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Robustness analysis of uncertain linear systems and robust stabilization of uncertain delayed systems

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Robustness analysis of uncertain linear systems and robust stabilization of uncertain delayed systems
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Basker, Varadharajan R., 1970-
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Linear systems ( jstor )
Mathematical robustness ( jstor )
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Thesis (Ph.D.)--University of Florida, 1998.
Bibliography:
Includes bibliographical references (leaves 141-147).
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Typescript.
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Vita.
Statement of Responsibility:
by Varadharajan R. Basker.

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ROBUSTNESS ANALYSIS OF UNCERTAIN LINEAR SYSTEMS AND
ROBUST STABILIZATION OF UNCERTAIN DELAYED SYSTEMS















By


VARADHARAJAN R. BASKER


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1998






































To my parents

Smt. Shyamala & Mr. Varadharajan Radhamani













ACKNOWLEDGMENTS

I express my deep and sincere gratitude to my advisor Professor Oscar Crisalle for

his continuing encouragement, guidance, and support throughout the course of my Ph.D.

program. Apart from introducing me to the exciting area of control, he enhanced the

spectrum of my research activities with a tremendous support system of collaborations

with other faculty and talented graduate students in theoretical as well as practical

projects, and provided me with outstanding facilities. He has always encouraged me to

acquire as diverse an education as possible. It is due to his constant encouragement that I

am also acquiring an extra degree from Electrical & Computer Engineering Department.

His hard-working nature, quest for perfection, unfailing courtesy, genuine care for the

aspirations of others, and outstanding ability to work successfully with people of various

temperaments, skills and backgrounds have taught me immensely. I think I could have

obtained as good an education in control theory from any good university. But, I

sincerely doubt whether I could have secured, as varied an exposure as I received here

under his guidance.

I wish to thank Professor Haniph Latchman (along with Prof. Crisalle) for teaching

me much of the robust control that I know. It was a truly a pleasure to interact with him

on the robustness research for uncertain systems. His enthusiasm has been contagious.

I cherish the opportunity that I had to work with Professors Fricke (and Crisalle) on

the design and construction of the UF pilot plant for the DOE project on evaluation of on-

line viscometers. Although our applied research work is not part of my dissertation and

are published elsewhere, I spent nearly two years of my stay here on the building of the

pilot plant and supporting the data acquisition and control operations. Without the

practical knowledge that I acquired during the course of this project, my education would







have been very much incomplete. His respect for labor, his width and depth of

engineering knowledge, and his enthusiasm for work at his age, are truly inspiring.

I would like to thank Professor Spyros Svoronos for serving in my supervisory

committee. I would also like to thank Professors Richard Dickinson and William

Edmonson for serving on the supervisory committee at a short notice. I would also like

to thank the support staff in the Chemical Engineering office; particularly Tracy, Janice,

Shirley, and Nancy for all the precious help over the years. The financial support from

DOE and NSF are gratefully acknowledged.

The friendship and support of my colleagues in the lab will always be appreciated.

Particularly, I would like to thank Kostas and Ke-Jian for collaborations in Variable

Structure control and Mike for endless and fruitful discussions over these years. It was a

pleasure to work with Tony Dutka, Tiago, Barbara and Rick Gibbs in the pilot-plant. The

presence, friendship and the assorted mischiefs of Jon over these years has certainly

lightened the stress of working in multiple projects. Many have contributed to my

personal growth and enjoyment over this period of my life; it is not possible that I

mention everyone by name. However, I would like to specifically mention a few. The

company of my best friend, Madhav Durbha, has been invaluable during this period. It

has been my pleasure and privilege to have known Professors Ranga and Vasudha

Narayanan. The kindness and the compassion that they show to the students and the

success and harmony that they have achieved in their lives, has provided me with a model

of modern Indian culture that I shall seek to follow in my own life.

My brother and sister-in-law have consistently supported my efforts and they are

always in my thoughts. Finally, after all these years of unlimited love and unexpecting

affection, thanking my parents would be like thanking myself. Neither of them went to

college, but it is due to their numerous sacrifices, small and big, mostly untold, that I

have been able to live some of their dreams. Mere words, in any of the languages that I

know, are inadequate to express my gratitude. I dedicate this work to them.








TABLE OF CONTENTS


ACKNOWLEDGMENTS .................................................................................. iii


A B ST R A C T ............................................................................................... viii


1. IN TR O D U CTIO N ......................................................................................... 1


2. THE NYQUIST ROBUST STABILITY MARGIN--A NEW METRIC
FOR THE STABILITY OF UNCERTAIN SYSTEMS ........................... 6

2.1 Introduction .......................................................................................... 6
2.2 Frequency-Domain Approach to Robust Stability Analysis ................... 7
2.3 The Critical Direction Method for SISO Systems ................................... 13
2.4 The Critical Direction Method for MIMO Systems ................................. 17
2.5 E xam ples ................................................................................................. 22
2.6 C onclusions .......................................................................................... 28

3. A NEW PERSPECTIVE ON COMPUTING ROBUST STABILITY
MARGINS FOR COMPLEX PARAMETRIC UNCERTAINTIES ...... 34

3.1 Introduction ............................................................................................. 34
3.2 Background and Preliminaries ................................................................. 35
3.3 The Critical Direction for a Characteristic Polynomial ........................... 39
3.4 M ain R esults .......................................................................................... 40
3.5 Parametric Robust Stability Margins for Highly Structured
U ncertainties ....................................................................................... 43
3.6 Connections with the Classical M A Formalism ................................ 45
3.7 C onclusions .......................................................................................... 47

4. VARIABLE STRUCTURE CONTROL DESIGN FOR REDUCED
CHATTER IN UNCERTAIN STATE DELAY SYSTEMS ................... 49

4.1 Introduction .......................................................................................... 49
4.2 Sliding Mode Control Design ................................................................. 51
4.3 Asymptotic Stability with Perturbation Compensation .......................... 58
4 .4 E xam ple ....... ................................................................................ 64
4 .5 C onclusions .......................................................................................... 65








5. SLIDING MODE CONTROL FOR UNCERTAIN INPUT DELAY
SY ST E M S ........................................................................................... 69

5.1 Introduction .......................................................................................... 69
5.2 Preliminaries and Problem Formulation ................................................. 71
5.3 The Design of the Sliding Mode Controller ............................................. 72
5.4 Analysis of the Perturbed System ............................................................ 77
5.5 Some Open Issues in Sliding Mode Literature ........................................ 86
5.6 Constraints on the Control Input ............................................................. 99
5.7 Extension to the Case of Multiple Delays ................................................... 101
5.8 Illustrative E xam ple ..................................................................................... 102
5 .8 C onclusion ............................................................................................... 103

6. FU T U R E W O R K ................................................................................................ 106

6.1 Robust Stabilization of Uncertain Input-Delay Systems-A Linear
Matrix Inequality Approach .................................................................... 106
6.2 Automatic Covariance Resetting for On-Line Recursive
Identification ........................................................................................... 119
6.3 Sliding Mode Control of Bilinear Systems .................................................. 124


APPENDIX A


APPENDIX B


APPENDIX C


APPENDIX D


APPENDIX E


APPENDIX F


APPENDIX G


...................................... .... ........... oo .............. ...... 12 6


.......... I .................................... .... ................................. 1 3 0


............................................................................................... 1 3 2


............... ............. ..................................... .............. ................ 1 3 3


............................................................................................... 1 3 4


............................................................................................... 1 3 6


............................................................................................... 1 3 9








BIBLIO G RA PH Y ............................................................................................... 141


BIO GRAPH ICAL SKETCH .............................................................................. 148











Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


ROBUSTNESS ANALYSIS OF UNCERTAIN LINEAR SYSTEMS
AND
ROBUST STABILIZATION OF UNCERTAIN DELAYED SYSTEMS


By


VARADHARAJAN R. BASKER

May 1998


Chairman: Dr. Oscar D. Crisalle
Major Department: Chemical Engineering

This dissertation focuses on two main aspects. One, developing new tools for the

robustness analysis of uncertain linear systems. Two, the synthesis of robust controllers

for uncertain delay systems.

Traditional methods to analyze the robust stability of linear systems have depended

on structured singular value bounds. However, this approach neglects phase dependence

of the uncertainty and results in undue conservatism. In this dissertation, the concept of

the Nyquist robust-stability margin, is introduced for characterizing the closed-loop

stability of uncertain systems. The approach makes direct use of Nyquist domain

arguments and is based on the analysis of the perturbed eigenvalue loci, hence avoiding

undue conservatism through the use of singular-value upper bounds. A key element in

the new approach is Critical Direction Theory applied to uncertainties in the Nyquist

plane. The critical direction method is based on recognizing that, at any given frequency

on the Nyquist plane, there is only one direction of perturbation of relevance to the

stability analysis. This allows the characterization of robust stability margins for







uncertain systems characterized by irregular perturbation templates, a problem that poses

significant challenges to other analysis methods. Examples of practical relevance are

given to illustrate the application of the new theory. Using the new approach, the

problem of assessing robust stability and computing stability margins for SISO systems

with affine complex parametric uncertainties is tackled successfully. Exact analytical

results are derived for geometrically simple uncertainty sets such as ellipses and

rectangles.

In the later part of the dissertation, the synthesis of robust controllers for state-

delayed and input-delayed systems is considered. In particular, Sliding Mode Control is

chosen as the technique of choice, as it possesses the combination of robustness and

performance guarantees that one seeks in a control system. Robust stability to the

chosen perturbation characterization is rigorously proven. Practical difficulties in

implementation are pointed out and ways to overcome these hurdles are presented. Some

open questions in the literature are brought out and theoretical analysis and analytical

answers are presented. Finally, ideas and interesting formulations are presented for

future work.













CHAPTER 1
INTRODUCTION

Modeling of any practical system for control design invariably involves uncertainty.

Since the exact model of a process may be difficult or impossible to determine, the

logical approach is to design a control strategy based on a suitable nominal (i.e., most

likely) model. Once a system has been identified and a nominal model and associated

uncertainty description is available, there are two main tasks. The first task is the

analysis of the uncertain system. The following issues arise naturally: The first, is the

question of robustness analysis. What is the measure of the maximum uncertainty that

the system can tolerate and still sustain stability and performance? Alternatively, given

the size of the uncertainty, one might wish to study the effect of the uncertainty on the

stability and performance of the system. A system is said to be robustly stable if it can

sustain stability in spite of suffering from uncertainty.

The second major issue is the synthesis of robust controller to ensure stability and

performance of the physical system. Over the past two decades, a plethora of robust

controller methods have been investigated and implemented in a wide array of industries.

Although, robust analysis and synthesis is an intellectually stimulating and

mathematically challenging task, there are some valid criticisms that can be leveled

against this approach. Is it a good idea to design a system for worst case uncertainty? If

the probability of worst case occurring is very low, why should we sacrifice the

performance of the system by designing for the worst case? Should the uncertainties be

simply bounded by norms (or) should there be probabilistic ideas inherently associated

with the uncertainty descriptions? These are areas of intense interest and importance and

the interested reader is referred to the research monograph by Smith and Dahleh [1994]







and the references therein.

In this dissertation attention will be focused on both the analysis and the synthesis

aspects of robust control. The thesis is organized as follows: Chapters 2-3 address the

robustness analysis problem, while Chapters 4-5 deal with controller design for robustly

stabilizing a class of uncertain delay systems. In Chapter 6, future research directions are

presented.

In Chapter 2, the concept of the Nyquist robust-stability margin, for characterizing

the closed-loop stability of uncertain systems is introduced. The predominant robustness

analysis methods typically rely on arguments based on real singular value or magnitude
bounds. Such is the case of well known multivariable stability margin km [Safonov,

1982] and the structured singular value ku [Doyle, 1982]. In contrast, the proposed new

approach makes direct use of Nyquist arguments and is based on the analysis of the

perturbed eigenvalue loci, hence avoiding undue conservatism that may result through the

use of singular-value upper bounds. A key element in the new approach is Critical

Direction Theory applied to uncertainties in the Nyquist plane. The critical direction

method is based on recognizing that, at any given frequency on the Nyquist plane, there is

only one direction of perturbation of relevance to the stability analysis. This critical

direction is defined by the oriented line that has its origin at the location of the nominal

eigenvalue of the unperturbed system and passes through the critical point -l+jO. In fact,

from a stability point of view all points on the uncertainty eigentemplate that do not lie on

the critical line can be ignored. This allows the characterization of robust stability

margins for uncertain systems characterized by irregular perturbation templates, a

problem that poses significant challenges to other analysis methods. Other attractive

features of the Nyquist robust-stability margin theory are that its MIMO (Multiple Input-

Multiple Output) form is a natural extension of the SISO (Single Input-Single Output)

case, and that its versions for continuous and discrete systems are formally identical. A

SISO and a MIMO example are given illustrating the concepts.








Chapter 3 focuses on the complex parametric uncertainty problem from the

perspective of the critical direction paradigm. Considerable attention has been given to

the problem of assessing robust stability and computing stability margins for SISO

systems with parametric uncertainties. While the general robust stability margin

computation problem is known to be NP-hard [Braatz et al., 1994], its rank-one

counterpart results in a convex optimization problem that can be readily computed, and

can be often derived as an explicit analytical expression. It is now well known that affine

parametric uncertainty problems result in a rank-one y formulation. This chapter

addresses the case of element-wise complex parametric uncertainties where each

individual uncertainty lies in a highly structured domain that is convex along any line

from the origin. This is a mild convexity condition, since the actual domain could be

highly non-convex. Exact analytical results are derived for geometrically simple

uncertainty sets such as ellipses and rectangles. This approach however, is applicable to a

much wider class of systems than the ones considered here. The only information that is

needed is the phase dependent magnitude of the uncertainty. The approach presented

here is simple, intuitive and yet, mathematically rigorous. For the special case of disk-

bounded affine uncertainties, the approach recovers, in a transparent manner, relevant

results by previous methods.

Chapter 4 and Chapter 5 deal with the issue of stabilizing an uncertain time-delay

systems (both state-delay and input-delay are considered). Time delay exists in various

branches of engineering. From biological, optical, electric networks, to chemical reaction

systems, time delay occurs and affects the stability and performance of a wide variety of

systems [See Schell and Ross, 1986, Inamdar et al., 1991]. For example, input delays

occur as one of the common sources of time delay, such as in chemical processes,

transportation lags and measurement delays, etc. It is well known that the existence of

time delay degrades the control performances and makes the closed-loop stabilization

difficult. Hence, the study of time delay systems has been paid considerable attention







over the past years. The description of time-delay systems lead to differential-difference

equations, which require the past values of the system variables. There are no known

methods to get necessary and sufficient conditions for the robust stability of such

systems. There are many approaches that yield sufficient conditions with varying degrees

of sophistication.

In Chapter 4, a Variable Structure Control (VSC) design methodology is proposed

to robustly stabilize an uncertain state-delay system with nonlinear and possibly

mismatched uncertainties, utilizing the concept of perturbation compensation. The study

of uncertain state-delay systems has received much attention by researchers in the past

decade. One reason for this interest is that time delay is encountered in various

engineering systems and can be the cause of instability. Another reason is that practical

control systems unavoidably include uncertainty or disturbances due to modeling or

measurement errors and other approximations. The sliding mode in VSC possesses well-

known features that make it very attractive for control systems. These include fast

response, insensitivity to parameter variations, and decoupling design procedure, among

others. Sufficient robust stability conditions are derived which are independent of the

size of the delay; furthermore no matching conditions on the uncertainty are assumed.

The main advantages of the proposed approach are that i) a realizable control law is

obtained, ii) chattering reduction/elimination is achieved, and iii) guidelines are given for

the design of the control matrix involved in the specification of the switching function.

Finally the results are illustrated with an open-loop unstable example.

Chapter 5 proposes an approach for uncertain input-delayed systems with linear and

possibly mismatching uncertainties via Sliding Mode Control. Sufficient robust stability

conditions are derived. We consider a transformation to convert the original system into

an equivalent system without delay [Artstein, 1982]. This reduction enables the use of

known stabilizing methods for linear systems without input delay. The main advantages

of the proposed approach are that (i) a realizable control law is obtained, (ii) chattering




5


reduction is achieved since a continuous control law is used, (iii) guidelines are given for

the design of the switching function to ensure the asymptotic stability of the normal

system in the sliding mode. Certain open issues in the sliding mode literature are brought

out and possible solutions are discussed.


Finally, future research directions and possible extensions to the presented work are

discussed in chapter 6.













CHAPTER 2
THE NYQUIST ROBUST STABILITY MARGIN--A NEW METRIC FOR THE
STABILITY OF UNCERTAIN SYSTEMS


2.1 Introduction

Many of the now well-known results on robust stability can trace their origins to

the Nyquist stability criterion (SISO case), or the Generalized Nyquist stability criterion

(MIMO case) [MacFarlane, 1980). Such is the case of the well known multivariable
stability margin k,, [Safonov, 1982] and the structured singular value M [Doyle, 1982].

Although the generalized Nyquist criterion addresses the issue of stability using complex

variable (eigenvalue or transfer function) arguments, the predominant robustness analysis

methods typically rely on arguments based on real singular value or magnitude bounds.

One reason for this choice was clearly the more advantageous conditioning of singular

values for numerical calculations. However, another important factor was the historical

fact that uncertainty measurements were initially proposed for MIMO systems using

singular value bounds, which collapse to magnitude (disk) bounds for SISO systems.

In this chapter we introduce the concept of the Nyquist robust-stability margin,
kN, for characterizing the closed-loop stability of uncertain systems. The approach

makes direct use of Nyquist arguments and is based on the analysis of the perturbed

eigenvalue loci, hence avoiding undue conservatism that may result through the use of

singular-value upper bounds.

A key element in the new approach is Critical Direction Theory applied to

uncertainties in the Nyquist plane. A basic version of the critical direction concept was

first presented in [Guzzella, et al., 1985] as a tool for analyzing the robust stability of

polynomials with ellipsoidal uncertainties. In [Latchman and Crisalle, 1995] the critical







direction notion is redefined directly on the Nyquist plane to facilitate the derivation of

necessary and sufficient robust stability conditions for transfer functions subject to

arbitrary perturbations. In this paper we further extend the critical direction theory for the

analysis of MIMO systems by formulating the concept in the generalized Nyquist plane,

introducing new nomenclature to facilitate the seamless extension from the SISO to the

MIMO case, and defining the new concept of the Nyquist robust stability margin.

The critical direction method is based on recognizing that, at any given frequency

on the Nyquist plane, there is only one direction of perturbation of relevance to the

stability analysis. This critical direction is defined by the oriented line that has its origin

at the location of the nominal eigenvalue of the unperturbed system and passes through

the critical point -l+j0. In fact, from a stability point of view all points on the uncertainty

eigentemplate that do not lie on the critical line can be ignored. This allows the

characterization of robust stability margins for uncertain systems characterized by

irregular perturbation templates, a problem that poses significant challenges to other

analysis methods. Other attractive features of the Nyquist robust-stability margin theory

are that its MIMO form is a natural extension of the SISO case, and that its versions for

continuous and discrete systems are formally identical.

The chapter is organized as follows. Section 2.2 gives a succinct review of

robust-stability margin results for the SISO and MIMO cases. Section 2.3 formulates the

critical direction theory for analyzing the robustness of uncertain SISO systems, and

Section 2.4 gives the details of the new critical direction theory for the MIMO case. A

SISO and a MIMO example are given in Section 2.5, followed by final remarks and

conclusions in Section 2.6.


2.2 Frequency-Domain Approach to Robust Stability Analysis

The new Nyquist robust stability margin is an object derived using Nyquist

arguments. In order to provide a contextual background for the ensuing discussions, in








this section we review relevant SISO and MIMO results on robust stability analysis,
placing particular attention on the Nyquist arguments that also lie at the root of the new

developments presented here.
The transfer function g(s) of a SISO system can be written in terms of an additive

perturbation 8(s) about a nominal transfer function go (s)

g(s) = go (s) + 8(s), 8(s) r d (2.1)

where d represents the set of allowed perturbations. When. all 8(s) E d are considered,

then at each frequency the map g(jo) = go(jo) + (jco) defines a region denoted

uncertainty template or value set [Barmish, 1993]. It is then possible to visualize the

transfer-function uncertainty directly in terms of the classical Nyquist plot as uncertainty

templates about the locus of the nominal system go(jCo).

The mathematical description for the set of perturbations may be formulated in

different spaces. For example, the Laplace-function domain d is often defined implicitly

in terms of a frequency-domain description. Such is the case where the uncertainty

templates are specified to be circular at each frequency. This makes the analysis of

robust stability particularly straightforward. On the other hand, the templates for a

number of useful uncertainty descriptions are not circular but rather irregularly shaped,

exhibiting highly directional features. This is the case for parametric uncertainty
descriptions of the form 8(s) = g(s,p)-g(s,Po), where g(s,po) and Po, respectively,

denote the nominal transfer function and the nominal parameter set, and where g(s,p) is

characterized by m real, uncertain parameters p de c 91m. In this framework the set d

is defined implicitly through the Euclidean domain dp. Typical uncertainty

representations for the parameter uncertainty set dp are polytopes and ellipsoids. The

special case where the uncertainty templates are ellipses is an interesting example of

directional uncertainty templates which also has a tractable mathematical representation.

The representation (2.1) is general and adequately encompasses additive

perturbations as well as uncertainties that may appear as multiplicative perturbations of







the nominal system, as parametric variations on pole/zero locations, and/or variations of

the coefficients of the numerator and denominator polynomials in the transfer function.

The difference in each case is the shape and orientation of the uncertainty templates

associated with 8(s).

Of particular interest are three types of SISO uncertainty descriptions.

Specifically, we denote the set of uncertainties giving rise to unstructured (circular),
elliptical, and structured (arbitrary) templates, respectively, by the notation du, dE, and

ds.

In the case of uncertain MIMO systems, the matrix transfer function G(s)can be

written as an additive perturbation A(s) about a known nominal transfer matrix Go(s)

G(s) G, (s)+ A(s), A(s) E D (2.2)

where D is the set of allowed perturbations. The uncertainty set D is described in terms
of available information about the modeling errors. We use the designation DU to

represent Unstructured uncertainties with a single norm bound, DB to denote block-

diagonal norm-bounded uncertainties, Dc to represent element-wise circular (disk-

bounded) uncertainties and DE to represent element-by-element elliptical uncertainties.

Other uncertainty descriptions are defined in terms of errors in the elements of the state-

space matrices, or errors in the real parameters of the matrix transfer functions. In all

these cases the uncertainty description still can be mapped (via eigenvalue inclusion

regions) on to frequency-response uncertainty templates, in the Nyquist plane which will

then form the basis for defining the Nyquist robust stability margin.


Classical SISO Robust-Stability Results

The SISO robust stability analysis is concerned with the stability of the system

that results when the system g(s) in (2.1) is arranged in a unity negative-feedback

configuration. It is normally assumed that (i) the nominal system is stable under unity







negative-feedback, and that (ii) the nominal and uncertain system have the same number

of open-loop unstable poles. These assumptions are adopted throughout this paper.

The most studied case is perhaps that where the uncertainty description is
unstructured, a(s) e du i.e., when 8(jo) < W(jco), where W(s) is a known function

whose frequency-response magnitude defines the radius of the circular uncertainty

templates.

Invoking the Nyquist stability criterion leads to the necessary and sufficient

stability requirement that all uncertainty templates exclude the critical point -l+jO, i.e.,

g(jo)= go(jco)+ (jo) -i V6(jco) and Vo) (2.3)

This condition can be expressed through the inequality [Doyle et al., 1992]

W(j(o) + g(Jc-o) (2.4)

or equivalently,
lw(i(O) .< I
+Wg(jo)
A+-go (jo) <1 (2.5)

The robustness analysis for uncertainty descriptions with non circular templates is

more challenging. For this reason it is quite common for elliptical or arbitrarily shaped

uncertainty templates, arising for example from parametric uncertainties, to be

circumscribed by an appropriate circle [Bhattachrya et al., 1993]. Although this approach

yields sufficient conditions for robust stability, it is nevertheless inherently conservative.

Classical MIMO Robust-Stability Results

As in the SISO case, the robustness analysis for MIMO systems makes use of the

generalized Nyquist stability criterion [MacFarlane, 1980], where avoidance of the

critical point l+jO is at the center of interest from the point of view of absolute stability

assessment. This has been the major focus of much research interest in the development

of robust multivariable stability margins. Consider the uncertain transfer matrix G(s) in








(2.2) with nominal model Go(s) affected by an uncertainty A(s). Again it is assumed

that (i) the nominal system is stable under unity negative-feedback, and that (ii) the

nominal and uncertain system have the same number of open-loop unstable poles.

The conditions under which no eigenvalue of the uncertain system

Go(jco)+ A(jo) is equal to -1+jO, i.e.,

A(Go(jco)+A(jao)) -I VA(jco) and Vco (2.6)

is readily shown to lead to the determinantal condition

det(I + M(jco)A(jco)) 0 VA(jco) and Vco (2.7)

where M(jo) := (I + Go(jc))-1.

Condition (2.7) is of course equivalent to the eigenvalue condition

-(M(jco)lA(jo))#-1 VA(jo) and Vo (2.8)

The multivariable stability margin km and the structured singular value Y. were

independently defined by Safonov [1982] and Doyle [1982], respectively, in terms of the

determinantal stability condition (2.7), rather than the eigenvalue conditions (2.6) or
(2.8). For an uncertainty of a given description D, the multivariable stability margin km

is defined as the matrix 2-norm of the smallest destabilizing uncertainty in the given

class, namely:

k..(o)) = min{5(A(jo)): det(I + M(jo)A(jco)) =O}
AeD (2.9)
This definition corresponds to the reciprocal of the robustness measure Y. defined in

[Doyle, 1982], which is defined as the inverse of the norm of the smallest destabilizing

uncertainty in the class, namely:

u(co) = "minjU(A(j0)): det(I + M(jo))A(jo))=O}-1
AED (2.10)







Note that if no uncertainty in the allowable class destabilizes the system (i.e.,, makes an

eigenvalue of Go(jo)+A(jco) equal to -1 at any frequency) then km = and U =0.

These margins provide a measure of tolerable uncertainty size.

A number of powerful robust-stability results can be obtained in terms of the

robust stability margins (2.9)and (2.10). Consider, for example, the following case of

interest. An uncertainty description D is said to be closed under contraction and rotation

if for any A(s) E D, then yejoA(s) E Dfor all 0 < y < 1 and for all 0 < 0 < 27.. Starting

from the eigenvalue stability condition (2.8), we can state the following lemma.


Lemma 2.1

Consider an uncertainty description D that is closed under contraction and
rotation. Then the uncertain system G(s) = Go(s)+ A(s) is stable for allA(s) ED if and

only if

sup p(M(jo)A(joO)) < 1 VO
AED (2.11)

Proof: Since the eigenvalue condition(2.8) is necessary and sufficient for robust

stability, it is clear that (2.11) immediately constitutes a sufficient stability condition.

That (2.11) is also necessary is established by contraposition as follows: Consider an
uncertainty A,(s) c D such that p(MAo) = []-' > 1, then due to the closure under

contraction we can always find an uncertainty A1 c Dwith A = A4,; 0 < y < 1, such

that p(MA1 ) = 1. Since the phase of the uncertainty class is arbitrary, due to the closure

under rotation we can assign a scalar phase multiplier to 4 to get A2 such that

A(MA2)=-1, which of course implies instability. Thus, it follows that for the

description D specified in the Lemma, condition (2.11) is both necessary and sufficient

for robust stability. V

For the classes of uncertainties covered by Lemma 2.1, the spectral radius

condition (2.11) is in fact equivalent (after appropriate normalizations) to the definitions








of y and k.. and thus it naturally follows that u is equal to the left hand side of (2.11).

Hence the conclusion of Lemma 2.1 can be equivalently formulated as

km > 1 Vo (2.12)

or

'U
The k.. and a stability margins serve to provide valuable characterization of

destabilizing uncertainties. Specific procedures and results have been developed for

computing these stability margins (or at least good upper and lower bounds) for various

useful classes of uncertainties. Additionally, the u problem formulation has produced an

extensive set of results associated with the class of block diagonal bounded uncertainties,

and has thus attained wide acceptance for robustness analysis, and more recently, also for

controller synthesis using the u-synthesis method. In all these applications the

computation of the stability margin is effected by calculating singular-value upper bounds

which give sufficient, and in some cases necessary and sufficient, stability conditions.

Unfortunately, however, the use of norm-bounds, especially singular value

bounds, in the characterization of destabilizing uncertainties often causes the rich

structural properties, including phase and directionality to be ignored. On the other hand,

as we show below significant advantages can be gained by studying the robustness

problem directly from an eigenvalue point of view, especially since necessary and

sufficient stability conditions correspond precisely to an eigenvalue condition via the

generalized Nyquist criterion.


2.3 The Critical Direction Method for SISO Systems

In this section we present an exact robust stability result for the case of SISO

systems using the critical direction theory. We adopt a nomenclature that permits the

extension of the critical-direction concepts to the MIMO case. We also introduce the







definition of the Nyquist robust stability margin for SISO systems in a form which

readily extends to the MIMO case.

Figure 2.1 shows a typical Nyquist diagram for a SISO system illustrating the
nominal frequency response go(jco) and an irregularly shaped uncertainty template. We

define the critical line at a given frequency o as the directed line which originates at the
nominal point go(jo) and passes through the critical point -1+jO. The figure is also

useful for identifying the entities defined below:


1. The crtitical direction

d(jo)= 1 + go (jo)
l + g(jo)) (2.14)

which may be interpreted as the unit vector that defines the direction of the critical line.


2. The uncertainty template

T(co) := { g(jo) I g(jo) = go(jo) +6(jo), 8(s) e d} (2.15)


3. The critical template

T'c(cO):={zcT(o) z=go(jo) +cd(jco) for some aE9E+ } (2.16)

namely, the set of perturbed frequency-responses lying along the critical direction

d(jco).


4. The critical perturbation radius

pc(o) := max {a z = go (jco) +Mat(jo)E Tc)} (2.1
ac:91+ (2.17)

As an illustration of the previous definitions note that the critical line at frequency
co = o, is readily identified in Figure 2.1 as the directed line with origin at go (Jo, ) and

passing through the point -l+jO. The critical direction d(jcol) is simply the unit-length

vector that characterizes the direction of the critical line. In addition, the critical radius








Pc(col) is shown as the distance between the nominal Nyquist point go(Jwl) and the

point where the critical line intersects with the boundary of the template. Finally, the
critical template T(co) is also readily characterized in Figure 2.1 as the subset of the

uncertainty template T(co) that intersects with the critical line.

By definition Tc (co) is a subset of the critical line, thus it follows that T (Co) is

either a single straight-line segment, or the union of such segments. The critical template

may also contain isolated points should the boundary of the template T(w) be tangent to

the critical direction. Figure 2.1 shows the case where the critical template is a

continuous segment, and hence Tc (a)) is a convex set even though the entire template is

highly non-convex. The critical template is discontinuous when it is made up of the
union of distinct segments. In this case T(o)) is not convex, but each of its member

segments is a convex subset.

With these definitions in hand we can proceed to state the following robust-

stability theorem. For ease of exposition we assume that the critical template T(co) is a

convex set for all frequencies. This restriction can be relaxed through obvious
modifications to account for each of the convex segments of T (o).


Theorem 2.1
Let the nominal SISO system go(s)be subject to an uncertainty 8(s) : d. Then the

uncertain closed-loop system remains stable under unity feedback if and only if

P ( ) < 1
1 + /c (g(Juo))) (2.18)


The proof of the theorem is omitted since all details are given in [Latchman and

Crisalle, 1995]. V

Motivated by (2.18) we now propose the following definition of the Nyquist

Robust-Stability Margin for SISO systems:









kNC) =1+ A'C(goJc) (2.19)


From Theorem 2.1 it follows that

kN(CO) < 1 V0) (2.20)

is a necessary and sufficient condition for robust closed-loop stability. Furthermore, the
quantity y = [kN(CO)]-1 specifies the amount by which the uncertainty template should be

increased or decreased to attain the limiting case of stability.

The SISO critical direction result enables an exact assessment of stability for

systems with directional uncertainty templates, without having to circumscribe the

templates with larger circular uncertainties. The calculation of the critical perturbation
radius p,(o) involves determining the intersection of a straight line (the critical

direction) with a curve (the boundary of the template T(o)). Section 2.5 shows an

example where the critical radius can be calculated analytically.

It is also worth noting that the critical perturbation radius also gives a systematic

methodology for determining the uncertainty weights which can be used as one of the

inputs for various robust synthesis methods. Of particular interest in this regard are cases

where there exists a Critical Weighting Function, We(s), such that its magnitude satisfies

the interpolation condition

1Wc(s) = pc(0) Vo) (2.21)

When (2.2 1) can be exactly satisfied, Theorem 2.1 leads to the familiar H_ condition

WC(s) <
1+ go (s) (2.22)

which is of the form (2.4). If (2.22) cannot be satisfied exactly but can be approximated

using standard frequency-domain regression methods, then the resulting approximate

weight can still be used as the basis for practical robust-synthesis design for systems with

highly structured templates.







It should be noted that Theorem 2.1 recovers as a special case the situation where

the uncertainties are circular or disk-bounded. The interested reader is referred to

[Latchman and Crisalle, 1995] for further details on this point.

In the following section we show how the critical direction approach and the

Nyquist robust-stability margin may be generalized to the case of uncertain MIMO

systems.

2.4 The Critical Direction Method for MIMO Systems

For MIMO systems we focus attention on the effect of uncertainties on the

eigenvalue loci along the critical direction. This treatment has several desirable

properties. First of all we derive necessary and sufficient robust stability conditions

based upon eigenvalue relationships obtained directly from eigenvalue uncertainty

templates. Thus, in principle this method yields a stability margin measure even in cases

where singular value conditions fail to give necessary and sufficient stability conditions,

provided that a method can be found to define tight inclusion regions for the uncertain

eigenvalues. Furthermore by considering only the subset of the eigenvalue inclusion

region that lies along the critical direction, we significantly reduce the computation

involved in using the boundary of the eigentemplates (See E-contours [Daniel and

Kouvaritakis, 1985] ) for stability assessment.

Consider the generalized Nyquist plots in Figure 2.2 which (for illustrative

purposes only) show two irregularly shaped eigenvalue inclusion regions (templates)
about two of the n eigenvalues Xi(Go(jco)), i =1, 2,...,n. In analogy with the SISO case,

for each of the n eigenloci we define at each frequency an associated critical line, defined
as the directed line originating at the location of nominal eigenvalues 2Ai(Go(jao)) and

passing through the critical point -l+jO. Also in analogy with the SISO case discussed in

Section 2.3 we define the following entities:







1. The critical directions

di (jo) 1 + X i(Go (jco)) (.3
di(jc) := _1
l + Xi(Go(jco)) (2.23)

which may be interpreted as the unit vectors which define the direction of each critical

line.

2. The uncertainty eigentemplates

Ti (o) :={2L(G(jco)) Xi (G(jo)) = 2Li(Go(jco) + A(jo)), A(s) e D} (2.24)


3. The critical eigentemplates

T'-fi(o) := {z E T(co) I z = /i(G,(jco)) +xdi(jo) for some ax e 9I 1(2.25)


4. The critical perturbation radii

Pci(co):= max {ai z = Ai(Go (jo)) + cidi (jo) (= T (O)} (2.26)

The entities defined above have identical interpretations to their SISO

counterparts defined in Section 2.2. Hence, their geometrical interpretation can be

obtained directly from the SISO Nyquist diagram given in Figure 2.1, provided that the
nominal frequency-response plot go(jo) is substituted by an eigenvalue plot Go(jco).

Obviously, the tight E-contour templates for unstructured and structured uncertainty

descriptions utilized in [Daniel and Kouvaritakis, 1985; Kouvaritakis and Latchman,
1985] are respectively equivalent to the uncertainty templates TI(co).

Using these definitions we can now state the following robust stability theorem.
As in the SISO case, for simplicity of exposition we assume that the critical uncertainty
eigentemplates Tj (o) are convex sets at all frequencies.







Theorem 2.2
Let the nominal MIMO system G(s) be subject to an uncertainty A(s) e D. Then

the uncertain closed-loop system remains stable under unity feedback if and only if

max 1+ = PC (G )) < 1 (2.27)
i=1,2,...n-it (Go(jO)) 1 + 2y(G0(jcv))<

where pc () and Ac (.) are respectively used to denote the critical perturbation radius and

the eigenvalue associated with the eigenloci resulting from the maximization over all

i = 1,2,...n in (2.2 7).
Proof. Assuming nominal closed loop stability and that the nominal and perturbed

open loop systems have the same number of open loop unstable poles, the Generalized

Nyquist stability criterion guarantees that the uncertain closed loop system is stable if and

only if

I(Go(jwo)+A(jwo)) # -1 Vi and Vco (2.28)

Using the defining equation for the eigenvalues of G, (jco) + A(jco), namely

det(Go (jco) z + A(jo))= 0 (2.28)

at each frequency co we can parametrize the uncertain eigenvalues by

z = Xi (Go(jco)) + piejoi (2.29)

where Oi = Oi(co) varies in the range 0 < &i < 2r, and for each value of 0i the scalar

pi = pi (co) varies in the range 0 pi < i where k i(Go (jco)) + jpiej i is an eigenvalue

corresponding to the boundary of the eigentemplate '-i(co). Then condition (2.28) can be

written as

l+2Li(Go(jco))+ pieJ' # 0 Vi (2.30)

Since the term 1 + Xi (Go(jco)) is fixed, the only possibility for violating the stability

condition (2.30) is for piejei to be oriented along the critical direction, namely







pieJoi = ai(co)di(jo) (2.31)
where 0 < ao(o) <- Ac. Thus, for robust stability we have the necessary and sufficient

stability condition

l+_ i(Go(jo))+oai(o)di(jo).# 0 Vi and Vo (2.32)

Invoking definition (2.23) it follows that a sufficient stability condition is given by
a<(co)

1l+ i(Go(jo)) <1, O ci(co)
Furthermore, since along the critical direction a1 (o) < p_ (co) stability is ensured if

Pci() 1 + 2j(Go(jo)) (2.33b)

The proof is completed by taking the maximum over all i and noting that, because of the
convexity assumption on the critical templates 'T (co) conditions (2.33a)-(2.33b) are also

necessary for stability. V

In complete parallel with the SISO case

kN (0))= Pei 0))
N( 1) :1+ (Go (jo))j (2.34)

defines the MIMO Nyquist Robust-Stability Margin, and the scalar 7 = [kN(o)i]-' again

specifies the amount by which the uncertainty eigentemplate should be increased or
decreased to attain the limiting case of stability. Consequently, from Theorem 2.2 it
follows that a necessary and sufficient condition for robust stability is given by the

Nyquist-derived constraint

kN(w)) < 1 V co (2.35)

The Nyquist robust stability condition (2.35) provides an exact answer to the robust
stability margin problem, much in the same vein as the margins k (o) and i(co).







A most pleasing feature of the new stability measure is that the SISO version

(2.20) and the MIMO version (2.35) of the Nyquist robust stability margin are formally
identical. Much more importantly, the kN(o) formalism adheres to the classical

generalization of SISO system properties to the MIMO case via the vehicle of the

eigenvalues of the frequency response matrix.

Relationship between km (o),(O)), and kN(a))

In light of Theorem 2.2 and the remarks above, it is clear that there are strong
equivalences between the Nyquist stability margin kN(wO) and the margins k,, (co)) and

j,(co). To see this recall that the definition (2.9) for kin(co) is equivalent to finding at

each frequency the smallest destabilizing A such that

1+ ,i(M(jco)A(jco)) # 0 Vi (2.36)

which is precisely the condition exploited in Theorem 2.2 to get the necessary and

sufficient stability condition

l + ,i(Go(jo)))+Oi(o))di(jc))# 0 Vi and Vo (2.37)

For uncertainties which satisfy the condition of Lemma 2.1, from (2.12)-(2.13) it follows

that

kN(CO) < 1 <*/ (CO) < 1 #* ki,(o) > 1 sup p(MA) < 1
AED (2.38)
In fact, even for uncertainties for which Lemma 2.1 does not hold, it is still true that for

all A(s) E D

l+A (M(jco)A(jo)) 0-# kN(cO)
The practical utility of Theorems 2.1 and 2.2 requires the computation of the
critical perturbation radius pc(co). Several methods are presently being developed to

exploit the critical direction theory to compute P (o) and hence kN (co) for the case of

affine and multi-affine parametric uncertainties in SSO and MIMO systems. The results







thus far are very encouraging, and have shown significant computational and algorithmic

simplifications due to the focus on the critical direction. For the case of structured and

unstructured MIMO uncertainties, tight eigenvalue inclusion regions may be obtained

using the singular-value based E-contours method [Daniel and Kouvaritakis, 1993;

Kouvaritakis and Latchman, 1985], with similarity or non-similarity scaling deployed to

remove or reduce conservatism.


2.5 Examples

SISO Example

This section presents an example of a SISO system with an uncertainty

description motivated from statistical parameter-estimation techniques. The uncertainty

considered is an ellipsoidal parameter-space model which, in addition to its intrinsic

merits, facilitates the direct analysis in the Nyquist plane permitting an explicit
characterization of the Nyquist robust-stability margin kN(W0),thus illustrating the

application of the critical direction theory proposed in this paper. Furthermore, a

discrete-time representation of the model is deliberately chosen to emphasize the fact that

the critical direction method and its attendant Nyquist robust-stability margin concept are

applicable to both the continuous and discrete domains.

Let us consider a general SISO system model given by the discrete-time transfer

function
q -k
H(z;p) = Yhkz
k 1 (2.40)
with q parameters defined by h = [hj,h2,.. .hq ]T. The nominal system, denoted H,(z), is

obtained when h assumes the nominal values ho = h,h. h so that the real

system is modeled as H(z) = Ho (z) + 6H(z). Let h in 9jq be such that h = ho + 5h.

Then we define the uncertain ellipsoidal parametric uncertainty description








M = h ho E Dh (2.41)

Di, M E 9q s.t 6h TQh-(%; Q1h =QhT >} (2.42)

We argue that, in addition to the remarkable mathematical tractability which we

shall show later, the ellipsoidal parametric description of uncertainties is quite natural in

many applications and offers, in contrast to hyper-rectangular descriptions, the further

advantage of allowing the dependence among various system parameters to be taken into

explicit account. Ellipsoidal models often arise quite naturally, as for example, whenever

linear regression or least-squares analysis is used in model estimation [Guzzella, et al.,

1991; Kosut et al., 1992].

Clearly the nominal parameter vector ho defines the nominal transfer function

Ho(z) = YXhz-k. Consider now the frequency response H(e')
k = I
q kh o 0 o T b l n i g t t e
whereH(z) = Xhkz- contains parameters ho= ,..ha1 belonging toth
k-I

parameter ellipsoid. Under these conditions the following lemma shows that the

parameter space ellipsoid maps precisely to ellipses at all frequencies except for co = 0

and co = ir. Let Re (z) and Im (z) represent respectively the real and imaginary

components of a complex number z.


Lemma 2.2

The parameter space ellipsoid defined by (2.42) maps to the elliptical uncertainty

template

T(co) = {z(co)= x1 (co) )= 0,1 xl(cO)- xl0(o)< VR(O) QhVR(O)),CJ =0 c)= /}(2.43)

where

X(co)=[X(co) x2(co)]TE 2 Xo(Co):=[xO(o) X0(co)]T E {2
X((O:=[X(CO X2 IJ


such that









FRe He
X(CO)= Im H(eJw)] = V(co)h


Re H(eJC) V(o)h
Im Ho(eiW) =)



Q0o :=V(o)Qh1V(o)T e 12x2

F coso cos2co ... cosqo]

[ sinco sin2o ... sinqco_

At co = 0 and co = 7r, matrix V(co) is singular and the uncertainty template is entirely

real and given by

T(o) {Z(CO) = X1 ()) 1 X2 (CO) = 0, 1 1(CO)- 1(w) 1vR(O)QAvR(CO),CO= 0 0) }

where VR(CO) := [cos CO cos 20 ... cos qo



Proof. The proof makes use of a result given in [Guzzella, et al., 1985]. Details

are omitted. V

Our intention is to characterize the Nyquist robust-stability margin for the unity

feedback system comprised of the nominal discrete-time system and its associated

parametric uncertainty ellipsoid (2.42). Note that the robust stability of the closed-loop

system cannot be analyzed using the 12 results in [Tsypkin and Polyak, 1991] because

their method is applicable only to continuous-time systems and to ellipsoidal uncertainty

descriptions where the matrix Qh is diagonal (i.e., the principal directions are aligned

with the coordinate axes). In fact, none of the conventional methods for analyzing robust

stability of SISO systems appears capable of treating in a systematic fashion the case of
ellipsoidal uncertainties Dh considered in this example.







Theorem 2.3

Under the assumptions of nominal closed-loop stability and that the nominal and

perturbed systems share the same number of open-loop unstable poles, the unity negative

feedback system with open-loop transfer function (2.40) and uncertain parameters (2.42)

is stable for all h e Dh if and only if



kN(CO) < 1 V 09 (2.44)

where

kN(CO)1 for co E (0,zr)
N) T(c)Q-d (o) (2.45)


kN ()) dT for o = (0,r)
(2.46)

and


d (CO) := ]1 V(co)h o 2.7
L0J (2.47)

Proof. Inequality (2.44) is a direct result of applying the necessary and

sufficient condition (2.18) of Theorem 2.1 in the Nyquist robustness margin form (2.20).
It then suffices to prove (2.45) and (2.46). First consider the case where o E (0,7r).

Using two-dimensional vector analysis and the definitions given in Section 3, it is clear
that the components of d,.(o) given by (2.47) are, respectively, the unnormalized real

and imaginary parts of the critical direction d(jco) as defined in (4.1). Now from Lemma
2, at each co the frequency-response H(ejc) is located inside an elliptical template with

center at the point Ho(eJo). Clearly, the elliptical template is a convex set because it is

continuous along any ray. A fundamental result from two-dimensional coordinate

geometry gives the length of the line joining the center of an ellipse to the point of
intersection along the vector dc (co) as








d (co)d (co)
PC (O) TCO)Q0 ld (w)(2.48)

Recognizing that

l + Ho (eJO)j = dT (co)dc(o) (2.49)

it readily follows that

kN(co)= =~ w
1 + Ho(ej)) (2.50)


is of the form (2.45). Analogously, the proof for the singular cases -- 0, 7r is derived

noting that from Lemma 2.2 it follows that at these frequencies

pC(co) = 1vR(cO)T QhvR(co) from which (2.46) is readily established.

In summary, for the case of ellipsoidal parametric uncertainties, the critical

direction theory permits the exact characterization of the Nyquist robust-stability margin,

and the derivation of the exact necessary and sufficient condition (2.44) for robust

stability. V

MIMO Example

Consider the MIMO nominal transfer function

__30 -3 1
(s + 1)(s + 2)(s + 3) (s +4)
Gos)=_05_ 10
(s+5) (s+l) j

and the associated element-by-element elliptical uncertainty description.

DE = {A(s) s.t. Aik(jo)) e Eik(jwo) VO)}\ (2.51)

where the boundary dEik (JO) of each elliptical domain Eik (j) is given by the map


dEik(jwo) = 2AikeJOi cosOik + j 2BikeJtiksineik, i,k=1,2


(2.52)








and where the coefficients Aik and Bik are the elements of the frequency-dependent

matrices A(o) and B(co), and the major-axis orientation parameters qik are the elements

of the matrix 0()). The objective is to illustrate the calculation of the Nyquist robust

stability margin kN (co) at the frequency 01 = 1.21, where
A(col) = L0.1264 0.0359] = 0.0246 0.0278-0(0)1) = 2.1799 4.57591

1 0.0680 0.3185] L0.0537 0.2707 L1.4906 3.8865

Note that formulations based on the multivariable stability margin km(co) or the

structured singular value (co) cannot currently offer an obvious approach for calculating

the stability margins for the uncertainty description DE.

At each frequency of interest we propose the following calculation procedure.
First, the nominal eigenvalues 2 i(G,(j0))) and the critical directions di(O),i= 1,2 are

computed.
Second, for each eigenvalue, z satisfying zi = Xi + pidi is found. In the above

expression, D is a diagonal similarity scaling matrix, P is the diagonal matrix where

each diagonal element is the radius of the circle circumscribing the elliptical
uncertainties. E1 and E2 are derived from the diagonalization of the uncertainty

[Kouvaritakis and Latchman, 1985] and i represents the eigenvalues of Go. This yields

the upper bound for Xi (G, + A) in the critical direction. Then an optimization is carried

out over all Oik minimizing the function

2di -

11 + +i(Go A) (2.53)

Equation (2.53) gives the eigenvalue of (Go + A) in the critical direction, which is closest

to the upper bound z. The basic principle is similar to that described in [Kouvaritakis et

al., 1991].
At the selected frequency o), = 1.21, the nominal eigenvalues for the system
considered in this example are Al(Go(jcol))=-0.2005-jl.2313, and








112(Go(jwO1))= 1.9332-j2.4525 and have the associated critical directions

d1(jco1)=-0.5446+j0.8387 and d2(jC0)=-0.7672+j0.6414. Then using the

structured E-contour method for the fixed critical directions d1 (j1) and d2 (Jc01) yields
the critical radii p,1 (o1) =0.3140 and P'2 (co) = 0.6445. Finally, from (2.27) and (2.34)

it is found that
k{E(c =max{ Poic(o) Pc2(0)1) max{0.2139,0.1686} (2.54)



Hence, the Nyquist robust stability margin is kNE (ot1) < 1, and it is concluded that the

system satisfies the necessary and sufficient condition for robust stability at this particular

frequency.

Figure 2.3 shows a Nyquist diagram with the eigen-plots for the nominal system
considered in this example. The eigentemplates T1(0)1) and T2(c)1) are not shown in the

figure because for the purpose of determining the Nyquist robust-stability margin it is not

necessary to calculate the entire templates. However, for reference the figure shows the

Optimal D-scaling bounds obtained using the scalings given in [Kouvaritakis and

Latchman, 1985]. The actual eigentemplates are bounded by these Optimal D-scaling
contours. The points denoted zl(Jwl) and z2(J)l) are on the boundary of the critical
templates T,1 (o1) and T,2 (co1), respectively, and were obtained using the E-contour

method. Note that for this example the Optimal D-scaling bounds provide a good
estimate for the boundary point z, (j)i), but yield a poor estimate for z2 (J0)y). Note also

that the critical eigentemplates T,, (o)1) and T,, (o),) can be readily identified from the

figure as the straight-line segments joining each nominal eigenvalue ,i(G, (j0)1)) with its

corresponding boundary point zi (J01).


2.6 Conclusions
In this paper we have proposed the Nyquist Stability Margin, kN (o) as a new

metric for robustness analysis of SISO and MIMO systems. The definition of new







stability margin is based on the Critical Direction Theory which provides a single

framework for robustness analysis for SISO and MIMO systems. The analysis

methodology makes direct use of the generalized Nyquist diagram, and in contrast to the

prevalent approaches which emphasize singular-value perturbations, it focuses attention

on eigenvalue perturbations.

The main advantage of the critical-direction theory is that it provides necessary

and sufficient conditions for robust stability in the presence of highly structured

uncertainties with phase and directionality constraints. Other approaches to these

problems either do not have the inherent capability to deal with these structural details, or

the directionality and phase constraints are deliberately ignored, giving rise in either case

to sufficient-only conditions such as those associated with singular-value theory. On the

other hand the new method explicitly exploits the detailed directionality and phase

constraints of the uncertainties as these are manifested in the frequency domain

uncertainty templates. Thus, the new method is applicable to a number of uncertainty

descriptions for which other methods fail, such as the case of element-by-element

ellipsoidal uncertainties in the transfer-function matrix, and other uncertainty descriptions

with highly directional frequency-domain templates.

The new critical-direction technique opens up new avenues for robustness

analysis and could lead to novel approaches for robust control synthesis. There is

significant promise for fruitful new results in this area where the computational efforts

are concentrated on a single and well-defined frequency-dependent directed line.









Nomenclature for Chapter 2


s Laplace variable
dc(jo) direction in the complex plane

g0(jo) Nominal SISO transfer function

g(jo) Uncertain SISO transfer function
Go(jo) Nominal MIMO transfer function

G(jco) Uncertain MIMO transfer function

kN(o)) Nyquist robust stability margin

km (co) Multivariable robust stability margin

M(jco) Augmented system matrix

W (s) SISO weight

Greek Letters

w) Frequency
Aji(G(jo)) Tth eigen value

A(s) Uncertainty description

PC (co) Critical perturbation radius
Y(wo) Structured singular value

dF(.) Maximum singular value

p(.) Spectral radius

T(co) Eigen template
T, (co) Critical eigen template




31



















critical line Imz g(jo))



____' Re g(jco)
go,(J),)+pc(t,)d(J ) (d




















Figure 2. 1. An irregularly shaped uncertainty template at frequency 0),
(shaded area) and its critical perturbation radius p,(w1). The critical
template 'i.(o)) (solid line) is the subset of template points lying on the
line segment with end points g0 (w1) and g0 (co) + p, (o1 )d(jcol).














d(Jc1)


Im (Go(jo))


/LI (G, (jco))


Figure 2.2. Nyquist plot of the eigenvalues of a 2x2 MIMO system
showing two irregularly shaped eigen templates Tl (wo) and T2(coj)
(shaded areas) at frequency CO = col.





























-3o ......


-4


-5 .
-2 -1 0 1 2 3 4 5

Re 1I(G(ja))











Figure 2.3. Nyquist plot of the MIMO Example. The contours represent
the optimal D-scaling bounds for the eigentemplates T1 (col) and T2 (o1) at
the frequency o) = 1.21 The points zi(o1) and z2(co1) lie on the
intersection of the boundary of their respective eigentemplates with the
critical direction













CHAPTER 3
A NEW PERSPECTIVE ON COMPUTING ROBUST STABILITY MARGINS FOR
COMPLEX PARAMETRIC UNCERTAINTIES

3.1 Introduction


Considerable attention has been given to the problem of assessing robust stability

and computing stability margins for SISO systems with parametric uncertainties. It is

now well known that affine parametric uncertainty problems result in a rank-one 9

formulation. While the general robust stability margin computation problem is known to

be NP-hard [Braatz et al., 1994], its rank-one counterpart results in a convex optimization

problem that can be readily computed, and can be often derived as an explicit analytical

expression. This aspect of the problem is amply documented in the literature [Qiu and

Davison, 1989]-[Hinrichsen and Pritchard, 1992] [Chen et al., 1994a]-[Chen et al.,

1994b]. Of particular interest are references [Chen et al., 1994b] and [Young, 1994],

which establish connections between frequency-based stability conditions and

polynomial conditions in the spirit of Kharitonov's theorem [Kharitonov, 1979].

This chapter focuses on the complex parametric uncertainty problem from the

perspective of the recently proposed critical direction paradigm [Latchman and Crisalle,

1995] [Latchman et al., 1997]. The approach followed in critical direction theory

consists of first mapping the parameter space uncertainties into the Nyquist plane in the

form of value sets or uncertainty templates, and then invoking specific directionality and

phase properties to infer stability. The method is based on recognizing that at any given

frequency on the Nyquist plane, the only uncertainties of relevance to the stability

analysis are those which lie on a well defined critical direction. In fact, from a stability

perspective all points on the uncertainty template that do not lie in the critical direction








can be ignored. This approach makes it possible to solve problems with highly

directional uncertainties, such as SISO elliptical uncertainties, for which, at this time,

approaches based on the more traditional structured singular value methods are not

obvious.

This chapter addresses the case of element-wise complex parametric uncertainties

where each individual uncertainty lies in a highly structured domain that is convex along

any line from the origin. This is a mild convexity condition, since the actual domain

could be highly non-convex. Exact analytical results are derived for geometrically simple

uncertainty sets such as ellipses and rectangles. For the special case of disk-bounded

affine uncertainties, the approach recovers in a transparent manner relevant results by

previous methods [Hinrichsen and Pritchard, 19921][Chen et al., 1995b] .

Section 3.2 of the chapter provides background information and mathematical

preliminaries, presents the uncertainty descriptions considered, and gives a brief review

of the main elements of the critical direction theory. The main results are presented in

Section 3.3, and connections with the classical approach based on M-D structures is

explored in Section 3.4. Concluding comments are given in Section 3.5.

3.2 Background and Preliminaries


Consider a SISO system with an open-loop transfer function
g(jco,q) = n(jco, q) (3.1)
d(jco, q)

where n(jco,q) and d(jco,q) are complex polynomials that depend on an uncertainty

vector

q = [q(jw),q2(00) j E Q c (3.2a)

whose components qk (jco) belong to complex domains Qk, i.e.,


qk(jwo) G Qk c C,


(3.2b)








and hence the uncertainty domain is the artesian product space Q = Q1 x Q2x... x Qm"

Figure 3.1 illustrates a case of a radially convex domain Qk. Note that the entire domain

is non-convex.

The uncertainty class considered in this s are described by element-wise

uncertainty sets Qk, k = 1, 2, ..., m, that are closed under contraction (i.e., are radially

convex), and are unconstrained in phase. Letting 9Qk denote the boundary of domain Qk,

then the boundary elements dqk(jo) E dQk are of the form


dqk= 7k(CO, Ok)ejk, k = 1, 2 ..., m (3.2c)

where 0 < 0k < 2r is an unconstrained phase angle, and Yk (CO Ok) = qk(jo)) 0 is the

frequency- and phase-dependent magnitude of the boundary element.
Im qk

Qk r(t) y

Qk Re q A,



Figure 3.1 Radially convex Figure 3.2 Unity negative

uncertainty domain Qk for feedback configuration.

complex component qk

The elements of the uncertainty vector q are assumed to appear in an affine

fashion in (3.1), i.e., the uncertain model is of the form
m
n(jc, O) + nk (jco)qk (jo)
g(jco,q)= k=1
m
d(jo, O) + dk (jCo)qk (jCO)
k=1 (3.3)

Under the unity negative-feedback arrangement shown in Figure 3.2, the

characteristic polynomial for system (3.3) is given by the polynomial sum


p(jo, q) = n(jco,q) + d(jw, q)


(3.4)








It follows that the robust stability of the closed loop with respect to the uncertainty set Q

can be evaluated by studying the uncertain polynomial (3.4). It is convenient to
decompose (3.4) into its nominal and uncertain parts p(jco,q)= po(jco)+S(jco,q),

where

po (jo) = n(jco, 0) + d(jco, 0) (3.5a)

and
m
3(jco,q):= Ypk(jo))qk(jCo) (3.5b)
k=1

where

pk(jo)) := nk(Co) + dk(jco) = pk(jco) ejePk (jC) (3.6)
The variable 0Pk (jco) is simply the frequency-dependent phase of polynomial Pk (ico).

Hence,

p(jco,q) = po(jco) + Y pk(jO)qk(jco) (3.7)
k=1

Definition. The Parametric Robust Stability Margin is the entity defined by the

minimization expression

a (co)= min 1OEp(jco,xq) for some q E Q}
a + (3.8)

The parametric robust stability margin is a non-negative real scalar that can be

interpreted as the minimal magnification a*(co)> 1 or contraction a*(co) < 1 of the

uncertainty set Q that brings the closed-loop system to the edge of stability.

Geometrically, the parametric robust stability margin represents the minimum tolerable

blow-up factor. Note that the parametric robust stability margin is defined for each

frequency.








Specific Parameter Uncertainty Descriptions Considered

Three types of uncertainty descriptions of the form (3.2c) are considered, namely

the (i) circular, (ii) elliptical, and (iii) rectangular element-wise complex domains

illustrated in Figure 3.3.
(a) hn qk (b) Im1qk (c) Imqk
)k Bk Ak
Ak Ak \
~kBk
Reqk > Reqk Re qk



Figure 3.3 Three types of element-wise complex parametric uncertainty

regions with unconstrained phase: (a) circular, (b) elliptical, and (c)

rectangular domains.

Circular Uncertainties

As depicted in Figure 3.3a, this is the familiar case of circular (disk-bounded)
uncertainties of radius Ak (c) where


Yk(coOk)>Ak(co) Vok (3.9)

Elliptical Uncertainties

Let Ak (co) > 0, Bk (co) > 0, and Ok (co) respectively represent the frequency-

dependent semi-major axis, semi-minor axis, and the orientation with respect to the real

axis of the an ellipse centered at the origin of the complex plane as shown in Figure 3.3b.

Then the boundary of each elliptical domain is given by the map

dqk (cO) = Ak (co)ejik (0))cos( Ok Ok (co)) + jBk (co)eJ 'k (C0)sin( Ok Ok (cv))

from which it follows that


Yk(CO, Ok) =[Ak(Co)2COS2(Ok k(o)) + Bk(cO)2sin2(Ok -k(c)))]2 (3.10)








Obviously, the circular uncertainty case (3.9) is recovered by (3.10) after setting
Ak(co) = Bk(co).

Rectangular Uncertainties

Let Ak((o) > 0 and Bk(o) > 0 respectively represent the frequency-dependent half-

width and the half-height of a rectangle centered at the origin of the complex plane as

shown in Figure 3c. Consider the orientation-phase definition
k (co) := arctan Bk (co) (3. 1 a)

Ak(O)

and the associated phase sets

0ak :=[0, Ok) U [-k' )r + k) U"[21r- Ok,27r] (3.11b)

'9k:= [kIr-O)U7r+ O,2r-k)(3.1 Ic)

The shaded and the plain areas in the Figure 3.3c corresponds to Obk and Oak,

respectively. Then an analytical expression for the magnitude of any point on the

boundary of the rectangular regions can be compactly written in the form

[ Ak (w0) for Ok E Oak

"k(oek) = COS(0) (3.12)
7k(Colt~k)Bk (() for Ok E= 'bk

sin(Ok)

3.3. The Critical Direction for a Characteristic Polynomial


The critical direction theory proposed for rational systems in [Latchman et al., 1997] can

be readily modified for the case of polynomials through the adoption of the definitions

given below.

(i) The Critical Direction

d(jo)(:= jO) e_ jec(o)) (3.13)
P (J) _=







is the unit vector which defines the direction from the nominal point

Po(jw) towards the origin. The critical direction is uniquely identified by the

critical phase angle e, (co).

(ii) The Uncertainty Template (or value set)

T(o):= {p(jco,q) e C Ip(jco,q) =po(jco) +8(jco,q), q EQ } (3.14)

(iii) The Critical Template

Tc (o) := { p(jo,q) T(co) I p(jco,q) = po(jo) + r d(jco), for some r e 91+}(3.15)

(iv) The Critical Perturbation Radius

p,(co):= max {r Z = po(jco) +rd(jo) T (co)} (3.16)

(v) The Nyquist Robust Stability Margin

kN(0o) := PC ()) (3.17)

Note that the Nyquist robust stability margin kN(o) defined in (3.17) is a metric

that characterizes the distance on the Nyquist plane to the point of instability (i.e., the

point l+jO). In a complementary fashion, the parametric robust stability margin a*(Co)

defined in (3.8) characterizes the "distance" on the parameter space Q to the limit of

stability.

3.4. Main Results


This section develops a technique for deriving analytical expressions for the

parametric robust stability margin. The approach makes use of the critical-direction

theory elements presented in Section 3.2. The developments are specialized here for the

case of uncertain polynomial systems.

Lemma 3.1. Consider the uncertain characteristic polynomial (3.4), and assume

that nominal stability is attained under unity feedback, and that the critical templates








T(wO) are convex at all frequencies. Then the uncertain system remains stable for all

parameter uncertainties q ( Q if and only if

kN(t))< 1 VO) (3.18)

Proof. The proof follows from an application of the zero exclusion principle

and is analogous to the development given in [Latchman et al., 1997] for transfer

functions, and details are omitted for brevity. V

Note that Lemma 3.1 is quite general and holds for multiaffine, and complex
and/or real uncertainties. The major issue is the calculation of the critical radius p,(o))

needed for the characterization of kN(Co) in (3.17); however, as shown below in

Theorem 3.1, the computation of the critical radius for complex affine parametric

uncertainties is particularly straightforward.

Theorem 3.1. Consider the uncertain system (3.3) with the uncertainty description

(3.2a)-(3.2c), and its corresponding closed-loop uncertain characteristic polynomial

(3.7). Then the critical perturbation radius is given by the expression


Pc())= I Pk(Jfl)) Yk(C,OC(0)- 0,k (c)) (3.19)
k=1

Proof. The points p(jo),q) on the value set T(co) that belong to the critical

template T, (co) must satisfy the condition


p(jco,q) = po(jco) + 8(jco,q) = po(jco) + rd(jco) (3.20)

or

i
YPk (jo)) qk (o)) = reJec) (3.21)
k=1
Due to the radial convexity of the uncertainty set, it suffices to consider only boundary
uncertainties dqk(CO); then, utilizing (3.2c) the preceding equality can be rewritten in the

form








Pk (ico)) Yk (c, Ok)eJ(Ok +OPk-Oc(cv)) e (3.22)
k=1
From the definition (3.16) it follows that p,(o) is equal to the maximal value of r in

(3.22) over all possible uncertainty phases 0 < Ok < 21r, i.e.,

p~cO)= 0 max k(jCO) )k(c ,Ok) eJ(Ok +01k -e(co)) (3.23)


Clearly, the maximum is obtained for Ok = (c) Op (no), leading to the expression

(3.19). V

Theorem 3.2. Consider the uncertain system (3.3) with the uncertainty description
(3.2a)-(3.2c), and the corresponding critical perturbation radius pc(CO). Then the

parametric robust stability margin is given by


a *(C) Po (Jo) (3.24)

Proof. From equation (3.4)

p(jco,q) = po(jo))+ IXPk(jco) qk(jco), q eQ (3.25)
k=1

and
m
p(jc9,aq) = po(jco)+ t XPk(jcO) qk(jco), q e Q (3.26)
k=1
Note that the map (3.25) yields the uncertainty template T(co) defined in (3.14). As for

the question of stability, note that only those uncertainties that give rise to points

p(jco,q) that are aligned with the critical direction are relevant. Then, using the

definition (3.13) for the critical direction, all the points of relevance for the stability

analysis must be of the form

p(jco,q) = p,(jo) + rd(jco) (3.27)


Equating (3.25) and (3.27) yields








m
mYPk(jCo) qk(jo) = rd(jco) (3.28)
k=1
Note that if qk E Q is such that p(jco,q) is aligned with the critical direction, then for

any scalar a > 0, (3.26) implies that p(jo,aq) also is aligned with the critical direction.

Hence, making use again of the definition of the critical direction given in (3.13), and

from (3.25) and (3.27) it is concluded that

p(jco, a q) = p,(jco) + a r d(jco) = d(jo)(ar-p,(jo) ) (3.29)

from which if clearly follows that
p(jo,aq)=O a- Po (jo) (3.30)

r
Since by definition the maximum possible value of r is p,(co) ,it follows that a*(co) is

obtained when r = p,(co) and hence a* (co) = po(jco)l/p,(co). V

Combining Theorem 3.1 and Theorem 3.2, it follows that the parametric robust

stability margin (3.8) can be written in the form

)= p (JCO)l (3.31)

X Pk(JCO) Yk(0), O,(CO)-OPk (co))
k=1
Hence, (3.31) gives an analytical expression for the parametric robust stability margin for

all complex element-wise parametric uncertainties that are closed under contraction of the

form (3.2a)-(3.2c) for systems with the affine structure (3.3).

3.5. Parametric Robust Stability Margins for Highly Structured Uncertainties


The connection between the Nyquist robust stability margin kN(aO) and the

parametric robust stability margin a* (wo) is the reciprocal relationship


=(co) = 1 (3.32)
kN (CO)

which follows from Theorem 3.2 and the definition (3.17). Invoking Lemma 3.1, it then

concluded that the closed-loop system is robustly stable if and only if a**(co) < 1, VcO.








This result holds for all element-wise complex parametric uncertainty descriptions that

are closed under contraction. This section focuses on obtaining analytical expressions

for parametric robustness margin for the three specific types of uncertainty descriptions

depicted in Figure 3.3

Theorem 3.3 Consider the uncertain system (3.3) with the characteristic

polynomial (3.7), and the circular, elliptical, and rectangular complex parametric

uncertainty descriptions whose element-wise magnitudes are respectively given by (3.9),

(3.10) and (3.12). The robust stability margins for each of these descriptions are given

by the analytical expressions:

(3. i) For the case the circular uncertainty description


a (0)) = m P() (3.33a)
I Ak(wO) pk(jo)l
k=1
(ii) For the case the elliptical uncertainty description





(CO) = [po (jco)-
I[Ak ())2Cos 2 (OC(0)-_pk (O) -Ok((O))+ Bk(W)2sin2(Oc(O))-Ok (cO) -O k())) 12 Pk(jo)[
k=1

(3.33b)

(iii) For the case the rectangular uncertainty description


a* ( ko) = [(J) ) iOe ) (3.33c)
'n Ak (0)) IN (J C0) Bk (0)) PkJA 0 CO]
k=-1 jcos(Oc(CO)-OPk (CO) ) o +sin(O (CO))- ON (CO) )


where (ak = 1 if Ok G (ak, otherwise 8ak = 0, and where 3bk = 1 if Ok E (lbk,

otherwise abk = 0.








Proof. Equations (3.28b)-(3.28a) simply follow by substituting the expressions

for the phase-dependent uncertainty magnitudes (3.9), (3.10), and (3.12) into the

analytical expression (3.27) for the parametric robust stability margin. V

The generalized structured singular value approach [Chen et al., 1995b] has

previously been used to derive the result (3.33a) for the case of circular uncertainties. In

principle, the general formalism presented in [Chen et al., 1995b] can also be used to

derive the results (3.33b) and (3.33c); however, the critical direction theory extracts exact

analytical results through very simple and straightforward geometric arguments.

3.6 Connections with the Classical M-A Formalism


In this section expression (3.31) for the parametric robust stability margin is

rederived utilizing the framework of the classical M A formalism where robust stability

is characterized by a determinantal constraint. The approach follows well-established

developments, except that we simplify the derivations by invoking a critical-direction

concept, namely, that a specific orientation in the Nyquist plane is the only direction of

relevance for stability analysis

Lemma 3.2. The unity negative-feedback configuration of the SISO uncertain system

(3.3) with affine uncertainty elements can be transformed into an equivalent MIMO

M- A structure via a linear fractional transformation [Zhou et al., 1996] where the
uncertainty matrix A(s) is diagonal and the structural matrix M = (Mik) is of rank-one

and has rational elements given by Mik(s) Pk(S) = 1, 2, ..., m; k = 1, 2, ..., m.
po(S)'
Proof. The proof is an exercise in block-diagram algebra. Details are given in

the Appendix. V

For the MIMO M- A structure in question it is typical to define the parametric

robust stability margin in terms of a determinantal condition, as follows

a *(co) min {a det(I + aM(jco)A(jco))=O } (3.34)
a 9 +







which is formally analogous to the definition of the parametric robust stability margin for

a SISO system given in (3.8). In fact, the theorem below shows that the definitions (3.8)

and (3.34) lead to identical expressions for the parametric robust stability margin for the

case where the uncertainties appears affinely in the model.

Theorem 4. Consider the system (3.3) with its associated affine uncertainty

description (3.2) and suppose that the nominal system is stable under unity negative-

feedback, and that the nominal and the uncertain systems have the same number of open-

loop unstable poles. Then the parametric robust stability margin at each frequency is

given by


= P (Jco) (3.35)
1 Yk(O C~O) p,(0)) -7r) jk0)l
k-I
Proof. From Lemma 3.1, the uncertain system with unity negative feedback can

be transformed into an equivalent MIMO M A structure. Using standard arguments, the

robust stability condition for the transformed MIMO system is given by the determinantal

condition

det(I M(j o)A(j0o))) # 0 Vco (3.36)

Introducing the parametric robust stability margin a c 9 + interpreted as a blow-up factor

for the uncertainties, the determinantal condition is rewritten as
det(I aM(jo)A(jco)) 0. Since from Lemma 3.1, matrix M is of rank one, it is

straightforward to establish the inequality


det(I aM(jco)A(jcO)) = 1 + I Pk(j(0) qk (jw)] (3.37)

Since the aim is to find the minimum scalar a ( 3Z+ that makes (3.37) identically zero,

equating (3.37) to zero and solving for a yields









a MPk(JO) k(co,Ok)e k+Ok) (3.38)

k=1

Clearly, the minimum real non-negative value of (3.37) will be realized when all terms in

the summation in the denominator (3.37) are aligned with respect to each other and with
respect to the numerator, i.e., using the notation po (jo) = p, (jco) ejepI and recalling the

fact that the uncertainty phase is 0k unrestricted, the minimum is attained when

Ok = Opo(o))-Op, (co) +7r, k = 1, 2,...,m (3.39)

From the definition (3.13) it follows thatOp (co) = 0,(co) r, hence the minimizer is

realized at

Ok = 09 ((o) ON (Co), k = 1, 2,.., m (3.40)

Substituting (3.40) into (3.38) gives the minimal value a = a*(co) which is of the form

(3.30). V

3.7 Conclusions


In this chapter we demonstrated the significance and ease of application of the

critical direction theory to the complex affine uncertainty problem. The development is

simple and intuitive and for the special cases of interest recovers the exact results by the

previous approaches. The case of multiaffine, complex and/real uncertainties may also be

handled using the critical direction arguments. However, for all these cases the

computation of the critical radius remains the major theoretical challenge; fortunately, for

a number of uncertainty descriptions, such as the ones considered in this chapter it is

possible to obtain analytical expressions for the critical radius.









Nomenclature for Chapter 3

S Laplace variable

q Uncertainty vector

n(jo,q) Uncertain numerator polynomial

d(jo),q) Uncertain denominator polynomial

p(jco,q) Uncertain characteristic polynomial

d, (jo) Critical direction in the complex plane

kN (Co) Nyquist robust stability margin

k, (co) Multivariable robust stability margin



Greek Letters

a (o) Blow up factor

0) Frequency

P, (a)) Critical perturbation radius
Y(co) Structured singular value

dF(.) Maximum singular value

p(.) Spectral radius

T(o) Eigen template

T, (co) Critical eigen template













CHAPTER 4
VARIABLE STRUCTURE CONTROL DESIGN FOR REDUCED CHATTER IN
UNCERTAIN STATE DELAY SYSTEMS

4. 1 Introduction


The study of uncertain state-delay and input-delay systems has received much

attention by researchers in the past decade [cf. Zaveri and Jamshidi, 1987 and Hale and

Lunel, 1993 for background and extensive references]. One reason for this interest is that

time delay is encountered in various engineering systems and can be the cause of

instability. Another reason is that practical control systems unavoidably include uncertainty

or disturbances due to modeling or measurement errors and other approximations. Several

authors have investigated the problem of stabilization of uncertain state-delay systems. The

main strategies for stabilization of delay systems include designing state or output feedback

controllers (using pole assignment, Lyapunov or LQ theory), and variable structure

controllers (VSC). Cheres et al., [1989] construct a min-max controller from the

knowledge of the upper bound on the delay. Shen et al., [1991] propose a memoryless

linear state feedback based on Riccati equation approach [also see Wang and Lin, 1988].

Phoojaruenchanachai and Furuta [1992] also construct a memoryless state feedback, but

consider a larger class of systems. Trinh and Aldeen, [1994] consider interconnected

systems and propose memoryless state feedback controllers to stabilize the uncertain state-

delay systems. Niculescu et al., [1996] design memoryless state feedback controllers to

stabilize an uncertain state delay system with constrained input via ricatti approach.

Recently, sliding mode control has been employed as a tool for stabilization of

uncertain state-delay systems. The sliding mode control possesses well-known features

that make it very attractive for control systems. These include fast response, insensitivity







to parameter variations, and decoupling design procedure, among others. Recently Shyu

and Yan [1993] and Oucheriah [1995] have used VSC to guarantee the stability of uncertain

delay systems, deriving sufficient conditions that depend on the size of the delay.

However, the Shyu-Yan method suffers from the disadvantage that the control synthesis

procedure involves unknown matrices; hence the control law can not be implemented.

Oucheriah proposes a dynamic switching-surface control scheme based on pole-

assignment. Both of the above approaches involve strong discontinuous control across the

switching surface to overcome the effects of the uncertainty, and hence suffer from severe

chattering. Chattering is undesirable because it might excite the unmodeled high-frequency

components of the system, and it may lead to premature wear or tear of the actuators. The

recent technique of perturbation compensation has been shown to provide effective chatter

suppression for delay-free systems [Kim, 1992; Chan, 1996]. This method exploits the

specific structure of VSC to estimate and compensate for the effect of perturbation, and

thus results in reduction or removal of chattering.

This chapter proposes a VSC design methodology to robustly stabilize an uncertain

state-delay system with nonlinear and possibly mismatched uncertainties, utilizing the

concept of perturbation compensation. Sufficient robust stability conditions are derived

which are independent of the size of the delay; furthermore no matching conditions on the

uncertainty are assumed. The main advantages of the proposed approach are that (i) a

realizable control law is obtained which does not need the bounds on the system

uncertainty, (ii) chattering reduction/elimination is achieved, (iii) delay- independent

sufficient robust stability conditions which can accommodate uncertainty in the delay are

derived, and (iv) guidelines are given for the design of the control matrix involved in the

specification of the switching function. Finally the results are illustrated with an open-loop

unstable example.








4.2 Sliding Mode Control Design


Consider an uncertain state-delay system subject to uncertainty and/or external

perturbations governed by the equations

i(t) = Ax(t) + Adx(t h) + Bu(t) + f(x(t), t) + fd (x(t h), t)
x(O)=qO) 0e[-h, 0] (4.1)

where x(t) r R' is the state vector which is assumed measurable and has the initial state

x(o) = xo, u(t) e Rm is the input vector; A e RlX and B E Rnxm are constant matrices;

O(t) is a continuous vector-valued initial function; and h > 0 is the time delay. The vector

functions f(x(t),t)( R' and fd(x(t-h),t)c R' represent nonlinear perturbations that

depend on the current state x(t) and on the delayed state x(t-h) of the system,

respectively. It is assumed that the modeling uncertainties satisfy the bounds

f(x(t),t) < k Ix(t)ll (4.2)
lfd (x(t-h),t)JJ k llx(t-h)ll

where k and kd are known positive real constants, and 11 II is any vector norm or

corresponding induced matrix norm. Notice that no matching assumptions are used. It

should be noted that these bounds are required not for controller design, but for robust

stability analysis.

Two definitions are useful for the ensuing developments. The matrix measure

[Vidyasagar, 1993] is defined as the function
#:C"' ---> R" /(A) = lim III + eAll 1
E---)O+ E

The matrix measure is also known as the logarithmic norm. For an excellent discussion

on logarithmic norms, see [Strom, 1975].
Also, a class of matrix pairs (A, B) is introduced for the stabilization problem,

defined through the set membership
U := {(A,B) 3 F E Rnxm: u(A+ BF) < 0} (4.3)








where U.I clearly represents the class of systems for which there exists a state-feedback law

u(t) = Fx(t) such that !u(A + BF) < 0.

The objective is to design a control law that assures reachability of the sliding mode

within a finite time and asymptotic stability of the system in the sliding mode, even in the

presence of uncertainties and/or external disturbances. The design involves two major

phases: The first phase is the selection of a switching surface manifold that has desirable

dynamics, and the second phase is the determination of an appropriate control law that

guarantees the existence and reachability of the sliding mode and the effective rejection of

disturbances.

Switching Function and Control Design

This section discusses the selection of an appropriate switching function. In the

sliding mode the system must not only be stable but also be robust to any kind of bounded

perturbations as specified in (4.3). The procedure is as follows. First, select matrices F
and Fd such that matrix A = A BF is a stable matrix, and that matrix Ad = Ad BFd

satisfies AdJ < AdJ At this point we anticipate that the role of Fd is to reduce the

effect of delay, since the stability conditions derived later in this chapter involve AdJ

Using the matrices described above the system equation (4.1) can be rewritten in the form

i(t)=( + BF )x(t)+ (Ad + BFd )x(t-h)+ Bu(t)+ f(x(t),t)+ fd(x(t-h),t)(4.4)

An integral switching function [Fernandez and Hedrik, 1987] is adopted in the modified

form proposed by Shyu and Yan [1993] namely

S(t) = Cx(t) f{C~ix(z)+CAdx(z-h)}d'r
0 (4.5)

where matrix C c R" is a control design matrix that is chosen such that the product CB

is nonsingular. The reason for choosing such a switching function instead of a traditional

combination of the states is that, this choice results in a desirable switching dynamics. This

will be clear after the design of the controller.







The goal is to find a VSC controller that drives the system to the switching surface

and ensures a sliding mode condition. The controller structure adopted is of the form

U(t) = Ueq(t) + Ur(t)+ up(t)

which contains three terms. The first term in turn is of the form

Ueq(t) = -(Fx(t) + Fdx(t-h)) (4.6)

and thus represents the model-based equivalent control for the nominal system in the

absence of perturbations, obtained by setting i(t) = 0 [DeCarlo et al., 1988]. The second

term

Ur(t) = (CB)"1 (Ks(t) +d sgn[s(t)]) (4.7)

is the reaching control law where the symmetric positive-definite matrix K and the scalar

d > 0 are design parameters. The final term is intended to cancel the effect of perturbations

on the dynamics of the switching function [Chan, 1996] and is given by

up(t) = (CB)-Tp(t) (4.8)

with

Tp(t) =(t) CB (Ur(t) +up(t -T)) (4.9)

where Tp(t) is designed to reject the effects of the perturbation signal. In order to avoid an

algebraic loop in calculation, a deliberately delayed value up(t T) is used. The value of

the computational time delay T is chosen to be small. As long as the variations in the

system dynamics are slower than the computational delay, this will be good approximation.
However, it should be admitted that regardless of the value of T, up (t) up (t T).

However, as long as the estimation error is less than the actual perturbation, we only gain

by using perturbation compensation. Also a finite difference scheme is used to estimate

i(t). This could also be a source of errors if the measurements are noisy.

The ideal VSC approach assumes that it is possible to switch infinitely fast across

the switching surface. The phenomenon of non-ideal but fast switching is labeled as








chattering [DeCarlo et al., 1988]. Interestingly, the name comes from noise generated by

the relays. Chattering is undesirable for two reasons: (i) It might excite the unmodeled high

frequency plant dynamics which could lead to unforeseen dynamics. Note that in certain

applications it is not of concern as the frequency of switching is far above the structural

frequencies of the mechanical system [DeCarlo et al., 1988]. However, in most

applications this will be a major concern, (ii) It might lead to the premature wear and tear of

the actuators. It should be noted here that it is the discontinuous control, which in turn

leads to chattering, enables one to guarantee the robustness features of sliding mode

control. In a sense, we gain robustness but pay for it with chattering.

To alleviate chattering, many approaches have been proposed in the literature.

Here, we give a brief review of the past efforts. Although, they have not been used for

sliding mode control of state-delayed systems, they are nevertheless useful as a concept.

One of them is the "boundary layer approach" [see Slottine and Sastry, 1983] which uses

the continuous control law and time varying sliding surfaces. Although this succeeds in

eliminating chattering, it is no longer possible to assure the asymptotic stability of the

system. Rather, one has to be content with uniform ultimate boundedness of the

trajectories within a neighborhood of the origin depending on the boundary layer.

Ambrosino et al., [1984] cascaded a low-pass filter after the VSC to attenuate the high

frequency switching component. In this case chattering is alleviated at the expense of

robustness. Espana et al., [1984] propose a zone in the vicinity of the switching surface in

which the feedback gain is adjusted in order to avoid chattering. Morgan and Ozguner

[1985] propose to decrease the control gain in order to minimize chattering, but the

disadvantage is the increased time that it takes to achieve sliding mode. Chang et al.,

[1990] proposed an adaptive alleviation algorithm which requires recursive prediction at

each step. Their procedure involves computing the derivative of the states, but they assume

that they are either available or can be computed without errors. Elmali and Olgac [1992]

do an on-line estimation of the effect of the perturbations based on the measurements of the







state dynamics. They assume that the error in estimation is bounded within a scalar

multiple of the estimation. Also see an experimental verification of their approach in Elmali

and Olgac [ 1996]. Chan [ 1996] proposed an perturbation compensation approach utilizing

the specific structure of the variable structure control. The approach seeks to estimate the

effect of perturbations on the switching dynamics with the assumption that the relative error

in the estimation of the switching dynamics is bounded and known. One of the major

advantages of the perturbation compensation schemes is that one need not have a priori

bounds on the uncertainties for controller design.

Note that with the application of the control law (4.6) (4.9) and the choice of the

switching function (4.5), the derivative function (t) is given by

(t) = CB (Ur (t) + up (t)) + C(f(x(t), t) + fd (x(t h), t))


i(t)= -(Ks(t) + d sgn[s(t)])+[C(f(x(t),t)+ fd(x(t-h),t))- Tp]

It is clear that i(t) is driven by the estimation error, and the purpose of control is to

counter the effect of perturbations on the switching surface. The usual approach is to

overpower the uncertainty by discontinuous control. For example, in this case the value of
d should be chosen such that d > Cl(k Jlx(t)l + kdllx(t h)11). In fact, a choice of this

type is made by [Oucheriah, 1995]. For proving the reaching condition, this is indeed a

clever selection of parameters. However, there are some significant problems with this

choice. Firstly, we are assuming it is always the worst case that is affecting the system.

Secondly, there is no method to estimate even conservatively, the norm of the states when

the system oscillates near the switching surface. Hence, while we are guaranteed to suffer

from chattering, we have no available worst case upper bound.
In contrast, the effect of up (t) is to ease the burden on the discontinuous control by

canceling the effect of perturbations on the switching dynamics. The discontinuous control

is now utilized, to combat the residual error in estimation rather than the whole

perturbation. Irrespective of the actual value of the perturbation, as long as the estimation







scheme works well, this scheme will outperform the conventional VSC. One point needs
to be clarified here. Estimating i(t) is in fact, equivalent to saying that we are estimating

the perturbation. Our contention here is that i(t) can be estimated from the measurable

states of the system via (4.5). It goes without saying that we are introducing a new

parameter here, namely the bound on the estimation error. Apart from the measurement

error in the states, there is also the danger of noise. Since s(t) depends on x(t), it is

assumed that proper filtering strategies will be employed in practice. In the worst case, one

has to cut off the perturbation compensation and has to live with the chattering. By worst

case, we mean that the computed perturbation signal is greater than the maximum norm of

the allowed perturbation, i.e.,

jT(t) < C(f(x(t),t)+ fd(x(t-h),t)) < Cl(klx(t)l+kdlx(t-h)l)

However, this decision can only be made depending on the application at hand. For

theoretical development, in order to prove the reaching condition within a finite time, we

need to assume certain error bounds on the derivative estimation. We assume that the

relative error in the estimation of i(t) is bounded,

~(t) est (t)<
~<3
Sest(t)

with the positive scalar bound 8 being known. An estimate of 3 can be obtained via

simulation studies with appropriately assumed perturbations. Then with the choice
d(t) > &est(t) + e, e > 0, and K the reaching condition can be satisfied and in addition,

the transient behavior can be shaped.

Theorem 4.1. The uncertain delay system (4.1) with model-uncertainty and

bounded measurement uncertainty given by (4.2) and (4.10) achieves the desired sliding
mode within a finite time if the control law u(t) = Ueq(t) + Ur(t)+ Up(t) is applied,


where the individual components are given by (4.6)-(4.9).







Furthermore, the sliding mode is attained within t = tr, where

tr 2 min() In + srmin (K) ( (O)s(O)


and the upper bound on the norm of the switching surface is given by


Is min (K)] A min (K)


Proof: Choose a positive definite function V(t) = 1 s(t) Ts(t). Then
2
differentiating V(t) along the trajectories of the system and using (4.5)-(4. 10) yields:

V(t) =s(t)Ti(t) =s(t)T [Ks(t) -ds(t) + (c(f(x(t),t)+fd(x(t-h),t))- Tp(t))](4.11)

In the rest of the proof the time argument is dropped for convenience of notation. Noting
that Ai = C(f(x(t), t) + fd (x(t h), t)) TP (t) and using the bound on the estimation
error (4.10) the equation (4.11) can be used to derive a bound on V(t) as follows:

V(t) = sT Ks dsT sign(s) + sT Ai

We note that since s(t) is a vector, s Tsign(s) = Isl > Is12. Hence,

V(t) Ks dlsl +lslJlAJ

Since d = est(t)l + E, and AJ 5[iest (t)l

V(t) < (-Amin(K))[s12 E[SI

V(t) (- min(K))V- eV

The above is a scalar differential equation and the solution is given by


WV(t) I (0) +Vkmin(+)] 8
Amin (K) Amin (K)








1-E )] Ain(K)t 2e
S (t)l+_ s(0) min (K) e 2 Amin (K)

Note that first term is always positive and decreasing, and the second term is a positive

constant and hence, their difference will attain zero within a finite time. It is

straightforward to compute the upper bound on the reaching time. A detailed proof is given

in Appendix F. Thus the system is guaranteed to reach the sliding mode within a finite time

since the reaching condition s(t)T(t) < 0 for s(t) # 0 is satisfied. V

Remarks:

1. The rate at which the switching surface is reached is determined by the

controller parameters K and d. Therefore by adjusting these parameters the

designer may select an appropriate reaching-transient behavior.

2. Chattering in the input variable is unavoidable for any positive value of d,

because ur(t) + _d when the system oscillates around s = 0. To reduce

chattering, the parameter d must be chosen as small as possible. In some cases

.(t) is measured directly, therefore d can be set equal to zero without any loss.
In other cases Sst (t) can be compared with simulated i(t) (for a given set of

perturbations) to aid in choosing a suitable the parameter d. Alternatively, one

can choose to set d = 0 and specify a large K to overpower the estimation

errors if any are present. However in the last case the penalty to be paid is high

initial controller gains.

4.3 Asymptotic Stability with Perturbation Compensation


Once in the sliding mode the system equations reduce to

i(t) = Ax(t) + ,dx(t h) + f(x(t),t) + fd(x(t h),t) B(CB)-1Tp(t) (4.12)







which is obtained from (4.4) after substituting (4.6)-(4.8). The intent now is to derive

conditions that establish the asymptotic stability of system (4.12). First, we note that if

perturbation compensation is not employed, then (4.12) reduces to

.i(t)= Ax(t) +Adx(t h)+f(x(t),t) +fd(x(t- h),t) (4.13)

With perfect perturbation compensation, the system equations become

i(t) = Ax(t)+-Adx(t- h) + y(f(x(t),t)+ fd(x(t h),t)) (4.14)
where y = [I B(CB)-'c]

There are basically two approaches in the literature for deriving conditions for the

asymptotic stability of these systems. One is to follow Lyapunov-type of arguments [Hale

and Lunel, 1993], and the other is to use norm-based inequalities [Mori et al., 198 1]. Both

approaches result in sufficient-only stability conditions.


Lemma 4.1. The delay system (4.13) with nonlinear uncertainties (4.2) is

asymptotically stable if the following condition is satisfied

y(A)+ AIjd +k+kd
Proof. See Appendix B. V


Theorem 4.2 The delay system (4.14) with nonlinear uncertainties (4.2) is

asymptotically stable if the following condition is satisfied

(A7) + d I +klll+kdjl j< 0 (4.16)

Proof. The proof is identical to the proof of the previous lemma. V








Remarks

1. It is possible that different choices of a given norm and the corresponding

matrix measure give different conclusions about stability. This is consistent

with the fact that the conditions derived are only sufficient.

2. In order to improve the sharpness of the stability condition, a similarity

transformation z(t) = T-lx(t) can be introduced so that the system equation

(4.14) becomes

i(t) = T-1ATz(t) + T-1AdTz(t h) + y (f(x(t), t) + fd (x(t h),t)) (4.17)
where T(I B(CB)-'c). In this case the stability condition reads


,-(T-AT)+ 1 TAdT +kd lTl y1 +kjjTj j; < 0 (4.18)

The usual approach is to find a transformation matrix T to diagonalize

T-XAT.

4. Under perfect estimation and the assumption of matched uncertainties [Shyu
and Yan, 1993] the perturbation error is (f+fd)+BuP = 0. The stability
condition then reduces to y(T-1A T) + rT-dT < 0. Of course, there is a

penalty for imperfect estimation and this is reflected in the reduction of the size

of the perturbations the system can withstand and still remain stable.

5. The conditions (4.15) and (4.16) are delay independent. This could be

advantageous in cases where delay-dependent stability conditions cannot assure

stability. The example in the chapter illustrates this fact. However, since all

conditions reported to date in the literature (including this one) are sufficient

only, no general claims can be made. It is also possible to derive delay-

dependent conditions for the problem considered in the chapter.







Optimal Choice of C

The stability condition (4.18) and the condition derived in Shyu and Yan [1993]
involves the matrix y = T-1 (I-B(CB)-1C). Then the matrix C is chosen to

minimize T-1(I B(CB)-'C) F" There appears to be no documented systematic

method for optimizing the left hand side of (4.18) (or similar stability conditions for other

approaches) over all allowable matrices T and C. To get the tightest possible stability

condition for the result in Shyu and Yan [1993] the matrix C must be chosen to minimize
the norm of y. We propose a new systematic approach to design matrix C.

The following definition and inequalities concerning Frobenius norm are useful.
Let A E= 9t'xn, and let II and ar represent the Frobenius norm and 'i'th singular

value, respectively. Then [Golub and VanLoan, 1989]




The following relationships are useful
IIA112 F = U2 + +. Co2; p =minfm,n}
I + }(4.19)
Ilal12 --[[AIIF -- Vn1II12



Lemma 4.2. Let N E 91 nPm and M E 9'. Then the best approximate

solution to the problem

min 11 M NX F
X F mxn
is Xopt = (NTN)-I NTM.

Proof: See Barnett [1971].


The definition of Xopt implies that for all X, X E 91'mxn

either

11 M NX IIF > M NXopt F







or

M NX IF = M NXopt F and ]X, o

It is instructive to note that (NTN) IN is the Moore-Penrose generalized inverse of

matrix N.


Theorem 4.3. The best approximate choice of matrix C for the following problem

min T-I(I- B(CB)-1C) F
C E 911LX )
is Copt =a(BTT-TT-1), where a isascalar.

Proof. Define the auxiliary matrix X = (CB)-'C. The minimization problem

reduces to
min T-I(I BX)
X C gr7tmxn F
Substituting M = T-1 and N = T-1B in Lemma 4.2 the best approximate solution is

found to be

Xopt= (BTT-TT-1B)-lBTT-TT-1 =(CCptB/-1
Copt

It is obvious that Copt = BTT-TT-1. Note that multiplying by a scalar a does not change
the value of the functional being minimized, hence Copt = a(BTTTT -1) V

Remarks:

1. Note that Theorem 4.3 gives the optimal solution in the sense of the Frobenius

norm. Using (4.19), it is easy to see that for the case where B E 9'x<, the
choice Cpt = a(BTT-TT-1) is optimal in the sense of the 2-norm as well.

2. The Matrix C enters in the last three terms of the stability condition (4.18). An

alternative method for finding the optimal matrix C is to carry out a numerical

optimization. However, noting that the contribution from the last term will be

small for a well performing estimation strategy, the last term can be ignored for

analysis purposes. Then, from triangular inequality it is easy to verify that the








result of Theorem 4.3 provides a sub optimal upper bound in the sense of the

Frobenius norm.

Implementation Issues

1. The estimation error depends on the approximation scheme used. Although

there is no a priori information on the estimation error, different estimation

strategies can be compared by assuming appropriate perturbations and tracking

the estimation error.

2. Also, note that the main challenge in the proposed approach is that of

approximating i(t) by Sest(t). In general, a finite difference estimate is used.

However, this may cause a problem near the switching surface where some

chattering combined with a small step length may contribute to large estimation

errors. Also, there is the unavoidable nuisance of noise, and the danger of

differentiating it. Hence it is necessary to employ filtering. Although the

switching dynamics with perturbation compensation is complex, one can still
perform a low pass filtering on s(t) for the purpose of estimating Sest(t) by

finite difference [Elmali and Olgac, 1996]. Note that the actual value of s(t) can

be used elsewhere in the controller. Hence, this does not destroy the guarantee

of asymptotic stability. The cutoff frequency for the filter can be chosen based

on the maximum frequency component that we wish to process in the state

dynamics. For example, if temperature in a reactor is the state variable that is

used in the construction of the switching function, a cut-off frequency of 100

Hz is acceptable, whereas if pressure in a flowing line is the variable then one

needs to go for higher frequencies. With the advances in digital signal

processing these decisions can be made scientifically depending on the

application at hand.








4.4 Example


Consider the uncertain time-delay system studied by Shyu and Yan [1993]

1.5] x(t)+ Lo -1 x(t-h)+ L u + f(x(t),t) + fd(x(t-h),t), t>O
[=0.3 -20 011]

The delay is h = 2, and the unknown non-linear perturbations are of the form
f(x(t),t) = 0.2x(t)sin(x(t)) and fd(x(t-h),t) = O.lx(t-h)sin(x(t-h))

The design objective is to asymptotically stabilize the uncertain system. The initial

condition is specified as x(t) = [1,3]T -h < t < 0. Note that for the delay considered,

the sufficient stability condition of Shyu and Yan [1993] is violated.

Also note that the homogeneous part of the system is unstable because A has a

positive eigenvalue A = 1.1432. Choosing the same control matrices investigated in
Shyu and Yan (1993), namely F =[0.3 0.15] Fd = [0 -0.1] leads to


= Li -2.15' Ad = -0.]

The particular choice F adopted makes A diagonal (and hence T = I), and the
choice of F,1 ensures Ad < Ad1. As per Theorem 4.3, Cpt = a (BTT-2). In this

design we choose K = 2, and d = 0. Then choosing a = 0.1 leads to C = [1 0.1]. If a

positive value for d is chosen, it will result in some chattering.

From (4.16) we can find the maximum tolerable estimation error. For the 1-norm

and infinity norm we find that asymptotic stability is assured if e < 1.4445, while for the

2-norm asymptotic stability is assured if e 1.6085. The delay introduced in the
computation of up(t) is 0.01 seconds.

The simulation results are presented in Figures 4.1 4.4. Figure 4.1 shows the

evolution of state norm as a function of time, reflecting the asymptotic stability of the

system. The switching function and its evolution towards the sliding mode is shown in

Figure 4.2. The sliding mode is reached rather quickly, and most importantly, a chattering-








free behavior is observed. Figure 4.3 shows that the input variable is also smooth and free

of chattering. The jump in the input around t = 2 reflects the fact that control variable is

responding to the delayed states coming into effect. Figure 4.4 shows the evolution of the

estimation error as a function of time, which also reaches the value of zero asymptotically.

From Figure 4.4, it is obvious that the estimation error in the sliding mode is much less

than 1.6085, thus implying robust stability via equation (4.16). In this example, there is

no chattering, and the system is asymptotically stable.

4.5 Conclusions


This chapter considered the robust stabilization of uncertain time-delay systems

with mismatched nonlinear uncertainties. A practically implementable control design

methodology is proposed and an illustrative design example is given. The present method

can significantly reduce or completely eliminate chattering as compared to previous

approaches.











3.5
3

2.5
S2

1.5



0.5

0


0 1 2 3
t (see)


4 5 67


Figure 4.1. State norm Vs
Time


4
3.5
3
2.5
, 2

1.5
1

0.5
0


0 1 2 3 4 5 6
t (sec)
Figure 4.2. Switching function Vs
Time


0 1 2 3 4 5 6 7
t (sec)
Figure 4.4. Compensation error Vs
Time


0.2

0

-0.2


-0.6

-0.8


0 1 2 3 4 5 6 7
t (see)


Figure 4.3. Control input Vs
Time


I,








Nomenclature for Chapter 4

A State matrix
Ad State delay matrix

B Control matrix

C Switching function matrix

d Magnitude of discontinuous control
f State dependent nonlinear perturbation

fd Delayed state dependent nonlinear perturbation

F State feedback matrix
Fd Delayed state feedback matrix

h delay

K Positive definite matrix

k Bound on state dependent uncertainty
kd Bound on state delay dependent uncertainty

s(t) Switching function

T Transformation matrix
Tp Estimated perturbation signal

u(t) Control input

x(t) State

z(t) Integral equation with deviating argument

Greek Letters

(5 Bound on the derivative estimation error

e Derivative estimation error in the sliding mode

ig Bound on the derivative estimation error in the sliding mode

a Scalar multiplier

0 time index




68


0(0) Initial condition

y(A) Matrix Measure of A













CHAPTER 5
SLIDING MODE CONTROL FOR UNCERTAIN INPUT DELAY SYSTEMS

5. 1 Introduction


Time delay exists in various branches of engineering. From biological, optical,

electric networks, to chemical reaction systems, time delay occurs and affects the stability

and performance of a wide variety of systems [See Schell and Ross, 1986, Inamdar et al.,

1991]. Input delays occur as one of the common sources of time delay, such as in

chemical processes, transportation lags and measurement delays, etc. It is well known that

the existence of time delay degrades the control performances and makes the closed-loop

stabilization difficult. Hence, the study of time delay systems has been paid considerable

attention over the past years. The description of time-delay systems lead to differential-

difference equations, which require the past values of the system variables. There are no

known methods to get necessary and sufficient conditions for the robust stability of such

systems. There are many approaches that yield sufficient conditions with varying degrees

of sophistication.

The case of perfectly known (i.e., no uncertainty) time-delay system is well studied.

It is known that a stabilizable time-delay system [see Kamen et al., 1985, Logemann,

1986] can always be stabilized by a finite dimensional controller. The study of state

uncertain delay systems has been paid much attention. However, with regard to the control

of uncertain input delay systems, only a few studies have been reported in the literature.

Based on Riccati-equation approach, a robust controller is derived in Kojima and Ishjiama

[1995]. Yan et al., [1997] propose a functional observer and state feedback mechanism

using the factorization approach. For a LQG/LTR method see Lee et al., [1988]. Wu and







Chou [1996] propose a control algorithm from input-output feedback linearization by

means of a parametrized coordinate transformation. Recently, an LMI based approach was

given in Niculescu et al., [1997] for delay-dependent closed loop stability of input-delay

systems. However, the authors allowed uncertainty only in the input delay.

One of the common methods used in chemical engineering applications, for processes

with time-delay, is to use the idea of the Smith-predictor scheme to cancel the effect of time

delay [Stephanopoulos, 1984]. However, if the model is imperfect (uncertain) then Smith-

predictor is known to give poor performance. Some approaches have been developed for

time-delay compensation based on prediction strategies [See Henson and Seborg, 1994].

In this chapter we propose an approach based on the Sliding Mode Control [Hung et

al., 1993; Utkin, 1977, Decarlo et al., 1988]. The sliding mode control approach

possesses many advantages, e.g., fast response, good transient performances, robustness

to the variation in plant parameters and external disturbances. The authors in [Shyu and

Yan, 1993; Oucheriah, 1995; Luo and Sen, 1993, Basker et al., 1997] study the sliding

mode control for uncertain systems with state delay. Also see Luo et al., [1997] for a

sliding mode approach for uncertain time delay systems with internal and external point

delays via various types of feedback.

This chapter proposes a SMC controller to robustly stabilize an uncertain input-delay

system with linear and possibly mismatching uncertainties. Sufficient robust stability

conditions are derived. No matching conditions on uncertainties are assumed.

The rest of the chapter is organized as follows: In Section 5.2, a transformation is

adapted to convert the original system with bounded input delay into one without any

delay. The stabilization of the modified system implies the stability of the original system

as long as the control law is bounded [see Kwon and Kim, 1980; Artstein, 1982]. Section

5.3 gives the guideline to design the sliding mode controller which consists of the

switching function and the control law. In Section 5.4, the analysis of the perturbed

system is carried out, and the reaching condition and the asymptotic stability of the sliding








mode are proven. In Section 5.5 some open issues in the sliding mode literature are raised

and answers are provided. Section 5.6 considers constraints on the controller input.

Section 5.7 gives an illustrative example with an open loop unstable plant. Conclusions

and future work are discussed in Section 5.8.

5.2 Preliminaries and Problem Formulation


Consider the uncertain input delay system

i(t) = Ax(t) + Bu(t) + Bdu(t h) + f(x,t) (5.1)

where x(t) E R is the state, u(.) : Rm is the control, A, B and Bd are constant matrices

with appropriate dimensions, h is a known constant time delay. The total plant uncertainty

is bounded by

If (x, t)l !lx (q1 (5.2)

where 1.1 denotes a vector-norm and Il1 denotes the associated induced matrix-norm.

Without loss of generality, the vector 2-norm 11.112 is used. The positive scalar boundfi is

assumed to be known. In rest of the development, without loss of generality, we assume

that m < n. Note that we do not assume any matching condition on the uncertainty. The

method described here can be trivially extended to the case of multiple known delays.
Assumption 1: The pair [A, B] is controllable and all the states are measurable.

Transformation into a delay free system

We consider a transformation to convert the original system (5.2) into an equivalent

system without delay. This reduction enables the use of known stabilizing methods for

linear systems without input delay. See Artstein [1982], for extensive theoretical

discussions on the validity and the applications of this type of reduction and also see

Fiagbedzi and Pearson [1986], for further generalizations.







Define the transformation

Z(t) = x(t) + -h e A(t--r)Bdu(T)dt (5.3)

Assumption 2. The solution Z(t) is not affected by a change in u(t) on a set of zero

Lebesgue measure [Artstein, 1982].

For an arbitrary bounded control u(t), the system (5.2) is transformed to (see

Appendix D for details)


i(t) = Az(t) + Bu(t) + f(x, t) (5.4)

Where -B = B + e-AhBd. It is assumed that B has full rank m. Then, via PBH rank test

[Kailath, 1980] it is trivial to show that the system (5.4) is also controllable. Note that the

system described by equation (5.4) is an uncertain linear system without input delay.

5.3 The Design of the Sliding Mode Controller


In the absence of perturbations, from (5.4), the nominal system is given by


i(t) = Az(t) + Bu(t) (5.5)

In the following we will design the switching function parameters on the basis of system

(5.5). The design consists of two phases. The first phase is the construction of the

switching function, so that, the nominal system restricted to the switching surface is

asymptotically stable. The second phase involves the development of a control law, which

must satisfy a sufficient condition for the existence and reachability of a sliding mode

within a finite time period.

Design of the switching function

Since system in (5.5) is a typical linear system, choose the switching function to be


s(t) = Cz(t) (5.6)

where C is an m x n matrix to be determined. The idea is to design the switching function

so that the nominal system is asymptotically stable in the sliding mode [Woodham and







Zinober, 1993]. Note that other types of switching function such as an integral switching

function can also be used to facilitate desirable switching dynamics. Since the matrix B

has full rank m, it is possible to reduce (n m) of the rows to zero, i.e., there exists a

nonsingular matrix T such that

TB=L (5.7)


where B1 is an m x m nonsingular matrix. In fact, we remark that such a T has a simple

form for rn = 1 and is given by


(n-1)


(n)
1


The transformed states are given by


w(t) = Tz(t)


(5.8)


and (5.5) becomes


-4(t) = TAT-1w(t) + T-Bu(t)


(5.9)


while the switching function becomes


s(t) = CT-'w(t) = Qw(t)

We introduce the following partitions for later use.


W(t)=W([] (m) Q= [
w(t)] (in) (n-m
(1)


(5.10)


Qg](m)
)(in)


(5.11)


T=[T T2]n
(n-m) (m)


TAT- =All A12](n-m)
[A21 A22/ (M)
(n-n) (m)


equation (5.9) can be rewritten in the form








W(t) = A1 W(t) + A12w1(t) (5.1 la)

ii'1(t) = A2lW(t) + A22wl(t) + Blu(t) (5.1 lb)

Furthermore, (5.10) can be written in the form

s(t) = QW(t) + QwI(t) (5.12)
where Q is an m x (n m) matrix, and Q, is an m x m nonsingular matrix.

Differentiating (5.12) and substituting (5.11) in to the dynamic equation yields

W(t) = (AI A12Q-,I)W(t)+ A12Q-ls(t) (5.13)



= [(QAI + QIA21) (-A12 + QIA22)QI-lQ]W(t) + (QA12 + QiA22)Ql-ls(t) + QiBlu(t)

(5.14)
When s(t) # 0, equations (5.13) and (5.14) describe the reaching mode. On the other

hand, when the system is in sliding mode, i.e., s(t) = 0 and (t) = 0, (5.13) yields

W(t) = (A,, A12QI-la)W(t) (5.15)

wI(t) = -Q-QW(t) (5.15a)

Since the pair (A11, A12) is completely controllable (see the Appendix E for a proof), there

exists an m x (n m) matrix K that satisfies

)L(A1 A12K)=A (5.16)

where V, represents the desired poles in the sliding mode. Since A11 and A12 are known,

a set of linear equations can be solved to yield K. Comparing (5.15) and (5.16), we have

QIK (5.17)
Choosing Q, = I yields


Q = [Q QI] = [K I]


(5.18)








Substituting (5.18) into (5.10) results in

C=QT=[K Im]T (5.19)

Therefore, the switching function has the form


s(t) = Cz(t) = [K I]Tz(t) (5.20)

It is instructive to recast the dynamics of the system in terms of our choice for C, i. e., K.

By using (5.11), it is easy to show the following:

W(t) = (A,1 A12K)W(t) + A12s(t) (5.21)


= [K(Al1 A12K) + (A21 A22K)]W(t) + (KA12 + A22)s(t) + Biu(t) (5.22)


w1(t) = s(t) KW(t) (5.23)

We also remark that in the sliding mode, wI(t) = -KW(t) and hence, the asymptotic
stability of (5.15) ensures that lim w, (t) -4 0.

Remarks:

(1) The above also illustrates an interesting system-theoretic point. We are seeking

to control an n-dimensional system via a m-dimensional input. From (5.21)-

(5.23), it is obvious that the input explicitly affects only a m-dimensional subset

of this realization and the remaining (n-m) dimensional subsystem is affected

via the feedback of the states as determined by the system. This information is

not new. In fact, by recasting the system in the controllability realization, this

fact can be trivially seen. But, in contrast to the controllability canonical form,

(5.21)-(5.23) also guide in choosing a controller to exploit the fact that the

matrix BI is of rank 'm'. More about this will follow in later sections.

(2) Note that with the choice of C, (CB)-1 = B1-' and since BI is nonsingular,

(CB)-1 always exists.







Design of the Control Law

The control law chosen is a combination of the traditional state feedback and a time

varying discontinuous control element

u(t) = -(CB)-I[CAz(t) + kjs(t)j" sgn[s(t)] + d(t)sgn[s(t)]] (5.24)

where the parameters are chosen such that k > 0, 0!_ a < 1.0 and d(t) = PC[Ix(t)l + E,

with e > 0. When s(t) is small, a faster approach rate than the linear approach algorithm

is realized. The state feedback is designed to cancel the effect of the state matrix A in the

switching dynamics. Once the system is in the vicinity of the switching surface, the

magnitude of the discontinuous control is linearly proportional to the norm of the states.

Since the states x(t) and z(t) are available, realizing the above control law is straight

forward. Using the partitions (5.11), it is also easy to verify that (5.24) is equivalent to

u = -(B,)-1{[K(A11 A12K) + (A21 A22K)]W(t) + (KA12 + A22)s(t)}


+

-(B,)-'[kjs(t)jasgn[s(t)] + d(t)sgn[s(t)]] (5.25)

Comparing (5.25) and (5.22), it is obvious that we exploit the fact B1 is non-singular

m x m matrix, and obtain u by equating (5.22) to zero.

With the chosen controller, the switching dynamics (5.22) is of the form

(t)= -kls(t)l' sgn[s(t)] d(t)sgn[s(t)] + Cf(x(t),t) (5.26)

Remarks:

(i) From (5.26), it is obvious that the discontinuous control is used to overpower

the effect of uncertainty. Since the only information we have about the

uncertainty is the norm bound, we can not use any lower value of d(t) and be

guaranteed to reach the sliding mode within a finite time. From (5.25), it is

clear that there will always be chattering for this choice of parameters.








(ii) For the nominal system d(t) can be chosen arbitrarily close to zero.

(iii) The parameter a can be used in a time varying fashion to guarantee a faster rate

of approach to the switching surface than the normal choice c = 1.0. For
example, c = 1.0 for ls(t)l 1.0 and (x = 0.5 for ls(t)l = 1.0 would assure a

faster rate of approach than the proportion control (i.e., a = 1.0, Vt > 0 ).

5.4 Analysis of the Perturbed System


In this section, the robust stability of the uncertain system (5.1) is analyzed. First, the

analysis is carried out for system (5.4), in order to give conditions for reaching the sliding

mode and the asymptotic stability in the sliding mode. Since system (5.4) is related to

system (5.1) via (5.3), conditions are given under which robust stability can be inferred for

the original system (5.1).

For an uncertain system to be robustly stable under sliding mode control, the

following conditions have to be met: (i) The system should attain the sliding mode within

a finite time; that is, the state trajectories should reach the switching surface and should stay

on the switching surface for all time thereafter; and (ii) the system in the sliding mode

should be asymptotically stable.

Reaching Condition for Sliding Mode.


Theorem 5.1: The uncertain delay system (5.4) with model-uncertainty and

bounded measurement uncertainty given by (5.2) achieves the desired sliding mode within

a finite time if the control law (5.25) is applied.

Furthermore, it can be shown that the value of oc chosen 1, then the sliding mode is

attained within t = tr, where


tr < (2 ln[l +k T{s (o)s(O) J] (5.27)
k 2s)


and the upper bound on the norm of the switching surface is given by







8
-s(t)- k e k (5.28)

Proof: We choose a positive definite Lyapunov function

V(t) = 1 (ST(t)S(t)) (5.29)

Our aim is to show that '(t) < 0 for the choice of parameters k and F. The development

here is fairly standard.

V(t) = sT (t)(t)
utilizing (5.26) for (t)

= s T (t)[ks(t)ja sgn[s(t)] d(t)sgn[s(t)] + Cf (x, t)] (5.30)

From the available bounds on the uncertainty

Cf (x,t) < [CIf(x,t)l 13 C[x (t)[ <_ d (5.31)

Using (5.31) in (5.30)

-(t)_-Is (t)[ ksT(t)sgn[s(t)]] d(t)[ST (t)sgn[s(t)]] + sT (t)Cf(x,t) (5.32)

We observe that

sT (t)sgn[s(t)] = ys1T(t)sgn[si (t)]

2 si(t)l[sgn[si (t)]2

= X si(t) = s(t)1 (5.33)

Using (5.33) in (5.32)

V (t) < -kls(t)als(t)l1 Es(t)l (5.34)

Since both the terms on the right hand side are negative, the system is guaranteed to attain
the sliding mode in finite time. Furthermore, for the choice of a = 1.0, (5.34) becomes

V(t) < -kjs(t)12 EIS(t)I1 (5.35)


since for a vector 1.12 <1.11'







Using (5.5.29)

V (t) -kV(t)- eV"12(t) (5.36)

The above is a scalar difference equation, and from Appendix F,


V < +- e 2-- (5.37)
k

I s(t)l< s(0)l+ e2 (5.38)

[ kj k

Equating the right hand side to zero we get the result. V

Corollary:

For the choice 0 < a < 1.0, a tighter result can be obtained. Following the same steps as in

the proof of the above theorem (see Appendix F), it can be shown that for a control law
(5.24) with d(t) = I3Cjjx(t)!, the sliding mode is achieved within t= tr, where

tr- 1 S T.5(1S-( )
r

and



IS(t)l F (l a( 2 a)X(0)]0l kt] 0!< t:5 tr

This is advantageous because, we avoid an extra discontinuous control element, 8.

Asymptotic Stability in the Sliding Mode

Note that when the system is in the sliding mode, the control law reduces to a linear

state feedback. In this chapter, we follow a Lyapunov-Razhamikhin type approach to

derive sufficient stability condition.

The dynamic equation in sliding mode is given by

w W)= (A1 A12K)W(t) + Tlf(x,t) (5.39)

Since in the sliding mode, wl(t) = -KW(t), it suffices to show the asymptotic stability of







(5.39). The first task is to convert the uncertainty description in terms of W(t).

z(t) = T-[ W "(t) ] = T-'[_ Wm(t) T1 Im(5.40)
LW1(t). -KwI ()K

Next we seek an expression relating x(t) and z(t). From (5.5.3)

Ix(t) Iz(t)l + h ea (t-h- )Bdu(,r)dT (5.41)

where u(t) = -(CB)-' CAz(t).

Thus

Jx(t)J: z(t)J+ max eA8 ] Bd (CB)CAfhhz(Jdr (5.42)

The main hurdle at this point is to express z('r)j in terms of Jz(t)l. To obviate this

difficulty, we take Lyapunov--Razhamikhin type approach. First of all we show that for

the case of matched uncertainties, proving asymptotic stability is straight forward.

Definition: Matching condition [Hung et al., 1993]. If the uncertainties satisfy

f(x, t) = Bf(x, t)
where f(x, t) e 91rex1, then the matching condition is satisfied.

This means that the perturbation is in the image of B.


Tf(x, t) = TBf(x, t)


=[T2Bf(xt)]

Hence, in the sliding mode, the system equation becomes

W(t) = (AlI Al2QI-1Q)(t) (5.43)

Note that (5.43) is of the same form as the equation describing sliding mode for the

nominal system (5.15). This implies that the uncertainties will have no effect on the

system in the sliding mode. This is one of the major advantages of the sliding mode

design. The state feedback is simply used to place the poles of the nominal system in the








sliding mode. However, the assumption of matched uncertainties is a highly restrictive

one, as for example in the m = 1 case, we are constraining the perturbation to be a scaled

version of B. One should note that the situation represented in (5.43) is an ideal case.

That is, if the controller is able to exactly cancel the perturbation, then the sliding mode is

independent of the perturbation. This obviously suggests that if one could estimate and

cancel the perturbation (or) the effect of the perturbation on the sliding mode, then the

system in sliding mode can be made to be independent of the matching perturbations.

These kinds of approaches are commonly called perturbation compensation and have been

applied to linear state delay systems [Chan et al., 1996]. Even if mismatched uncertainties

are present, perturbation compensation can be employed to reduce the magnitude of the

required discontinuous control.

In the mismatched case, it is not possible to completely cancel the perturbation and

there will be a residual effect of the perturbations in the sliding mode and the state feedback

should be such that the overall system is asymptotically stable. Theorem 5.2 gives a

sufficient condition for the mismatched case.

Before we state and prove Theorem 5.2, we explain about a difficulty that commonly

arises in proving asymptotic stability in state-delayed systems and input-delayed systems

(as the control usually involves state feedback, the input-delay problem has relevance and

resemblance to state-delayed problem). There are two classical approaches in the literature

for proving asymptotic stability for delayed systems. Over the years there have been many

incremental improvements, but the message is that all methods are conservative. This is

because the differential equations that determine the dynamics of the system have two

difficult elements. One, the perturbations. Two, the delayed states. We know only the

norm bound on the perturbation, and while using this sufficiency creeps in. The problem

with delayed states is more interesting. In using Lyapunov-Razhamikhin type approaches,

a restriction is imposed on the dynamics of the system.







Consider a Lyapunov candidate


V(t) = WT(t)pW(t)

where P = PTP > 0.

At this point, we introduce the usual Razhamikhin type restriction on the system. We

restrict that

V(O)1

It is important to understand, how restrictive the above assumption is. The reason for

using such an assumption is that it allows one to bound the norm of the delayed states of

the system in terms of the norm of the present states of the system. Obviously any

condition based on this will only be sufficient. Since the following condition holds for any

positive-definite matrix P = pT > 0, and W = 91n-m

1min(P)WTW _! j-TpW _< ',min(P)WTW

it follows that



where a 4 )min(P)
X'min (P) '

The above condition is not very restrictive as it is true for some (51 > 85 > 1, for all

stable systems whose state norm continuously decreases (and it is easy to show that for

unstable systems, the above is true for any positive 8 ). The actual values of 8 and a

(and hence P) quantify the decay rate of the system. This condition, however could be

violated for a short durations, for example, when a delayed disturbance hits the system

(i.e, the norm of the states could increase). However, the controller will act against the

disturbance and restore the performance. Hence, temporary violation of this constraint is

not fatal, and will not affect the stability conclusion.







Theorem 5.2: The system (5.4) in sliding mode with the mismatched uncertainties

(5.2) is asymptotically stable if the following condition is satisfied

T-r I-fl 1+ha max e 16 Bd(CBY1CA ]} <'min(Q) (544)
-K L -h ij 211PI1TI l

where

a =ImxjP) and P > 0 is the solution of
min (P)

P(A _A12K) + (Aj1 _A12K)T p = _Q, Q >0 (5.45)



Proof. Under the similarity transformation (5.8) the system (5.4) transforms to

(5.11). If the uncertainties are present, using (5.11) equation (5.1 la)-(5.1 lb) adopts the

form

W(t) = A1 W(t) + A12Wl(t) + Tf(x,t) (5.46)

i1(t) = A21W(t) + A22Wl(t) + Btu(t) + T2f (x,t) (5.47)


Consider a Lyapunov candidate

V(t) = WT(t)PW(t) (5.48)
where P = pT > p = pT > ,P (n-)(n

Taking the derivative of (5.48)

V,(t) = WT(t)[P(A11 A12K) + (Al1 A12K)Tp]wT (t) + 2wT (t)PTf (x,t)(5.49)

At this point, we make the usual Razhamikhin type restriction on the system. We restrict

that

V(o)!<_ N(t), t- h<__0 ___t, V(5>1 (5.50)


it follows that,







Ii(o)1- (! 4 m (P)l (t)I (5.51)
Amin (P)
Denoting a = x} using this in (5.42)

Ix(t)l<- T-1[:-] {l+h8a[ max eAO Rd(CB)-) (5.52)

Therefore,
If (Xt) <- P T-1[Im] {i + haa[_max eAO Bd(CB)-1CA ]}iv(t)I (5.53)

We combine all the terms to get

f3l = / T-l[In-m] {1 + amax eAO1 Bd(CB)-IcA

if(,q P wtj(5.54)
hence
V(t) = WT(t)[P(A11 A12K) + (A, I A12K)T P]w(t) + 2/, 11PJ II W(t)12(5.55)

Choose P > 0, Q > 0, such that

P(A11 A12K) + (AI A12K)TP = -Q

<-w T(t)QW(t) + 2/ llPjTI (t)12 (5.56)

V7(t) < [-Amn(Q) + 2P,111l] W(t)12 (5.57)
Hence a sufficient condition for asymptotic stability in the sliding mode is given by

P 1in(Q)
211PJJIJTalI

i.e.,

T 1[-I-MlI+h'a[ max e A 1nCd(cA)-ca] < min(Q) (5.58)
-K il L -heO 211PI11T1







Note that if the above expression is true for 8 = 1 then there exists a 3 > 1, sufficiently

small that it will still hold true. Therefore, a sufficient condition for stability is given by the

above expression, with 8 = 1. V

Remarks:

(i) In the sliding mode, the derivative of the Lyapunov function is given

V (t) -< [-,Imin (Q) + 2pa, 1P[T1 2 JjW(t)j2 <0.

Denoting g := -(-/min (Q) + 231 11PIIT1 [), we can obtain

<(t) __


I /Iin () 1(5.59)

This explains why we can get only asymptotic stability guarantee, that is, we

can only state that the norm of the states will reach zero as time goes to infinity.

In the next section we will elaborate more about this issue.

(ii) For a given system description, once the poles of the closed-loop nominal

system are chosen, everything except P and hence Q is known. Hence P is

an optimization parameter. We remark that the best choice is to make sure that

the condition number of P is minimized while satisfying (5.44)-(5.45). For

large systems, this constrained optimization problem can be cast as an convex

LMI problem and easily solved through available commercial software

[Niculescu et al., 1997].

Theorem 5.3. If the uncertain system (5.4) is asymptotically stable, it implies the

asymptotic stability of system (5.1).

Proof. From the transformation (5.3), we have

x(t)<-lz(t)j+ h max eAO]. Bd. ut (5.60)

From Theorem 5.2, the transformed system (5.4) is asymptotically stable. Hence,








lim Jz(t)l -- 0. Since u(t) = -(CB)- CAz(t), in the sliding mode, lim [u(t)l -- 0 From
f--->-t-400
(5.60) the result follows. V

Implementation Issues

Due to inherent limitations such as the switching delay, system inertia, etc., the sliding

motion does not take place in any real system. Instead, the trajectories chatter in a

neighborhood of the switching surface. The chattering phenomenon is undesirable because

it may adversely affect the actuating mechanism and excite the high-frequency unmodeled

plant dynamics, leading to instability. The discussion about chattering and the ways to

alleviate this problem, as outlined in Chapter 4, applies to the present case of stabilizing

systems with input-delay as well.

5.5 Some Open Issues in Sliding Mode Literature


Although the sliding mode control approaches have been used for a long time for a

variety of systems, there are nevertheless some open issues. We raise the following issues

of interest:

(1) Is it possible to guarantee that the norm of the states continuously decrease

during the reaching phase?. If so, what is the non-trivial upper bound, on the

norm of the states?

(2) Is it possible to get an upper bound the norm of the states when the system is
in the reaching phase (other than of course, the trivial estimate Jx(0)[) ?

(3) We are able to prove that the sliding mode will be attained within a finite time.

At the same time, the best we can prove for the system in the sliding mode is

asymptotic stability. We also seek to explain why this is so.

We also remark that these questions apply to all sliding mode control schemes (for

linear uncertain, Uncertain linear systems with state delay and so on..) and not just for the

input-delay case that we consider here. Note that the usual approach in the literature for







sliding mode systems is to prove the reaching condition within a finite time (as we did via

Theorem 5.1), and then prove asymptotic stability for the reduced system in the sliding

mode (as in Theorem 5.2). However, there are problems with this approach. For

example, one does not have an estimate of the norm of the states, when the system reaches

the sliding mode. Hence, we do not know the proper initial condition for the scalar

Lyapunov differential equation given by (5.59). One only hopes that it is less than the

norm corresponding to the initial condition. Even assuming so, it is still highly

conservative. What we need is a way of estimating a tight upper-bound on the norm of the

states, during the reaching phase. If we obtain such an expression, then we will be able to

answer questions (Ql)-(Q2).

Before we seek to get an analytical expression to the above questions, it is instructive

to ponder over the significance of Theorem 5.1. What (5.27)-(5.28) imply is that the norm

of a linear combination of the states continuously decreases and attains zero within a finite

time. But, a linear combination of two time-dependent functions (here states of the system)

may be decreasing towards zero, while the individual elements may be diverging in

opposite direction, making their sum (or difference) approach zero. Hence, it seems like

that we do not even have a guarantee that the norm of the states of the system will

continuously decrease, when the norm of the switching function continuously decreases

and attains the sliding mode. In other words, the central question here is, can one obtain

sliding mode when the system is unstable.

However, upon reflection, one simple fact emerges. Consider the linear uncertain

system (5.4) and bounded perturbation (5.2). The solution will be driven by the interaction

of the control and perturbations and the rate at which the states converge (or) diverge will

depend on the state transition matrix, and whether or not it is stable enough to overcome

the perturbations. Ultimately the form of the solution will contain complex exponentials,

apart from other terms. Since, the perturbation is bound by the norm of the states, an

unique solution does exist. It is not our intention to derive it. In fact, it is impossible to







derive it, unless the form of the perturbation is exactly known. Now, Theorem 5.1 implies

that the norm of a linear combination of the complex exponential terms is continuously

decreasing. If the system is unstable, it is the real part of the complex exponentials that

will dominate the system dynamics. In that case, the resulting linear combination of the
complex exponentials will diverge. Hence, it is not possible for js(t)j to continuously

decrease and attain zero within a finite time, if the system is unstable.

This however, does not necessarily mean that the norm of the states will continuously

decrease. For example, for a system with state delay, when the delayed states first come

into action, there could be a temporary increase in the norm. Alternately, there could be a

non-linear uncertainty that satisfies all our constraints and still has a spike leading to an

increase in the norm of the states. Of course, the controller will seek to react to this

situation and bring the system under control. Therefore, although mathematically not

precise, it is nevertheless acceptable to assume in general, that the norm of the states

decrease in the reaching phase. This conclusion is important to assess the behavior of the

state norm in the reaching phase. For example, without such a guarantee, it is impossible
to bound the discontinuous control signal d(t). From (5.24), d(t)= /31Cjx(t)j + E. Hence,

the value of the discontinuous control is linked to the norm of the states. In proving

asymptotic stability in the sliding mode, we obviate this difficulty, as sgn(s(t)) = 0. With

the above conclusion, however, one could make a confident assumption on the expected

value of d(t), much in the same vein of the Razhamikhin type assumption, that seeks to

restrict the norm of the states.

The following lemma is needed for the ensuing developments.

Lemma 5.1

Consider the following scalar differential equation

k(t)< -ax(t)+(bexp(-ct/2)+e exp(-82t/2)-d)x12(t) (5.61)

with the initial condition x(O) and the following constraints:

(1) The constants (a,b,c,d,e) > 0 and a # c and a # 52








(2) x>0, Vt >0 and Vx>0, Vt >0

Then,


x+ bexp t) + exp t exp t
a (a c) (5.62)
e [(s 2 ( xpa t)] d
+----2)exp ex t i--
+(a -382)[ 2) (_ 2 a

Proof: See Appendix G V

Remark:

In (5.62) the first three terms are positive, while the third term is a negative constant.

Furthermore, the first term continuously decreases from the initial value, while the second

and terms always have a maxima. As a parallel we remark that the second and third terms

in (5.62) look like the response of an under damped second order system to unit impulse

input.

Upper Bound on the Norm of the States During the Reaching Phase


In this section, the aim is to find an upper bound on the norm of the states, for the stable

closed-loop system. The analysis for the nominal system is straightforward and is

presented in the sequel. However for the perturbed system, the analysis is quite involved.

In the reaching phase, the dynamic equations are given by

W(t) = (A,1 Al2K)W(t) + Tlf(x, t) + Al2st) (5.63)


wi(t) = s(t) KW(t) (5.64)


In complete parallel to the proof of Theorem 5.2, consider a Lyapunov candidate


V(t) = WT(t)pW(t) (5.65)
where P = PT > 0, p = PT > 0, p = 9 (n-m)x(n-in) and impose the restriction


V(O)_8V(t) t-h<_O<_t,V8>1


(5.66a)







It is straightforward to show from (5.66a) that

/IZmin (P ) ITF(t)[ 56b


Choose P > 0, Q > 0, such that

3(A11 A12K) + (A,1 A12K)Tp= -Q (5.67)
Taking the derivative of (5.65) yields

V(t) WT(t)[P(A A12K) + (A,1 A12K)TP]wT(t)
+2w (t)PTf(x,t) + 2wT(t)PA12s(t)

Converting the Uncertainty in Terms of W(t)

Again, the challenge is to convert the uncertainty description in terms of W(t). Since
f(x(t),t)l < Ix(t)l, we run into a problem, since the stated aim is to find an upper bound
on Ix(t)l In the ensuing development, we will pinpoint this difficulty and propose an
additional assumption on the perturbation that will enable us to obviate this hurdle.. From
Theorem 5.1, we have

s(t) [S(O)I-+ e 2 (5.69)
I 1t~ ks0 l --- k

For simplicity, we denote
/[=s(0)+-- 2--] d -qe
=k, d--

Hence s(t) < be 2 -da.
In the reaching mode,

z(t) = T-'w(t) = T- W =) T [ Wt
Lw = K(t). + T (t)
Z(t) = T-' 4-, W(t) + T-1 0"-M]Lst






1' n 1 (q +
where T2 is a matrix of dimension n x m, partitioned out of T-.
Since we used the transformation (5.3), from (5.42)
Ix(t) < z(t)l + t-h a(t-h-BU()d
where the controller is given by
u(t) = -(CB)-[CAz(t) + kjs(t)jsgn[s(t)] + d(t)sgn[s(t)]]
since Isgn[s(t)] < -m, we take an upper bound

u(Ol- (cn)-'CAz(t) + m (&)1 [, kd + d(t)]
Ix(t)l-Iz(t)j+I max e A] Bd(C-B)-1CA Jj hl(T)Id


+ -Mmaxe eAO8 ] Bdjjj(Cffh] ft-h &kb2 0+ d()jdT
L-h<_O<_O I

Substituting for lz(t)l in terms of IW(t)l and [s(t)j, and using Razhamikhin type assumption
for the delayed states, via (5.66b), results in
Ix(01_< T-1[ IW(t)+11T21 be d

+ h~a T-l[In-ml max eA0 Bd(C-B)-ICA W(t)I
LK-oJ -hO _o

+ mx aoe Bd(CB)-1CA ft-h be--2 d dz (5.70a)

+ -m max eA1] JBd1 (CB jk)-1 b 2 -kd+d r)1dr
k-h<_5<_O5 "Jt-h kbI d+ (c '

In the above equation, the only unknown quantity is d(t). According to our control law,
d(t) = P13 Icllx(t)l + E, and our stated aim is to find an upper bound on the norm of the




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