l/e) large, as is to be expected ofthe output of the high-gain amplifier in Fig. 1.9.

The last case study illustrates a state scaling by e compatible with therange of the actual physical variable. A state scaling could have been usedin the airplane model (7.19) by treating v\ as x and v2 as ez. However, theinitial condition for z would then be z° = (\/e)vQ

2, and, in order to havethe properties of a standard form, v\ would not be allowed to be largerthan O(e). The methodology of Proposition 6.1 has avoided this restriction.

Although we have given guidelines for modeling two-time-scale systemsin a standard singular perturbation form, it is appropriate to close thissection and the whole chapter with the conclusion that, in general, thismodeling task requires considerable a priori knowledge about the physicalnature of the system.

1.8 Exercises

Exercise 1.1

Generalize the derivations (6.18)-(6.23) to include the case of v -dependent5-matrices, B = B(v), and apply them to the system (2.11), (2.12) ofExample 2.2 with v\= x- r, v2 = ez - 1, y = v\ + v2 and u = y/e.

Exercise 1.2

A scalar Pi-controller


of the type analyzed in Case Study 3.1, is now applied to a nonlinear plant

The error e = y - r between its scalar output y = h(z) and a constant set-point r is to be regulated to zero. Introducing an additional state variablex via the equation

represent this nonlinear feedback system in a singular perturbation form.When will this form satisfy Assumptions 2.1, 3.1 and 3.2?

Exercise 1.3

Extend the invariant manifold methodology of Sections 1.4 and 1.5 to allowa dependence of/andg in (4.1), (4.2) on e. Using this methodology, obtainan improved slow model and an improved fast model for the system (2.11),(2.12) of Example 2.2.

Exercise 1.4

In the coupled circuit in Fig. 1.10 the ratio of leakage inductances /1} /2 andthe self-inductances L}, L2 is assumed to be the same small parameter

Using the flux linkages

Fig. 1.10. Coupled circuit of Exercise 1.4.

1.9 NOTES AND REFERENCES 45

as the state variables and eliminating /1? i2 from

show that the following non-standard model is obtained:

where

Apply the methodology of Section 1.6 to transform this model into astandard form and interpret the meaning of its slow and fast systems.

Exercise 1.5

Paralleling Case Study 7.1, perform a scaling to obtain a dimensionlessform of the 7?C-circuit model (2.15), (2.16) of Example 2.3. Show that ecan be introduced as the dimensionless ratio of two time constants.

1.9 Notes and References

Basic references for an in-depth study of the method of asymptotic expan-sions applied to the standard singular perturbation model are the articlesby Vasil'eva (1963) and Hoppensteadt (1971), and the books by Wasow(1965) and O'Malley (1974c), in English, and by Vasil'eva and Butuzov(1973), in Russian. Our presentation is self-contained in that it does notassume familiarity with the above sources, which are recommended asadditional reading, especially for higher-order expansions and proofs oftheir asymptotic validity. A broader view on applications to systems andcontrol problems can be obtained from surveys by Kokotovic, O'Malleyand Sannuti (1976), Saksena, O'Reilly and Kokotovic (1984) and Kokotovic(1984). These surveys contain references to singular perturbation models ofelectromechanical networks and power systems, chemical kinetics, nuclearreactors, heat exchangers and to some of the modeling issues dealt with inthis chapter. A selection from these references is reprinted in a volumeedited by Kokotovic and Khalil (1986).


A class of models, not explicitly covered in this book, are Markov chainsin which low transition probabilities lead to slow phenomena and hightransition probabilities to fast phenomena. Gaitsgori and Pervozvanskii(1975), Delebecque and Quadrat (1981) and Phillips and Kokotovic (1981)develop two-time-scale models of Markov chains and apply singular per-turbation techniques to queueing networks. As shown by Kokotovic (1981),such problems are analogous to problems of aggregate modeling of largeelectromechanical networks, treated by the methodology of Section 1.6.This methodology was developed and applied to power systems in themonograph edited by Chow (1982). It was further extended to a hierarchyof time scales by Delebecque (1983), Coderch et al. (1983) and Khalil(1984b). Its application to a nonlinear model in economics is found inPeponides and Kokotovic (1983), and to networks with dense and sparseconnections in Chow and Kokotovic (1985).

In our presentation, the modeling methodology of Section 1.6 follows adiscussion of slow manifolds. This notion is introduced in Section 1.4 andused as a tool for construction of appropriate models in Section 1.5, ageneralization of the iterative modeling procedure proposed in Kokotovicet al. (1980). Although the slow manifold concept is implicit in everyseparation of slow and fast models, its explicit use as a geometric frameworkfor a singular perturbation analysis is recent; see Fenichel (1979) andSobolev (1984). Invariant or integral manifolds more commonly appear inthe averaging method of Bogolyubov and Mitropolsky (1961) and Hale(1980) and in the reduction method of Pliss (1966a, b), which extends theclassical conditional stability results of Lyapunov and Perron. Monographsby Mitropolsky and Lykova (1973), in Russian, and Carr (1981), in English,dedicate special sections to integral and center manifolds of singularlyperturbed systems.

Only single-parameter perturbations are considered in this book. A classof multiparameter perturbations of different orders of magnitude can betreated as nested single-parameter perturbations giving rise to multipletime scales. They were analyzed by Tikhonov (1952), Hoppensteadt (1971)and O'Malley (1974c), and applied to a rolling mill problem by Jamshidi(1974). Multiparameter perturbations of the same order of magnituderepresent a new class of problems. As shown by Khalil and Kokotovic(1978, 1979a, b), they are of interest in decentralized control when eachcontroller uses a different simplified model of the same large-scale system.

2 LINEAR TIME-INVARIANT SYSTEMS

2.1 Introduction

Linear time-invariant models are of interest in "local" or "small-signal"approximations of more realistic nonlinear models of dynamic systems. Let*e, ze be the equilibrium state of the nonlinear singularly perturbed system

for a constant control we, that is

and consider the small deviations

about the equilibrium (*e, ze, we). These small deviations are approximatelydescribed by the linear time-invariant model

where the partial derivatives df/dx, df/dz, etc., evaluated at x = xe, z =ze, u = we, are constant matrices. Most of the local properties of thenonlinear system (1.1), (1.2) can be deduced from the linear system (1.5),(1.6). For example, if the linear system (1.5), (1.6) is asymptotically stable

47

48 2 LINEAR TIME-INVARIANT SYSTEMS

with 5u = 0, then the equilibrium xe, ze of the nonlinear system (1.1), (1.2)is asymptotically stable. If this is not so for du = 0, but a feedback controldu = Fidx + F26z can be found to make the linear system (1.5), (1.6asymptotically stable, then the same feedback control will make the equi-librium jce, ze of the nonlinear system (1.1), (1.2) asymptotically stablThese and similar properties explain why this chapter and the next twochapters of this book are devoted to a separate study of the system

which differs from (1.5), (1.6) only in the obvious change of notation. Inaddition to (1.7), (1.8), our study will also include the system output ydefined by

All the matrices of the model (!.?)-( 1.9) are assumed to be constant andindependent of e. The latter assumption is for convenience only and leadsto a minor loss of generality, which we avoid as follows. If some of thematrices depend on e, say A22 = A22(£), then we assume that they aresufficiently many times continuously differentiable with respect to e at e =0, and that for e = 0 they possess the properties required from the matriceswhose dependence on e is suppressed. For example, when we assume thatA22 is nonsingular, then this can be extended to mean that A22(e) isnonsingular at £ = 0 and for e = 0 its derivative dA22/de exists and isbounded. We will also use the Laplace transform of (1.7)-(1.9), namely,

and represent the model by the block diagram in Fig. 2.1. The deterministicand stochastic designs of feedback controls for (1.7)-(1.9) are presented inChapters 3 and 4 respectively. In this chapter, we concentrate on theanalysis of two-time-scale properties of the model (!.?)-( 1.9) in the timedomain, and the two-frequency-scale properties of the model (1.10)-(1.12)in the frequency domain. In Sections 2.2-2.5, we disregard the input u andthe output y and study the properties of the free system (1.7), (1.8) withu = 0. The so-called actuator and sensor forms derived in Section 2.2 makesome of the time-scale properties of (1.7), (1.8) not only easy to analyze,

2.2 BLOCK-TRIANGULAR FORMS 49

Fig. 2.1. Block diagram of standard open-loop singularly perturbed model.

but also more clearly related to applications where fast transients aredue to physical actuators and sensors. Among these properties are theeigenvalue properties analyzed in Section 2.3. A second step in the trans-formation of (1.7), (1.8) brings it to a block-diagonal form in which all thfast phenomena are described by one block and all the slow phenomenaby another block. Since each block corresponds to an invariant subspaceof (1.7), (1.8), this step gives a geometric interpretation of the time-scaleproperties alluded to in the discussion of the slow manifold in Chapter 1.In Section 2.5 we show that the results of Sections 2.2, 2.3 and 2.4 offer agreat flexibility in constructing and validating approximate models withknown asymptotic properties. In Section 2.6 we return to the model (1.7)-(1.9) with the control input u and the output y and study its controllabilityand observability properties. Also, this model provides the basis for acharacterization of approximate models in the frequency domain in Section2.7.

2.2 The Block-Triangular Forms

In order to study two-time-scale properties of the system

or, equivalently, to study two-frequency-scale properties of its Laplacetransform


we introduce two different sets of coordinates in which the system appearsin two distinct block-triangular forms. For ease of reference and with someintuitive appeal, these forms are called the actuator form and the sensorform, and are represented in Fig. 2.2. In the actuator form the fast bloc

Fig. 2.2. The two block-triangular forms of (2.1), (2.2).

whose state is the new variable r/, "drives" the slow block, whose state isthe original variable x. In the sensor form the roles are reversed: the slowblock, whose state is the new variable £, is followed by the fast block,whose state is the original variable z, which can be thought of as the sensorinformation about the slow variable £. To avoid repetition of similarderivations, we deal mostly with the actuator form and relegate the detailsabout the sensor form for problems and exercises.

The actuator form is the result of the change of variables

introduced in Section 1.4, that is, the similarity transformation

which transforms (2.1), (2.2) into

The requirement that the matrix R(L, e) be zero, that is, the requirementthat the m x n matrix L(e) satisfy the algebraic equation


results in the system (2.7) assuming the upper block-triangular form

which is the actuator form of Fig. 2.2(a). What has been achieved is thatthe system (2.1), (2.2) has been partially decoupled in (2.9) to provide aseparate fast subsystem

rewritten in the fast time scale r as

with, by (2.5), the initial condition

The block-diagram of the actuator form (2.9), shown in Fig. 2.3, is adetailed representation of the diagram in Fig. 2.2(a). Note that the feedbacloop of the fast subsystem is a high-gain loop, that is, its gain tends toinfinity as e—> 0. This is a characteristic of the fast part of every singularlyperturbed system in the standard form, that is, when A22 is nonsingular.This would not be so if A22 were singular, because then in some componentloops of the vector loop the gain 1/e would be multiplied by e coming fromthe term eLA12. Before proceeding with a discussion of the properties ofthe actuator form, we first need to establish that an L(e) satisfying (2.8)exists.

Fig. 2.3. Actuator form.


Lemma 2.1

If A22 is nonsingular, there is an e* > 0 such that for all e G [0, e*], thereexists a solution L(e) to the matrix quadratic equation (2.8) which isapproximated according to

where

Proof The unique solution of (2.8) for e = 0 is

Differentiating (2.8) with respect to e, we obtain

and see that the unique solution of this equation at e = 0 is

which proves that (2.13) represents the first two terms of the MacLaurinseries for L(e). D

It should be pointed out that L(e) defined by (2.13) is unique in the sensethat, although equation (2.8) may have several real solutions, only one ofthem is approximated by (2.13). It is sometimes useful to have an upperbound, however conservative, on the value of the small parameter e forwhich the approximation (2.13) of Lemma 2.1 holds.

Lemma 2.2

Under the conditions of Lemma 2.1, the approximation (2.13) is valid forall £ in the interval 0 ̂ e < e}, where


Proof Denoting L(0) = A^A^ = L0 and introducing D = L - L0, wewrite (2.8) as

and show that, for sufficiently small c = H ^ A ^ H ' /(£*) is a contractionmapping. Let

provided that

provided that

Thus, by the contraction mapping principle, (2.19) has a unique solutionin y which can be reached by successive approximation starting with anyD0 satisfying ||D0|| ̂ (0//fr)1/2. The bound (2.18) follows from (2.20). D

A by-product of Lemma 2.2 is an iterative scheme for solving (2.8),namely,


After k iterations the exact solution L(e) is approximated to within O(ek)error. For the purposes of the model development and design, the firstiteration giving (2.13) is usually sufficient.

Examination of Lemma 2.2 and its proof reveals that the small parametere is associated with A^ only: it does not appear in L0, AQ and A12. Werewe to subsume e in A22, the proof of Lemma 2.2 would go through asbefore with the bound (2.18) replaced by

This bound guarantees that the system

has a two-time-scale property, analogous to that of a singularly perturbedsystem. In other words, it can be transformed to the upper block-triangularform

where the smallest eigenvalue of A22 + LAU is larger than the largesteigenvalue of An - A12L, that is,

since (2.22) can be shown to imply

As we shall see later in this section, the parameter e represents the orderof magnitude of the ratio of the large and small eigenvalues (2.25). As atime-scale factor, e conveniently allows the designer to determine reduced-order models by solving L iteratively to within "order of 0(e*+1) error. Interms of the condition (2.22), where explicit use of e is suppressed, theerror diminishes as ll^l^1! an^ \\A0\\ decrease; that is, as the ill-conditioningof the system (2.23) increases.


Finally, we note that Lemma 2.2 furnishes us with a bound on the rangeof e, albeit a conservative one, for which the validity of the theorems insucceeding sections is guaranteed.

Example 2.1

The matrix in the DC-motor model of Case Study 7.1 in Chapter 1, andthe corresponding L-equation (2.8) are

and hence L0 = 1, A0 = -1, and the approximate solution (2.13) is

which can be readily verified by evaluating the exact solution. Note in thissimple example that the L-equation (2.26) also has an 0(l/e) root. Theactuator form for the DC motor is as in Fig. 2.4, where A! is 0(e2) and A2

is 0(e3).

Fig. 2.4. The DC-motor model in actuator form.

Let us now transform (2.1), (2.2) into a lower triangular form, that is,the sensor form, by the change of variables

and the requirement that the matrix M(e) satisfy the algebraic equation

Then the system (2.1), (2.2) becomes

216 Introduction to Matrix Analysis

The basic formula is

for Re (s) > Re (a). It follows that if we know that/(0 has the form

then the equality

means that c\ = c^ and \\ — a.Similarly, from the additivity of the integral, it follows that if f(t) is

of the form given in (1), and if g(s) has the form

then we must have

26. An Example. Consider the problem of solving the linear differen-tial equation

Since the characteristic equation is

with roots X = 1 and X = 2, we see that the solution of (1) will havethe form

where c\ and Cz will be determined by the initial conditions in (1). Thuswe obtain the linear system

yielding

so that the solution is

In place of this procedure, let us use the Laplace transform. From (1),for Re (s) > 2, we obtain the relation

2.3 EIGENVALUE PROPERTIES 57

original system (2.1), (2.2), the first n eigenvalues {A l5 . . . , AM} are theroots of the slow characteristic equation

and the remaining m eigenvalues are

where n-} are the roots of the fast characteristic equation

The roots of the slow characteristic equation at e = 0, namely,

are the eigenvalues of the matrix AQ defined by (2.14), and the roots of fastcharacteristic equation at £ = 0,

in the high-frequency scale p are the eigenvalues of the matrix A22. Thisleads to the issue as to how closely the slow eigenvalues and the fasteigenvalues, evaluated at e = 0, approximate the eigenvalues of the actualsystem (2.1), (2.2). The next theorem provides an answer to this question.

Theorem 3.1

If A 22 exists, then as £ — » 0 the first n eigenvalues of the system (2.1), (2.2)tend to fixed positions in the complex plane defined by the eigenvalues ofAQ, namely,

while the remaining m eigenvalues of the system (2.1), (2.2) tend to infinity,with the rate 1/e, along asymptotes defined by the eigenvalues of A22,namely,

Furthermore, if the n eigenvalues A,(^40) are distinct and the m eigenvaluesAy-(y422) are distinct, where A,(/10) = A;(y422) is allowed, then the eigenvaluesof the original system (2.1), (2.2) are approximated as


Proof The limiting behavior of (3.9) and (3.10) follows from the con-tinuity of the coefficients of the polynomials (3.2) and (3.3) with respect toe. The approximation (3.11) follows from the total derivative of the exactcharacteristic equation (3.4) with respect to e:

Since at e = 0 and s = A,-(^40) the fact that A,-, i = 1, . . . , n, are distinctguarantees that dtys/ds i~- 0, the derivative dA(/de exists at e = 0, whichproves (3.11). Analogously, taking the total derivative of the exact charac-teristic equation (3.6) and using d tyf/dp ̂ 0 at e = 0 and/? = Ay(^422) proves(3.12).

The presence of two disjoint sets of eigenvalues of different orders ofmagnitude, 0(1) and O(l/e), is an inherent property of singularly perturbedlinear time-invariant systems. The smaller the value of e, the wider the"eigenvalue gap" and the greater the separation of time scales in the originalsystem (2.1), (2.2). When, as in the model (2.23), the parameter e is notexplicit, but the presence of eigenvalues of different orders of magnitudeis known, this should alert us to the two-time-scale or singularly perturbednature of that system.

The eigenvalue approximations of Theorem 3.1 make no assumption asto the stability of the system (2.1), (2.2). On the other hand, stability is acrucial system requirement in control system design and other fields. Inthis respect, Theorem 3.1 affords us the following useful two-time-scalestability result.

Corollary 3.1

If A 22 exists, and if A(] and A22 are Hurwitz matrices, then there exists ane* > 0 such that for all e E (0, £*] the system (2.1), (2.2) is asymptoticallystable.

The fact that the stability of the actual system (2.1), (2.2) can be inferredfrom the analysis of lower-order systems in separate time scales is of greatpractical significance for analysis and design.

It is sometimes useful to achieve closer eigenvalue approximationsthrough consideration of corrected models in separate time scales. This isparticularly true when e is not very small or when the eigenvalues of A 0 orA22 are close to the imaginary axis of the complex plane. For an 0(e2)approximation, we substitute (2.13) into the characteristic equation (3.2),

o

2.3 EIGENVALUE PROPERTIES 59

and obtain a corrected slow characteristic equation

where

Also, substituting (2.14) into (3.3), we obtain a corrected fast characteristicequation

where

Then, in analogy with Theorem 3.1, an O(e2) approximation to the eigen-values of the original system (2.1), (2.2) is provided by

Notice that (3.18) and (3.19) can be used in combination to providedifferent orders of approximation to the system eigenvalues. For example,an O(e) accuracy of the "fast" eigenvalues may be sufficient in a particularapplication, in which an O(e2} accuracy of the slow eigenvalues is desirable.Higher-order approximations of the eigenvalues can be obtained by wayof evaluating more terms of the MacLaurin series beyond those of (2.13),and substituting the higher-order approximation for L(e) into the charac-teristic polynomials (3.2) and (3.3).

Example 3.1

Continuing with the DC-motor Example 2.1, where A0 = -1 and A22 =-1, we see that its eigenvalues are approximated by

where A ] is slow and A 2 is fast. Should greater accuracy be required, thecorrected eigenvalue approximations (3.18) and (3.19) yield


As a check, the roots of the exact characteristic equation eA2

are

which, in view of (1 - 4e)1/2 = 1 - 2e - 2f2 + O(e3), coincides with (3.21).The bound from Lemma 2.2 is in this case e = 1/4. For e = 1/6 the approxi-mate and the correct expressions (3.21) and (3.22) yield respectively A! =-1.17 instead of A : = -1.27 and A2 = -5 instead of A2 = -4.73. As it

happens, typical values of motor constants in Example 2.1 result in asingular perturbation parameter of e = 0.1 for which the corrected eigen-values A j = -1.10,A2= -9 compare favorably with the actual eigenvaluesof-1.13 and -8.87.

2.4 The Block-Diagonal Form: Eigenspace Properties

From either the actuator or the sensor block-triangular form, one morechange of variables leads to a complete separation of the fast and slowstates of the system (2.1), (2.2). Starting with the actuator form (2.9), thechange of variables

results in

and the requirement that the n x m matrix H satisfy the linear algebraicequation

allows the system (4.2) to finally assume the block-diagonal or decoupledform

As in the actuator form, (2.11) defines the exact fast subsystem, while thenew block

2.4 BLOCK-DIAGONAL FORM: EIGENSPACE PROPERTIES 61

defines the slow subsystem with, by (4.1) and (2.12), the initial condition

That the unique solution H(e] of (4.3) exists and is approximated by

for all £ for which (2.13) holds, is readily established by the linearity of(4.3).

Taken together, the change of variables (2.5) leading to the actuatorform and the present change of variables (4.1) constitute a similaritytransformation T:

Among many conveniences that this transformation offers is that for all Land //, and not only those satisfying (2.8) and (4.3), the inverse trans-formation is explicitly given by

This is extremely helpful when, instead of the exact solution L(e) of (2.8)and H(e] of (4.3), we use their approximations. Although with approximatesolutions of (2.8) and (4.3) the transformed system will not be in the exactblock-diagonal form (4.4), it will be close to it because its off-diagonalterms will be O(e) or smaller. To illustrate this point let us use theapproximations

to form the transformation

When T(] is applied instead of T in (4.8), the system (2.1), (2.2) is trans-formed to


where the subscript "0" distinguishes the state variables in (4.12) from theexact slow and fast variables £ and rj in (4.4), although the system (4.12) isstill an exact similarity transformation of (2.1), (2.2). If the O(e) off-diagonal terms and the O(e2) diagonal term e2LoAQH0 are neglected, andL() and H0 are as expressed in (4.10), the approximate slow and fast modelsobtained from (4.12) are

We have already encountered these slow and fast corrected models inChapter 1 (page 24), and we will have more to say about them in the nextsection. In the discussion above, the emphasis is on the use of different Land H in the transformation (4.8) as additional tools for the constructionof approximate models. In Chapter 5, a time-varying generalization of themodel (4.12) is used to investigate the stability and robustness propertiesof an adaptive system. A simplified time-invariant version of the samesystem is briefly analyzed in the following example.

Example 4.1

Equations of an adaptive system are

where k is an adjustable parameter and the goal of adaptation is to achievey(t) - ym(t) —> 0 for arbitrary bounded reference inputs r(t). This goal canbe achieved if the order of the model and the plant are the same, which isnot the case here. The model-plant mismatch is due to a parasitic timeconstant e, which can destroy the stability properties of the ideal scheme.While a general time-varying analysis is postponed until Chapter 5, weinvestigate the stability of (4.16)-(4.18) when r = const. The homogeneouspart of (4.16)-(4.18) is


To develop the approximate model (4.13), (4.14) we use

and the change of variables r] = z - rk, ̂ = y + er], |2 = k- To simplify

notation we let £, = | and keep A: instead of £2- The exact transfmodel (4.12) is

and the approximate model (4.13), (4.14) is obtained by neglecting theO(e) off-diagonal term and the O(f?) term in (4.22), (4.23), that is,

From the slow model's characteristic equation

we see that the slow model will be unstable if eyr2 > 1. Thus the parasiticparameters e sets a limit on the amplitude of the input r and the adaptationgain y. Although extremely simple, this example illustrates the use ofapproximate models in discovering destabilizing effects of parasitics.

Whenever a similarity transformation T brings a system to a block-diagonal form, the two sets of columns of T corresponding to the diagonalblocks constitute bases of the eigenspaces represented by the blocks. In thecase of the transformation (4.8), this means that the first n columns of Tare a basis for the slow eigenspace of (2.1), (2.2), and the remaining mcolumns are a basis for the fast eigenspace of (2.1), (2.2). Let us make thisprecise in the folowing lemma.

Lemma 4.1

Let Vt G R" be an eigenvector of the matrix As of the slow system (4.5) andVf) G Rm be an eigenvector of the matrix^ of the fast system (2.11), where,as in (4.4),


Then the corresponding slow eigenvector usER"+m and the fast eigen-vector V{ G Rn+m of the original system (2.1), (2.2) are

Proof The application of the transformation T in (4.8), with L and Hsatisfying (2.8) and (4.3) respectively, results in

or, in the expanded form.

Noting that ASL>S = A sus and Afi>,; = A ff,,, where As and Af are the eigen-values of As and Af corresponding to the eigenvectors vs and v^ respectively,we postmultiply (4.29) by us and (4.30) by v,} to prove (4.27). D

An immediate consequence of this lemma and the eigenvalue approxi-mation theorem is the following useful eigenvector approximation.

Corollary 4.1

Under the conditions of distinct eigenvalues in Theorem 3.1, the slow andit eigenvectors of the system (2.1), (2.2) are approximated accordingthe fast eigenvectors of the system

to

where U| is an eigenvector of A0 = An —vector of A22.

21 and 0^ is an eigen-


When defining approximations (4.31), we should take into account thefact that for an eigenvector only the angle is specified, while its length isfree. Therefore the O(e) refers to an angular displacement of an eigenvectorwhose length is normalized to one. Comparing ys from (4.27) and (4.31)with the expressions (4.3) and (4.7) of Chapter 1 for the slow manifold M£

and the equilibrium manifold M0, we see that vs G Me, that is, the sloweigenspace is the slow manifold ME as expected, while the approximation(4.31) of ys lies in M0. As for the fast eigenspace spanned by the eigenvectorsUf, the exact expression (4.27) shows that its angle with respect to thesubspace of the original z-variables is O(e), and the approximation (4.31)of yf is in the z-subspace, that is, orthogonal to all the x coordinates. UsingH0 = A12A22 , a closer O(e2) approximation of the angle that uf in (4.27)makes with the z-subspace is achieved by

Since we have not corrected the term Im in (4.32) by a corresponding e-correction from (4.27), it is not obvious that the approximation (4.32) isO(e2) as claimed. This is left as Exercise 2.6.

In summary, the exact slow eigenspace Me and the exact fast eigenspaceF0 of (2.1), (2.2) and their approximations are sketched in Fig. 2.6.

Fig. 2.6. Exact and approximate eigenspaces.


Recalling the discussion of fast manifolds Fx from Section 1.4 of Chapter1, we see that, in the case of linear time-invariant systems, they are thetranslates of the fast eigenspace F0 to the value of x prescribed by the initialcondition x° = x(t0). Fast manifolds are represented in Fig. 2.6 by linesparallel to F0 and hence inclined with respect to the z-axis by an O(e) angle.It is observed that the fast state r](t) moves rapidly along Fx towards oraway from the equilibrium manifold M£ depending on whether ME is stableor unstable. Fast manifolds Fx appear as "foliations" off the unique slowmanifold Me.

Example 4.2

For the DC motor of Example 2.1 described by

the slow and the fast eigenspaces, in this case the slow and the fasteigenvectors vs and y f , are approximated by

As a check, the exact expressions are

and coincide with (4.34) in view of the expression (1 - 4C)1/2 =1 - 2e + O(e2). Let a and a be the angles that

respectively make with the z-axis, where 1 is corrected by the e-correctionce in vf:

2.5 VALIDATION OF APPROXIMATE MODELS 67

It is clear that & is an O(e2} approximation of the exact angle of uf in (4.27)and both the exact angle and a are O(e) with the z-axis. An O(e2}approximation is also achieved by a since

whereas the angle of the eigenvector [0 1]T of (4.31) is zero and approxi-mates the angle of vf by O(e}. In this example c = 1.

2.5 Validation of Approximate Models

The results of preceding sections, as well as offering a repertoire of approxi-mate models, also justify some common ad hoc model simplification pro-cedures. One of those is the reduction of the (n + m)th-order model (2.1),(2.2) to the nth order model

by formally setting s = 0 in (2.1), (2.2) and disregarding the initial conditionfor z. When A 22 exists

is the unique solution of (5.2), whose substitution into (5.1) results in theunique nth order uncorrected slow model

An ad hoc derivation of the "fast" subsystem assumes that the slow variablesare constant during fast transients, that is, zs = 0 and x = jcs = const., andthat the only fast variations are the deviations of z from its quasi-steady-state zs. Denoting these fast variations by

one obtains from (2.2), (5.3), (5.5) and ezs = 0 the rath-order uncorrectedfast model

rewritten in the fast time scale T as

This ad hoc procedure was justified in Chapter 1 by Theorem 3.1, estab-lishing that if Re A(A22) < 0 then the solution x, z of the original system


(2.1), (2.2) is approximated, for e sufficiently small, by

where zf(r) is expressed in the t time scale. Here we establish the validityof these uncorrected state approximations by way of a direct comparison ofthe uncorrected candidate models (5.4) and (5.7) with the exact decomposetwo-time-scale model (4.4).

Theorem 5.1

If Re A(^2z) < 0, there exists an e* > 0 such that, for all e E (0, e*], thestates of the original system (2.1), (2.2) starting from any bounded initilconditions x° and z°, \\x°\\ ̂ c}, j|z°|| ̂ c2, where cl and c2 are constantsindependent of f, are approximated for all finite t ̂ t0 by

where x^(t) and zf(r) are the respective states of the slow model (5.4) andthe fast model (5.7). If also ReA(,40) < 0 then (5.10) and (5.11) hold foral l fe[f 0 ,*>) .

Moreover, the "boundary layer" correction zf(r) is significant only duringthe initial short interval [t0, r,], f, - t0 = O(e\ne), after which

Proof From (4 . 8) and (4 . 4) , the exact solution of the system (2 . 1 )-(2 . is

where, by (4.9)


First, note that the initial conditions x° and z°, and hence £° and 77°, are nolarger than 0(1). Accordingly, it follows from the continuous dependence ofthe matrix exponential function eM on the coefficients of the matrix M that(5.13) to (5.16) and (2.13) imply

where ReA(A2 2)<0. This proves (5.10) and (5.11). If in additionReA(^0) < 0, (5.17) and (5.18) hold for all t£[t0, oc). To show that the"thickness" r, - t0 of the boundary layer is O(eln £), we note that

where k and ex are some positive constants independent of e, and thate-(/-r0)/e < £ for tl- to> e\ne.

It is observed that for the eigenvalue approximations of Theorem 3.1 tohold well, all that was assumed was that detv422 =£ 0. For the time-domainstate approximations of Theorem 5.1 it is assumed that the fast system (5.7)also be asymptotically stable. This condition, in addition to being a sufficientone, is almost a necessary one since if any Re A(A22) > 0 then Theorem 5.1no longer holds in that e^22(f~'()) / /e grows without bound, invalidating the0(e) approximation in (5.17) and (5.18). As before, the slow subsystemmay be either stable or unstable.

Returning to the model (4.12), obtained using the transformation (4.11),Theorem 5.1 can be used to justify neglecting £ in the right-hand side of(4.12). In doing so, we retain e multiplying the derivative of r]0, so that asbefore, £r/0 = dr/o/dr. This is a characteristic step when, in the trans-formation (4.8), the matrices L and H do not exactly satisfy (2.8) and (4.3)respectively, and the exact transformed models are not in the block-diagonal form (4.4). On the other hand, note that the proof of Theorem5.1 uses the block-diagonal form (4.4) and the fact that its solution £ in thetime scale t, and YI in the time scale T, are continuous and differentiablefunctions of e. Observe that the same is not true about 77 as a function oft, because as e—> 0 in f-scale r] experiences a discontinuity ("boundary layerjump") at t = t(]. This jump distinguishes singular perturbations from regularperturbations. The transfer from t to T has made the fast subsystem of (4.4)regularly perturbed by e. A convenience provided by the block-diagonalform (4.4) is that the transfer from t to T can be performed in the fastsubsystem while letting the slow subsystem remain in the ?-scale. For models

o


where the fast and slow parts interact, such as nonlinear models, the proofof more general versions of Theorem 3.1 of Chapter 1 is more complicated.However, the spirit of the proof is the same in that the discontinuity in thefast variable is removed by analyzing it in the fast time scale.

An algebraic counterpart of the removal of the time-domain discontinuityis the removal of the ill-conditioning in the system matrix of (2.1), (2.2).In other words, the ill-conditioning due to the presence of the singularperturbation parameter e,

is alleviated in the lower-order calculations associated with AQ and A22 forthe slow and fast models (5.4) and (5.7) respectively.

Example 5.1

The dynamic behavior of the PID control system in Case Study 3.1 ofChapter 1 with d = 0 is described in the time scale of the integral action bythe state-space model

where x(t) is the integral action state variable, z\(i) and z2(t) are the systemstate variables, and the parameter e scales the integral action. As in (3.34)of Chapter 1, the uncorrected slow model (5.4), obtained by setting e - 0in (5.21), is

with eigenvalue

The uncorrected fast model (5.7) is


In the fast frequency scale p = es, the characteristic equation of (5.24) is

PD control design is carried out on the uncorrected fast system model(5.24) by choosing kl and k2 so as that (5.25) is equal to the desiredcharacteristic equation

for some specified damping ratio £ and natural frequency <wn. Equatingcoefficients in (5.25) and (5.26),

Also, in s = p/e scale, the desired eigenvalues from solving (5.26) are

Integral action control design is completed on the uncorrected slow model(5.22). Since, in the fast design (5.27) kl > 0, it is necessary to chooseA;3 > 0 so as to obtain a stable slow system (5.22).

In a numerical design, £, coa and e were chosen to be

The gains k\ and k2 of (5.27) and the desired fast system eigenvalues arerespectively

and

Given /:, in (5.30), the choice in (5.22) of

results in a stable uncorrected slow system with the eigenvalue

But what of the actual performance of the full closed-loop system (5.21)using the uncorrected design values of k { , k2 and A:3 in (5.30) and (5.32)?The eigenvalues of the full closed-loop system (5.21) are


Fig. 2.7. Uncorrected and corrected state approximations for a PID control system.

The state trajectories and their O(e) approximations of Theorem 5.1,based on the uncorrected slow and fast models of Theorem 5.1, are givenin Fig. 2.7. Evaluating (2.18) using the induced norm

we have an upper bound on e given by

Clearly, this bound is a conservative one, as the uncorrected slow and fasteigenvalues and state trajectories are still within O(e) approximation of thetime eigenvalues and state trajectories for the design value of e = 0.2.

In general, the reasons for seeking corrected models are threefold. First,the parameter e may not be very small. Second, the presence of eigenvaluesclose to the imaginary axis may mean that more reliable stability information


about the model is necessary. Last, more accurate analysis of steady-statesystem behavior requires a more accurate model.

Again, the instrument at hand is the exact block-diagonal form (4.4).Typically, the most important, and often sufficient, approximation is thecorrection which includes the e-terms and neglects O(e2) and higher-orderterms. This has been done in the model (4.13), (4.14), which is rewrittenhere as

It should be recalled that the roots of the corrected "slow" characteristicequation (3.14) and the roots of the corrected "fast" characteristic equation(3.16) are the eigenvalues of A^ in (5.36) and A22c in (5.37) respectively.

In analogy with the O(e) approximations of Theorem 5.1, an O(e2)approximation to the actual system states is obtained through the use ofthe corrected slow and fast models (5.36) and (5.37) in separate time scales.

Theorem 5.2

Under the conditions of Theorem 5.1, the approximations

hold for all finite time t ̂ t0 where xsc(t) and zfc(r) are the respectivesolutions of the corrected slow model (5.36) and fast model (5.37). If AQ

is also Hurwitz, then (5.38) and (5.39) hold for all re [r0, °°).

Proof Similarly to the proof of Theorem 5.1, (5.13)-(5.16) and (2.13)imply

74

where

etc.

2 LINEAR TIME-INVARIANT SYSTEMS

Comparing Theorem 5.2 with Theorem 5.1, it is seen that the approxi-mation to the actual state response of the system (2.1), (2.2) is improvedfor two reasons. First, corrective fast terms in z{c(t — t0)/e are introducedon the right-hand sides of (5.38) and (5.39) which are, however, only activeduring the initial short interval [t0, f j . Second, and more important fromthe point of view of application, the "slow" state xx(t) is itself the state ofa model (5.36) which corrects the reduced model (5.4), through the additionof the term £Al2A22A2iA0.

Example 5.2

Pursuing the PID control design of Example 5.1, from (5.36) the correctedslow system model is

whereas, from (5.37), the corrected fast (normal) system model is

The eigenvalue

of the corrected slow system model (5.44) indicates that for k3 > 0, byuncorrected design, the stability of corrected slow system enjoys an 0(s)improvement over that of the uncorrected model (5.22). Consider now thecharacteristic equation of the corrected fast system model (5.46) inp-scale,

0

2.6 CONTROLLABILITY AND OBSERVABILITY

where s = p/e as in (3.16):

which, using (5.27), is rewritten as

The roots of (5.50), in s = p/e scale, are the corrected fast eigenvalues

Since &, > 0 and &3 > 0, by uncorrected design (5.27), the effect of theintegral action gain Ek3 in (5.51) is seen to be destabilizing. Therefore, ourlast requirement for a successful PID design is to keep the integral actione/c3 small by choosing e small.

For the design values (5.29) and (5.32) the corrected eigenvalues (5.48)and (5.51) are

Comparisons of the state trajectories of the corrected slow and fast modelswith those of the uncorrected slow and fast models, and with the statetrajectories of the actual full system are made in Fig. 2.7. Moreover, thecorrected slow and fast models enable us to judge, without recourse to theanalysis of the full model (5.21), that the original uncorrected design (5.29),(5.30) and (5.32) is satisfactory vis-a-vis application to the full system(5.21).

2.6 Controllability and Observability

Let us now return to the analysis of the linear system (1.7)-(1.9) with acontrol input u(t) E Rr and an output y(t) G Rp; namely,

75


It should be recalled that the system (6.1) is controllable if there exists acontrol u(f) that transfers x(t), z(t) from any bounded initial state *(%),z(t0) to any bounded terminal state x(T), z(T) in a finite time T— 10.Similarly, the system (6.1), (6.2) is observable if the initial state *(f0), z(t0)can be determined from the measurement of the input u(t) and the outputy(f) over the period [t0, T\. The presence of the two-time-scale structureoffers the possibility of deducing the controllability and observabilityproperties of the full system (6.1), (6.2) in terms of the controllability andobservability properties of lower-order models in separate time scales.First, we transform the system (6.1), (6.2), using the similarity trans-formation (4.8), to the equivalent system

which represents a parallel connection in Fig. 2.8 where

and

Fig. 2.8. Decomposed open-loop system.

2.6 CONTROLLABILITY AND OBSERVABILITY 77

Making use of the well-known fact that controllability and observabilityare invariant with respect to similarity transformation, we analyze theseproperties of the equivalent system (6.3)-(6.6). For linear time-invariantsystems, a common characterization of controllability and observability isvia the controllability and observability of the eigenvalues.

Lemma 6.1

A necessary and sufficient condition for the tth eigenvalue A, of the system(6.3) to be controllable is

and that for the ith eigenvalue A, of the system (6.3), (6.4) to be observableis

Furthermore, the system (6.3), (6.4) is completely controllable [com-pletely observable] if and only if (6.7) [(6.8)] is satisfied for all its eigenvaluesA,, / = 1, . . . , n + m. The system (6.3), (6.4) is stabilizable [detectable] ifand only if all its eigenvalues with nonnegative real parts are controllable[observable].

A crucial property of the parallel system configuration of Fig. 2.8 is thatthe eigenvalues of A^ are distinct from those of A^/e. Consequently, if aneigenvalue of the slow subsystem is controllable [observable], it will remainso in the parallel connection (6.3)-(6.6). Similarly, if an eigenvalue of thefast subsystem is controllable [observable], it will remain so in the parallelconnection (6.3)-(6.6). In view of the fact that this property will not belost for small perturbations of e, we have the following result.

Theorem 6.1

Let A •* be the eigenvalue of A0 = A n - A 12^-22^-21 approximating the sloweigenvalue A, of A in (6.3). If

where 50 = Bl - AUA22B2, then there exists an e* > 0 such that A, iscontrollable for all e E (0, e*]. Similarly, let JTy be the eigenvalue of A22

approximating the fast eigenvalue A, of A according to A, = ;ry/e, i = n+ j.If

then there exists an e* > 0 such that A, is controllable for all e E (0, e*].


Proof In view of Theorem 3.1 and the preservation of the full rank ofa matrix under regular perturbation, there exists an e* > 0 such that (6.9)and (6.10) respectively imply, for all e E (0, e*],

where A, is the fth eigenvalue of As(e), and

At(e)where JT, is the yth eigenvalue of Af(e). By Theorem 3.1, As(e) and ——

have no eigenvalues in common, and hence

if A, is the ith eigenvalue of As(e) and if A, = jr;-/e, / = n + j, is the ;theigenvalue of A{(e)/e. D

The observability counterpart of Theorem 6.1 is as follows.

Theorem 6.2

Let A? be the eigenvalue of AQ approximating the slow eigenvalue A, of Ain (6.3). If

where C0 = C] - C2^ 22^21 • Then there exists an e* > 0 such that A, isobservable for all e G (0, e*].

Similarly, let ;r° be the eigenvalue of A22 approximating the fast eigen-value A, of A according to A, = XJ/E, i = n + j. If

then there exists an e* > 0 such that A/ is observable for all e E (0, e*].

Proof The proof is analogous to that of Theorem 6.1.

Complete controllability [observability] and stabilizability [detectability]then follow as corollaries.

o


Corollary 6.1

If the pair (AQ, B0) [(C0,A0)] and the pair (A22, B2) [(C2,A22)] are eachcompletely controllable [completely observable], then there exists an e* > 0such that the original system (6.1), (6.2) is completely controllable [com-pletely observable] for all e E (0, e*].

Corollary 6.2

If the pair (A0, B0) [(C0,A0)] and the pair (A22, B2) [ ( C 2 , A 2 2 ) ] are eachstabilizable [detectable], then there exists an e* > 0 such that the originalsystem (6.1), (6.2) is stabilizable [detectable] for all e E (0, e*].

A conclusion of practical importance is that instead of testing the e-dependent triple (C, A,B) of the full-order system (6.1), (6.2) for con-trollability and observability (stabilizability and detectability), one needonly test the e-independent triples (C0, A0, B0) and (C2, A22, B2) of lower-order subsystems.

Example 6.1

The model of a two-tank chemical reactor system, described in Section 3.3,is

where x E R2 and z E R2 respectively represent the slow reactor and actu-ator dynamics. The slow system model is described by the triple


while the fast system model is described by the triple

For controllability we check that

for A? = -i X\ = -i and that

for ;TI, Ji2 = -1. Thus, by Theorem 6.1 all four eigenvalues of (6.16) arecontrollable for e sufficiently small.

Testing for observability, we note that

for A? = -i, A? = -i However,

for ji°i,jT2 = -1. As we would expect from C2 = 0 both n\ and n\ of thefast model are unobservable, but stable, and the fast model pair (C2,A22)is detectable.

In fact, the original system (6.16), (6.17) is completely observable for£ > 0. It only loses observability of the fast system modes as e tends tozero. It illustrates the fact that Theorems 6.1 and 6.2 are only sufficient,but not necessary conditions for controllability and observability near e = 0.In other words, the converse of Theorem 6.1 (and its dual Theorem 6.2)may not hold: setting e - 0 in (6.11) and (6.12) does not necessarily imply(6.9) and (6.10) respectively. This motivates us to call an eigenvalue A, ofA in (6.1) weakly controllable [weakly observable] if it is controllable[observable] for e arbitrarily small, but the corresponding sufficiency con-dition (controllability at e = 0) (6.9) or (6.10) [(observability at e = 0)(6.14) or (6.15)] is not satisfied. Conversely, if the corresponding sufficiencycondition (6.9) or (6.10) [(6.14) or (6.15)] is satisfied, the eigenvalue A, ofA is said to be strongly controllable [strongly observable].


Example 6.2

The two systems shown in Fig. 2.9 have the same eigenvalues: one sloweigenvalue A j = -1 and one fast eigenvalue A2 = -1/e. They are, however,

Fig. 2.9. (a) Actuator form, (b) Sensor form.

structurally different: system (a) is in actuator form while system (b) is insensor form. The state realizations of the systems in Fig. 2.9 are

while their modal forms

are parallel configurations as in Fig. 2.8. For both systems, the sloweigenvalue Xl = -1 and the fast eigenvalue A2 = —l/e are controllable andobservable for e > 0.

The realization (a') displays the following structural characteristics of thesystem (a):

(al) the slow eigenvalue A j = -1 is strongly controllable and stronglyobservable;


(a2) the fast eigenvalue A2 = -1/eis strongly controllable, but is weaklyobservable through an e-term.

Also, the realization (b') displays the following structural characteristics ofthe system (b):

(bl) the slow eigenvalue A j = -1 is strongly controllable and stronglyobservable;

(b2) the fast eigenvalue A2 = -1/£ is strongly observable, but is weaklycontrollable through an e-term.

It is left as Exercise 2.8 to show that, in general, for systems in sensor formthe fast eigenvalues are always weakly controllable, whereas for systems inactuator form the fast eigenvalues are always weakly observable.

Weak controllability and weak observability are either a blessing or acurse when it comes to linear feedback design. They are a blessing in thatstable parasitic (fast) modes, which are either weakly controllable orweakly observable, will not be affected by more than O(e) through outputfeedback. Accordingly, output-feedback designs, based on reduced-orderslow models, are robust in the sense that stability of the fast subsystemis preserved when these designs are applied to the actual full system. Onthe other hand, should it be desired to shift a weakly controllable mode,0(l/e) high state-feedback gains are necessary to accomplish this task.More will be said about these problems in Chapter 3.

Two other tests for complete controllability and complete observabilityoccasionally used, equivalent to (6.7) and (6.8), are the original rankcriterion and Grammian tests due to Kalman (Kwakernaak and Sivan,1972). Treating controllability only, since observability follows by duality,the rank criterion states that the system (6.3), (6.4) or the pair (A, B) iscompletely controllable if and only if

Thus, again in view of the parallel structure of the system (6.3), (6.4) andthe preservation of the rank of a matrix under regular perturbation, Cor-ollary 6.1 can be restated in terms of separate lower-order rank criteria forthe slow system pair (A0, B0) and the fast system pair (A22, B2).

Collollary 6.3

If the following two conditions


are satisfied, then there exists an e* > 0 such that the original system iscompletely controllable in the sense of (6.26) for all e E (0, e*].

Even for linear time-varying systems encountered in Chapters 5 and 6,the controllability condition for the fast system is of the algebraic form(6.28) where (A22(i), B2(t)) is defined at a fixed t in r-scale. In Chapter 7,composite control problems nonlinear in x but linear in z and u, require asimilar algebraic controllability criterion only this time (A22(x), B2(x}} isdefined for each fixed A: in a certain domain.

For controllability and observability properties defined in terms of Gram-mian matrices, let us consider the observability Grammian

where the usual finite interval of integration is extended to [0, *>] on accountof assuming A to be a Hurwitz matrix. Then, as is well known (Kwakernaakand Sivan 1972), P is the solution of the Lyapunov equation

where

Premultiplying* (6.30) by TT and postmultiplying by T, where Tis definedby (4.8), we have

The solution of this Lyapunov equation is

Equation (6.33) shows the scaling for P that occurs in Chapters 3, 4 and 6,where we study Lyapunov equations and regular Riccati equations forsingularly perturbed systems. It follows from

* As a superscript T denotes matrix transposition, otherwise T is denned by the similaritytransformation (4.8).


that /* in (6.33) is scaled as

The same scaling is valid for the original P = T TPT l of (6.29) and willbe used in Chapters 3, 4 and 6. Corollary 6.1 can then be restated in termsof lower-order observability Grammians, and allied Lyapunov equations,for the slow and fast system pairs (C0,^40) and (C2,A22), respectively.Finally, it is left as Exercise 2.9 to show that the (dual) controllabilityGrammian W, satisfying the dual Lyapunov equation

is scaled as

2.7 Frequency-Domain Models

Singularly perturbed systems display further interesting properties whenexhibited in the complex domain. The starting point in our analysis, as inthe previous section, is the system (6.1), (6.2), transformed to the system(6.3), (6.4) by the similarity transformation (4.8).

The Laplace transform of (6.3), (6.4), omitting initial conditions, is

which gives

2.7 FREQUENCY-DOMAIN MODELS 85

Thus the transfer-function matrix G(s, e) from the input u(s) to the outputy(s, E) is given by the sum of the slow and fast transfer-function matrices

The slow transfer-function matrix is a function of s:

while the fast transfer-function matrix is a function of es:

Their parallel connection is illustrated in Fig. 2.10. The two frequencyscales corresponding to the time scales t and t/e are 5 andp = es respectively.

Fig. 2.10. Two-frequency decomposition.

We have already used this scaling of frequencies in Section 2.3 whendiscussing the characteristic polynomials tys(s, e) and i/;f (/?,$), associatedwith Gs(s, e) and Gf(p, e) respectively. In this scaling, the complex fre-quency p = es becomes significant, that is, 0(1), only when the origincomplex frequency s is high, that is, O(l/e) or higher. The decompositioninto the slow and fast systems now acquires the meaning of a decompositioninto low-frequency and high-frequency blocks. To see this decompositionmore clearly, let us restrict s to the imaginary axis, i.e., s = jco, and assumethat As(e = 0) and Af(e = 0) have no eigenvalues on the imaginary axis.We observe that Gs(s, e) in (7.4) is a strictly proper transfer-function matrix,and thus has the low-pass property

for some positive fixed constant col. On the other hand, for low frequenciess = 0(1), that is, p = O(e), the fast transfer-function matrix Gf(/?, e) in(7.5) can be approximated by its DC-gain, that is,


for some positive fixed constant o>2- This suggests that a low-frequencyapproximation of G(s, e) in (7.3) is

and is depicted in Fig. 2.11(a). The corresponding high-frequency approxi-mation is

and is depicted in Fig. 2.11(b). In the state-space representation (7.2), thelow-frequency model (7.8) is

Fig. 2.11. (a) Low-frequency approximation; (b) high-frequency approximation; (c) low-frequency approximation; (d) high-frequency approximation; (e) low-frequency approxi-

mation; (f) high-frequency approximation.

2.7 FREQUENCY-DOMAIN MODELS 87

as shown in Fig. 2.11(c), and the high-frequency model (7.9) is

as shown in Fig. 2.11(d). It should be recalled from Section 2.1 that everymatrix in (7.10) and (7.11) is a continuously differentiable function of e,and, by (2.13) and (4.7), for e sufficiently small can be approximated byits value at e = 0. Thus a further simplification of the low-frequency model(7.10) is

as shown in Fig. 2.11(e), while the corresponding simplification of the high-frequency model (7.11) is

as shown in Fig. 2.11(f).

Example 7.1

Let us return to the system (6.24a) in actuator form of Example 6.2. TakingLaplace transforms of the model realization (a') in (6.25), the exact slowand fast transfer functions (7.4) and (7.5) are

and

in the parallel configuration of Fig. 2.10. From (7.15), the low-frequency"equivalent" of Gf(p, e) is

Taking Cf(0, e) in (7.16) and combining it with the exact slow transferfunction Gs(s, e) in (7.14), we have

as the low-frequency approximation of the complete transfer function in


Fig. 2.11(a). Setting e = 0 further simplifies (7.17) to

which is the low-frequency model obtained by neglecting the actuatordynamics. It can be seen that, in this particular example, the model (7.17)yields O(e2) approximation at low frequencies, while (7.18) is only an O(e)approximation. The presence of an unstable zero in (7.17) is significant asit shows the destabilizing effect of e.

2.8 Exercises

Exercise 2.1

Complete the block-diagonalization of the system (2.1), (2.2) by applicationof the change of variables r] = z + N(e)% to the sensor form (2.30) whereN(e) satisfies a Lyapunov-type equation which is the adjoint of (4.3).Although the eigenvalues of the sensor form and the eigenvalues of theactuator form are the same examine in what sense the state variables aredifferent.

Exercise 2.2

Derive the corrected slow and fast models (4.13) and (4.14) from (2.1),(2.2) using the following separation of time scales procedure. Set e = 0 toobtain the quasi-steady-state z; derive the fast subsystem in terms of thefast state 77j = z — z; remove the fast part from x to obtain the slowsubsystem in ̂ = x - sA^A^^i • Is this procedure equivalent to L0 - H0

transformation in (4.10), (4.11)? Suggest how it might form part of aniterative scheme to obtain corrected models to any order of accuracy (seealso Kokotovic et al., 1980).

Exercise 2.3

For the adjoint quadratic equation (2.29), deduce the analogs of Lemma2.1, Lemma 2.2 and equations (3.14)-(3.19).

Exercise 2.4

Consider the discrete-time system

2.8 EXERCISES 89

where Im — A22 is nonsingular. Use the transformation rj(n) = z(n) + Lx(n)to show that the system is asymptotically stable if AH + A\2(I — A22)"1 A2i isHurwitz in a continuous-time sense; i.e. its eigenvalues have negative real parts,and A22 is Hurwitz in a discrete-time sense; i.e. its eigenvalues are inside the unitcircle (for more details see Litkouhi and Khalil, 1984).

Exercise 2.5

Compute the corrected slow and fast eigenvalues and eigenvectors for thesystem

and compare the exact slow and exact fast eigenvalues and eigenvectorsfor a value of e less than that given by the upper bound of Lemma 2.2.

Exercise 2.6

Show that the angle of the fast eigenvector

in (4.32) makes with the z-subspace is an O(e2) approximation of the anglethat Vf in (4.27) makes with the z-subspace by considering the effect of ane-correction of the term Im on its angle.

Hint: Proceed similarly to Example 4.2 by calculating the angulardifference

Exercise 2.7

Analyse the controllability and observability properties of the system

Exercise 2.8

Show that, in general, the fast eigenvalues of systems in the actuator form


of Fig. 2.3 [sensor form of Fig. 2.5] are strongly controllable and weaklyobservable [weakly controllable and strongly observable], lending furtherweight to the adjectives actuator and sensor.

Exercise 2.9

Establish Corollary 6.1 in terms of lower-order controllability Grammians,while at the same time verifying that the controllability Grammian

satisfying the Lyapunov equation

where A is a Hurwitz matrix, must be scaled as

Exercise 2.10

In Example 7.1, alternatively determine the low-frequency "equivalent" ofthe system (6.24a) through performing a partial-fraction expansion of themodel in Fig. 2.9(a).


The transformation of a singularly perturbed system model into block-triangular form, in its more general time-varying version, is due to Chang(1972). Following Kokotovic (1975), Lemma 2.1 and Lemma 2.2 specializeChang's result to linear time-invariant systems. In Avramovic (1979),Kokotovic et al. (1980) and O'Malley and Anderson (1982), the deter-mination of the matrix L is iterative. When the separation of time scales islarge (e is small), the convergence rate is quite rapid, typically a couple ofiterations. Even when e is not small, as for example in the power systemdecomposition of Winkelman et al. (1980), the iterative scheme is stillapplicable, provided that the original choice of slow and fast states in themodel is valid. The iterations for solving the matrix quadratic equation(2.8) are closely related to the eigenspace power iterations of Avramovic


(1979) and the quasi-steady state iterations of Kokotovic et al. (1980)and Winkelman et al. (1980). Phillips (1983) establishes the equivalencesbetween these iterative procedures and the earlier asymptotic seriesmethods of Vasil'eva and Butuzov (1978).

Theorem 3.1, which characterizes the essential property of singularlyperturbed systems, is a time-invariant version of a theorem of Klimushchevand Krasovskii (1962) (see also Wilde and Kokotovic, 1972a), and has beenapplied to networks with parasitics by Desoer and Shensa (1970) and tocontrol systems by Porter (1974) and Kokotovic and Haddad (1975a). Amore detailed stability analysis leads to an upper bound on e (Zien, 1973),different from that of Lemma 2.2, while another upper bound using singularvalues is given by Sandell (1979).

The second change of variables in Section 2.4 completes the exactdecomposition of the system into slow and fast states. Determination of an//-matrix, satisfying the linear equation (4.3), like that of L is also iterative(Avramovic, 1979, Kokotovic et al., 1980). Lemma 4.1 on the eigenspaceproperties of linear time-invariant systems is based on Avramovic (1979)and Kokotovic (1981) and can be extended to include generalized eigen-vectors as well. It forms the basis of a grouping algorithm for power systems(Avramovic et al., 1980; Chow, 1982) and for Markov chains (Delebeque,Quadrat and Kokotovic, 1984; Kokotovic, 1984; Saksensa et al., 1984).

The initial value theorem, Theorem 5.1, for linear time-invariant systems,is contained in Kokotovic and Haddad (1975a) and Kototovic et al. (1976).As Example 4.1 illustrates, corrected models such as (5.36) and (5.37) areuseful in pinpointing the destabilizing effects of parasitics in adaptivecontrol (loannou and Kokotovic, 1983). Corrected models have also beenused to advantage in demonstrating the stabilizing effects of the damperwindings in synchronous machine models (Sauer, Ahmed-Zaid and Pai,1984a, b) and improving the stability of large-scale power system models(Pai, Sauer and Khorasani, 1984). The O(e2) state approximations ofTheorem 5.2 for corrected models are new. Higher-order approximationscan readily be obtained through the use of higher-order terms in the power-series expansions of L and //.

The controllability result of Theorem 6.1 was introduced in Kokotovicand Haddad (1975a). Analogous results apply to continuous-time anddiscrete-time linear two-time-scale systems (Saksensa et al., 1984). Theobservability conditions follow from duality with those of controllability.The notions of weak and strong controllability (observability), introducedin Section 2.6, are due to Chow (1977b).

In Section 2.7 the introductory discussion of singularly perturbed systemsfrom a frequency-domain point of view is new. An early paper by Porterand Shenton (1975) uses a transfer-function approach, while Luse and


Khalil (1985) and Luse (1984), more recently, develop a two-frequencyanalysis of transfer-function matrices and their properties. A suggestion fortwo-frequency design is presented in Fossard and Magni (1980), and afrequency-domain robustness condition appears in Sandell (1979). Other-wise, considering how much is presently known about frequency-domainmethods for general multivariable systems, surprisingly little is knownabout two-frequency-scale systems. It is expected that future analysis will,among other things, extend our understanding of system robustness in themid-to-high frequency (parasitic) range. Steps in this direction haverecently been taken by Silva-Madriz and Sastry (1984) and O'Reilly (1985).

3 LINEAR FEEDBACK CONTROL

3.1 Introduction

Our attention in this chapter is focused on linear feedback design fordeterministic linear time-invariant systems containing both slow and fastdynamic phenomena. Normally, any feedback design, like the system itseeks to control, will suffer from the higher dimensionality and ill-con-ditioning resulting from the interaction of slow and fast dynamic modes.

In the singular perturbation or, more generally, the two-time-scaleapproach to feedback design, these stiffness properties are taken advantageof by decomposing the original ill-conditioned system into two subsystemsin separate time scales. State-feedback design may then proceed for eachlower-order subsystem, and the results combined to yield a composite state-feedback control for the original system. At the same time, the compositecontroller is required to achieve an asymptotic approximation to the closed-loop system performance that would have been obtained had a state-feedback controller been designed without the use of singular perturbationmethods.

This is the substance of the two-time-scale design approach of Section3.2, adopted for eigenvalue assignment, and the optimal linear regulatorin Sections 3.3 and 3.4. A feature of these composite state-feedbackcontrollers is that they do not require knowledge of the singular per-turbation parameter e. Hence, they are applicable to systems where erepresents small uncertain parameters.

Should e be known, closer "order of e2" approximations are obtainedusing composite state-feedback control designs based upon corrected slowand fast models. A related corrected linear-quadratic design is presentedin Section 3.5. In Section 3.6, it is shown how two-time-scale methods ofstate-feedback control provide a convenient framework within which to

93

94 3 LINEAR FEEDBACK CONTROL

approach the analysis and solution of high-gain feedback and cheap controlproblems.

Finally, the effect of designing feedback control strategies using a modelthat neglects the unknown parasitic elements is considered. It is shown inSection 3.2 that state-feedback controllers are robust in the sense ofretaining overall system stability, provided that the disregarded fast modesare asymptotically stable. Unlike state feedback, it is of fundamentalimportance to note that, in general, no such robustness property holdsfor output-feedback controllers. In Section 3.7, however, conditions areidentified whereby closed-loop instabilities resulting from output-feedbackdesign are avoided.

3.2 Composite State-Feedback Control

It is desired to construct a state-feedback control for the singularly per-turbed linear time-invariant system model analyzed in Section 2.6:

Preliminary to any separation of slow and fast designs, the system (2.1),(2.2) is approximately decomposed into a slow system model with n smalleigenvalues and a fast system model with m large eigenvalues, as in Section2.6. The wth-order slow system is

where

and the vectors xs, zs and ws are the slow parts of the corresponding variablesx, z and u in the original system (2.1), (2.2). Also, the mth-order fastsystem is

where zf = z - zs and uf=u-u^ denote the fast parts of the correspondingvariables in (2.1), (2.2).

It is appropriate to consider the following decomposition of feedbackcontrols where

3.2 COMPOSITE STATE-FEEDBACK CONTROL 95

are separately designed for the slow and fast systems (2.3) and (2.6). Acomposite control for the full system (2.1), (2.2) might then plausibly betaken as

However, a realizable composite control requires that the system states jcs

and zf be expressed in terms of the actual system states x and z. This canbe achieved by replacing ;cs by x and zf by z - zs, so that the compositecontrol (2.8), in view of (2.4), takes the realizable feedback form

where

The above design procedure is a decomposed one in that the gain matricesGO and G2 are separately designed according to slow and fast modeperformance specifications, resulting in the physically realizable compositecontrol law (2.9) as shown in Fig. 3.1. It remains to establish the asymptoticvalidity of this composite control (2.9), as e—»0, when applied to thesystem (2.1), (2.2). This we now do in the following theorem.

Fig. 3.1. Composite state feedback control.

Theorem 2.1

Let G2 be designed such that ReA(^422 + B2G2) < 0. Then there exists anE* > 0 such that if the composite control


is applied to the system (2.1), (2.2), the state and control of the resultingclosed-loop system, starting from any bounded initial condition x° and z°,are approximated according to

for all finite t > t0 and all e E (0, £*]. If in addition G0 is designed such thatRe A(A0 + BQGo) < 0, there exists an e* > 0 such that the resulting closed-loop system is asymptotically stable and (2.12)-(2.14) hold for alle E(0, e*] and t e f t o , °°).

Proof Consider a feedback control of the form (2.9) for any G l5 andnot only GI given by (2.10), applied to the system (2.1)-(2.3). The closed-loop system may be written as

where

The closed-loop system (2.15) is in the standard singularly perturbed formwhere, as a Hurwitz matrix, F22 is nonsingular. Hence a complete separationof (2.15) into slow and fast systems is achieved by the nonsingular trans-formation into block-diagonal form introduced in Section 2.4:

where for e 6 (0, e*], e* > 0, the matrices L = L(e) and H = H(e) respec-tively satisfy

and

3.2 COMPOSITE STATE-FEEDBACK CONTROL

In the new coordinates, the exact system is

97

with the initial conditions

Instead of using G! and G2 as design parameters, we use Gs and G2 asdesign parameters with G^ given by

Then substitution of (2.21) and (2.16) into the quadratic equation (2.18)results in the equation

It is observed that (2.22) is independent of the fast gain matrix G2. Rearrang-ing (2.22) as

and substituting (2.23) into (2.20), using (2.16) and (2.21), results in theexact closed-loop system

with the same initial conditions as (2.20), where

Since L in (2.23)-(2.26) is independent of G2, the matrix As + BSGS isindependent of G2. The design parameters Gs and G2 are chosen to shapethe slow and fast models to the desired accuracy. For O(e) approximationslet Gs = G0 and approximate A& and Bs by A0 and B0. Then G0 is chosento control the closed-loop matrix A0 + B0G0. Similarly, Af and B{ areapproximated by A22 and 52, and G2 is chosen to control the closed-loopmatrix A22 + B2G2. In calculating G! using (2.21), it suffices to use an O(e)


approximation of G!. Hence, by (2.23) GI is approximated by G0 + G2LQ,where

and this results in the expression (2.10) for Gj. When this value of G: isused and the analysis leading to (2.24) is applied to the closed-loop system,the resulting closed-loop slow model will be O(e) close to A0 + B0G0, andthe resulting closed-loop fast model will be O(e) close to A22 + B2G2. Theremainder of the proof then follows from the stability of F22 = A22 + B2G2,in complete analogy with that of Theorem 5.1 of Chapter 2.

It is noted that an upper bound e* for e can be obtained by applyingLemma 2.2 of Section 2.2 to the closed-loop system (2.15), where for An

we write Fn, etc. The preceding two-time-scale design procedure has twimportant features: first, a reduction in computational requirements isachieved through solving two lower-order control problems in separatetime scales; second, the resulting composite feedback control (2.11) doesnot require knowledge of the singular perturbation parameter e, which mayrepresent small uncertain parameters. Since the control of the fast system(2.6) occurs in an 1/e faster time scale than that of the slow system (2.3),the response of the full system (2.1), (2.2) will be dominated by the reducedsystem state (2.3) after the decay of fast-system transients; that is, after aninitial short boundary layer interval [0, ti], zf(t) ~ 0 and uf(t) ~ 0 in (2.12)-(2.14) for f s^ .

Also, if A22 is a Hurwitz matrix, one need only stabilize the reducedsystem (2.3). This is simply effected by setting G2 = 0 in (2.11) to obtainthe following corollary to Theorem 2.1.

Corollary 2.1

If A22 is a Hurwitz matrix and the control

is applied to the system (2.1), (2.2), then there exists an e* > 0 such thatthe relations (2.12)-(2.14) hold for all finite t > t0 and all e E (0, e*]. If G0

is designed such that A0 + B0G0 is also Hurwitz, then there exists an e* > 0such that the closed-loop system is asymptotically stable for all e E (0, e*].

Corollary 2.1 is a fundamental robustness result in that it states that thesystem zero-order approximations of Theorem 2.1 remain valid for a controlusing feedback of the slow state x(t) only, provided that the neglectedfast modes are asymptotically stable. The corollary provides a theoreticaljustification for disregarding the parasitic elements of the full-system model

o


(2 . 1 ) , (2 . 2) in the design of the control (2 . 29) by showing that , for sufficientsmall e, the actual performance of the closed-loop system is arbitrarily closeto that predicted by the lower-order model (2.3)-(2.5): that is, the controlstrategy is robust with respect to the neglect of parasitic elements repre-sented by the small scalar e > 0. This is in contrast with the output-feedbackcontrol problem of Section 3.7, where we shall see that such a strategy maybe destabilizing even when the neglected system modes are asymptoticallystable.

In the same way, and for similar reasons, as in Section 2.5, corrected orO(e2) approximations of the state trajectories for the system (2.1), (2.2under linear state-feedback control (2.9) can be achieved through the useof separate corrected slow and fast designs. The key to this development,as in the uncorrected state-feedback design of Theorem 2.1, is the exacttwo-time-scale decomposition of the closed-loop system in (2.24).

For the corrected design, we choose

where

Also, using (2.19) and (2.16), let us define

In analogy with (5.36) of Chapter 2, the corrected slow model is (cf. (2.25)and (2.26))

with the initial condition

Similarly, in analogy with (5.37) of Chapter 2, the corrected fast model inthe T=(t- t0)/£ scale is (cf. (2.27))


with the initial condition

In analogy with the O(e) approximations of Theorem 2.1, O(e2) approxi-mations are obtained by way of a composite state-feedback controller basedon separate lower-order corrected feedback designs

for (2.34) and (2.38) in different time scales.

Theorem 2.2

If the composite control

is applied to the full system (2.1), (2.2) and if A22 + B2G2 is a Hurwitzmatrix, then there exists an e* > 0 such that the state and control of theresulting closed-loop system, starting from any bounded initial conditionsx° and z° are approximated according to

for all finite t > t0 and all e E (0, £*], where L0, L^ and H0 are defined by(2.28), (2.32) and (2.33) respectively. If A0 + B0G0 is also Hurwitz thenthere exists an E* > 0 such that the closed-loop system is asymptoticallystable and (2.44) -(2.46) hold for all £ e (0, e*) and t E [f0, »).

It is immediate from (2.25), (2.26) and (2.35), (2.36) that ̂ s andBs are approximated to within O(e2} according to

Similarly, by (2.27) and (2.39), (2.40),

The gain matrices G0c and G2 are designed to control AQC + B^G^ andA22c + B2cG2 respectively, where, by (2.30), (2.47) and (2.48),

Pro of


and

Also, substituting (2.30) in (2.21), we have the composite control (2.43),where

Since A22 + B2G2 is Hurwitz, the remainder of the proof proceeds inanalogy with that of Theorem 5.2 of Chapter 2. D

Similar to Corollary 2.1, one has the following useful robustness result,this time using a corrected reduced control.

Corollary 2.2

If A22 is a Hurwitz matrix, and the control

is applied to the full system (2.1), (2.2), then there exists an e* > 0 suchthat the relations (2.44)-(2.46) hold for all finite t > t0 and all e £ (0, e*].If A + BQGQ is also Hurwitz, then there exists an e* > 0 such that theclosed-loop system is asymptotically stable and (2.44)-(2.46) hold for alle£(0, e*] a n d f £ [ f 0 , °°).

Another design possibility is a composite control law based upon acorrected slow design and an uncorrected fast design. All feedback controlstrategies, involving corrected slow designs, yield the same O(e2) approxi-mation after the decay of fast-system transients in the initial short boundarylayer interval.

It is observed in Theorem 2.2 and Corollary 2.2 that the corrected systemmodels (2.34) and (2.38) depend on the uncorrected slow gain matrix G0.This disadvantage is not as grave as it might appear, since it is reasonableto first assess the uncorrected state approximations of Theorem 2.1 orCorollary 2.1 before proceeding to Theorem 2.2 or Corollary 2.2. Byconstruction in (2.30), the corrected slow gain G0c is related to the uncor-rected one GO by G0c(e) = G0 + £G'0 for some matrix GO that is O(l).When the design problem has more than one solution, as in the multi-input eigenvalue assignment problem of Section 3.3, a solution for theuncorrected design G0 may not be the limit of the solution for the correcteddesign G0c unless the designer carefully picks G0c such that lim G0c(e) =G e-»0

0-


Corollary 2.3

If A22 is nonsingular, the exact slow model (As, Bs) of (2.24) is invariantwith respect to the class of fast feedback controls u = v + G2z.

Proof It is sufficient to note that for the exact slow model pair (^4S, fis)in (2.24) the solution L(e) of the quadratic equation (2.22) or (2.23) doesnot depend on the fast feedback gain matrix G2 of the original linearfeedback control law (2.9), applied prior to L-H transformation. D

It follows from Corollary 2.3 that not only are the slow designs inde-pendent of fast feedback design, but the controllability properties of theuncorrected and corrected slow models are unaffected by fast statefeedback.

3.3 Eigenvalue Assignment

We observed in Theorem 6.1 of Chapter 2 that if the eigenvalues of theslow system (2.3) and the eigenvalues of the fast system (2.6) are control-lable, then the corresponding eigenvalues of the original system (2.1),(2.2) are controllable for e sufficiently small. Given the well known fact(Kwakernaak and Sivan 1972) that under linear state feedback, con-trollability is equivalent to the arbitrary assignability of the eigenvalues ofthe closed-loop system matrix, it follows that the separate stabilization ofthe systems (2.3) and (2.6) implies the stabilization of (2.1), (2.2).

Suppose it is required to assign n + m closed-loop eigenvalues of thesystem (2.1), (2.2) to arbitrary positions {Af, A|, . . . , A£+m} in the openleft-half complex plane by way of the linear feedback control (2.9). Giventhe separation in the open-loop eigenvalue spectrum of (2.1), (2.2), it isnatural to assign n small eigenvalues to the slow system and to assign mlarge eigenvalues to the fast system.

Theorem 3.1

exists and if the slow system pair (A0, B0) and the fast system pair(A22, B2) are each controllable, and G0 and G2 are designed to assigndistinct eigenvalues A,-, / = 1, . . . , n and A;, ;' = 1, . . . , m, to the matricesA0 + 50G0 and A22 + B2G2 respectively, then there exists an e* > 0 suchthat for all e E (0, e*] the application of the composite feedback control

if A22

3.3 EIGENVALUE ASSIGNMENT 103

to the system (2.1), (2.2) results in a closed-loop system containing n smalleigenvalues {Ai, A 2 , . . . , A£} and m large eigenvalues A£ + 1 , . . . , A£+ m},which are approximated by

Complete controllability of the pairs (A0, B0) and (A22, B2)implies the existence of gain matrices G0 and G2 which arbitrarily assigncorresponding eigenvalues to the matrices A0 + B0G0 and A22 + B2G2

respectively (Kwakernaak and Sivan 1972). Since the closed-loopeigenvalues A f , / = 1,. . . , n + m, are precisely those of (2.24), the prooffollows from the fact that AQ + B0G0 and A22 + B2G2 are O(e) regularperturbations of those of (2.24) and the application of Theorem 3.1 ofChapter 2.

The above separation property suggests the following two-time-scaledesign procedure for an O(e) approximate eigenvalue assignment of thesingularly perturbed linear system (2.1), (2.2): construct G0 so as to placethe n small eigenvalues of A0 + B0G0; separately construct G2 so as toplace the m large eigenvalues of A22 + B2G2; then form the compositefeedback control (3.1). It is important to observe that the success of thecomposite control idea, here applied to eigenvalue assignment, hinges onthe implicit requirement that the design objective be compatible with theslow-fast nature of the open-loop system (2.1), (2.2). In other words, thedesign objective should allow 0(1) feedback to be applied in such a wayas to keep what is slow slow and likewise what is fast fast.

In the light of Theorem 3.1 of Chapter 2, should one remove theassumption that the assigned eigenvalues A,-(yl0 + B0G0) are distinct andthe assigned eigenvalues A,(A22 + B2G2) are distinct, the result (3.2), (3.3)of Theorem 3.1 is replaced by the weaker one: namely, as £—» 0, the firstn eigenvalues of the closed-loop system tend to fixed positions in thecomplex plane defined by A,(y40 + BOGQ) while the remaining m eigenvaluesof the closed-loop system tend to infinity with the rate 1/e along asymptotesdefined by Ay(A22 + B2G2), that is to [^(A22 + B2G2)}/e.

As an illustration of the preceding two-time-scale design procedure,consider the following example.

Example 3.1

Recall for the DC motor of Case Study 7.1 in Chapter 1 that the normalized

Proof

o

262 Introduction to Matrix Analysis

ential and Difference Equations, Office of Naval Research, Depart-ment of the Navy, January, 1949.

R. Bellman, Methods of Nonlinear Analysis, Academic Press, Inc.,New York, 1970.

These questions, as well as the corresponding questions for the differ-ential equation dx/dt = (A + B(t))x, belong more to the theory of lineardifferential and difference equations than to the theory of matrices per se.

For a discussion of the stability of linear differential systems withrandom coefficients, see

0. Sefl, On Stability of a Randomized Linear System, Sci. Sinica,vol. 7, pp. 1027-1034, 1958.

S. Sninivasan and R. Vasodevan, Linear Differential Equations withRandom Coefficients, American Elsevier Publishing Company, Inc.,New York, 1970 (to appear).

J. C. Samuels, On the Mean Square Stability of Random LinearSystems, IRE Trans, on Circuit Theory, May, 1959, Special Supple-ment (Transactions of the 1959 International Symposium on Circuitand Information Theory), pp. 248-259.

where many additional references to earlier work by Rosenbloom andothers will be found.


is given by the e-dependent control law

Finally, noting that the fast mode is stable, and experiences an exponen-tial decay of the order of e~'/e, the reduced control, based upon the slowmodel (A0, BQ) = (-1,1), is robust and takes the form

which results in a stable closed-loop system with the characteristic equation

As a check, let us perform a frequency analysis of the characteristicequation (3.15) along the lines of (3.8)-(3.11). In the slow frequency-scale,the characteristic equation (3.15) or eA2 + A + 3 = 0 tends to A + 3, ase—> 0, which is associated with the desired slow eigenvalue A = -3. Also,in the fast frequency scale let p = eA, so that (3.15) may be rewritten as

As e—»0 the characteristic equation (3.16) tends to p2 + p = p(p + 1)= 0, the nonzero root of which, p = -1, yields the desired fast eigenvalueA = -1/e. Observe that in the limit as e — > 0 the fast eigenvalue is notshifted by the reduced control (3.14), but remains at its open-loop valueof A = -1/e.

A closer placement of the eigenvalues of the system (2.1), (2.2), to within0(e2) of desired closed-loop values, is achieved by way of the correctedcomposite control (2.43) based on eigenvalue placements for the correctedslow and fast models (2.34) and (2.38) in separate time scales.

Theorem 3.2

If A22 exists and if the slow system pair (A0, B0) and the fast system pair(A22, B2) are each controllable then there exists an e* > 0 and gain matricesG0c and G2 which arbitrarily assign distinct eigenvalues A(, / = 1,. . . , n andAy ,y = 1,. . . ,m to the matrices A0c + fi0cG0candA22c + B2cG2, respectively,such that for all e E (0, e*] the application of the composite feedbackcontrol


where

to the system (2.1), (2.2) results in a closed-loop system containing n smalleigenvalues {Aj, A 2 , . . . , A£} and ra large eigenvalues {A£+ 1 , . . . , A£+m},which are approximated by

Proof The proof is similar to that of Theorem 3.1. As in the proof ofTheorem 6.1 of Chapter 2, (A0, B0) and (A22, B2) controllable imply that(AQC, B0c) and (A22c, B2c) respectively are controllable for e E [0, e*]. Thusthere exist gain matrices GOC and G2c which arbitrarily assign eigenvaluesto AQC + B0cGQc and A22c + B2cG2 respectively. The proof then follows from(2.24) and the application of equations (3.18) and (3.19) of Chapter 2. D

Again, as in Theorem 2.2, we note the dependence of the corrected slowand fast models (2.34) and (2.38) on the uncorrected slow gain matrix G0.Thus, the price of achieving a closer eigenvalue assignment than in Theorem3.1 is that it is necessary to first solve an uncorrected slow eigenvalueproblem to obtain G0. This disadvantage is mitigated by the fact that anuncorrected slow eigenvalue assignment will usually be attempted beforeproceeding to the corrected design. As remarked in Section 3.2, it isimportant to choose G0c such that lim Goc(e) = G0. An illustration of the

e—»0

approximation in eigenvalue assignment and the resulting state trajectoriesachieved using both uncorrected and corrected composite control is pro-vided by the next example.

Example 3.2

Consider the two-tank chemical reactor system referred to in Example 6.1of Chapter 2, where it is required to maintain the concentration of liquidin the second tank at a desired level, in spite of variation of inlet con-centration to the first tank, by the addition of reactant through a controlvalve.

A linearized model of the control scheme is depicted in Fig. 3.2, wheredeviations from desired steady-state values are described by

Cr = 0 = deviation in desired concentration,Xi = deviation in concentration of second tank,


Fig. 3.2. Two-tank chemical reactor system.

x2 = deviation in concentration of first tank,u = actuator control value signal.

By inspection, the open-loop system has four poles at {-5, -5, -4, -4}, ofwhich the last two are associated with the actuator dynamics. The actuatorblock in Fig. 3.2 may be represented, through scaling the complex frequencys by e = ?, as

with the initial conditionn

where x £ /?2 and z E.R2 respectively describe the slow reactor and fastactuator dynamics. Since the actuator poles {-4, -4} are stable, or equiv-alently, from Example 6.1 of Chapter 2, A22 is Hurwitz, the control problemis approached as one of designing the slow state-feedback control (G2 = 0)

so that the complete system model takes the singularly perturbed form


such that the dominant reactor system poles {-5, -5} are shifted to near(-0.707 ± jO.707}, corresponding to a damping ratio of £ = 0.707 and anatural frequency of a>n = I rad/s.

Let us first perform the uncorrected slow design of Corollary 2.1, whereA0 and B0 are as defined in equation (6.18) of Chapter 2. The uncorrectedgain, GO = [-go ~^oL chosen such that

where £ = 0.707 and con = 1 rad/s, is given by

The eigenvalues of the actual full system (3.21) under this feedbackcontrol (3.23), (3.25) are

of which the slow eigenvalues are observed to be within O(i) of the desiredeigenvalues {-0.707 ± jO.707}. The corresponding slow states xl and x2 ofthe original system (3.21) under feedback control (3.23), (3.25) are com-pared with those of the slow closed-loop system (2.3), (3.25) or

in Fig. 3.3.For a closer slow eigenvalue assignment and closed-loop system response,

we proceed to the corrected slow design of Corollary 2.2 for the slow modelpair (A,*, B0c) of (2.35), (2.36) or

The corrected gain G0c, chosen such that

for the design values of £ = 0.707 and con = 1 rad/s, is


Fig. 3.3. (a), (b) Reactor system response.

The eigenvalues of the actual full system (3.21) under this feedback control(3.23), (3.30) are

of which the slow eigenvalues are this time within 0((?)2) of the desiredeigenvalues {-0.707 ± jO.707}. The corresponding responses of the slowstates of the original system (3.21) under corrected feedback control (3.23),(3.30) are compared with those of the uncorrected design (3.21), (3.23),(3.25), and with those of the desired slow closed-loop system (3.27) in Fig.3.3.

Example 3.3

As an illustration of weak controllability, consider again the state realization(b) of Example 6.2 of Chapter 2, where it is desired to shift the eigenvalues{-1, -1/e} of the open-loop system

to the positions {-2, -2/e} of the complex plane. Although the system(3.32) is completely controllable for s > 0, its fast eigenvalue is but weaklycontrollable, as noted in Example 6.2. Consequently, although a reducedslow design is possible, the composite design procedure of Theorem 3.1


cannot apply because of our inability to shift the "fast" eigenvalue -1 inr-scale.

On the other hand, the linear state feedback that assigns the desiredcharacteristic polynomial (A + 2)(A + 2/e) exactly, for e > 0, is given bythe high-gain control

The gains in (3.33) are large on account of the fact that the control u needsto force its effect on the fast transient of z through the predominantly slowstate x of (3.32).

3.4 Near-Optimal Regulators

The infinite-time optimal linear regulator problem for the linear time-invariant system

where x E Rn, z E Rm, u E Rr and y E Rp, is defined as the problem offinding a control u(f) E Rr, t E [0, °°], so as to regulate the state [*($] to theorigin by way of minimizing the quadratic performance index

where the r x r weighting matrix R penalizes excessive values of control

The solution of the linear quadratic regulator problem (4.1)-(4.3) is theoptimal linear feedback control law (Kwakernaak and Sivan 1972)

where satisfies the algebraic matriz Riccati equation

u(t).

3.4 NEAR-OPTIMAL REGULATORS 111

with

Moreover, the optimal cost is given by

Conditions providing for the existence of a solution (4.4)-(4.8) to theoptimal linear regulator problem (4.1)-(4.3) are as follows.

Theorem 4.1

If the slow system triple (M0,A0, B0) and the fast system triple (M2, A22,B2) are each stabilizable and detectable [observable] then there exists anE* > 0 such that the original optimal linear regulator problem (4.1)-(4.3)has a solution (4.4)-(4.8) where the (n + m) x (n + m) matrix K = K(E)exists as the unique positive-semidefinite [positive-definite] stabilizing solu-tion of the algebraic Riccati equation (4.5) for all e E (0, e*].

Proof By Corollaries 6.1 and 6.2 of Chapter 2, the conditions of Theorem4.1 imply that the full system triple (M, A, B) is stabilizable and detectable[observable]. The theorem then immediately follows from standard linearregulator theory (Kwakernaak and Sivan 1972).

In Theorem 4.1 advantage has been taken of the singularly perturbednature of the system (4.1), (4.2) to give existence conditions for a solutionto the full regulator problem in terms of stabilizability-detectability con-ditions on lower-order systems, (M0, A0, B0) and (M2, A22, B2), in separatetime-scales. It is natural to suspect that the original regulator problem(4.1)-(4.3) itself can be decomposed into two separate regulator problems:one for the slow system (M0,A0, BQ) of (2.3), and the other for the fastsystem (M2, A22, B2) of (2.6). Indeed, Theorem 2.1 suggests that from thesystem optimal controls «s and wf a composite control

can be formed for implementation on the original system (4.1), (4.2).

o


First, let us see how the original regulator problem (4.1)-(4.3) may bedecomposed into two regulator problems in separate time scales. Equation(4.2) may be decomposed, using (2.12) and (2.13), into a slow part and afast part; that is,

where, separating slow and fast variables, the output

with

is associated with the slow system (2.3), and the output

is associated with the fast system (2.6).Now, after the decay of fast-system transients in the initial short boundary

layer interval [0, rj, we have from (2.14) and (4.10) that u(t) ~ us(t) andy(t) ~ ys(t) for t ̂ ̂ . By the same token, for t ̂ t1 or in the slow time-scale,the original regulator problem (4.1)-(4.3) is described by the followingone.

Slow regulator problem

Find M so as to minimize

for the slow system (2.3), (4.11).In terms of xs and ws, (4.14) becomes

where


Lemma 4.1

If the triple (M0, A0, 50) is stabilizable and detectable [observable] thenthere exists a unique positive-semidefinite [positive-definite] stabilizingn x n solution ^s of the algebraic matrix Riccati equation

and the optimal control for (2.3) and (4.14) is

Proof It is well known (Anderson and Moore 1971) that if the triple(M0, AQ - BQR^NQMQ, BQ) is stabilizable and detectable [observable],where M^M0 = Mj (/ - NQR^NQ^MQ, there exists a unique positive-semi-definite [positive-definite] stabilizing solution K. Since stabilizability isunaffected by linear state feedback, the pair (A0 - BORQINQMO,BQ) isstabilizable. From the matrix identity, proved by multiplying both sides by

it follows that there exists a nonsingular Q0 such that Q^Qo=I-NQR^NQ. Hence, as a dual of the preceding stabilizability result, the pair(A0 - B0Ru lNoM0, <2oM0) is detectable if and only if the pair (M0, AQ) isdetectable. Thus the stabilizability and detectability [observability] of thetriple (M0, A0 - BoR^NoMo, B0) is equivalent to that of the triple(M0,,40,fl0).

In the fast time scale r= t/e only the fast variations zf(r), wf(r) and ^f(r)of (2.13), (2.14) and (4.10) are active; *s, ws and ys are constant during theboundary layer interval [0,^]. Thus, in the fast time scale, the originalregulator problem (4.1), (4.3) is described by the following one.

Fast regulator problem

Find wf so as to minimize

for the fast system (2.6), (4.13).

o


Lemma 4.2

If the triple (M2, A22, B2) is stabilizable and detectable [observable] thenthere exists a unique positive-semidefinite [positive-definite] stabilizingm x m solution Kf of the algebraic matrix Riccati equation

and the optimal control for (2.6), (4.13) and (4.20) is

Proof The lemma is a special case of Lemma 4.1.

The controls «s and w f, defined by (4.18) and (4.22), are optimal only forthe slow and fast systems. According to (2.11), (4.18) and (4.22), a com-posite control wc, operating upon the actual states x and z, 4s given by

Again, the composite control (4.23) does not explicitly depend upon e,which might represent small unknown parameters. Given that the gains G0and G^ are only optimal for their respective slow and fast regulator prob-lems, the question arises as to the optimality or near-optimality of thecomposite feedback control law (2.11) or (4.23).

3.4.1. Near-Optimality of the Composite Control

In order to compare the composite control «c with the exact optimal control(4.4), it is convenient to express uc in a more suggestive form.

Lemma 4.3

The composite control (4.23) is identical with the composite control

where B is as in (4.6) and

o


Proof Consistent with the identity

and (4.22), define

H and H'1 are well-defined provided A22 and A22 - B2R~lB2Kf are non-singular. Using (4.21),

which, upon inversion and rearrangement, yields the identity

or

Hence, using (4.21), (4.23) can be rewritten as

or (4.24), where Km is given by (4.25).

The composite control uc in (4.24) is now in a form that invites directcomparison with the exact optimal control wopt of (4.4). It is of interest toanalyze the relationship between K(e) and Mc in (4.24) for e small andpositive. For this purpose, we construct a power series expansion of K(e),where K(e) is assumed to be scaled, as in equation (6.36) of Chapter 2 toavoid unboundedness of coefficients in the feedback gain matrix (4.4).

Theorem 4.2

Under the conditions of Theorem 4.1, the positive-semidefinite stabilizingsolution K = K(e) of the algebraic Riccati equation (4.5) possesses a powerseries expansion at e = 0, that is,

o


Furthermore, the matrices Kf> , Kf , Kf satisfy the identities

where Ks, Kf and Km are defined by (4.17), (4.21) and (4.25) respectively.

Proof It is convenient to rewrite the original Riccati equation (4.5) inthe Lyapunov form

where G is as denned in (4.4). Then, the substitution of (4.26) into (4.28)yields at e = 0 the equations

where G(0) = [GJ0) G(2

0)] and the partitions of the closed-loop systemmatrix Fn = An + B1G

(^} , etc., are defined in analogy with (2.16). Underthe conditions of Lemma 4.2, (4.21), (4.22) and (4.29c) imply thatKf = Kf. Also, by Lemma 4.2, F22 4 A22 + B2Gf is a Hurwitz matrixand so F22 exists.

Therefore, (4.29b) can be rewritten as

whereupon substitution of (4.30) in (4.29a) yields

The remainder of the proof is devoted to showing that (4.31) is independentof the fast Riccati solution Kf = Kf (and Gf) and is quite the same as(4.17). Now, equations (2.18) and (2.23) of Section 3.2 imply the followingidentity:


(Alternatively, as a simple exercise the reader may verify (4.32) by pre-multiplication of (4.32) by F22.) Also, in view of (4.32) and (2.10), we have

Substituting (4.33), (4.29c) and (2.10) in (4.31) yields

Noting (4.32) and performing the indicated cancellations in (4.34), we have

Since, by (4.32),

we finally arrive, after a rearrangement of terms in (4.35), at the expression

which is the Lyapunov form of the Riccati equation (4.17), where G0 isdefined by (4.18). Thus, under the conditions of Lemma 4.1, (4.17) and(4.36) imply that tf<°> = Ks. Also, substituting (2.16) and (4.22) in (4.30),we have, upon comparison with (4.25), that K^ = Km. The existence ofthe series (4.26) then follows from the implicit function theorem (Dieu-donne, 1982).

Theorem 4.2 establishes that the composite feedback control wc in (4.23)is stabilizing and is O(e) close to wopt of (4.4). This implies that/c, the valueof the performance index / of the system (4.1), (4,2) with wc, is at leastO(e) close to the value of the optimal performance index 7opt. The followinganalysis of Jc and 7opt reveals that MC in fact yields an O(e2) approximationof /opt.

o


Since

where Pc is the positive-semidefinite solution of the Lyapunov equation

with 5 = BR~1BT, the following theorem holds.

Theorem 4.3

Under the conditions of Theorem 4.1, the first two terms of the powerseries of Jc and /opt at e = 0 are the same, that is,

and hence the composite feedback control (4.23) is an O(s2) near-optimalsolution to the complete regulator problem (4.1)-(4.3).

Proof Adding (4.5) to (4.37) and rearranging, we obtain a Lyapunovequation for Pc - K = W:

By an application of the implicit function theorem to (4.37), it can be shownthat Pc possesses a power series at e = 0. Thus W can also be expanded asfollows:

From (4.24), (4.26) and (4.27) we have

and, since the matrices A0 - B0RQ1(NoMQ + B^KS) and (A22 -B2R~lB%Kf) are Hurwitz, the substitution of (4.40) into (4.39) yieldsWJ0) = 0 and W,0) = 0, ; = 1, 2, 3. Hence W= O(e2}, which proves(4.38).

It has been shown that the composite control (4.23), even though it doesnot contain e explicitly, guarantees an O(e2} approximation of the optimalperformance. Knowledge of e would, however, be needed for a higher-order approximation. This is considered in Section 3.5.

o


3.4.2 Near-Optimal Reduced Control

Similar to the stabilizing reduced control of Corollary 2.1, let us nowconsider the approximation achieved by optimizing only the slow system.If A22 is a Hurwitz matrix and G2 = 0, then (4.23) reduces to

The value JT of the performance index J with ur in (4.41) applied to thesystem (4.1), (4.2) is

where Pr is the positive-semidefinite solution of the Lyapunov equation

In contradistinction to the composite control uc, the reduced control «r of(4.41) is not O(e) close to uopt owing to the absence of the boundary layerterm wf in (2.14). After the decay of the boundary layer term uf away fromr = 0, ur is O(e) close to wopt, so that its performance 7r does approximate7opt to within O(e).

Theorem 4.4

If A22 is a Hurwitz matrix, then the constant terms of the power series ofJT and /opt at e = 0 are equal, that is,

and hence the feedback control ur in (4.41) is an O(e) near-optimal solutionto the complete regulator problem (4.1)-(4.3).

Proof Using (4.21) and the identity

ur in (4.41) can be expressed as

3 LINEAR FEEDBACK CONTROL120

where

as

where 5 = BR~1BT and PT possesses a power series in e. Adding (4.5) to(4.47) and rearranging, we obtain a Lyapunov equation for Pr - K = V:

Substituting the power series

into (4.48) and evaluating at e = 0 yields

where, with S

Since A22 is Hurwitz, there exists a unique positive-semidefinite solutionVf of (4.52). Expressing Vf in terms of Vf and Vf and substitutinginto (4.50), we obtain, using (4.52), that

Rearrangement of (4.46) yields

and hence the right-hand side of (4.55) is identically zero. Furthermore,

hence, (4.43) can be rewritten


An - A12A221A21 = A0- B0R^(NlM0 + B$KS), which is Hurwitz. Thus

the solution of (4.55) is V<°> = 0, implying V = O(e) and (4.44).

Comparing Theorem 4.4 with Theorem 4.3, it is obvious that the majorportion of the cost / is contributed by the slow system, while the contributionof the fast system is 0(e).

3.4.3 Design Procedure

The preceding two-time-scale approach, applied to the optimal linearregulator problem (4.1)—(4.6), is summarized in the following designprocedure.

For an O(e2) near-optimal regulator, solve the separate lower-orderalgebraic Riccati equations (4.17) and (4.21), which are independent of e,for Ks and Kf respectively. The composite feedback control to beimplemented on the original system (4.1)-(4.6) is given by (4.23).

For an O(e) near-optimal regulator when A22 is Hurwitz, solve the(n x «)-order Riccati equation (4.17) for A"s and implement the reducedcontrol given by (4.41).

Example 4.1

Returning to the DC-motor control problem of Example 3.1, let us thistime design near-optimal composite and reduced controls for the motorsystem

with the performance index

where ql > 0, q2 > 0 and r > 0.

The slow Riccati equation (4.17) is

of which the unique stabilizing positive root, under the conditions of Lemma4.1, is

o


resulting in the slow regulator gain given by (4.18):

The reduced control (4.41) is simply

Similarly, the fast Riccati equation (4.21) is

of which the unique stabilizing positive root, under the conditions of Lemma4.2, is

resulting in the fast regulator gain matrix given by (4.22)

By (4.25), (4.60) and (4.64),

while from (4.24) or (2.11), (4.61) and (4.65), the composite control is

It is of interest to compare the composite control (4.67) with the optimalsolution of the original regulator problem (4.57), (4.58). The e-dependentRiccati equation (4.5) takes the partitioned form

3.4 NEAR-OPTIMAL REGULATORS

The positive root of (4.68a) is

123

Given K2(e) in (4.69), we solve (4.68c) for its positive root

Substituting (4.69) and (4.70) in (4.68b) and rearranging terms,

Upon performing Maclaurin series expansions of K^e) in (4.71) and K3(e)in (4.70) at e = 0, it is observed that

Also, from (4.66) and (4.69),

Equations (4.72) and (4.73) mean that equations (4.26) and (4.27) ofTheorem 4.2 are satisfied. Evaluating the optimal feedback control law(4.4), it is readily shown that

It is left as an exercise to demonstrate that the composite cost

where Pc is the solution of (4.37), is near-optimal in the sense that

and that the reduced cost

where Pr is the solution of (4.43), is near-optimal in the sense that


Example 4.2

For the singularly perturbed system

it is desired to regulate the output y by way of minimizing the quadraticperformance index

A near-optimal solution of the linear quadratic regulator problem (4.77)-(4.79) is sought through the independent solutions of lower-order slow andfast regulator problems in the manner of Lemma 4.1 and Lemma 4.2.

Slow problem: The triple

is completely controllable and completely observable in the sense of Lemma6.1 of Chapter 2. Consequently, by Lemma 4.1 there exists a uniquepositive-definite stabilizing solution Ks of the algebraic matrix Riccatiequation (4.17), which is

Substitution of (4.80), R0 = 1.316 and N0 =

slow regulator gain

in (4.18) gives the

3.4 NEAR-OPTIMAL REGULATORS

Fast problem: The triple

125

is completely controllable and completely observable. Consequently, byLemma 4.2 there exists a unique positive-definite stabilizing solution of thealgebraic Riccatic equation (4.21), which is

By (4.82) and (4.22), the fast regulator gain is

Composite control: By (4.23) or (2.11), (4.81) and (4.83) there resultsthe composite control

Reduced control: Since the eigenvalues of A22 are -0.47 and -1, A22

is a Hurwitz matrix and the two fast modes experience an exponential decayof the order of e~°-47'/£ and e~'/f respectively without any fast regulatorcontrol (G2 = 0). Accordingly, we would expect satisfactory state regulationto be achieved through the use of the reduced control (4.41), viz

obtained from the solution of the slow Riccati equation (4.80).

Comparison with the optimal solution: For e = 0.1, the optimal feed-back solution to (4.77)-(4.79) is

obtained from solving a fourth-order e-dependent Riccati equation (4.5)or ten scalar equations. For the initial condition [*OT z°T] = [1 0 1 0],the values of the performance index are


Hence the loss of performance compared with the optimal cost, with uc lessthan 0.052% and ur less than 0.24%, is practically negligible in both cases.By contrast with the optimal solution, the separate sets of three scalarequations associated with the "slow" and "fast" Riccati equations areindependent of e and are readily solved by hand.

Example 3.3 highlighted the fact that strong controllability [stabilizabilityis an essential structural property without which no composite feedbackdesign procedure is possible. If, instead of employing eigenvalue assignmentfor the weakly controllable singularly perturbed system (3.32), we were toconsider the optimal linear regulator, similar difficulties would occur:namely, the exact optimal control (4.4) would be of the order of 1/e largeand the near-optimal composite control (4.23) would not exist. In thecase of two-time-scale regulator design, it is not, however, necessary thatunstable eigenvalues of the system (4.1), (4.2) be strongly observable (inthe sense of Section 2.6) with respect to the performance index (4.3). Thenext example explores this issue.

Example 4.3

Consider the singularly perturbed optimal linear regulator problem

The system (4.90) is completely controllable for e > 0, and as e-*0 theslow system *s = -us and the fast system zf = zf + «f are each completelycontrollable; by definition (Section 2.6), the system (4.90) is stronglycontrollable. Also, the system (4.90) is seen to be strongly observable withrespect to the performance index (4.91). So far, all is well for a two-time-scale decomposition of regulator design along the lines of Lemmas 4.1 and4.2.

Our attention is now focused on the detectability of the system (4.90),(4.91) as q-* 0, that is, on the detectability of the pair

This pair is detectable for e > 0, but loses its detectability as e-» 0 since


the fast subsystem pair (C2, ^22) = (0,1) is undetectable. In other words,the fast unstable eigenvalue A = 1/8 of the system (4.90) is weakly observ-able, as < ? — » 0 , with respect to the performance index (4.91) and a two-time-scale separation of regulator design is inadmissible.

It is interesting, nonetheless, to examine the asymptotic behavior of thesolution to the full regulator problem (4.90), (4.91) as q and e both tendto zero. This we now do by solving the algebraic Riccati equation (4.5) for(4.90), (4.91) or the three scalar equations

to obtain the unique solution

The associated optimal linear feedback control (4.4) for (4.90), (4.91) is

which has the unique limit, as q-* 0 and e— » 0, of

Consider now the asymptotic behavior of the composite control (2.11)or (4.23). The Riccati equation (4.21) of the fast regulator is

which, as g—» 0 thereby losing observability of the fast system eigenvalue,has not one but two positive-semidefinite solutions

Picking K{ = 2 results, by (4.22), in the stabilizing fast regulator gain2 = -2, while it is left as a simple exercise to show that as q—> 0 theG


slow regulator gain is G0 = -1 and the resulting composite control isidentical with that of (4.95); that is, the composite control exists as q—> 0and asymptotically approaches the optimal control as e—>0. However, ifthe other positive-semidefinite optimal solution K{ = 0 of (4.97) is pickedthen the fast regulator gain G2

0) is not stabilizing and no stabilizing com-posite control exists in this case.

3.5 A Corrected Linear-Quadratic Design

The fact that the algebraic Riccati equation (4.5) possesses a power serieexpansion (4.26) at e = 0 suggests that a closer approximation to the optimalperformance index (4.8) can be obtained through evaluation of higher-order terms in the expansion. What we seek is a method that will result ina performance superior to the performance achievable by the uncorrectedcomposite control (4.23).

The method adopted here is to approximate the optimal feedback gainmatrix G(e) by

The analysis is facilitated by rewriting the exact solution (4.4), (4.5) in thefollowing partitioned form:

3.5 CORRECTED LINEAR-QUADRATIC DESIGN 129

where 5t = B^^Bf, S2 = B2R~1B^ and 512 = B^R^B^ as before, andthe partitioned matrices K^e), i= 1,2,3 correspond to those of (4.26).

Setting e = 0 in (5.4), we obtain, as in Subsection 3.4.1, the zeroth-ordersolution given by (4.29). In particular, by (5.2), (5.3), Lemma 4.3 andTheorem 4.2,

Gl(Q) = (Ir + GfA^B2)GQ + GfA^A21, G2(0) = G<°>, (5.5)

where G0 and G(2

0) are the uncorrected slow and fast regulator gain matricesdefined by (4.18) and (4.22) respectively.

Next, it is required to compute the gain correction e[dG(e)/de]e=0 in(5.1). Evaluating the derivatives of (5.2) and (5.3) with respect to e, at£ = 0 ,

where the derivatives K^ = [dX,-(e)/de]£=0, i =1,2,3, satisfy the fol-lowing equations:

Observing that the corrected state-feedback gain matrix G in (5.1)approximates the optimal gain matrix G to within O(e2), as the solution

approximates the Riccati equation solution in (4.26), it is clear that thecorresponding cost Jc is at least within O(e2) of the optimal cost /opt of(4.8).


Theorem 5.1

Under the conditions of Theorem 4.1, the first four terms of the powerseries of Jc and /opt at e = 0 are the same, that is,

and hence the corrected feedback control (5.1) is an O(e4) near-optimalsolution to the regulator problem (4.1)-(4.3).

Proof The proof (Sannuti and Kokotovic, 1969a) is a lengthier versionof that of Theorem 4.3 and is omitted.

It should be noted (and checked) that (5.8) in the corrected solutionpartitions K(l\ i = 1, 2, 3, is also obtained by formal substitution of (5.9)in the original matrix Riccati equation (5.4) and equating coefficients of e.

In order to solve the linear equations (5.8) for K^ , K^> and K^ , it isuseful to first rewrite (5.8c) as

where

are the partitions of the uncorrected closed-loop system matrix, as in (4.29).Rearrangement of (5.8b) using (5.12), where under the conditions ofLemma 4.2 F22 Hurwitz implies F22 exists, yields

Similarly, (5.8a) may be rewritten as

whereupon substitution of K^ from (5.13) in (5.14) gives


Using (5.11), (5.15) becomes

Using the identities (4.32) and (4.33) , (5 . 16) finally assumes the form, linear

In summary, the corrected near-optimal design consists of the followingsteps.

1. Solve the slow and fast regulator equations of Lemmas 4.1 and 4.2,with K,= K<°), Kf=Kf and Km = Kf in (4.25), to obtain theuncorrected gain matrix G(0) = [G}(0) G2(0)] of (5.5).

2. Solve \n(n + 1) linear scalar equations (5.17) for K^ .

3. Solve \m(m + 1) linear scalar equations (5.11) for K^ .

4. Obtain K™ from (5.13).

5. Compute the corrected gain G(e) in (5.1) from G(0) (step 1), (5.6)and (5.7).

The next example indicates the significant improvement that the cor-rected design can have over the uncorrected design of Section 3.4 when eis known.

Example 5.1

In a voltage regulator problem, the plant is defined by the block diagramof Fig. 3.4, where

Now, the small time constants can be expressed as multiples of a singlesmall parameter e = 0.1:

in


Fig. 3.4. Block diagram of a voltage regulator.

The associated state model accordingly assumes the explicit singularlyperturbed form

Taking into account the fact that A22 is a Hurwitz matrix, it is desired toregulate the output (voltage) y = x^ by way of minimizing the quadraticperformance index

where it is observed that (5.23) is of the form (4.2), (4.3) with M2 = 0.Consequently, by (4.21) Kf = Kf> = 0, and by (4.22) G<0) = 0. CombiningG2(0) = 0 with the solution Gj(0) = G0 of (4.17) and (4.18), we have, by(5.5), that

Also, solving (5.17), (5.11) and (5.13) for K™, #0) and K^ respectively,and substituting in (5.6) and (5.7),

For the values of T3, T4 and T5 given in (5.18), e = 0.1, and in view of(5.24) and (5.25), the corrected near-optimal gain matrix in (5.1) is givenby


It is of interest to compare the gain matrix G(0.1) with the optimal gainmatrix G(0.1) obtained from solving the full-order problem (5.21)—(5.23)using (4.4) and (4.5):

The comparison of the responses of the optimal (solid line) and thecorrected near-optimal (dotted line) systems in Fig. 3.5 indicate the close-ness of the approximation.

It should be pointed out that the corrected near-optimal design may serve

Fig. 3.5. Responses of optimal (solid line) and corrected near-optimal (dotted line) systems.


not only for e = 0.1, as required in this example, but also for a range ofvalues of e. To see how large this range is, the corrected near-optimaldesign (5.1) is compared with the optimal design (4.4), (4.5) and with theuncorrected near-optimal design (4.18). Figure 3.6 gives the values of theperformance index / obtained by these three designs, with the initialcondition ^(0) = -10, *2(0) = z^O) = z2(0) = z3(0) = 0, for a range ofvalues of e (similar results are obtained for other initial conditions). For

Fig. 3.6. Performance indexes for the three designs.

this example it is seen that the range of e in which the uncorrected near-optimal design (4.18) can be used is as narrow as 0 < e ̂ 0.025, while thecorrected near-optimal design (5.1) can be applied in a range about twelvetimes larger, 0 ̂ e ̂ 0.3. The responses in Figs. 3.7 and 3.8 indicate that the(voltage) output of the corrected near-optimal system is close to the optimalone, while the results of the uncorrected design are unstable even at e =0.07.

Finally, we note that the full-order design consists of solving a systemof f(5 + 1) = 15 quadratic equations (4.5), while the |(3 + 1) = 6 linearequations (5.8c) and the 1(2 + 1) = 3 linear equations (5.8a) are easilysolvable by hand.


Fig. 3.7. State *,(*) for the three designs when £ = 0.07.

Fig. 3.8. State x,(/) for the three designs when £ = 0.20.


3.6 High-Gain Feedback

A fundamental property of high-gain systems, as we have seen in Example6.2 of Chapter 1, is their relationship with singularly perturbed systems. Sofar in our discussions in this chapter, the singular perturbation propertiesof the system to be controlled were not altered by the control law. However,even if the original system is not singularly perturbed, a strong controlaction can force it to have fast and slow transients, that is, to behave likea singularly perturbed system. In feedback systems the strong control actionis achieved by high-gain feedback.

Consider, for simplicity, the system model

where x E R", z E Rm and B2 is a nonsingular m x m matrix. No loss ofgenerality is incurred, since the system model can always be transformedto (6.1) provided B is of full rank m. Since B2 is nonsingular the pair(A22, B2) is completely controllable for any A22. Also, the next lemma isimmediate from the application of Lemma 6.1 of Chapter 2 to the system(6.1), where B2 is nonsingular.

Lemma 6.1

If the system pair (A, B) of the system (6.1) is completely controllable[stabilizable], then the pair (AU,A12) is completely controllable[stabilizable].

Suppose that the high-gain feedback control

where £ is a small positive parameter and Gj and G2 are as yet unspecifiedgain matrices, is applied to the system (6.1). The resulting feedback system(6.1), (6.2) has a singularly perturbed form

which can be analyzed by the methods of Chapter 2 under the followingassumption.

3.6 HIGH-GAIN FEEDBACK 137

Assumption 6.1

B2G2 is a nonsingular matrix.

If, under Assumption 6.1, G2 is chosen such that

then Theorem 5. 1 of Chapter 2 holds and a two-time-scale design is possibleby designing the slow system, obtained by setting e = 0 in (6.3),

and the fast system

Taking G{ = G2G0, (6.5) becomes

Thus, invoking Theorem 3.1 of Chapter 2 and Lemma 6.1, the two-time-scale high-gain design method is encapsulated by the following eigenvalueassignment theorem.

Theorem 6.1

Let the system (6.1) be completely controllable, and let the m x n matrixG0 and the m x m matrix G2 be chosen such that

where pt and qj/e are the prescribed locations of the distinct slow anddistinct fast eigenvalues respectively. Then, there exists an e* > 0 such thatfor all e G (0, e*] the application of the high-gain feedback control

assigns the eigenvalues of the system (6.1) to /?, -I- O(e) and [gy + O(e)]/e.

The foregoing theorem, in analogy with Theorem 3.1, suggests a two-time-scale design procedure for the assignment of the closed-loop systemeigenvalues using high-gain feedback: construct G0 so as to place the 0(1)


eigenvalues of Au — A12G0; separately construct G2 so as to place theO(l/e) large eigenvalues of B2G2', then form the composite feedback control(6.10) for application to the system (6.1). Although the two separate lower-order eigenvalue assignment problems (6.8) and (6.9) do not depend on e, thecomposite control (6.10), in contradistinction to (3.1), does depend on e. Thesingularly perturbed form of the closed-loop system (6.3) corresponds to thefrequently observed fact that systems (6.1) under high-gain feedback control(6.10) exhibit a two-time-scale response in that the initial fast transient(boundary layer) is followed by a slow time evolution. The fast transient

and the slow evolution are due respectively to the eigenvalues — (B2G2)

and A,(v4n - ^4i2G0) specified in Theorem 6.1.

Corollary 6.1

If B2G2 and An - A12G0 are Hurwitz matrices, and the high-gain feedbackcontrol (6.10) is applied to the system (6.1), then there exists an e* > 0such that the resulting closed-loop system (6.1), (6.10) is asymptoticallystable for all ee(0,e*].

Proof Although the proof is immediate from Theorem 6.1, the fol-lowing alternative proof is of independent interest. Any high-gain controlof the form

when applied to the open-loop system (6.1) induces it to assume thesingularly perturbed form

Since eA22 is singular at e = 0, application of the fast feedback control

to (6.12) is required to convert (6.12) to the standard singularly perturbedform

To stabilize this system we apply the composite feedback control of Section3.2. Upon setting e = 0 in (6.14), the slow system of (6.14), described by


the pair(An, -AuG2i), is stabilized by the control

since An — A12GQ is Hurwitz. With B2G2 also Hurwitz, it is enough to usethe reduced control (6.15). The closed-loop system (6.14) is asymptoticallystable by Corollary 2.1, while (6.15), (6.13) and (6.11) yield the overallstabilizing control (6.10).

Example 6.1

Consider a system in the form of (6.1):

First, suppose that a high-gain feedback control is to be found to placethe slow eigenvalue near Xl = — 3 and the fast eigenvalues near A2i3 =(-1 ± jl)/e. We solve the two lower-order eigenvalue assignment prob-lems (6.8) and (6.9) and obtain

The high-gain composite feedback (6.10) is

The exact eigenvalues are computed for comparison purposes and theresults for values of e of 0.01 and 0.001 are summarized in Table 3.1. It is

TABLE 3.1

Gain\!B

100

1000

Exacteigenvalues

-2.69-95.15

±j!07.96

-2.97-995.02±j!008.0

Specifiedeigenvalues

-3.0

-100±jlOO

-3.0

- 1000 ± j 1000

Percentageerror

14

~1

o


observed that as e— » 0, Aj 2,s tend to the specified values. For designpurposes the gain l/e does not have to be very high, since with 1/e = 100the accuracy of 14% in eigenvalue location is often acceptable. Finally,should a closer eigenvalue assignment be desired, a corrected high-gainfeedback design based upon Theorem 3.2 may be pursued (see Exercise3.9).

3.6.1 The Cheap Control Problem

High-gain feedback systems also result from the optimization of the system(6.1) with respect to a quadratic performance index having a small penaltyon u:

where Q ̂ 0, R > 0 and e is a small positive scalar.Using the high-gain control (6.11), the so-called "cheap" control problem

(6.19), (6.1) is transformed to the equivalent linear-quadratic controlproblem

for the system (6.12) where it is observed that the singularity at e = 0 hasbeen transferred from the performance index / to the dynamic systemmodel (6.12). Under the conditions of Theorem 4.1, the solution to thisequivalent optimal linear regulator problem (6.20), (6.12) for any givene>0 is of the form (4.4), (4.5). Although the system model (6.12) is anonstandard singularly perturbed model, the corresponding algebraic Ric-cati equation (4.5) nonetheless possesses a power series expansion at e = 0of the form of (4.26) where tf<°>, X<°> and Kf satisfy (4.28)-(4.30) withA21 = 0, A22 = 0, Si = 0 and 512 = 0:

Assumption 6.2

Q22 is a positive-definite matrix.


Under this assumption, a two-time-scale decomposition of the problem(6.20), (6.12) proceeds in the manner of Section 3.4. Rewriting (6.23) as

and taking the square root of both sides of (6.24), we have, under Assump-tion 6.2 where B2 is nonsingular, that the unique solution of (6.23) is

Rearranging (6.22) as

and substituting (6.26) in (6.21), noting (6.23), results in the nth-orderalgebraic Riccati equation

Lemma 6.2

If the pair (A, B) is stabilizable and the pair (D,An) is detectable, where

then there exists a unique positive-semidefinite stabilizing solution of thealgebraic Riccati equation (6.27).

Proof The proof follows from Lemma 6.1 and Lemma 4.1.

Consider the control

Substituting (6.26) in (6.30), and using (6.23), we have

or

where

and G2 is as denned in (6.30). By (6.30)-(6.33), and Theorem 4.2, the


original optimal control (4.4) for the regulator problem (6.20), (6.12) isapproximated to within O(e) by the composite control

It is observed that (6.34) is of the same structure as the high-gain compositecontrol (6.10). This time, G0, given by (6.33) and (6.27), is the solution ofa reduced-order regulator problem for the slow system (6.7), while G2,given by (6.30) and (6.25), is the solution of a reduced-order regulatorproblem for the fast system (6.6).

Beyond the fact that the composite control (6.34) is within O(e) of theoptimal one, it can be proved using Theorem 4.3 that the composite controlis near-optimal in the sense that the performance J of the feedback system(6.1), (6.34) is O(e2) close to its optimum performance.

To regulate the output

we set Q = M^M as in Section 3.4.

Assumption 6.3

M2B2 is a nonsingular matrix.

By assumption 6.3 and the nonsingularity of B2, M2 is nonsingular, andan 0(e2) near-optimal control can be found by the preceding decompositionprocedure. Since Qu =M|Af1, Q12 = M]M2, Q22 = M^M2 and M2 isnonsingular, then D = 0 in (6.28), and the detectability condition of Lemma6.2 is replaced by

Also, with D = 0 the solution of (6.27) is

By (6.37), k(An - A12M21M}) are the eigenvalues of the slow system

(6.7) under the slow control (6.33). In fact, X(An - A12M21M1) are the

transmission zeros of the open-loop system (6.1) with output (6.35). Con-dition (6.36) requires that the transmission zeros of the system (6.1), (6.35)should be in the open left-half plane (Young, Kokotovic and Utkin, 1977).

3.7 ROBUST OUTPUT-FEEDBACK DESIGN 143

3.7 Robust Output-Feedback Design

It was established in Corollary 2.1 that a state-feedback control law,designed to stabilize the reduced-order system model is robust in the sensethat it will stabilize the actual full-order system model for sufficiently small£ provided the neglected fast modes are asymptotically stable. Unfor-tunately, a similar conclusion for output-feedback control does not hold ingeneral.

In this section, we discuss the robustness of output-feedback controldesign for linear time-invariant models to a common class of structureduncertainties, that of high-frequency parasitics. As before, the structuringof high-frequency uncertainties is achieved through the use of singularperturbation methods. The actual system model P£ is represented as

where x E R" is the dominant state vector, z G Rm is the state vector of theparasitic dynamics, y G Rp is the output vector, and £ > 0 is a small positivescalar representing the parasitics (e.g. small inductances, capacitances,inertias). Neglect of the parasitic elements by setting e = 0 in (7.1) resultsin the reduced-order design model P0, assumed to be stabilizable anddetectable:

where

and where A22 is assumed to be nonsingular.The robustness issue under discussion is whether an output-feedback

controller, designed to stabilize the nominal system model in (7.2), will infact stabilize the actual system model P£ in (7.1) for e sufficiently small.

A fairly general output-feedback control scheme for the design model(7.2), depicted in Fig. 3.9, is

where (7. 5 a) is a dynamic compensator of order /; Q < I < n. The closed-


Fig. 3.9. Closed-loop output feedback control system,

loop design model (7.2), (7.5) is given by

where, assuming F2 is chosen such that / - F2D0 is nonsingular

The actual closed-loop model (7.1), (7.5) is given by

where [x1 VT]T is the dominant state, z is the fast state and


Performing a standard two-time-scale decomposition in the manner ofSection 2.4, the closed-loop singularly perturbed model (7.7) assumes, intransformed coordinates, the decoupled form

provided F4 is nonsingular to ensure

is well defined.Let us verify that F4 is in fact nonsingular. Use of the identity

yields

Rearranging (7.11), we have

so that

Premultiplication of both sides of (7.13) by (/ - F2D0) l results in

Hence, substituting (7.14) into (7.11), we have

and so F4 is nonsingular.Similarly, it can be shown after lengthy manipulation that

Equation (7.16) is an intuitively pleasing result in that it implies that theslow system associated with the full singularly perturbed closed-loop system(7.7) is identical with the system (7.6) obtained from applying output-feedback control to the open-loop slow system (7.2).

By design, the controller parameters /, G1? G2, G3, Fj and F2 are chosen


such that the closed-loop design model (7.6) is asymptotically stable, thatis

However, even if ReA(v422)<0, there is no guarantee that the secondcondition

for overall closed-loop system stability in (7.8) is observed. What is providedin the next theorem are conditions that ensure (7.18) is satisfied.

Theorem 7.1

If Re A(,422) < 0, and any one of the following three conditions is satisfied:

(a) (A22, B2) is uncontrollable: weak controllability of high-frequencyparasitics;

(b) (C2, A22) is unobservable: weak observability of high-frequencyparasitics;

(c) F2 = 0: no static feedback from output to input;

then there exists an e* > 0 such that the feedback controller (7.5) stabilizesthe actual system (7.1) for all e G (0, e*].

Remark 7.1: If none of the conditions (a), (b) and (c) are observed,stability of the actual closed-loop model (7.7) or (7.8) cannot be guaranteed.

Remark 7.2: It is recalled from Section 2.6 that weak controllability andweak observability of the fast eigenvalues (high-frequency parasitics) meansthat they will not be affected by more than O(e) through output feedback.

Remark 7.3: For a system model (7.1) with high-frequency parasitics,conditions (a) and (b) may be replaced by

(a) B2 = 0: weak control of parasitics;^b) C2 = 0: weak observation of parasitics.

Remark 7.4: The state-feedback control u = GQX of Corollary 2.1 isalways robust since C\ = /„ and C2 = 0.

A General Robust Design Rule

Avoid static output feedback for systems with unmodeled high-frequencydynamics.


In the light of Theorem 7.1 and the above design rule, one may expectan instability problem in the case of static output feedback, as shown bythe following simple example.

Example 7.1

Consider the second-order system

for which ReA(A22) = -1 < 0. Upon neglecting the parasitic element bysetting e = 0, one obtains the reduced-order slow model

Using this first-order model as the design model, a designer has chosen theoutput-feedback control

so as to place the eigenvalue at -3. When the same feedback control (7.21)is applied to the actual system (7.19), the resulting closed-loop system is

which indeed has a slow eigenvalue -3 + O(e), as desired. However, italso has an unstable fast eigenvalue (1 -I- O(e))/e; that is, the design isnonrobust. The source of instability is the presence of z fed from the input(B2 =£ 0: strong controllability of parasitics) and the presence of z in theoutput equation of (7.19) (C2 =£ 0: strong observability of parasitics) andstatic feedback from output to input (F2 =£ 0).

3.7.1 Frequency-Domain Robustness Design

In order to further assess the robustness of output feedback using thefrequency-domain models of Section 2.7, the actual open-loop system


model Pe of (7.1) is described in the block diagonal form of Section 2.4 by

where the matrix triple {A0, BQ,C0} is as denned in (7.3) and (7.4). If A0

and A22 have no eigenvalues on the imaginary axis, the open-loop systemmodel P£ has a transfer-function matrix G(s, e) which is approximated forall frequencies s = }o) according to

where

is the transfer-function matrix of the design model (7.2) and is the asymp-totic form of G(s, e) in the low-frequency scale s = 0(1), and

is the transfer-function matrix of the parasitic block and is the asymptoticform of G(s, e) in the high-frequency scale s = p/e, p = 0(1).

Consider the application of a controller C(s), designed on the basis ofthe nominal model Gs(s), to the actual system G(s) as depicted in Fig. 3.10.Given that the controller (7.5) has the transfer-function matrix

the robustness condition F2 = 0 of Theorem 7.1 requires that the controllertransfer-function matrix C(s) in (7.27) be strictly proper. In other words,the robustness condition (c) of Theorem 7.1 is equivalent to requiring thatthe controller C(s) have the low-pass property


Fig. 3.10. Robust output feedback control.

Example 7.2

The robustness of output-feedback design based upon the full-order stateobserver (O'Reilly, 1983b) for (7.2),

follows from Theorem 7.1, since in this case F2 = 0. In a frequency-domaincontext, the observer-based controller design (7.29) is robust by virtue ofthe fact that its transfer-function matrix

has the low-pass property (7.28).

Consider again another example of static output feedback.

Example 7.3

The actual model (7.1) is given by

The nominal design model (7.2) is given by


It is readily shown that, although the nominal plant model (7.32) is stabilizedby

for all k > i, the same feedback applied to the actual model (7.31) leads to

which has unstable eigenvalues for all k > 2.This potential for instability of the fast system modes may also be inferred

from a two-frequency-scale decomposition of the transfer-function matrixof the open-loop system (7.31) along the lines of (7.24):

The source of instability is signal transmission through the high-frequencyblock as in Fig. 3.10; that is,

or inp-scale, p = es,

Example 7.4

As a means of overcoming the problem of instability in Example 7.3,consider instead of the static feedback controller (7.33), the class of con-troller C in Figure 3.10 described by

This first-order lag controller is robust by virtue of the fact that it blockshigh-frequency signal transmission in the sense of (7.28).

Another interpretation of the robustness property of the controller (7.37)is as follows. For a fixed T> 0, the low-pass filter l/(Ts + 1) is describedby

3.8 EXERCISES

Augmenting the actual model (7.31) with (7.38), one has

In the filtered outputs yfilt of (7.39c) the system parasitics are now weaklyobservable (C2 = 0). Hence, in view of condition (b) of Theorem 7.1,stability of the actual closed-loop system is ensured by the application ofany feedback loop

for any k that stabilizes the nominal design model (7.2) obtained from(7.39) by setting e = 0.

In conclusion, a frequency-domain interpretation of our earlier robustdesign rule is as follows.

Frequency-Domain Robust Design Rule

Avoid closing high-frequency plant loops by using strictly proper controllersor controllers with the low-pass property (7.28).

3.8 Exercises

Exercise 3.1

Using the state transition matrix method or any other method, solve thenormalized motor equations (3.4) of Example 3.1 under (a) the exactcontrol w(e) of (3.13), (b) the composite control uc of (3.6) and (c) thereduced control ur of (3.14). Hence, for the system initial condition *(0) =4, z(0) = 3, compare graphically the closed-loop responses x(t) and z(f)under controls (a), (b) and (c) for e = 0.1 and e = 0.01. Comment uponthe accuracy of the state trajectory approximations. Also, comment on

151


the possible significance of the singular perturbation parameter e beingunknown except that it is in the range 0.01 < e < 0.1.

Exercise 3.2

Deduce an upper bound on e for Example 3.1 such that the closed-loopstability of the motor system is maintained (a) under the composite controlwc of (3.6) and (b) under the reduced control UT of (3.14).

Hint: Use Lemma 2.2 of Chapter 2.Would such a value of e result in an acceptable approximation to the

desired eigenvalue placement of {-3, — 2/e}?

Exercise 3.3

Derive discrete-time analogues of Theorem 2.1 and Theorem 3.1 for thelinear system model

where Im - A22 is nonsingular.Hint: The slow system, evolving in the slow time scale en, is

which upon division by £, letting e —> 0, and eliminating Z, is of thecontinuous-time form (2.3) in the slow variables xs(t) and us(t). The fastsystem is defined by the variables x{(n) = x(n) - xs(t), zf(n) = z(ri) - zs(t)and u{(n) = u(n) - us(t). See Litkouhi and Khalil (1984) for more details.

Exercise 3.4

The linear composite control design (2.11) presumes A22 exists. Constructa composite feedback control for the system (2.1), (2.2) when A22 issingular.

Hint: Use a preliminary fast feedback control, recalling Corollary 2.3,u = Kz + v, where A22 + B2K is nonsingular. (See Sannuti (1977).)

3.8 EXERCISES 153

Exercise 3.5

Consider the weakly controllable system of Exercise 2.7:

Why is it neither possible to design a reduced feedback control nor acomposite feedback control? Calculate and comment on the exact controlnecessary to place the system eigenvalues at {-3, -1/e}.

Exercise 3.6

Construct a corrected linear quadratic design for the DC-motor system(3.4) with e = 0.2 and the performance index

Compare the design and the associated performance index with the uncor-rected near-optimal design.

Exercise 3.7

Derive a two-time-scale optimal linear regulator design for the discrete-time system model of Exercise 3.3 with the performance index

Prove its near-optimality along the lines of Theorem 4.2 and Theorem 4.3.Hint Scale the exact Riccati equation solution as

(See Litkouhi and Khalil (1984) for more detail.)

Exercise 3.8

Design a high-gain composite control to assign the eigenvalue spectrum


(-3, -100, -150} to the system

Comment on the accuracy of the scalar gain factor chosen.

Exercise 3.9

Derive a corrected high-gain feedback design, corresponding to the uncor-rected one of Theorem 6.1, so as to assign the system eigenvalues toPi + 0(f2), i = 1, . . . , n, and fa + O(e2}]/e, j = 1, . . . , m.

Hint: Determine slow and fast corrected models so as to apply theeigenvalue assignment procedure of Theorem 3.2.

Exercise 3.10


By selective use of a low-pass filter, a designer has the choice of meas-urements y = x + zory=x. Which measurement should he use if he is toemploy a robust output feedback control u = ky, and why? Having madehis choice, what design value of k is required in order to shift the dominanteigenvalue of the system to -6?

Exercise 3.11

Deduce conditions similar to those of Theorem 7.1 such that the linearsystem

with direct input-output transmission

is robust with respect to neglect of parasitics when under the static outputfeedback control u = Ky.



The complete decomposition of linear state-feedback design according toslow mode and fast mode performance specifications, and the subsequentrecombination in the shape of an "order of e" accurate composite control,is due to Chow and Kokotovic (1976a). The composite state feedbackcontrollers for corrected models in Theorem 2.2 and Corollary 2.2 are new,as is the invariance of the slow model to fast feedback in Corollary 2.3.The use of fast feedback in overcoming possible singularity of the matrixA22 was first noted by Kokotovic and Haddad (1975a) and is furtherdiscussed in Sannuti (1977) and Porter (1977).

The eigenvalue assignment results of Theorem 3.1, implicit in Chow andKokotovic (1976a), are developed by Porter (1974, 1977), and inde-pendently by Suzuki and Miura (1976), through the application of a stabilitytheorem of Klimushchev and Krasovskii (1962) to the closed-loop system.Theorem 3.2 on eigenvalue assignment for corrected models is new. Anearly general stabilizability condition for linear time-varying systems wasformulated by Wilde (1972). Extensions of these eigenvalue assignmentand stabilization procedures to time-varying systems, general two-time-scale systems, discrete-time systems, eigenstructure assignment and mul-tivariable tracking are surveyed in Kokotovic (1984) and Saksensa et al.(1984). More recently, composite control strategies for discrete-time sys-tems have been developed by Litkouhi and Khalil (1985) for the cases ofsingle-rate sampling and multirate sampling.

Since the publication by Sannuti and Kokotovic (1969a), one of the mostactively investigated singularly perturbed control problems has been thelinear-quadratic (LQ) regulator problem. Approaches, in the main, havebeen either via the singularly perturbed Riccati equation or boundary-valueproblems (Kokotovic et al. 1976). The particular scaling of the regulatorRiccati equation was first proposed by Sannuti and Kokotovic (1969a),while the first complete decomposition of a more general linear regulatorproblem into separate slow and fast regulator problems, upon which Section3.4 is based, is achieved in Chow and Kokotovic (1976a). Two-time-scaledecomposition of near-optimal regulators for discrete-time linear systemsin Litkouhi and Khalil (1984) parallels the earlier continuous-time resultsof Chow and Kokotovic (1976a). A practical application of the near-optimalcontrol to a drum boiler power plant is presented in Cori and Maffezzoni(1984).

The corrected linear quadratic design of Section 3.5 is an extension ofthe original paper by Sannuti and Kokotovic (1969a) to the general casewhere M2 ^ 0 and A22 may not be Hurwitz. Higher-order approximations


to the optimal performance are possible through matched asymptotic expan-sions. Numerous other references on asymptotic expansions, multiple timescales, distributed parameter systems, quasi-conservative oscillatorysystems, and large space structures for the optimal linear problem are givenin Kokotovic (1984) and Saksensa et al. (1984). The discussion in Sections3.3 and 3.4 of the deleterious effect of weak controllability on linear statefeedback control is based on Chow (1978b). Dual conclusions obtain forweak observability in the state observer design briefly introduced in Section3.7.

High-gain feedback control problems have been investigated from manydifferent standpoints: chiefly, cheap control, multivariable root loci, almost-singular state estimation, variable structure systems and the theory ofalmost invariant subspaces. See Saksensa et al. (1984), Kokotovic (1984)and Saberi and Sannuti (1985, 1986) for more detailed discussions andnumerous references therein. The unifying framework provided by thesingular perturbation methods of Section 3.6 is based in part upon theexposition of Young et al. (1977) under the assumption that B2G2 isnonsingular.

Our detailed examination of the robustness of general output feedbackcontrol strategies in Section 3.7 is based on Khalil (1981b); see also Khalil(1984a), Kokotovic (1984) and O'Reilly (1985). Earlier conditions for thecases of static output feedback and reduced-order observer design also takethe form of "weakly observable parasitics" (Kokotovic, 1984; Saksensa etal. 1984). Robust output feedback control strategies, based upon the full-order state observer O'Reilly (1983b), date from Porter (1974); see alsoBalas (1978), O'Reilly (1980) and Javid (1982). An analysis of the inter-action of actuator and sensor parasitics—frequently neglected in feedbackdesign—with the fast system modes is undertaken in Young and Kokotovic(1982). A recent complementary result of Vidyasagar (1984, 1985) showsthat the actual system model Pe approaches the design model P0 in thegraph topology as the parasitic elements e tend to zero if the high-frequencyblock Gfas) = 0 in Fig. 3.10. The design principle of avoiding closed-loophigh-frequency signal transmission for robust feedback control carries overto adaptive systems and nonlinear systems. For instance, loannou andKokotovic (1983) establish the robustness of an adaptive observer throughthe use of a first-order filter. The two-frequency-scale description of generaltransfer-function matrix models by Luse and Khalil (1983) provides astarting point for more general studies of robustness within the feedbackconfiguration of Fig. 3.6 as, for instance, described in Khalil (1985).

4 STOCHASTIC LINEAR FILTERING ANDCONTROL

4.1 Introduction

It is to be expected that the linear-quadratic (LQ) designs of Sections3.4 and 3.5 have their stochastic counterparts for systems where modeluncertainties are by no means negligible and partial measurements of statesare accompanied by measurement noise. As with the deterministic problemsof Chapter 3, advantage may be taken of the singularly perturbed natureof the system model to decompose the stochastic filtering and controlproblems into well-conditioned subproblems in separate time scales.Additional care must, however, be exercised in the slow-fast decompositionof systems driven by white noise, since, as discussed in Section 4.2, thewhite noise process "fluctuates" faster than the fast states no matter howsmall the singular perturbation parameter e > 0 is. Taking the variance ofthe fast states into account, state approximations are achieved which areof the order of e1/2 rather than of the order of £ previously obtained fordeterministic systems in Section 2.5.

This two-time-scale decomposition for stochastic system models isextended in Section 4.3 to that of the Kalman-Bucy filtering problem. Itis shown that the estimates of the slow and fast states, provided by Kalman-Bucy filters in separate time scales, are within an order of e1/2 of thoseprovided by a Kalman-Bucy filter for the original system. The associatedslow and fast filter gain matrices are independently determined by well-conditioned lower-order Riccati equations which are dual to those of thenear-optimal regulator of Section 3.4.

Section 4.4 proceeds with the stochastic version of the near-optimalregulator solution of Section 3.4 for systems with incomplete state infor-mation. In the spirit of Chapter 3, the essence of the approach is the

157

158 4 STOCHASTIC LINEAR FILTERING AND CONTROL

construction of a near-optimal composite control from the solutions oflower-order stochastic control problems in separate time scales. By the wellknown separation theorem of stochastic control (Kwakernaak and Sivan,1972), these slow and fast solutions are provided by the slow and fastregulators of Section 3.4 in tandem with the corresponding slow and fastKalman-Bucy filters of Section 4.3. The near-optimality of the compositestochastic control is illustrated in Section 4.5 by way of application to theaircraft autopilot Case Study 7.2 of Chapter 1.

As we have seen in Chapter 3, an asset of the singular perturbationapproach to control system design is the ability to construct correcteddesigns should the performance of the initial uncorrected design proveinadequate. A higher-order approximation to the optimal strategy isachieved in Section 4.6 through the use of a corrected linear-quadraticGaussian (LOG) design that parallels the corrected near-optimal regulatordesign of Section 3.5. As before, the key to the corrected design is theparameterization of the problem solution in e.

Finally, the case of scaling the contribution of the white noise disturbancesby a fractional power of e is discussed in Section 4.7. It is shown that whenall white noise inputs are scaled by the same factor, the approximation anddesign schemes of Sections 4.2-4.6 remain valid.

4.2 Slow-Fast Decomposition in the Presence of White-NoiseInputs

The stochastic counterpart of the singularly perturbed system model studiedin Chapter 2 is

where x, z and w are «-, m- and s-dimensional vectors respectively. Theinput w(f) is zero-mean stationary Gaussian white noise with intensitymatrix V>0, i.e. E{w(t)wT(s)} = V6(t — s). The initial conditions x0, z0

are jointly Gaussian random vectors independent of w. It is assumed, asusual, that A22 is nonsingular.

Our objective is to define a slow subsystem that describes the slowdynamics and a fast subsystem that describes the fast dynamics; then usetheir solutions to approximate x and z. We start by formally decomposingthe system by extending the deterministic ideas of Chapters 1 and 2 to thestochastic case. To define the slow subsystem, we neglect the fast dynamics,which is equivalent to setting e = 0 in (2.2), and dropping the initial

4.2 SLOW-FAST DECOMPOSITION WITH WHITE NOISE 159

condition z0. The resulting equations are

Unlike deterministic systems with slow driving inputs, z defined by (2.4) isnot a valid approximation of z as e — » 0. In fact, the variance of z — z isinfinite for all e because of the white noise component of z. However, zcan still be used to approximate z as an input to a slow system. This is animportant point that is worth more explanation. Let r/(f) satisfy

where F is a Hurwitz matrix and w(t) is white noise. As we explained above,-r^w^) is not a valid approximation of 77 (t) as e—> 0.

Let £(0 = Jo r/(r) dr, which represents the dynamics of a slow system(an integrator in this case). As e-» 0, |(f) can be approximated by replacingr/(r) by -T"1w(r). To see this, define f(0 = -J0 T~ I>V(T) dr and calculatethe variance of

Reversing the order of integration in the first term, we obtain

Now

Therefore

for some positive constants a and /C.


Thus |(0 is a valid approximation of |(f) as e— » 0 in the mean-squaresense. Actually, |(r) is an 0(£1/2) approximation of £(f) according to thefollowing definition, which will be adopted throughout the chapter.

Definition 4.1

A random variable f(e) is said to converge to / in a mean-square sense(m.s.s.) if E(/(e) -/)2-» 0 as e-* 0. Furthermore, /(e) -/is said to beO(ey) for some y > 0 if there exist e* > 0 and K > 0 such that for all

Coming back to (2.3), (2.4), we see that as far as x is concerned (2.4)can be used to approximate z in (2.3). Substituting (2.4) in (2.3) anddenning xs(t) = x(t), we obtain the slow subsystem

where

The slow state xs(t) is a candidate for approximating x(t).To define the fast subsystem, we introduce the stretching time scale T =

t/e and the notation X(T)=X(ET), Z(r) = Z(ET) and W(T) = e1/2w(eT),where the e1/2 factor is used so that W(r) and w(t) have the same intensitymatrix. In the r-time scale (2.1) and (2.2) are given by

For sufficiently small e, X(T) is much slower than Z(r), and the ^lA^r)term on the right-hand side of (2.8) can be viewed as a constant inputwhose presence results in a bias in Z(r). To remove the bias we define

Taking the derivative of Zf(r) with respect to T, and using (2.7) and (2.8),it follows that the new variable Zf(r) satisfies the differential equation

with initial condition


Neglecting coefficients that are O(e1//2) or smaller, we get

Equation (2.11) is the desired fast subsystem. Setting zf(f) = Zf(f/e), thefast subsystem can be expressed in the Mime scale as

Now, based on (2.9), z f(f) - A^^i-^sW is taken as a candidate forapproximating z(t).

Fig. 4.1. Slow-fast decomposition of a singularly perturbed system driven by white noise.

In summary, the candidate decomposition, shown in Fig. 4.1, is given by

It is important to notice that in the proposed decomposition the slowcorriponent of z is its component due to A2lx and not its component due tothe input w, as it is customary in singularly perturbed systems driven byslow inputs. No matter how small e is, the fluctuations of the white noiseinput are faster than the fast dynamics. Therefore, the fast subsystemaccounts not only for boundary layer transients but also for the componentof z due to the white noise input. The following example illustrates thedecomposition (2.13).

Example 2.1



Fig. 4.2 Second-order stochastic system.

where w is white noise with intensity a2. A block diagram representationof this system is shown in Fig. 4.2.

Let us consider only the steady-state behavior of * and z. From Fig. 4.2,we see that the power spectra of x and z are given by

which are sketched in solid lines in Fig. 4.3. Setting E = 0 in (2.14) isequivalent to replacing H2(es) = 1/(1 + ES) by the identity transferfunction. The corresponding spectrum of z, shown in dotted lines, is flat

Fig. 4.3. Power spectra of AT and z.


for all co, i.e. white noise. It does not make sense to approximate z bywhite noise, because no matter how small e is, the difference between thespectrum of z and the white noise spectrum will yield infinity when inte-grated over all CD. However, Fig. 4.3 shows that whether we use the exacspectrum Sz(co) or its idealized white version, we still end up with the samespectrum Sx(a>). In other words, the error in approximating z by whitenoise is filtered out by the slow subsystem. That is why in defining the slowsubsystem we take z = w and *s = —xs + w. On the other hand, to deter-mine the wideband spectrum of z we have to use the equation ez = -z+ w, which is the fast subsystem called for in the decomposition (2.13).

Asymptotic validity of the decomposition (2.13) is established by the followingtheorem, which is the stochastic counterpart of the initial value theorem, Theorem5.1, of Chapter 2. Notice, however, the order of the state approximation is 0(£1//2)rather than 0(e).

Theorem 2.1

Suppose that A0 and A22 are Hurwitz matrices, then there exists an e* > 0such that for all e E (0, e*) (2.15) is satisfied for all t > 0:

Proof The proof is complicated by the fact that the variances of bothzf and z are of order l/e. The order of the variance of zf can be easily seenfrom (2.12), while the order of the variance of z follows from showing(Exercise 4.3) that the variance matrix of (XT, ZT) takes the form

The proof comprises three steps. First, the decoupling transformation ofSection 2.4 is applied to the system (2.1), (2.2) to decouple the slow andfast dynamics. It is verified that the decomposition (2.13) is obtained fromthe decoupled system as perturbations of right-hand-side coefficients andinitial conditions. Second, we apply an approximation result given inAppendix A for singularly perturbed systems driven by white noise. Finally,the inverse of the decoupling transformation is employed to show (2.15).

Application of the decoupling transformation (see Section 2.4) to (2.1),


(2.2) yields

Using the fact that L(s) and H(e) are analytic at e = 0 and L(0) =A22A2i , it can be easily seen that the right-hand-side coefficients and initialconditions of (2.5) and (2.12) are O(e) perturbations of (2.16) and (2.17),respectively. Hence, by Theorem A.I of Appendix A,

Now, using the inverse of the decoupling transformation, we have

where we have used the fact that var(r;(0) = O(l/e).Notice that the error in approximating the slow variable £ by *s is O(e),

but the error in approximating the slow variable x by *s is O(e1//2) owing tothe O(l/e) order of variance of 77.

Similarly,

Hence

The use of the decoupling transformation in the proof of Theorem 2.1provides a basis for obtaining higher-order approximations. Equations(2.16), (2.17), together with


and

comprise an exact slow-fast decomposition of (2.1), (2.2). The decom-position (2.13), which yields O(e1/2) approximation, is obtained from(2.16)-(2.19) by 0(e) perturbations of coefficients and initial conditions.Higher-order approximations can be obtained from (2.16)-(2.19) by usinghigher-order approximations of coefficients and initial conditions. In thisrespect the reader may consult our model building discussions of Chapter 2.Exercise 4.4 gives a slow-fast decomposition for an O(e3/2) approximation.

Example 2.2 Highly damped Langevin equation

The Langevin equation

describes the motion of a particle in a liquid (Papoulis, 1965). The processy(t) is the position of the particle in one of its rectangular coordinates. Theparticle is subject to three forces: a collision force represented by the whitenoise term (2KTf)l/2w(t) , a friction force represented by -fy and anexternal force proportional to its displacement represented by —ky. Theconstants m,f,K and T are mass, friction coefficient, Boltzmann constantand absolute temperature respectively, and w(t) has intensity one (i.e. w(t)is the derivative of a standard Wiener process). Iif2>km the poles of(2.20) are given by

Let us suppose that the smallness of km/f2 is due to the smallness of m orthe largeness of/, but not due to the smallness ol k (as k-* 0 the externalforce —ky disappears and the particle moves freely rather than harmoni-cally bound; this case is not considered here). Rescaling time by settingt = (f/k)s and setting x(s) = y(fs/k) and w(s) = (f/k)1/2w(fs/k), (2.20) canbe rewritten as

Taking e = mk/f2 = \k^l\k2 and z = dx/ds, we get the singularly perturbed


system (2.21), which is of the form (2.1), (2.2):

Here A22 = -1 andA0 = An - A12A22A21 = -1, so that the stability con-ditions of Theorem 2.1 are satisfied. Therefore, by Theorem 2.1, x and zcan be approximated for sufficiently small e by xs and zf — xs, respectively,where *s satisfies (2.22) and zf satisfies (2.23):

4.3 The Steady-State Kalman-Bucy Filter

In Section 4.2 we have seen how to obtain slow-fast decompositions ofsingularly perturbed systems driven by white noise, and how to analyze theasymptotic behavior of such systems as e—> 0. This analysis is now extendedto an examination of the time-scale properties of Kalman-Bucy filters(Kwakernaak and Sivan, 1972) for stochastic singularly perturbed systems.To avoid the complication of having time-varying filter gains in the case offinite-time estimation, we limit our study to the steady-state estimationproblem. This problem is one of generating an estimate

in a least-square sense (Kwakernaak and Sivan, 1972) for the linear time-invariant system

on the basis of the observed output {y(cr); o< t}, given by the outputequation

where w and v are independent, zero-mean, stationary Gaussian white

4.3 STEADY-STATE KALMAN-BUCY FILTER 167

noise processes with intensities /s (identity) and V > 0 respectively. Sincethe estimation problem will be studied at steady state (as f-»°°), it isassumed that

so that, by Corollary 3.1 of Chapter 2, the system is asymptotically stablefor sufficiently small E. The solution of this problem is the well-knownsteady-state Kalman-Bucy filter (Kwakernaak and Sivan, 1972) given by

where the filter gain matrix Q is specified byt

whereas P = P(e) > 0 is the solution of the algebraic Riccati equation

Here A, F and C stand for overall system matrices, i.e.

Because of the l/e terms in A and F, the solution of the Riccati equation(3.7) takes the form

which is similar to the solution of the Lyapunov equation (6.38) of Chapter2. Substituting (3.8) in (3.7) results, after partitioning, in

t The matrix X1 denotes the transpose of the matrix X.

164 8 SYSTEMS OF NONLINEAR EQUATIONS

sequence {;cfc}, / = ! , . . . ,« of {x*} is monotone decreasing and boundedbelow by xf and thus has a limit y-t ; that is, the vector sequence {xk} has alimit y. From the continuity of F' it follows that

Now suppose that x* and y* are any two solutions of Fx = 0. Then, using8.3.3, we have

so that if we multiply through by F'(x*)-1 we obtain y* < x*. It follows byreversing the roles of x* and y* that x* < y* and therefore x* = y*. Hence,x* is the unique solution and therefore equal to the limit vector y.

The inequality (5) is very useful because if we stop the iteration at any kwe know that x* is an " upper bound " for the solution x*. In fact, by meanof a second sequence

we may obtain lower bounds, and hence the inequality

which provides the basis for an excellent stopping criterion for the itera-tion. The situation is depicted for n = 1 in Figure 8.3.2. Note that F' isevaluated in (7) at the Newton iterate x* and not at x*.

Figure 8.3.2

8.3.5 Assume that the conditions of 8.3.4 hold and that in addition F'satisfies the monotonicity condition F'(x) < F'(y) whenever x < y. Sup-


From Lemma 3.1 we can see that

Thus the calculation of the filter gains <2i and Q2 can be greatly simplifiedby employing Lemma 3.1. Solving the high-dimensional ill-conditionedRiccati equation (3.7) is replaced by solving two lower-order e-independenRiccati equations (3.10) and (3.11). Furthermore, higher-order approxi-mations can be obtained as in Section 3.5. Simplification of off-line Riccatiequation computations is not the only simplification that can be achievedhere. Further simplification can be achieved by decomposing the Kalman-Bucy filter into two lower-order filters: a slow filter to estimate the slowvariables and a fast filter to estimate the fast variables. Such restructuringof the system gives a deeper insight into the problem, and leads to two-time-scale hierarchical structures.

The desired decomposition will be derived by formally defining slow andfast subsystems, as was done in Section 4.2. Slow and fast filters are designedto estimate the slow and fast variables respectively. Then slow and fastestimates are used to recover estimates of the original variables x and z.

A slow subsystem is defined by replacing z in (3.1) and (3.3) by its steadystate component, which is the solution of (3.2) with e = 0. The slowsubsystem is given by

The corresponding steady-state Kalman-Bucy filter is

where

and Ps > 0 is the stabilizing solution of the slow Riccati equation (3.10).A fast subsystem is defined by treating x as a constant variable. The bias

caused by x in z and y can be removed by defining

and

The fast subsystem is given by


The corresponding steady-state Kalman-Bucy filter is

where

and Pf 2: 0 is the stabilizing solution of the fast Riccati equation (3.11). Inimplementing the fast filter, y{ is replaced by y - CQXS. Thus we haveproposed the following decomposition:

A block-diagram representation of the decomposition (3.18) is shown inFig. 4.4, and its near-optimality is established in Theorem 3.1.

Fig. 4.4. Two-time-scale hierarchical state estimation.

Theorem 3.1

If A0 and A22 are Hurwitz matrices then the decomposition (3.18) is near-optimal for sufficiently small e. In particular, there exists an e* > 0 suchthat, for all e 6 (0, e*), (3.19) holds at steady state (as t-> <»):

Furthermore, if the optimal filter is initialized at certain initial conditions


jc0 and f0, and the near-optimal filter (3.18) is initialized at £,(0) = jc0 andff(0) = z0 + AvlA2i£Q, then (3.19) holds for all t > 0.

Prao/ The idea of the proof is similar to that of Theorem 2.1. First, adecoupling transformation is used to transform the optimal filter into newcoordinates where it can be verified that the decomposition (3.18) followsfrom the optimal filter by perturbation of right-hand-side coefficients.Then Theorem A. 2 of Appendix A is employed to prove near-optimality.Although the decoupling transformation was discussed in Chapter 2 forsingularly perturbed systems whose right-hand-side coefficients are inde-pendent of e, extension to the case where the coefficients are analyticfunctions of e at e = 0 is straightforward (see Exercise 4.6). This extensionis needed here because the Kalman-Bucy filter gains <2i(e) and Q2(£) are

analytic at e = 0. In the proof of Theorem 2.1, the matrices L and H of thedecoupling transformation were chosen exactly as in Chapter 2. This time,the choice of L and H is more involved. So, let us derive the transformationstep by step. A comparison of the fast filter (3.18b) with the second equationof the Kalman-Bucy filter, (3.14), shows that when both equations areviewed as systems driven by the innovation process v — QJC - C2z ~y —CQXS - C2zf, the essential difference between them is the term A2lx on theright-hand side of (3 . 14) . That term can be eliminated via the transformation

where L satisfies

This equation is the same as equation (2.8) of Chapter 2. Hence, forsufficiently small e, L(e) exists and L(Q) = A22A2l. Using (3.20), theKalman-Bucy filter (3.13), (3.14) is transformed into

It is apparent that

which confirms that the right-hand-side coefficients of the fast filter (3.18b)are O(e) perturbations of the right-hand-side coefficients of (3.23). Con-


sider now the slow equations. Comparison of the slow filter (3.18a) with(3.22) shows that the term (An ~ (2iQ>)*7 on the right-hand-side of (3.22)should be eliminated. This can be achieved by the transformation

where HI is chosen to satisfy

Setting £ = 0 in (3.25) shows that

which is well-defined because A 22 - PfC2V~1C2 isaHurwitzmatrix. Appli-cation of the implicit function theorem (Dieudonne, 1982) shows theexistence of H^e) satisfying (3.25) for sufficiently small e. Combining(3.20) and (3.24), we conclude that the transformation

transforms the Kalman-Bucy filter (3.13), (3.14) into the new form

It is important to notice that there is no approximation involved in deriving(3.28). It is merely a similarity transformation of the Kalman-Bucy filterequations, of which the estimates x(t) and z(t), provided by (3.28), are theoptimal estimates.


Lemma 3.2

The right-hand-side coefficients of (3.18) are O(e) perturbations of theright-hand-side coefficients of (3.28).

The truth of this lemma can be easily seen by inspection of (3.18) and(3.28) and use of the expressions An - A12L(Q) = A0, <22(0) = Qt,C, - C2L(0) = C0 and ^(0) - //i(0)Q2(0) = Qs, where the last one isshown in Appendix B. To apply Theorem A.2, we augment both (3.18)and (3.28) with the state equations (3.1), (3.2) to obtain systems ofequations driven by white noise. For the optimal filter (3.28) we get

The corresponding equation for the proposed filter (3.18) is

The matrices s4,y, SQ(-, si^ and 98, are obtained in an obvious way. Moreover,by Lemma 3.2 we have

To apply Theorem A.2 we need to verify that ^n(O) - s^l2(G)s^22'and <s$22(0) are Hurwitz matrices. The matrix .s$22(0) is given by the lowerblock-triangular matrix

which is Hurwitz since both A22 and A22 — QiC2 are Hurwitz matrices. The

0)A21(0)


matrix 64n(0) - ^((^^(OJ.s^O) is given by

Again this matrix is Hurwitz since both A0 and A0 — QSC0 are so. ApplyingTheorem A.2, we conclude that (as t—> <»)

Equation (3.19) follows by writing

and applying the triangle inequality. If the slow and fast filters are initializedat *s(0) = x0 and if(0) = z0 + A^A2\xQ then Theorem A.I implies that(3.19) holds for all f s» 0 since *s(0) - §(0) = O(e) and ff(0) - r)(0) = O(e).

D

One important application of Theorem 3.1 lies in the establishment ofthe validity of reduced-order filtering. Notice that in the decomposition(3.18) the estimate of the slow variable x depends only on the output ofthe slow filter (3.18a); it is independent of the fast filter (3.18b). Recallthat the slow filter (3.18a) is derived on the basis of the reduced-order slowsubsystem, which has been obtained by neglecting the fast dynamics of thesingularly perturbed system (3. l)-(3.3). Thus, if our objective is to estimatethe slow variable x only, then in designing the Kalman-Bucy filter the fullmodel (3.1)-(3.3) can be replaced by the reduced-order slow model (3.16).Theorem 3.1 shows that such reduced-order filtering is near-optimal as faras the slow variable x is concerned.

4.4 The Steady-State LOG Controller

Chapter 3 introduced the composite control method where the control issought as the sum of a slow control us and fast control wf. The slow controlis designed using the slow model and the fast control is designed using thefast model. In this section we extend the near-optimal composite controlmethod of Section 3.4 to the linear-quadratic Gaussian (LQG) optimalregulator. For this problem, the system model takes the stochastic time-

4.4 STEADY-STATE LOG CONTROLLER 175

invariant singularly perturbed form

where x,z,y,w and v are as defined in Section 4.3 and u E Rr is the controlinput. The steady-state linear-quadratic Gaussian control problem isdefined as the problem of finding u(f) as a function of the observed output[y(o); o < t] so as to minimize the performance index

where _yc G /?p is the controlled output to be regulated to zero, which isgiven by

This control problem is the stochastic version of the optimal regulatorproblem of Section 3.4, except that the feedback solution is sought underimperfect and partial observations instead of perfect and full state feedback.Under standard stabilizability-detectability conditions, which are guaran-teed to hold for sufficiently small e if the conditions of Theorem 4.1 ofChapter 3 and Lemma 3.1 are satisfied, the optimal control law is given by(Kwakernaak and Sivan, 1972)

where x(t) and z(t) are the steady-state optimal estimates provided by theKalman-Bucy filter:

The regulator gains G^e) and G2(e) are givent by

t Note the notational difference of sign between the regulator gain matrices Gl and G2 of(4.9) in this chapter and (5.2), (5.3) of Chapter 3.


where K^ K2 and K3 comprise the solution of the regulator Riccati equation(4.5) or (5.4) of Chapter 3. The filter gains Ql and Q2 are given by (3.15).

A composite feedback control approach to the LQG control problemcan be developed in a way similar to the composite control of the deter-ministic LQ control problem of Section 3.4. The composite control is soughtin the form

where us is a slow control component and uf is a fast control component.In Chapter 3, the slow control «s is a feedback function of the slow variable*s, and the fast control u{ is a feedback function of the fast variable zf. Itis reasonable to seek the composite control (4.10) in the same form, butwith xs and xf replaced by their estimates. Applying (4.10) to the system(4.1), (4.2) yields

To define a slow subsystem we set £ = 0 in (4.12), solve the resultingalgebraic equation in z and eliminate z from (4.11) and the output equation(4.3) to get

where D0 = -C2A22B2. In other words, the slow subsystem is defined byreplacing z by its steady-state component. A fast subsystem is defined byremoving the slow bias from z and the output y. The slow term on theright-hand-side of (4.12) is A21x + B2us. Therefore, roughly speaking, theslow bias of z is given by

The fast component of z, denoted by zf, and the fast component of v,denoted by yf, are defined by

Computing the derivative ezf with zs(/) treated as a constant, i.e. ezs(f)


0, we obtain the fast subsystem

The next task is to associate with each subsystem an appropriately selectedperformance index. Using (4.15), the controlled output yc(t) in (4.5) canbe expressed as

which decomposes as the sum of a slow component M^x + N0us and a fastcomponent M2zf. Thus for the slow subsystem (4.13) we consider theperformance index

and for the fast subsystem (4.17) we consider the performance index

The performance indices /s and Jf are stochastic versions of those used inChapter 3 (compare (4.18) and (4.19) with (4.14) and (4.20) of Chapter 3respectively). Thus we have replaced the original LQG problem (4.1)-(4.4) by two LQG problems: the slow problem is defined by (4.13), (4.18)and the fast problem is defined by (4.17), (4.19). The slow LQG problemis a standard one except for the presence of the input uf(t). However, sinceby construction uf(t) is a function of y(t), the well known separation principleof stochastic control (Kwakernaak and Sivan, 1972) yields the optimalsolutions

where, for any wf(f), xs(t) is the optimal estimate provided by the slowKalman-Bucy filter


Fig. 4.5. Block diagram representation of the composite stochastic control.

The fast LQG problem is a standard one whose optimal solution is givenby

The filter gains Qs and Qf are as defined in Section 4.3 and the regulatorgainst Gs and Gf are as defined in Section 3.4. In implementing the fastfilter (4.23), yf(t) is replaced by y(t) - C^s(r) - D0us(t). Thus the composicontrol takes the final form (4.24), whose block-diagram representation isshown in Fig. 4.5:

The near-optimality of the composite control is established by Theorem4.1.

Theorem 4.1

Suppose that the conditions of Theorem 4.1 of Chapter 3 and Lemma 3.1are satisfied. Let*(f) and z(t) be the optimal trajectories and/be the optimalvalue of the performance index. Let x(t), z(t) and / be the correspondingquantities under the composite control (4.24). Then there exists an e* > 0

t It should be noted, as a matter of notation, that the regulator gains Gs and Gf replace-Go and -G^0) of Section 3.4.

4.4 STEADY-STATE LOG CONTROLLER

such that for all e £ (0, *•*), (4.25)-(4.28) hold:

179

Furthermore, if the optimal Kalman filter (4.7), (4.8) is initialized at certaininitial conditions x0 and z0, and the slow and fast filters of the compositecontrol are initialized at

then (4.26) and (4.27) hold for all t > 0.

Proof The steps of the proof are similar to those of the proof ofTheorem 3.1. We show that the coefficients of composite control (4.24) areO(e) perturbations of the coefficients of the optimal control (4.6)-(4.8)when the latter is expressed in appropriately selected coordinates. Iden-tifying the appropriate coordinates is done exactly as in the proof ofTheorem 3.1. Omitting details, the required transformation is

where L2 satisfies an equation similar to (3.21) with Ay replaced byAy - BiGj, and //3 satisfies an equation similar to (3.25) with L replacedby L2. It can be verified that the use of (4.29) transforms the optimalcontrol into the new form


Lemma 4.1

The right-hand-side coefficients of the composite control (4.24) are O(e)perturbations of the right-hand-side coefficients of the optimal control(4.30).

This lemma can be seen by inspection of (4.24) and (4.30) and use of thefollowing expressions, which follow either from Section 3.4 or from Section4.3:

as well as the expression

and its dual

Next, the closed-loop system under optimal control is represented as asystem driven by white noise similar to (3.29) except for different «$#,•,• , 95,matrices. It can be verified that the use of the composite control results inO(e) perturbations in ̂ and 2ft,. The stability requirements on s£n(0) -

i (0) and ^22(0) follow by showing thatA12(0)A22(0)A


and

where A = B0G<(A22 - B2Gfr1A22(A22 - QfC2)-1Q fC0.

The use of Theorem A.2 of Appendix A gives (4.26) and (4.27). TheoremA.3 leads to (4.25) after verifying that when / and J are expressed asquadratic forms in (x, f, z, fj) and (x, xs, z, ff) respectively, the matrices inJ are O(e) perturbations of the matrices in /. Finally, Theorem A.I shows"that if *(0) and f(0) are chosen as in (4.28) then (4.26) and (4.27) hold forall t > 0 since Jt(0) - f(0) = O(e) and if(0) - ?)(0) = O(e). D

The composite control (4.24) and Theorem 4.1 extend, in an intuitivelyappealing way, to the LQG problem the composite control method of thedeterministic LQ problem of Section 3.4. There are, however, differencesbetween the LQ and the LQG results worthy of note. First, the use of thecomposite control in the LQ problem results in A/// = O(e2} versusA/// = O(e) in the LQG problem. Second, if x and z are the states ofthe system under optimal control, then in the LQ problem x - * = O(e)and z-z = O(e), whereas in the LQG problem x - x = O(e) andz — z = O(e^2}, in a mean-square sense. Third, and this is an importantdifference, in the LQ problem it is shown (Theorem 4.4 of Chapter 3) thata reduced control law derived by solving only the slow optimal controlproblem, i.e. w r=-G s jc, is near-optimal with A///=0(e). This factjustifies using the slow reduced-order model as a basis for solving the LQproblem. In the LQG problem a similar reduced control will not be near-optimal. This point can be explained, loosely speaking, in the followingway. In the LQ problem the optimal value of / takes the form/ = /s + O(e)(recall Theorem 4.3 of Chapter 3); so optimization of the slow controlproblem alone is enough to achieve near-optimality. In the LQG problemthe optimal value of / takes the form J = Jf/e + 0(1) (see the proof ofTheorem A.3 of Appendix A and replace a by /); so optimization of thefast control problem is necessary to the achievement of near-optimality.The use of the reduced control will be near-optimal only in the special casewhen J{ = 0, since in that case J = Js + O(e) (see Exercise 4.7). Exampleswhere /f = 0 are the case M2 = 0 (the fast variables are not penalized inthe performance index) and the case F2 = 0 (no white noise input in fastequation).


4.5 An Aircraft Autopilot Case Study

The near-optimality of the composite stochastic control of Theorem 4.1will now be demonstrated on the autopilot control of the longitudinalmotion of an F-8 aircraft introduced in Case Study 7.2 of Chapter 1. Thenumerical data are taken from Elliott (1977). The equations of motion aregiven by

where all the variables and parameters, except orN, are defined in Section1.7; ON is the nose angle of attack due to wind disturbance. It is usuallymodeled as exponentially correlated noise with power spectral density ofthe form

Although it is not realistic to assume that the bandwidth of this noise ismuch wider than the "short-period" frequency, for the sake of simplifyingthe presentation of the example we model orN as white noise with intensityo2I/JtVl. The numbers used in our calculations are representative forflight of the F-8 aircraft at 20000ft with V0 = 620ft/s (Mach number = 0.6)and aQ = 0.078rad. They are

The angles a, 6, 6 and <7N are in rad, q in rad/s and v is dimensionless(v = u/V0 where u is the incremental velocity in ft/s). Together with theequations of motion we have to choose a set of measurements. Supposewe measure the pitch angle d and the pitch rate q. Then the measurement

4.5 AIRCRAFT AUTOPILOT CASE STUDY 183

equations are

Each of the measurement noise processes v[ is modeled as an exponentiallycorrelated process with power spectral density 2a2T;/(l + co2r2), wherea, and r, are given by

pitch angle sensor:

pitch angle gyro:

Thus v{ and v2 can be reasonably modeled as white noise processeswith intensities 2o\iv = 12.184697 x 10"6rad2.s and 2o2

2T2 = 3.8991 x10"6rad2/s respectively.

Finally, the performance index is chosen as

We give results for the case vmax = 0.1, 0max = amax = 6max = 0.1 rad, ^rmax =O.lrad/s and p = 40.

Now, the LQG problem is completely defined, and our first task is tobring it into the singularly perturbed form (4.1)-(4.5). This modeling stepis done as in Section 1.7. The angle Q is replaced by

the time scale is changed from t to t' = tcoQ and the singular perturbationparameter is chosen as e = O)Q/O)I, where co0 and coj are the undampedfrequencies of phugoid and short period modes and are given, respectively,by (7.25) and (7.31) of Chapter 1. The numerical values for these quantitiesare

To take care of the effect of changing the time scale on the white noiseprocesses, and to match the problem statement (4.1)-(4.5), the white noise


w(t') is defined as

and the intensity matrix of the measurement noise, V, is taken as

The matrices appearing in (4.1)-(4.5) are given by

R = 4000.Numerical results for both the optimal and composite controllers are

presented in Tables 4.1-4.3. Table 4.1 compares the solutions of the fullRiccati equation with the solutions of the slow and fast Riccati equations.Table 4.2 compares the closed-loop eigenvalues, while Table 4.3 comparesthe steady-state root-mean-square values of the state variables and thevalues of the performance index for closed-loop optimal and composite

4.5 AIRCRAFT AUTOPILOT CASE STUDY 185

TABLE 4.1Comparison of Solutions of Riccati Equations

TABLE 4.2Open-Loop and Closed-Loop Eigenvalues

Slow Fast

Regulator poles Filter poles

Slow

composite

Fast-1.071 ± J2.13

Slow-0.02 ± jO.02

Fast-0.572, -66.561

-1.055 ± J2.084 -0.021 ± jO.019 -0.589, -66.536

TABLE 4.3Performance Comparison

rms vrms yrms arms qJ

Optimal

0.0043630.0057660.0344710.0644820.370759

Composite

0.0048220.0059910.0351330.0649890.372158

Error

0.459 x 1Q-3

0.225 x 10~3

0.662 x ID'3

0.507 x 1Q-3

1.399 x 10^3

Relativeerror

0.10520.0390.01920.00790.0038

optimalClosed-loop

losed-loop

open-loop

-0.084, -0.137

-0.0 83 , 0 .14

-0.0069 + j0.0765 -0.6656 + j2.1821


control systems. In order to improve the conditioning of the Riccatiequations in computations, the performance index is multiplied by 10~2

while the noise intensities are multiplied by 104. Such scaling does not affectthe feedback control law; see Section 4.7 for further discussion of scaling.The Riccati equations solutions of Table 4.1 are given for the scaledproblem, while the r.m.s. values and performance index of Table 4.3 arethe actual ones. The eigenvalues of Table 4.2 are given in the original timescale t.

4.6 Corrected LQG Design and the Choice of the DecouplingTransformation

Should a closer approximation than that provided by the LQG design ofTheorem 4.1 be desired, a corrected LQG design may be pursued in themanner of Section 3.5. In the proof of Theorem 4.1 we have shown thatthe composite control (4.24) is nothing but an O(s) perturbation of thecoefficients of the optimal control (4.30), in which the Kalman-Bucy filtervariables are restructured by the decoupling transformation (4.29). ByTheorems A.I -A. 3 of Appendix A, all that is needed to obtain closerapproximations of the system trajectory and performance index is to retainhigher-order terms in the expansions of the e-dependent coefficients of theoptimal control (4.30). We will illustrate this by presenting the case of an0(e2) approximation of coefficients.

The optimal control (4.30) is given in terms of six matrices: the regulatorgains Gi(e) and G2(e), the filter gains Q\(e) and Q2(e), L2(e), whichsatisfies (3.21) with A^ replaced by Ay — 5,-Gy(f) and //3(e), which satisfies(3.25) with L replaced by L2. Each of these six matrices is analytic ate = 0. Writing their Maclaurin series as G^e) = G<0) + eG(^ + . . . , withsimilar expressions for the other matrices, substituting in (4.30) and drop-ping 0(e2) coefficients, we arrive at the corrected design

4.6 CORRECTED LOG DESIGN AND DECOUPLING 187

The computation of the matrix corrections G^, G^, Q^\ Q(f, L(^> and^ proceeds as follows. First, the regulator gain corrections are specified

as in Section 3.5 with K2 = Km and a notational change of sign by

where K^ satisfies the Lyapunov equation

K^ satisfies the Lyapunov equation

and K^\ which is dependent on K^ and K^\ is given by

As the dual of the expressions (6.2) and (6.3), the filter gain correctionsare specified by

where P^, P^ and P^ respectively satisfy the linear equations

h


Thus we are left with the calculation of L^ and H^. Substituting Maclaurinexpansions for the respective variables in the equations of L2 and H3 andmatching coefficients of like powers of e, we obtain the expressions for

and H^\ which are given by

The following theorem gives the order of approximation of the statetrajectories and the performance index.

Theorem 6.1

Suppose the conditions of Theorem 4.1 hold. Let x(t) and z(t) be theoptimal trajectories and let J be the optimal value of the performanceindex. Let x^(t), Zi(t) and Jv be the corresponding quantities under thecorrected controller (6.1). Then there exists an e* > 0 such that for alle£(0, £*), (6.8)-(6.10) hold:

Proof Similar to the proof of Theorem 4.1.

An initial condition statement similar to that of Theorem 4.1 is alsopossible.

Example 6.1

As a numerical illustration of Theorem 6.1, let us consider the second-

4.6 CORRECTED LOG DESIGN AND DECOUPLING

order example

189

where both w and y are white noise processes with intensity one. Table 4.4compares values of the performance index / under the optimal, the com-posite (uncorrected) and corrected designs for five different values of e.The reader may verify that the numerical results agree with the orders ofapproximation established by Theorems 4.1 and 6.1. As e—»0 both designsyield acceptable approximations. For a specified value of e one may haveto correct the design to get acceptable results. For example, at e = 0.2 theuncorrected design results in 19.23% error which may not be acceptable.One step of correction brings this error down to 1.598%. The numericalresults demonstrate also the fact that / is O(l/e).

TABLE 4.4Performance Index Comparison

e

0.20.10.050.0250.01

Optimal

59.03677294.468821

165.65066308.182981735.918503

Composite

70.39106398.890743

167.638049309.128464736.285953

Corrected

59.98031294.663604

165.718247308.212322735.929465

Percentageerror

composite

19.234.681.20.310.05

Percentageerror

corrected

1.5980.2060.040.010.001

Let us conclude this section by re-examining the role of the decouplingtransformation used in deriving the composite control (4.24) or the cor-rected control (6.1). Arriving at (4.24) or (6.1) can be described as a two-step process. In the first step, the decoupling transformation

with the particular choice L = L2 and H = H3, was used to restructure theKalman-Bucy filter. In the second step, the coefficients of the optimal


control law are approximated; an O(e) approximation yields (4.24), whilean O(e2) approximation yields (6.1). The choice L = L2 and H = H3 wasmotivated by our desire that the O(e) approximation coincides with thecomposite control that has been intuitively derived. A question that couldbe posed here is whether there are choices of L and H that could restructurethe Kalman-Bucy filter in a different way. In fact, there are several choicesof L and H that could be given a physical interpretation. For example, Land H could be chosen to block-diagonalize any of the matrices A, theopen-loop matrix A - QC, the homogeneous part of the Kalman-Bucyfilter when viewed as a system driven by both the control input u and theobserved output y, and A - BG, the homogeneous part of the Kalman-Bucy filter when viewed as a system driven by the innovation process.Each of these matrices corresponds to a standard singularly perturbedsystem model since det (A22) =£ 0, del (A 22 - Qz(G)C2) =£0 anddet(^422 ~~ B2G2(0)) =£0. Furthermore, we could have mixed situations.For example, the reader may verify that the choice of L = L2 and H = H 3

corresponds to the following situation. In choosing L2 the Kalman-Bucyfilter is viewed as a system driven by the innovation process and L2 ischosen to put it in the actuator form. Then, in choosing H3 the Kalman-Bucy filter is viewed as a system driven by u and v and H3 is chosen to putit in the sensor form. Obviously, there are several meaningful choices of Land H. Therefore, instead of studying the effect of using a particulartransformation, we consider a class of transformations of the form (6.11),where the matrices L and H are required only to be analytic functions ofe at e = 0. Each member of the class corresponds to a choice of coordinatesystem for representing the optimal Kalman-Bucy filter (4.7), (4.8). Sincein this class either L or H or both could be zero, the class includes theoriginal coordinate system in which the Kalman-Bucy filter (4.7), (4.8) isrepresented.

To gain some insignt into the effect of using the transformation (6.11)on the optimal Kalman-Bucy filter, we compute the power spectral densityof (|, 77), the state vector of the transformed filter. The power spectraldensity of (f , fj) can be determined by studying the equation

Since the innovation process v is white noise, the power spectral densityof (|, 77) is determined by the transfer-function matrix

4.7 SCALED WHITE-NOISE INPUTS 191

To compute the resolvent matrix (si - A + BG)~l, the transformation(6.11), with appropriately selected L = L2 and H = H2, is used to block-diagonalize A - BG, resolvent matrices of the diagonal blocks arecomputed, and then the inverse transformation is applied to recover(sI-A + BG)"1. Multiplying (si - A + BG)~l from the right by Q andfrom the left by T, where T is any member in the class, we get

Examination of expressions (6.14) and (6.15) reveals that all trans-formations of the form (6.11) share two properties. First, the fast variablefj has a slow (low-frequency) bias. Second, the fast (high-frequency) com-ponent in W^s) is multiplied by e; hence, for sufficiently small e, | ipredominantly slow. Inspection of (6.15) shows also that the choice L =L2 (which was adopted in Section 4.4) eliminates the slow bias of r).The final word on the choice of the appropriate coordinates is problem-dependent, and implementation factors, which are not considered here,will affect the choice. It is clear, however, that, irrespective of whichtransformation is used, near-optimum controllers can be obtained by trun-cating coefficients of Maclaurin expansions since L and H are analytic at

4.7 Scaled White-Noise Inputs

Frequently in applications we are faced with problems where the whitenoise input to a singularly perturbed system is scaled by a fractional powerof the singular perturbation parameter, e.g. e1/2. As an example of amechanism through which such scaling factors arise, let us consider a zero-mean stochastic process with correlation function R(T) = a2e~'T ' /6>, whereo is of order one and 6 is small. One can model this process as the state ofa first-order differential system driven by white noise:


where w has intensity o. Taking 6 = e, we see that we have a singularlyperturbed equation similar to the one we studied earlier except for thee1/2 factor multiplying the white noise input. In this section, we study theeffect of scaling the white noise input on the results obtained throughoutthe chapter. We start with the simple case studied in Section 4.2. Considerthe singularly perturbed system

where all the variables are as defined in Section 4.2 and /j, is a positiveparameter that may depend on e. It is only required that // be a boundedfunction of e in the neighborhood of e = 0. Typical forms of ^u are ju = ea

for aE. [0, 1]. In view of the analysis of Section 4.2, it should not be asurprise to see that n has no effect on the slow-fast decomposition (2.13),except of course, for scaling w by p. In particular, defining xs and zf as

we have the following theorem.

Theorem 7.1

Suppose that A0 and A22 are Hurwitz matrices and fi is as defined above.Then there exists an e* > 0 such that for all e £ (0, e*), (7.5) is satisfiedfor all t > 0:

Moreover, at the steady state (as t —> °o) we have

The proof of this theorem is the same as that of Theorem 2.1, and makesuse of Theorems A.I and A. 2 of Appendix A. Notice that (7.6) gives asharper bound on the approximation error whenever // = ea with or > 0.

Moving now to the estimation and control problems of Sections 4.3 and4.4, we notice that scaling the white noise inputs may have a crucial role,since it will affect the filter Riccati equation and the filter gain. There is

4.7 SCALED WHITE-NOISE INPUTS 193

one significant special case, however, where the results of Sections 4.3 and4.4 remain invariant with respect to white noise scaling. This is the casewhen all the white noise inputs are scaled by the same factor jU. Let usexamine the LQG controller. The system equations are taken as

where all the variables are as defined in Section 4.4 and ^ is defined above.The performance index is defined by (4.4) and (4.5). The factor \i has noeffect on the solution of the regulator part of the problem. It affects thefilter part. However, since both the state excitation noise and the obser-vation noise are scaled by the same factor \i, the effect of \i will be to scalethe solution of the filter Riccati equation but it will not affect the filter gain.In particular, the solution of the filter Riccati equation takes the form

where P1, P2 and P3 satisfy (3.9). The filter gains are given by (3.15). Inother words, |U does not affect the optimal control but affects the optimalvalue of/ and the variances of the states variables. As a consequence, thecomposite control (4.24) is near-optimal, as stated in the following theorem.

Theorem 7.2

Suppose that the conditions of Theorem 4.1 hold and ju is as defined above.Let x(t), z(i) and/ be the optimal trajectories and cost of the LQG problem(7.7)-(7.9) and (4.4), (4.5). Let x(t), z(t) and J be the correspondingquantities under the composite control (4.24). Then there exists e* > 0such that for all e E (0, e*), (7.12)-(7.14) hold:


Again, the proof of this Theorem is similar to that of Theorem 4.1 anduses Theorem A. 2 and A.3 of Appendix A.

The more general cases when w and v are scaled by different factors orwhen the white-noise inputs to the slow and fast equations are scaled bydifferent factors, e.g. jU^ and /*2^2» are more difficult and less understood.The difficulty arises from the fact that the solution of the filter Riccatiequation and the filter gain will depend on ratios between various scalingparameters. There has been some effort to study such problems which willbe surveyed in the notes and references section.

4.8 Exercises

Exercise 4.1

Consider the second-order singularly perturbed system

where v (t) is a zero-mean stationary Gaussian exponentially correlatedprocess with power spectral density 2o26/(l + <w202). Three cases are ofinterest here:

(a) In which of these cases can v(t) be modeled as white noise? If v(f)cannot be modeled as white noise, model it as the solution of a differentialequation driven by white noise.

(b) For each case, write the augmented state equation in the singularlyperturbed form and obtain slow-fast decompositions.

Exercise 4.2

The noise voltage sources in the J"?C-circuit of Fig. 4.6 represent the thermalor Johnson noise associated with the resistors R^ and R2, due to the randommotion of charges, which is independent of their mean or average motion.For an Ohmic resistance R at thermal equilibrium, the thermal noise has aconstant spectrum 4kTR, where k is Boltzmann's constant and T is tem-perature in Kelvin. Let Rl = R2 = R, Cl = eC, C2 = C.

(a) Taking x = v2 and z = u l 5 show that the state equations are in thesingularly perturbed form.

(b) Obtain a slow-fast decomposition using (2.13).

4.8 EXERCISES 195

Fig. 4.6. RC-circuit with noise voltage sources of Exercise 4.2.

(c) Suppose now that e = 10~5, which is a reasonably small number, andR = 1000 Q. Taking into consideration that k = 1.38 x 10"23 J/K and T (atroom temperature) = 300 K, the constant (4kTR)1/2 ~ 4.07 x 10~9 is muchsmaller than e. How would this fact affect the validity of the approximationsobtained in part (b)?

Exercise 4.3

Let

where w(i) is a zero-mean stationary white noise process with intensitymatrix V, and x0 is a zero-mean random vector independent of w withvariance Q0.

(a) Show that (2(0 > tne variance of x(t), is given by

(b) Show that Q(i) satisfies the Lyapunov equation

(c) Taking

where del ^ show that Q(t, e) takes the form

Use the decoupling transformation of Section 2.4.hint


Exercise 4.4

Consider (2.16), (2.17) and let

be the Maclaurin series of L and H respectively.(a) Show that

(b) Let x(t), z(f) be the solution of (2.1), (2.2), and let x(t), z(f) be givenby

where |(r), fj(t) respectively satisfy

Show that

Exercise 4.5

(a) Define and solve the slow and fast steady-state filtering problems for

where w(0 and v(t) have intensity one.

4.8 EXERCISES 197

(b) Solve the original steady-state filtering problem, parameterized in e.(c) Compute the steady-state root-mean-square estimation errors in both

cases for e = 0.5, 0.1 and 0.05 and compare them.

Exercise 4.6

Consider the singularly perturbed system

where A/y(e) are analytic at e = 0 and detA22(0) =£ 0. Consider the similaritytransformation (4.8) of Chapter 2, where L and H satisfy equations similarto (2.8) and (4.3) respectively, of Chapter 2, but with Atj replaced by A^e).

(a) Using the implicit function theorem, prove that there is an e* > 0such that Ve G (0, e*) there exist L(e) and H(e) satisfying their respectiveequations with L(0) = A£(Q)A2l(ty and //(O) = Al2(fyA£(Q).

(b) Show that the similarity transformation decouples the above systeminto slow and fast parts.

Exercise 4.7

With reference to the variance analysis of Appendix A, consider the system(A 1) and the quadratic cost (A 18). Let as and of be defined by

where Qs and Qf respectively satisfy

and where

(a) Verify that Go = Gn(0) and Qf = Q33(Q) (G,y(0) is denned in Appen-dix A).

It is shown in Appendix A that a = crf/e+ 0(1) and


(b) Show that

(c) Show that of = 0 => o = crs + O(e).

Exercise 4.8

Consider the LQG control problem

where V = 72.(a) Determine the composite control design.(b) Determine the corrected design of Section 4.6.(c) Determine the optimal design parameterized in e.(d) Find / for each of the three designs for the following values of e: 0.5,

0.25, 0.125, 0.0625 and 0.03125.


Research in filtering and control problems for singularly perturbed modelswith white-noise inputs has revealed difficulties not present in deterministicproblems. This is due to the fact that, as we have seen in Section 4.2, theinput white-noise process "fluctuates" faster than the fast dynamic variables,which as e—»0 themselves tend to white-noise processes. Because ofthese difficulties, other formulations of stochastic problems for singularlyperturbed systems have been investigated. Blankenship and Sachs (1979)studied a linear system of the form (2.1), (2.2) where the white-noise inputis replaced by wideband noise. Their work deals with two asymptotic


phenomena; one due to small time constants and the other due to smallcorrelation times.

Section 4.3 is a steady-state treatment of a singularly perturbed Kalman-Bucy filtering problem of Haddad (1976). The singularly perturbed LQGcontrol problem of Section 4.4 is a steady-state version of Haddad andKototovic (1977); see also Teneketzis and Sandell (1977). The treatmentof the filtering and control problems of Sections 4.3 and 4.4 is differentfrom the original papers. It is based on a more recent approach by Khaliland Gajic (1984), which makes use of the decoupling transformation ofSection 2.4 and performs a variance analysis of singularly perturbed systemsdriven by white noise by studying singularly perturbed Lyapunov equations.This variance analysis is included in Appendix A. The corrected LQGdesign of Section 4.6, which appears in the literature for the first time, isbased again on Khalil and Gajic (1984), where general procedures forobtaining higher-order approximations are discussed.

The issue of scaling the white noise input disturbances, which is addressedin Section 4.7 for the special case of uniform scaling, has been motivated bytwo considerations. First, it arises in physical models through mechanismssimilar to the one discussed in Section 4.7. Second, it has been called uponas a way to reformulate LQG control problems so as to avoid divergent(O(l/e)) performance indices. Various alternative formulations of thelinear stochastic regulator problem, where several scaling parameters areused to scale white-noise contributions and the fast part of the performanceindex, have been reported in the literature. Interested readers may consultthe surveys by Kokotovic (1984) and Saksena et al. (1984). A nonlinearversion of the problem is treated in Bensoussan (1981).

The treatment of this chapter is limited to standard linear singularlyperturbed models driven by white noise. It does not cover stochasticaveraging and small-noise problems; sources for which are the surveys byBlankenship (1979) and Schuss (1980) and the book by Kushner (1984).

5 LINEAR TIME-VARYING SYSTEMS

5.1 Introduction

For linear time-invariant models the only source of multiple time-scaleproperties, as we have witnessed in Chapter 2, is the separation of thesystem eigenvalues into two (or more) groups of eigenvalues of differentorders of magnitude. In the case of linear time-varying systems, anothersource of multiple time-scale properties is the variation of modelparameters. The simplest example of this, studied in Section 5.2, is thelinear system

where the model coefficient matrix A(t) is a function of time t oversome time interval of interest. Section 5.2 establishes conditions, includingbounds on e, for the asymptotic stability of the system (1.1).

Later sections parallel the time-invariant development of Chapter 2through dealing with the more general linear time-varying singularly per-turbed system model

Section 5.3 provides a transformation for decomposing the system (1.2),(1.3) into two subsystems in separate time scales, while Section 5 A estab-lishes conditions for system asymptotic stability. An important class ofsystems with time-varying parameters is the class of adaptive controlsystems. The robustness issue in Section 5.5 is whether the consideredadaptive system remains stable in the presence of unmodeled high-frequency dynamics.

Proceeding to linear time-varying singularly perturbed systems with

201

202 5 LINEAR TIME-VARYING SYSTEMS

inputs, Section 5.6 develops state approximations similar to those of Section2.5 for time-invariant systems. Controllability properties of such systemsare explored in Section 5.7. In anticipation of the trajectory optimizationproblems of Chapter 6, this controllability study is extended to the analysisof the asymptotic behavior of a control which transfers the system statefrom a given initial point to a desired final point in the state space. Finally,observability properties are established in Section 5.8.

5.2 Slowly Varying Systems

In this section, we study time-varying systems described by (1.1). When\\A(t)\\ = 0(1), the parameters are slowly varying relative to the dynamicsof the system. One would intuitively expect that for e sufficiently small thedynamic behavior of the system could be predicted by studying a family oftime-invariant systems with parameters frozen at a particular time instantt. In particular, when Re k(A(i)} < -c < 0 for all f, intuition suggests thatas e— » 0 the above system should be asymptotically stable.

Consider the system

where the time interval / could be a finite interval [t0, ff] or an infiniteinterval [?0, °°)- We make the following assumption.

Assumption 2.1

This assumption guarantees that the family of frozen-time systems associ-ated with (2.1) is asymptotically stable. Hence the exponential matrix exp-[A(t)6] satisfies

where a > 0 is independent of t , but K(A) > 0 depends on t . To extend theexponentially decaying bound (2.2) from a frozen-time system to the actualtime-varying system (2.1), the inequality (2.2) should hold uniformly in t.Towards that goal we assume the following.

Assumption 2.2

5.2 SLOWLY VARYING SYSTEMS 203

Lemma 2.1

Under Assumptions 2.1 and 2.2, there exist positive constants a^ and KIsuch that

Proof The matrices ^4(f) satisfying Assumptions 2.1 and 2.2 form acompact set, which we denote by S. Consider

Using (2.2), we get

Application of Gronwall's Lemma (Vidyasagar, 1978) yields

By (2.4), there exists a positive constant al < a and a neighborhood N(A)of A such that if C G N(A) then

Since S is compact it is covered by a finite number of these neighborhoods,from which (2.3) follows.

With the uniform inequality (2.3) established, we are now ready toestablish the asymptotic stability of the time-varying system (2.1) when eis small and A(i) = 0(1). The last assumption, which corresponds toassuming that the variation in model parameters is slow, takes the followingexplicit form.

Assumption 2.3

Lemma 2.2, below, shows that the transition matrix of (2.1), denoted by0(r, 5), can be uniformly approximated by the exponential matrixexp[A(s)(t - s)/e], corresponding to a system in which the parameters of

o


A(i) are frozen at t = s. The lemma gives an exponentially decaying boundon the error

Lemma 2.2

Under Assumptions 2.1-2.3 there exist positive constants e*, a2 and K2

such that, for all

Proof The definition of W(f, 5) implies

Applying the variations of constants formula to (2.7), we obtain

Multiplying through by ? where 0 < or2 < arl5 and letting rj(t, s) = we obtain

Viewing r](t, s) as a functional on [5, t] with norm||r;|| = sup \\rj(t, r)||, (2.8)is rewritten in the form

We shall prove that T( •) is a contraction mapping over a set of the form

5.2 SLOWLY VARYING SYSTEMS 205

{rj : \\rj\\ ^ ep}, where p will be chosen later. By Assumptions 2.1-2.3, wehave

where we have used the fact that x2e~ax ^ 4/a2e2 V* ̂ 0. For ||r/|| < ep wehave

Combining (2.9) and (2.10), we get

Choose

where 0 < y < 1. Then, for all e < e*, \\T(rj)\\ < ep. Moreover,

Thus T(-) is a contraction mapping, and the unique solution of (2.8)satisfies

Lemma 2.2 confirms the intuitively expected result that the system (2.1)is uniformly asymptotically stable for sufficiently small e. The proof of the

o


lemma gives an estimate e* = (a{ — a2)2Y/fiKl such that for all e < e*

uniform asymptotic stability holds. The arbitrary positive constants a2 < &iand y < 1 can be adjusted to improve e* while maintaining a usefulexponentially decaying bound on \\<f>(t, s)|| (see Exercise 5.2). It is obvious,however, that within the analysis of Lemma 2.2 the best bound on eguaranteeing uniform asymptotic stability can be obtained by letting a2 — » 0and y-* 1 to get e* = ai/ftK^

An alternative bound on e guaranteeing uniform asymptotic stability canbe obtained by using a quadratic Lyapunov function v(t, z) = zTP(t)z. LetP(f) be the solution of the algebraic Lyapunov equation (2.11), para-meterized in f.

It can be shown (Exercise 5.3) that Assumptions 2.1-2.3 on /= [/o,00)guarantee that for every t > t0, (2.11) has a unique positive-definite solutionand that P(t) is bounded from above. Let

Therefore V is negative-definite for all e < (sup||P||2) *. The calculation of

these bounds is illustrated by the following example.

Example 2.1

Consider ez = A(t)z, where

The characteristic equation of A(t) is

For all a < 2 the eigenvalues of A(t) satisfy Re A < 0. Moreover, A(f) andA(i) are bounded for all t. Thus A(t) satisfies Assumptions 2.1-2.3. This,however, does not guarantee asymptotic stability if e is not small. For

5.2 SLOWLY VARYING SYSTEMS

example, when e = 1, 0(r, 0) is given by

207

which shows that for 1 < a < 2 the system is unstable even though Assump-tions 2.1-2.3 are satisfied. As e — » 0 Lemma 2.2 guarantees exponentialstability for all e< e*, where e* is estimated by Lemma 2.2 as £* =a\/flKl and by the Lyapunov function (2.11) as ef = (supHP^L)"1- Tocalculate e* we evaluate the exponential matrix

where

The matrix A(0 is given by

and ^ «V2. Therefore

taking it can be shown that


On the other hand, P(t) satisfying the Lyapunov equation (2.11) is givenby

Then

where tan 0 = 1(2 - fl), and

Therefore

The expressions for £* and ef vanish as fl—> 2, which is expected since theeigenvalues of A(f) tend to the imaginary axis as a—> 2. Comparison ofe* and e* shows that ef > e* for all 1 < a < 2. For example, at a = 1.1,e* = 0.042 and ef =0.671. The superiority of ef °ver £* is typical.

We conclude this section by showing that regular perturbations of theright-hand-side of (2.1) do not destroy the uniform exponential boundestablished by Lemma 2.2. Consider the system

where \\T(t, e)\\ ^ c3 Vr G /, and y > 0. Let 0^, 5) be the transition matrixof (2.13).

Lemma 2.3

Under Assumptions 2.1-2.3, there exist positive constants e*, a3 and K3such that for all e e (0, e*)

where 0(f, s) is the transition matrix of (2.1), which, by Lemma 2.2, satisfiesthe bound \\<b(L s}\\ < KQ-*'-^*.

5.3 DECOUPLING TRANSFORMATION 209

Proof

From (2.15) we have

Taking norms,

Multiplying throughout by ear3('-i)/e } where 0 < ar3 < a, and setting r/(r, 5)

fas) - 0(f, s)|| yields

By Gronwall's Lemma (Vidyasagar 1978), we have

5.3 Decoupling Transformation

In this section, we consider the singularly perturbed system

where A22(t) satisfies Assumptions (2.1)-(2.3) and Ay are smooth functionsof t; smoothness assumptions will be spelled out later. It is shown that, for


e sufficiently small, there exists a Lyapunov transformation that transforms(3.1) into a block-diagonal form, decoupling the slow and fast dynamics incomplete analogy with the similarity transformation (4.8) of Chapter 2.The decoupling transformation is derived in two steps exactly as in Chapter2. First, rj(f) = z(t) + L(t)x(t) is substituted, where L is chosen to satisfy

so that the lower left corner block is eliminated. Second, %(i) =x(i) — £H(t)r)(t) is substituted, where H is chosen to satisfy

so that the upper right corner block is eliminated. If there exist boundedcontinuously differentiable matrices L(t) and H(t), whose derivatives arebounded on /, which satisfy (3.2) and (3.3), then the transformation

is a Lyapunov transformation that transforms (3.1) into

It is left to the reader to verify that det T~l(t) = 1 and

Our task now is to show that, for sufficiently small e, there exist L(f) andH(t) with the abovementioned properties. It helps to notice that there areno boundary conditions imposed on (3.2) and (3.3). We are going to makeuse of this to simplify the solution of these equations. In particular,boundary conditions will be chosen to eliminate the boundary layers ofL(t) and H(t) so that as £ —> 0 they converge, uniformly on /, to the reducedsolutions obtained by setting £ = 0 in (3.2) and (3.3). These reducedsolutions are L0(?) = A22(t)A2l(t) and H0(t) = An(t)A22(t). Let us startby seeking the solution of (3.2) in the series form

Substituting (3.6) into (3.2) and matching coefficients of like powers in e,


we obtain equations determining L0, . . . , LN_l in the form

and so on. The remainder term RL satisfies an equation of the form

By imposing appropriate differentiability and boundedness conditions onthe matrices A^ and their derivatives, it can be established that (3.9) has abounded solution RL. This then shows the existence of L satisfying (3.2)

N-l

and gives ^ e'Lj as a uniform O(^} approximation of L. Equation (3.3)/=o

can be treated in the same manner. Its solution is sought in the form

HQ, . . . , //M_! are determined from the equations

etc., and RH satisfies an equation of the form

which under appropriate conditions has a bounded solution.The above discussions show that obtaining higher-order approximations

for the solutions of (3.2) and (3.3) is a mechanical repetition of the case offirst-order approximations. Therefore the next theorem finalizes theresult for first-order approximations. The reader is encouraged to work outthe case of second-order approximations (Exercise 5.4).


Theorem 3.1

Assume that A22(f) satisfies Assumptions 2.1-2.3, that A^t) are con-tinuously differentiable and bounded on / and that An, A2l and A22 arebounded on /; then there exists an e* > 0 such that for all e E (0, e*)there exist continuously differentiable matrices L and //, bounded on/, satisfying (3.2) and (3.3). Moreover, L(t) = LQ(i) + O(e) and H(f) =0(e).

Proof Continuing our discussion, we take N = M = 1. Then /•(•) and£,(•) of (3.9) and (3.13) are given by

We start by studying (3.9). Consider the integral equation RL(t) = SRL(t),where

where 0(f, r) is the transition matrix of ez = A22(i)z. Let !£ ={RL : \\RL\\ < p}, where ||jRL|| = sup||^L(Ol| and p will be chosen later. Usingthe matrix identity 7

and the boundedness of An, A12 and L0 on /, it follows that for RL1,RL2 G X

Using Lemma 2.2, there exists e * > 0 such that for e < e2

5.3 DECOUPLING TRANSFORMATION

Hence

Also, from the boundedness of An, A12, L0 and L0 we have

and for every RL£:!£

Employing Lemma 2.2 once again, we conclude that for e < e*

Choose p = 2KK2/o and e4* = mm(K3/2K4p, o/3KK3), then for all£<£*

It follows from (3.15) and (3.16) that, for e<e* = min e* , SRL is a1<(<4

contraction mapping on ££. Therefore by the contraction principle theintegral equation RL(t) = SRL(t) has a unique solution in \\RL\\ < p. It iseasily seen by differentiation of (3.14) that RL(t) is a solution of (3.9).Obviously, L = L0 + eRL is continuously differentiable and bounded on /.

We turn now to (3.13). Consider the integral equation RH(t) = TRH(t),where

Let 3t = {RH : \\R^\ < pj, where \\RH\\ = sup ||/?//(Oll- It can be easily verifiedthat 7

213


Hence, for any RHE.^i the use of Lemma 2.2 yields

Choose P! = 2KK6/o and £g = o/2KK5, then

Moreover,

Thus, for e < e* = min e* , 77?// is a contraction mapping on 26, and the

integral equation Rn(t\= TRH(f) has a unique solution in \\Rf^\ <= Pl. Noticethat in showing that TRH is a contraction mapping all the constants areindependent of f f, so t{ —>• o° is allowed. Finally, by differentiation of (3.17)it is seen that Rn(t) is a solution of (3.13).

The decoupling transformation derived here provides us with a usefultool that simplifies the analysis of singularly perturbed systems. In the restof this chapter, this transformation is employed to study stability, asymptoticapproximations, controllability and observability. In developing theseresults, the transformation (3.4) is used with L and H satisfying (3.2) and(3.3) respectively. Recalling the discussion of Section 2.4, we notice againthat even when L and H are functions of t the transformation (3.4) isnonsingular, independent of L and H. While in proving theorems it is moreconvenient to use L and H satisfying (3.2) and (3.3) so that the transformedsystem has zero off-diagonal blocks, in solving singular perturbation prob-lems it is sometimes more convenient to use (3.4) with L and H that areapproximations of the solutions of (3.2) and (3.3). For example, thetransformation

o


results in a system whose diagonal blocks are O(e) perturbations of (3.5),while the off-diagonal blocks are multiplied by e. Such a transformation isan exact one and holds for all e. This idea will be illustrated in the adaptivecontrol example treated in Section 5.5.

In all cases studied in this chapter A22(t) will be required to satisfy thestability Assumption 2.1, which is a reasonable requirement since boundarylayers must be asymptotically stable for the time-scale decomposition to beuseful. The stability Assumption 2.1, however, can be replaced by a moregeneral conditional stability assumption. That is, the eigenvalues of A22(f)are bounded away from the imaginary axis with a fixed number of them inthe right half-plane. In Chapter 6 a situation will arise where A22(t) is theHamiltonian matrix associated with optimal control problems, which isconditionally stable. We give here one possible generalization of Theorem3.1 that will be used in Chapter 6, where we restrict our attention tocompact time intervals. First, we make the following assumption.

Assumption 3.1

There exists a nonsingular continuously differentiable matrix W(t) such that

where ReA^^f)) < -c1 and ReA(Z)2(0) — C2-

Corollary 3.1

Suppose that Assumption 3.1 is satisfied and all the assumptions of Theorem3.1 hold on [t0, ?f] except that A22(i) does not satisfy Assumption 2.1. Then,there exists an e* > 0 such that for all e E (0, e*), there exist continuousdifferentiable matrices L(t) and H(t) satisfying (3.2) and (3.3). Moreover,L(f) = L0(0 + 0(e) and H(t) = H0(t) + O(e) Vr E [f0, f f].

Proof Substitute L = WL and H = HW~l in (3.2) and (3.3)respectively.

From this point on, the proof is a step-by-step repetition of that of Theorem


3.1, with the exception that the integral (3.14) is replaced by an integral ofthe form

where 0j( • , •) is the transition matrix of ez\ = Dl(t)zl and 02(' > ') is thetransition matrix of £Z2 = D2(t)z2. Notice that, by Lemma 2.2,||02(f, r)|| < Ke~a(r-')/£ for r> f. The integral (3.17) is modified similarly.

In fact, the transformation W(t) called for in Assumption 3.1 existswhenever the eigenvalues of a continuously differentiable matrix A22(t) arebounded away from the imaginary axis. For more details see Coppel (1967).

5.4 Uniform Asymptotic Stability

Uniform asymptotic stability of the singularly perturbed system

when £ is sufficiently small, is implied by the uniform asymptotic stabilityof the slow (reduced) system

and asymptotic stability, uniformly in t, of the time-frozen fast (boundarylayer) system

where t is treated as a parameter. Namely, if the transition matrix 0s(f, 5)of the slow system (4.2) satisfies the inequality ||0s(f,s)\\^ Kse~a^'~s),t^s, and the eigenvalues of A22(t) are in left half-plane uniformly in t, i.eReA(v422(0) — ~Ci < 0, then one would intuitively expect that for wellbehaved coefficients A^t), the singularly perturbed system (4.1) will beuniformly asymptotically stable as e—» 0. Theorem 4.1 states this result.

Theorem 4.1

Let all the assumptions of Theorem 3.1 hold on [f0, oo) and suppose that theslow system (4.2) is uniformly asymptotically stable. Then, there exists ane* > 0 such that for all e G (0, e*) the singularly perturbed system (4.1) isuniformly asymptotically stable. In particular, (f)(t, s), the transition matrix

5.4 UNIFORM ASYMPTOTIC STABILITY 217

of (4.1), satisfies

where K > 0 and a > 0 are independent of e.

Proof Transform the system into decoupled form (3.5). Let 0X( • , • ) bethe transition matrix of | = (An — AUL)^ and 0 2 ( - , • ) be the trmatrix of erj - (A22 + eLAu)r]. Then

The system | = (An - A12L)% is a regular perturbation of the uniforasymptotically stable slow system (4.2), so \\(f>i(t, s)\\< A'1e"(Tl('~5). ByLemmas 2.2 and 2.3, \\(^2(t,s)\\< K2&~a^''^£ . Thus, for sufficientlysmall e,

Theorem 4.1 is a fundamental result which holds asymptotically as e— » 0.In addition to this conceptual result, we calculate bounds e* such that theuniform asymptotic stability of the singularly perturbed system is guaran-teed for all e< e*. We can also correct the slow and/or fast models byretaining higher-order terms of the expansions of L or H. In the rest of thissection, two e* bounds are given and illustrated by an example, while themodel correction process is illustrated by an adaptive control example inthe next section.

Bounds e* will be calculated using Lyapunov methods. The first boundwill be calculated using a converse Lyapunov method with conceptualLyapunov functions. The second bound uses a direct Lyapunov methodwith quadratic Lyapunov functions.

Lemma 4.1

Suppose the assumptions of Theorem 4.1 hold, and let ||0S(M)||2 —K.e-'*'-'), 4>f(t, 5)||2 < K fe-<"<<-'>/*, M12(0l|2 ̂ Mlt ||L0(fMi2(Oll2 * M2

and || L0(f) -I- L0(t)A0(t) \\ < M3, where 0S( • , • ) and 0f( • , • ) are the transitionmatrices of (4.2) and (4.3) respectively. Then the singularly perturbesystem (4.1) is uniformly asymptotically stable for all e < e* , where

o


Proof Consider v(t, x,z) = alV(t,x) + a2W(t, x, z) as a conceptualLyapunov function candidate, where a^ > 0, a2 > 0,

and

whereIt can be shown (Exercise 5.5) that there are positive constants y1? y2, y3

and y4 such that

and

Thus v(t, x, z) is positive-definite and decrescent. Now,

Similarly,

5.4 UNIFORM ASYMPTOTIC STABILITY 219

Using (4.6) and (4.7), we obtain

v will be negative-definite if or: and a2 can be chosen such that

which is possible if

or

Lemma 4.2

Suppose the assumptions of Theorem 4.1 hold. Let Ps(f) >the Lyapunov differential equation

/„ > 0 satisfy

o

c1


and let Pf(t) > c2 Im > 0 be the solution of the algebraic Lyapunov equation

Then the singularly perturbed system (4.1) is uniformly asymptoti-cally stable for all e < el = l/(y + ^£2), where [\\Pft)\\2 +2\Pf(t)L0(t)Al2(t)\\2] < y, 2\\P,(t)A12(t)\\2 < ft and

The proof of Lemma 4.2 appears later as a special case of a general resultfor nonlinear systems in Section 7.5 (see Example 5.1 of Chapter 7).

Example 4.1

Consider the singularly perturbed system (4.1) with

The matrix A22(t) is a special case of Example 2.1 with a = 1.1.

and

The slow system is given by

We calculate a bound on e using Lemma 4.1:

Hence 0

l0(t)a0(t)

with

5.5 STABILITY OF LINEAR ADAPTIVE SYSTEM 221

Using expressions from Example 2.1, it can be shown that in ||-||2e\p[A22(t)9] satisfies inequality (2.3) with (xv = 0.45 and Kl = 3.0533,and ^22(0 satisfies Assumption (2.3) with 13 = 1.1. Thus, from Lemma2.2 (Exercise 5.2), (}>f(t,s) satisfies the inequality | 0f(f, s)\ ^K^-°^-s}le for all e< £0* = (cxl - a2)

2r//3Kl, where

The choice of a2 and y affects the e-bound in two ways. First, it affects £f

and crf, which are used to determine e* . Second, it affects EQ > which affectsthe final bound which is min(eo , £*)• Because of this, several choices of&i and y may be tried in order to improve the bound. Let us take <x2 -Q.5a} and take y = 0.787 such that Kf = 2K^ The corresponding value ofEQ is 0.0119. Calculating the matrix norms indicated in Lemma 4.1 in || • ||2,we obtain Af j = 1, M2 = 2.0495 and M3 = 11.6974; hence e? = 0.0045< e(* . Thus the singularly perturbed system is uniformly asymptoticallystable for all e< 0.0045. Since EQ is much larger than e* we mayrechoose y and/or a2 allowing a smaller value of EQ in order to improvee* . Let us take a2 as before, but take y = 0.48105 such that £f = 1.25 .̂The corresponding values of EQ and e* are 0.00724 and 0.00715 respec-tively. Hence the system is uniformly asymptotically stable for all£ < 0.00715. Now, we calculate a bound on e using Lemma 4.2.

From Example 2.1 we have expressions for Pf(t) and Pf(t), and calculatingtheir norms we obtain ||Pf||2 = 2.2292 and |Pf||2 = 1.4892. The constants y,/3t and fi2 can now be estimated as y = 10.6267, /^ = 0.5294 and /32 =52.152, leading to e* = 0.0267, which is more than thrice the boundobtained by Lemma 4.1.

5.5 Stability of a Linear Adaptive System

Problems with time-varying parameters naturally arise in schemes withcontinuous adjustment of parameters that are used for identification of, oradaptation to, unknown plants. Among those particularly popular are theso-called "model reference" schemes in which the same reference inputr(t) is applied to the system and to a reference model, and the error betweentheir output signals e(t) = y(t) — ym(t) is regulated to zero by parameteradjustment. For identification the parameter adjustment is performed on


the model, while in adaptive control the adjustable parameters are con-troller gains in the feedback loop.

A fundamental result of the adaptive control theory is that the iden-tification or adaptation process is uniformly asymptotically stable (u.a.s.)and e(f) —>• 0 as t —> o°, if (a) the reference input r(t) is persistently exciting,and (b) the order of the model is not lower than that of the plant. Thesecond assumption, however, can seldom be satisfied in applications wherethe order of the plant is unknown. A more realistic assumption is that onlythe order of the dominant (slow) part of the plant is known. Then themodel-plant mismatch can be attributed to the unmodeled parts of theplant, the so called high-frequency parasitics, and the effect of this mismatchcan be analyzed by singular perturbation methods. An example presentedhere will illustrate this possibility on a simple linear time-varying adaptivesystem. The result can be extended to most identification schemes, becausethey are also linear. Although extensions to other adaptive control systems,which are almost always nonlinear, are technically involved, the instabilitymechanisms revealed in this simple system are an inherent property of mostdirect adaptive control schemes.

The adaptive system shown in Fig. 5.1 is linear because the only adjust-able parameter is the feedforward gain k, and, ideally, the only unknownparameter is the plant gain g. The goal of adaptation is to adjust the overallgain kg to be equal to that of the model, gm. The ideal plant has the poles = -1 identical with that of the model, but the real plant has an additionalparasitic pole at s = — e . Without this parasitic pole, that is when e = 0,the ideal adaptation is exponentially stable with any nonzero input r(t),such as r(t) = R sin cot or even r = const ^ 0. Then e(t) —» 0, k(t)g—> gm asf-» 30. The robustness issue is whether this stability property will be pre-served in the presence of the parasitic pole, at least when this pole is large,

Fig. 5.1. A linear adaptive system.


that is when e > 0 is small. If the stability is lost at some e* and beyond,then it would also be useful to give an estimate of e*, and show how itdepends on system and signal properties. These issues are now addressedusing the results of the preceding sections in this chapter.

The equations of the adaptive system in Fig. 5.1 with g = 1 are

where y > 0 is the adaptation gain, which determines the speed ofadaptation: high for fast adaptation, and low for slow adaptation. It is clearthat the equations (5.2)-(5.4) constitute a third-order linear time-varyingsystem, with the signal ym(t) as its forcing term. Owing to linearity thestability analysis is performed for the unforced part of the system, rewrittenin the form of (4.1) with the matrices An, An, A21 and A22 as follows:

Since A22 = -1 is both nonsingular and Hurwitz, the corresponding assump-tions in the preceding sections are satisfied and we can immediately applythe decoupling transformation (3.18). We first evaluate L0 = A22A2l andH0=AUA22

1:

and substituting for L0 and for H0 in (3.18), we obtain

To simplify notation, instead of %2 = k we use k, and in |T = y + er] wedrop the subscript, £ = y + er]. In terms of f, A: and r\ the system (5.5),(5.6) becomes


Its advantage over (5.5), (5.6) is that (5.9), (5.10) is decoupled in the sensethat 77 appears in (5.9) multiplied by e and so do £ and k in (5.10). Had wchosen an O(e2} approximation of L and H, rather than L0 and H0, thisdecoupling property would be enhanced to O(e2). However, the O(e)decoupling of (5.9), (5.10) is sufficient to reveal the instabilities that canoccur in fast and slow adaptation due to the presence of e.

By Theorem 4.1 the u.a.s. of (5.9), (5.10) for £ small follows from theu.a.s. of the slow and fast systems. The fast system is ezf = — zf, which isobviously u.a.s. The slow system obtained from (5.9) by letting e = 0,

can be analyzed using the Lyapunov function

whose derivative along its solutions is i>s = -y2. This proves that (5.11) isuniformly stable and, moreover, ys(t)-*Q as t—»°°. To prove that alsoks(i)-> 0, we note from the first row of (5.11) that if i>s = -yl = 0 thenalso rks = 0. Therefore, if after ys(t) has converged to zero, the signal r(f)remains different from zero ("persistently exciting"), the fact that i)s < 0can only be satisfied by ks = 0 establishes that (5.11) is u.a.s. if r(f) is notidentically zero. Our conclusion from Theorem 4.1 applied to the adaptivesystem (5.5), (5.6) is as follows.

Corollary 5.1

If r(t) and f(i) are bounded for all t ̂ 0 and r(t) is different from zero foralmost all t ̂ 0, then there exists an e* such that the unforced system (5.5),(5.6) is exponentially stable for all e E (0, e*], and hence all the variablesin the adaptive system (5.2)-(5.4) are bounded.

We see an impact of parasitics already. When e > 0 in (5.2)-(5.4), thetracking error e(t) = y(t) - ym(t) will, in general, remain nonzero as t —> <»,because the responses of a first-order model and a second-order systemcannot exactly match. A further analysis would show that for small e thetracking error will be small, if also \r\ is O(l) or less. A more significantimpact of the parasitic pole is that it can lead to instabilities in both fastand slow adaptation. These phenomena will now be demonstrated by ananalysis of (5.9) with rj = 0.


Proposition 5.1

The system (5.9) with rj = 0 will be unstable if

The proof follows from the classical result that the determinant A(r) ofthe state transition matrix of the system x = A(f)x satisfies the equationA = (trace^4(r))A. The trace of the matrix in (5.9) is -1 + eyr2, andwhen (5.13) is satisfied the corresponding A(?) tends to infinity. Anotherconclusion from (5.9) for r = const, r = 0, is simply derived from its charac-teristic equation

in which case the condition

is both necessary and sufficient for the exponential stability of (5.9).Treating y7?2 as the average, and yr2(f) as the instantaneous adaptationgain, we interpret (5.13) and (5.15) as the bounds on the adaptation speedimposed by the parasitic pole -e~l. Hence the mechanism of the fastadaptation instability is triggered by high gain yr2.

A remarkable property of this adaptive system is that it can be unstableat a very low gain yr2, if r is sufficiently large. A detailed analysis showsthat for r = sin cot and with y sufficiently small, the stability bound for slowadaptation is

that is, the input frequency must be lower than the square root of theparasitic pole.

The results of this analysis are illustrated by a simulation of the responsesof the system (5.5), (5.6) with r = sin cot for three different sets ofparameters. In all cases the parasitic pole is —10, that is, e = 0.1.

The response in Fig. 5.2(a) for y = 1 and co = 1 is exponentially stable,that is, the bound (5.13) is not violated and the Corollary 5.1 holds.However, when y is increased an important instability occurs in Fig. 5.2(b),where y = 24, co = 0.5 violate the bound (5.13). If, instead, y is kept low,but co is increased, a creeping instability is noticed along the axis y = 0; seFig. 5.2(c), where y = 1, co = 4 violate the bound (5.16). In conclusion, letus point out that it was the transformation of the original system (5.5),(5.6) into its decoupled form (5.9), (5.10) which led to the discovery of the


Fig. 5.2. Response of adaptive system.

two types of instability. Even a cursory inspection of the slow system matrixin (5.9) points to the two causes of instability: the terms eyr2 and er, which,in fact, appear in the stability bounds (5.13), (5.15) and (5.16). A similarqualitative analysis suggested that the instabilities can be counteracted byadding a term -ok in the adaptive law (5.4). In Fig. 5.2(d) the responseof Fig. 5.2(c) is stabilized by choosing o = 0.05. In Exercise 5.8 the readeris asked to repeat our analysis with this modification.

5.6 State Approximations

Paralleling the time-invariant treatment of Section 2.5, let us consider thesingularly perturbed system

5.6 STATE APPROXIMATIONS 227

where the input u(f) is a continuous bounded function of t £ / and theinitial state is given by x(tQ) = x° and z(/0) = z°. The solution of (6.1) forsufficiently small e can be approximated by employing the decouplingtransformation (3.4). Application of (3.4) transforms (6.1) into

The solution of (6.2)-(6.5), when transformed back using the inverse of(3.4), yields the exact solution of (6.1). Asymptotic approximations to thesolution of (6.1) with O(eN) error can be obtained by retaining O(sN)approximations of the right-hand-side coefficients of (6.2)-(6.5). In otherwords, the decoupling transformation of time-varying singularly perturbedsystems, similar to its time-invariant counterpart of Section 2.5, reducesthe approximation problem to a problem with regular perturbations ofright-hand-side coefficients. Recalling from Section 5.3 that MacLaurinexpansions for L and H can be easily constructed, we have now an efficientmethod for deriving asymptotic approximations for the solution of (6.1).The derivation of O(e) approximations is particularly simple, since in thatcase L and H can be approximated by L0 = A22A2i and H0 = AUA22 ,and all O(e) terms on the right-hand sides of (6.2)-(6.5) can be dropped.This process yields

where (6.7) is written in the fast time scale r= (t — t0)/e. The reader shouldnotice that (6.6) and (6.7) are the slow and fast systems derived by formallyneglecting E in (6.1).

Theorem 6.1

Suppose for all t E [t0, tf] that the assumptions of Theorem 3.1 hold andBi(t), /= ! , 2, and v(t) are continuous. Then there exists an e* > 0


such that for all e E (0, £*] the following expressions hold uniformly on[*o,'f]:

Moreover, if the conditions of Theorem 4.1 hold and B^t) and v(t) arecontinuous and bounded, then the above expressions hold uniformly on

Proof From Lemmas 2.2 and 2.3 the transition matrix of the homo-geneous part of (6.3) satisfies the inequality ||0(f, s)\\ < Ko.'a(t's^E. Usingthis, it is obvious that the contribution of the input term eLB^u is 0(e).

Using again Lemmas 2.2 and 2.3, we see that the homogeneous part of(6.3) can be approximated by A22(t^)r](i), which yields

Comparison of (6.2), (6.4) with (6.6) and use of regular perturbation results(Hale 1980) yields

Using the inverse of the decoupling transformation (3.4), we get

which complete the proof for finite-time intervals. The case of infinite timefollows immediately upon utilizing Theorem 4.1.

The derivation of O(e2) approximations is given in Exercise 5.9.

o

5.7 CONTROLLABILITY 229

5.7 Controllability

In Section 2.6 we have seen that controllability of the slow and fast systemsimplies controllability of the full singularly perturbed time-invariant systemwhen e is sufficiently small. In this section we show that the same resultholds for time- varying systems. We actually go beyond that and study theasymptotic behaviour of the steering control that moves the state of thesystem from a given initial point to a given final point in the state space.

Returning to the singularly perturbed time- vary ing system (6.1), definedon a finite-time interval [f0, ff], let us associate with this system the slowsystem

and the fast system

The fast system is a time-invariant system at a frozen time t* E [tQ, tf]. Theslow system (7.1) is controllable on [t1, t2] C [t0, tf] if for any given jc° andxl there exists a finite energy control ws(f) that moves the state of the systemfrom *s at t = tl to xf

s at t = t2. This is equivalent to the following assumption.

Assumption 7.1

where 0 S ( - , - ) is the transition matrix of (7.1) and Ws(ti,t2) is the con-trollability Grammian on [t1, t2]. The control that achieves this task withminimum energy is given by

Similarly, the fast system is controllable if the following assumptionholds.

Assumption 7.2

Rank[B2(t*),A22(t*)B2(t*),...,A>?2-i(t*)B2(t*)] = m,

and the minimum energy control that moves the state of the system from


Notice that [r1? T2] could extend to infinity, which is permissible since weare working with Re A(A22(f*)) < 0 (a precise statement of conditions isgiven in Theorem 7.1 below).

To study the controllability of the full singularly perturbed system (6.1),we use the decoupling transformation (3.4) to transform it into

where Bs = B1 - eHLBl - HB2 and Bf = B2 + eLB^ Let 0^ • , • ) be thetransition matrix of (7.6) and let 02( • , • ) be the transition matrix of (7.7).Using Lemma 2.3, a modified version of Lemma 2.2 which is given inExercise 5.15, Theorem 3.1 and the continuous dependence of 0j( • , • ) onparameters, we have

Furthermore, assuming that #i(-) and fi2(-) are continuously different-,iable, we have

The controllability Grammian of (7.6), (7.7) is given by

where


We study the components of the matrix W:

232

Therefore

5 LINEAR TIME-VARYING SYSTEMS

which is positive for sufficiently small e. Thus we have proved the followingtheorem.

Theorem 7.1

Suppose that the assumptions of Theorem 3.1 hold on [t0, ff], that Assump-tions 7.1 and 7.2 (at t* = t2) are satisfied, and that Bfa) are continuouslydifferentiable. Then there exists an e* > 0 such that for all e E (0, £*) thesingularly perturbed system (7.1) is controllable on the interval [tl, t2].

It is worth emphasizing that the controllability of the fast system isrequired only at the terminal time t2.

Let us continue with our analysis to find the minimum energy controlthat steers the state of the system from (*°, z°) at t = ti to (*f, zf) at t = t2.This control is given by

where

Seeking v in the form


it can be easily verified (Exercise 5.12) that

Substituting back in (7.13) yields

Examining (7.15) shows that the dominant part of u(t) comprises twoterms. The first term contains 0S(J2, t) as a multiplier while the second termcontains exp{^422(r2)(r2 - t)/e} as a multiplier. The second term creates aboundary layer at t = t2 which is significant only within an O(e) interval oft2. Denoting the first term by ws for slow control, and the second term bywf for fast control, the expression for w, dropping O(e), can be rewrittenas

where

It is interesting now to see that the slow and fast controls can be intuitivelyderived from the slow and fast systems in the following manner. Postulatingthe control as the sum of the slow and fast components as in (7.16) wherewf decays exponentially as (t2 - t)/e-> «>, we see that the effect of wf onslow dynamics is O(e). Neglecting fast dynamics and dropping wf, we arriveat the slow model (7.1). The solution of the slow control problem, givenby (7.3), with x°s = x° and x[ = xf yields the slow control MS as in (7.17).


To see the effect of the postulated control on the fast dynamics we expressthe second equation of (6.1) in the fast time scale r= (t — t2)/e and neglectterms O(e) to get

Removing the slow bias by defining zf as

we arrive at the fast model (7.2) with t* = t2. Notice now that, without thepresence of the fast control, the terminal value of z is determined by itsslow bias. The role of the fast control is to force z to terminate at zf.Therefore the fast control problem is solved to steer zf from an initial valuez? = 0 to a final value zf = zf + A^ (t2)[A2l(t2)x

f + B2(t2)us(t2)].Moreover, since the fast control problem is solved in the fast time scaleT = (t - t2)/e, the terminal time is i2 = 0, while the initial time —> -°° as£—> 0 so that we can take TJ = -°°. With these values the solution of thefast control problem, given by (7.4), yields «f as in (7.18). Figure 5.3 givesa pictorial representation of the trajectory. The trajectory consists of threeparts, which, using guidance terminology, we called climb, cruise anddescent arcs. The cruise arc is determined by the solution of the slowcontrol problem to move the slow variable from x° to x{. The climb arc isan initial boundary layer due to the deviation of z° from the slow valueof z determined by the cruise arc. Since the fast system is open-loopasymptotically stable, the initial boundary layer decays exponentially in thefast time scale (t - t^/e. If xi is the only imposed terminal condition wecan allow the trajectory to continue with the cruise arc until it hits the value

Fig. 5.3. State trajectory for minimum energy control.


x = xf. However, when z(ff) = zf is imposed as well, a fast control in thefast time scale (t - t2)/£ is applied forcing the trajectory to follow thedescent arc towards the target point. Of course, owing to O(e) perturbationsin deriving the control, there is no guarantee that the trajectory will hit thetarget exactly. It will be, however, within an O(e) neighborhood of thetarget which is represented by a dotted circle in Figure 5.3.

The calculation of the steering control is illustrated by a simple time-invariant example.

Example 7.1

Consider again the normalized DC-motor model of Example 3.1 of Chapter3:

where x, z and u are the speed, current and controlling voltage respectively.It is desired to steer the state from *(0) = z(0) = 0 to *(1) = 1, z(l) = 0.The steering control problem corresponds to achieving a desired speed ata given time while regulating the current to zero at that time.

The slow control problem is defined by

and the slow control is given by

The fast control problem is defined by

and the fast control is given by

The total steering control is given by

The application of the control (7.19) to the system brings (x(t), z(f)) towithin O(e) from the target point (1,0), for sufficiently small e. To geta better feeling for the deviation from the target point we calculate(*(1), z(l)) for e = 0.1, 0.05, and 0.01 (Table 5.1).

The results confirm that the final point is within an O(e) neighborhood


TABLE 5.1

e = 0.1 e = 0.05 e = 0.01

x(l) 0.804 0.903 0.98z(l) 0.039 0.017 0.003

of the target. If 10% error is tolerable then e = 0.05 is small enough forthe control to be successful. If 2% error is tolerable then e = 0.01 is smallenough. The results also show that, in this example, for e = 0.1 the errormight not be tolerable. In this case the control will have to be corrected toaccount for O(e ) terms that have been neglected throughout the derivations.To account for 0(e) terms the slow and fast models will have to be correctedby including higher-order terms of L, and the effect of boundary conditionswill be corrected by including higher-order terms of H. Also, the effect ofthe fast control on the slow system will have to be taken into consideration.Since correction by including higher-order terms of L and H is more orless straightforward, let us illustrate the correction by taking into con-sideration the effect of the fast control on the slow system. This correction,alone, might be enough to achieve the required accuracy and it is par-ticularly interesting since it does not change the structure of the control.

In calculating the slow component of the postulated control us + u{ weneglected the effect of wf as O(e). In the current example, when e = 0.1,the application of «f = -2.626er to the slow system results in a deviationof *s(l) that is -0.269. The error in jcs(l) may be compensated for byshifting the boundary condition x{ to 1.269. Recalculating the slow controlfor boundary conditions jcs(0) = 0, *s(l) = 1.239, we obtain

The fast control problem is resolved with boundary conditions xf(-*) = 0and ;cf(0) = -(-1.239 + «s(l)) = -1.627 to obtain

The total steering control is given by

Application of (7.20) to the system (e = 0.1) results in

which improves *(1) a lot without affecting z(l) very much.


So far we have required that A22(t) satisfies Assumption 2.1, whichguarantees that the fast system is open-loop asymptotically stable. If thiscondition is not satisfied but the fast system is controllable for everyt E [ t l , t2] and the fast state z is available for feedback, then it is possibleto apply a preconditioning state feedback to stabilize the fast system. Thisresult is summarized in the following corollary.

Corollary 7.1

Suppose that all the assumptions of Theorem 7.1 hold except that Assump-tion 7.2 holds for all t* E [ f l 5 1 2 ] instead of t* = t2 only, and ^22(0 satisfiesdet [^22(0] ̂ 0 instead of Re A(/122(?)) =£ -q. Moreover, suppose that z(f)is available for feedback. Then there exists an e* > 0 such that for all£E(0, e*) the singularly perturbed system (6.1) is controllable on theinterval [^, t2].

Proof There exists a continuously differentiable matrix K(t) such that

Application of the state-feedback control law

results in

The state-feedback control (7.21) does not affect the controllability of thefull singularly perturbed system nor the controllability of the slow system.To see the latter claim notice that the slow system of (7.22) is given by

where

Using algebraic manipulations similar to those of Chapters 2 and 3, it canbe shown (Exercise 5.13) that


where

Thus (7.23) is related to (7.1) via the transformation

where Y is nonsingular. Since controllability of (7.1) is invariant withrespect to the transformation (7.25), it follows that (7.1) is controllable ifand only if (7.23) is. Therefore the system (7.22) satisfies all the assumptionsof Theorem 7.1 and the result follows.

Example 7.2

Consider the system

where A22 = 1 is not Hurwitz. It is desired to steer the state from *(0) =z(0) = 0 to jc(l) = 1, z(l) = 0. The first step is to stabilize the fast systemby applying the feedback control

so that the closed-loop system is

Taking K = -2, the problem becomes identical with Example 7.1, wherev is given by (7.19). Thus

5.8 Observability

In view of the observability properties studied in Section 2.6 and thecontrollability properties of time-varying systems established in the previoussection, it should come as no surprise that the observability of a time-varying singularly perturbed system is implied, for sufficiently small e, bythe observability of the slow and fast systems. Such a result is given inTheorem 8. 1 below. First, let us recall the singularly perturbed time- varying

5.8 OBSERVABILITY 239

system (6.1) with u = 0 and the observed output

on a finite time interval [r0, tf]. The slow system is obtained, as usual, bysetting e = 0 and eliminating z:

The fast system at some time instant t* E [t0, tf] is obtained by expressing(6.1), (8.1) in a fast time scale r = (t - t*)/e, setting e = 0 in coefficients(hence x = constant), and eliminating the constant bias from z and y bydefining

The resulting fast system is given by

which is a time-invariant system at a frozen time t*. Let us make thefollowing assumptions on the slow and fast systems.

Assumption 8.1

where 0S(- , •) is the transition matrix of (8.2) and Ms(- , •) is its observ-ability Grammian on [f1? t2]>

Assumption 8.2

Theorem 8.1

Suppose that the assumptions of Theorem 3.1 hold on [f0, rf], that Assump-tions 8.1 and 8.2 (at t* = tj are satisfied, and that C^r) and C2(0 arecontinuously differentiable. Then there exists an e* > 0 such that for all

rank


eG (0, e*) the singularly perturbed system (6.1), (8.1) is observable on

Proof The proof is dual to that of Theorem 7.1. Application of thedecoupling transformation (3.4) to (6.1), (8.1) yields

where Cs = Q - C2L and Cf = C2 + ̂ Q - C2L)#.The observability Grammian of (8.6)-(8.8) on [tlt t2], as in Section 2.6,

takes the scaled form

where

and

(8.9)

is the observability Grammian of the fast system (8.7) on [0, 7]. Hence

which is positive for sufficiently small e.

As a consequence of Theorem 8.1, we know that, for sufficiently smalle, the initial states Jt^) = x° and z(^) = 2° can be obtained from theobservations y(t), ti^t<t2. In particular, using the decoupling trans-formation (3.4), we have

[t1, t2].

o

5.8 OBSERVABILITY

and (Kwakernaak and Sivan, 1972)

241

where (f)^-, •) and 02(-, •) are transition matrices of (8.6) and (8.7)respectively. Repeating arguments from Section 5.7, it can be shown that

where y{(i) = y(tv + er) - C0(ri)|0.

Defining #° to be the initial state of xs, as determined from the slowmodel (8.2) using y-observations on [f1? r2], we get

Now, replacing A:^) in the definition of the fast system (8.5) and of thefast variables (8.3) and (8.4) by its estimate x®, and defining z\ to be theinitial state of zf as determined from the fast model (8.5) and y{ observationson [0, (t2 - O/e], we get


Thus from (8.13), (8.14) and (8.16)-(8.19) it is obvious that

What we have just described is an intuitive procedure to obtain O(e)approximations of the initial state x° and z° from the ^-observation on[ t l , t2]. First, to obtain the initial state x°, we neglect the fast dynamics anduse the slow model to obtain x®, an O(e) approximation of x°. Second, thefast model is defined by assuming that x is constant with value x^, and isused to determine the initial state of the pure fast variable zf. Finally, anO(e) approximation of z° is obtained by adding to z° the initial value ofthe slow component of z. The procedure is illustrated by the followingexample.

Example 8.1

Let us again return to the familiar DC-motor model

where y is observed on [0,1].The slow system, obtained by setting e = 0, is

The observability Grammian Ms(0,1) is

and an O(s) approximation of x° is given by

== ®Using x(er) == x® Vr, the fast system is given by

5.9 EXERCISES 243

where zf = z + x® andyf = y + x®. The observability Grammian Mf(0,20; 0)is

and

Notice that in calculating z° we have replaced Mf(0,1/e; 0) by Mf(0, °°; 0),which results in an O(e) error. The initial states x° and z° are approximatedby

The errors in calculating x° and z° for A:° = z° = 1 and e = 0.1, 0.05 and0.01 are shown in Table 5.2. The numerical results in Table 5.2 show anO(e) error (in this case the error ^2e).

TABLE 5.2

0.1 0.05 0.01

0.805 0.902 0.9800.195 0.098 0.0200.805 0.902 0.9800.195 0.098 0.020

Exercise 5.1

Let

5.9 Exercises

(a) Verify that ^4(0 satisfies Assumption 2.1 but not Assumption 2.2.(b) Calculate Qxp[A(t)6] as a function of t and show that an inequality

like (2.3) does not hold.


Exercise 5.2

Let (j>(t, s), KI, a2, y and e* be as in the proof of Lemma 2.2. Show thatfor all E < e*

Exercise 5.3

Let P(t) = Jo [eAWT]T[eXWr] dr, and assume thaM(f) satisfies Assumptions2.1-2.3 on [r0, °°). Show that for all t > tQ

Exercise 5.4

Consider (3.2) and (3.3) and let L = L0 + eL^ + E2RL and H = H0 +EHl + e2RH, where L0, L l5 HQ and H1 are defined by (3.7), (3.8), (3.11)and (3.12) respectively.

(a) Verify that RL and RH satisfy equations of the form (3.9) and (3.13).Give expressions for /\, /2, gi and g2-

(b) Similarly to the proof of Theorem 3.1, show that the integral equations(3.14) and (3.17) have bounded solutions for sufficiently small e. Hence

Exercise 5.5

Verify inequalities (4.4) and (4.5).

Exercise 5.6

This is an extension of Lemma 4.2. Suppose the assumptions of Theorem 4. 1

5.9 EXERCISES 245

hold. Let Fs(f) > c/n > 0 and Pf(f) > clm > 0 satisfy the Lyapunov equations

where <2s and Qi are constant nonsingular matrices. Let y,defined by

and /?2 be

Show that the singularly perturbed system (4.1) is uniformly asymptoticallystable for all e < l/(y + ̂ 2).

Hint: Apply the state transformation

to (4.1), and use Lemma 4.2.

Exercise 5.7

Consider the singularly perturbed (4.1) with

where y, cr and to are positive constants.(a) Verify that the assumptions of Theorem 4.1 hold.(b) Verify that the Lyapunov equations of Exercise 5.6 are satisfied with

(c) Using Exercise 5.6, calculate a bound e* such that the system isuniformly asymptotically stable for all e < e*.


Exercise 5.8

Instead of (5.4), perform the analysis of Sections 5.5 for the adaptation lawk= -y(y - ym)r - ok and explain the plot of Figs. 2(c), where o = 0.05.

Exercise 5.9

The purpose of this exercise is to derive an O(e2} approximation of thesolution of (6.1). For convenience set u = 0.

(a) Seeking the solutions of (6.2) and (6.3) in the series form

show that £o, £1, r/o and rjl are given by

(b) Show that, under the assumptions of Theorem 6.1,

Notice that H l need not be calculated; only L0, Ll and H0 are needed tocalculate an O(e2) approximation. Will this observation extend to the case


Exercise 5.10

Apply the results of Exercise 5.9 to the singularly perturbed system ofExercise 5.7. Take JCOT = [1 1], z° = 1.

Exercise 5.11

Apply the results of Exercise 5.9 to the singularly perturbed system ofExample 4.1. Take x° = 1 , z°T = [0 0] .

Exercise 5.12

Verify expressions (7.14).

Exercise 5.13

Verify expressions (7.24).

Exercise 5.14

Use the results of Section 5.7 to calculate a control that steers that state ofthe system

from x° = 0, z? = 0, z°2 = 0 to an O(e) neighborhood of xf = 1, z{ = l,z\ = 1.

Exercise 5.15

In Lemma 2.2 we saw that <f>(t,s), the transition matrix of (2.1), can beapproximated by exp[A(s)(t - s)/e\. In this exercise it is shown that <j>(t, s)can also be approximated by exp[A(t)(t - s)/e]. Let W(r, 5) = 0(f, s)-exp[A(t)(t,s)/e\. Show that W(r, s) satisfies inequality (2.6).

Hint show that *V(t, s) is given by

and proceed as in the proof of Lemma 2.2.



Stability properties of slowly varying systems have been studied in variousreferences. Versions of Lemma 2.2 are given in the books by Coppel (1965),Brockett (1970) and Vidyasagar (1978). Our treatment is patterned afterCoppel's, although the contraction mapping proof is different from his.One aspect that is emphasized in our treatment is the calculation of boundson e. Slowly varying systems have also been studied under conditionalstability assumptions (e.g. Assumption 3.1); for this, the reader may consultCoppel (1967), Chang and Coppel (1969) and Skoog and Lau (1972).

The main sources for the decoupling transformation of Section 5.3 areChang (1969, 1972). The contraction mapping proof is similar to Chang'salthough the technical nature of the result is slightly different. Changemploys a conditional stability assumption, while Theorem 3.1 assumesthat the eigenvalues of A22 have negative real parts. The uniform 0(e)approximation in Theorem 3.1 is stronger than Chang's results because ofstronger differentiability assumptions. A multiparameter version of thedecoupling transformation is given in Khalil and Kokotovic (1979a, b) andLadde and Siljak (1983) and Abed (1985a). The uniform asymptotic stabilityresult of Theorem 4.1 is as in Wilde and Kokotovic (1972a). It also followsfrom various results on uniform asymptotic stability of nonautonomousnonlinear singularly perturbed systems, similar to those of Chapter 7. Theuse of the decoupling transformation results in the present short proof.The stability bound of Lemma 4.1 is due to Javid (1978b), although ourderivation is different from his. Also, the stability bound of Lemma 4.2 istaken from Saberi (1983).

The use of singular perturbations to analyze robustness of adaptivesystems with respect to unmodeled high-frequency parasitics is treatedagain in Section 7.4. For a broader perspective, the reader may consultloannou and Kokotovic (1983, 1984a, b).

The controllability Theorem 7.1 can be found in Sannuti (1977). Thecalculation of the asymptotic form of the steering control is new, althoughKokotovic and Haddad (1975a) give similar calculations for time-invariantsystems. The feedback control and observer designs for linear time-varyingsystems, which are not covered here, can be found in O'Reilly (1980).

6 OPTIMAL CONTROL

6.1 Introduction

In optimal control problems on sufficiently long intervals, boundary layerphenomena occur even if the system is not singularly perturbed. In thiscase, the time scales are due to the different nature of the control tasksnear the ends of the interval and over the rest of the interval. In a minimumenergy problem, for example, the requirement to satisfy the initial andend conditions dominates at the ends of the interval, while the energyminimization requirement is dominant over the rest of the interval. Optimaltrajectories of this kind have a familiar "take-off", "cruise" and "landing"pattern, with the boundary layers being the "take-off" and "landing" parts.For fast subsystems of singularly perturbed systems the intervals are alwayslong compared with their dynamics. Their boundary layers are super-imposed on the solution of the slow subsystem.

In this chapter, optimal control problems for singularly perturbed systemsare approached by a preliminary study in Section 6.2 of boundary layersappearing in systems that are not singularly perturbed. This section iscrucial for the understanding of the rest of the chapter. After a definitionof the reduced problem in Section 6.3, we apply the results of Section 6.2to near-optimal linear and nonlinear control in Sections 6.4 and 6.5, andto "cheap control" and singular arc analysis in Section 6.6.

6.2 Boundary Layers in Optimal Control

When a control problem is formulated over a finite time interval t G [0, T],the determination of its slow and fast parts depends on the length of theinterval T. In this section, we consider that T is long by comparison with

249

A

the dynamics of the system. By this we mean that the time needed totransfer the state of the system through a distance of 0(1), using a control0(1), is small compared with T. In this case, the solutions of many optimalcontrol problems consist of boundary layers, and their properties are similarto those of singularly perturbed systems. Let us display these properties byanalyzing a terminal control problem for the system

which is not singularly perturbed. A control v(t) is sought so as to drivethe state f](f) from initial state ^0 at r = 0 to a prespecified state 77 1? at aprespecified time t = T, while minimizing the cost functional

Treating T as the unit of a new time variable t and denoting the functionsof t without the tilde,

If T is long with respect to the system dynamics, that is, if

is small, then (2.4) is a singularly perturbed system analyzed in Section 5.1as a "slowly varying system."

In the standard optimality condition for (2.4), (2.5), the Hamiltonian

is minimized by the control

50 6 oPtimal contral

6.2 BOUNDARY LAYERS IN OPTIMAL CONTROL

where p(t) satisfies the adjoint equation

251

Substitution of (2.8) into (2.4) and (2.9) results in the boundary valueproblem

where 5(0 = B(t)R-l(i)BT(t) and Q(t) = <?(i)C(t}. An analysis of this prob-lem is made under the following assumptions.

Assumption 2.1

The derivatives A(i), B(i), C(t) and R(t) are continuous for all t G [0,1].

Assumption 2.2

The eigenvalues X% of the Hamiltonian matrix,

lie off the imaginary axis; that is, there exists a constant c such that

Assumption 2.3

The pair A(t), B(t) satisfies the "frozen t" controllability condition, that is,

Under these conditions, the solution of (2.10) has the boundary layerproperties established in Theorem 2.1. We approach this theorem byproving a sequence of three lemmas which restate some well known resultsfrom linear optimal control theory.

Lemma 2.1

Under Assumptions 2.1-2.3, the time-dependent algebraic Riccatiequation

252 6 OPTIMAL CONTROL

has a root K = P(t) > 0 and root K = N(t) < 0 such that the relationships

hold for alUe[0, 1].

Proof Noting that P(t) > 0 with its property (2.16) is a stablizing solu-tion of (2.14) for each fixed t, the proof follows from established results oflinear control theory (e.g. Anderson and Moore, 1971). To see that thesame results apply to N(f) < 0, and to prove (2.15), (2.17) and (2.18),consider the equation

which is the replica of (2.14) with -A(t) used instead of A(t). Since (2.13)remains valid for -A(t), there exists a stabilizing solution PM(f) > 0 of(2.19). Letting K = -M, we identify N(f) = -PM(t)<0 as the matrixsatisfying (2.14), (2.15) and (2.17), noting that (2.17) and (2.18) expressthe well known fact that the eigenvalues A'^ appear in pairs symmetric withrespect to the imaginary axis.

Lemma 2.2

Under Assumptions 2.1-2.3, there exists an et > 0 such that for alle E (0, e^] two matrices, Pe(t, e) and Ne(t, e), satisfy the differential Riccatiequation

and are O(e) close to the solutions P(t) and N(f) of the algebraic equation(2.14), that is,

Proof Substituting

o

6.2 BOUNDARY LAYERS IN OPTIMAL CONTROL 253

into (2.20), we obtain

where N(t), as a solution of (2.14), is bounded because of Assumption 2.1.The existence of N£(t, e) and its property (2.21) are then established usingthe same argument as in the proof of Theorem 3.1 in Chapter 5. The proofis based on the fact that -A(t) + S(i)N(i) is Hurwitz for each tE [0,1].The only difference from the proof of Theorem 3.1, Chapter 5, is that,instead of the state transition matrix for -A + SN, in this proof we use theproduct of the state transition matrices for -A + SN and (-A + SN)T. Toprove the existence of P£(t, e] and its property (2.21), we repeat the sameargument considering that (2.20) is integrated in reverse time 1 - t startingat 1 - t = 0, and making use of the fact that A(f) - S(f)P(t) is Hurwitz foreach/£ [0,1]. D

Lemma 2.3

Under Assumptions 2.1-2.3, there exists an £2 > 0 such that for alle E (0, e2] the transformation

is nonsingular for each t E [0, 1] and transforms the system (2.10) into twoseparate systems

which are an expression of the exponential dichotomy of (2.10). Namely,there exist constants c> 0 and k > 0 such that for all t E [0, 1], e E (0, e2]the solutions of (2.25) and (2.26) are bounded by

Proof The nonsingularity of (2.24) follows from (2.15) and the propertyof the determinants of partitioned matrices, which, when applied to (2.24),yields


In view of (2.15) and (2.21), there exists an e2 > 0 such that det[Nf - PE]=£0 for all £6(0, e2]. The fact that (2.24) applied to (2.10) results in(2.25), (2.26) is verified by direct substitution. Finally, replacing P£(t, e) byP(t) + O(e) and Ne(t, e) by N(i) + O(e) in (2.25) and (2.26) respectively,and using Lemmas 2.1-2.3, Chapter 5, proves (2.27).

We are now ready to display the boundary layer behavior of the solutionYI, p of (2.10) for e sufficiently small, that is, for T sufficiently large.

Theorem 2.1

Under Assumptions 2.1 and 2.3, there exists an e3 >0 such that for alleG (0, £3] the solution r/(r), p(f) of (2.10) and the optimal control v(f)satisfy

where T and a are the fast time variables

and T]I(T) and r]r(a) are the solutions of the systems

referred to as the initial and terminal boundary-layer systems respectively.

Proof By Theorem 6.1, Chapter 5, and Lemmas 2.1-2.3, the solution/(r) of (2.25) for /(O) = r?0 and ^(T) defined by (2.33), and the solution r(t)of (2.26) for r(l) = r]l and r/X^7) defined by (2.34), satisfy for all t E [0, 1]and all e G (0, e3]

Furthermore, P(t)r],(T) = P(0)rjfa) + O(e) and N(t)rjr(o) = N(l)r]r(o)+ O(e), Analogous expressions hold for R~l(t)BT(t)pi(T) and

o


R-l(t)BT(t)pr(o). In view of (2.24), this proves (2.29) and (2.30), while(2.31) follows by the substitution of (2.30) into (2.8).

In the time-invariant case, that is, when the matrices A, B, C and R areconstant, the roots P and N of the algebraic equation (2.14) satisfy thedifferential equation (2.20), and (2.21) reduces to

Consequently, the system (2.25) is identical with the system (2.33), and thesystem (2.26) is identical with the system (2.34), and it follows from (2.24)that

Using (2.27), we obtain from (2.37) at t = 0 and t = 1

Hence, the error of the approximation rj(t) ~ ///(T) + r/r(cr) in this case isonly O(e~c/e), and it is due solely to the approximation of the boundaryconditions (2.38).

The importance of Theorem 2.1 is that it approximates a time-varyingboundary value problem by two much simpler time-invariant initial valueproblems, namely the initial and terminal boundary layer regulator prob-lems. The initial regulator problem is

and the optimal state-feedback control

stabilizes the initial boundary layer of the original problem (2.4), (2.5).The terminal regulator problem is

and the optimal state-feedback control

o


stabilizes the terminal boundary layer of the original problem (2.4), (2.5)in reverse time as o varies from o = 0 to a—> =». This means that in realtime t the terminal layer is totally unstable, which is an inherent propertyof the Hamiltonian boundary value problem (2.10).

Example 2.1

Various aspects of the above analysis are illustrated by way of the scalarproblem

through considering four cases with different values of a(f) and q.

Case 1: q = 0, a(f) = -(1 + at), \ a < 0.5. In this case, (2.14) and itssolutions P(t) and N(t) are

Thus VI(T) = 0 and the initial layer is r//(r) = e~Trjo, because the open-loopsystem (2.45) is asymptotically stable and q = 0, that is, 77 is not observablefrom /. The terminal layer r\r(o) and the corresponding control vr(o) are

and hence the approximate optimal trajectory (2.29) is

In this case it is possible to obtain the exact solution and to compare it with(2.48). Approximate controls and trajectories are sketched in Fig. 6.1(1).

Case 2: q = 1, a = 0. In this case, the open-loop system (2.45) is stable,but not asymptotically stable, while 77 is observable in /, so that (2.14) andits solutions P and N are

Both initial and terminal layers are controlled,

and the approximate trajectory (2.37) is

and is shown in Fig. 6.1(2).


Case 3: q = 0, a = 1. The open-loop system is unstable and 77 is unob-servable from /, so that (2.14) and its solutions P and N are

In this case, only the initial layer is controlled by V,(T) = -2^/(r), whilethe terminal layer is the solution of (2.42) with vr(o) = 0. The expressionfor the approximate trajectory is the same as in (2.51); see also Fig. 6.1(3).

Case 4: q = 0, a = 0. The system is as in Case 2, but now r\ is unob-servable in / and both eigenvalues of the Hamiltonian matrix (2.11) arezero:

Fig. 6.1. The boundary layer behavior of the solution to problem (2.45) for e = 0.2 in cases1, 2, 3. In case 4, the solution does not exhibit boundary layer behavior.


The boundary layer approximation does not apply because the condition(2.11) is violated. In fact, the exact optimal solution

shown in Fig. 6.1(4), does not exhibit a boundary layer behavior.

It is clear from this example that the presence of boundary layers dependson the observability of individual eigenvalues and, in particular, on theobservability of the zero eigenvalues. In a higher-order problem the partof the solution corresponding to the unobservable zero eigenvalues will notpossess boundary layers, but the remaining part will be in the boundarylayer form. For unobservable eigenvalues away from the imaginary axis,both layers will appear, but only one of them will be controlled: theinitial layer will be controlled if Re A > 0, while the terminal layer will becontrolled if Re A < 0. For observable eigenvalues both layers will becontrolled.

Thus far, we have discussed the problem with the terminal state 77 fixed.Similar conclusions can be drawn from Theorem 2.1 for other types ofboundary conditions. For example, if the terminal state 77(1) is free, thenp ( l ) = 0 and we see from (2.30) that

where the last equality applies to the time-invariant case, along with (2.37),(2.38). It follows that 77(1) will be regulated close to zero if ./Vis nonsingular,which is the case when the pair (A, C) is detectable, that is, when theunstable eigenvalues are observable in /.

Let us conclude this section with a glimpse at the boundary layerphenomena occurring in a more general nonlinear problem, namely,

This problem, a current topic of research, is sketched here and illustratedby an example. Proceeding as in (2.1)-(2.5) we write (2.56), (2.57) as


Motivated by the state-feedback interpretation of the boundary layer regu-lators (2.39)-(2.44), we examine the Hamilton-Jacobi optimality conditionfor (2.56), (2.57), that is,

which, rescaled for (2.58), (2.59), becomes

where J*(r],t) is Bellman's optimal value function (Athans and Falb,1966). A nonlinear generalization of Theorem 2.1 would establish that thecondition (2.61) can be approximately satisfied by two boundary layerregulator solutions obtained from the simplified condition

As in the linear problem, the initial-layer feedback control vl = VI(T)I) wouldstabilize the system (2.58) in real time, while the terminal-layer feedbackcontrol vr = vr(r]r) would stabilize it in reverse time.

Example 2.2

For the nonlinear problem

the minimizing control for (2.62) is v = -dJ*/dr], which, when substitutedback into (2.62), yields

The roots of this equation, which is a nonlinear counterpart of (2.14), are

and the decision is to be made as to which of them is vi and which of themis iv Since v1 is stabilizing in real time and v2 is stabilizing in reverse time,the correct choice is


A nonlinear analog of Theorem 2.1 would then establish that the solutionof (2.58)-(2.59) is approximated by

where the corresponding terms are obtained from the boundary layersystems of (2.63) controlled respectively by vt and vr given by (2.67).

6.3 The Reduced Problem

Let us now consider the problem of finding a control u(t) that steers thestate x, z of the singularly perturbed system

while minimizing the cost functional

This problem can be simplified by neglecting e in two different ways. First,an optimality condition can be formulated for the exact problem andsimplified by setting e = 0. The result will be a reduced optimality condition.Second, by neglecting e in the system (3.1), (3.2) the same type of optimalitycondition can be formulated for the reduced system. When the obtainedoptimality conditions are identical, it is said that the reduced problem isformally correct. Since this may not be obvious in a Hamiltonian formu-lation, we first formulate a necessary optimality condition for the exactproblem, using the Lagrangian

where p and q are the multipliers (costates or adjoint variables) associatedwith x and z respectively. The Lagrangian necessary condition is

from that is

6.3 THE REDUCED PROBLEM 261

The reduced necessary condition is now obtained by setting e = 0 in (3.6)-(3.10) and disregarding the requirement that z(0) = z0, z(l) = z l 5 whichmust be dropped because (3.7) and (3.10) have degenerated to 0 = g and0 = Vz££, and are not free to satisfy arbitrary boundary conditions.

On the other hand, the reduced problem is obtained by setting e = 0 in(3.1), (3.2) and (3.4), and dropping the requirement that z(0) = z0, z(l) =Z T ; that is, the reduced problem is defined as

(The subscript "s" intimates our intention to use the solution of the reducedproblem for an approximation of the slow ("outer") part of the optimalsolution.) The question of formal correctness is answered by comparingthe Lagrangian necessary condition for the reduced problem (3.11)-(3.13)with the reduced necessary condition (3.6)-(3.10).

The Lagrangian for the reduced problem (3.11)-(3.13) is

and the corresponding necessary condition is


subject to xs(0)=xQ, xs(l)=x}. By setting e = 0 in (3.6)-(3.10), andcomparing the result with (3.15)-(3.19), we come to the followingconclusion.

Lemma 3.1

The reduced problem (3.11)-(3.13) is formally correct.

We have used the Lagrangian necessary condition because it convenientlyincorporates both differential and algebraic constraints. The more commonHamiltonian form is obtained from (3.6)-(3.10) via the Hamiltonianfunction

Rewriting (3.6)-(3.10) in terms of 2C, we get

It is important to observe that in the definition of 3 use is made of g ratherthan of g/e. This means that the Hamiltonian adjoint (costate) variableassociated with z is eq. Let us apply the Hamiltonian condition to theproblem

Using the Hamiltonian

6.3 THE REDUCED PROBLEM

we obtain from (3.23), (3.24) and (3.25)

263

The substitution of (3.31) into (3.26), (3.27) results in the boundary valueproblem consisting of the system

and the boundary conditions

where <2,y = CjCj and 5,y = BfR 1BJ, /, j = 1, 2. Wherever convenient,

the system (3.34), (3.35) will be rewritten in the more compact notation

with the obvious definitions of Fn, Fn, ¥2\ and F22. This system is in thefamiliar singularly perturbed form analyzed in Chapters 2 and 5. To defineits reduced system, we need F22. However, anticipating that F22 will be theHamiltonian matrix for a fast subproblem, we make an assumption anal-ogous to Assumption 2.2.

Assumption 3.1

The eigenvalues of F22(t) lie off the imaginary axis for all t G [0,1]; that is,there exists a constant c such that


Under this assumption F22 exists and the reduced necessary condition is

subject to

From Lemma 3.1, we know that the same necessary condition is obtainedby setting e = 0 in (3.26)-(3.29), even if A22 does not exist. Accepting aloss of generality, we assume that A22 exists, and define the reducedproblem as

where zs is obtained from (3.27) with ez = 0 and substituted into (3.26),(3.28) and (3.29), that is,

whence

Application of the necessary condition 0 = V ws^s, ps = -V*S3 S to theHamiltonian for (3.43)-(3.45),

and substitution of the control «s minimizing 3^s,

into (3.43) and ps = -V*S3 S, results in the boundary value problem

Specialized to this case, Lemma 3.1 reads as follows.

Lemma 3.2

The matrix Fn - Fl2F22F2l in (3.41) and the Hamiltonian matrix in are identical, that is,

If one were to prove this fact without the help of Lemma 3.1, one wouldhave to go through manipulations as in Chapter 3, involving an expressionfor F22 in terms of A22 . While in the Lagrangian formulation, it isalmost obvious that the reduced problem is formally correct, this propertyis obscured in situations in which algebraic constraints are explicitly solvedand eliminated from the problem. A special case in which (3.51) can readilybe verified by algebraic manipulations is when C2 = 0. In this case,Z)() = 0, R0 = R, C0 = C1? an

O n the other hand, the substitution of

into (3.41) gives

and confirms (3.51). Fortunately, Lemma 3.1 and its consequence, Lemma3.2, render such calculations superflous.

All the results in the remaining part of this chapter will be obtainedunder the following assumption.

Assumption 3.2

The solution of the reduced problem exists and is unique.

In other words, our analysis assumes that the reduced solution is given

o

6.3 The reduced probem 265


and uses it to construct two-time-scale approximations of the exact solution.Depending on the particular problem, this assumption implies several otherassumptions discussed in detail in standard texts on optimal control. Forexample, in a fixed-end-point problem Xi must be reachable from XQ, whichin the linear case implies the controllability of the reduced system (3.43).Another example is a convexity assumption frequently made to avoid theso-called conjugate points in the Hamiltonian problem (3.50). Since thematrices appearing in (3.50) have already been examined in Chapter 3,when dealing with the composite control, we know that this assumption isautomatically satisfied because the (2,l)-block of F0 in (3.51) is nonpositive,

owing to R> 0; see (4.19) of Chapter 3. Let us conclude this section witha discussion of the reduced problems for a linear example and a nonlinearexample.

Example 3.1

Suppose that a DC motor without load, modeled by

where x is its angular velocity and z and u are the armature current, andvoltage respectively, is to be accelerated from x0 = 0 to jc1? while the initialand terminal values of the current are to be kept at z0 = 0, Z] = 0. Thecontrol accomplishing this task should also minimize the energy function

The Hamiltonian ̂ for (3.56), (3.57) and the result of VUW = 0 are

which, when substituted into (3.24), (3.25) and (3.56), give

Letting e = 0 in (3.59), one obtains the reduced necessary condition

6.3 THE REDUCED PROBLEM 267

which is also the necessary condition for the reduced problem

as predicted by Lemma 3.2. We will continue the analysis of this problemin the next section.

Example 3.2

The nonlinear optimal control problem

occurs in chemical processes, where x represents the concentration of aningredient whose rate of change is proportional to the flow z. The flow zis controlled by u through a first-order delay with the time constant e. Thepenalty x4 in (3.63) attempts to reduce large increases in the concentration.The Hamiltonian 2C and the result of VU3C = 0 are

and their substitution into (3.24), (3.25) and (3.62) yields the necessarycondition in the form of the nonlinear boundary value problem

The reduced problem obtained by setting e = 0 in (3.62), (3.63) is

and its necessary condition is


The same reduced necessary condition is obtained by setting e = 0 in (3.65).We shall return to this problem in Section 6.5 in the context of near-optimalnonlinear control.

6.4 Near-Optimal Linear Control

Our understanding of boundary-layer phenomena and the relationshipbetween the exact and the reduced necessary conditions serve as a basisfor a two-time-scale approximation of the exact solution. To begin with,let us consider the linear boundary value problem (3.34)-(3.37), assumingthat all of its coefficient matrices are constant. Using the more compactnotation, we apply our block-diagonalizing transformation, developed inChapter 2, to this singularly perturbed system; namely, we let

where L and H are respectively the solutions of the matrix equations (2.8)and (4.3) of Chapter 2, with F/,- appearing instead oiAij, i, j = 1,2. For ourpurposes it is sufficient to remember from (2.13) and (4.7) of Chapter 2that L and H can be approximated by

where F0 = Fn - FUF22F2[, as in (3.41). Using (4.1), the system (3.38),(3.39) is transformed into two separate subsystems; namely, the slowsubsystem

and the fast subsystem

To obtain the boundary conditions for the new variables, we apply theinverse transformation to the boundary conditions (3.36), (3.37). For theslow subproblem we get

6.4 NEAR-OPTIMAL LINEAR CONTROL 269

It is important that, allowing an error of O(e), the conditions for £ can betaken to be equal to those for x. For the fast subproblem, we obtain from

that the boundary conditions for 77 depend on the values of all four originalvariables ;c, z, p, q at t = 0 and t = 1. This suggests that we should firstconsider the slow subproblem, which, taking into account (4.2) and (4.5),can be written as

The limit of this problem as e-*0 is the reduced problem (3.41), (3.42)for which in Assumption 3.2 we have assumed the existence of a uniquesolution. In (4.7) this reduced solution is subjected to both an O(e)perturbation of the system matrix F0 and an O(e) perturbation of itsboundary conditions. In theory such perturbations will produce only anO(e) perturbation of the solution, but in practice care must be exercisedin implementing this result because F0, being a Hamiltonian matrix, has nunstable eigenvalues. Whenever the problem (4.7) is solved by a methodinvolving numerical integration, it must be ensured that its unstable part isintegrated in reverse time. With this caveat the solution to the reducedproblem will be an O(e] approximation of the exact solution |(f), n(f) ofthe slow subproblem (4.7), that is,

£(0 = *s(0 + 000, *(0=/> s(0 + 000 Vre[0, i ] . (4.8)This result furnishes the data for an approximate solution of the fast

subproblem, owing to the approximation of p ( t ) , q(t) in (4.6) byps(r), q^f).At t = 0 and t = 1 this gives

and the data to solve (4.4) as a pure initial value problem at either end ofthe interval seem to be complete. However, if we were to do so, we wouldbe disregarding our knowledge of the boundary layer phenomena andignoring the caveat about the unstable eigenvalues. Remembering (2.27),


it is, however, expected that the unstable modes of (4.4) diverge at anexponential rate 0(ec'/£). At this rate an O(e) error at t = 0 would become0(eec//f) at t = 1, when evaluated analytically, and would certainly be muchworse numerically. Therefore one must make sure that the boundary valuesfor the unstable part of the fast subsystem (4.4) are prescribed at the endof the interval, t = 1, to allow the unstable modes to be integrated in theirstable direction, that is in reverse time 1 - t from 1 - f = 0 to I- t=l.This can be accomplished by using the Riccati machinery of Section 6.2 toexploit the exponential dichotomy of (4.4). With the obvious change ofnotation, A^>A22, #—» B2, C—> C2, S—» S22, etc., the results of Section6.2 are now applied to the fast subproblem (4.4)-(4.9).

Lemma 4.1

If (A22, B2) is a controllable pair and Assumption 3.1 holds, then

where P ^ O and N^O are as in Lemma 2.1 and A(f) — » 0 as e— »0.Therefore there exist constants cl > 0 and e4 > 0 such that for all e G [0, £4]

This lemma is a direct consequence of Lemma 2.1 and the continuousdependence of the eigenvalues on matrix coefficients. Then, by Lemma2.3, a dichotomy transformation for (4.4) exists and is O(e) close to (2.24);that is,

and constants cl>0 and kl > 0 exist such that for all t G [0, 1] and£ G (0, £4]

A conclusion analogous to that of Theorem 2.1 is that the solution to the


fast subproblem (4.4), (4.9) is approximated by the sum of the boundarylayers

where rji(i) and r]r(o) can be interpreted as the solutions of the boundarylayer regulator problems (2.39)-(2.41) and (2.42)-(2.44) respectively.

So far our discussion in this section has been restricted to the case whenall the coefficient matrices are constant. The only new step to be made inorder to state the analogous results for the time-varying problem is to treat(4.1) as a time-varying transformation, allowing for the fact that m of the2m eigenvalues of F22(t) have positive real parts for each r £ [0,1]. Thisproblem is addressed in Chapter 5, where Theorem 3.1 and its Corollary3.1 establish the existence of matrices L(t, e) and H(t, e) with the sameproperties as in (4.2) for each t G [0,1] and all e G (0, e5], where e5 > 0 isa constant. The results of the two preceding sections in this chapter arealso valid for the time-varying case. In this way, we obtain the followingapproximation result.

Theorem 4.1

Consider the optimal control problem (3.26)-(3.29) in which the timederivatives of all the matrices are continuous and bounded and R(t) > 0 forall r G [0, 1]. If Assumptions 3.1 and 3.2 hold and if

rank

then there exists an e5 > 0 such that for all e G (0, e5] and all t G [0, 1] theoptimal trajectory x(t, e), z(f, e) and the corresponding optimal controlu(t, E) satisfy

where xs(t), zs(f), us(t) is the optimal solution of the reduced problem(3.43)-(3.50), while I//(T) and vr(o), solve the boundary layer regulatorproblems (2.39)-(2.41) and (2.42)-(2.44) respectively, for r]Q and r]l givenby (4.9), and with A,B,C replaced by A22, B2, C2.


Proof The result (4.18) follows from (4.5) and (4.7), (4.8). To prove(4.19), we note from (3.39) with e = 0 that the reduced solution satisfies

Subtracting this zero quantity from (4.6) and taking (4.8) into account, thatis, the fact that x - xs and p - ps are O(e), we obtain

which, in view of our previous discussion on boundary layers, proves (4.19)and (4.20),

This theorem is extremely simple to apply. First, we solve the reducedproblem (3.43)-(3.50). Next, we determine the initial conditions rj,(0) and7/r(0) from (4.19) at t = 0 and t = 1 as

Their meaning as the correctors of the discrepancies in the boundaryconditions for z caused by the reduced solution zs(f) is now clear. Usingthese initial conditions, the optimal stabilizing solutions of the layer regu-lator problems (2.39)-(2.41) and (2.42)-(2.44) are evaluated with, ofcourse, A, B, C replaced by A22, B2, C2.

Example 4.1

Continuing the discussion of Example 3.1, the solution of the reducedproblem (3.60) is

and hence the corrective initial conditions are

In this case A22 - -1, B2 = 1, C2 = 0, R = 1, 522 = 1. The Riccati equation(2.14) and its solutions P > 0 and N < 0 are

o


and hence the stabilizing solutions of the boundary layer regulators are

The approximation to the solution of the optimal control problem for theDC motor (3.56), (3.57) can now be composed as in (4.18)-(4.20). Onlythe terminal layer is controlled by the control vn whose task is to regulate

Fig. 6.2. (a), (b), (c) Near-optimal control of a DC motor.

the current from zs(l) to zero, z2 = 0. In practice, this would prepare theunloaded DC motor for a constant speed operation at x(t) = xl, aftert = 1. Exact and approximate solutions are plotted in Fig. 6.2. The readershould note that the same problem was solved using the controllabilityGrammian in Chapter 5, Example 7.1.


6.5 Nonlinear and Constrained Control

The linear results of the preceding sections are readily extended to thenonlinear problem of the form

where all the functions of x possess continuous and bounded derivatives withrespect to x and A 2(x) is nonsingular for all x . Some of these assumptions , aswell as the form of the problem, are unnecessarily restrictive, and are madeonly to simplify the presentation. The Hamiltonian

is minimized by the control

and when this control is substituted into (5.1), (5.2) and (3.24), (3.25), theresulting nonlinear boundary value problem is

The reduced problem obtained by setting s = 0 in (5.2) and substituting

into (5.1) and (5.3) is

6.5 NONLINEAR AND CONSTRAINED CONTROL 275

where

The Hamiltonian

is minimized by

and the corresponding reduced boundary value problem is

It is assumed that this problem has a unique solution. Anticipating thatxs(f) will be O(e) close to the optimal solution x(t, e), we introduce thequantities

and substitute them into (5.2), taking into account that zs is defined by(5.10). In this way, one obtains

where A incorporates the terms 6a2, dA2z, 6B2u and zs. By analogy withthe linear problem, we treat

as the variables for the boundary layer regulator problems, at t = 0, inforward time r,


and at t = 1, in reverse time a,

Then, under conditions analogous to those in Theorem 4.1, the sameapproximation (4.18)-(4.20) holds for the nonlinear problem (5.1)-(5.3).The proof of this result makes use of the fact that the linearization of (5.6)-(5.9), along the approximate solution, has the form of the linear necessaryconditions in (3.34), (3.35). For details of the proof and higher-orderasymptotic expansions the reader is referred to Chow (1979).

Example 5.1

Proceeding with the chemical process problem of Example 3.2, it is observedthat (3.62), (3.63) is in the form (5.1)-(5.3). The reduced problem (3.67),although nonlinear, can be solved analytically:

and the initial conditions for the boundary layer problems are

The Riccati equation and its solutions P > 0 and N ^ 0 are

Hence the boundary layer regulators are

When the solutions for (5.28) are combined with the reduced solution(5.25), the near-optimal solution (4.18)-(4.20) for this problem is


These approximations are compared with the optimal solutions obtainedfor £ = 0.1 numerically in Fig. 6.3. A satisfactory agreement is achieved inparticular for x(t, e), while the outer approximation of z(r, e) can be furtherimproved by a first-order correction of the slow part of the solution.

Fig. 6.3. (a), (b), (c) Near-optimal control of nonlinear chemical process example.

All the optimal control problems considered thus far have been withoutcontrol constraints. The additional difficulty with constrained problems isthat both the slow (reduced) control «s(f) and the fast controls V/(T) andvr(o) share the same constraints. Nevertheless, the separation of time scalesand the insight gained by the analysis of the boundary layer phenomenais helpful in constructing asymptotic approximations similar to those ofTheorem 4.1. As an illustration, consider the time-optimal problem for thelinear time-invariant system

in which the scalar control is constrained by


and is to steer x(t), z(t) from x0, z0 at t = 0 to zero in minimum time T.The results for this special case are readily extended to the time-optimalcontrol problem for the nonlinear system (5.1), (5.2) with a constrainedcontrol vector u.

The solution of the time-optimal control problem (5.32)-(5.34) is anexercise in the application of the results of Chapter 2. First we use thetransformation (4.8) of Chapter 2 to represent the system (5.32)-(5.34) inthe block-diagonal form

where %0, r]0 are known from the inverse transformation (4.9) of Chapter

2, and u and its constraint (5.34) are unaltered. Obviously, a control u thatsteers f, 77 from £0> r)0 to zero in minimum time T will steer JCQ, z0 to zeroin the same minimum time T. The Hamiltonian for (5.35), (5.36)

is minimized, subject to (5.34), by the control

where p and q are the solutions of

Remembering that

let us assume that A22 is a Hurwitz matrix, then so is Af, for e sufficientlysmall; that is,

Substituting into (5.38) the solution of (5.39) for some end condition, yetto be found,

we arrive at the conclusion that the switching law (5.38) has a slow-plus-fast behavior, namely,


The switches due to the second term in parentheses occur only near t—Tbecause of (5.41). They are of a high frequency, while the switches due tothe first term are of a low frequency and occur throughout the interval[0, T]. If we approximate A& and bs by As « A0 and £s = bQ, we will discoverthat the first term is the solution of the time-optimal control problem forthe reduced system. Suppose the minimal time for the reduced problem isTQ, and hence xs(TQ) = 0. However, z(T0) « zs(r0) is in general differentfrom zero, and the task of the high-frequency switches near the end of theinterval is to steer z to zero. Again we see that, because A22 is Hurwitz,only the terminal layer is controlled. One way to use (5.43) is to separatelycalculate the slow and the fast switching instants. Let us illustrate this byan example.

Example 5.2

Suppose that a double-integrator plant is controlled through an actuatorwith two small time constants £ and 0.5£ (see Fig. 6.4). In this case, Xi =

Fig. 6.4. Time-optimal control of a double-integrator plant with two parasitic time constants.

The reduced solution is well known,

and the one switch time is ts = —u^c2Q + [0-5^20 ~ Ws*io]^2> while the reducedminimum final time is Ts = -usx20 + 2[0.5^2o ~ "s^io]1^2- At r = Ts the fastvariables are not zero:

The terminal layer control vr is to steer z(Ts) to zero in minimum time.This is also a textbook example, for which the solution is

and the reduced system is


The concatenation of us(t) and vr((t - rs)/e) will steer x, z to an O(e)neighborhood of the origin. This result is obtained with As « A0, bs « b0

and

A further improvement is possible by a closer approximation of As, bs, Af

and b.

6.6 Cheap Control and Singular Arcs

An unconstrained optimal control that satisfies the necessary optimalitycondition

but does not satisfy the strengthened Legendre-Clebsch conditiond23C/dul, that is, such that

is referred to as a singular control and the corresponding part of thetrajectory as a singular arc (Bell and Jacobson, 1975; Bryson and Ho,1975). A well known necessary condition for optimality in this case is

A direct use of this condition in optimal control computations is complicatedby the need to connect a singular arc with boundary points which may notbe on the arc.

Some singular control problems can be treated as the limits of simpler"cheap control" problems, such as the following class of problems analyzedby O'Malley and Jameson (1975):

6.6 CHEAP CONTROL AND SINGULAR ARCS 281

The term "cheap" indicates that e is small, and hence the cost of control islow. There is no loss of generality in considering (6.4), (6.5), because everylinear system with a full-rank control input matrix B can be represented inthis form after a similarity transformation. Although the system (6.4), (6.5)is not singularly perturbed, our (jc, z) notation anticipates that a large uwill force the z-variable to act as a fast variable. The Hamiltonian functionfor this problem is

and it is obvious that for £ = 0 the problem is singular, since then 3 islinear in u and (6.2) holds. For e > 0 and R > 0, the control minimizing itis

Since its substitution into (6.5) would lead to a multiplication of z by e2,we rescale the adjoint variable q as q = q/e and write the adjoint equationsin the form

Combining (6.4), (6.5), (6.8) and (6.9), we obtain the singularly perturbedboundary value problem

which is a special case of the problem (3.34)-(3.37). The analysis of Sections6.3 and 6.4 is applicable if Q22 > 0. Under this assumption we set e = 0,solve (6.11) for


and, substituting into (6.10), obtain the reduced slow system

with the boundary conditions

The expressions (6. 12) -(6. 15) define the same singular arc *s(f), zs(f)that would have been obtained using the condition (6.3), if such an arcexists. Conditions for its existence can be established by an analysis of(6.14), (6.15) as a linear boundary value problem. To prove that thisproblem is the limit, as e—>Q, of the necessary condition for optimality(6.10), (6.11), we need to show what happens with the fast transients whichoccur near t = 0 and t = T because, in general,

The analysis in Section 6.4 shows that these transients are governed by theboundary layer regulators (4.13). In the case of (6.11), the Hamiltonianmatrix of these regulators is

and the corresponding Riccati equation is

Its positive- and negative-definite solutions P and N are respectively

The boundary-layer regulators (4.13) are therefore

Thus, by Theorem 4.1, a near-optimal solution to the "cheap control"problem (6.4)-(6.6) is

The singular arc xs(t), zs(f) and the fast transients away from it are clearlyexhibited.

6.6 CHEAP CONTROL AND SINGULAR ARCS 283

Example 6.1

In order to compare the singular perturbation approach to singular controlproblems with a more common use of the necessary condition (6.3), weconsider the problem

Using (6.3), it is shown by Bryson and Ho (1975) that a singular arc forthis problem lies on the curve

and that impulsive transients occur at both ends of the arc. However, ananalysis based on (6.3) does not reveal the nature of these transients.

To apply the methodology of this section we first use x = y\ + y2, z = y2

to transform (6.24), (6.25) into

We also add a "cheap control" term to (6.26) as

The boundary value problem (6.10), (6.11) is in this case

Since <222 = 1 > 0, the reduced slow system (6.14) is obtained by settinge = 0 and solving (6.34), (6.35) for

which, substituted into (6.31), (6.32), give


The singular arc *s(r), zs(r) is thus obtained by solving (6.37), (6.38) for*s(0> Ps(0 an(* then evaluating zs(f) from (6.36). To compare this resultwith (6.27), we note from (6. 36) -(6. 38) that zs=-xs + ps = xs andxs = -zs. Therefore

which, in view of x = y1 + y2, z = —y2, is the same as (6.27).In order to complete the approximation (6.22), (6.23), we calculate the

boundary layers. Using the Riccati solution

we obtain

As can be seen, this analysis identifies not only the curve on which asingular arc can be found, but also both ends of the arc. Moreover, itcharacterizes the singular behavior near these ends.

6.7 Exercises

Exercise 6.1

Consider the optimal control problem

where constants a\,a2,b\ and b2 are given and T = 1/e > 1. Following themethodology of Section 6.2, find an O(e) approximation to the solution ofthis problem and compare it with the exact solution when e = 1/7" = 0.1.

Exercise 6.2

(a) Verify the formal correctness of the singularly perturbed optimal controlproblem

6.7 EXERCISES 285

where a0, b0 and b\ are given constants. First, write the Hamiltoniannecessary optimality condition and obtain its reduced form by settinge = 0. Next, set e = 0 in the problem statement and write the Hamiltoniannecessary condition for the reduced problem thus obtained. Compare thetwo necessary conditions.

(b) Paralleling the derivations in Example 2.2, find an O(e) approxi-mation of the optimal solution. Explain the form of its slow part by anuncontrollability property.

Exercise 6.3

(a) Verify that the optimal control problem

where a0, a l y a2, bQ, bl and b2 are given constants, satisfies Assumptions3.1 and 3.2 and that the matrices P and N of Lemma 4.1 are the same asthose found in Exercise 6.1.

(b) Using Theorem 4.1, find an O(e) approximation to the optimalsolution.

Exercise 6.4

Using Theorem 4.1, find an O(e) approximation to the solution of the time-varying optimal control problem

in which x(t) and z(t) are fixed at both ends,

where al, a2, bl and b2 are given constants. Discuss the implementation ofthis approximate solution.


Exercise 6.5

Applying the procedure (5.37)-(5.47), find a near-optimal control for theminimum-time problem

where a = -I. Observe that the fast switch occurs near the end of theinterval. Can a near-optimal control be found and implemented if a = 1?


Trajectory optimization problems were among the first problems to attractcontrol engineers to singular perturbations. Kokotovic and Sannuti (1968)introduced the formulation presented in Section 6.3, and proposed acorrection of the reduced solution. Kelley and Edelbaum (1970) and Kelley(1970a, b, 1971a, b) demonstrated the applicability of reduced-order model-ing and boundary layer corrections to aircraft manoeuver optimization.With this impetus, earlier works on singularly perturbed boundary valueproblems became relevant. Among them are results of Levin (1957) onstable initial manifolds, Tupchiev's two-point boundary value analysisreported by Vasil'eva (1963), and the dichotomy decomposition of Chang(1972).

Our presentation of the exponential dichotomy as a characteristic ofHamiltonian boundary value problems in Section 6.2 and its use in Section6.4 are patterned on the exposition of Wilde and Kokotovic (1972b, 1973).Their advantage is that a boundary value problem is treated as a compositionof initial-value problems. The nonlinear extension of this approach inSection 6.5 is due to Chow (1979) and is related to the nonlinear stabilityand regulator problems of Chapter 7.

Asymptotic expansions of more general boundary value problems, notcovered in this book, are found in Bagirova, Vasil'eva and Imanaliev(1967), Hadlock (1970), O'Malley (1972a, b, 1974a, b). Sannuti (1974a, b,1975), Freedman and Kaplan (1976), Vasil'eva and Dmitriev (1980) andothers. The relationship of the assumptions made by the two approacheswas investigated by O'Malley (1975). The assumptions differ in how theystate the "matching" conditions, that is, the conditions that guarantee thestability of the initial layer in forward time and of the terminal layer in


reverse time. Geometrically, they require a saddle point property, that isthe existence of two manifolds, one stable and the other totally unstable.To avoid "turning points", this saddle point property must hold throughoutthe interval. While for general boundary value problems this requirementis restrictive, further research may show that most optimal control problemssatisfy it owing to their Hamiltonian structure.

Asymptotic analysis of problems with constraints and singular arcs hasbeen much more difficult. Only for linear time-optimal problems of thetype considered in Section 6.5 have simple near-optimal solutions beenproposed—by Collins (1973), Kokotovic and Haddad (1975a, b), Javidand Kokotovic (1977), Javid (1978a) and Halanay and Mirica (1979).Approximate solutions to nonlinear constrained problems have beenobtained by application-specific techniques. They are particularly promisingin flight dynamics and guidance, where results of Ardema (1976, 1979),Calise (1976,1978,1979,1980,1981), Sridhar and Gupta (1980) and Shinar(1983) extend earlier results of Kelley (1973). Analytical elucidation ofsome heuristic concepts emerging from these applications is expected tobroaden the applicability of the time-scale methodology to optimal controlproblems. The use of a "cheap control" problem for providing a limitinganalysis of a more difficult singular optimal control problem in Section 6.6is discussed in O'Malley and Jameson (1975), and, for time-varying systems,in O'Malley and Jameson (1977).

7 NONLINEAR SYSTEMS

7.1 Introduction

An advantage of the two-time-scale approach to system modeling andcontrol systems design is that it is not restricted to linear systems. In thischapter we exploit this advantage in our study of the stability propertiesand the composite state feedback control of nonlinear systems.

First, Section 7.2 addresses a fundamental question of this book, andone that has long interested singular perturbation researchers: assumingthat the associated slow and fast systems are each asymptotically stable,what additional conditions will guarantee the asymptotic stability of theoriginal nonlinear singularly perturbed system for a sufficiently small singu-lar perturbation parameter el Using Lyapunov stability methods, a com-posite Lyapunov function is constructed in a step-by-step manner, startingwith the stability of a slowly varying system of the type considered inSection 5.2. It is shown that under mild conditions, any weighted sum ofquadratic-type Lyapunov functions for the slow and fast systems is aquadratic-type Lyapunov function for the singularly perturbed system whene is sufficiently small. The choice of the weights of the composite Lyapunovfunction involves a trade off between obtaining a large estimate of thedomain of attraction and a large upper bound on the singular perturbationparameter e. Analysis developed in Section 7.2 for autonomous systems isextended in Section 7.5 to nonautonomous systems.

The application of this method is illustrated in two case studies presentedin Sections 7.3 and 7.4. The method is applied in Section 7.3 to estimatethe domain of attraction of the stable equilibrium point of a synchronousgenerator connected to an infinite bus. The second case study, Section 7.4,illustrates an important application to the analysis of robustness of adaptivecontrol systems with respect to parasitics. This is illustrated in Section 7.4by a second-order example.

289

290 7 NONLINEAR SYSTEMS

In Sections 7.6 and 7.7 we extend the composite state feedback controlmethod of Chapter 3 to nonlinear system models including those thatcannot be stabilized through linearization. The design is sequential ingeneral, since the fast control design depends on the slow control design.However, should the system be linear in the fast states and control inputs,as in the nonlinear regulator problem of Section 7.7, the fast-slow designsare separated as before. The near-optimal regulator design is radicallydifferent from the finite-interval trajectory optimization results of Chapter6 because of the stability and boundedness requirements fundamental ininfinite-time problems which require feedback solutions.

7.2 Stability Analysis: Autonomous Systems

We consider a nonlinear autonomous singularly perturbed system

which has an isolated equilibrium at the origin x = 0, z = 0. It is assumedthroughout that / and g are smooth enough to ensure that, for specifiedinitial conditions, (2.1) has a unique solution. Stability of the equilibriumis investigated by examining the reduced (slow) system

where z = h(x) is an isolated root of 0 = g(x, z), and the boundary-layer(fast) system

where x is treated as a fixed parameter. Theorem 2.1 states, essentially,that if x = 0 is an asymptotically stable equilibrium of the reduced system(2.2), z = h(x) is an asymptotically stable equilibrium of the boundary-layer system (2.3), uniformlyt in jc, and / ( • , • ) and g( • , •) satisfy certaingrowth conditions, then the origin is an asymptotically stable equilibriumof the singularly perturbed system (2.1).

In Theorem 2.1, asymptotic stability requirements on the reduced andboundary-layer systems are expressed by requiring the existence of Lyapu-nov functions for each system. The growth requirements on / ( • , • ) andg( - , •) take the form of inequalities satisfied by the Lyapunov functions.

t That is, the e-6 definition of Lyapunov stability and the convergence z —> h(x) are uniformin x (Vidyasagar, 1978).

7.2 STABILITY: AUTONOMOUS SYSTEMS 291

Most of the analysis in this section is devoted to motivating the conditionsof Theorem 2.1 and relating them to the previous analysis of Section 5.2.We start by studying the slowly varying system

where x(t) satisfies

It is assumed that for all t > 0, x(t) E S, where 5 is a closed bounded subsetof R". This assumption does not require the trajectory x(t) to converge toany limit as t—> °°. It merely requires that x(t) remains bounded as ?-» °°.System (2.4) is a special case of the linear slowly varying system studied inSection 5.2. Instead of having A as an explicit function of t, A now dependson x, which depends on t. Therefore, in view of Lemma 2.2, it is seen that

imply that z(r)—»0 as r—»°°. The same conclusion can be reached byemploying Lyapunov methods. In view of (2.6), for every x E S there existsa symmetric positive-definite matrix P(x), bounded on S, satisfying theLyapunov equation

Moreover, P(x), the derivative of P(x) along the trajectory of (2.5), isbounded on 5. This follows from the explicit form of the solution of (2.7),given by

With P(x) we can form a Lyapunov function candidate! for (2.4) as

The function W(x, z) can be viewed as a Lyapunov function of (2.4) whenx is treated as a fixed parameter which may take any value in 5. If * werefixed, the derivative of W along the trajectory of (2.4) would have beengiven by -(l/e)zTz. But rather than being a constant, x is slowly varyingaccording to the solution of (2.5), and hence the derivative of W(x, z) along

t Following the terminology of Vidyasagar (1978), a function W is called a Lyapunovfunction candidate if it satisfies the usual differentiability and positive definiteness require-ments. If, for a particular system, W *£ 0 then W is referred to as a Lyapunov function.

292 7 N O N L I N E A R SYSTEMS

the trajectory of (2.4), (2.5) is given by

Thus W(x(t), z(r))-»0 as *-»«>, implying that z(t)^>0 as *-»». In theabove inequality, and throughout this chapter, || • || denotes the Euclideannorm ||-||2.

Let us point out that the Lyapunov function W(x, z) has two properties

These two properties will carry over to Lyapunov functions of the moregeneral system (2.1).

Now consider the system

When x is treated as a fixed parameter (2.11) has an equilibrium h(x) =-A~l(x)b(x) instead of the zero equilibrium in (2.4). The existence ofA~l(x) is guaranteed by (2.6). In addition, we assume that dh/dx exists inS. A natural Lyapunov function candidate for (2.11) is

This function has the following properties for every x E 5:

Comparing (2.12), (2.13) with (2.9), (2.10) shows that ||z - h(x)\\ replaces|| z ||, reflecting the fact that now the equilibrium is h(x) instead of zero.More significantly, there is an extra term in (2.13) not appearing in (2.10)which is due to the slow motion of the equilibrium. When the derivative


of W is computed along the trajectories of (2.11) and (2.5) we get

It is important to see that now we cannot assure that z(t) —> h(x(t)) as r—»°cwithout additional requirements on the asymptotic behavior of x(t) ast^> oo. This brings us to the next step of our study, where we consider thefull singularly perturbed system

Let Bx C Rn and Bz C Rm denote closed sets.

Assumption 2.1

The origin is an isolated equilibrium of (2.15), (2.16) in Bx x Bz, i.e.

and z = h(x) is the unique root of

We study the asymptotic stability of the origin using Lyapunov methods.To construct a Lyapunov function candidate for the singularly perturbedsystem (2.15), (2.16) we consider each of the two equations separately.Starting with (2.16), we make the following assumption.

Assumption 2.2

There exists a Lyapunov function candidate W(x, z) such that for all(x, z) E Bx x Bz


where i//( • ) and 0( • ) are scalar functions of vector arguments which vanishonly when their arguments are zero, e.g. ty(x) = 0 if and only if x = 0. Theywill be referred to as comparison functions.

Conditions (2.20) and (2.21) look exactly like (2.12) and (2.13) with|| z - h(x)\\ replaced by 0(z - h(x)) and \\(dh/dx)f(x)\\ replaced by ty(x).Working with comparison functions more general than norms allows us tohandle a wider class of nonlinearities, as will be shown later by examples.The function W(x, z) is a Lyapunov function of the boundary layer system(2.3) in which x is treated as a fixed parameter.

We turn now to (2.15). Adding and subtracting f ( x , h(x)) to the right-hand side of (2.15) yields

The term/X*, z) - f ( x , h(x)) can be viewed as a perturbation of the reducedsystem

It is natural first to look for a Lyapunov function candidate for (2.23) andthen to consider the effect of the perturbation term/(jc, z) - f ( x , h(x)). Inthis spirit, we assume the existence of a Lyapunov function V(x) thatsatisfies

Condition (2.24) guarantees that x = 0 is an asymptotically stable equi-librium of (2.23). It is important to observe here that the comparisonfunction in (2.24) is the same as in (2.21). Computing the derivative of Valong the solution of (2.22) yields

We assume that

so that

Condition (2.25), in a sense, determines the allowed growth of /in z. In


typical problems, verifying (2.25) reduces to verifying the inequality

which means that the rate of growth of/in z cannot be faster than the rateof growth of < / > ( • ) •

The conditions on V(x) are put together in the following assumption.

Assumption 2.3

There exists a Lyapunov function V(x) that satisfies inequalities (2.24) and(2.25) for all x E Bx.

With the Lyapunov functions V(x) and W(x, z) to hand, we considerv(x, z), defined by a weighted sum of V and W,

as a Lyapunov function candidate for the singularly perturbed system(2.15), (2.16). To explore the freedom we have in choosing the weights,we take d as an unspecified parameter in (0, 1). The derivative of v alongthe trajectories of (2.15), (2.16) is given by

Using inequalities (2.20), (2.21), (2.24) and (2.25), we get


The right-hand side of the above inequality is a quadratic form in ip( •) and0(-)- This quadratic form is negative-definite when

which is equivalent to

The inequality (2.28) shows that for any choice of d the corresponding vis a Lyapunov function for the singularly perturbed system (2.15), (2.16)for all e satisfying (2.28). Notice that in the above development only o^and a2 are required by definition to be positive. The other three parametersPi, f}2 and y could, in general, be positive, negative or zero. In mostproblems, however, one arrives at inequalities (2.21) and (2.25) using norminequalities leading automatically to nonnegative values for f$i, /32 and y.Therefore we continue our discussion assuming /^ ^ 0, f$2 > 0 and y > 0.The inequality (2.28) can be written as

The dependence of the right-hand side of (2.29) on the unspecified para-meter d is sketched in Fig. 7.1.

It can easily be seen that the maximum value of ed occurs at d* =

Fig. 7.1. Stability upper bounds on e.


and is given by

It follows that the equilibrium point of (2.1) is asymptotically stable for all£ < £* . The number e* is the best upper bound on e that can be providedby the above stability analysis. Theorem 2.1 summarizes our findings.

Theorem 2.1

Let Assumptions 2.1-2.3 be satisfied. Then the origin is an asymptoticallystable equilibrium of the singularly perturbed system (2.1) for alle E (0, e*), where e* is given by (2.30). Moreover, for every numberd£(0 , 1)

is a Lyapunov function for all e G (0, ed), where ed < e* is given by (2.29).

Corollary 2.1

Let the assumptions of Theorem 2.1 hold for all x, z E R" x Rm and letV(x) and W(jc, z) be radially unbounded (i.e. V(*)— » « = as * — » ac andW(x, z )— » °° as ||z - /I(A:)||^- °°). Then, the equilibrium (jc = 0, z = 0) isglobally asymptotically stable for all e < e* .

Corollary 2.2

Let all the assumptions of Theorem 2.1 hold with ^(jc) = ||jc|| and 0(z — h(x)) =||z — h(x) || and suppose, in addition, that v(x) and W(x, z) satisfy the inequalities

where e 1 ? . . ., e4 denote positive constants. Then, the conclusions ofTheorem 2.1 hold, with exponential stability replacing asymptotic stability.

Corollary 2.3

Let /, g and h be continuously differentiable. Suppose that x = 0 is anexponentially stable equilibrium of the reduced system (2.2) and z = h(x)is an exponentially stable equilibrium of the boundary layer system (2.3),


uniformly in *, i.e.

where a and K are independent of*. Then, the origin is an exponentiallystable equilibrium of the singularly perturbed system (2.1), for sufficientlysmall e.

Proof This follows from Corollary 2.2 with the conceptual Lyapunovfunctions

where X(t, x) is the trajectory of the reduced system starting at initial pointx at t = 0 and Z(r, z;*) is the trajectory of the boundary layer systemstarting at initial point z at T — 0. It is left for the reader (Exercise 7.3) toverify that V and W satisfy all the conditions of Corollary 2.2 D

The stability analysis that led to Theorem 2.1 delineates a procedure forconstructing Lyapunov function candidates for the singularly perturbedsystem (2.1). One starts by studying the equilibria of the reduced andboundary layer systems, searching for Lyapunov functions V(x) and W(x, z)that satisfy (2.24) and (2.20) respectively. Then inequalities (2.21) and(2.25), which may be called interconnection conditions, are checked. Severalchoices of V and W may be tried before one arrives at a choice forwhich (2.21) and (2.25) hold. At this point, the asymptotic stability of theequilibrium (x = 0, z = 0) is guaranteed for e sufficiently small. However,for a given value of e, say e0> one wants to improve the bound e* byexploiting the freedom that usually exists in verifying inequalities (2.21)and (2.25) (e.g. freedom in defining boundaries of the sets Bx and Bz). TheLyapunov function v(x, z) is known to be valid when e0 < e*. If so, oneuses the fact, illustrated in Fig. 7.1, that for a given eQ there is a range(d1, d2) such that any dE(di, d2) results in a valid Lyapunov function. Inother words, v(x, z) = (1 - d)V(x) + d(W(x, z), dE(dlt d2), defines aparameterized family of Lyapunov functions for (2.1). The freedom inchoosing d can be used to achieve other objectives like improving estimatesof the domain of attraction. To illustrate this point, let c be any positivenumber such that


Then L is included in the domain of attraction of the equilibrium. If Ls andLf(x) are estimates of the domains of attraction of the equilibria of thereduced and boundary layer systems, defined by

then c can be chosen as

which is maximum if d can be chosen as d = cs/(cs + c{).

Example 2.1

The second-order system

has a unique equilibrium at the origin. We shall investigate the stability ofthe equilibrium in three steps.

Step 1: Reduced system Setting e = 0 in (2.32b), we get

which when substituted in (2.32a) yields the reduced system

Taking V(x) = ix4, we have dV/dx = x2 and

Hence inequality (2.24) is satisfied with a{ = I and

Step 2: Boundary layer system Treating x on the right-hand side of(2.32b) as a constant and expressing (2.32b) in the fast time scale r = t/e,we obtain the boundary layer system

We take W(x, z) = %(x + z)2 so that inequality (2.20) is satisfied witha2 = 1 and (£>(x + z) = x + z .


Step 3: Interconnection conditions It remains now to verify inequalities(2.21) and (2.25):

Hence, inequalities (2.21) and (2.25) are satisfied with fi} = /32 = y = 1.

All the inequalities hold globally and V and W are radially unbounded.Therefore, by Corollary 2.3, the equilibrium is globally asymptoticallystable for all £ <^e* = 0.5. To see how conservative this bound is, let usnote that the characteristic equation of the linearized system is

which shows that the equilibrium is unstable for £ > 1 .In Step 1 above, we could have chosen V(x) = \x2, which would satisfy

(2.24) with al = 1 and ip(x) = x2. This choice, however, would not satisfyinequality (2.25) since \8V/dx = x . This observation serves as a guide inour search for Lyapunov functions that would satisfy the interconnectionconditions. We should search for Lyapunov functions that satisfy (2.20),(2.24) and the inequalities \dV/dx\ < K r f , \f(x, h(x))\ < K2^> and8W/dx\ ^ /C30. We conclude the example by the following remark.

Remark 2.1 Since all the conditions of Theorem 2.1 are satisfied withV(x) = lx4 and W(x, z) = i(* + z)2, we know that

is a Lyapunov function for the singularly perturbed system (2.32) fore < e*d Since our example is a simple second-order system, we may try tocompute the derivative of v with respect to (2.32) and see if we can get anupper bound on e less conservative than the one provided by Theorem 2.1:

7.3 STABILITY OF A SYNCHRONOUS MACHINE 301

It is apparent that the choice d = \ cancels the cross-product terms andyields

which is negative-definite for all e < 1 and is the actual range of e for whichthe origin is asymptotically stable.

Motivated by Remark 2.1, let us explore another angle of using singularperturbation decompositions in Lyapunov stability analysis. In the stabilityanalysis of Theorem 2.1, singular perturbation decompositions have beenemployed for two different purposes. First, they were used to defineseparate reduced and boundary layer systems whose Lyapunov functionswere used to form a Lyapunov function candidate for the full system. Thisprocess simplifies the search for Lyapunov functions since it is done at thelevel of the lower-order systems. Second, the singularly perturbed form ofthe system together with inequalities (2.20), (2.21), (2.24) and (2.25) wereused to show that the derivatiye of the Lyapunov function candidateis bounded by a negative upper bound. However, even if some of theabovementioned inequalities are not satisfied, the same Lyapunov functioncandidate may still be used to investigate the stability of the full system bycalculating its derivative along the trajectory of the full system.

The stability analysis of this section will be illustrated by two case studiespresented in subsequent sections. In the synchronous machine case studyof Section 7.3, we start by choosing functions V and W that satisfy all theconditions for the application of Theorem 2.1. Then, in the spirit of Remark2.1, we try another choice of W whose derivative with respect to theboundary layer system is only negative-semidefinite, so that (2.20) is notsatisfied. Nonetheless, we form v = (\ — d)V + dW, calculate its derivativewith respect to the full singularly perturbed system and show that d can bechosen such that v is negative-semidefinite.

The adaptive control case study of Section 7.4 is particularly interestingbecause it illustrates an extension of the method when the system has anequilibrium set rather than an equilibrium point. Subsequently, one triesto establish stability rather than asymptotic stability. A stability analysissimilar to the one preceding Theorem 2.1 is performed to construct aLyapunov function candidate and show that its derivative is negative-semidefinite.

7.3 Case Study: Stability of a Synchronous Machine

The stability analysis of Section 7.2 will be employed to estimate the domainof attraction of the stable equilibrium of a synchronous generator connected


to an infinite bus through a transmission line. The generator is representedby the one-axis model (Anderson and Fouad, 1977; Pai, 1981) given by

where

TdQ = direct-axis open-circuit transient time constant,xd = direct-axis synchronous reactance of the generator,x'd = direct-axis transient reactance of the generator,£q = instantaneous voltage proportional to field-flux linkage,

6 = angle between voltage of infinite bus and £q,D = damping coefficients of the generator,

EFD = field voltage (assumed constant),Pm = mechanical input to the generator (assumed constant),

a) = slip velocity of the generator,xe = reactance of the transmission line,V = voltage of infinite bus.

The variable E'q is usually slower than 6 and to as a result of a relativelylarge value of the time constant Td0. Viewing Eq as a slow variable and6 and CD as fast variables, we study the domain of attraction of thestable equilibrium using time-scale decomposition. The_first task is tobring (3.1) into the singularly perturbed form. Let E'q, 6 and w denotethe stable equilibrium of (3.1). The state variables are taken as x =(Eq - £q)/Eq, Z j = 6 - d and z2 = (o - d), so that x will be the slowvariable. In order to have the singularly perturbed form (2.1), we define eas E — 1/Tdo and change the time scale from T to t = r/rd0 to obtain

where

7.3 STABILITY OF A SYNCHRONOUS MACHINE

The reduced system is obtained by setting e = 0 in (3.2) to get

which when substituted in (3.2a) yields

The boundary layer system is defined by

303

In order for the reduced and boundary layer systems to be well defined, aswell as for other reasons that will be apparent later, we restrict x and Z j toa region defined by the following inequalities:

where the positive numbers r\ and 6 will be specified later; 6 should satisfy

To study the stability of the equilibrium of the reduced system (3.4), wechoose

as a Lyapunov function candidate. From the definition of N(x) it followsthat W(0) = 0, xN(x) > 0 and

where


We assume that the positive number 6 can be chosen such that KN < a/b.It follows that V(x) is positive-definite. The derivative of V(x) along thetrajectory of (3.4) is given by

Therefore inequality (2.24) is satisfied with ty(x) = \(ax - bN(x))\ andoy= 1. Next, we consider the boundary layer system (3.5). This systemhas an equilibrium at zl = h^x) and z2

= 0- We choose a Lur'e-typeLyapunov function candidate W(x, z) given by

where q > 1 and

Inequalities (3.6a, b) guarantee that W(x, z) is positive-definite. The deriva-tive of W(x, z) along the trajectory of (3.5) is given by

Using inequality (3.6c), we get

A comparison function for the boundary-layer system (3.5) is taken as

where § is an arbitrary positive number to be chosen later to improve theupper bound on e. With the above choice of the comparison function andwith the choice q = 1 + c(l - 0)r/£2, inequality (2.20) is satisfied with<x2 = c(l — 9)r]. It remains now to verify the interconnection conditions,


inequalities (2.21) and (2.25). For (2.21) we have

Using the inequality

where

and K} is an upper bound on dhjdx (i.e. \dhjdx\ ^ K{), which is given by

So, inequality (2.21) holds. Finally, for (2.25),

Thus all the conditions of Theorem 2.1 hold and we conclude that theequilibrium of (3.2) is asymptotically stable for all e < ed for any d e (0, 1),where£d =


Moreover, v(x, z), given by

is a Lyapunov function of (3.2).In (3.9) and (3.10) there are four unspecified positive parameters,

namely, d, 6, 77 and £. The first three should be chosen to satisfy certainrestrictions that were imposed throughout the derivation, while the fourthparameter | is arbitrary. In making these choices, we are guidedrequirement that ed (given by (3.9)) should be greater than £, and by ourdesire to obtain a large estimate of the domain of attraction. Let us illustratethat by a numerical example. Consider the typical numerical data (Siddiqee,1968): M = 147 x 10~4, x'A = 0.3, xe = 0.2, *d = 1.15, T'M = 6.6, Pm =0.815,V = 1 and £FD = 1.22, where M is in (per unit power) x second2/radian,Tdo is in seconds and all parameters are per unit. These data yield 6 =0.42 rad, a = 2.7, b = 1.7 and c = 2 x 104/147. In addition, we take A =£)/M = 4s~1. Choosing d = 0.01, 0 = 0.42, 77 = 0.35 and | = 0Ed- 0.1844, which is acceptable since 1/1̂ = 0.1515. Hence v(x, z) isgiven by

An estimate of the domain of attraction can be obtained by means of closedLyapunov surfaces as in (2.31). However, a better estimate can be obtainedby means of open Lyapunov surfaces (Willems, 1969). Let C and D be twopoints on the line x = - 6 such that for any point on the segment CD thederivative x is positive and let Cmax = min{^(*c, zc), V(XD, ZD)}. It can beshown that the set

is included in the domain of attraction. A sketch of the set t£ is shown inFig. 7.2.

Let us now use this example to elaborate on Remark 2.1 and thediscussion thereafter. Another Lyapunov function candidate for the bound-ary layer system (3.5) is the energy-type function given by


Fig. 7.2. Estimates of the domain of attraction.

This function, though, does not satisfy the conditions of Theorem 2.1, sinceits derivative along the solution of boundary layer system is only negative-semidefinite, so that inequality (2.20) is not satisfied. However, we maystill consider

as a Lyapunov function candidate. It is interesting to note that computingthe derivative of v along the trajectory of the full system (3s2) shows thatthere is a unique choice of d, namely d = b/(b + c), that makes v negative-semidefinite. Moreover, it can be checked that the trivial solution is theonly solution of v = 0 in the neighborhood of the origin. Thus v is aLyapunov function for all e > 0. In comparing the two Lyapunov functionsv and v, we note that in v there is no freedom in forming the compositeLyapunov function, while in v the parameter d is free and can be used togenerate a family of composite Lyapunov functions. The freedom in choos-ing d was used to enlarge the domain of attraction estimated by v in thedirection of the <5-axis (zraxis), at the expense of its extension in thedirection of the *-axis, compared with the domain of attraction obtainedby v (see Fig. 7.2). It is usually more beneficial to extend the domain ofattraction in the direction of the (5-axis (Saberi and Khalil, 1984). Otherestimates of the domain of attraction can be obtained by different choicesof the parameters d, 6,77 and f. The price paid for the freedom in choosing


d is the restriction of the range of eto(0, ed). Getting an acceptable ed

hinges on having A = D/M not too small. With the numerical values weused here we need A 5= 1, which is not unrealistic. Although for uncontrolledmachines when D accounts only for field damping a typical value of A couldbe as low as 0.2 (Anderson and Fouad, 1977), for a feedback-controlledmachine in which D accounts for damping introduced by power systemstabilizers typical values of A could be as high as 10 (Demello and Concordia,1969).

7.4 Case Study: Robustness of an Adaptive System

We consider a second-order plant

with unknown constant parameter a and a "parasitic" time constant e. It isrequired to design an adaptive control law to regulate the output y (i.e. fory(r) — » 0 as r — > sc). The design problem can be simplified by neglecting the"parasitic" time constant £ to obtain a reduced-order model

An adaptive control for (4.3) can be taken as

with the adaptive law proposed in Narendra et al. (1980):

Application of the adaptive control (4.4) to the reduced-order model (4.3)results in

To show that y(t)—> 0 as r — > =*, consider a Lyapunov function candidate

The derivative of V along the trajectory of (4.5), (4.6) is given by

7.4 ROBUSTNESS OF AN ADAPTIVE SYSTEM 309

Moreover, V - 0 only at the equilibrium sety = 0, K = constant. It is easilyseen that as t—> «=, V converges to a finite value and V converges to zero.Therefore y(t)-+ 0 and K(t)-+ constant as f — > =». This convergence holdsfor all bounded initial conditions y(O), K(Q), and lim K(f) depends on theinitial conditions.

We employ singular perturbation stability analysis to study the effect ofthe model-plant mismatch due to the parasitic time constant e. Applicationof the control law (4.4), (4.5) to the plant with parasitics (4.1), (4.2) resultsin the closed-loop system

which takes the form (2.1) with x = (y, K)J. Following the procedure ofSection 7.2, we start by analyzing the reduced and boundary-layer systemsand then investigate interconnections. The reduced system, obtained bysetting £ = 0 in (4.11), is given by

which is the closed-loop system obtained in the design stage by neglect-ing parasitics. We have already seen that V(x) = V(y, K) = iy2 +(l/2b)(K — K*}2 is a Lyapunov function for this system. We rewrite (4.8)as

The boundary-layer system is given by

where Ky is treated as a constant parameter. This linear system has aquadratic Lyapunov function W(x, z) = \(z + Ky)2. Furthermore,

Finally, to study interconnections, we analyze, in the notation of Section


7.2, (dV/dx)[f(x, z) -/(*, Ky)] and (dW/dx)f(x, z):

where y and /32 are positive constants such that for all values of interest ofy and K the following inequalities hold:

Thus the Lyapunov functions V(x) and W(x, z) satisfy inequalities (2.20),(2.21), (2.24) and (2.25) of Theorem 2.1. The only assumption of Theorem2.1 that does not hold is the requirement that y(x) = 0 if and only ifx =0. In the current problem y(x) = \y\, which vanishes at y = 0 for all K.The asymptotic stability conclusion of Theorem 2.1 does not hold, buttaking v = (1 — d)V + dW as a Lyapunov function candidate, we obtain

where

Consider now the set for some

7.4 ROBUSTNESS OF AN ADAPTIVE SYSTEM 311

c > 0. It can be easily verified that in !£(d, c), inequalities (4.20) and (4.21)are satisfied with

Choosing d = d* = P\/(P\ + /82), which maximizes ed, we obtain

Therefore for all (y, K, z) G 5£(c) and for each £ < e* z> ̂ 0, and z> = 0 onlyat the equilibrium set y = 0, /£ = constant, z = 0. Any solution of (4.9)-(4.11) starting from ££(c) remains inside it. Furthermore, v(y,K,z) is anonincreasing function of time which is bounded from below and henceconverges to a finite value vx. Therefore v(y(t), K(t), z(t)) — > 0 as t— * *=,which implies that y(t) -» 0, /£(?) — > constant and z(t) — > 0 as t — » =*.

For any bounded initial conditions y(0), K(Q), z(0), we can choose clarge enough so that (y(Q), ^(0), z(G)) G ££(c). Therefore we conclude thatfor any bounded initial conditions there exists an e^ > 0 such that for eache G (0, ej) the solution (y(t), K(t), z(t)) is bounded and y(f)-> 0, K(t)-+constant and z(0~~^ 0 a§ ^^ oc. This shows that the adaptive control law(4.4), (4.5) is robust with respect to the presence of a small parasitic timeconstant. On the other hand, if e is not small enough, the stability propertiesof (4.9)-(4.11) may be lost. In particular, linearization of (4.9)-(4.11) atthe equilibrium (y = 0, K = K0, z — 0) gives the characteristic equation

which has an unstable root for s > l/(a + KQ). Since K ( t ) is nondecreasing, itis clear that instability occurs if s > I /(a + AT(O)).

As an illustration of the stability properties established above, let a = 4,K* = 7, b = 5, y(0) = 1, K(0) = 3, z(0) = -2. Taking c = 4.23, it can beverified that (_y(0), K(0), z(0)) G ££(c) and e* = 0.012. Hence stability isguaranteed for each e< 0.012. On the other hand, instability occurs for£ > l/(a + /C(0)) = 0.143. Simulation results (loannou and Kokotovic,1984a, b) show that at e = 0.05 the response is stable, while at £ = 0.07 itis unstable. This shows that the bound e* = 0.012 estimated by our stabilityanalysis is not overly conservative. Finally, we use the estimates £* and!£(c) to observe that increasing the adaptive gain b for a fixed e reducesthe size of J£(c), and the stability properties can no longer be guaranteedif b > O(l/e). This is seen from the fact that e* depends on the product cband not on c and b individually. Increasing b can be offset by reducing c,


which reduces the size of !£(c). Moreover, for sufficiently large b and fixedc we have y = 0(61/2), /?2 = 0(6), and hence £* = 0(1/6), which showsthat the inequality £ < e* is violated if b > 0(l/e).

7.5 Stability Analysis: Nonautonomous Systems

In Section 7.2, we studied autonomous systems where / and g are notfunctions of e. In this section, it is shown that dependence of /and g on tand £ can be allowed as long as inequalities similar to (2.20), (2.21), (2.24)and (2.25) hold uniformly in t and e. Consider the system

Assume that the following conditions hold for all (r, jc, z, e) G [r0, °°) xflv x 5. x [0, e,].

Assumption 5.1

The origin (x = 0, z = 0) is the unique equilibrium of (5.1), i.e.

Moreover, the equation

has a unique root z = h(t, x) and there exists a class K function! p( • ) suchthat

Assumption 5.2: Reduced system

There exists a Lyapunov function V(t, x) satisfying(i) V(t, x) is positive-definite and decrescent, that is,

for some class K functions q \ ( - ) and qi(-}',

t A continuous function a:R—>R is said to be class K if (i) <*(•) is nondecreasing,(ii) or(0) = 0 and (iii) a(p) > 0 whenever p > 0.

7.5 STABILITY: NONAUTONOMOUS SYSTEMS 313

where ip( • ) is a continuous scalar function of x that vanishes onlyat x = 0.

Assumption 5.3: Boundary-layer system

There exists a Lyapunov function W(t, x, z) satisfying for some class

K functions q^( • ) and q4( • )

where $(•) is a continuous function of an Rm vector zf whichvanishes only at zf = 0.

Assumption 5.4: Interconnection conditions

V and W satisfy the following inequalities:

For simplicity the constants fa, fa, fa, y l 5 y2 and y2 are assumed to benonnegative. The term ey^2 in (5.7) is added to allow for more generaldependence of /on e. It drops out when/is independent of e. Similarly,(5.8) drops out when g is independent of e. In arriving at the aboveinequalities, one might need to obtain uniform bounds on e-dependentterms like Kl + eK2\ that is why e may be restricted to an interval [0, ej.

Consider a Lyapunov function candidate


From the properties of Vand Wand inequality (5.4) it follows that v(t, x, z)is positive-definite and decresent. Computing v with respect to (5.1) andusing (5.5)-(5.9), we obtain

where /32 = /32 + (32 and y2 = y'2 + y'2. It can be easily seen that for all

the right-hand side of (5.11) is a negative-definite quadratic form in ty and0. Hence there exists a positive constant K (possibly dependent on e andd) such that

Since </>2(zf) is a continuous positive-definite function of zf, there exists aclass *• function z such that

Therefore

Using (5.4), it can be shown that i^2(x) + q(\\z - h(t,x)\\) is positive-definite.Therefore v is negative-definite. Recalling Fig. 7.1, it is seen that ed ismaximum with the choice d = /V(/?i + Az)> which yields

Our conclusions are summarized in Theorem 5.1.

Theorem 5.1

Suppose that Assumptions 5.1-5.4 hold. Then the origin (x = 0, z = 0) isa uniformly asymptotically stable equilibrium of the singularly perturbedsystem (5.1) for all eG (0, min(e*, f j ) ) , where e* is given by (5.13).Moreover, for every d G (0,1), v(t, x, z) = (1 - d)V(t, x) + dW(t, x, z) is aLyapunov function for all e E (0, min (ed, e,)), where ed is given by (5.12).

7.6 COMPOSITE FEEDBACK CONTROL 315

Example 5.1

As an illustration of Theorem 5.1, we prove Lemma 4.2 of Chapter 5. Weuse the same notation as in Chapter 5. Choosing V and W as

where Ps and Pf are respectively the solutions of the Lyapunov equations(4.8) and (4.9) of Chapter 5. It can be easily verified (Exercise 7.4) thatV and W satisfy Assumptions 5.2-5.4 with y(x) = \\x\\2, <p(z - h(t,x)) =\\z + L0(t)x\ 2, a i = a2 ~ 1> 7i = 0 and )3i, /32 and y2 are upper boundson the time functions 2||Ps(0^i2(Oll2, 2||Pf(0(L0(0 + L0(0^o(0)ll2 and||Pf(0 + 2Pf(r)L0(0^ iiCOlb respectively. Hence the linear system (4.1) ofChapter 5 is uniformly asymptotically stable for all

which is the upper bound of Lemma 4.2 of Chapter 5.

7.6 Composite Feedback Control

The composite feedback control method, introduced in Chapter 3 for linearsystems, is now extended to nonlinear autonomous systems

where u E Rr is a control input. Let us assume that the open-loop system(6.1) is a standard singularly perturbed system for every u G Bu C Rr, thatis to say, the equation

has a unique root z = h(x, u) in Bx x Bz x Bu. The composite controlmethod seeks the control u as the sum of slow and fast controls,

where us is a feedback function of x,


and «f is a feedback function of x and z.

The fast feedback function Ff(jt, z) is designed to satisfy two crucialrequirements. First, when the feedback control (6.3) is applied to (6.1),the closed-loop system should remain a standard singularly perturbedsystem, i.e. the equation

should have a unique root z = H(x) in Bx x Bz. This requirement assuresthat the choice of Ff will not destroy this property of the function g in theopen-loop system. This is best illustrated in the linear case, where

and

It is clear that, when (A22, B2) is controllable, the matrix A22 - B2Gf canbe singular even when A22 is nonsingular. We did not need to explicitlyrequire A22 — B2G{ to be nonsingular in the linear case, since it is impliedby the stronger stability requirement ReX(A22 ~ B2Gf) < 0. The secondrequirement on Tf(x, z) is that it be "inactive" for z = h(x, «s), i.e.

The importance of (6.7) can be seen from the closed-loop equation

The requirement (6.7) guarantees that z = h(x, FS(JC)) is a root of

By (6.6), equation (6.9) has a unique root z = H(x). Thus, in view of (6.6)and (6.7),

holds as an identity. With (6.7) and (6.10), the reduced model of the closed-loop system (6.8) is given by

which is independent of Tf and is the same reduced model obtained from


the open-loop system (6.1) when u is taken as ws. Thus the design of theslow control us = Ts(x) can be carried out independently of the fast designrf.

Once Fs(;t) has been chosen, the boundary layer model of the closed-loop system is defined as

where x is treated as a fixed parameter. The requirement (6.7) is nowinterpreted as a requirement on the feedback control u{ = Ff(jc, z) not toshift the equilibrium z = h(x, Ys(x)) of the boundary layer system (6.12).The design of wf must guarantee that z = h(x, rs(*)) is an asymptoticallystable equilibrium of (6.12) uniformly in x. This alone may not be enough,and stronger technical conditions may be needed. A procedure based onTheorem 2.1 to design Ts and Tf such that the origin (x = 0, z = 0) becomesan asymptotically stable equilibrium of the closed-loop system is givenbelow.

Stabilization design

Step 1: Design a slow control «s = Ts(x) for the reduced system (6.11) suchthat x = 0 is its unique asymptotically stable equilibrium in Bx, and aLyapunov function V:Rn->R+, exists, guaranteeing for all x £ Bx

Step 2: With the knowledge of ws = Fs(;t), design a fast control uf =rf(jc, z) such that

(ii) g(x, z, rs(*) + rf(*, z)) = 0 has a unique root z = h(x, Ts(*)) inBx x Bz-

(iii) z = h(x, Ys(x) ) is an asymptotically stable equilibrium of the closed-loop boundary layer system (6.12), and there is a Lyapunov func-tion W: Rn x Rm^ R+ such that for all (x, z)EBxx Bz


Step 3: Verify the interconnection conditions

The comparison functions t/> and </> are as denned in Section 7.2.The composite feedback control

designed in these three steps has the following stabilization property.

Theorem 6.1

If uc of (6.17) is designed following the stabilization design, the origin is anasymptotically stable equilibrium of the closed-loop system

for all e < £d, where ed is given by (2.29) and d E (0,1). Moreover, v(x, z) =(1 - d)V(x) + dW(x, z) is a Lyapunov function of this system.

Proof This follows from Theorem 2.1.

Example 6.1

Consider

and let Bx = [-1,1] and Bz = [—%, $]. It is desired to design a feedbackcontrol law to stabilize the system at the equilibrium point x = 0, z = 0.Notice that the desired stabilization cannot be achieved by linearization atthe equilibrium point because of an uncontrollable pole at the origin. Wego through the design procedure as follows.


Step 1: The reduced system is given by

We design «s = ;t4/3. The closed-loop reduced system is x= —x5, and withthe Lyapunov function V(x) = £;t6 we get ijj(x) = x\5 and a^ = 1.

Step 2: The boundary layer system is defined by

We design «f = -3(z + *4//3). The closed-loop boundary layer system isdz/dr = -2(z + *4/3). The choice W = \(z + *4/3)2 yields ̂ = z + ^:4/31 andOf2 = 2.

Step 3: It is straightforward to verify that the interconnection conditionshold with Pi = I, /32 = I and y = |.

The composite control is formed as

Now the choice d = §7 yields ed = f • Therefore the origin is an asymptoticallystable equilibrium of the closed-loop system

for all e < f.

Remark 6.1 In this particular example, the estimate ed is conservativesince the origin is asymptotically stable for all e > 0, as can be seen fromthe Lyapunov function

whose derivative with respect to (6.18) is given by

and v(i) = 0 => z(t) = 0 ̂ x(i) = 0. Theorem 6.1 guarantees only localstability with an estimate of the domain of attraction, and only for e lessthan the given bound.


We conclude this section by observing that the boundary layer system(6.12) is dependent on the slow control Ts(x). This implies that the designof the fast control rf(jc, z) will, in general, depend on rs(*). An importantclass of systems for which such dependence can be eliminated is the casewhen g is linear in z and u, i.e.

In this case, the existence of a unique root of 0 = g is equivalent to thenonsingularity of A2(x), and the boundary layer system (6.12) takes theform

Shifting the equilibrium of (6.21) to the origin via the change of variables

yields

which is independent of Ts(x). The design of the fast feedback control «f =rf(*, z) is greatly simplified. In fact, all the requirements on Tf stated inStep 2 of the stabilization design will be satisfied if Ff is taken as

where Gf(jt) is designed such that

The existence of Gf(x) satisfying (6.25) is guaranteed if the pair (A2(x),B2(x)} is controllable uniformly in x, V* E Bx, i.e. the correspondingcontrollability Grammian is bounded from below by a positive-definitematrix. If, in addition, /is linear in z and u, i.e.

the design problem becomes much more tractable, since the reduced system(6.11) will be linear in ws.

7.7 NEAR-OPTIMAL FEEDBACK DESIGN 321

7.7 Near-Optimal Feedback Design

The composite control method proposed in the previous section is nowshown to yield near-optimal feedback design for a class of singularlyperturbed nonlinear regulators. The composite control stabilizes the desiredequilibrium and produces a finite cost which tends to the optimal cost fora slow problem as e — » 0.

The problem considered is to optimally control the nonlinear system

with respect to the performance index

where x E Rn and z G Rm are states, and u E Rr is the control. We makean assumption which, in addition to the differentiability and positivityproperties of terms in (7.1), (7.2), also guarantees that the origin is thedesired equilibrium.

Assumption 7.1

There exists a closed set Bx C Rn, containing x = 0 as an interior point,such that for all x E Bx functions a^, a2, A^, A2, A2

l, B1} B2, p, s, R andQ are continuously differentiable with respect to x; a1? a2, p and s are zeroonly at x = 0; Q and R are positive-definite matrices for all x G Bx; matricesAi,A2, B I , B2, R, Q, A2

l, R~l and Q~l are uniformly bounded on Bx; thescalar p + sjz + zTQz is a positive-definite function of its arguments, thatis, it is positive except for x = 0, z = 0, where it is zero.

Notice that the boundedness of Alt A^B^B^R, Q, A2l, R~l and Q~l

will hold on any bounded set Bx by the continuity of these matrices withrespect to x.

We define two, slow and fast, optimal control problems and combinetheir solutions to form a composite control. The slow problem is definedby taking u = ws, setting e = 0 in (7.1b) and eliminating z from (7.la) and


(7.2) using

The resulting slow problem is to optimally control the slow system

with respect to

where

We note that x = 0 is the desired equilibrium of the slow system (7.4), sincein view of Assumption 7.1, a0(0) = 0, and L(x, h(x, «s), MS) > 0, for x =/= 0and MS =£ 0.

Our next assumption imposes some growth conditions on the slowproblem.

Assumption 7.2

There exists a scalar positive-definite differentiable function ty(x), whosederivative dty/dx is bounded on Bx, such that the following inequalitieshold for all x G B:

From optimal control theory (Athans and Falb, 1966), it is well knownthat if V(x) is a unique positive-definite solution of the Hamilton-Jacobi


equation

then the minimizing control of the slow problem is given by

and the optimal value of 7S is V(XQ) .Assumptions 7.1 and 7.2 are checked on the given data of the problem.

Now we make a mild assumption (mild in view of (7.6) and (7.7)) on thesolution of the Hamilton-Jacobi equation (7.9).

Assumption 7.3

For all x G Bx (7.9) has a unique differentiable positive-definite solutionV(x) which satisfies

(When* is scalar and #0^0 ̂ o — cs > 0 V* G Bx, Assumption 7.3 followsfrom Assumptions 7.1 and 7.2.)

Assumption 7.3 allows us to use V(x) as a Lyapunov function for theclosed-loop slow system

Let us verify that V(x} satisfies inequality (6.13) of the stabilization designof the previous section. Using the Hamilton-Jacobi equation (7.9), wehave

Inequalities (7.6) and (7.11) imply that (dV/dx)aQ satisfies (7.14):


The right-hand-side inequality shows that V(x) satisfies (6.13); the left-hand-side inequality will be used later on in studying the performanceindex.

The fast problem is defined by freezing the slow variable x in a stretchedtime scale r and shifting the equilibrium of the fast system to the origin.The fast system is given by

where zf = z — h(x, «s). With (7.15) we associate the performance index

This problem has the familiar linear-quadratic form for each fixed x G Bx.

Assumption 7.4

The pair [A2(*), B2(x)] is controllable uniformly in x for all x £ Bx.

Recalling also that R(x} > kilm and Q(x) ^ k2Im for all x 6 Bx, we obtain,for each x £ Bx, the optimal solution of the fast problem

where K(x) is the positive-definite solution of the ^-dependent Riccatiequation

The control (7.17) is stabilizing in the sense that

It is straightforward to verify that wf satisfies requirements (i) and (ii) ofStep 2 of the stabilization design of Section 7.6. To show that requirement(iii) is also satisfied, consider the Lyapunov function


Thus inequality (6.14) is satisfied with 0 =The composite control is now formed as

zf

We shall establish that the composite control (7.22) possesses three impor-tant properties. First, it is a stabilizing control. Second, the full system (7.1)controlled by uc results in a bounded value Jc(x, z) of the performanceindex. Third, we obtain upper and lower bounds on Jc(x, z) in terms ofV(x), y(x) and W(x,z).

The fact that (7.22) is a stabilizing control will follow from Theorem 6.1if V(x) and W(x, z) are shown to satisfy the interconnection conditions(6.15) and (6.16). To verify (6.15) we note that

Assumptions 7.3 and (7.23) yield

Furthermore, in view of (7.7), (7.23) and Assumption 7.3, we have

Also, from (7.20)

If K is independent of x, and 5 .̂ is chosen such that (d/dx)[A2l(a2 +

B2ws)] is bounded, which is always possible at least by taking Bx to be abounded set, it will follow that | B W/dx ^ c60. If K is a function of x it isstill possible to show that \8W/dx is bounded by <p by restricting (x, z) toa bounded set Bx x #2 where zf is bounded. Thus it is always possible to


show that |aw/6*| < c60, which together with (7.25) imply (7.26):

With inequalities (7.14), (7.21), (7.24) and (7.26), all the conditions ofTheorem 6.1 are satisfied and the origin is an asymptotically stable equi-librium of the closed-loop system

for all e < ed, where ed is given by (2.29) for some d E (0,1). Furthermore,an estimate of the domain of attraction can be obtained by using theLyapunov function v = (1 - d)V + dW. Let D C Bx x Bz be such an esti-mate. From now on, all our discussions are limited to (XQ, z0) E D.

Asymptotic stability of an equilibrium point at the origin is not sufficientto guarantee that an integral of the type (7.2) will be finite along thetrajectories asymptotically converging to this equilibrium (for an examplesee Exercise 7.8). To qualify as a candidate for near optimality, MC mustproduce a bounded cost for all (*<,, z0) e D. To show that this is the casewe need first to verify the inequality

which is implied by Assumptions 7.1-7.3 since

The boundedness of /c is established in Theorem 7.1.

Theorem 7.1

Suppose that Assumptions 7. 1-7 A hold and let aQ, tf1? a2, /3l5 (32, y, dl

and 62 be defined by inequalities (7.14), (7.21), (7.24), (7.26) and (7.28).Let d and e be positive numbers such that d G (0, 1) and e > aQ/ai(l - d).Then there exists an e*(d, e) < ed such that for all e E (0, e*(d, e)), (7.29)holds:


Proof It can be seen that by inequalities (7.14), (7.21), (7.24), (7.26)and (7.28) we have

where

Hence, for all

Integrating (7.31) from 0 to o° and using the fact that v(x, z) —» 0 as f —» <*,the result follows. D

Notice that e*(rf, e) -^ ed as e —» =», which means that we can have £*(d, e)arbitrarily close to £d and still have bounded Jc.

Having established that Jc is bounded, we move now to compute upperand lower bounds on Jc - V. Let us define the constants b3 > 0, b4 > 0 andb5 > 0 by means of the following inequalities, which follow from ourAssumptions 7.1-7.4:

Let (x(o), z(o)} be the trajectory of the closed-loop system (7.27) startingat (t, ;c,z). For convenience, in the sequel we shall adopt the notationfc(x, z) = f ( x , z, MC) and Lc(x, z) = L(x, z, wc).

To obtain an upper bound on Jc(x, z) - V(x) we consider the function


where b6, b-, and b8 will be specified later. If we can show that q(x, z) > 0V(*, z) e D, it will follow that JC-V< e(b6V + b7W+ bsyW1/2). Thefunction q can be written as

The time derivative of V can be expressed as

Using (7.13), which holds as an identity in x, yields

where L0(*) = L(x, h(x, «s(*)), us(x)). Substituting (7.36) into (7.35), weobtain

Using inequalities (7.14), (7.21), (7.24), (7.25), (7.26), (7.28), (7.32) and(7.33), it can be shown that


where

Choosing £>6, b1 and b8 to satisfy

it can easily be seen that the right side of (7.38) will be nonnegative forsmall £. Indeed, it is nonnegative for all e G (0, ej where

A lower bound on /c - V is obtained, similarly, by showing that

is nonnegative for all (x, z) E D. It is interesting that this is true fore G (0, ej), with the same EI as in (7.40). Now, taking r = 0, x = x0 and z =z0, we arrive at the following theorem.

Theorem 7.2

Suppose that Assumptions 7.1-7.4 hold and let the constants aQ, or l5 a2,0i, 02, 7> 51? 62 and ̂ to fc5 be defined by inequalities (7.14), (7.21),


(7.24), (7.25), (7.26), (7.28), (7.32) and (7.33). Choose positive numbersd, e, b6 and b7 that satisfy

Then, for all eE(0,mm{e*(e,d),ei} and for all initial conditions(*0, z0) E D, we have

where e*(e, d), el and b8 are given by (7.30), (7.40) and (7.39) respectively.

As a consequence of Theorem 7.2 we have

Corollary 7.1

Under Assumptions 7.1-7.4 and for all (x0, z0) E D

Corollary 7.1 shows that MC is near-optimal in the sense that as £—> 0 theperformance of the system under the composite control tends to the optimalperformance at e = 0.

In calculating the bounds on e and the performance deviation bound(7.42), one does not have to start from Assumption 7.2, 7.3, and workone's way through verifying the various inequalities as we have just donein our analysis. A more direct and less conservative procedure is to obtainthe Lyapunov functions V(x) and W(x, z) and then proceed to verify theinequalities listed in the statement of Theorem 7.2. The fact that this groupof inequalities is indeed what is needed to establish the bound (7.42) bringsout a fundamental point in our analysis: neither optimality of MS and uf northe linearity in z and u is crucial for the analysis. What is really needed areMS, Mf and V, W for which that group of inequalities hold. For furthertreatment of the problem in that general framework the reader is referredto Saberi and Khalil (1985).

Example 7.1

The steps of the composite control design and the calculation of the bounds


will be illustrated by a simple example:

Assumption 7.1 is clearly satisfied VxE R. The slow problem is given by

Assumption 7.2 is satisfied with ip(x) = x 3. For dty/dx = 3*2sgn.x to bebounded on Bx, we have to take Bx as a bounded set. Let Bx = [-£, 5]. TheHamilton-Jacobi equation is

whose positive-definite solution satisfies

Thus

Obviously, Assumption 7.3 is satisfied. Moreover,

which shows that (7.14) holds with a0 = al = f. The fast problem is givenby

where z{ = z - us = z + k3, which yields


the positive root of which is K = i, so that

Therefore, W(x, z) = iz2 and

so that (7.21) is satisfied with a2 = 1. It is now straightforward to verify(7.24), (7.25), (7.26), (7.28), (7.32) and (7.33) to obtain the constants ft =1, b, = 1, b2 = f, r = A, ft = M, <$! = 1, 62 = 4, 63 = 64 = i and 65 = }.Let us start by obtaining an estimate of the domain of attraction. Weneed to choose d. Because the fast problem is linear-quadratic, we canarbitrarily extend the domain of attraction in the direction of the z-axis bychoosing d sufficiently small. However, we know from Fig. 7.1 that ed-* 0as d—> 0, so d should be kept large enough to produce an acceptable bounded. Let us suppose that e takes values less than one, so that any ed > 1 willbe acceptable. We choose d = i, which, using (2.29), yields ed = 1.13.Taking into consideration that the domain of attraction should be containedin Bx x R, where Bx = [-£, 5], we take D as

So as to apply Theorem 7.2, we start by choosing e. As we mentionedbefore, the choice of e is not crucial. Typically e will be chosen large enoughto yield £*(d, e) very close to ed. Taking e large increases the upper boundon Jc given by Theorem 7.1, but this is not important since a sharper boundon Jc is given by Theorem 7.2. Taking e = 1000 and using (7.30), we obtaine*(d, e) = 1.123. Now we choose 66 and b7, which should satisfy

Let us take b6 = 1 and b7 = 8, then using (7.40) we have £l = 1.127. Thuswe conclude that for all e E (0, 1.123) and for all (jt0, z0) £ D the followingbound holds:

In particular, using the fact that for any (x, z) G Z), *|^0.5 andz + 0.5*3| < JV3, we obtain

7.8 EXERCISES 333

7.8 Exercises

Exercise 7.1


which has an isolated equilibrium point at the origin.(a) Show that asymptotic stability of the origin cannot be established by

linearization.(b) Using Theorem 2.1, show that the origin is asymptotically stable for

all e E (0, e*). Obtain estimates of E* and the domain of attraction.Hint: Let Bx = {x E R: x < 1}, Bz = {z e R: |z < a}, where a is less

than, but arbitrarily close to, \n\ V(x) = lx4, W(x, z) = i(z - tan"1*3)2.

Exercise 7.2

Consider the tunnel diode circuit shown in Fig. 7.3. The nonlinearity £(u)is sketched in Fig. 7.4. Suppose the values of E and R are such that there

0, v0) as shown in Fig. 7.4.

Fig. 7.3. Tunnel diode circuit of Exercise 7.2.

(a) Choosing x = (v - VQ)/VO and z = (i - /o)Ao as dimensionless statevariables, t' = t/CR as dimensionless time and E = (L/R)/CR = L/CR2,show that the state equation takes the singularly perturbed form


Fig. 7.4. Tunnel diode nonlinearity.

where

and

(b) Show that the reduced system is given by

and verify using Fig. 7.4 that x[r](x) + x] > 0 V* =£ 0.(c) Verify that the functions

and

sa(2

itisfy all the assumptions of Theorem 2.1. In particular, show that (2.20),:.21), (2.24) and (2.25) hold V(*, z) e R2 as equalities with a} = a2 = 1,t = a, fa= -I/a, ?= I ,b

7.8 EXERCISES 335

and

(d) Show that the derivative of

along the trajectory of the singularly perturbed system is given by

(e) Show that1 the equilibrium is globally asymptotically stable for all

Exercise 7.3

Verify that V and W given in the proof of Corollary 2.3 satisfy all theassumptions of Corollary 2.2.

Exercise 7.4

Verify that Kand W of Example 5.1 satisfy Assumptions 5.2-5.4 as claimed.

Exercise 7.5

The stability analysis of Sections 7.2 and 7.5 employs quadratic-type Lyapu-nov functions. In this exercise, an alternative analysis that employs linear-type Lyapunov functions is developed. Consider the system (5.1) and letAssumption 5.1 hold. Let V(t,x) and W(t,x,z) be Lyapunov functionssatisfying part (i) of Assumptions 5.2 and 5.3 as well as the followinginequalities:


where ip(-) and 0(-) are positive-definite functions. For convenience let0i» 71,02, 02, 72 and y2' be nonnegative.

Let v(t, x, z) = (1 - d)V(t, x) + dW(t, x, z). Show that for all

where 02 = 02 + 02 and y2 = 72 + 72, d can be chosen such that thederivative of v along the trajectory of (5.1) is negative-definite. Give therange of permissible values of d.

Exercise 7.6

Apply the stability analysis of Exercise 7.5 to Example 2.1. Take V(x) =(*2)1/2 and W(x, z) = [(x + z)2]1/2. Calculate a bound on £ and show yourchoice of d.

Exercise 7.7

Let all the assumptions of Corollary 2.2 hold with ty(x) = ||*|| and0(z - /*(*)) = ||z - h(x)\\, and take aE. (0, (Xi/2e2). Show that d inv(x, z) = (1 - d)V(x) + dW(x, z) can be chosen such that j>^ -lav for all

Hence a is an estimate of the degree of exponential stability. Estimate thedegree of exponential stability of the system

when £ = 0.1.

Exercise 7.8

Consider the system x = x2 + u and the cost J = /Q (x4 + |w2) dt. Let u =


(a) Show that the closed-loop system has an asymptotically stable equi-librium point at x = 0.

(b) Show that for *(0) = x0 * 0, x(t) = sign(*0) [4f + *o4]~1/4.(c) Show that / is infinite.

Exercise 7.9

(a) Consider the linear-quadratic regulator defined by (4.1)-(4.3) ofChapter 3. Suppose that [Mt M2] is a square nonsingular matrix, and theslow system pair (A0, B0) and the fast system pair (A22, B2) are eachcontrollable. Verify that Assumptions 7.1-7.4 are satisfied.

(b) Suppose that (A0,'B0) and (A22, B2) are only stabilizable. ClearlyAssumption 7.4 is not satisfied. Verify, however, that all the inequalitiesstated in Theorem 7.2 are satisfied. Hence Theorem 7.2 holds.

Exercise 7.10

Apply Theorem 7.2 to Example 4.1 of Chapter 3 to calculate a bound onJc(x0, z0) - V(x0)\ as in (7.42).


Stability properties of nonlinear singularly perturbed systems have beenextensively studied in the literature. The exposition of Sections 7.2,7.3 and7.5 using quadratic-type Lyapunov functions follows the recent treatment ofSaberi and Khalil (1984). For further reading consult Klimushchev andKrasovskii (1962), Hoppensteadt (1967), Habets (1974), Chow (1978a) andGrujic (1981). The adaptive control example is patterned on loannou andKokotovic (1984a,b). A key point in the composite control developmentof Section 7.6 is the design of the fast control to be inactive on the slowmanifold of the singularly perturbed system under slow control. As aconsequence of this design, the slow system is invariant with respect to thefast design. This invariance property, which was observed by Suzuki (1981),has been shown in Chapter 3 for linear systems using matrix manipulations.The presentation of Section 7.6 is based on Saberi and Khalil (1985).The near-optimal regulation of Section 7.7 was first given in Chow andKokotovic (1981). Out treatment is based on Saberi and Khalil (1985),where the restriction of the Hnear-quadratic form of the regulator problemwith respect to the fast states and control inputs is removed.

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high-gain feedback systems. IEEE Trans. Autom. Control 22, 931-938.Zien, L. (1973). An upper bound for the singular parameter in a stable, singularly perturbed

system. J. Franklin Inst. 295, 373-381.

References Added in Proof

Kokotovic, P. and Riedle, B. (1986). Stability bounds for slow adaptation: integral manifoldapproach. Proc. 1986 American Control Conf., Seattle.

Visser, H. G. and Shinar, J. (1986). First-order corrections in optimal feedback control ofsingularly perturbed nonlinear systems. I.E.E.E. Trans Automatic Control 31, 387-393.

APPENDIX AAPPROXIMATION OF SINGULARLYPERTURBED SYSTEMS DRIVEN BY WHITENOISE

Consider the linear time-invariant singularly perturbed system

where x E R", y E Rm, w E Rr and 0 < ^ = ju(e) < C. The system matrices areanalytic functions of e at e = 0, i.e.

The input w(/) is zero-mean stationary Gaussian white noise with intensitymatrix V>0. The initial conditions x0(e) and Zo(e) are jointly Gaussian randomvectors independent of w with moments that are analytic at e = 0. T0 guaranteeasymptotic stability of (A 1) for sufficiently small e, it is assumed that

The purpose of this Appendix is to study approximations u(t) and v(t) of x(f)and z(t) respectively for small e, where u(i) and v(t) are defined by

The matrices >4,y(e) and 5,(e) are analytic functions of e which are O(eN) closeto the corresponding matrices Atj and Bh where N is a positive integer, i.e.

357

358 APPENDIX A

The initial conditions u0(e) and v 0(e) are jointly Gaussian with moments that areanalytic at e = 0. It is assumed that

In other words, the approximation (A3) is obtained by making O(eN) pertur-bations in the matrix coefficients and initial conditions of (Al), where the per-turbations in initial conditions are taken in a mean-square sense. In order tovalidate approximating x(t) and z(t) by u(i) and v(f), we study the means andvariances of the approximation errors ex(t) = x(t) - u(t) and ez(t) —z(i) - v(t)for small E. The mean of a stochastic process will be denoted by a bar over thevariables, e.g. x(t) is the mean of x(t). We begin by stating a lemma that will beused repeatedly in our study. Lemma A.I follows from the decouplingtransformation of Chapter 2 in an obvious way.

Lemma A.1

Consider the singularly perturbed system

where

where /3 is a nonnegative number. Then there exists an e* > 0 such that for alle E (0, e*), £(f) and r](t) exist for all / > 0 and satisfy

where O(f /j) holds uniformly for all t > 0.

The means jc(^) and z(f) are governed by

The stability criterion (A 2) and Lemma A.I guarantee that, for sufficientlysmall £, x(t) and z(0 exist and are uniformly bounded in e and t for all ? > 0.Employing (A 4) and (A 5), it is easy to see that ex(f) and ez(f) satisfy

APPENDIX A 359

Again, application of Lemma A.I and the use of (A2) yields

To study the variances of ex(t) and ez(t), we consider the following system of dif-ferential equations driven by white noise:

The above equation is rewritten as

with obvious definitions for y, si and S&. Let Q° be the variance matrix of theinitial condition y(O). The matrix Q° is partitioned, compatibly with thepartitioning of y, as

The initial condition closeness assumption (A 5) and the use of the Schwartzinequality imply

Let (2(0 be the variance matrix of y(t); then 0(r) satisfies the Lyapunov matrixdifferential equation

Because of the presence of 1/e terms in si and S6, Q is sought in the form

360 APPENDIX A

Substituting (All) in (A 10) and partitioning the Lyapunov equation, we obtainthe following ten equations:

Thus, by seeking 0 in the form (A 11) the study of the variances of ex and ez hasbeen replaced by the study of the solution of the linear time-invariant singularlyperturbed system (A 12). We divide (A12a)-(A12j) into three groups ofequations and apply Lemma A.I successively. Consider first (A12a, c, h). Thesethree equations form a singularly perturbed system in Qn, <213 and <333. Thissystem of matrix equations can be rearranged in the form of vector equations.Employing Lemma A.I and the stability condition (A 2) shows that, for suffi-ciently small e, 0n, Qn and 033 exist for a l l f > 0 and are 0(1). Next, consider(A12b, d,f , i). These equations form a singularly perturbed system in 012, Q14,023 and 034 with driving terms dependent on (?,,, <2,3 and <233. Equations (A 4)and (A 9) imply that the driving terms and initial conditions of these fourequations are O(eN}. Thus by applying Lemma A.I we conclude that for suf-

APPENDIX A

ficiently small e,

Finally, consider (A12e,g, j) , which form a singularly perturbed system in Q22,Qu and Q^. Equations (A 4), (A 9) and (A 13) show that the driving terms andinitial conditions of these three equations are O(e2N). Thus Lemma A.I implies

Combining (A 8) with (A 14) and recalling the form of the variance matrix(All) completes the proof of the following theorem.

Theorem A.1

Consider (A 1) and (A3) and assume that (A 2), (A 4) and (A 5) are satisfied. Thenthere exists an e* > 0 such that, for all £ £ (0, e*), (A 15) holds for all t > 0:

Theorem A.I establishes that (u(t), u(t)) is a valid approximation of (x(t), z(t))as e^ 0. It is interesting to observe that the mean-square order of approximationof the slow variable x is the same as the order of perturbation of parameters andinitial conditions, while for the fast variable z the order of approximation is less by2 owing to the effect of the O(\/E) variance of z.

Frequently, the solution of (Al) is of interest only at the steady state, i.e. ast-* °°. In the steady-state situation the initial condition closeness assumption (A 5)is not required since the part of the solution due to initial conditions decays to zeroas f — » oo, in view of the asymptotic stability of the system. Furthermore, sharperbounds on the approximation errors can be obtained for the case jU < 1. TheoremA. 2 summarizes our conclusions for the steady-state case.

Theorem A.2

Consider (A 1) and (A3) and assume that (A 2) and (A 4) are satisfied. Then thereexists an e* > 0 such that, for all e £E (0, e*) (A 16) holds as t — » °c:

Proof Consider yT(t) = (xj(t), ef(t), zT(f), ej(0) as before. At steady state, thevariance of y(t) satisfies an algebraic Lyapunov equation that is the equilibrium of(A 10). Seeking Q in the form (All), it follows that the Q,y satisfy algebraicequations that are the equilibrium of (A 12). Writing Ql} • = ^2Qij and dividing theequations by jW2 shows that the Qtj satisfy the equilibrium of (A 12) with ^ = 1. Theexpression (A 17) follows as a limiting case of (A 13) and (A 14) when f — » o°.

Equation (A 17) implies (A 16).

361

o

362 APPENDIX A

We conclude this Appendix by studying the effect of employing the approximation(A 3) on evaluating a quadratic form in x and z given by

where C { ( e ) and C2(e) are analytic at e = 0. Such quadratic forms appear inthe steady-state LQG control problem (Section 4.4). We examine the followingapproximation of o by a:

where

Theorem A.3 shows the order of magnitude of the error in approximating o by a.

Theorem A.3

Consider (Al), (A3) and the scalar quadratic forms (A 18) and (A 19). Assumethat (A 2) and (A 4) are satisfied. Then there exists an e * > 0 such that, for allf e (0. f*),

Proof a and o are given by

where the ()// are as defined in the proof of Theorem A. 2. Using (A 17) and (A 20)we get

where Q(^ is the Mh-order term in the Maclaurin series of (?34(£). It can be verifiedthat <233(0) and Q(^ are respectively the unique solutions of the algebraic Lyapunovequations

APPENDIX A 363

On the other hand

If tr[Cl(0)C2(0)Q33(0)]^0, dividing (A22) by (A25) yields (A21). Iftr[CT(0)C2(0)(?33(0)] = 0 such division will only show that Aa/a = O^'1);further analysis is needed in this case to establish (A 21). Suppose that

and recall that Qw(ty satisfies (A 23); hence it is given by

Substituting (A 27) in (A 26) yields

which implies

Hence

Since, by (A24),

using (A 27) and (A 28) it can be shown that

Equations (A 29) and (A 30) imply that Aa = O(eN), which completes the proof ofTheorem A. 3.

The above analysis shows that Aa is O(l) if and only if (A 28) is satisfied, whichis equivalent to

That is to say, Aa is 0(1) if and only if the fast transfer matrix Gf(p, e)from the input w to the output yc = C\x + C2z is identically zero at e = 0.

APPENDIX B

We verify the matrix identity <2i(0) - //i(0)Q2(0) = <2S, which is used in the proofof Theorem 3.1 of Chapter 4.

The matrix identity (B 1) follows by dualizing the coefficients of x in (4.4) and(4.23) of Chapter 3:

Now

But

Hence

Using (B 1), we obtain

Observing that

shows that

365

INDEX

Actuator form, 50-54, 81Adaptive systems

approximate models, 25-28block-diagonal form, 62-63model-plant mismatch, 62, 221, 307robust stability, 220-225, 306-310

Approximate models, 22-28, 62validation, 67-75

Approximationscorrected eigenvalue, 59corrected state, 73eigenvalue, 57frequency-domain, 85-88matrix H, 61, 212, 214matrix L, 52, 212, 214state, 9, 10, 68, 96, 100state estimate, 170stochastic control, 178, 193stochastic systems, 353-359time-varying state, 225-227, 270valid stochastic state, 158-166, 192

Asymptotic expansions, 2, 11, 12, 59, 128,164, 165, 169, 186, 210, 275, 276, 285

two-time-scale, 11, 12Averaging techniques, 28, 199

Block-diagonal form, see also L-H two-timescale transformation

closed-loop, 97stochastic, 164time-invariant, 60-67, 267time-varying, 209

Block-triangular form, 49-56actuator form, 50-54, 81sensor form, 50, 55, 56, 81, 88

Boundary-layeraircraft climb or landing, 32, 233, 248correction, 10, 235initial, 9, 248-259jump, 69model, 2nonlinear optimal control problem, 257-

259, 265-267, 273-279regulator problem approximation, 253-

259, 270-272series, 2stability, 10system, 10, 19terminal, 248-259

Bounds, see Singular perturbationparameter bounds

Characteristic polynomialfast, 56, 57, 104slow, 56, 57, 104

Cheap control, 140-142, 279-283Composite state feedback control

asymptotic validity, 95eigenvalue assignment, 102-110linear time-invariant, 94-102near-optimal, 110-128, 319-330nonlinear system, 313-318stochastic system, 176-178

Controllability, 75-84complete, 79, 110, 113, 136eigenvalue, 77, 78Grammian, 84, 228invariance to fast feedback, 102, 152rank criterion, 82, 228, 250strong, 80, 125

367

368 INDEX

Controllability continuedtime-varying systems, 228-237uniform, 322weak, 80, 109, 126, 146

Correctedeigenvalue assignment, 105linear quadratic design, 128-135linear quadratic Gaussian (LOG) design,

186-191models, 58, 62, 99-102PID design, 74, 75state feedback, 100-102

Decoupled form, see Block diagonal formDecoupling transformation, see L-H two-

time scale decompositionDetectability, 77, 78, 111, 113, 126, 141,

175Dichotomy of optimal control, 252, 269Discrete-time systems, 88, 152, 153

Eigenspacefast, 63properties, 60-67slow, 63

Eigenvalueapproximation, 57assignment, 102-110closed-loop approximation, 103corrected approximation, 59corrected assignment, 105gap, 58high-gain assignment, 137properties, 56-60ratio, 54

Eigenvectorapproximation, 64

models, 84-88robustness design, 147-151

Frequency scales, 85, see also High-frequency and

Low-frequency scales

Hamiltonianfunction, 261, 266, 273, 280matrix, 250, 262, 264, 280, 281

Hamilton-Jacobi optimality condition, 258,320-321

High-frequencyapproximations, 85-88models, 84-88periodic inputs, 28scale, 56, 71, 85, 148signal transmission, 150

High-gainadaptive instability, 224amplifier, 5, 12, 43cheap control, 140-142, 279-283feedback, 32, 82, 126, 136loop, 51

Ill-conditioning, 2, 70, 93Implicit function theorem, 172Initial value

problem, 10, 268theorem, 11, 68, 73, 95, 100, 163, 170,

178, 188, 192, 193, 270Inner series, 2

Kalman-Bucy filter, 166-174valid reduced order, 174

Fastinput, 27manifold, 15-22transient, 9, 14

Filtering, 166-174reduced-order, 174

Finite-time optimal control, 248-286Frequency-domain

approximations, 85-88

Lagrangian, 262Langevin equation, 165-166L-H two-time-scale decomposition

Kalman-Bucy filter, 172LOG systems, 189-191linear time-invariant feedback systems, 96

linear time-invariant systems, 61, 88linear time-varying systems, 208-215, 267

Linearization, 47, 275, 309

INDEX 369

Linear quadratic control, 110-128corrected design, 128-135

Linear quadratic Guassian (LQG) control,174-181

corrected design, 186-191Low-frequency

approximations, 85-88circuit equivalent, 7models, 84-88scale, 56, 148

Low-passfilter, 150property, 85, 148

Lyapunov equationscaling, 84stability, 215-220, 288-299, 310-313

corrected stochastic control, 188-189linear time-varying control, 267-272nonlinear systems, 319-330reduced control, 119-121

Nonstandard singular perturbation model,8, 28-35, 140

Nonlinear systemsapproximate models, 22-28composite control, 313-318finite-time optimal control, 257-261,

265-267, 273-279slow and fast manifolds, 17-21stability analysis, 288-299, 310-313stabilization design, 315-316standard singular perturbation model, 2,

3,5time-scale properties, 9-17

Manifoldcondition, 18, 22equilibrium, 19, 29fast, 15-22, 29invariant, 18slow, 15-22

Minimum energy control, 231-235, 248Model

actuator form, 50-54, 81approximate, 22-28, 67-75approximate fast, 23approximate slow, 23block diagonal form, 60-67, 97, 164, 209,

267block triangular form, 49-56corrected, 58, 62, 99-102exact slow, 19frequency-domain, 84-88linear, 21-22, 28-31linearized, 47, 275linear time-invariant, 47-92nonlinear, 2, 9, 17-21plant mismatch, 62, 221, 307sensor form, 50, 55, 56, 81, 88stochastic, 158-166

Near-optimalitycheap control, 142composite control, 114-118composite stochastic control, 178-181corrected control, 128-135

Observability, 75-84complete, 79, 111, 113eigenvalue, 77, 78Grammian, 83, 239rank criterion, 82, 238strong, 80, 126time-varying systems, 237-242weak, 80, 126, 146, 151

Observer, 149Optimal control

Bellman optimal value function, 258boundary layers, 248-259boundary-layer regulator approximation,

253-259, 270-272constrained, 276-279dichotomy, 252, 269finite-time, 248-286Hamiltonian

function, 261, 266, 273, 280matrix, 250, 262, 264, 280, 281

Hamilton-Jacobi optimality condition,258, 320-321

Lagrangian, 260linear quadratic, 110-128nonlinear, 273-279reduced problem, 259-267Riccati differential equation, 251singular, 279-283stochastic, 176-178, 186-191time-optimal, 276-283

370 INDEX

Output feedback design, 143-151Outer series, 2

Parasitics, 1adaptive systems, 25, 62circuit capacitance C, 8circuit resistance R, 9destabilizing effects, 63, 143, 146, 151,

223-225model-plant mismatch, 62, 221, 307motor inductance L, 4time constant, 1, 26, 278

PID control, 15-17

Quasi-steady-state, 9model, 3

Reduced-order model, 3validity, 13

Regulatorsboundary-layer, 253-259, 270-272linear time-invariant, 110-128linear time-varying, 267-272near-optimal, 110-128

Riccati algebraic equationcheap control, 141, 281Kalman-Bucy filter, 167linear quadratic control, 110, 113, 114,

128linear quadratic Gaussian (LOG)

control, 175Riccati differential equation, 251Robust stability

adaptive system, 26, 63, 220-225, 306-310

output feedback design, 82, 99, 143-151state feedback design, 98, 143, 146

Scaling of singular perturbation parameter,35-43

airplane model, 37DC motor model, 36Lyapunov equation, 84states, 42system parameters, 38

voltage regulator, 40white noise, 183-184, 191-194

Sensor form, 50, 55, 56, 81, 88Separation of time scales, see also Two-

time-scalenonlinear, 9-15, 274slow and fast manifolds, 17-22time-invariant, 267time-varying, 208-215

singular arcs, 279-283Singular control, 279Slow

exact model, 19inputs, 27manifold, 17-22model invariance to fast feedback, 102,

152nonlinearities, 25transient, 16

Slowly varying systems, 201-208, 249, 290-292

Stabilityadaptive system, 220-225, 306-310boundary-layer, 10closed-loop, 98, 101conditional, 19exponential, 295high-gain, 138nonlinear autonomous systems, 288-299nonlinear nonautonomous systems, 310-

313output feedback, 146robustness, see Robust Stabilityslowly varying, 203synchronous machine, 299-306uniform asymptotic, 215-220

Stabilizability, 77, 78, 111, 113, 136, 141,175

Standard singular perturbation model, 2, 3,5

Stateapproximation, 9, 10, 68closed-loop approximation, 96, 100estimation, 166-174time-varying approximation, 225-227',

270trajectories, 14, 17, 72, 109valid stochastic approximation, 158-166,

192Stiffness, 2, 70, 93

INDEX 371

Stochasticcontrol, 176-178, 186-191model, 158-166, 353-359state estimation, 166-174

Synchronous machine stability case study,299-306

Three-time-scale model, 34, 35Tikhonov's theorem, 11Time constants

parasitic, 1, 26, 278Time-optimal control, 276-283Time-scale, see also Two-time-scale

model, 2properties, 9-17stretched, 10

Trajectory optimization, 248-286Transmission zeros, 142Two-time-scale

cheap control, 140-142, 279-283decomposition with white noise, 158-166eigenvalue assignment, 102-110feedback design, 93geometric view, 17-22

high-gain design, 137L-H transformation, 61, 88, 96, 172,

189-191, 208-215, 267output feedback, 145PID control, 15-17property, 54, 58stability, 58

Uncertaintiesstructured, 143white-noise, 158-166wideband noise, 198

Validation ofapproximate models, 67-75stochastic models, 158-166

White noiseinputs, 158-166, 183power spectra, 162scaled inputs, 183-184, 191-194

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