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Robust stability analysis methods for systems with structured and parametric uncertainties

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Robust stability analysis methods for systems with structured and parametric uncertainties
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ROBUST STABILITY ANALYSIS METHODS FOR SYSTEMS WITH
STRUCTURED AND PARAMETRIC UNCERTAINTIES
















By

CHARLES THOMAS BAAB


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

2002





























Copyright 2002

by

Charles Thomas Baab
































To Holly and C.J.













ACKNOWLEDGMENTS

I would like to express my sincere appreciation to my advisor, Oscar Crisalle,

without whose support this dissertation would not have been possible. I'm also grateful

for the opportunity he has given me to obtain a master's degree in electrical engineering.

I wish to thank Professors Richard Dickinson, Dinesh Shah, Spyros Svoronos, and

Haniph Latchman for serving on my supervisory committee. It was reassuring knowing

that no matter how mathematically intense my research became Dr. Latchman always

understood where I was and where I needed to go.

I thank V. R. Basker, Jon Engelstad, Serkan Kincal, and H. Mike Mahon who

have led the way and Chris Meredith and Brian Remark who will follow. They all have

not only contributed greatly to my research but have been good friends.

Finally, I wish to thank my family. In particular, my loving wife, Holly, and my

wonderful son, C. J., for their unending support.














TABLE OF CONTENTS

page

ACKNOWLEDGMENTS ........................................................................................... iv

A B STR A C T ............................................................................................................... viii

CHAPTER

1 INTRODUCTION ......................................................................................................

1.1. M otivation ................................................................................. .............................
1.2. Objective and Structure of Dissertation............................................. ..............3

2 A DUALITY PROOF FOR THE MAJOR PRINCIPAL DIRECTION ALIGNMENT
PR IN C IPLE ................................................................................................................... 6

2.1. Introduction ............................................................................. ..............................
2.2. Mathematical Background ...............................................................................8
2.2.1. The Singular Value Decomposition and Eigenvalue Decomposition .............8
2.2.2. Dual Norms and Dual Vectors................................................................ 10
2.2.3. Dual Eigenvector Result ........................................................................... 11
2.2.4. Eigenvector-Singular Vector Equivalence Result ........................................12
2.3. Statement of the Major Principal Direction Alignment Property ........................13
2.4. Modified Statement of the Major Principal Direction Alignment Principle..........14
2.5. Exam ples.................................................................. ........................................... 17
2.5.1. Exam ple 2.1 .............................................................................................. 17
2.5.2. Exam ple 2.2 ..............................................................................................21
2.6. Conclusions...................................................................................................... 24

3 MAJOR PRINCIPAL DIRECTION ALIGNMENT WHEN THE MAXIMUM
SINGULAR VALUE IS REPEATED AND ITS RELATIONSHIP TO OPTIMAL
SIMILARITY SCALING ............................................................................................25

3.1. Introduction......................................................................................................25
3.2. Mathematical Background ...................................................................................27
3.2.1. The Singular Value Decomposition............................................................27
3.2.2. Statement of the Major Principal Direction Alignment Principle ................29
3.2.3. Affine Sets, Convex Sets, and Convex Functions ........................................29
3.2.4. Differential Theory .................................................................................32
3.2.5. Expression for the gradient when the maximum singular value is distinct. ...36







3.3. Main Result Characterization of the Subdifferential when the Maximum
Singular V alue is Repeated ......................................................................................39
3.3.1. General Expression for the Subdifferential...................................................39
3.3.2. Characterization of the Subdifferential as an Ellipsoid ................................42
3.4. Determining the Steepest Descent Direction and Conditions for a Minimum ......50
3.5. Attainability of MPDA when the maximum singular value is repeated ..............52
3.6. Reconciling the Results with the PDA Results...................................................54
3.7. E xam ples..........................................................................................................54
3.7.1. Exam ple 3.1 ..............................................................................................55
3.7.2. Exam ple 3.2 .............................................................................................. 58
3.7.3. E xam ple 3.3 .............................................................................................. 59
3.8. C onclusions....... ......... .......................................... ........... .............................. 61

4 SPECTRAL RADIUS MAXIMUM SINGULAR VALUE EQUIVALENCE
UNDER OPTIMAL SIMILARITY SCALING..............................................................62

4.1. Introduction .................................................................................................... ...... 62
4.2. M them atical Background ........................................ ...........................................64
4.2.1. Dual Norms and Dual Vectors.....................................................................64
4.2.2. Positive M atrix Result.............................................................. .................... 65
4.2.3. Major Principal Direction Alignment Property ............................................69
4.2.4. MPDA as a Control Theory Application ................................................70
4.3. Main Result Extension of the Positive Matrix Result to General Complex
M atrices................................................................................................... ..................72
4.4. Exam ple 4.1 ..................................................................................................... 76
4.5. C onclusions.................................................. ....................................................... 77

5 GENERALIZATION OF THE NYQUIST ROBUST STABILITY MARGIN AND
ITS APPLICATION TO SYSTEMS WITH REAL AFFINE PARAMETRIC
UN CERTA IN TIES ...................................................................................................... 78

5.1. Introduction ............................................................................................................ 78
5.2. Generalization of the Critical Direction Theory .......................................... ..80
5.2.1. Prelim inaries ............................................................................................. 80
5.2.2. Analysis of Robust Stability ................................... .....................................84
5.3. Systems with Affine Uncertainty Structure ...................................................88
5.4. Robust Stability and Uncertainty Value-Set Membership...................................89
5.5. Computation of the Critical Perturbation Radius............................. ............ 95
5.6. Intersection of a Ray and Arcs in the Complex Plane .........................................96
5.7. Exam ples................................. .......... ................................................................ 100
5.7.1. Example 5.1 Convex Critical Value Set ..................................................100
5.7.2. Example 5.2 Nonconvex Critical Value Set............................................104
5.8. C onclusions........................................................................................................ 109







6 ROBUST CONTROLLER SYNTHESIS FOR SYSTEMS WITH NONCONVEX
VALUE SETS USING AN EXTENSION OF THE NYQUIST ROBUST STABILITY
M A R G IN ......................................................................................................................... 10

6.1 Introduction.................................................................................................... 110
6.2. D esign M ethodology............................................................................... ..........112
6.3. D esign Exam ple ...................................................................................................116
6.4. C conclusion ..................................................................................................... 121

7 ROBUSTNESS OF CLASSICAL PROPORTIONAL-INTEGRAL CONTROLLER
D ESIG N M ETH O D S.................................................................................................122

7.1. Introduction.......................................................................................................... 122
7.2. Prelim inaries ..................................................................................................125
7.2.1. Process Model and Uncertainty Description...................................................126
7.2.2. Proportional-Integral Control and Controller Tuning Rules.................... 127
7.3. Analysis of Robust Stability .............................................................................. 129
7.3.1. Conditions for Robust Stability ..................................................................129
7.3.2. Parametric Boundaries for Robust Stability ............................................. 133
7.3.4. Stability M argins.................................................................... .................. 39
7.4. Results of Num erical Studies.......................................................................... 143
7.4.1. Region of Stable Perturbations for the ITAE Regulation Tuning Rule........ 143
7.4.2. Stability Margins Computation for Each Tuning Rule............................... 148
7.5. C onclusions............................................................................................. .......... 153

8 CONCLUSIONS AND FUTURE WORK ................................................................154

APPENDIX

A PRO O F O F LEM M A 2.1........................................................................................... 156

B PROOF OF THEOREM 2.1..................................................................................... 158

C PROOF OF THEOREM 7.1.................................................................................. 161

D PROOF OF LEMMA 7.3....................................................................................163

E PROOF OF THEOREM 7.2..................................................................................... 167

F SIGN CHANGES IN EQUATIONS (7.12A) AND (7.12B) ....................................171

LIST OF REFEREN CES .................................................................................................176

BIOGRAPHICAL SKETCH ...................................................................................180













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

ROBUST STABILITY ANALYSIS METHODS FOR SYSTEMS WITH
STRUCTURE AND PARAMETRIC UNCERTAINTIES

By

Charles Thomas Baab

December 2002


Chairman: Oscar D. Crisalle
Major Department: Chemical Engineering

The major principal direction alignment principle is investigated in detail for the

case when the maximum singular value is repeated. A first result is a new proof based on

duality theory for the necessary and sufficient conditions that ensure equality of the

spectral radius and maximum singular value of a matrix; namely, that there must exists at

least one aligned pair of major input-output principal-direction vectors. A second result

is the development of a novel numerical optimization algorithm to solve the optimal

similarity-scaling problem that yields an upper bound for the structured singular value.

The algorithm provides a systematic procedure for identifying the steepest-descent search

direction even for the case when the singular value is repeated and the underlying

optimization problem is locally nondifferentiable. The key theoretical element is the

characterization of the subdifferential at every point of nondifferentiability.


viii







The critical-direction theory is extended to include nonconvex critical uncertainty

value sets through the introduction of a general definition of the critical perturbation

radius. The Nyquist robust stability margin is calculated for systems with affine

parametric uncertainty using an explicit map from the parameter space to the Nyquist

plane. A practical design approach based on parameter space methods is introduced.

First the controller parameters that result in robustly stable closed-loop systems are

determined. Then, a performance objective is optimized over the set of robustly

stabilizing controller parameters, resulting in a robustly stabilizing controller with some

optimal performance characteristics.

A formal robustness analysis of popular proportional-integral controller tuning

rules for systems approximated by a first-order-plus-time-delay model is presented. The

uncertainty in the process model is represented by multiplicative parametric perturbations

in the process gain, process time constant, and process time-delay. The robustness of the

uncertain system is characterized in terms of the set of all perturbations that result in

stable closed-loops. This set is used to calculate the standard gain and phase margins,

and the parametric stability margin which is a metric of robustness to simultaneous

variations in all three system parameters. These margins are used to compare the relative

robustness properties of several disturbance-rejection and tracking tuning rules in

widespread use.












CHAPTER 1
INTRODUCTION

1.1. Motivation

Uncertainty is a fact of any real-world system. This uncertainty inherently

translates to the model of the system used for control design, and is most often in the

form of neglected dynamics or variations in model parameters. An important

requirement of any control system is that it be robust (i.e., it functions satisfactorily under

these uncertainties), and the design of such control systems is known as Robust Control.

An important aspect of the robust control problem is the robust analysis problem which is

determining if a control system satisfies stability and performance requirements given an

admissible set of uncertainties.

Robust stability is obviously a necessary requirement for robustness, and has been

studied since the earliest days of feedback control, which originated to desensitize control

systems to changes in the process as well as stabilize unstable systems. The classical

design techniques focused on frequency domain methods such as those based on Bode

plots and Nyquist plots (Nyquist, 1932) and resulted in the gain and phase stability

margins. With the advent of the space race of the 1960's, the focus of control engineers

was shifted away from frequency domain robust stability methods to the field of optimal

control. In fact, the linear quadratic regulator (LQG) design appeared to give controllers

with good stability properties, but in the late 1970's it was found that LQG and other

prevailing methods of control design such as state feedback through observers lost their

stability guarantees under uncertainty.







As a result, H. optimal control (Zames, 1981; Francis and Zames, 1984; Doyle,

1983; Safonov and Verma, 1985; Doyle et al., 1989) was introduced as a framework to

effectively deal with robust stability and performance problems. The theory provides a

precise formulation and solution to the problem of synthesizing an output feedback

compensator that minimizes the H. norm of a prescribed system transfer function. The

method considers unstructured uncertainties where the only information known about the

uncertainty is a norm bound. Typically, more information about the uncertainty is known

than a simple norm bound. As a consequence, several robust analysis methods have been

developed to consider these structure uncertainties.

Possibly the most effective and comprehensive result is the structured singular

value (p ) analysis method introduced by Doyle (1982), which considers the problem of

robust stability for a known plant subject to a block-diagonal uncertainty structure under

feedback. In general, any block-diagram interconnection of systems and uncertainties

can be rearranged into the block-diagonal standard form.. The value of p corresponds to

the smallest uncertainty that will destabilize the system. Unfortunately, calculating p is

not trivial; in fact the underlying optimization problem has been proven to be NP-hard

(Braatz et al., 1994). However, there is a convex optimization that gives a conservative

upper bound for p,. In addressing the existence of solutions to the proposed convex

optimization, Kouvaritakis and Latchman introduce the major principal direction

alignment (MPDA) property (1985), which gives necessary and sufficient conditions for

p to equal its upper bound, thus eliminating the conservatism.

Another robust stability analysis method for structure uncertainties is the critical-

direction theory developed by Latchman and Crisalle (1995) and Latchman et al. (1997)







which addresses the problem of robust stability of systems affected by uncertainties that

are characterized in terms of arbitrary frequency-domain value sets that are convex. The

critical direction theory proposes the Nyquist robust stability margin as a measure of

robust stability which has obvious connections to the Nyquist stability criteria. The

advantage of the critical direction theory over the structured singular value theory is that

for several common structured uncertainty types there is an analytical expression for the

Nyquist robust stability margin. Also, even if there is not an analytical expression,

determining the Nyquist robust stability margin is a tractable problem.

Another type of structured uncertainty is real parametric uncertainty in the

process model. The robust stability problem under parametric uncertainty began to

receive renewed attention with the seminal result of Kharitonov (1979) on the stability of

interval polynomials, and is considered the most important development in the area after

the Routh-Hurwitz criterion. The theory makes it possible to determine if a linear time

invariant control system, containing several uncertain real parameters remains stable as

the parameters vary over a set (Bhattacharyya et al., 1995). Accordingly, the parametric

stability margin is defined as the length of the smallest perturbation in the parameters

which destabilizes the closed loop. The parametric stability margin is useful in controller

design as a means of comparing the performance of proposed controllers.

1.2. Objecive and Structure of Dissertation

The first goal of this dissertation is to revisit the MPDA principle to strengthen

the result when the maximum singular value is repeated. Chapter 2 introduces a revised

statement of the MPDA property that fully considers the case of a repeated maximum

singular value. An alternative proof is presented that is based on the theory of dual

norms and dual vectors which was the inspiration of the original result. The MPDA







results are also used to determine the upper bound on p given by the minimization over

a positive diagonal similarity scaling of the maximum singular value. When the

maximum singular value is distinct there exists an analytical expression for the gradient

of the objective function. The first order necessary condition for a minimum (i.e., the

gradient being indentically 0) is equivalent to MPDA; therefore the minimum is a tight

upper bound. Chapter 3 investigates this optimization problem when the maximum

singular value is repeated such that the gradient does not exist and the objective function

is non-differentiable. One result is a method for determining the subdifferential when the

maximum singular value is repeated where the subdiffemtial represents the set of all sub-

gradients. The necessary condition for a minimum is that zero is an element of the

subdifferential. Furthermore, it is shown that MPDA is still achievable when zero is on

the boundary of the subdifferential; otherwise, MPDA is not attainable and the upper

bound on p is conservative. Finally, Chapter 4 gives a necessary condition for the

optimal similarity scaling. The necessary condition requires that the vector of diagonal

elements of the similarity scaling be an element of the null space of a matrix formed from

the absolute values of the elements of the left and right eigenvectors of the matrix.

The second goal of this dissertation is the extension of the critical direction theory

to the more general case where the critical value-set is nonconvex. This work is

presented in Chapter 5. The key to extending the theory is the introduction of a

generalized definition of the critical perturbation radius in a fashion that preserves all

previous results. The nonconvexity of the critical value set is observed in a number of

interesting problems, including the case studied by Fu (1990) consisting of rational

systems where the uncertainty appears affinely in the form of real parameters that belong







to a known rectangular polytope. The generalized critical direction theory is applied to

this particular class of uncertain systems, and is used to calculate the required Nyquist

robust stability margin with high precision and in the context of a computationally

manageable framework. Finally, Chapter 6 proposes a practical design approach based

on parameter space methods (Siljak, 1989) to illustrate the utility of the Nyquist robust

stability margin as a measure of robust stability.

The final goal of this dissertation is to perform a complete robust stability analysis

of classical proportional-integral (PI) controller design techniques based on approximate

first-order-plus-time-delay models. The uncertain parameters for this problem are the

plant gain, plant time constant, and plant time delay. The region of all stabilizing

parameter perturbations is determined. By modeling the uncertainties as multiplicative

perturbations it is shown that the stability properties of the closed-loop system are only

dependent on the time-delay-to-time-constant controller tuning parameter. The results

include plots of the classical gain and phase margin and the parametric stability margin as

a function of the controller tuning parameter for the PI controller design methods

investigated.












CHAPTER 2
A DUALITY PROOF FOR THE
MAJOR PRINCIPAL DIRECTION ALIGNMENT PRINCIPLE

2.1. Introduction

The structured singular value, p(M), defined as the supremum of the spectral

radius of MU over diagonal unitary matrices U (Doyle, 1982), is a widely accepted tool

in the robust analysis of linear systems. It considers the problem of robust stability for a

known plant subject to a block-diagonal uncertainty structure under feedback. In general,

any block-diagram interconnection of systems and uncertainties can be rearranged into

the block-diagonal standard form. Calculating p is not trivial; in fact the underlying

optimization problem has been proven to be NP-hard (Braatz et al., 1994). The difficulty

is that the spectral radius is non-convex over the set of unitary matrix transformations.

One approach is to consider upper bounds for the spectral radius that can be calculated

easily, and ideally should be attainable to eliminate conservatism. The maximum

singular value is a reasonable choice for an upper bound because it is invariant under

unitary matrix transformations. In addition the maximum singular value upper bound can

be decreased by optimizing over similarity transformations, because the spectral radius is

invariant under such transformations. Ultimately, the problem becomes one of

conditioning a matrix through optimal similarity and unitary transformations to achieve

equality between the spectral radius and maximum singular value. Therefore,

determining the conditions under which the upper bound is attained is a significant issue

in the field of robust control.







In addressing the existence of solutions to the proposed optimization,

Kouvaritakis and Latchman introduce the major principal direction alignment (MPDA)

property (1985). The result states that the spectral radius of a matrix is equal to the

maximum singular value of the matrix if and only if the major input and the major output

principal direction of the matrix are aligned. MPDA is a strict condition for a matrix, but

can be used to determine the optimal positive diagonal matrix and unitary matrix that

result in equality between the aforementioned definition of p and the maximum singular

value upper bound. The proof of the MPDA principle is based on linear algebra

arguments, and considers separately the cases of a unique and a repeated maximum

singular value. For either case the proof of sufficiency is straightforward. The proof of

necessity for the case of a unique maximum singular value is precise but not as clear-cut.

On the other hand, the proof of necessity for the case of a repeated maximum singular

value is slightly ambiguous.

The inspiration for the MPDA principle is early work on determining when the

spectral radius equals the maximum singular value for positive matrices transformed by

non-negative diagonal matrices (Stoer and Witzgall, 1962). One motivation for the focus

on positive matrices is that they have good numerical properties (i.e., less round off error)

and therefore may be used for conditioning of matrices. In addition, positive matrices

remain positive under transformations by non-negative diagonal matrices leading to

connections to Perron-roots ;r(M) (positive eigenvalues of largest modulus) of positive

matrices M (Ortega, 1987). These results on positive matrices are based on the

mathematical concepts of dual norms and dual vectors utilized by Bauer (1962) which

lead to elegant proofs for many of the results.







It is the goal of this chapter to revisit the MPDA principle from the viewpoint of

duality theory. To facilitate the reading, the next section provides relevant mathematical

background including a discussion of the singular value decomposition, a summary of

dual-norm and dual-vector concepts, and a dual eigenvalue result. Section 2.3 introduces

the original MPDA theorem. Section 2.4 provides a modified statement of the MPDA

principle theorem that explicitly considers a repeated maximum singular value with a

proof based on the dual-norm and dual-vector theory. Several examples are given in

Section 2.5 and concluding remarks are made in Section 2.6.

2.2. Mathematical Background

2.2.1. The Singular Value Decomposition and Eigenvalue Decomposition

In this chapter only square matrices are being considered; therefore the definitions

are specialized for this type of matrices, but it is noted that the singular value

decomposition theory is applicable to generally rectangular matrices. The singular value

decomposition of an arbitrary matrix A e C"" is given by

A= X(A)Z(A)Y*(A) (2.1)

where E(A):= diag(a,(A),r2(A),..-,a,(A)) is the diagonal matrix of singular values

organized in descending order, and X(A) and Y(A) are unitary matrices. The singular

values of square matrix A e C"" are given by

ra(A):= A,(A'A), i=1,2,.--,n

where A,(A*A) represent the i-th eigenvalues of the matrix A'A and where the singular

values are ordered such that







The matrices X(A) and Y(A) are of the form

X(A)=[x,(A) x2(A) -.. x,(A)]

Y(A)=[y,(A) y2(A) -, y,(A)]

where the set of normalized left singular-vectors (input principal directions) {x,(A)} and

normalized right singular-vectors (output principal directions) {y,(A)} for i = 1,2, .-,n,

respectively constitute orthonormal eigenbasis of AA' and A*A. Furthermore, a pair of

singular vectors {x,(A), y(A)} is associated with each singular value a,(A) through the

relationship

Ay,(A) = o,(A)x,(A) (2.2)

The maximum singular value is denoted a(A). It must be noted that the

maximum singular value can be associated with a repeated singular value, i.e.

C(A) =a,(A)=o-2(A)=---. A maximum left/right singular vector pair (or major

output/input principal direction pair) {((A),y(A)} is any pair of left and right singular

vectors x,(A) and y,(A) that correspond to the maximum singular value and satisfy (2.2

). Necessarily, a major output principal direction and major input principal direction

respectively must be elements of the orthonormal eigensubspace of AA' and A'A

associated with the maximum singular value.

In this chapter the following definitions are used in relation to the eigenvalue

decomposition (Golub & Van Loan, 1983; Isaacson & Keller, 1966; Stewart, 1970). Let

A,(A) be an eigenvalue of A; then a right eigenvector v, of A is any non-zero vector

that satisfies


Avi = Avi







Furthermore, a left eigenvector wi of A is any non-zero vector that satisfies

w*A = AwJ

The reader is cautioned that some authors use the term left eigenvector for an eigenvector

of A'. Finally, an eigenvalue of maximum modulus is any eigenvalue A,(A) that

satisfies IAi(A) = p(A).

2.2.2. Dual Norms and Dual Vectors
In the theoretical development that follows, the mathematical concepts of dual

norms and dual vectors are utilized. These concepts are explained in a paper by Bauer

(1962) and are reviewed here to facilitate the theoretical development. Given a vector

norm |II its dual vector norm |II-lo is defined as

Rey'x
Iy := max Re y'x = max

For such dual norms the Holder inequality

IlyllJD |x Re yx

holds an is sharp, i.e., for any yo there exists at least one xo, and for any xo there exists

at least one yo such that the equality holds (Bauer, 1962). If such a pair (xo,yo) with

IIyollIllIoll = Reyoxo also satisfies the scaling condition

IIYO ollD 11oll

it is called a dual pair. Note that the dual vector of x is often written (x)D. A pair

(x,,yo) is strictly dual and is written yOll D if IyollDJ oO = =Yo =1. For strictly

homogenous norms (i.e., those satisfying Iax| = IaI I'xll for all complex scalars a) the

Holder inequality may be sharpened to (Bauer, 1962)







Ilylloll. >- ly'Xj

For a dual pair (Xo, y) under a homogenous norm it follows that

Rey;oxo = IIyollDllxoI > yoo which implies that Reyoxo = yoxo. Hence, for a strictly

homogenous norm every pair of dual vectors (xo,Yo) is also strictly dual pair. In

addition, there exist a strict dual yo for any xo 0 and a strict dual xo for any yo # 0.

Furthermore, the concept of approximately dual vectors is proposed such that a pair

(xoYo) is approximately dual if Ilyollolxoll = IYXo = .

In general, the dual norm of a p-norm Ix|1 := (Ix, j)')1 is the associated p-norm

Il-l where 1/ p+ / q = 1. So the infinity-norm and the 1-norm are duals, and the dual

norm of the 2 (Euclidean) norm is itself. For the 2-norm, a pair (xo,yo) is dual if

Yo = Xo / l||xo|, and approximately dual if yo = ejxo /I xo .

2.2.3. Dual Eigenvector Result
The basis of the following Lemma is a result of Bauer (1962) on the field of

values of a matrix.

Lemma 2.1. If the spectral radius of a matrix A e C""" is equal to the maximum

singular value of A, then for each normalized right eigenvector v, associated

with an eigenvalue of maximum modulus A,(A) there exists a normalized left

eigenvector w; = v, such that vi and w, form a dualpair wII vi.

Proof. Lemma 2.1 is a specialization of Bauer's result to the case of the

Euclidean norm, and is therefore in terms of the maximum singular value of the matrix.







The proof makes use of dual norm dual-vector theory presented earlier. Details are in

Appendix A. Q.E.D.

2.2.4. Eigenvector-Singular Vector Equivalence Result

The following Lemma is a consequence of Lemma 2.1.

Lemma 2.2. If the spectral radius of a matrix A e C"" is equal to the maximum

singular value of A, then each normalized right eigenvector v, of A associated

with an eigenvalue of maximum modulus A i(A) is also a right singular vector y,

of A associated with the maximum singular value rF(A).

Proof It suffices to prove that v, is a right eigenvector of A"A associated with

an eigenvalue whose square root is a(A), because by definition the rights singular

vectors y, are an orthonormal eigenbasis of A'A and the singular values are the square

roots of the eigenvalues of A*A. First, from Lemma 2.1, it follows that for each

normalized right eigenvector vi of A associated with an eigenvalue of maximum

modulus A,(A) there exists a normalized left eigenvector w, = vi of A. For each such

eigenvector vi

A'Av, = Av,A,(A)

= (vA)*i,(A)

= (w;A)'2,(A)

= (A,(A)wA)'A,(A)

= v '"(A)A2(A)


= I,(A)12 v







= A,(AA)v, (2.3)

Hence, from (2.3) if follows that v, is an eigenvector of A*A with eigenvalue 2i(A'A).

Finally, A/(AA) = FIl(A)I = V2(A) = (A) completing the proof. Q.E.D.

2.3. Statement of the Major Principal Direction Alignment Property

In solving various robust control problems it is necessary to determine the

conditions under which the spectral radius of a matrix attains its maximum singular-value

upper bound. The major principal direction alignment (MPDA) property addresses this

problem. Consider the singular value decomposition of a matrix A given by (2.1) where

(A) is the diagonal matrix of singular values organized in descending order, and X(A)

and Y'(A) are unitary matrices whose columns are the respective output and input

principal directions of A, arranged in an order conformal with the order of the singular

values. The major input principal direction y(A) and major output principal direction

Y(A) of a matrix A are defined as input and output principal directions respectively,

corresponding to the maximum singular value, r(A) of A. In addition, the major input

principal-direction y(A) and the major output principal-direction 1(A) are said to be

aligned if the exists a real scalar 0 R such that y(A)= e'el(A). The following

statement of the Major Principal Direction Alignment (MPDA) property is found in

Kouvaritakis and Latchman (1985).

Theorem 2.1. The spectral radius of any matrix A ECn"" is equal to the

maximum singular value of A if and only if the major input and output principal

directions of A are aligned.







Proof The proof consists of two cases, namely, when the maximum singular

value is distinct, and when it is repeated. The proof is taken directly from Kouvaritakis

and Latchman (1985) and is relegated to Appendix B. Q.E.D.

For the case of a distinct maximum singular value, Theorem 2.1 as stated is

entirely accurate and the proof rigorous. Unfortunately, when there is a repeated

maximum singular value, Theorem 2.1 as stated is not entirely accurate and the proof is

not rigorous. In the proof, Equation B.5 states that the variable z must assume a given

form (i.e., that z = Y(A)w must be at least one element of the form). This does not

mean that every major input and output principal direction pair results from

z= Y'(A)w; instead it should be interpreted as meaning that there is at least one such

pair that results from z= Y'(A)w. Hence, when the maximum singular value is

repeated, there may exist a major input and output principal direction pair that is not

aligned even when the spectral radius equals the maximum singular value.

Counterexamples are given in the examples section. A modified statement of MPDA

with a proof based on duality arguments is provided in the next section.

2.4. Modified Statement of the Major Principal Direction Alignment Principle

The following theorem is a modification of the MPDA Theorem 2.1 which

accurately takes into account the case of a repeated maximum singular value.

Theorem 2.2. The spectral radius of any matrix A e Cn" is equal to the

maximum singular value of A if and only if there exists a major input and major

output principal direction pair of A that is aligned.

Proof To prove sufficiency note that alignment of a major input and major

output principal direction pair of A implies







y(A) = ejI(A) (2.4)

Pre-multiplication of equation (2.4) by A gives

Ay(A) = e'AY(A) (2.5)

The singular value decomposition of A implies

Ay(A) = a(A)Y(A) (2.6)

Combining equation (2.5) and equation (2.6) gives

AY(A)= e-=je(A)Y(A)

so that A = e-'oa(A) emerges as an eigenvalue of A with eigenvector I(A). Noting

that the eigenvalues of A are always bounded from above by C(A), it follows that

IAI = p(A) = r(A)

To prove necessity, assume p(A) = _(A), then from Lemma 2.2 it follows that any right

eigenvector vi of A associated with an eigenvalue of maximum modulus Ai (A) is also

a right singular vector y, of A associated with the maximum singular value a(A).

From equation (2.2), the corresponding left singular vectors are

Ay, (A)
x, (A) =Ay,(A)
T(A )
c(A)

Av, (A)


SA,(A)v,(A)
jA, (A)I

=eJ'TKA',(A))yi(A)


= e9y,.(A)







Therefore, for each orthonormalized right eigenvector vi there is a major input/output

principal direction pair that is aligned. Namely

yi(A) = v,

and

x,(A) = ej'y,(A) (2.7)

where

0= arg(i.(A)) (2.8)

Finally, there is always at least one right eigenvector vi of A associated with an

eigenvalue of maximum modulus Ai(A); therefore, there must exist at least one major

input/output principal direction pair that is aligned, which completes the proof. Q.E.D.

Theorem 2.2 is a precise statement of the MPDA property. The theorem

eliminates any ambiguity that may result when applying the MPDA property as stated in

Theorem 2.1 to the case of repeated maximum singular values. In addition, the proof of

necessity makes well-designed use of the earlier work on dual vectors and dual norms,

and avoids the confusions associated with the earlier proof. This section is concluded

with a simple corollary that restates the MPDA property in the duality terminology,

namely

Corollary 2.1. The spectral radius of any matrix A EC"'" is equal to the

maximum singular value of A if and only if there exists a major input and major

output principal direction pair of A that is approximately dual with respect to the

Euclidean norm.

Proof. It suffices to show that approximate duality of a major input/output

principal direction pair with respect to the Euclidean norm is equivalent to alignment of







the pair. By definition, the pair are approximately dual with respect to the Euclidean

norm if and only if


I= ey/ly|2


(2.9)


Principal directions are always normalized; therefore (2.9) is equivalent to


which is exactly the condition for alignment completing the proof.


2.5. Examples


2.5.1. Example 2.1

Consider the matrix


-0.9026-1.0077i
A= 0.6086 +0.2053i
0.6487 + 0.2968i


02586-0.1506i
1.2588 -1.1670i
-0.5918- 0.4665i


0.1661+0.2372i
-0.6442 + 0.2239i
0.1641-1.4383i


with eigenvectors


-0.0687 + 0.1159i 0.1807+ 0.1816i 0.6834 0.0523i
I{v,, v,, v } -0.8719 + 0.2183i 0.3478 02425i -0.3628 0.1105i
0.3920 + 0.142 li 0.8670 + 0.0538i -0.4576- 0.4208i

and associated eigenvalues

{A 1,,A2,3 } = {2.0000e-.6000j ,1.2503e-1.6660 ,1.5996e-2.555i}

and singular value decomposition A = XYY", where


-0.0018 + 0.0876i
X = [x, x2 3]= 02741+0.8117i
0.2903- 0.4173i

0.1556
Y=[Yi Y2 Y3]= -0.2965+0.8731i
0.3367- 0.1101i


0.4121 + 0.3962i
-0.4542 + 0.0343i
-0.4806 0.4845i

-0.7481
-0.0272- 0.1299i
0.5404- 0.3615i


0.4426 + 0.6853i
0.2421- 0.0061i
02491 + 0.4624i

-0.6451
-0.0400 + 0.3612i (2.10)
-05455 + 0.3927i


Q.E.D.







and the singular values are

{o,,I 2,a'3} = {2,2,1}

The spectral radius equals the maximum singular value, i.e.

IA I = p(A) = =(A)= a =o 2

In this case the eigenvalue of maximum modulus is unique and non-repeated, and the

maximum singular value is repeated. An inspection of the left and right singular vectors

reveals that x, ejoy, and x2 # e'y,2 which appears to contradict the MPDA Theorem

2 which states that there must exist at least one major input/output principal direction pair

that is aligned. This apparent contradiction can be resolved by realizing (2.10) is only

one possible orthonormal eigenbasis of A*A whose vectors are right singular vectors.

Different orthonormal eigenbasis of A'A are achieved through unitary transformations

of the orthonormal bases of the eigenspaces of A'A associated with each particular

singular value. The eigenspace of A'A associated with a non-repeating singular values

is rank one; therefore an orthonormal basis consists of only one vector and the only

unitary transformation of this basis is of the form ei0. On the other hand, the eigenspace

of A'A associated with a repeating singular value has rank greater than one, and

therefore an orthonormal basis consists of more than one vector and a unitary

transformation of this basis is a unitary matrix whose size is the rank of the

corresponding eigenspace.

Hence, for this example, there must exist a unitary matrix that transforms the left

singular vectors x, and x2 into x, and x2 such that at least one of the transformed left







singular vectors is aligned with the corresponding right singular vectors y, and y,. The

problem becomes finding a matrix U such that

[x, x;]=[x, x2]U (2.11)

[y y'2]=[Yi y]U (2.12)

and

x' = ej;yi

with unitary constraint

U'U =I

The solution can be found by solving the system of equations that equates the moduli of

the elements of x, and y, and that constrains the arguments of elements of x' and y' to

differ by 0, where the unknowns are the elements of U and the variable 0. Although

this is a simple problem in complex algebra, the resulting set of equations have many

terms and are relatively cumbersome. Further theoretical work in this area is discussed in

Chapter 3. Therefore, an alternative method is used to solve the problem. First, from

Lemma 2.2 it follows that the right eigenvector v, is also right singular vector yi;

therefore, if U =[u, n2] then the first part of the problem becomes finding a

normalized u, such that

Y' = v, =[yi y2]u, (2.13)

The normalized least squares solution to (2.13) is

S[y y2]+v, [0.5548 + 0.8057i1
U [l y, y2 ]v1 0.2072 + 0.0126i







where [e]* denotes the Moore-Pinrose pseudo-inverse (Ortega, 1987). The second and

last part of the problem is to choose u2 such that U is unitary. One choice is

0.5548 + 0.8057i 0.2075
U =[' 2J 0.2072 + 0.0126i -0.6026 + 0.7705i

Now using the relationships (2.11) and (2.12) and defining X' =[x, x; x'] and

Y' = [y y2 Y3 yields an alternative singular value decomposition A= X'ZY'

where

0.0088 + 0.1345i -0.5540 + 0.0969i 0.4426 + 0.6853i
X =[x', x x3]= -0.5964 + 0.67251 0.3042 0.202 i 0.2421-0.006 li
0.4038 0.1041i 0.7232 0.1649i 0.2491 + 0.4624i

-0.0687+0.1159i 0.4831-0.5764i -0.6451
Y =[y; y; y,]= -0.8719+0.2183i 0.0549 + 0.2386i -0.0400 + 0.3612i
0.3920 + 0.1421i 0.0228 + 0.6114i -0.5455 + 0.3927i

and the singular values again are

{cr 1, 2 3} = {2,2,1}

Finally, the apparent contradiction of Theorem 2.2 is resolved by verifying

arg(A-) -o0.60 j y
x,; = e y, = e -0 y1

Note that x; ; e y; even though a2 is equal to the maximum singular value. A

reasonable question now is whether it is possible to choose u2 such that x, is also

aligned with y2 ? The answer is no. This is proved as follows. By construction y, is an

eigenvector of A corresponding to an eigenvalue of maximum modulus, namely v,.

Next, it can be shown (see, Theorem 2.1, proof of sufficiency) that alignment of major

input and major output principal directions implies that the major principal directions are







both necessarily eigenvectors of A. Now, assume that is possible to choose u2 such that

x2 and y; are aligned. Necessarily, y2 is also an eigenvector of A which implies that

there are two linearly independent eigenvectors, yl and y', associated with the

eigenvalue of maximum modulus. Hence, the eigenvalue's geometric multiplicity is

greater than one. Furthermore, for this case the eigenvalue's algebraic multiplicity is one,

but an eigenvalues geometric multiplicity can not exceed its algebraic multiplicity. This

is an obvious contradiction, therefore the assumption is false.

The result that it is not possible to achieve alignment of all the major input and

major output principal directions is not compatible with the original statement of the

MPDA property as given in Section 2.3. However, it is compatible with the revised

version of the MPDA property of Section 2.4, which allows for a major input/output

principal direction pair that is not aligned as long as at least one other input/output

principal direction pair that is aligned as is the case in this example.

2.5.2. Example 2.2

Consider the matrix

1.7907 0.8729i -0.0780 + 0.0482i 0.0085 + 0.151 li
A = 0.0827 0.0396i 1.6645 -1.1040i 0.0475 0.0001i
0.1225 + 0.0888i -0.0258 + 0.0399i 1.6883-1.0605i

with eigenvectors

0.8554- 0.0000i -0.0224- 0.4144i 0.0187 +0.1611i
{viv2,3 0.0145-0.2681i 0.0631+0.4205i -0.5177+0.7269i
0.4002 0.1901iJ 0.5880 + 05489i J 0.3707 0.1999i

and associated eigenvalues

{'l ,,,2 3} = {2.0000e "' ,2.0000e W O ,2.0000e .' }







and singular value decomposition A = XEY*, where

-0.0381 + 0.0008i 0.8954 0.4364i 0.0789 + 0.0124i1
X=[x, x2 3]= 0.9550-0.2457i 0.0414-0.0198i -0.0952-0.1281i
-0.1616 0.000i 0.0613 + 0.0444i -0.3854 0.9053i

0.0000 1.0000 0.0000
Y = [Y 2 Y3] = 0.9340+ 0.3268i 0.0000 -0.0243 0.1422i
-0.1138-0.0887i 0.0000 0.1537 0.9775i

and the singular values are

{O(l r,2, 3} = {2,2,2}

The spectral radius equals the maximum singular value, i.e.

AI I= 21 = l3 = p(A)= o(A)= a, = a2 = 3

where there are two eigenvalues of maximum modulus with one being non-repeated and

the other having a multiplicity of two. The maximum singular value is associated with a

repeated singular value of multiplicity three. Again, inspection of the left and right

singular vectors reveals that x, ejoy,, x,2 e'y,2, and x2 ejy2. From Theorem 2.2,

it is known that there is at least one major input/output principal direction pair that is

aligned, but it is not apparent if there are more than one. The possibility exists that all

three can be aligned through a unitary transformation, because there are three

independent eigenvectors associated with the eigenvalue of maximum modulus. The

unitary matrix that transforms all three singular vectors such that all input/output

principal directions pairs are aligned is not found by solving the resulting system of

complex algebraic equations, because the equations are even more cumbersome than

would for the previous example. In fact, in this example, the existing singular value

decomposition as not transformed at all. Instead, an alternative singular value







decomposition is constructed from the three eigenvectors associated with the eigenvalue

of maximum modulus. First, one right singular vector is obtained from the eigenvector

associated with the eigenvalue that is not repeated, i.e., y' = v, as dictated by Lemma 2.2

and the corresponding left singular is given by

x'; = ejazrg(')y = e-0.40ojy

according to (2.7) and (2.8) of Theorem 2.2. The remaining two right singular vectors

are obtained from v2 and v3, the eigenvectors associated with repeated eigenvalue of

maximum modulus. It can be shown that both of these eigenvectors are eigenvectors of

A'A, but that alone does not make them both right singular vectors, because singular

vectors are obtained from the orthonormal eigenbasis of A'A. It is easy to show that v,

is normal and orthogonal to v2 and v,. Therefore, the remaining step is to

orthonormalize v2 and v3. One such orthonormalization is

Y2 = V2

V3 V2V3 V2
V3 ;2V2 3 'V211

The corresponding left singular vectors are then given by

eiarg(A) -0.6000 '
X2 = eJ ( Y2 = e 2. O2

W(A,) y -.6000)
x3 = e J( y3 = e- 6jy3

The alternative singular value decomposition is A = X'EY'" where

0.7878 0.333ii -0.2525- 0.3294i 0.2144 + 0.2240i
X' =[x' x x;]= -0.0910-0.2525i 0.2895+0.3114i -0.1311 +0.8544i
0.2946-0.3310i 0.7952 + 0.1210i -0.0666 0.3902i








0.8554 0.0000i -0.0224- 0.4144i 0.0505 + 0.3059i'
Y'=[y, y'2 y;3]= 0.0145-0.2681i 0.0631+0.4205i -0.5907+0.6311 i
0.4002 0.190li 0.5880 + 0.5489i 0.1654 0.3597i

and the singular values are still

{oa,,a 2,a3} = {2,2,2}

By construction, the input/output principal direction pairs are aligned as follows

e -0.4000 j '
x, OO y,

= 6000jy,
x2=e 32

-0.6000 "j
x3 =- e 3

This result shows that it is possible for several pairs of input/output principal directions to

be aligned when there is singular value multiplicity. Note that the alignment factors,

however, are not necessarily identical.

2.6. Conclusions

This chapter clarifies the implications of the MPDA principle by explicitly

considering the case of a repeated maximum singular value. An alternative proof of the

necessity of the MPDA property is presented that is based on dual norm and vector

theory. This proof shows the ties the MPDA property has to the earlier duality work

which partly inspired it. Examples show that the alignment properties of the input/output

principal direction pairs associated with maximum singular value are directly related to

the eigenvectors associated with eigenvalues of maximum modulus in terms of both the

multiplicity and the amount of alignment.












CHAPTER 3
MAJOR PRINCIPAL DIRECTION ALIGNMENT WHEN
THE MAXIMUM SINGULAR VALUE IS REPEATED AND
ITS RELATIONSHIP TO OPTIMAL SIMILARITY SCALING

3.1. Introduction

The Major Principal Direction Alignment (MPDA) theory yields a necessary and

sufficient condition for the spectral radius of a matrix to equal its maximum singular

value (Kouvaritakis and Latchman, 1985). This has been proved using duality arguments

in Chapter 2 where it is shown that the results hold, even for the case of a repeated

maximum singular value. The primary reason for the development of the MPDA

principle is to solve the structured-singular-value / u problem, that is often written in the

form (Doyle, 1982)

sup p(MU) = p(M) < inf a(DMD-') (3.1)
Uel) DED

Where M eC"n', V.:= {diag(e'J,e'j' ,...,ej*)O< <, <2;,i =1,2,---,n} is the set of

diagonal unitary matrices and lD:= {diag(d,,d2,..-,d,)l d, > 0,i= 1,2,. -,n} is the set of

positive diagonal matrices. In equation (3.1), p represents the spectral radius, p the

structured singular value, and ~ the maximum singular value. The supremization over

V is known to be an NP-hard and non-convex optimization problem (Braatz, 1994);

therefore, when using standard optimization techniques there is always the problem of

local verses global optima. On the other hand, the infimization over D can be shown to

be a convex optimization problem (Safonov and Verma, 1985; Tzafestas, 1984;

Latchman, 1986) and the global minima can be determined via an appropriate







optimization technique. However, as (3.1) implies, in general this yields only an upper

bound on p .

The MPDA theory shows that if the maximum singular value is distinct for a

given D, then there is an analytic expressions for the gradient 8f(DMD-') / D. From

this expression for the gradient, the condition for a stationary point (i.e.,

a8(DMD-')/aD=0) implies that the moduli of the input and output principal

directions are elementwise equal. Therefore, if at the infimum the maximum singular

value is distinct, then the gradient exists and is identically zero, and the moduli of the

input and output principal directions are pairwise equal. In addition, a unitary

transformation matrix U (note the maximum singular value is invariant under unitary

transformations) can be determined that shifts the angles of the elements of the input or

output principal direction such that MPDA is achieved, and therefore the upper bound is

tight and the value of p is determined by solving a convex optimization problem.

In general the maximum singular value is not unique for a given scaling D. This

work investigates further the situations that arise when the maximum singular value is

repeated. There are two aspects of this problem that are investigated. The first aspect is

the effect the repeated maximum singular value has on the optimization over D, with

specific interest on gradient search methods. The second aspect is the attainability of

MPDA when the maximum singular value is repeated for the optimal scaling. Finally,

this work attempts to reconcile the results obtained with those of the principal direction

alignment (PDA) principle (Daniel et al., 1986).







3.2. Mathematical Background

3.2.1. The Singular Value Decomposition

The following definitions are associated with the singular value decomposition

(Ortega, 1987). In this chapter only square matrices are being considered, therefore the

definitions are specialized for the case of square matrices, but it is noted that the singular

value decomposition theory is applicable to rectangular matrices.

The singular value decomposition of an arbitrary matrix A eC".x is given by

A =X(A)E(A)Y'(A) (3.2)

where S(A):= diag(-a,(A),a2(A),-.-,-, (A)) is the diagonal matrix of singular values

places in descending order, and X(A) and Y(A) are unitary matrices. The singular

values of square matrix A eC"" are given by

0-/(A):= /A(A*A), i=1,2,...,n

where A (A A) represent the i-th eigenvalues of the matrix A*A and where the singular

values are ordered such that

cr-(A) >a2 (A) >.. -> -,(A)

The matrices X(A) and Y(A) are of the form

X(A)=[x,(A) x,(A) -. x,(A)]

Y(A)=[yt(A) y2(A) -.. y(A)]

where the set of normalized left singular-vectors (input principal directions) {x,(A)} and

the set of normalized right singular-vectors (output principal directions) {y,(A)} for

i = 1,2, **,n, respectively constitute orthonormal eigenbasis of AA* and A'A, such that

AA'x,(A) = (A)x,(A)








and

A'Ay,(A) = a (A)y,(A) (3.3)

Furthermore, a pair of singular vectors {xi(A),y,(A)} is associated with each singular

value ai(A) through the relationship

Ay,(A) = a,(A)x,(A) (3.4)

The maximum singular value is defined as F(A):= a,(A). The maximum

singular value can be associated with a repeated singular value, i.e. F(A)= ao(A)

= a2(A) =* *= ra,(A), where r < n is the multiplicity. A maximum left/right singular

vector pair (or major output/input principal direction pair) {Y(A),y(A)} is any pair of

left and right singular vectors that corresponding to the maximum singular value and

satisfy (3.4). Necessarily, a major output principal direction and major input principal

direction respectively must respectively be normalized elements of the eigensubspaces of

AA* and A'A associated with the maximum singular value. If {x (A)} for i = 1,2,- -,r

and {y,(A)} for i = 1,2,- -,r are orthonormal bases for these eigensubspaces that satisfy

(3.4), then any and all major output principal directions and major input principal

directions are respectively given by

r
Y(A) = [x,(A) x2(A) x,(A)]u= x, (A)u, (3.5)
i=1

and


y(A)=[y,(A) y2(A) ... y,(A)]u = y(A)u, (3.6)
i=e

where u E C satisfies


u u=1







that is, u must be on the unit ball in C".

3.2.2. Statement of the Major Principal Direction Alignment Principle

The following theorem is a modification of the MPDA principle as proposed in

Kouvaritakis and Latchman (1985) which takes into account the case of a repeated

maximum singular value.

Theorem 3.1. The spectral radius of a matrix A e C""" is equal to the maximum

singular value of A if and only if there exists a major input and major output

principal direction pair of A that is aligned, i.e. there exists a pair {Y(A),y(A)}

such that

y(A) = ej'x(A) (3.7)

for some 0 e[0,2;r).

Proof The proof is given in Kouvaritakis and Latchman (1985), and Chapter 2

offers an alternative proof based on the theory of dual vectors and dual norms. Q.E.D.

Given the optimal matrices UO and Do, Theorem 3.1 gives a necessary and

sufficient condition for the left hand side (spectral radius) and right hand side (maximum

singular value) of (3.1) to hold with equality. It is apparent that equation (3.7) requires

that the major input and major output principal directions have element-by-element equal

moduli and a constant element-by-element phase difference.

3.2.3. Affine Sets, Convex Sets, and Convex Functions

If x and y are different point in R", the set of points of the form

(1-A)x+Ay=x+A(y-x), A eR

is called the line through x and y. A subset M of R" is called an affine set if

(1- )x + Ay M Vx M,yeM,AeR







In general, an affine set has to contain, along with any two different points, the entire line

through those points. The intuitive picture is that of an endless uncurved structure, like a

line or a plane in space. The subspaces of R" are the affine sets which contain the

origin. The dimension of a non-empty affine set is defined as the dimension of the

subspace parallel to it (the dimension of the empty set is -1 by convention). Affine sets

of dimension 0, 1 and 2 are called points, lines, and planes, respectively. An (n-l)-

dimensional affine set in R" is called a hyperplane. Hyperplanes and other affine sets

may be represented by linear functions and linear equations. For example, given j3 e R

and a non-zero b e R", the set

H = {xxTb = } (3.8)

is a hyperplane in R". Moreover, every hyperplane may be represented in this way, with

p and b unique up to a common non-zero multiple. For any non-zero b e R and any

P e R, the sets

{xIx'b < fl, {xlxTb f}

are called closed half-spaces. The sets

{xlxTb /}

are called open half-spaces. These half-spaces depend only on the hyperplane H given

by (3.8). Therefore, one may speak unambiguously of the open and closed half-spaces

corresponding to a given hyperplane. Finally, the intersection of an arbitrary collection

of affine sets is again affine. Therefore, given any S c R" there exists a unique smallest

affine set containing S. This set is called the affine hull of S and is denoted affS.

A subset C of R" is said to be convex if







(l-,R)x+ Ay C Vx EC,y C, 1e(0,1)

All affine sets are convex, as are half-spaces. A vector sum

A,x, + A2x 2 -+A m~
is called a convex combination of x,x ,,x-. ,x, if the coefficients A, are all non-negative

and A, +l2 +.-.+Am =1. A subset of R" is convex if and only if it contains all the

convex combinations of its elements. The intersection of all the convex sets containing a

given subset S of R" is called the convex hull of S and is denoted convS. Necessarily,

convS is the smallest convex set containing S. In addition, for any S c R", convS

consists of all the convex combinations of the elements of S. In general, by the

dimension of a convex set C one means the dimension of the affine hull of C.

A supporting half-space to a convex set C is a closed half-space which contains

C and has a point of C in its boundary. A supporting hyperplane to C, is a hyperplane

which is the boundary of a supporting half-space to C. As such, a supporting hyperplane

to C is associated with a linear function which achieves its maximum on C. The

supporting hyperplanes passing through a given point a e C correspond to vectors b

normal to C at a, as defined by (3.8).

Let a function f(d) be de defined on a convex set S c R" (note, for the MPDA

problem f(d):= ao(DMD-') where D = diag(d) and S is the positive orthant such that

D e D). In what follows, it is assume that the function f(d) is finite on its domain of

definition. The function f(d) is said to be convex on S if

f(ad, +(1- a)d2) af(d,)+(- a)f(d,) Vdl,d2 e S, a E[0,1]







A concave function on S is a function whose negative is convex. An affine function on

S is a function which is convex and concave. The set {(d,f(d)) e R"' d eS} is the

graph of the of the function f(d) defined on the set S. The set

epif:= {(d,f) e R+ ld eS,p eR,p/ f(d)}

is called the epigraph of the function f(d) defined on the set S. The epigraph of a

convex function is a convex set.

3.2.4. Differential Theory

A vector 4 is said to be a subgradient of f:S c R" -> R at d e S if

f(g) 2 f(d)+4T(g-d), VgeS (3.9)

This condition, which is referred to as the subgradient inequality, has a simple geometric

meaning: it says that the graph of the affine function h(g) = f(d) + T(g- d) is a non-

vertical supporting hyperplane to the convex set epif at the point (d,f(d)). The set of

all subgradients of f at d is called the subdifferential of f at d and is denoted by

af(d). The multivalued point-to-set mapping af:d -> Of(d) is called the subdifferential

of f. Obviously, af(d) is a closed convex set, since by definition 4 e Of(d) if and only

if 4 satisfies a certain infinite system of weak linear inequalities (one for each g of

(3.9)). In addition, 8f(d) is also nonempty and bounded.

The directional derivative of f at d e S in the direction of g, denoted f'(d;g),

is defined by the limit

f'(d;g) = lir f(d + Ag) f(d)
f'(d;g)= him
J v w ^ ^







if it exists. Notably, for convex functions the directional derivative f'(d; g) exists for all

d e S and for all g e R". Dem'yanov and Vasil'ev (1985) show that the relation

f'(d;g)= max Vg (3.10)
SEff (d)

holds.

The function f is differential at de S if and only if there exists a vector Vf(d)

(necessarily unique), called the gradient, for which

f(g) = f(d) + VfT (d)(g d) + O(|g dl)

or, equivalently,

Sf(g) f(d) VfT (d)(g d) 0
g-d |g d|

If f is a convex function then f is differential at d e S if and only if the partial

derivatives exists. In addition, the gradient is given by

T
Vf(d) (d) Of(d) af(d)d
Sd, 8d, Md,

and f has only one subgradient, namely the gradient Vf(d), such that

8f(d) = {Vf(d)} (3.11)

Also,

f(g) > f(d)+ VfT(d)(g-d), Vg eS

That is, Vf(d) is the normal of the tangent supporting hyperplane of epif at d.

With the terminology of differential theory thus developed, several important

theorems are given that are used in the sequel. The first theorem describes the set of

points where f is differentiable. This theorem is used as the basis of the primary







assumption made in the results section, namely, that the function being considered (i.e.,

f(d):= o'(DMD-')) is nondifferentiable only at points in its domain. The second

theorem gives a characterization of the subdifferential that is used to construct an

expression for Of(d) when the function is nondifferentiable.

Theorem 3.2. Let f be a convex function defined on a convex set S c R", and

let D be the set of points where f is differentiable. Then D is a dense subset

S, and its complement in S, given by D, is a set of measure zero. Furthermore,

the gradient mapping Vf: d -> Vf(d) is continuous on D.

Proof Two different proofs are given in Dem'yanov and Vasil'ev (1985) and

Rockafeller (1972). Both proofs are based on measure theory, and show that there are

countable number of sets where f is not differentiable. Q.E.D.

Theorem 3.2 essentially states that f is differentiable almost everywhere in S.

Theorem 3.3. Let f be a convex function defined on a convex set S c R", and

let D be the set ofpoints where f is differentiable. Then

af(d)= conv{z eR"z= lim Vf(dk), d -> d, d, ED}

Proof Again, two different proofs are given in Dem'yanov and Vasil'ev (1985)

and Rockafeller (1972). Both proofs use the continuity of the gradient on D given in

Theorem 3.2 to show that the limit sequences exist and that they converge to exposed

points of 8f(d). Therefore the Of(d) is the convex hull of all such limit sequences. Q.E.D.

As presented Theorem 3.3 seems of little practical value, because to construct

af(d) from it requires the construction of an infinite number of limit sequences. This

not the case as is shown in the results section. The next to theorems deal with solving the







optimization problem of infimizing f(d). One gives conditions for an infimum, the

other gives an expression for the steepest desent direction.

Theorem 3.4. For the convex function f(d) to obtain its optimum value on S at

the point do, it is necessary and sufficient that

0 a f(do)

Proof. A detailed proof is given in Dem'yanov and Vasil'ev (1985). Basically,

the condition is sufficient, because epif is entirely above a horizontal supporting

hyperplane at do. The condition is necessary, because if 0 af(do) then it is possible to

find a direction that would decrease f(do) such that f(do) is not optimum. Q.E.D.

The steepest decreasing direction is given in the following theorem.

Theorem 3.5. If 0 o af(d), then the subgradient given by

Sd(d) = arg min (3.12)
Wef(d)

points in the opposite direction of the steepest descent direction. That is,

sd (d)
g(d) =-
I isd(d)II

is the steepest descent direction of f at d.

Proof. A detailed proof is given in Dem'yanov and Vasil'ev (1985). The proof is

based on finding the direction that gives the smallest directional derivative as given by

(3.10). Q.E.D.

It is obvious that Theorem 3.4 and Theorem 3.5 are of great utility for any

steepest descent nondifferentiable optimization algorithm.







3.2.5. Expression for the gradient when the maximum singular value is distinct.

The mathematical background will now focus on the problem at hand, namely

performing the infimization

inf c(DMD-') (3.13)
DED

The objective function f(d) = &(DMD-') (where D = diag(d) and the domain of the

objective function is the positive orthant such that D eD) is convex as was already

stated. Latchman (1986) has stated that when the maximum singular value is distinct, the

gradient exists and is given by a relatively simple expression. The following is the

derivation of this expression. After defining M:= DMD-' to simplify the notation, the

singular value decomposition and equation (3.3) give

S(DMD-1) = 32 (M) = *(M)M*y(M) (3.14)

If it exists, the partial derivative of (3.14) with respect to the diagonal element d, of

matrix D is given by

a2(M) a_ '(M)M' (M))
ad, ad,

which by the chain rule becomes

a2 (M) yO(M) M. (M) + ."(M).M By(M) + () (M'M)
M- 0-(=- M 1My0(M) + (MM () 1+Y (M) My(M)
od, adi a9d, di

Using (3.3) gives

d(M) -2 ( ) (M) )
8d, = ad, adi Jd,


which simplifies to








adi adi adi
2 1 -. a(M'M)
= (M)- + y (M) y(M)
Sd, Od,;

y (M) d (M)
ad,.

Expanding the partial derivative term now gives



a~'(M) a(D-'M*D2MD-')
adi (M) a(M)

[ aD-' .D2 -D2MD
= y (M) M'D2MMD-' + D-'M a MD-' + D-M'D2 y(M)


Y= (M1) -2 EiM'D2MD- + 2dD-'M*EiMD' D-'M'D2M E (M)
d d


where E, is a diagonal n x n matrix with a 1 in the (i,i) position, and zeros everywhere

else. Since E, = d,E,D-' = DE,D /d = dD-'E,, the above equation becomes



3M2 ( .()[-EiD-'M'D I MD-' + 2D-1MDEiDMD-' D-'M'D2MID-'E](M)
9d, d,

= ((M)[M'EMl -2(M)Ej]y(M)



Using equation (3.4) this becomes

aa2 (M) 22 (i) [ (M)E(M)-()EY )
di d
22 (M) (M
= d (1 ) (M


Now,







aQU2 aa(M)
8di di

such that an expression for the partial derivative of c(DMD-') with respect to the

diagonal element d, of matrix D is given by

8a(DMD-') (DMD') [DMD -(DMD (3.15)
adi di

When the maximum singular value C(DMD~') is distinct, then the major output

principal direction i(DMD-') and the major input principal direction y(DMD-') are

determined by a scaling factor ejo of the left singular vector x,(DMD-') and right

singular vector y, (DMD-'), respectively. Therefore, I, (DMD' ) and Iyi (DMD ')1 are

unique and the partial derivatives (3.15) exists for i = 1,2,..-,n such that the gradient is

given by

Vf(d) = Vr(DMD-') = (DMD1)D-'1Y(DMD-1 )2 (DMD-')1] (3.16)

where the absolute value I|* is considered an element-by-element operator when applied

to a vector. As the preceding development has verified, when the maximum singular

value is distinct the gradient of the objective function f(d) = C(DMD-') exists and is

given by (3.16). In addition, the subdifferential is given by (3.11). When the maximum

singular value is repeated the gradient no longer exists, but it is possible to determine the

subdifferential and therefore a steepest descent direction. This is the main theoretical

result of this paper and is given in the next section.







3.3. Main Result Characterization of the Subdifferential when
the Maximum Singular Value is Repeated

3.3.1. General Expression for the Subdifferential

When the maximum singular value is repeated the major output principal

direction x(DMD-') and the major input principal direction y(DMD-') are determined

by (3.5) and (3.6). As such, the expression for the gradient given by (3.16) may not be

uniquely determined which implies the objective function may not be differentiable.

When the function is non-differentiable then the subdifferential must be determined, as

opposed to the gradient. To characterize the subdifferential define the function

2 r 2"
Vf(d;u) = r(DMD-')D' x, (DMD-' )u, Zy,(DMD-')ui (3.17)
Ii=1 i=1

where u eC' satisfies u'u = and {x,(DMD-')) for i = 1,2, ,r is an orthonormal set

of left singular vectors and {y,(DMD-')} for i = 1,2,.*-,r is an orthonormal set of right

singular vectors corresponding to the maximum singular value a(DMD-') of

multiplicity r. Definition (3.17) represents the evaluation of the gradient function (3.16)

for possible values of i(DMD-') and y(DMD-'). For different u, the function

Vf (d; u) may give different values, such that the gradient is not unique and is therefore

undefined. The subdifferential is now characterized in the following theorem.

Theorem 3.6. The subdifferential of the function f(d) = T(DMD-') is given by

f (d) = conv{Vf,(d;u)ju'u = 1) (3.18)

where Vf, (d; u) is defined by (3.17).

Proof From Theorem 3.3, the subdifferential is given by







f(d) = conv{z e R" = lim Vf(d,), dk -+ d, d e D}
I k->00

where D is the set of points where f(d) = a(DMD-') is differentiable (i.e., the

maximum singular value is distinct). The gradients Vf(dk) are given by (3.16), and are

determined from the from the sequence of major output principal directions

xk(DkMD;') and major input principal directions yk(DkMD;'), which are uniquely

determined up a multiple of ejo. From the perturbation theory of matrices (Lancaster

and Tismenetsky, 1985), analytic perturbations on normal matrices (i.e.,

(DMD-' )(DMD-')) have continuous eigenvalues and eigenvectors in a neighborhood

of the perturbation. Now, the right singular vectors {y,(DMD-')} are an orthonormal

eigenbasis of (DMD- )'(DMD-'). Therefore, the right singular vectors are continuous

in D. This implies that for points where the maximum singular is non-differential each

major input principal direction y(DMD-') (all of which are given by (3.6)) is the limit of

a sequence of major input principal directions y,(DMD~') that correspond to a

maximum singular value that is differentiable. In addition the converse is true; that a

sequence of major input principal directions yk(DAMD~') that corresponds to a

sequence of maximum singular values k, (DMD ') that are differential converges to

major input principal direction y(DMD-') given by (3.6) if the sequence a, (DAMDi')

converge to the maximum singular value a(DMD-'). Similar arguments can be made

for the left singular vectors/major output principal directions. Therefore, for all dk D

as d, -d then k (DkMDA)->. (DMD-) and yk(DkMD') -+ (DMD-') which

are given by (3.5) and (3.6) with u'u = 1. In addition, for every point where the







maximum singular value is non-differential i(DMD-1) and y(DMD-') are given by

(3.5) and (3.6) and there exists a sufficiently small perturbation of d such that there

exists sequences Yk (D MD;')-> Y(DMD-') and Y, (DAMD ') y(DMD-') which

are uniquely determined up a multiple of eo' such that the gradients Vf(dk) exist. All

that is left is to define Vf,(d;u) by (3.17), which represents the limit of Vf(dk) as

dk d for some dk e D, where all u such that u*u = 1 represents all possible limit

sequences d, -> d for d, e D. Q.E.D.

Theorem 3.6 is the natural extension of the gradient result given in Section 3.2.5.

For the case when the maximum singular value is distinct, Vf.(d;u) = Vf(d) for all

u*u = 1 (i.e., u = u = eje) such that af(d) = conv{Vf(d)} = {Vf(d)} as expected. On

the other hand, when the maximum singular value is repeated Vf(d) does not exists.

Instead one has Vf,(d;u), which is an extension of equation (3.16) for Vf(d), in that

Vf (d;u) represents the vector obtain when equation (3.16) is evaluated at one of the

possible major output and input principal directions given by (3.5) and (3.6). Obviously,

Vf, (d; u) is a subgradent, since it is an element of 8f(d). In fact, Vf, (d; u) represents a

subgradent that is arbitrarily close to some Vf(dk) where d4 -> d. That is Vf (d;u)

for u'u = 1 represent the boundary of 8f(d). Note, that a repeated maximum singular

value does not necessary guarantee a non-differentiability. Consider the matrix M = I

where f(d) = a(DMD-') = a(DID-') = a(I) = 1 with multiplicity n independent of d.

The function Vf,(d;u) = 0 for all d and u, such that f (d) = {0} = {Vf(d)) where the

gradient exists and is identically zero. This is an extreme case where the set given by







(3.18) degenerates to a point. This and other degenerate cases should be taken into

consideration when using Theorem 3.6.

3.3.2. Characterization of the Subdifferential as an Ellipsoid.
When the maximum singular value is distinct, the subdifferential is given by the

point 8f(d) = Vf,(d;u) = Vf(d), where Vf(d) is given in Section 3.2.5. The next step

is to explore the case when the maximum singular value is repeated once. In this section,

it is shown that af(d) is an ellipsoid when the maximum singular value is repeated once

by examining the properties of the function Vf,(d;u). To simplify the notation define

the vector valued function g:C2 -* R" as g(u):= Vf(d;u) where d is a fixed point

such that maximum singular value f(d) = -(DMD-') has multiplicity 2. From

Theorem 3.6, this gives

f (d) = conv{g(u)lu'u = l,u e C2} (3.19)

To analyze (3.19), g(u) is expressed in terms of u = [Iulejz"' u2 lueiL ]T as

Ju, 12
g(u) = H cos(Lu, z2),u21 (3.20)
sin(Zu, Lu2 )uI, lu2 I
1212

where the elements of the constant matrix H are given by

h,. = (DMD-) (I I) (3.21)
di


h.2 = 2 (DM ) (IX, (,,2 I cos(LZ,, Lx,-)-y,1 y12 Icos(y,, Lyi2)) (3.22)

di -2 d2 (3.23)
,,.3 = -2 a(i -') (^Ill^,Isi -( ,-ZX,,,)-|y,|ll,|lsi'n(Z^L,-ZY,)) (3.23)
di







and

hi4 (= (D ) (Ixi2 -Y2 12) (3.24)


for i = 1,2,-**,n with


x21 X22
x,= X2= (3.25)

Xnl Xn2

an orthonormal set of left singular vectors and

Y1 Y12
Y21 Y22
Yl = Y2 = (3.26)

.Yn. _Yn2

an orthonormal set of right singular vectors corresponding to the maximum singular value

c(DMD-') of multiplicity 2. Equation (3.20) is obtained from equation (3.17) when the

maximum singular value has multiplicity 2, i.e.

g(u) = Vf,(d;u)= W(DMD-')D'l[Ixu +x2u212 y,u, +yu2 2]

Using (3.25) for {x,,x2} and (3.26) for {y1,y2}, and considering one element of g(u)

the law of cosines gives

S(DMD-') 1, 2,22
g, (u) = -( d [IxFiMu + 21x|iu, 11x12u2Icos(Z(xiu)- Z(x,2u2)) + jxu2 12

dUsing tri +gonom c ad cos(Z(yu,)x n-(yc,,x )) + y,,2s

Using trigonometric and complex number identities gives





44




a(DMD-') 2
g,(u) = (xl,, ly,,


d,
2 (IX(DM") (lx1Xi2 ICOS(LXr, Zr,.)- Yi ljYi2 I Cos(Ly, Ly,2 ))0os(LZ. Lu .)1 l I -

a(DMD-')
2 ((D I xIx,,2 I sin(x Z,, )-IY l I sin(Zy,, Zy,2)) sin(Zu, Lu )Iu, I2 I +

-(DMD-' (I2 12 I2 12 )1,
d,


which is of the form (3.20) where the elements of H are defined by (3.21)-(3.24)

respectively.

There are now three cases to consider. The first case is when n = 2. This is a

trivial case, in that the optimal similarity scaling is given by the Perron scaling.

Therefore, there is no need to further investigate the properties of the subdifferential

when n = 2. The other two cases are when n = 3 and when n > 3. As will be discussed

shortly, the case when n = 3 is a degenerate case of the more general case n > 3.

Therefore the case when n > 3 will be discussed next followed by the case when n = 3.

The first result is that the subdifferential given by (3.19) is contained within an affine set

of dimension 3. The result is stated in the following theorem.

Theorem 3.7. For n > 3 and d such that the maximum singular value

f(d) = i(DMD-') has multiplicity 2, the subdifferential, Of(d), is contained in

the affine set S = {z e R" Pz = q} where elements of the matrix

P1.1 P1,2 P1,3 1 0 *.. 0
SP2.1 P2.2 P2,3 0 1 ..- 0 (n()

Pn-3., Pn-3,2 P.-3,3 0 0 ... 1







and the vector q = [q q2 *. q_-3]T satisfy

Pi,I hil h2, h3, -1 -hi+3,
Pi,2 h,2 h22 h3,2 0 -hi+3,2
= (3.27)
Pi,3 h,3 h2,3 h33 0 -hi3,3
.q _h],4 h2,4 3,4 -1 .-hi+3,4

for i = 1,2,---,n 3, and where h,j are given by (3.21)-(3.24). Additionally, the

dimension of S is 3.

Proof. It is sufficient to show that every g(u) given by (3.20) with u'u = 1 is an

element of the affine set S = {z e R Pz = q}, because if a set (i.e., {g(u)lu'u = 1) ) is

contained within an affine set (i.e., S) then convex hull of the set (i.e.,

8f(d) = conv{g(u)lu'u = 1) is also contained within the affine set. Therefore, it must be

shown that for all u'u = 1, g(u) satisfies each of the n 3 linear equations that defines

the affine set. The first linear equation is

[Pi,, P,2 P,3 1 0 ... 0]g(u)=q=

which must hold for all u'u = 1. This becomes

Pl,.lg (u)+ ,2g(+ ,39 + Pg4 () = q1

which from equation (3.20) is equivalent to

p., (hA,, u 12 + h,,2 COS(Lu LZu )Iu, 1u2 I+ h,,3 sin(Zu, Lu2 )lu, u2 I + h,,4 u') +
P1,2 (h2.1 I, 2 + h2,2 cos(Lu, LZ2 )Ii IIu I + h23 sin(Z, Lu), IIu 2 I + h2 u,, I2) +
P,3 (h3,1 u, 12 + h,,2 cos(u, Zu2 )u, II2 I + h3. sin(Lu, Zu,)I, C2 I + h3.4 1u2 '2) +
(h4,, Iu, 12 + h4,2 Cos(Z Z2 )I~, IIU I + h4,. sin(Lu, ZL )u IIJ2 I + h.,4 I|2 2) = q,







(pl.h,, + pl.2h2, + I,3h3,1 + h4,. )u12 +
(plh1,2 + p,2h,2 + + ,3h3 4,2) COS(LZ LZ2 )Il, IU2 I +
(3.28)
(P,,h,,3 + PI.2h2,3 + 3h,3 + h4 3)sin(Lu, Lu )lu, 1U2 I +
(pl.h,,4 + P12h2,4 + P,3h34 + h4,4 )1,12 =q

Now, equation (3.27) gives

hI, h2,1 h3.i -1 Pil -hi+
h,2 h2,2 h3,2 0 Pi,2 -hi+,2
h,3 h2,3 h3,3 0 i,3 -hi+3,3
hI,4 h2,4 h3,4 -1L q -hi+3,4

such that (3.28) becomes

(q, h4, + h4,, )l 12 +
(-h4,2 + 4,2) cos(u, u,2 )lu, ll, +
(-h4,3 + h,3) sin(Zu, u,2 )lu, 1u2 I2 +
(q, h4,4 + h,4)u, 2 = q,

or

Iu 12 +u212 =1

which holds for all u'u = 1. Hence, for all uuu =1, g(u) satisfies the first linear

equation that defines S. In fact, the preceding arguments hold for all n-3 linear

equations that define S. Therefore, g(u) is contained within the affine set S for all

u'u = 1 implying that the subdifferential is contained within S. Finally, the n -3

linearly independent rows of P are a basis for the orthogonal complement of the

subspace parallel to S such that the dimension of the affine set S is 3. Q.E.D.

Theorem 3.7 implies that the last n-3 terms of g(u) can be expressed as an

affine functions of the first 3 terms of g(u) such that the subdifferential 8f(d) is a 3-







dimensional convex set in an n -dimensional space. This means that the first 3 terms of

g(u) (i.e., {g,(u),g2(u),g3(u)}) describe af(d). Therefore, to complete the

characterization of af(d) it is only necessary to investigate conv{g,(u),g (u),g,(u)}

for u'u = 1, and then translate this 3-dimensional set to the R" using the affine functions

given in Theorem 3.7.

The convex hull, conv{g,(ug),g2(u),g3(u)u'u = 1}, is now shown to be a 3-

dimensional ellipsoid, and thus Qf(d) is a 3-dimensional ellipsoid. Consider equation

(3.20), even though u = [lu lei"' u2 je"'2 ]T has 4 parameters (i.e., u, I Lu 1 ,and

Zu2 ), the function g(u) with u*u = 1 is a function of only 2 parameters. One of the

parameters is x, = u, and the other is 0, = Zu, Zu2. The reason Iu2 is not a third

parameter is that u'u = 1 necessarily requires u2 = 1lu, i Now consider a fixed

value of x,, the terms {g (u),g2 (u),g(u)} are of the form

g, (u) = e,, + e,,, cos(9,) + e,3 sin(O,)

g2 (u) = e2, + e2,2 cos( ) + e23 sin(, )

g3 () = e3, + e,2 cos(O,) + e3,3 sin(0,)

which is obviously a parametric representation of a 2-dimensional ellipse in a 3-

dimensional space centered at [e,1 e2, e3,]T. To satisfy u'u = 1, lu, must be an

element of [0,1]. Therefore, varying x, over its admissible range of 0 to 1 generates a

set of ellipses which form the surface of an ellipsoid. This ellipsoid is given by

E = {z eR 3(z- c) B(z- c) = 1} (3.29)

The center c of the E is given by








c] hA,2 + hI,4
c = c2 0.5 h 22 + h2,4 (3.30)
c3 h3,2+ h3,4

and the matrix B which characterizes the length of the axes of E and its orientation has

the form

bi b1,2 bl3
B= b,2 2 b23 (3.31)
b1,3 b2,3 b3,3

where the 6 parameters {b6 b2,2 ,b33 ,b2 ,b,3 ,b2, 3 } can also be expressed in terms of the

constants hi, 's. These expression can be obtained by picking six different values of u

with u'u = 1, setting z = [g (u) g2 (u) g(u)]T and then solving the resulting system

of six linear equations in terms of {b, ,b2,2,b,3 ,bl,2,b,,3,b2, 3 obtained from (3.29).

Unfortunately these expressions are vary cumbersome, and therefore in practice it is

easier to just solve the system of six linear equations resulting from the numerical data of

the particular problem.

The following theorem combines Theorem 3.7 and the above result that

{g,(u),g2(u),g3(u)} with uu = 1 is an ellipsoid to give a useful characterization of

af(d).

Theorem 3.8. For n > 3 and d such that the maximum singular value

f(d) = i(DMD-') has multiplicity 2 the subdifferential af(d) is given by

af(d)= {zR"IPz=q, ([z, z2 Z3] -T)B([z, Z2 z3 T ) 1)

where constants P and q are given by (3.27) and c and B by (3.30) and (3.31).







Proof. The subdifferential af(d) is just the convex hull of the 3-dimensional

ellipsoid E given by (3.29) translated to R" by making it an element of the affine set

given by Theorem 3.7. The convex hull of E is the union of itself and its interior which

is given by convE = {z e RI (z- c)TB(z-c) 1}. Q.E.D.

To complete the ellipsoidal characterization of af(d) when the maximum

singular value is repeated twice the case of n = 3 is now discussed. When n = 3, g (u)

is an affine function of g, (u) and g2 (u), such that aff9f(d) becomes a 2-dimensional

plane in 3-dimensional space. The effect is that the 3-dimensional ellipsoid E is

degenerate in that it has an axis of length zero, because it is required to be a subset of a 2-

dimensional plane. Consequently, convE = E such that af(d) becomes a 2-dimensional

ellipse including its interior in a 3-dimensional space. Also, 8f(d) has no relative

interior (i.e., there are no elements of Sf(d) that are not also on the boundary of af(d)).

Finally, note that degenerate cases are possible. Consider, the matrix

M = diag([1 1 0 O]T) such that the maximum singular value is repeated twice. The

above analysis gives H = 0 such that equation (3.27) is not meaningful. For this case

8f(d) is no longer contained within a 3-dimensional affine set, but is actually

af(d) = {0} which is a special ellipsoid whose axis are all length zero.

Theorem 3.8 and the preceding paragraph concerning the case of n = 3 give the

desired ellipsoidal characterization of af(d) when the maximum singular value is

repeated once. The next logical set is to extend the results of this section to the case

when the maximum singular value is repeated more than once. Unfortunately, the

preceding ellipsoidal characterization no longer holds and the only characterization of







af(d) is that given by Theorem 3.6. As is shown in the next section, this still has some

utility in determining a steepest descent direction.

3.4. Determining the Steepest Descent Direction and Conditions for a Minimum

When the maximum singular value is distinct the gradient exist and the steepest

descent direction is given by -Vf(d) / Vf(d)I. Furthermore, the necessary and

sufficient condition for a minimum is Vf(d) = 0. When the maximum singular value is

repeated the results of the previous section and Theorem 3.4 and Theorem 3.5 can be

used in a steepest descent optimization algorithm. First the case when the when the

maximum singular is repeated once is considered, because the ellipsoidal characterization

of af(d) results in a convex optimization problem for determining the steepest descent

direction. This is followed by the more general case when the maximum singular value is

repeated more than once.

Using Theorem 3.5 and the ellipsoidal characterization of af(d) given by

Theorem 3.8, the subgradient that gives the steepest descent direction is now given by the

optimization

S(d) = arg minIll11 (3.32)

such that

P4 = q (3.33a)

and

([Gr '2 ]-T)B([ 2 3 T ) 1 (3.33b)

Optimizaiton (3.32) with constraints (3.33a) and (3.33b) represent the minimum distance

from the origin to the ellipsoid Of(d). Obviously, the objective function of optimization







(3.32) is convex in the n parameters {(,2,-..,~4)} and the constraints (3.33a) and

(3.33b) are convex sets. This n -dimensional optimization can be reduced to a 3-

dimensional optimization by incorporating the equality constraints (3.32a) into the

objective function (3.32), because by Theorem 3.7, (3.33a) implies that {44,,"5,.t}

are affine function of {( r,2 3}. The optimization given by (3.32) and (3.33) becomes

Sd(d) = argminjI|112 = argmi [ + 2 + i~ ,( p,, p.2 Pi,3 3 (3.34)
'1 42 43

such that

([1 2 3 c)B([, 2 3]T -c) 1 (3.35)

where the terms {(, ,---,4,} of sd(d) are obtain from the affine functions of

{19,2~ 3}. The objective function of optimization (3.34) is a positive semi-definite

quadratic function and is therefore convex. In addition, the constraint (3.35) is a convex

set. Therefore, determining the steepest descent direction when the maximum singular

value is repeated once reduces to a simple 3-dimensional convex quadratic optimization

over a convex set. Finally, from Theorem 3.8 the necessary and sufficient condition for a

minimum, i.e. 0 e Of(d), reduces to

cTBcl1, q = 0 (3.36)

because [z, z2 z3 T =0 must be an element of the ellipsoid and when

[z, z2 3 ]T = 0, the terms {z4,z, ... ,z,} are zero only when q = 0 (i.e., the affine set

S = {z E R"IPz = q} must pass through the origin).

Now for the case when the maximum singular value is repeated more than once.

From Theorem 3.5, the steepest descent direction is obtained from the smallest






subgradient in the Euclidean norm. From Theorem 3.6, all subgradients are given by the

convex hull of Vf, (d; u) for u'u = 1, which means every subgradient can be expressed

as the linear combination AVf (d; u)+(1 )Vf (d; u2) with u, # u,, A = [0,1],

u;u, =1 and u u2 =1. Therefore the optimization problem given by (3.12) to

determine the subgradient used to obtain the steepest descent direction can be written in

the form

4sd(d) = arg minAVf (d; u,)+ (1 A)Vf,(d; u ,) (3.37a)
A,Ul,u2

with the constraints

u, u2, A=[0,1], u;u, =1 and uu2, =1 (3.37b)

Unfortunately, the objective function of optimization (3.37) is non-convex in the

components of the complex vectors u, and u2, and therefore has all of the associated

difficulties, like local versus global minimums. In addition, from Theorem 3.4 the

necessary and sufficient condition for a minimum is given by s (d) = 0.

3.5. Attainability of MPDA when the maximum singular value is repeated

When the maximum singular value is distinct, the necessary condition for a

infimum of (3.13) is Vf(d) = 0 where the gradient is given by (3.16). This implies that

the moduli of the major input and major output principal directions are elementwise

equal. Furthermore, a unitary transformation matrix U can be determined that shifts the

angles of the elements of the input and output principal directions such that MPDA is

achieved and the upper bound for p is non-conservative. In general MPDA is not

possible when the maximum singular value is repeated and the upper bound on p given

by is conservative. Therefore, the goal of this section is to determine the sufficient







conditions for which MPDA is attainable when the maximum singular value is repeated.

These conditions are important, because they result in a non-conservative upper bound

for p.

A sufficient condition for attainability of MPDA is that there exist a major input

and major output principal direction pair with elementwise equal moduli. This is

equivalent to the existence of u such that Vf (d;u)= 0. In contrast, the less stringent

sufficient condition for a minimum is 0 e8f(d), where as the condition Vf,(d;u) = 0 is

equivalent to 0 being an element of the surface of af(d). For the case when the

maximum singular value has multiplicity 2 this becomes the condition that 0 is on the

surface of the ellipsoid. In other words

cTBc = 1 (3.38a)

and

q = 0 (3.38b)

Equations (3.38a) and (3.38b) represent the sufficient conditions for attainability of

MPDA when the maximum singular value is repeated twice. When the maximum

singular value is repeated more than once the sufficient condition for attainability of

MPDA becomes

min Vf (d;u) = 0 (3.39)

with u'u = 1. Condition (3.39) is not as convenient as (3.38), but is still useful as a

method for determining attainability of MPDA and thus the conservatism of the upper

bound of .







3.6. Reconciling the Results with the PDA Results

The principal direction alignment (PDA) principle (Daniel et al., 1986) states the

infimum of (3.1) occurs at a stationary point of the largest singular value for which a

stationary point exists starting with the maximum singular value. If the maximum

singular value is repeated then there is no stationary point (the maximum singular value is

non-differentiable), and an attempt is made to find a stationary point of the second largest

singular value, and so on. This statement is not entirely accurate. Consider the case

when at the infimum, the singular value is repeated, and therefore the gradient does not

exist. As such the gradient can not be 0 and there is no stationary point, but it is possible

to have a repeated maximum singular value and still achieve MPDA as demonstrated by

Example 3.3. As such, the infimum occurs at a non-stationary point contradicting the

PDA theory.

The PDA theory can rectified as follows. First, a more accurate statement than

stating the infimum occurs at a stationary point (i.e. when all the partial are zero) of a

singular value is to state that the infimum occurs at a point where exist a left and right

singular vector pair that element wise equal moduli. The work of the previous section

gives the conditions for under which it is possible to equate the moduli when a singular

value is repeated. If the moduli can be equated, then MPDA achieved, otherwise it is

necessary to use the PDA algorithm by infimizing the next singular value.

3.7. Examples

The following three examples demonstrate the results of the previous sections.

The first example shows how to determine the steepest descent direction. The second

example demonstrates the conditions for a minimum. The third example illustrates the

conditions for which MPDA is attainable.







3.7.1. Example 3.1.

Let M = AB', where

-0.1582 0.3074i 0.3252 + 0.3078i
0.4198 0.5890i 0.0843 0.0067i
A= 0.2182 + 0.0182i 0.7031 + 0.1455i
0.1039 0.4891i -0.4090 + 0.0507i
-0.0765 + 0.2315i -0.3256 0.030 li

and

-0.3681-0.3181i 0.2366 + 0.271li
-0.2708 + 0.0371i 0.0536 + 0.3304i
B = -0.4548 + 0.5280i -0.0931 0.2255i
0.3127 0.1501i -0.0013- 0.0917i
0.2842 + 0.0444i 0.8244 + 0.1044i

In performing the infimization infc(DMD-'), consider the point d = [1 1 1 1 1]T

corresponding to D = I. The maximum singular value F(DMD-') = a(M) is repeated

(i.e., a, (M)= a-2(M)= with a, (M) = a (M) = r5(M) = 0). Therefore, the

objective function f(d) = F(DMD-') is non-differentiable at d = [1 1 1 1 1]T and

the results of the this chapter are used to efficiently solve the optimization by either

determining a steepest descent direction from the point d = [1 1 1 1 1]T or by

determining if the point satisfies the optimality and MPDA conditions.

First, the ellipsoidal characterization of the subdifferential is obtained using the

method of Section 3.3.2. An orthonormal set of right singular vectors corresponding to

the repeated maximum singular value is








0.1106 0.1679i
-0.1893 0.4442i
0.3526 0.6134i
-0.2971 + 0.1422i
-0.0002 + 0.3428i


0.5229 + 0.0786i
0.0843 + 0.5385i
0.1979 0.1544i
0.0567 + 0.5552i
-0.2160- 0.0469i


and an orthonormal set of left singular vectors corresponding to the repeated maximum

singular value is


0.0000 + 0.0000i
0.2151+ 0.1955i
-0.0322 + 0.3960i
-0.0690 0.2280i
0.3825 0.7447i


0.6051 + 0.0000i
0.3141 -0.0597i
-0.1384 0.6067i
-0.1529 + 0.2204i
0.1730- 0.2060i


Using these sets of left and right singular vectors and equations (3.21)-(3.24) gives


H =


0.0404
0.1486
0.3427
0.0517
-0.5834


0.0893
-0.6221
0.8006
0.2036
-0.4713


-0.1930
-0.0195
0.0148
0.2458
-0.0481


-0.0865
0.1949
-0.3243
0.2395
-0.0235


From Theorem 3.8 the ellipsoidal characterization of the subdifferential is given by


af(d)= {zeR"IPz=q,


([z, z2 Z3] -T)B([z, Z2 z3]T -c) 1}


where the elements of the matrix


1.2180
1 -0.2180


0.6214
0.3786


0.0927
0.9073


1.0000
0.0000


and the vector


0.2251
q= [-0.2251


are obtained from (3.27), the matrix


{XIX2}=


{yI,y2}


0.00001
1.0000







106.6882 -13.8175 -21.7949
B= -13.8175 32.1680 17.6270
-21.7949 17.6270 15.3504

is obtained by the method mentioned after equation (3.31), and the vector

[-0.02311
c = 0.1718
0.0092

is given by (3.30). The point d = [1 1 1 1 1]T is obviously not optimal, because the

necessary optimality condition q = 0 of (3.36) is not satisfied. Consequently, MPDA

does not hold either. Therefore, the next step is to find a steepest descent direction in

order to decrease the objective function in the next step of an iterative optimization

algorithm. The subgradient that gives the steepest descent direction is obtained by

solving the simple 3-parameter optimization given by (3.34) and (3.35) and is determined

to be

0.0264
0.1069
~d(d)= -0.0795
0.1338
-0.1877


Finally, the steepest descent direction is

-0.0988
-0.3996
g(d) = (d)= 0.2970
d(d) -0.5002
0.7015







3.7.2. Example 3.2.

The following example is taken from Daniel et al. (1986). Let M = AB, where

0.65012 + 0.00000i 0.00000 + 0.00000i
0.45970 + 0.00000i 0.45970 + 0.00000i
A=
0.45970 + 0.00000i 0.00000 + 0.45970i
-0.39322 + 0.00000i -0.53729 + 0.53729i

and

0.00000 + 0.00000i 0.65012 + 0.00000i
0.45970 + 0.00000i -0.45970 + 0.00000i
B=
0.45970 + 0.00000i 0.00000 0.45970i
0.53729 0.53729i 0.39332 + 0.00000i

Again, in performing the infimization infa(DMD-1) the point d=[l 1 1 1]T

corresponding to D = I has a maximum singular value F(DMD-') = 1(M) that is

repeated (i.e., o, (M) = a2(M)= 1, with 3(M)= a4 (M) = 0). Therefore, the

objective function f(d) = F(DMD-') is non-differentiable at d = [1 1 1 1]T and the

results of the this chapter are used to solve the optimization by either determining a

steepest descent direction from the point d = [1 1 1 1] or by determining if the point

satisfies the optimality and MPDA conditions.

The ellipsoidal characterization of the subdifferential is given by

af(d)= {zeR= Pz=q, ([z, z2 z,]-cT)B([z, z2 z3T -- )l}

where

P=[I 1 1 1]

q=0








3.5575 1.0750 0.00001
B = 1.0750 6.0773 1.0750
0.0000 1.0750 3.5576

and

S0.0548
c = -0.0749
0.0548

The point d = [1 1 1 I] is optimal, because the necessary optimality

conditions q = 0 and CTBc = 0.0378 < 1 of (3.36) are satisfied implying 0 af(d). This

means the upper bound inf (DMD-') is 1.0000. On the other hand, the MPDA

attainability condition cTBc = 1 is not satisfied. Therefore, MPDA is not attainable and

the upper bound is conservative, i.e. p(M) < inf F(DMD-') = 1, and either the principal

direction alignment (PDA) method proposed in Daniel et al. (1986) or a direct attempt at

solving the lower bound supp(MU) must by used to obtain an exact value of the

structured singular value.

3.7.3. Example 3.3.

This last example shows that even though the maximum singular value is repeated

at the optimum it may still be possible to attain MPDA and thus eliminate the

conservatism in the upper bound of p/. Consider the matrix

-0.0274 + 0.2253i -0.0622 + 0.0571i -0.0597 + 0.0705i -0.0147 + 0.0149i 0.1624 0.1333i
0.2201 0.2277i 0.2355 0.0394i 0.1303 + 0.0643i -0.0632 + 0.1792i -0.3688 0.1437i
M = -0.4758 + 0.2550i -0.1977 0.198 i -0.1025 + 0.0008i 0.1533 + 0.1583i 0.2666 + 0.1838i
0.1192 0.0574i -0.2418 0.0274i 0.1239 0.2037i 0.3778 0.3278i -0.0824 + 0.3762i
-0.0974 0.3482i 0.1610 + 0.1308i -0.1589 0.0976i -0.4272 0.1706i 0.1610 + 0.0723i







The point d = [1 1 1 1 1]' corresponding to D = I has a maximum singular value

&(DMD-') = a(M) that is repeated (i.e., 0a,(M) = a2 (M) = 1, with

a3 (M) = 4 (M) = a0 (M) = 0). Therefore, the objective function f(d) = 5(DMD-') is

non-differentiable at d = [1 1 1 1 1]T and the results of the this chapter are used to

efficiently solve the optimization by either determining a steepest descent direction from

the point d = [1 1 1 1 1]T or by determining if the point satisfies the optimality and

MPDA conditions.

The ellipsoidal characterization of the subdifferential is given by

af(d)={zER "nPz=q, ([z, z2 z3]-cT)B([z, z2 z ]T -c)
where

[0.0828 -0.9106 0.4221 1.0000 0.00001
0.9172 1.9106 0.5779 0.0000 1.0000

[0.0000
S[0.0000

37.4471 -7.5773 25.1289
B= -7.5773 28.1213 -6.3079
25.1289 -6.3079 26.8994

and

[-023981
c= 0.0632
0.2010

The point d = [1 1 1 1 1] is optimal, because it satisfies the necessary

optimality conditions (3.36). Furthermore, the MPDA attainability conditions (3.38) are







also satisfied, namely q = 0 and cTBc = 1. Therefore, the upper bound

inf (DMD-) = 1.0000 is tight and the structured singular value is exactly

p(M) = 1.0000 even though the maximum singular value is repeated such that the

objective function is nondifferentiable.

3.8. Conclusions

The MPDA principle approach to solving the structured singular value problem is

investigated. In the infimization that gives an upper bound to mu, a repeated maximum

singular value results in a non-differentiablity of the objective function. Therefore,

efficient gradient descent optimization algorithms that use the analytical expression for

the gradient must be modified. The first result of this paper is characterization of the

subdifferential which represents the set of all sub-gradients or generalized gradients. In

addition, for the case of a once repeated maximum singular value it is shown that the

subdifferential is in fact a 3-dimensional ellipsoid in and n-dimensional space. Using

results from non-differential optimization theory, the steepest descent direction is obtain

from this characterization of the subdifferential to facilitate the optimization.

Furthermore, conditions for optimality are presented which are based zero being an

element of the subdifferential. Finally, attainability of MPDA at the optimum is shown to

be equivalent to zero being on the boundary of the subdifferential enhancing the PDA

results when themaximum singular value is repeated.











CHAPTER 4
SPECTRAL RADIUS MAXIMUM SINGULAR VALUE EQUIVALENCE UNDER
OPTIMAL SIMILARITY SCALING

4.1. Introduction

It is well known that the maximum singular value of a matrix is an upper bound of

the spectral radius (i.e., p(M) 3(M) where M eC""*). Determining the conditions

under which the upper bound is attained is a significant issue in the field of robust

control. One approach is to seek properties of matrices that are necessary and sufficient

for equality of the spectral radius and the maximum singular value. Another approach

uses optimization to condition the matrix through similarity and unitary transformations

in order to increase the spectral radius and decrease the maximum singular value upper

bound so that equality is achieved.

Previous work deals with the optimal conditioning of matrices from a numerical

accuracy stand point (Bauer, 1963) and focuses on similarity transformations using

nonnegative diagonal matrices. The scaling problem for non-negative matrices yields a

very elegant and precise result. It provides a closed form expression for the optimal

similarity scaling matrix for which the Perron-root (largest positive eigenvalue of a

positive matrix) equals the least upper bound subordinate to an absolute norm. In

addition there are analytical expressions for the elements of the optimal diagonal matrix

that involve the Perron-eigenvectors of the given positive matrix (Stoer and Witzgall,

1962). The relationship to the present work is that the least upper bound of the matrix

subordinate to the Euclidean norm is the maximum singular value of a matrix. The







previous results are based on earlier work that derive a necessary condition for the least

upper bound of a matrix to equal the modulus of an eigenvalue of the matrix, namely, that

the corresponding right and left eigenvector are dual (Bauer 1962). Unfortunately, for

the general case of complex matrices, there are no equivalent analytical results on optimal

scaling by positive diagonal matrices, although there exist several numerical algorithms.

From a robust control perspective, the structured singular value, p (defined as

supp(MU) where tV:= diag(eJ't,eJ ',..-,ejo*)0<5 U eV

is a widely accepted tool in the robust analysis of linear systems. It considers the

problem of robust stability for a known plant subject to a block-diagonal uncertainty

structure under feedback. In general, any block-diagram interconnection of systems and

uncertainties can be rearranged into the block-diagonal standard form. Calculating p is

not trivial; in fact the problem has been proven to be NP-hard (Braatz et al., 1994). The

difficulty is that the spectral radius is non-convex over the set of unitary matrix

transformations. One approach is to consider upper bounds for the spectral radius that

can be calculated easily, and ideally should be attainable to eliminate conservatism. The

maximum singular value is reasonable choice for an upper bound because it is invariant

under unitary matrix transformations. In addition, the maximum singular value upper

bound can be decreased by optimizing over similarity transformations because the

spectral radius is invariant under such transformations. Ultimately, the problem becomes

one of conditioning a matrix through optimal similarity and unitary transformations to

achieve equality between the spectral radius and the maximum singular value.

In addressing the existence of solutions to the proposed optimization,

Kouvaritakis and Latchman introduce the major principal direction alignment (MPDA)







property (1985). The result states that the spectral radius of a matrix is equal to the

maximum singular value of the matrix if and only if a major input principle-direction and

a major output principal-direction of the matrix are aligned. MPDA is a strict condition

for a matrix, but can be used to determine the optimal positive diagonal matrix and

unitary matrix that results in equality between the afore mentioned definition of p and

the maximum singular value upper bound for the case when the maximum singular value

is distinct.

It is the goal of this work to establish relationships between results obtained from

different perspectives of the same spectral-radius/maximum-singular-value equivalence

problem. To this end, the earlier work by Bauer (1963) on positive matrices is extended

to the class of general complex matrices. The results are necessary conditions for

equality that are used to improve the calculation of p through its upper bound.

4.2. Mathematical Background

4.2.1. Dual Norms and Dual Vectors

In the theoretical development that follows the mathematical concepts of dual

norms and dual vectors are utilized. These concepts are explained in a paper by Bauer

(1962) and are reviewed here to facilitate the theoretical development. Given a vector

norm I||- its dual vector norm -|Ile is defined as

Rey*x
IIYI:= max Re y x = max


For such dual norms the Holder inequality

IlyllD lxll 2 Rey'x




65

holds an is sharp, i.e., for any yo there exists at least one xo, and for any xo there exists

at least one yo such that the equality holds (Bauer, 1962). If such a pair (xo,yo) with

IlyoDJllollo = Reyoxo also satisfies the scaling condition

IlyollD 1ol 0 =
it is called a dual pair. Note that the dual vector of x is often written (x)D. A pair

(xo,yo) is strictly dual and is written yolloD if IIyollDI0ll=lo =y 1' For strictly

homogenous norms (i.e., those satisfying axll = lal lxI for all complex scalars a) the

Holder inequality may be sharpened to (Bauer, 1962)

IIYID1.114- ly'xl

For a dual pair (Xo, y) under a homogenous norm it follows that

Reyoxo = IIYollollDXI L yoX0 which implies that Re yoo = Yoxo. Hence, for a strictly

homogenous norm every pair of dual vectors (x,,yo) is also strictly dual pair. In

addition, there exists a strict dual yo for any xo 0 and a strict dual xo for any yo # 0.

In general, the dual norm of a p-norm Ix l:=l (Zlx')1', is the associated

p-norm |II I where / p + / q = 1. So the infinity-norm and the 1-norm are duals, and the

dual norm of the 2 (Euclidean) norm is itself. For the 2-norm, a pair (xo,Yo) is dual if

yo = Xo/|lxoll(

4.2.2. Positive Matrix Result
Early work on determining when the spectral radius equals the maximum singular
value is concerned with positive matrices transformed by non-negative diagonal matrices,
because they have good numerical properties (i.e., less round off errors) and therefore







may be used for conditioning of matrices. In addition, positive matrices remain positive

under transformation by non-negative diagonal matrices leading to connections with

Perron-roots r(P) (positive eigenvalues of largest modulus) of positive matrices

P e RX"n (note, R. is the set of positive real numbers). From this perspective, Stoer and

Witzgall (1962) show that for the positive matrix P and non-negative diagonal

matrices D


,z(P) = min lub(D-'PD)
DED


(4.1)


where D:= {diag(d,,d2,...,d.)I d, > 0,i = 1,2,.--,n}, and


lub(A):= max Ax max|iAx|I
0 |x||' II-1=1

is the least upper bound norm of a matrix A e C""' subordinate to the vector norm II. It

is noted that the least upper bound norm is equivalent to the induced matrix norm, and

that when the subordinating norm is the Euclidean norm then lub(A) = C(A) .

In developing the result it is necessary to make use of a result from Bauer (1962)

that states that if A is an eigenvalue of A, then

1A = lub(A) (4.2)

is only possible if a right and left A -eigenvector are dual with respect to the norm to

which the bound norm is subordinate, where by definition a left A -eigenvector w of A

satisfies the relation w'A = Aw* (Golub & Van Loan, 1983; Isaacson & Keller, 1966;

Stewart, 1970). The reader is cautioned that some authors use the term left eigenvector

for an eigenvector of A'.







Let Do be the minimizing D of(4.1), then

71(P) = r(DO'PDo) = lub(Do'PDo) (4.3)

From the result of Bauer (1962), (4.3) is only possible if the right and left eigenvectors of

Do'PDo are dual. Now, if v > 0 and w > 0 are the right and left Perron vectors of P

(note that it is implied that greater than operator ">" is an element wise operation when

applied to a vector), then it is straightforward to show that D-v > 0 and Dw > 0 are the

right and left Perron vectors of D-'PD. Therefore, any Do that minimizes (4.1) such that

equality is achieved must also make the vectors D0'v and Dow dual, where v > 0 and

w > 0 are the right and left Perron vectors of P.

The problem now reduces to transforming the positive vectors v > 0 and w > 0

to dual vectors Do'v and D0w where Do is a non-negative diagonal matrix. Stoer and

Witzgall (1962) state that for absolute norms (i.e., norms that only depend on the moduli

of their components (Bauer et al., 1961)) there exists one, and up to positive multiples

only one, non-singular non-negative diagonal matrix Do such that Do'v and D0w form a

dual pair. For a p-norms which are necessarily absolute norms, the positive vectors y > 0

and x > 0 are dual if

(y,)q =(x,)P, i=1,2,...,n

and

1 1
-+-=
P q







Therefore, the matrix

Do =diag,- I (4.4)

makes Do'v and Dow a dual pair for any right and left Perron-eigenvectors v > 0 and

w>0.

Duality is only a necessary condition for (4.2). Therefore, to show (4.1) holds it

suffices to show (4.3) holds for those matrices Do'PDo whose right and left Perron-

vectors Do'v and Dow are dual, where Do is given by (4.4). Using the definitions of

eigenvalues and eigenvectors it can be shown that

Re{(w*Do)(Do'PDo)(Do'v)} = r(Do'PDo) Re{(w*Do)(Do'v)} (4.5)

and from the definition of duality of vectors it is true that

Re{(w*Do)(D;'v)}
=1 (4.6)
IIDo WIiD Do'vl

Combining (4.5) and (4.6) gives

Re{(w'Do)(Do'PDo)(Do'v)}
IjjDow HDv = (Do'PDo) (4.7)
DoWI Io ll

Using of the bilinear characterization of the least upper bound

Re(y Ax}
lub(A):= max ReyAx (4.8)
Xyro I1yllI.X4

Stoer and Witzgal (1962) show there is a maximizing pair for (4.8) in the positive

orthant, and that the only maximizing pair in the positive orthant for lub(Do'PD,) is the

pair Do'v and Dow such that








Re{(w'Do)(Do'PDo)(Do'v)}
lub(DO'PDo) =
IlDowllollDD'vli

which from (4.7) equals ;r(DoPD0). Therefore, (4.1) holds where the minimizing D is

given by (4.4).

The relationship of Stoer and Witzgall's positive matrix result to the spectral-

radius/maximum-singular-value problem can be shown by specifying the least upper

bound norm to be subordinate to the Euclidean norm, i.e.

lub(A) = o(A) (4.9)

where A e Cx"". Combining (4.9) and the fact that the Perron-root of a positive matrix is

the spectral radius, (4.1) becomes

p(P)= min a(D-'PD) (4.10)
DeB

for positive matrices P and positive diagonal matrices D. In addition, from (4.4), there

is an analytical expression for the optimizing Do given by

( /2 1/2 1/2 (4.11)
DO = diag ,--, (4.11)


where v > 0 and w > 0 are right and left Perron-vectors of P. Clearly, (4.10) shows that

for positive matrices there is a simple similarity transformation for which the spectral

radius attains its the maximum singular value upper bound.

4.2.3. Major Principal Direction Alignment Property

In solving various robust control problems it is necessary to determine the

conditions under which the spectral radius of a matrix attains its maximum singular value

upper bound. The major principal direction alignment (MPDA) property addresses this







problem (Kouvaritakis and Latchman, 1985). Consider the singular value decomposition

of a square matrix A e C"x" given by

A= X(A)E(A)Y'(A)

where E(A) is the diagonal matrix of singular values placed in descending order, and

X(A) and Y'(A) are unitary matrices whose columns are the respective output and

input principal directions of A, arranged in an order conformal with the order of the

singular values (Lancaster and Tismenetsky, 1985). Now, define a major input principal

direction y(A), and a major output principal direction Y(A), of a matrix A respectively

as normalized input and output principal directions, corresponding to the maximum

singular value, U(A) of A. The MPDA property is given in the following theorem.

Theorem 4.1. The spectral radius of any matrix A eC""" is equal to the

maximum singular value of A, if and only if there exists a major input principal

direction and a major output principal direction of A which are aligned such

that

x(A) = ejoy(A)

Proof. Given by Kouvaritakis and Latchman (1985). An alternative proof based

on dual norms and dual vectors is given in Chapter 2. Q.E.D.

4.2.4. MPDA as a Control Theory Application

One area in the field of robust control that makes use of the spectral-

radius/maximum-singular-value equivalence problem is the stability analysis of

multivariable feedback systems in the presence of structured uncertainties. Of particular

interest is the stability of diagonally perturbed systems for which the uncertainty is

represented by the complex diagonal matrix







A ediag(,,82, ..., 1I < P,, P eR i= 1,2,-.,n

This class of systems is especially amenable to spectral radius-preserving similarity

scaling, and through simple transformations is representative of the more general class of

full structured uncertainties.

Using Nyquist arguments in the complex plane, it can be shown that

sup p(MA) <1 (4.12)
A

is a necessary and sufficient stability condition, where the complex matrix M is function

of the system's transfer function matrix evaluated a particular frequency. The

optimization problem (4.12) is non-convex, but it can be simplified by introducing the

following positive diagonal similarity scaling

p(MA) = p(D-'MDA) 5 5(D-'MDA)

Furthermore, using geometric arguments based on the MPDA principle, it can be shown

that the supermizing diagonal-matrix A,, has the form

Aop, = QU

where Q = diag(q,,q2,.*,qn) with q, eR, and

U eV.:= diag(e'e",e j',...,ejo )0 c0, <2;r,i= 1,2,-.,n}

The optimization problem (4.12) becomes equivalent to

sup p(MA) = sup p(MQU) 5 inf -(D-'MQD) (4.13)
A Uet De

and the necessary and sufficient stability condition becomes

inf(D-'MQD) <1 (4.14)
DelD







Furthermore, using MPDA arguments it can be shown that the optimizing Do in (4.14)

results in the equality

sup p(MQU) = F(D'MQDo)

when the maximum singular value is distinct at the infimum.

4.3. Main Result Extension of the Positive Matrix Result to
General Complex Matrices

The positive matrix result of Stoer and Witzgall as stated by (4.1) and specialized

to the Euclidean norm by (4.10) gives a positive diagonal similarity scaling (4.11) that

results in equality of the spectral radius and maximum singular value of a positive matrix.

When applied to robust control problems that involve complex matrices, the positive

matrix result is usually only sub-optimal. Therefore, it is necessary to extend the result to

the class of complex matrices. Unfortunately, much of the theoretical development is

dependent on the characteristic properties of positive matrices. Therefore, when

generalizing the result to complex matrices it is not possible to explicitly state that there

exists a similarity scaling that will result in equality of the spectral radius and maximum

singular value of a matrix. Nevertheless, it is possible determine the necessary conditions

for the existence of a positive diagonal similarity scaling that leads to equality. The result

is given in the following theorem.

Theorem 4.2. Let A e C""U have a right eigenvector v and a left eigenvector w

associated with an eigenvalue A(A) of maximum modulus such that

IA(A)I = p(A), and let Do = diag(do,,, d,2, ...,d ,) be define as

Do:= arg min (D-'AD)
DeD

where D:= {diag(d,,d2,...,d,) d > 0,i= 1,2,---,n}. Then if







p(A) = min o-(D-'AD)
DeD

the following three conditions hold

i)


[do, d,2


S1, 11 W 12 -I
Iv211w, 12
1IH211


-.. do] Te null(N) = ker(N)


IV1 1 12
IV21'W"1 -IW2


... IVll MI


arg(v,)= arg(w) i= 1,2,-* ,n


(4.18)


and either

iii-a)


X Iv,= lw,=
i=1


(4.19a)


or

iii-b)

Iw, = for at least onei 1,2,-*.,n (4.19b)

Proof. Assume (4.15) holds where Do is an optimizing D such that

IA(A)I = 5(Do'ADo) (4.20)

where A(A) is an eigenvalue of maximum modulus. Following the development of the

positive matrix result, a necessary condition for (4.20) to hold is that the corresponding

right and left eigenvectors of Do'ADo be dual with respect to the Euclidean norm. Given


(4.15)


where


(4.16)


(4.17)






that v and w are a pair of right and left eigenvectors of the A(A) eigenvalue of A,

then Dolv and Dow are corresponding right and left eigenvectors of Do'ADo.

Therefore, the necessary condition is that Do'v and Dow are dual with respect to the

Euclidean norm. In the mathematical background section it is stated that this is
equivalent to requiring

Do'v = w (4.21)
IIDowIl:

Using the notation

u = [IulleaT I)i, Iu2eaT( u,) J ,* e I

v = [vle arg(,) j ,v2 Jearg(Y2) j ,IvearB(v")j ]T

the necessary condition (4.21) is equivalent to the set of scalar qualities

SIv, le arg(,) dO, w'l I:"l ) (4.22-1)
do, d 1Iw, 2 + d.2w, 2 + ...+ + doJ, w, 2


Iv2 learg(v,)j d.lw21w2 le ) (4.22-2)
do,2 d, w, IWI2 + do2 W22 + + d. Iw, 12



1 Iv.le'ar"gS.)i = d, l~("" )i (4.22-n)
1o, WI12 +d2 W212 + +d 2 2 2
d0,o dw, 2 +do,2 '2+ -+do.Iw.l2

which leads directly to necessary condition (4.18). Given that (4.18) is satisfied, (4.22)

can be rearranged as







IV 1I1 I12 _IWI V I IW212 ... IVj I IJV.12 dr2'l
I lI I -,I 2 i I I I -2 I ,llw,2 d 2
2I I Iv 11Xw'-w I2 2 Iv~ w2 ,2 0 (4.23)

.I IwI2 IW22 ... Iw 2. I -Iw.I jV _dO

from which necessary condition (4.16) becomes apparent. For the null space of the
matrix given in (4.23) to be non-trivial, its determinant must be 0 (i.e. the matrix must be

rank deficient). First, note that if any Iw, =0 then the corresponding column i is

composed of only zeros making the matrix rank deficient, resulting in part iii-b) of

necessary condition (4.19). For the case when no Iw, =0 for i=1,2,-..,n the

determinant can be determined using elementary row and column operations to obtain a

matrix that is sparse and has the same determinant. Multiplying column 1 by -Iw,2 /1w11

and adding the result to each column i for i = 2,3,.* ,n gives the matrix


II W I w, IwII
Iv w-w, OI -Iw~l 0 ... 0
v3 ll 1 0 w3l
: .. 0
IvI Iw, 2 0 -.. 0 -w.

Now, for i= 2,3,---,n, multiplying row i by Iwi/Iwl1 and adding the result to row 1


gives the matrix

(IIIwI- +Iv21IW21 +v I31 w3+...+IV. IwI)w,II


IV.3I Iw I
INllI2


0
-I2 o
0 31

0


for which the determinant is


0
0

0
0 -|w,|








(lv, I Iwl I+lv, I112 I+ ..+ IV .1% I, I- 1)Iw I y1 I w2l... Iw ,I

So, for the case when no [wl = 0 the determinant is identically zero and the null space is

non-trivial only when part iii-a) of necessary condition (4.19)


Y1vil IwI=1 (4.19)
i=1

is satisfied. Q.E.D.

4.4. Example 4.1

The following example demonstrates the result of Theorem 4.2. Consider the

matrix

-0.5259 + 0.6358j 0.3090-1.3791j 0.2031 +0.2317j -0.1016 + 0.9524j
0.4712 +0.1832j 0.7383-0.5966j -0.3174-0.1128j -0.2840-0.2127j
A=
-0.0290-0.1034j -0.7906 +2.0522j 0.4991-0.6463j -0.0584 + 0.1540j
0.0925 0.2759j 0.5359 + 0.7832j 0.2490 + 0.083 Ij 0.0694 0.1919j

where the eigenvalue of maximum modulus is 2(A)=1.6507-1.1293j such that

I2(A)I = p(A) = 2. Performing the minimization on the right-hand side of (4.15) gives

min _(D-'AD)= 2
Del

with

10 0 0
0 0.5 0 0
Do:= argmin-(D-IAD)=
DeD 0 0 1.2 0
0 0 0 0.7

such that equation (4.15) holds. Therefore, the three necessary conditions of Theorem

4.2 must be satisfied. First, the right and left eigenvectors of A associated with the

eigenvalue ,(A) = 1.6507-1.1293j of maximum modulus respectively are








-0.0845- 0.2857j
0.3690- 0.2780j
V=
-0.1441 + 0.7943j
0.0167 + 0.2140j

-0.0567 -0.1918j
0.9912- 0.7467j
W=
-0.0672 + 0.3704j
0.0229 + 0.2932j

For condition i) the matrix N given by (4.17) is

-0.1881 0.4589 0.0422 0.0258
0.0185 -0.5294 0.0655 0.0400
N=
0.0323 1.2432 -0.2620 0.0698
0.0086 0.3305 0.0304 -0.2756

and

d02,1 1
d2 0.25
N 0,2 =N =0
d 1.44
_d024. 0.49

such that condition i) is satisfied. Finally, it is easy to shows that condition ii) and iii-a)

of Theorem 4.2 are satisfied.

4.5. Conclusions

In this paper we recover the dual-norm arguments for the case of complex A and

obtain an exact and closed form expression for the optimal D matrix. This result has

independent value in terms of the mathematical completeness of the extension of the case

of complex matrices as well as potential algorithmic improvements in computing the

optimal scaling matrices.












CHAPTER 5
GENERALIZATION OF THE NYQUIST ROBUST STABILITY MARGIN AND ITS
APPLICATION TO SYSTEMS WITH REAL AFFINE PARAMETRIC
UNCERTAINTIES

5.1. Introduction

The critical-direction theory developed by Latchman and Crisalle (1995) and

Latchman et al. (1997) addresses the problem of robust stability of systems affected by

uncertainties that can be characterized in terms of frequency-domain value sets. The

approach introduces the Nyquist robust stability margin kN (o) as a scalar measure of

robustness analogous to the structured singular value p (Doyle, 1982) and the

multivariable stability margin km (Safonov, 1982) within the value-set paradigm. This

chapter extends the critical direction theory to the more general case where the critical

value-set may be nonconvex. The key to extending the theory is the introduction of a

generalized definition of the critical perturbation radius in a fashion that preserves all

previous results. The nonconvexity of the critical value set is observed in a number of

interesting problems, including the case studied by Fu (1990) consisting of rational

systems where the uncertainty appears affinely in the form of real parameters that belong

to a known rectangular polytope. The generalized critical direction theory is applied to

this particular class of uncertain systems, and is used to calculate the required Nyquist

robust stability margin with high precision and in the context of a computationally

manageable framework.

The robust stability problem studied by Fu is part of an extensive literature on

systems where the uncertainty appears in the form of parameters that vary in prescribed







real intervals, a situation of relevance to many engineering problems. Early advances in

this field are due to Kharitonov (1978, 1979) who derived necessary and sufficient

conditions for the robust stability of interval polynomials, that is, polynomials with

independent coefficients that take values in closed real intervals. An extension of

Kharitonov's theorem to rational interval plants is proposed in Chapellat et al. (1989),

where the objective is to assess the stability of a family of plants by testing a subset of

extreme plants or extreme segments. The number of extreme plants required to determine

robust stability depends on the functional relationship between the uncertain parameters

and their bounding interval-sets. Comprehensive results based on extreme plants or

segments are known to exist only for a restricted set of uncertainty structures. A detailed

account of Karitonov-like methods can be found in Barmish (1994) and in the references

therein. For contextual value, it is worth mentioning that many of the methods proposed

are based on determining the stability of a set of Kharitonov plants (or extreme plants)

derived from the interval bounding-set description. For example, Chapellat et al. (1989)

and Bartlett et al. (1990) give conditions that use 32 Kharitonov segments or edges.

Barmish et al (1992) prove that when using first-order compensators it is necessary and

sufficient that sixteen of the extreme plants be stable; furthermore, under certain

conditions only eight or twelve plants are necessary.

In this chapter the generalized critical direction theory is applied to systems with

affine parametric uncertainty and exploits earlier results of Fu (1990) regarding the

mapping of the uncertain parameters from their polytopic domain to the Nyquist plane to

develop a computationally tractable algorithm for calculating the Nyquist robust stability

margin. The chapter is organized as follows. Section 5.2 generalizes the critical direction

theory for systems with nonconvex critical value sets. Sections 5.3 through 5.8 are







concerned with the application of the generalized theory to the case of affine uncertain

rational systems with real polytopic parametric uncertainties. Section 5.3 introduces a

precise definition of the uncertain system considered, and Section 5.4 derives two robust-

stability theorems for these types of systems. Section 5.5 presents a systematic method

for calculating the critical perturbation radius, and Section 5.6 provides two examples of

the analysis method, including the case of a convex and the case of a nonconvex critical

value set. Overall conclusions are given in Section 5.7.

5.2. Generalization of the Critical Direction Theory

5.2.1. Preliminaries

Consider the single-input single-output linear time invariant system

g(s) = go(s) + 8(s) (5.1)

where go(s) is a known nominal transfer function, and 8(s) A is an unknown

perturbation belonging to a known perturbation family A. The focus of this analysis is

on the robust stability of the closed-loop system that results when the uncertain system

(5.1) is configured in the unity negative feedback control structure shown in Figure 5.1.

g (s)




Figure 5.1. Unity feedback control scheme for an uncertain plant g(s).

The following standard assumptions are made throughout this chapter:

(Al) The nominal transfer function go(s) is stable under unity negative

feedback.








(A2) The set of allowable perturbations A is such that g(s) and go(s) have the

same number of open loop unstable poles.

The robust stability analysis is based on a frequency domain description of the uncertain

perturbations using value sets. The uncertainty value set of g(s) at frequency to is

defined as

V(co):= {g(jo) Ig(o) = go((jw)+ S(jo), 8(s) e A}

and V(o() is said to lie on the Nyquist plane. A generic uncertainty value set is shown in

Figure 5.2.


r(w,)
r
'4
\4
\4


go(jw,) + d,(w,)

go(jco,) +P,(w,)dc(wo,)


Img(jw)


Re g(jw)


Figure 5.2. Schematic of an uncertainty value set ((w,) (shaded area),
and the critical perturbation radius pc(wo) at a frequency tow. Also
shown in the figure are the critical line r(co) (dashed line); and the
nonconvex critical uncertainty value set V(w,) which in this case is the
union of two disjoint straight-line segments (shown by the dotted lines).







The critical-direction theory advanced in Latchman and Crisalle (1995) and in

Latchman et al. (1997) is based on the observation that the smallest destabilizing

perturbations occur along the critical direction

1 + go(jw)
1+ g0(jco)

which is interpreted as the unit vector with origin at the nominal point go(joe) and

pointing towards the critical point -1 + j (cf Figure 5.2). This direction in turn defines

the critical line r(w) := go(jw) + ad,(jw) a e R where R' denotes the nonnegative

real numbers. The critical line r(ao) is interpreted as a ray that originates at the nominal

point go(jo) and passes through the critical point -+ j0. The intersection of the

uncertainty value set with the critical line determines the critical uncertainty value set

S(co) := (c(o) n r(w) which may be (i) a single straight-line segment or a single isolated

point (in which case ((co) is a convex set) or (ii) a union of disjoint straight-line

segments and isolated points (in which case V((w) is a nonconvex set). Figure 5.2 shows

the case of a nonconvex critical uncertainty value set. Finally, the boundary of the

uncertainty value set is denoted V8(co), and the set of critical boundary- intersections

Bc (w) is defined as

B, (w): = {89(w)) n r(co)} \ go (ja)

where "\" is the set-difference operator. For the special case where 9V(o) n r(co)

contains go (jw) as its only element, the following definition is applied:

%(w):= {go(jw)}







Note that to determine c (to) it is necessary to have knowledge of the uncertainty value

set boundary only along the critical line. Clearly, B (co) contains a single element if

4(ow) is a convex set, and contains at least two elements if V'(w) is nonconvex.

When the critical value set V((m) is convex (as in the case of star-shaped value

sets with respect to the nominal point, for example), the critical perturbation radius is

defined as (Latchman and Crisalle, 1995; Latchman et al., 1997)

pc ():= max {a z=go(jo)+ ad,(j j) C EV() } (5.2)

Definition (5.2) states that the critical perturbation radius for the case of a convex set

T((j) is simply the distance along the critical direction between the nominal point

go(j0 ) and the uncertainty value set boundary V8(w). Note also that the perturbation

radius captures the "size" of the uncertainty that is relevant for stability analysis.

Definition (5.2) is not suitable, however, for the case of nonconvex critical value sets

V,(to). In this chapter the following generalization of the definition of the critical

perturbation radius is proposed, which is applicable to both the convex and nonconvex

cases:


p(w):= +g(j ) if-l+j0 (w ) (5.3)
11+ go(jU ) j+ (w) otherwise

where

(w) = eai)1 + zI (5.4)

represents the distance from -1+ jO to the point in B, (t) that is closest to the critical

point -1+ jO. The upper statement in definition (5.3) states that when -1+ j0 is not an

element of V(w), the critical perturbation radius p,(w) is defined as the difference







between two distances, namely, the distance from the critical point -1+j0 to the

nominal point go(jw) (represented by 1+ go(jo|) ) and the distance from the critical

point -1+ jO to the closest critical-boundary intersection (represented by ((o)). On the

other hand, when -1+ j0 is an element of V(c), the lower statement in (5.3) states that

the critical perturbation radius is taken as the sum of the two distances in question.

Observe that when the critical uncertainty value set is convex, B, (o) has only one

element (i.e. there is only one critical boundary intersection), and definition (5.3)

becomes equivalent to definition (5.2). Note also that to compute the critical perturbation

radius from (5.3) it is necessary to have full knowledge of the set of critical boundary

intersections B (co) and to be able to evaluate whether the set membership condition

-l+ jO ef'(w) holds; both of these issues are completely resolved in Section 5.3 and

Section 5.4 of this chapter for the case of systems with real affine parametric

uncertainties. For either definition it can be shown that Pc (o) > 0 for all frequencies.

Finally, the Nyquist robust stability margin

kN () := P()) (5.5)(6)
+ g(jo) I

is defined as the ratio of the critical perturbation radius to the distance between the

nominal point go(jc) and the critical point -l+ jO measured along the critical

direction. Note that kN (c)) > 0 for all frequencies.

5.2.2. Analysis of Robust Stability

The analysis of the robust stability of the uncertain feedback system being

considered can be resolved in terms of the following theorem.







Theorem 5.1. Consider the uncertain system g(s) given in (5.1) with

assumptions (Al) and (A2). Then, the closed loop system is robustly stable under

unity feedback if and only if

-1+ j0 V V(w) Vw (5.7)

Theorem 5.1 is simply a restatement of the well-known zero-exclusion principle

(Barmish, 1994), and it gives a necessary and sufficient condition for the robust stability

of the closed loop in question. However, Theorem 5.1 does not provide a measure of the

degree of robust stability of the loop, a quantity that would be most useful as the basis for

the synthesis of optimally robust controllers or for the assessment of the relative merits of

alternative control schemes. The critical direction theory seeks to quantify the robust

stability of such systems in terms of the Nyquist robust stability margin (5.5), which plays

a role analogous to that of the structured singular value (Doyle, 1982) and of the

multivariable stability margin (Safonov, 1982). Efficiency in the analysis is obtained

through the realization that it suffices to verify condition (5.7) only for value-set points

that lie along the critical direction; more precisely, the set membership condition (5.7)

holds if and only if -1+ jO V%(w) holds. These observations lead to the following key

result of the critical direction theory.

Theorem 5.2. Consider the uncertain system g(s) given in (5.1) with

assumptions (Al) and (A2). Then the closed loop system is robustly stable under

unity feedback if and only if

k (o) < Vwc (5.8)

Proof A complete proof is given in Latchman and Crisalle (1995) for the case

where 1((w) is convex. For the non-convex case in which the generalized definition







(5.3) of pc(w) is utilized the proof is extended as follows. From Theorem 5.1 the

uncertain closed loop system is stable if and only if -l + jO 0 V() Vw Therefore, to

prove that (5.8) is sufficient for robust stability, we must show that if kN(c)) <1 VW

then -l+jO 0 (w) Vw. To prove by contradiction, assume that kN(w) <1 Vow and

that 3c such that -l+jE0 c(w). Then applying definitions (5.3) and (5.5) for a

frequency at which -1 + jO e 9V() gives

p,(w) 1+g0(jw) +4(w)1 (()
kN (o)= PC I + ) = 1)
I+ go(jw) I I1+ g0(jc) I 1+ go0(J) I

where 4(w) is the nonnegative real scalar given by (5.4). Hence, kN (c) 2 1 for at least

one frequency, which contradicts the assumption. Therefore, if kN,(o)<1 V'd then it

follows that -1+ j0 0 V(w) Vw To prove that (5.8) is necessary for robust stability,

one must show that if -1+ jO 4(co) Vco then kN(c))< 1 Vco. To establish this, note

that if -1 + jO 4 V(c) Vo then by definitions (5.3) and (5.5)

kPC () ( 1 + lgo (jo) I ()) =()
kN(W)= ( =1 (w
S+ go(jc 1+ g() I 1+ go() 1 +go

where (wc) is given by (5.4). In this case, however, since -1 + jO 9V(w) it follows that

-1 + jO0 Qc (c), and thus (co) must necessarily be a positive number. Using this fact in

the above equality leads to the conclusion that kN (co) <1 'V Q.E.D.

From Theorem 5.2 it follows that the scalar kN (c) serves to quantify the robust

stability of the closed-loop system. The computation of k (ct) requires knowledge of the

critical perturbation radius pc(w) defined in (5.3). The challenging task in a given

problem is in fact the calculation of the critical perturbation radius.







When C((w) is convex, definition (5.2) indicates that pc(w) represents the

distance between the point go(ji) and the (unique) point where the critical line

intersects the boundary of V(co). On the other hand, when V (w) is nonconvex there are

multiple points where the critical line intersects the boundary of V(co). In such cases,

definition (5.3) indicates that pc(w) is a function of the distance between go(jiw) and

the boundary-intersection point that is closest to the critical point -1+ j0. Since in many

cases the convexity of Vc(w) at any given frequencies may not be known a priori, the

generalized critical radius definition allows the application of the critical direction theory

without conservatism to a more general class of uncertain systems, including the case of

real affine uncertain systems discussed in ensuing sections. The Nyquist robust stability

margin k,(w) computed using the general definition (5.3) for p(co) is attractive from

an analysis standpoint because through Theorem 5.2 it gives necessary and sufficient

conditions for robust stability. On the other hand, if kN (c) is computed using equation

(5.2) for p, (w), then the condition kN,() < 1 Vw is only sufficient for robust stability

when the set V (w) is nonconvex. From a control design point of view, however, it may

be advantageous to adopt the computationally simpler definition (5.2) even for the case

where V1(w) is nonconvex, and accept the result as a suboptimal design, as is done in the

context of the structured singular value paradigm where control design is based on an

upper bound rather than on the exact value of the structured singular value. It must be

remarked, however, that when V'(co) is in fact convex, using definition (5.2) for pc(o)

makes the resulting condition kN(w))
stability; and in such cases the results are not conservative. It must also be emphasized


i.,'








that the uncertainty value set V(co) itself does not have to be convex for the critical

uncertainty value set V (co) to be convex

5.3. Systems with Affine Uncertainty Structure

In this section the generalized critical direction theory is specialized to systems

with real parametric uncertainties that appear in an affine fashion, namely, an uncertain

rational function of the form

p
no(s) + Zqin (s)
g(s,q)= ,= qeQ (5.9a)
do(s) + qid,(s)
1=1

where


no(s):= noksk
k=0

and


do(s):= d0k k
k=0

are known nominal polynomials,


n,(s) = nis
k=O

and


d,(s) = dik
k=0

are known perturbation polynomials, and q = [q, q2 ... q,]T RP is a vector of real

perturbation parameters belonging to the bounded rectangular polytope


Q={qeRP' q,






where q[ and q i = 1, 2, ..., p are finite real bounds. Equations (5.9a)-( 5.9b) define a

class of finite-dimensional, linear, time-invariant, real systems with affine parametric

uncertainties. For completeness, the perturbation family A is implicitly understood to be

the set A: = {((s, q) = g(s, q) g(s, q) q e Q} for this class of uncertainties.

The value set V9(w) at a given frequency w is defined as the set of the Nyquist-

plane points g(jw,q) obtained for all q e Q. Let 89(co) represent the boundary of the

uncertainty value set '(co) and E(Q) represent the 2P-'p edges of the bounding set Q.

Furthermore, let g(jw, E(Q)) represent the frame of the value set, namely, the image of

the edges of Q on the Nyquist plane under the mapping g(jw,q). Two important

properties of the value sets generated by systems with affine uncertainty are the following

(Fu, 1990): (i) at each fixed frequency the boundary 8V(co) of the uncertainty value set

V(co) is spanned by the image of the edges of Q, e.g., 89(co) is spanned by the frame of

the value set; (ii) the image of each edge of Q is either a circular arc or a line segment that

can be easily calculated analytically. The second property is a consequence of the affine

structure of the uncertainty which induces a linear fractional mapping. In the following

sections we exploit these properties to develop a computational approach to find the

Nyquist stability margin for affine uncertain systems. The results allow the efficient

verification of the set membership (5.7) invoked in Theorem 5.1 via a linear feasibility

problem, and permit the calculation of the robust stability margin invoked in Theorem 5.2

via a systematic algorithm.

5.4. Robust Stability and Uncertainty Value-Set Membership

The first step in the computation of the generalized critical perturbation radius for

the uncertain system (5.9a)-(5.9b) is to determine whether the critical point -1 +jO








belongs to the uncertainty value set V(m)). The more general problem of determining if

an arbitrary point w e C belongs to V(w) is solved in this section, and the results are

then utilized to reformulate Theorem 5.1 in terms of computable quantities.

The affine uncertain system (5.9) can be written in the vector-matrix form
no0 n10 n20 ... npO q

's s'' s']
S'noI n,, n21 pl p q2 C 1
S5 +1 "
/Lo, Lnj nO..! 2, .. npt l.,J qj S.T~no+NPq),.


Idoo d- d20 ... d, -q, s (do + D, q) '

m-oi d c 0 d,, d2,, .. d,, q2
SS "... S S +

do,- d,. d2m "" d,P qp

where s, and sd are vectors of lengths + and m+1, containing powers of the Laplace


variable s, and where no e R'', do e R"', N, e R(t+')xp and D, eR('+I)"+ are constant

vectors and matrices that represent the structure of the affine parametric uncertainty. The

value set at frequency c is obtained by evaluating (5.10) at s = jc for all q eQ to yield


+ (no.R + NRq)+ js,(no,, + N,,q)
g(jm,q)= -,R( + De.qe, + DQ, (5.11)
SdR (do,a + D,,q) + jSA, (d + Dp,,q)


where


--02 W4


-a3 05

-0)2 CO4
-Oy2 4C


-3 s5
-0) 0)


...]eRt/2+1








..] Rr(mt+)/21
... eRr-,,,


\'


IV)


ST =[1


= =[c

s,R = [1

ST =[W
Sd, =m


=


=r I









noR = N2 ER F12+1



Sn"'2 0 n2 o
P R = n12 An22 p '0



0,/ No3 o R ((T't)r2
no,' f ERrrf'ifI)V



S"021 n- "F

L E J


doR = J:J21 e Rrm/2+


d'O d20 ... d
D,.x d12 d do









di d2, d
D d P




23 d23 d- 3 eR(r(m )2)xp


Where f67 represents the reatestinteg
greatest-integer function..




Full Text
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS iv
ABSTRACT viii
CHAPTER
1 INTRODUCTION 1
1.1. Motivation 1
1.2. Objective and Structure of Dissertation 3
2 A DUALITY PROOF FOR THE MAJOR PRINCIPAL DIRECTION ALIGNMENT
PRINCIPLE 6
2.1. Introduction 6
2.2. Mathematical Background 8
2.2.1. The Singular Value Decomposition and Eigenvalue Decomposition 8
2.2.2. Dual Norms and Dual Vectors 10
2.2.3. Dual Eigenvector Result 11
2.2.4. Eigenvector-Singular Vector Equivalence Result 12
2.3. Statement of the Major Principal Direction Alignment Property 13
2.4. Modified Statement of the Major Principal Direction Alignment Principle 14
2.5. Examples 17
2.5.1. Example 2.1 17
2.5.2. Example 2.2 21
2.6. Conclusions 24
3 MAJOR PRINCIPAL DIRECTION ALIGNMENT WHEN THE MAXIMUM
SINGULAR VALUE IS REPEATED AND ITS RELATIONSHIP TO OPTIMAL
SIMILARITY SCALING 25
3.1. Introduction 25
3.2. Mathematical Background 27
3.2.1. The Singular Value Decomposition 27
3.2.2. Statement of the Major Principal Direction Alignment Principle 29
3.2.3. Affine Sets, Convex Sets, and Convex Functions 29
3.2.4. Differential Theory 32
3.2.5. Expression for the gradient when the maximum singular value is distinct. ...36
v


107
Furthermore, since g0(jco) -0.4140-70.6277 at this frequency, it follows that £n()
= 0.3475 / |l 0.4140 0.6277y| = 0.4047 < 1, a result that is consistent with the
conclusion reached by invoking Theorem 5.4 that the value set does not include the
critical point at this frequency.
10 10 10 10 10
co
Figure 5.7. Critical perturbation radius pc(co) for the system considered
in Example 5.6.2. An observed discontinuity is shown with a dashed line;
a second discontinuity is not visible in the plot due to the tight scale of the
ordinate.
Figures 5.7 and 5.8 respectively show the values of pc(co) and kn(co) calculated
for a sequence of 250 frequency points equally spaced in a logarithmic scale in the range
[0.001, 10]. From Figure 5.8 it is readily concluded that &N( feedback loop is robustly stable. Also notice from Figures 5.7 and 5.8 that pc{co) and
kN(co) are discontinuous at frequencies approximately equal to co = 0.020 and
co 0.9417 The presence of such discontinuities is not surprising since it is well known
that other stability margins (such as the real-^ robust stability margin) have also shown
discontinuities. The observed discontinuities can be explained by examining Figure 5.6,


21
both necessarily eigenvectors of A. Now, assume that is possible to choose u2 such that
x2 and y2 are aligned. Necessarily, y2 is also an eigenvector of A which implies that
there are two linearly independent eigenvectors, y and y2, associated with the
eigenvalue of maximum modulus. Hence, the eigenvalues geometric multiplicity is
greater than one. Furthermore, for this case the eigenvalues algebraic multiplicity is one,
but an eigenvalues geometric multiplicity can not exceed its algebraic multiplicity. This
is an obvious contradiction, therefore the assumption is false.
The result that it is not possible to achieve alignment of all the major input and
major output principal directions is not compatible with the original statement of the
MPDA property as given in Section 2.3. However, it is compatible with the revised
version of the MPDA property of Section 2.4, which allows for a major input/output
principal direction pair that is not aligned as long as at least one other input/output
principal direction pair that is aligned as is the case in this example.
2.5.2. Example 2.2
Consider the matrix
A =
1.7907 0.8729/
0.0827 0.0396/
0.1225 + 0.0888/
-0.0780 + 0.0482/
1.6645 -1.1040/
-0.0258 + 0.0399/
0.0085 + 0.1511/
0.0475-0.0001/
1.6883 -1.0605/
with eigenvectors
"0.8554-0.0000/"
"-0.0224-0.4144/"
"0.0187 + 0.1611/T
V1>V2>V3} = <
0.0145 0.2681/
5
0.0631 + 0.4205/
9
-0.5177 + 0.7269/
0.4002-0.1901/
0.5880 +0.5489/_
0.3707-0.1999/ _
and associated eigenvalues
{1,A2,A3} = {2.0000e-4000J,2.0000e'O6OOOj ,2.0000e 60007}


76
(lvilhl+lv2lKI+---+klKI-1)KIKI'-'KI
So, for the case when no |w(.| = 0 the determinant is identically zero and the null space is
non-trivial only when part iii-a) of necessary condition (4.19)
SKI h|=i (4.i9)
i=i
is satisfied. Q.E.D.
4.4. Example 4.1
The following example demonstrates the result of Theorem 4.2. Consider the
matrix
-0.5259 + 0.6358./
0.4712+ 0.1832y
-0.0290-0.1034y
0.0925 0.2759j
0.3090-1.3791y
0.7383-0.5966y
-0.7906 + 2.0522j
0.5359 + 0.7832j
0.2031 + 0.231 Ij
-0.3174 -0.1128y
0.499 l-0.6463y
0.2490 + 0.083 \j
-0.1016 + 0.9524j
-0.2840-0.2127j
-0.0584 + 0.1540y
0.0694-0.1919y
where the eigenvalue of maximum modulus is /1(A) = 1.6507 -1.1293y such that
|T(A)| = p( A) = 2 Performing the minimization on the right-hand side of (4.15) gives
min cr(D-1AD) 2
DeB
with
D:= arg min Del)
1
0
0
0
0
0.5
0
0
0
0
1.2
0
0
0
0
0.7
such that equation (4.15) holds. Therefore, the three necessary conditions of Theorem
4.2 must be satisfied. First, the right and left eigenvectors of A associated with the
eigenvalue A(A) = 1.6507 -1.1293j of maximum modulus respectively are


80
concerned with the application of the generalized theory to the case of affine uncertain
rational systems with real polytopic parametric uncertainties. Section 5.3 introduces a
precise definition of the uncertain system considered, and Section 5.4 derives two robust-
stability theorems for these types of systems. Section 5.5 presents a systematic method
for calculating the critical perturbation radius, and Section 5.6 provides two examples of
the analysis method, including the case of a convex and the case of a nonconvex critical
value set. Overall conclusions are given in Section 5.7.
5.2. Generalization of the Critical Direction Theory
5.2.1. Preliminaries
Consider the single-input single-output linear time invariant system
g(s) = go(s) + S(s) (5-1)
where g0(s) is a known nominal transfer function, and A(s)eA is an unknown
perturbation belonging to a known perturbation family A The focus of this analysis is
on the robust stability of the closed-loop system that results when the uncertain system
(5.1) is configured in the unity negative feedback control structure shown in Figure 5.1.
Figure 5.1. Unity feedback control scheme for an uncertain plant g(s).
The following standard assumptions are made throughout this chapter:
(Al) The nominal transfer function g0(s) is stable under unity negative
feedback.


148
PM = 38.61 degrees, which agrees with the previously calculations.
Finally, using (7.17) the parametric stability margin is = 0.2395. This
represents the smallest additive perturbation in the nominal parameters that will
destabilize the system. In fact, the perturbed system given by K = (1 + pKt6)K0 = 1.2395,
r = (1 PKt9)t0 = 0.7605, and 6 = (1 + pKzg)0Q = 0.6198, is critically stable. This is
represented in Figure 7.4, as the intersection of a x =0.7605, ccg = 1.2395, and
f(aT,ad) = 01.2395. The utility of the parametric stability margin is now apparent.
Consider the gain margin, GM = 1.816, and phase margin, PM = 38.61 degrees, both are
within the recommend range for a well-tuned controller (i.e., 1.7 < GM < 2.0 and
30 Pkto =0-2395, reveals that in fact only a relatively small additive perturbation of the
three process parameters results in an unstable closed-loop. Therefore, this example
demonstrates the observation suggested earlier about the gain and phase margin. Namely,
that a good gain margin and a good phase margin are necessary for robustness, but are
not sufficient. To ensure robust stability, even for this relative simple system of a PI
controller acting on a first-order-plus-time-delay system, it is necessary to consider a
margin that considers uncertainties in all the process parameters, such as the parametric
stability margin.
7.4.2. Stability Margins Computation for Each Tuning Rule
The results of the previous section are specialized to only one tuning rule and for a
specific value of the tuning-parameter ratio, namely the ITAE regulation tuning rule and
0q/ t0 = 0.5. These results can be repeated for other tuning rules or other values of


5
to a known rectangular polytope. The generalized critical direction theory is applied to
this particular class of uncertain systems, and is used to calculate the required Nyquist
robust stability margin with high precision and in the context of a computationally
manageable framework. Finally, Chapter 6 proposes a practical design approach based
on parameter space methods (Siljak, 1989) to illustrate the utility of the Nyquist robust
stability margin as a measure of robust stability.
The final goal of this dissertation is to perform a complete robust stability analysis
of classical proportional-integral (PI) controller design techniques based on approximate
first-order-plus-time-delay models. The uncertain parameters for this problem are the
plant gain, plant time constant, and plant time delay. The region of all stabilizing
parameter perturbations is determined. By modeling the uncertainties as multiplicative
perturbations it is shown that the stability properties of the closed-loop system are only
dependent on the time-delay-to-time-constant controller tuning parameter. The results
include plots of the classical gain and phase margin and the parametric stability margin as
a function of the controller tuning parameter for the PI controller design methods
investigated.


116
performance for a desired level of stability robustness. The advantage of this approach is
that the performance measure does not need to be calculated for all values of the
controller parameters.
An alternative approach to robust controller design is to perform a direct
parametric optimization of the performance objective with robust stability conditions
applied as constraints. Since in general this optimization problem is non-convex in the
objective function and the constraints are non-convex and not connected over the
controller parameters the solution will also required an exhaustive search of the parameter
space (Luenberger, 1984).
6.3. Design Example
Consider the uncertain system
P(s,q) =
s~ + (4 + 0.4g, + 0.2q2)s + (20 + q^ q2)
(s,q)
(6.1)
where
d(s, q) = s4 + (9.5 + 0.5<7, 0.5q2 + 0.5<73 )s3
+(27 + 2q^ + q2Js~ +(22.5 q¡ +^3)5 + 0.1
and
(q},q2, System (6.1) is a modified version of the model investigated by Fu (1990). Note that the
transfer function coefficients depend affinely on the parameters q e Q. Using the
proposed synthesis technique, a robust stabilizing PI controller will be designed. A unity
feedback structure is obtained by defining g(s,q) = c(s)/?(s,q). The first step is to
determine the nominal and robust stability regions in terms of the controller parameter


30
In general, an affine set has to contain, along with any two different points, the entire line
through those points. The intuitive picture is that of an endless uncurved structure, like a
line or a plane in space. The subspaces of Rn are the affine sets which contain the
origin. The dimension of a non-empty affine set is defined as the dimension of the
subspace parallel to it (the dimension of the empty set is -1 by convention). Affine sets
of dimension 0, 1 and 2 are called points, lines, and planes, respectively. An (n-1)-
dimensional affine set in Rn is called a hyperplane. Hyperplanes and other affine sets
may be represented by linear functions and linear equations. For example, given ft g R
and a non-zero b g Rn, the set
H = {x|xTb = /?} (3.8)
is a hyperplane in Rn. Moreover, every hyperplane may be represented in this way, with
P and b unique up to a common non-zero multiple. For any non-zero b g Rn and any
P g R, the sets
/?}
are called closed half-spaces. The sets
{x|xTb P)
are called open half-spaces. These half-spaces depend only on the hyperplane H given
by (3.8). Therefore, one may speak unambiguously of the open and closed half-spaces
corresponding to a given hyperplane. Finally, the intersection of an arbitrary collection
of affine sets is again affine. Therefore, given any S a Rn there exists a unique smallest
affine set containing S This set is called the affine hull of S and is denoted affS .
A subset C of Rn is said to be convex if


169
co,
co.
co] cos(i1)-y sin^j) cox cos(cox) y sin(cox)
(E.3a)
co,
CO.
ycostco^ + co^ sin^) y cos(cox) + cox sin(ux)
(E.3b)
So, to prove there is no solution cox and cox to (E.la) and (E.lb) it must be shown that
there is no solution to (E.3a) and (E.3b) for all y > 0. To prove by contradiction, assume
that there exists an cox> 0, cox > cox, and y > 0 such that (E.3a) and (E.3b) hold. Since
co, and cox are considered to be in the intervals given by Theorem 7.2, the denominators
of (E.3a) and (E.3b) are nonzero such that equations (E.3a) and (E.3b) become
fK(cox,cox,y):= coxcox cos(cox)-ycox sin{cox)-coxcox cos(cox) + ycox sin(t>,) = 0(E.4a)
fT(col,cox,y): = yco\ cos(cox) + co]cox sin^) yco\ cos(U!) coxco\ sin(t>,) = 0 (E.4b)
Now, equations (E.4a) and (E.4b) hold simultaneously if and only if
The function fKr(cox,cox,y) is obviously a binomial in y of the form
fKr(>icXr) = fyo>l,cox)y2+f/((ox,cox)y + fr0(col,)x) (E.5)
where the functions f^(cox,cox), f^(cox,cox), and f^{cox,cox) are simply products of
powers of cox and cox and sins and coss of cox and cox. For there to be a positive real
solution y to (E.5) the discriminant of must be non-negative. The
discriminant is
fdiScMxY= ffaxV.)-fya^cox)fy0{cox,cox)
and after much algebraic and trigonometric manipulation fdiscr(cox,cox) is given by


164
)dl,)d2,--- are the frequencies at which the sign of aK changes. Starting at co = 0+,
T 0
aK is positive when 1 represents a frequency at which aK changes between positive and negative infinity.
Therefore, when 1 < < oo, aK is positive for
Ti to
(o e^,:=(0,^1)u(ffl2)fflj3)u(ffl4,rf5)u-
For ag <1, aK is negative at co = 0+ and again the sign changes over at each point
*/ r0
codi as the frequency increases. Therefore, when ag < 1, aK is positive for
TI T0
co eClKb.= (cod],cod2)(cod:},cod4)yj(cod5,cod6)YJ-
These frequency ranges for which a K is positive exclude the end points codi, because at
these frequencies the denominator of (7.12a) is zero and aK becomes discontinuous.
To determine the frequency ranges for which a r is positive first consider
t 6
cos{agco) + cosin(agco)
Y tJoA
lim aT = lim ,
0+ co-*0+ I
\TI T0
AToJ
\ToJ
Tn 0,
cocos(agco) sm(agco)
To
01
= oo
CO
where the sign of the denominator 0* is determined from
cocos(agco) cos(agco)-coagsin(agco) 1
en 7= 5?* T^J0 7 7 = ~V~9
s\n(agco)
T, To
TI T0
ag cos(agco)
0
*1 *0
a,
so that


26
optimization technique. However, as (3.1) implies, in general this yields only an upper
bound on ju.
The MPDA theory shows that if the maximum singular value is distinct for a
given D, then there is an analytic expressions for the gradient 5ct(DMD"')/ 5D. From
this expression for the gradient, the condition for a stationary point (i.e.,
dcr(DMD_1) / 3D = 0) implies that the moduli of the input and output principal
directions are elementwise equal. Therefore, if at the infimum the maximum singular
value is distinct, then the gradient exists and is identically zero, and the moduli of the
input and output principal directions are pairwise equal. In addition, a unitary
transformation matrix U (note the maximum singular value is invariant under unitary
transformations) can be determined that shifts the angles of the elements of the input or
output principal direction such that MPDA is achieved, and therefore the upper bound is
tight and the value of /j is determined by solving a convex optimization problem.
In general the maximum singular value is not unique for a given scaling D. This
work investigates further the situations that arise when the maximum singular value is
repeated. There are two aspects of this problem that are investigated. The first aspect is
the effect the repeated maximum singular value has on the optimization over D, with
specific interest on gradient search methods. The second aspect is the attainability of
MPDA when the maximum singular value is repeated for the optimal scaling. Finally,
this work attempts to reconcile the results obtained with those of the principal direction
alignment (PDA) principle (Daniel et al., 1986).


173
Let codx, co*d2, be the positive solutions to (F.lb), arranged in increasing order. The
7T 3
tangent curve is positive only when co* e(0)u(;r,Also, in each of these
ranges the tangent curve goes from 0 to co. When, 0 *0
that intersects the tangent curve is greater than 1, therefore,
n 1 3
0 T 6
If 1 < a0 < co then the slope of the line that intersects the tangent curve is less than
*/ r0
7T
1 and in the interval (0,y) there is no intersection with the tangent curve, therefore
. 3 .5
TZ<(od,<-n, 27r Therefore, if ct)nl, con2, are the positive zeros of f(a>) and codx,cod2,--- are the
positive zeros of fd(co) arranged in increasing order, then for 0 < ae < 1 it follows
TI To
that
0 1 7T
2 aQ
d2 <
CCa
3 n ^ n
~<>n2<2 <
2 a* a*
(F.2a)
and for 1 *1 To
1 71
2 aa
K
3 n
71
at
'd 1
2 CL,
n2
a.
'd 2
5 71
2 aa
(F.2b)
Similarly, the positive zeros of and are given by the positive
dco
dco
solutions to


APPENDIX D
PROOF OF LEMMA 7.3
Proof. First the frequencies that give aK >0 are determined, followed by the
frequencies that give ar> 0. The intersection of these two sets of frequencies is the
desired set. Starting with co = 0, the solution is aK = 0 and aT arbitrary, and therefore
co = 0 is excluded from the range of frequencies that yield aK > 0 and aT> 0. After
using LHopitals rule, the limit as co > 0+ of the solution (7.12a) is
lim a K, = lim
>-0+ (u->0+
-1
f
K0KC
0,
\
cos(a0co) coad sin(a0co) ae cos(a0co)
T, *0
-1
( T 9 ^
I _0._0
K0KC
at
V Ti To
so that when \ O T0
considered. When aa = 1
7/ r0
lim a K = lim = +oo
o+ CO-+0+K0Kc{-coa e sin(a 0co))
Tn 6,
o d0
Therefore lim aK > 0 when 1 < ae < oo, and lim aK < 0 when ag < 1. The
co-+0
*1 T0
0
*1 T0
next step is to determine the positive frequencies at which the sign of aK changes. As
discussed Appendix F., for positive frequencies the sign of the numerator of (7.12a) never
changes and the sign of the denominator of (7.12a) changes at codl, cod2, . Therefore,
163


97
(the frame) in the complex plane. The equations needed for characterizing the
intersection of the critical line with straight-line segments are trivial and are therefore
omitted for brevity. The determination of all the intersections of the critical line with a
finite number k of arcs of circle is somewhat more subtle and is therefore discussed in
greater detail.
The basic geometric objects of interest are defined as follows. A line passing
through two points p0,px eC can be represented by
LCPo A) :={zeC|z = /?0 +t(Pi ~ Po) > ieR} (5.17)
The same relation can be used to represent a ray, r(p0, px) with origin at p0 and pointing
towards pl by restricting the parameter t to adopt only non-negative values. The line
L(PoPi) is said to be the supporting line for r(p0,px). The critical direction r{co) is
therefore represented by the ray r(p0, px) with p0 = g0(jco) and px =-l + j0.
Consider a set k of circles with center at points z¡ e C and radii ri gR ,
i = 1,2, where each circle satisfies the relation
C, := { ZC| (z-z,.)(z-z,.) = i;2> r,>0} (5.18)
Therefore, two parameters are sufficient to define each circle. On the other hand, three
parameters are required to describe an oriented circular arc that passes through three
points a0,a{,a2 eC, in that order, and such arc will be denoted implicitly as
a ¡(a0, al,a2) if it belongs to a supporting circle C,.
The relative position of a set of points and the orientation of arcs and rays in the
plane can easily be determined invoking the cross product operation. Let p0, px eC, then
the cross product of p0 and px can be defined as


15
y(A) = ej0x(A) (2.4)
Pre-multiplication of equation (2.4) by A gives
Ay(A) = eJ0Ax(A) (2.5)
The singular value decomposition of A implies
Ay(A) = Combining equation (2.5) and equation (2.6) gives
Ax(A) = e^a^A^A)
so that A = e~J0a(A) emerges as an eigenvalue of A with eigenvector x(A). Noting
that the eigenvalues of A are always bounded from above by |A| = p(A) = 5(A)
To prove necessity, assume p{ A) = ct(A), then from Lemma 2.2 it follows that any right
eigenvector v(. of A associated with an eigenvalue of maximum modulus A¡(A) is also
a right singular vector y(. of A associated with the maximum singular value cr(A).
From equation (2.2), the corresponding left singular vectors are
X|(A) = ^)
a(A)
Av,(A)
H a)I
^,(A)v,.(A)
K-(a)|
= eyarga'(A))y.(A)
= ej0yi(A)


67
Let D0 be the minimizing D of (4.1), then
*(P) = ^(D'PDo) = lub(D0-PD0) (4.3)
From the result of Bauer (1962), (4.3) is only possible if the right and left eigenvectors of
D¡'PD0 are dual. Now, if v > 0 and w > 0 are the right and left Perron vectors of P
(note that it is implied that greater than operator > is an element wise operation when
applied to a vector), then it is straightforward to show that D'v > 0 and Dw > 0 are the
right and left Perron vectors of D~PD. Therefore, any D0 that minimizes (4.1) such that
equality is achieved must also make the vectors Dv and D0w dual, where v > 0 and
w > 0 are the right and left Perron vectors of P .
The problem now reduces to transforming the positive vectors v > 0 and w > 0
to dual vectors D¡v and D0w where D0 is a non-negative diagonal matrix. Stoer and
Witzgall (1962) state that for absolute norms (i.e., norms that only depend on the moduli
of their components (Bauer et al., 1961)) there exists one, and up to positive multiples
only one, non-singular non-negative diagonal matrix D0 such that D'v and D0w forma
dual pair. For a p-norms which are necessarily absolute norms, the positive vectors y > 0
and x > 0 are dual if
(y,r=(x,y, i=i,2,---,
i+i=i
p q
and


52
subgradient in the Euclidean norm. From Theorem 3.6, all subgradients are given by the
convex hull of V/u(d;u) for iTu = 1, which means every subgradient can be expressed
as the linear conbination TV/u(d;u1) + (l-A)V/u(d;u2) with u,*u2, A = [0,1],
u*u, = 1 and u2u2=l. Therefore the optimization problem given by (3.12) to
determine the subgradient used to obtain the steepest descent direction can be written in
the form
£sd(d) = argmin||AV/u(d;u1) + (l-T)V/u(d;u2)||
A,Uj,u2
(3.37a)
with the constraints
u, ^u2, A = [0,1], 11*11, =1 and u2u2 =1 (3.37b)
Unfortunately, the objective function of optimization (3.37) is non-convex in the
components of the complex vectors u, and u2, and therefore has all of the associated
difficulties, like local versus global minimums. In addition, from Theorem 3.4 the
necessary and sufficient condition for a minimum is given by £sd (d) = 0.
3.5. Attainability of MPDA when the maximum singular value is repeated
When the maximum singular value is distinct, the necessary condition for a
infimum of (3.13) is V/(d) = 0 where the gradient is given by (3.16). This implies that
the moduli of the major input and major output principal directions are elementwise
equal. Furthermore, a unitary transformation matrix U can be determined that shifts the
angles of the elements of the input and output principal directions such that MPDA is
achieved and the upper bound for /u is non-conservative. In general MPDA is not
possible when the maximum singular value is repeated and the upper bound on // given
by is conservative. Therefore, the goal of this section is to determine the sufficient


17
the pair. By definition, the pair are approximately dual with respect to the Euclidean
norm if and only if
* = e'ey1 IMIS
Principal directions are always normalized; therefore (2.9) is equivalent to
x = ejey
which is exactly the condition for alignment completing the proof.
2.5. Examples
(2.9)
Q.E.D.
2.5.1. Example 2.1
Consider the matrix
A =
-0.9026-1.0077/
0.6086 + 0.2053/
0.6487 + 0.2968/
0.2586-0.1506/
1.2588 -1.1670/
-0.5918-0.4665/
0.1661 + 0.2372/
-0.6442 + 0.2239/
0.1641-1.4383/
with eigenvectors
-0.0687 + 0.1159/
-0.8719 + 0.2183/
0.3920 + 0.1421/
0.1807 + 0.1816/
0.3478 0.2425/
0.8670 + 0.0538/
0.6834-0.0523/
-0.3628 0.1105/
-0.4576 0.4208/
and associated eigenvalues
{A,, A 2, A 3} = {2.0000e 0 60007,1.2503e 666oy ,1.5996e'2 25557}
and singular value decomposition A = XZY*, where
X = [x, x2 x3] =
-0.0018 + 0.0876/
0.2741 + 0.8117/
0.2903-0.4173/
0.4121 + 0.3962/
-0.4542 + 0.0343/
-0.4806-0.4845/
0.4426 + 0.6853/
0.2421-0.0061/
0.2491 + 0.4624/
Y = [y, y2 y3]
0.1556 -0.7481
-0.2965 + 0.8731/ -0.0272 0.1299/
0.3367-0.1101/ 0.5404-0.3615/
-0.6451
-0.0400 + 0.3612/
-0.5455 + 0.3927/
(2.10)


102
In order to quantify the degree of robust stability at this frequency, it is useful to
calculate the value of the Nyquist robust stability margin. It is possible to calculate pc(co)
following the procedure discussed in Section 5.5. The first step consists of finding the set
of points F= { P¡, i = 1, 2,..., k} that define all the intersections of the critical line r(co)
with the frame g(jco,E(Q)). It follows that F = {-0.5185 -j 0.9523, -0.5494-j 0.8913,
-0.5660 j 0.8584}.
Re
Figure 5.3. Frame of the uncertainty value set for the system of
Example 5.6.1 at the frequency co = 0.7 The critical point -1+/0 and the
nominal point g0(ja>) are represented by the "x" markers, the three
intersections of the arcs with the critical line are represented by the "+"
markers, and the intersection that defines the boundary point used in the
calculation of pc(co) is represented by the marker. The critical value
set T'(y) is convex at this frequency, and it is represented by a single
straight-line segment.


13
= A,(A*A)v, (2.3)
Hence, from (2.3) if follows that vf is an eigenvector of A*A with eigenvalue A,. (A* A).
Finally, ^A¡(A*A) = ^|A(.(A)|~ = ^/or2(A) = 2.3. Statement of the Major Principal Direction Alignment Property
In solving various robust control problems it is necessary to determine the
conditions under which the spectral radius of a matrix attains its maximum singular-value
upper bound. The major principal direction alignment (MPDA) property addresses this
problem. Consider the singular value decomposition of a matrix A given by (2.1) where
£(A) is the diagonal matrix of singular values organized in descending order, and X(A)
and Y*(A) are unitary matrices whose columns are the respective output and input
principal directions of A, arranged in an order conformal with the order of the singular
values. The major input principal direction y(A) and major output principal direction
x(A) of a matrix A are defined as input and output principal directions respectively,
corresponding to the maximum singular value, principal-direction y(A) and the major output principal-direction x(A) are said to be
aligned if the exists a real scalar 0 e R such that y(A) = eJ0x(A) The following
statement of the Major Principal Direction Alignment (MPDA) property is found in
Kouvaritakis and Latchman (1985).
Theorem 2.1. The spectral radius of any matrix A eCnxn is equal to the
maximum singular value of A if and only if the major input and output principal
directions of A are aligned.


6 ROBUST CONTROLLER SYNTHESIS FOR SYSTEMS WITH NONCONVEX
VALUE SETS USING AN EXTENSION OF THE NYQUIST ROBUST STABILITY
MARGIN 110
6.1. Introduction 110
6.2. Design Methodology 112
6.3. Design Example 116
6.4. Conclusion 121
7 ROBUSTNESS OF CLASSICAL PROPORTIONAL-INTEGRAL CONTROLLER
DESIGN METHODS 122
7.1. Introduction 122
7.2. Preliminaries 125
7.2.1. Process Model and Uncertainty Description 126
7.2.2. Proportional-Integral Control and Controller Tuning Rules 127
7.3. Analysis of Robust Stability 129
7.3.1. Conditions for Robust Stability 129
7.3.2. Parametric Boundaries for Robust Stability 133
7.3.4.Stability Margins 139
7.4. Results of Numerical Studies 143
7.4.1. Region of Stable Perturbations for the ITAE Regulation Tuning Rule 143
7.4.2. Stability Margins Computation for Each Tuning Rule 148
7.5. Conclusions 153
8 CONCLUSIONS AND FUTURE WORK 154
APPENDIX
A PROOF OF LEMMA 2.1 156
B PROOF OF THEOREM 2.1 158
C PROOF OF THEOREM 7.1 161
D PROOF OF LEMMA 7.3 163
E PROOF OF THEOREM 7.2 167
F SIGN CHANGES IN EQUATIONS (7.12A) AND (7.12B) 171
LIST OF REFERENCES 176
BIOGRAPHICAL SKETCH 180
Vll


56
" 0.1106-0.1679/ '
'0.5229+ 0.0786/T
-0.1893-0.4442/
0.0843 + 0.5385/
0.3526-0.6134/
9
0.1979-0.1544/
-0.2971 + 0.1422/
0.0567 + 0.5552/
[-0.0002 + 0.3428/_
-0.2160-0.0469/J
and an orthonormal set of left singular vectors corresponding to the repeated maximum
singular value is
{yi>y2}
0.0000 + 0.0000
0.2151 + 0.1955
< -0.0322+ 0.3960
-0.0690-0.2280
0.3825-0.7447
0.6051+ 0.0000
0.3141 -0.0597
-0.1384-0.6067 >
-0.1529 + 0.2204
0.1730-0.2060
Using these sets of left and right singular vectors and equations (3.21)-(3.24) gives
0.0404
0.0893
-0.1930
-0.0865
0.1486
-0.6221
-0.0195
0.1949
0.3427
0.8006
0.0148
-0.3243
0.0517
0.2036
0.2458
0.2395
-0.5834
-0.4713
-0.0481
-0.0235
From Theorem 3.8 the ellipsoidal
0/(d) = {zei?n Pz = q,
where the elements of the matrix
characterization of the subdifferential is given by
(IX z2 z3]-ct)B([z, z2 z3]t c) < 1}
P =
1.2180
-0.2180
0.6214 0.0927 1.0000 0.0000
0.3786 0.9073 0.0000 1.0000
and the vector
0.2251
-0.2251
are obtained from (3.27), the matrix


142
crossover frequency. Letting (7.12) when q = [1,1, pd(90 / r0)]T then the standard gain-crossover frequency is given
by cog a>g(0o / t0) / 0O. Note that the dimensionless gain-crossover frequency
cog(90 / t0) is only dependent on the tuning ratio 0O / r0 and the tuning rule. Whereas
the standard gain-crossover frequency is dependent on the tuning ratio 90 / r0, 90 and the
tuning rule. The standard phase margin is determined as follows. First, the only
difference in open loop response of the time-delay perturbed system and the open-loop
response of the nominal system occurs in the exponential term. At the gain-crossover
frequency the open loop response of the time-delay perturbed system is -1 + j0. By
definition (Ogata, 1990), the standard phase margin PM is the phase shift that will rotate
the open-loop response of the nominal system to -1 + jO. Therefore at the gain-
crossover frequency the exponential term of the time-delay perturbed system must equal
the exponential term of the nominal system phase shifted by PM. That is
e~a0eazgj e~e0a>gje-PMj
or
-ae90a>g = -60)g PM
This gives rise to the following definition of the standard phase margin
PM(<90 / T0) = pe(0o / T0)cog(90 / r0) cog(0o / r0) (7.20)
which is in terms of the dimensionless gain-crossover frequency cog{0o / r0) to show its
sole dependency on the tuning ratio 90 / r0 and obviously the tuning rule.
Because the classical gain and phase margins consider perturbations in only one of
the parameters at a time, they may not capture the true robust stability characteristics of


39
3.3. Main Result Characterization of the Subdifferential when
the Maximum Singular Value is Repeated
3.3.1. General Expression for the Subdifferential
When the maximum singular value is repeated the major output principal
direction x(DMD ') and the major input principal direction y(DMD ') are determined
by (3.5) and (3.6). As such, the expression for the gradient given by (3.16) may not be
uniquely determined which implies the objective function may not be differentiable.
When the function is non-differentiable then the subdifferential must be determined, as
opposed to the gradient. To characterize the subdifferential define the function
V/U(d;u) = x,(DMD>,
;=i
y,(DMD '),
(3.17)
where u eCr satisfies iTu = 1 and {x,.(DMD ')} for i = l,2,*--,r is an orthonormal set
of left singular vectors and {y(DMD ')} for i = 1,2is an orthonormal set of right
singular vectors corresponding to the maximum singular value a(DMD ') of
multiplicity r Definition (3.17) represents the evaluation of the gradient function (3.16)
for possible values of x(DMD ') and y(DMD '). For different u, the function
V/U(d;u) may give different values, such that the gradient is not unique and is therefore
undefined. The subdifferential is now characterized in the following theorem.
Theorem 3.6. The subdifferential of the function /(d) = o:(DMDl) is given by
df(d) = conv{V/M(d;u)|u*u = 1} (3.18)
where V/U(d;u) is defined by (3.17).
Proof. From Theorem 3.3, the subdifferential is given by


70
problem (Kouvaritakis and Latchman, 1985). Consider the singular value decomposition
of a square matrix A e C"x" given by
A = X(A)D(A)Y*(A)
where Z(A) is the diagonal matrix of singular values placed in descending order, and
X(A) and Y*(A) are unitary matrices whose columns are the respective output and
input principal directions of A, arranged in an order conformal with the order of the
singular values (Lancaster and Tismenetsky, 1985). Now, define a major input principal
direction y(A), and a major output principal direction x(A), of a matrix A respectively
as normalized input and output principal directions, corresponding to the maximum
singular value, Theorem 4.1. The spectral radius of any matrix A eCnxn is equal to the
maximum singular value of X, if and only if there exists a major input principal
direction and a major output principal direction of A which are aligned such
that
x(A) = eJ0y( A)
Proof Given by Kouvaritakis and Latchman (1985). An alternative proof based
on dual norms and dual vectors is given in Chapter 2. Q.E.D.
4.2.4. MPDA as a Control Theory Application
One area in the field of robust control that makes use of the spectral-
radius/maximum-singular-value equivalence problem is the stability analysis of
multivariable feedback systems in the presence of structured uncertainties. Of particular
interest is the stability of diagonally perturbed systems for which the uncertainty is
represented by the complex diagonal matrix


APPENDIX E
PROOF OF THEOREM 7.2
Proof. Based on the arguments in the brief proof following the statement of the
theorem, it is sufficient to show that the curve obtained from the first frequency range Q,
of Lemma 7.3 gives the smallest value of aK for each value of ar. This is done in two
steps. First it is shown that there is at least one value of aT for which Q, gives the
smallest value of a K. Second, due to the continuity of the curves over each frequency
interval, the only way a curve of a different frequency range can give a lower value
of aK for some other value of aT is if the curve intersects the curve obtained from Q,.
It is then shown that the curves never intersect, such that a different frequency range
cannot give a lower value of a K.
., 1 n 5 n 9 n _
First, consider cox co2 = , co3 = , etc. From (F.2) it is easy to
2 at
2 a,
2 a.
show that cox eQ,, co2 e Q2, co3 e Q3, independent of the value of ae. Now from
*/ To
(7.12) these frequencies yield
af),) = aT(co2) = afeo,) ==
rT V'
^ r
\Ti J
and
aK(col) =
1
n
2 K0KA-a
TI T0
167


95
Theorem 5.4 represents a systematic method for determining the robust stability of
systems with real affine parametric uncertainties.
5.5. Computation of the Critical Perturbation Radius
The computation of the Nyquist robust stability margin &n() for the affine
uncertain system (5.9) requires that the critical perturbation radius pc(co) be calculated
first. As indicated by (5.3) and (5.4), this in turn requires determining the set <2f( Once that all the elements of pc(co) from the applicable formulas. For the case of affine-uncertain systems of the form
(5.9a), the critical boundary-intersections set <2c() can be effectively identified using a
two-step strategy.
The first step consists of finding the set of points F= {/), i = 1, 2,..., k) that
correspond to all the intersections between the critical line r(co) with the frame
g(jco,^(Q)). This reduces to a simple problem in two-dimensional computational
geometry after recognizing that the critical line is a ray and that the frame is a collection
of arcs of circles and straight-line segments. Further details are given in Section 5.6.
Note that all the points in the frame-intersection set F are elements of Fc(co) because
each point in turn belongs to r(co). It is straightforward to conclude that arguing that some elements of the frame-intersection set F may not lie on the value-set
boundary dF(co), and that all elements of (Bc(co) must be elements ofF.
The second step consists of constructing the set of critical intersections selecting from F all the points that also belong to c(). For the special case where
F = {g0(jco)} it follows that (Bc(co) = (g0(jco)}. For the more general case where
g0(jco) is not the only element of F, then (Bc(co) is constructed by selecting all the points


75
V, w, w,
NN
NW2
hr,
|v2||w2| -\w2\
hr K1F2I
V2 K
w 1 -\wJ
n /?
i2
0,1
2
0,2
l0,n
= 0
(4.23)
from which necessary condition (4.16) becomes apparent. For the null space of the
matrix given in (4.23) to be non-trivial, its determinant must be 0 (i.e. the matrix must be
rank deficient). First, note that if any |w(| = 0 then the corresponding column i is
composed of only zeros making the matrix rank deficient, resulting in part iii-b) of
necessary condition (4.19). For the case when no |w(.| = 0 for i = the
determinant can be determined using elementary row and column operations to obtain a
matrix that is sparse and has the same determinant. Multiplying column 1 by -|u>. I2/hi
and adding the result to each column i for i = 2,3,, gives the matrix
w,
w,
h ri ~r>
w, w,
V3 W1
- W,
0
0 w.
w,
0
0
KIM2 0 0 -\wn\_
Now, for / = 2,3,,, multiplying row i by |wi|/|w,| and adding the result to row 1
gives the matrix
(k hl-1+lv2|kl+
vi||w,| + + |v||wD|wI|
0
...
... 0
|V2
hf
-k|
0
... 0
|V3
k.l2
0
-N
*. 0
|V"
Ihf
0
...
1
1
0
for which the determinant is


20
where []+ denotes the Moore-Pinrose pseudo-inverse (Ortega, 1987). The second and
last part of the problem is to choose u2 such that U is unitary. One choice is
0.5548 + 0.8057/ 0.2075
0.2072 + 0.0126/ -0.6026 + 0.7705/
Now using the relationships (2.11) and (2.12) and defining X = [x, x2 x3J and
Y = [yj y2 y3j yields an alternative singular value decomposition A = X XY *
where
0.0088 + 0.1345/ -0.5540 + 0.0969/ 0.4426 + 0.6853/
-0.5964 + 0.6725/ 0.3042-0.2021/ 0.2421-0.0061/
0.4038-0.1041/ 0.7232-0.1649/ 0.2491 + 0.4624/
Y=[yi y2 y3]
-0.0687 + 0.1159/
-0.8719 + 0.2183/
0.3920 + 0.1421/
0.4831-0.5764/
0.0549 + 0.2386/
0.0228 + 0.6114/
-0.6451
-0.0400 + 0.3612/
-0.5455 + 0.3927/
and the singular values again are
{o-,,ct2,o-3} = {2,2,1}
Finally, the apparent contradiction of Theorem 2.2 is resolved by verifying
X; =^'arg(A*)y; =e--6000jy\
Note that x2 + ejdy'2 even though a2 is equal to the maximum singular value. A
reasonable question now is whether it is possible to choose u2 such that x, is also
aligned with y2 ? The answer is no. This is proved as follows. By construction y, is an
eigenvector of A corresponding to an eigenvalue of maximum modulus, namely v,.
Next, it can be shown (see, Theorem 2.1, proof of sufficiency) that alignment of major
input and major output principal directions implies that the major principal directions are


108
which shows the frame of a value set at the frequency co = 0.9500. At this frequency the
critical line intersects the boundary of the uncertainty value set at three points (one point
near the nominal point and two points closer to -1 + j0).
Figure 5.8. Nyquist robust stability margin kN(a>) for the system
considered in Example 5.6.2. Observed discontinuities are shown with a
dashed line.
The critical uncertainty value set Vc{co) is composed of two disjoint line segments: one
segment joining the nominal point with its nearest boundary point, and a second segment
joining the other two boundary points. The numerical value of pc{co)is in this case
calculated using the boundary point closest to -1 + j0. As the frequency decreases the
uncertainty value set rotates in a counterclockwise direction. This causes a progressive
reduction in the length of the line segment formed by the two boundary points that are
closest to -1 + j0. Eventually this line segment collapses into a single point in such a
way that the critical line passing through the point is locally tangent to the uncertainty
value set. When this situation arises, the critical value set Vc(co) is composed of the
tangent point just mentioned and one line segment (namely, that joining the nominal point


74
that v and w are a pair of right and left eigenvectors of the A(A) eigenvalue of A,
then D'v and D0w are corresponding right and left eigenvectors of D¡'AD0.
Therefore, the necessary condition is that D¡'v and D0w are dual with respect to the
Euclidean norm. In the mathematical background section it is stated that this is
equivalent to requiring
D'v = -^
"owi;
(4.21)
Using the notation
u = [|m, |earg(')j,\u2 |earg(2 )y', ,\un |earg("); ]T
v = ||v1|earg(v')y,|v2|earg(V2)y,---,|v|earg(v")7]
;-iT
the necessary condition (4.21) is equivalent to the set of scalar equalities
vAe
arg(v,)y _
o>i|earg(>Vl)y
O.* '
^0,l|Wl| + ^0,2 1^11 + +^0,n|VVl|
(4.22-1)
l0,2
\vAe
arg(v2)y _
do,2
^0,l|Wl| + ^0,2 1^2 | + +
(4.22-2)
0,n
arg(v)y _
d \w leargK)y
u0,n\yvn r
0,1 M + <^0,2 1^2 I +-"+hhl
(4.22-n)
which leads directly to necessary condition (4.18). Given that (4.18) is satisfied, (4.22)
can be rearranged as


137
interval. Hence, each curve divides the first quadrant of the ar-aK plane into two
regions. For ag = 1, the lower region always contains the nominal point; therefore, (7max
is the open set for which a r ranges from zero to infinity and a K ranges from zero to the
lowest value of given by (7.12) over all frequency intervals. This implies that the
robust stability-boundary dQm.lx is the curve that gives the lowest value of aK for each
value of aT. Hence, it is sufficient to show that the curve obtained from the first
frequency range Q, of Lemma 7.3 gives the smallest value of aK for each value of aT.
Further details of the proof are given in Appendix E. Q.E.D.
Theorem 7.2 defines the region of stable parameter perturbations (7max in terms of
its boundary for a fixed perturbation in time delay, ae. For the nominal value a0 = 1,
the above results give the stability boundary when there is no uncertainty in the time
delay. To characterize the effect of arbitrary perturbations in the time delay, the above
analysis can be performed over a range of values ae> 0. The result is a different
stability boundary curve for each value of ag, and hence in the aT-ag-aK space, the
stability boundary becomes a surface. As such, every point on the stability boundary
T
surface q = [a^ aT ag] must satisfy
aK = f(aT,a0) (7.15)
where /(aT,a0) depends implicitly on the tuning rule adopted from Table 7.1 because it
features the product K0KC and the ratio r0 / z,. Given a pair of boundary coordinates aT
and a0, the mapping (7.15) if found as follows. First, for notational simplicity let
aK = fK(co,a0) and aT = fT(o),ag) respectively represent equations (7.12a) and (7.12b)


32
A concave function on S is a function whose negative is convex. An affine function on
S is a function which is convex and concave. The set {(d,/(d)) eR gS} is the
graph of the of the function /(d) defined on the set S The set
epi/:= {(d,/?) 6 Rn+l d e Stp e R,/3 > /(d)}
is called the epigraph of the function /(d) defined on the set S. The epigraph of a
convex function is a convex set.
3.2.4. Differential Theory
A vector £ is said to be a subgradient of f:S a Rn > R at d e S if
/(g)>/(d) + 4T(g-d), Vg g5
(3.9)
This condition, which is referred to as the subgradient inequality, has a simple geometric
meaning: it says that the graph of the affine function h(g) = /(d) + £,T(g d) is a non
vertical supporting hyperplane to the convex set epi/ at the point (d,/(d)). The set of
all subgradients of / at d is called the subdifferential of / at d and is denoted by
df (d). The multivalued point-to-set mapping df:d - 3/(d) is called the subdifferential
of / Obviously, df (d) is a closed convex set, since by definition £ e df(d) if and only
if ^ satisfies a certain infinite system of weak linear inequalities (one for each g of
(3.9)). In addition, df (d) is also nonempty and bounded.
The directional derivative of / at d eS in the direction of g, denoted /'(d;g),
is defined by the limit
/'(d;g)= lim
7 6 A->0+
/(d +Ag)-/(d)
A


128
Some of the historically and practically most important tuning correlations are
given in Table 7.1, where the prescribed values of control gain and integral time-constant
are expressed through equations that require knowledge of the known nominal process
parameters K0, t0 and 0O of the nominal model (7.4). The tuning correlations proposed
by Ziegler and Nichols (1942) and those by Cohen and Coon (1953) shown in the table
are both based on the goal of achieving a quarter-decay ratio in the regulation response of
the loop. Lopez et al. (1967), and Rovira et al. (1969) developed the tuning rules shown
in the table that seek to minimize specific integral-error criteria of the servo or regulation
response, including the IAE: = |^|e(i)|2 dt, ISE:= e(t)2dt, and the
ITAE: = t\e(t)\dt performance measures. Note that the entries in the table depend on
the time-delay-to-time-constant ratio 0O/ r0, a fact that is exploited later in the paper.
These tuning correlations have been developed and tested for nominal models in the
range 0.1 <0O/ t0< 1.0, which represents a very wide range of practical processes of
interest. After invoking any of these tuning correlations, the parameters of the PI
controller (7.6) could be written in the form Kc = Kc(Kq,0q / Tq) and
tj = Ti(tq,0q / tq) where the specific functional dependencies are given by the table
entries, and hence it follows that c(s) = c(s;qo) in a very specific sense. For simplicity
of notation, however, in the sequel the controller is denoted as c(s) and its parameters are
denoted as Kc and r¡, since their dependence on the nominal process parameters is
implicitly understood.
It is well known that in the absence of parametric uncertainty, the Ziegler-Nichols
and the Cohen-Coon methods tend to yield aggressive and oscillatory responses, a


159
w*A* = e JV Multiplying equation (B.3) on the right by equation (B.2), and introducing the singular
value decomposition of A, we derive
wA'Aw w*Y(A)Z2(A)Y\A)w _2/ ^
* *; (7 (A)
ww w Y (A)Y(A)w
which for z = Y*(A)w implies
z z
If we assume, without loss of generality that w is normalized then this last equality can
only be attained for z = eJ0ex, where ex is the first standard basis vector, with a 1 in the
first postion and 0s everywhere else. Thus w = Y(A)z = Y(A)e10ex = ejeyA which is
next substituted in equation (B.2) to give
Aej0yA =a(A)eJ0xA = ej'l/a(A)eJ0yA
or
*a =eJy/yA
and this completes the proof.
(ii) Repeated Maximum Singular Value:
A simple modification of the above arguments, applied to the subspace spanned
by the principal directions associated with the repeated singular value, caters for the
general case. Let the maximum singular value be repeated with multiplicity q, and let
x¡, y¡, 1 repeated singular values cr = a(A). Then the major principal directions of A will no
longer be unique, but will be given by


100
5.7. Examples
Consider the system with affine uncertainty structure
g(s,q)
(0.3s3 + 2.2s1 + 10j + 20) + (0.12.?2 + 0.7s + 1 )qx + (0.06s2 + 0.2s)q2 + (-0.35 \)q,
(s4 + 9.5s3 + 27s2 + 22.5s + 0.1) + (0.5s3 + 2s2 s)qx + (-0.5s3 + s2 )q2 + (0.5s3 + s)q3
(5.19)
where the parameter vector q-[qxq2q3]T is an element of a rectangular polytope
QaR\ and where the nominal system g0(s) = g(s,q0) is obtained with qJ =[000].
System (5.19) is a modified version of the model investigated by Fu (1990). The
objective is to analyze the robustness of the unity-feedback structure shown in Figure 5.1.
Two examples are given, namely, a case where the critical uncertainty value set is convex
and one where it is nonconvex.
5.7.1. Example 5.1 Convex Critical Value Set
Consider the system (5.19) where the parametric uncertainty vector belongs to the
square polytope
Q={qeR3 -3< qt < 3,i = 1,2,3}
(5.20)
Figure 5.3 shows the frame g(ja>,E(Q)) of the uncertainty value set at the frequency
(o- 0.7 At this frequency it is readily concluded by inspection that the critical value set
Vc(co) is convex, since it consists of a single line segment. Note that, in contrast, the
entire value set V{co) is nonconvex.
In order to apply Theorem 5.4, consider the frequency co = 0.7 and define the
following elements:
sTnR =[1.000 -0.4900], si, = [0.7000 -0.3430] (5.21b)
sj* =[1.0000 -0.4900 0.2401],


4
results are also used to determine the upper bound on / given by the minimization over
a positive diagonal similarity scaling of the maximum singular value. When the
maximum singular value is distinct there exists an analytical expression for the gradient
of the objective function. The first order necessary condition for a minimum (i.e., the
gradient being indentically 0) is equivalent to MPDA; therefore the minimum is a tight
upper bound. Chapter 3 investigates this optimization problem when the maximum
singular value is repeated such that the gradient does not exist and the objective function
is non-differentiable. One result is a method for determining the subdifferential when the
maximum singular value is repeated where the subdiffemtial represents the set of all sub
gradients. The necessary condition for a minimum is that zero is an element of the
subdifferential. Furthermore, it is shown that MPDA is still achievable when zero is on
the boundary of the subdifferential; otherwise, MPDA is not attainable and the upper
bound on ju is conservative. Finally, Chapter 4 gives a necessary condition for the
optimal similarity scaling. The necessary condition requires that the vector of diagonal
elements of the similarity scaling be an element of the null space of a matrix formed from
the absolute values of the elements of the left and right eigenvectors of the matrix.
The second goal of this dissertation is the extension of the critical direction theory
to the more general case where the critical value-set is nonconvex. This work is
presented in Chapter 5. The key to extending the theory is the introduction of a
generalized definition of the critical perturbation radius in a fashion that preserves all
previous results. The nonconvexity of the critical value set is observed in a number of
interesting problems, including the case studied by Fu (1990) consisting of rational
systems where the uncertainty appears affinely in the form of real parameters that belong


3.3. Main Result Characterization of the Subdifferential when the Maximum
Singular Value is Repeated 39
3.3.1. General Expression for the Subdifferential 39
3.3.2. Characterization of the Subdifferential as an Ellipsoid 42
3.4. Determining the Steepest Descent Direction and Conditions for a Minimum 50
3.5. Attainability of MPDA when the maximum singular value is repeated 52
3.6. Reconciling the Results with the PDA Results 54
3.7. Examples 54
3.7.1. Example 3.1 55
3.7.2. Example 3.2 58
3.7.3. Example 3.3 59
3.8. Conclusions 61
4 SPECTRAL RADIUS MAXIMUM SINGULAR VALUE EQUIVALENCE
UNDER OPTIMAL SIMILARITY SCALING 62
4.1. Introduction 62
4.2. Mathematical Background 64
4.2.1. Dual Norms and Dual Vectors 64
4.2.2. Positive Matrix Result 65
4.2.3. Major Principal Direction Alignment Property 69
4.2.4. MPDA as a Control Theory Application 70
4.3. Main Result Extension of the Positive Matrix Result to General Complex
Matrices 72
4.4. Example 4.1 76
4.5. Conclusions 77
5 GENERALIZATION OF THE NYQUIST ROBUST STABILITY MARGIN AND
ITS APPLICATION TO SYSTEMS WITH REAL AFFINE PARAMETRIC
UNCERTAINTIES 78
5.1. Introduction 78
5.2. Generalization of the Critical Direction Theory 80
5.2.1. Preliminaries 80
5.2.2. Analysis of Robust Stability 84
5.3. Systems with Affine Uncertainty Structure 88
5.4. Robust Stability and Uncertainty Value-Set Membership 89
5.5. Computation of the Critical Perturbation Radius 95
5.6. Intersection of a Ray and Arcs in the Complex Plane 96
5.7. Examples 100
5.7.1. Example 5.1 Convex Critical Value Set 100
5.7.2. Example 5.2 Nonconvex Critical Value Set 104
5.8. Conclusions 109
vi


91
n0 ,R
n
oo
02
gR
ff/2]+l
NP,R =
!10
20
" nPo
12
22
" np2
gR
(r u2\+\)*p
0 ,/ =
01
03
G R
N,I =
i. n21
13 23
n
p i
e Rr(w)>p
gR
["m/2"|+l
\o
^20
dpo
n
d22
dp2
eR
([" m/2"|+l)x p
do,i
*01
03
gR
f(m+1)/2l
and
G Rr(m+i)/2ixp
where |~] represents the greatest-integer function.


166
These frequency ranges for which aT is positive exclude the end points, because at coni
the numerator of (7.12b) is zero and aK =0, and at codi the denominator is zero and
aK = oo, and therefore aT is not strictly positive at these points.
Finally, the frequencies for which both aK and aT are both positive is given by
Qa=QA.flnQra for 1< ad *1 To
t, To
becomes all
where
co eQ, uQ2 uQ3u-
Q,:= <
Q3: =
r
if
1 < < 00
otherwise
'f
(/2
if
1 <.a¡) <00
otherwise
'
if
\ T0
(>d5> otherwise
completing the proof.
Q.E.D.


147
given by the contour curve passing through the *+* marker that represents the point
aT- 1 and ag = 1, again giving pK{0.5) = GM(0.5) = 1.816.
Figure 7.4. The contour plot of the stability-boundary surface for the
ITAE regulation tuning rule when 0Q/ r0 = 0.5. The marker shows the
parametric time delay margin pe(0.5) = 1.781.
Figure 7.4 also shows that the /(ar,a0) = 1 contour intersects the vertical line aT = 1 at
a9 = 1.781 consistent with Figure 7.3. Therefore, the parametric time-delay margin is
p9{05) = 1.781, because by the definition (7.19) this is the smallest value of ae at which
/(1, ae) = 1. This means that when only considering perturbations in the process time
delay, a multiplicative perturbation greater than or equal to 1.781 destabilizes the closed-
loop system. In addition, the frequency solution to (7.12) at which aK = 1, at = 1, and
ae = pe (0.5) = 1.781 is the gain-crossover frequency, which for this example is


81
(A2) The set of allowable perturbations A is such that g(s) and g0(s) have the
same number of open loop unstable poles.
The robust stability analysis is based on a frequency domain description of the uncertain
perturbations using value sets. The uncertainty value set of g-(s) at frequency co is
defined as
V(co) := [g(jco) | g(jco) = gfjco) + 5{jco), 8{s) e A}
and V(co) is said to lie on the Nyquist plane. A generic uncertainty value set is shown in
Figure 5.2.
r(o),)
Im g(Ja>)
goUx) + dc
goO'^i) + Pc (.K
Figure 5.2. Schematic of an uncertainty value set (shaded area),
and the critical perturbation radius pc{cox) at a frequency cox. Also
shown in the figure are the critical line r{(ox) (dashed line); and the
nonconvex critical uncertainty value set Vc(a>x) which in this case is the
union of two disjoint straight-line segments (shown by the dotted lines).


Ill
Available results for interval plant descriptions enable the development of
computationally manageable methods for solving a wide range of robust stability analysis
problems. However, it is usually the case that each plant coefficient depends on more
than one uncertain parameter. One such uncertainty description occurs when the plant
coefficients are affine in the uncertain parameters. Unfortunately, the extension of
Kharitonovs approach to the case of affine uncertainties is not straightforward.
Nevertheless, Fu (1990) presents comprehensive results that are useful for quantifying an
entire uncertainty value set in the Nyquist plane for plants with affine parametric
uncertainties. A remaining challenge is to utilize these results to produce a scalar
measure of robustness analogous to the well known structured singular value (Doyle,
1982) and the multivariable stability margin (Safonov, 1982) paradigms.
The previous chapter proposes an alternative method that is applicable to both
interval and affine perturbations and is based on using the Nyquist robust stability margin
kN(a>) as a measure of robust stability. The technique extends the critical-direction
theory developed by Latchman and Crisalle (1995) and Latchman et al. (1997) by
considering nonconvex critical uncertainty value sets. A general definition of the critical
perturbation radius pc{co) used in the calculation of kN(a>) is proposed to take into
account nonconvex critical uncertainty value sets. The new general theory is applied to
the case of systems with real parametric affine uncertainties. Earlier results of Fu (1990)
are combined with an explicit map from the parameter space to the Nyquist plane to
calculate the required critical perturbation radius with high precision and efficiency.
In this chapter, a practical design approach based on parameter space methods
(Siljak, 1989) is proposed to illustrate the utility of the Nyquist robust stability margin as


152
A comparison of Figures 7.6 and 7.7 reveals the expected connection between the phase
margin and the parametric time delay margin based on the similarity in the trends of the
results.
Finally, the parametric stability margin is shown in Figure 7.8. This margin is the
most representative measure of robust stability because it considers simultaneous
uncertainties in all the parameters.
Figure 7.8. Parametric stability margin of the tuning rules vs. the tuning
parameter.
For higher values of 60 / r0, the figure shows that the order of increasing robustness for
the tuning rules is ISE-reg., CC, IAE-reg., ITAE-reg., ZN, LAE-servo, and ITAE-servo.
For lower values of 0o/ro, the CC and IAE-reg. tuning rules change their order of
robustness. The ZN and ITAE-reg. tuning rules also change their order of robustness for
lower values of d0 / r0. The figure is interpreted as follows. For 00/ r0 = 0.1 the ISE-


113
controller with a guaranteed level of performance. Note that the parameter space
considered here refers to the controller parameters rather than to the uncertain plant
parameters.
The set of robustly stable controllers in the parameter space is implicitly
determined by its associated robust stability boundary, that is, the level sets in the
parameter space that separate robustly stable and unstable controllers. These are
manifolds where the Nyquist robust stability margin kN is equal to one. In general, it is
not possible to find closed form expressions that describe these regions in the space of
controller parameters. Therefore, an efficient search over the parameter space is often
necessary. Since nominal stability is necessary for robust stability this search can be
restricted to the subspace of nominally stable controllers.
Since nominal stability is necessary for robust stability and it is therefore possible
to parameterize all the nominally stable systems about a specified stabilizing controller, it
is possible to initialize the search with a feasible point. In this paper we propose a simple
search strategy where robust stability is evaluated at a representative set of values of the
controller parameters, and the robust stability for the points lying in between is
determined by interpolation. Since the regions describing robustly stable controllers are
not simply connected, this may lead in general to an exhaustive search of the controller
parameters space
One of the main limitations of parameter space methods is that it is difficult to
characterize regions of nominal and robust stability when the number of controller
parameters is greater than three. Therefore, we will focus on fixed order controllers that
have at most three free parameters. An important class of controllers widely used in
industry that can be effectively designed using this approach are the PID controllers. To


CHAPTER 3
MAJOR PRINCIPAL DIRECTION ALIGNMENT WHEN
THE MAXIMUM SINGULAR VALUE IS REPEATED AND
ITS RELATIONSHIP TO OPTIMAL SIMILARITY SCALING
3.1. Introduction
The Major Principal Direction Alignment (MPDA) theory yields a necessary and
sufficient condition for the spectral radius of a matrix to equal its maximum singular
value (Kouvaritakis and Latchman, 1985). This has been proved using duality arguments
in Chapter 2 where it is shown that the results hold, even for the case of a repeated
maximum singular value. The primary reason for the development of the MPDA
principle is to solve the structured-singular-value / p problem, that is often written in the
form (Doyle, 1982)
sup p(MU) = p(M) < inf Ue DeB
Where MeCnxn, V:= ^diag(ej9' ,ej2 < 0¡ < 2k,i 1,2,,} is the set of
diagonal unitary matrices and := {diag(/j,£/2,--*,Jn)| d¡ > 0,i = 1,2,,} is the set of
positive diagonal matrices. In equation (3.1), p represents the spectral radius, p the
structured singular value, and V is known to be an NP-hard and non-convex optimization problem (Braatz, 1994);
therefore, when using standard optimization techniques there is always the problem of
local verses global optima. On the other hand, the infimization over D) can be shown to
be a convex optimization problem (Safonov and Verma, 1985; Tzafestas, 1984;
Latchman, 1986) and the global minima can be determined via an appropriate
25


175
cot
\
*0
0O
r
1-
To
0 )
(Xg
(Xg
T,
T0
Tj
ro )
Tp eo
T1 *0
at
(F.4a)
and
tan
V
*0
0o
(
1-
1 ^
Io
Io
\
(Xg
(Xg
T,
To
Ti t0
y
Tp Op
Ti *o
at
(F.4b)
respectively. A frequency given by (F.3) is real only when ag < 1, and it is easily
Ti To
verified that there is no solution ae strictly less than 1 to (F.4a) or (F.4b).
Tl To
T / \
Therefore, there is no frequency at which f(a>) and are simultaneously zero,
dco
/ \
and there is no frequency at which fd(co) and - are simultaneously zero, implying
dco
that the sign of the numerator of aT changes at the frequencies conX, con2, , and the sign
of the denominators of aK and aT changes at the frequencies codx, cod2, .


77
v =
-0.0845-0.28577
0.3690-0.27807
-0.1441 + 0.79437
0.0167 + 0.21407
w =
-0.0567 0.1918j
0.9912-0.7467j
-0.0672+ 0.3704j
0.0229 + 0.2932]'
For condition /') the matrix N given by (4.17) is
-0.1881
0.4589
0.0422
0.0258
0.0185
-0.5294
0.0655
0.0400
0.0323
1.2432
-0.2620
0.0698
0.0086
0.3305
0.0304
-0.2756
and
1
d¡2
0.25
0,2
= N
<3
1.44
1
K>
U
1
_0.49_
such that condition i) is satisfied. Finally, it is easy to shows that condition ii) and iii-a)
of Theorem 4.2 are satisfied.
4.5. Conclusions
In this paper we recover the dual-norm arguments for the case of complex A and
obtain an exact and closed form expression for the optimal D matrix. This result has
independent value in terms of the mathematical completeness of the extension of the case
of complex matrices as well as potential algorithmic improvements in computing the
optimal scaling matrices.


10
Furthermore, a left eigenvector w( of A is any non-zero vector that satisfies
w*A = Aw]
The reader is cautioned that some authors use the term left eigenvector for an eigenvector
of AT. Finally, an eigenvalue of maximum modulus is any eigenvalue A¡(A) that
satisfies |A,.(A)| = p(A).
2.2.2. Dual Norms and Dual Vectors
In the theoretical development that follows, the mathematical concepts of dual
norms and dual vectors are utilized. These concepts are explained in a paper by Bauer
(1962) and are reviewed here to facilitate the theoretical development. Given a vector
norm ||| its dual vector norm ||| is defined as
For such dual norms the Holder inequality
IMUMMey'*
holds an is sharp, i. e., for any y0 there exists at least one x0, and for any x0 there exists
at least one y0 such that the equality holds (Bauer, 1962). If such a pair (x0,y0) with
||y0||D||xo|| = Rey¡x0 also satisfies the scaling condition
it is called a dual pair. Note that the dual vector of x is often written (x)D. A pair
(x0,y0) is strictly dual and is written y0||Dx0 if ||y0||D||x0|| = y*x0 = 1 For strictly
homogenous norms (i.e., those satisfying ||ox| = \a\ ||x|| for all complex scalars a) the
Holder inequality may be sharpened to (Bauer, 1962)


CHAPTER 6
ROBUST CONTROLLER SYNTHESIS FOR SYSTEMS WITH NONCONVEX
VALUE SETS USING AN EXTENSION OF THE NYQUIST ROBUST STABILITY
MARGIN
6.1. Introduction
The stability analysis of feedback control systems in the presence of modeling
uncertainty is the subject of extensive studies; in particular, the class of structured
uncertainties where the parameters of a transfer function vary in prescribed real intervals
is relevant to many engineering applications. Early advances in this field are due to the
well-known theorem by Kharitonov (1979) that gives conditions for the stability of
polynomial systems with coefficients that belong to a rectangular polytope. Extensions of
Kharitonovs work to rational functions make use of standard frequency-domain
techniques such as Nyquist plots and the small gain theorem. These methods are based
on determining the stability of a set of Kharitonov plants (or extreme plants) derived from
an interval plant description where each coefficient of the numerator and denominator
polynomial varies in a fixed interval. The number of extreme plants required varies with
the technique utilized. Chapellat et al. (1989) suggest a method which involves checking
the stability and calculating H^ norms along a finite number (at most 32) of extreme
segments called Kharitonov segments. Barmish et al. (1992) prove that it is necessary
and sufficient that sixteen of the extreme plants be stable, and under certain conditions
only eight or twelve are necessary. Bartlett et al. (1990) give conditions that use 32 one
dimensional subsets of the interval plant.
110


104
all the frequencies investigated, and it can then be concluded from Theorem 5.2 that the
closed loop system is robustly stable.
Figure 5.5. Nyquist robust stability margin kN(a>) for the system
considered in Example 5.6.1.
5.7.2. Example 5.2 Nonconvex Critical Value Set.
Now reconsider the system (5.19) for the case where the parametric uncertainty
vector belongs to the rectangular polytope
Q={qeR3 -10 <#,<10, 0.3 < g2 < 0.3, 0.3 < <73 < 0.3}
(5.25)
With this modification to the example discussed in Section 5.6.1 the system now has a
nonconvex critical uncertainty value set Vf( = 0.95, as can be
determined by inspection of the frame g(jco,E(Q)) of the value set shown in Figure 5.6.
The figure clearly shows that as one follows the critical-direction ray from the nominal
point goU) to the critical point, the ray realizes three intersections with the value set
boundary. The ray leaves the value set immediately after the first intersection with the


93
Proof. A finite point we C belongs toV(a>) if and only if there exists a vector
q e Q that satisfies equation (5.12). Since the denominator on the left-hand side of (5.12)
is non-zero due to the finite magnitude of w, the equality can be rationalized by
multiplication by the denominator, and after isolating the real and imaginary parts leads to
the equivalent set of equations
+ WRSd,R^O,R WISdjd0i
+ ws^,dnr + w,sT. DdnD
R^d.R^O.R
R^dJ1* 0,1 ^ rvIJd,RU 0,R
which becomes equality (5.13a) after the matrix A(w) and the vector b{w) are defined as
given in (5.14) and (5.15), respectively. From (5.9b), the restriction that q eQ can be
described by the linear inequality (5.13b). Hence, it follows that w eC is an element of
V(cd) if and only if there exists a feasible solution to the linear equality/inequality
Q.E.D.
problem (5.13).
Theorem 5.3 poses the uncertainty value set membership problem as a standard
linear equality/inequality feasibility problem whose solution can be found in classical
linear programming references (see Luenberger, 1984, for example). The linear map
given by (5.13)-(5.15) is obtained for the rational function (5.1) with the affine
uncertainty structure given by (5.9). A formally analogous linear map based on the zero-
exclusion principle has been developed by Bhattacharyya et. al. (1995) for the case of
affine uncertain polynomials. Equations (5.13)-(5.15) constitute an extension of that
approach to the case of rational systems, and a generalization to the case of an arbitrary
point on the complex plane.
To calculate the critical perturbation radius using (5.3) it is necessary to determine
if -1 + jO is an element of the uncertainty value set V{(o). From Theorem 5.1, the


37
a?2() a2(M) +y (M)M^IkM)
dd;
ddi
ai ~ a(M*M)
= (M) + y (M)
ddt
= y (M) v 'y(M)
Od
a.
dd;
y(M)
Expanding the partial derivative term now gives
d dd:
_v^a(D-lM*D2MD-,)_ ~
y (M) y(M)
dd:
= y*(M)
= y*(M)
aD
-i
r. dD
MDMD 1 +D M* MD 1 +D M D M
dd, dd;
aD1
~dd:
-^EMDMD 1 +2JD M EMD 1 -D M*D2M^-E,
i/.2 11 /2 '
y(M)
y(M)
where E(. is a diagonal n x n matrix with a 1 in the (i,i) position, and zeros everywhere
else. Since E(. = d^iD"1 = DE,D / df = d¡D_1Ef, the above equation becomes
da (M) =i_y*(M)r E[D-iM*D2MD-i +2D M*DE,DMD 1 -D M*D2MD E^M)
/H/7 L J
dd,. d,
= y^MM'EjM-^CMEjlyM)
d: L J
Using equation (3.4) this becomes
ao^M) 2 2 ,
ddt d,
, 2
[x (M)EjX(M) y*(M)E,y(M)]
_ 2a\M)T
d: L
x,.(M) -y;.(M)
Now,


85
Theorem 5.1. Consider the uncertain system g(s) given in (5.1) with
assumptions (Al) and (A2). Then, the closed loop system is robustly stable under
unity feedback if and only if
-\ + jO£V(co)\fco (5.7)
Theorem 5.1 is simply a restatement of the well-known zero-exclusion principle
(Barmish, 1994), and it gives a necessary and sufficient condition for the robust stability
of the closed loop in question. However, Theorem 5.1 does not provide a measure of the
degree of robust stability of the loop, a quantity that would be most useful as the basis for
the synthesis of optimally robust controllers or for the assessment of the relative merits of
alternative control schemes. The critical direction theory seeks to quantify the robust
stability of such systems in terms of the Nyquist robust stability margin (5.5), which plays
a role analogous to that of the structured singular value (Doyle, 1982) and of the
multivariable stability margin (Safonov, 1982). Efficiency in the analysis is obtained
through the realization that it suffices to verify condition (5.7) only for value-set points
that lie along the critical direction; more precisely, the set membership condition (5.7)
holds if and only if -1 + y'O <£Vc(co) holds. These observations lead to the following key
result of the critical direction theory.
Theorem 5.2. Consider the uncertain system g(s) given in (5.1) with
assumptions (Al) and (A2). Then the closed loop system is robustly stable under
unity feedback if and only if
/:N( Proof. A complete proof is given in Latchman and Crisalle (1995) for the case
where Vc(co) is convex. For the non-convex case in which the generalized definition


3
which addresses the problem of robust stability of systems affected by uncertainties that
are characterized in terms of arbitrary frequency-domain value sets that are convex. The
critical direction theory proposes the Nyquist robust stability margin as a measure of
robust stability which has obvious connections to the Nyquist stability criteria. The
advantage of the critical direction theory over the structured singular value theory is that
for several common structured uncertainty types there is an analytical expression for the
Nyquist robust stability margin. Also, even if there is not an analytical expression,
determining the Nyquist robust stability margin is a tractable problem.
Another type of structured uncertainty is real parametric uncertainty in the
process model. The robust stability problem under parametric uncertainty began to
receive renewed attention with the seminal result of Kharitonov (1979) on the stability of
interval polynomials, and is considered the most important development in the area after
the Routh-Hurwitz criterion. The theory makes it possible to determine if a linear time
invariant control system, containing several uncertain real parameters remains stable as
the parameters vary over a set (Bhattacharyya et al., 1995). Accordingly, the parametric
stability margin is defined as the length of the smallest perturbation in the parameters
which destabilizes the closed loop. The parametric stability margin is useful in controller
design as a means of comparing the performance of proposed controllers.
1.2. Objecive and Structure of Dissertation
The first goal of this dissertation is to revisit the MPDA principle to strengthen
the result when the maximum singular value is repeated. Chapter 2 introduces a revised
statement of the MPDA property that fully considers the case of a repeated maximum
singular value. An alternative proof is presented that is based on the theory of dual
norms and dual vectors which was the inspiration of the original result. The MPDA


150
characteristics, with a gain margin always less than 1.5. The ITAE servo (ITAE-servo)
and IAE servo (IAE-servo) tuning rules can be considered highly conservative with gain
margins always greater the 2.25. For the ZN, CC, ITAE-reg., LAE-reg., and ISE-reg.
tuning rules the general trend is that the gain margin increases with increasing values of
the ratio 60/ r0. This implies that the tuning rules yield inherently more conservative
controllers as the time-delay-to-time-constant ratio increases. This trend is appropriate
and reasonable considering the difficulties usually associated with processes having large
time-delays. As for the ITAE-servo and IAE-servo the gain margin decreases as the time-
delay-to-time-constant ratio increases, but this is not considered a problem given that the
tuning rules are already considered to be overly conservative.
Figure 7.6. Phase margin of the tuning rules vs. the tuning parameter.
Controller manufacturers recommend that a well-tuned controller have a phase
margin between 30 and 45 degrees (Seborg, 1989). Figure 7.6 shows that none of the


33
if it exists. Notably, for convex functions the directional derivative /'(d;g) exists for all
d g S and for all g e Rn. Demyanov and Vasilev (1985) show that the relation
/'(d;g)= max £Tg
£ed/(d)
(3.10)
holds.
The function / is differential at d e S if and only if there exists a vector V/(d)
(necessarily unique), called the gradient, for which
/(g) = /(<) + V/T( d)(g d) + 0(||g d||)
or, equivalently,
lim/(g)~/(d)~V/T(d)(g~d) 0
g->d
g-d
If / is a convex function then / is differential at de5 if and only if the partial
derivatives exists. In addition, the gradient is given by
V/(d)=
df(d) df(d) dfid)
dd. dd7 ddn
i n
and / has only one subgradient, namely the gradient V/(d), such that
df(d) = {V/(d)} (3.11)
Also,
/(g) /(d) + V/T(d)(g d), Vg e S
That is, V/(d) is the normal of the tangent supporting hyperplane of epif at d.
With the terminology of differential theory thus developed, several important
theorems are given that are used in the sequel. The first theorem describes the set of
points where / is differentiable. This theorem is used as the basis of the primary


126
7.2.1. Process Model and Uncertainty Description
For the purposes of this study, it is convenient to represent the uncertain first-
order plus delay model as
Ke~Os
P(s;q) = (7.2)
7S + 1
where the process gain K, the time constant r > 0, and is the time delay 0 > 0 are real
parameters, and where the uncertainty in the parameters is expressed in the multiplicative
form
K = aKK0
(7.3a)
ii
R
**
o
(7.3b)
ii
a
Ob
o
(7.3c)
where Kq*0, tq> 0, and Oq > 0 are the known nominal values of the process
parameters, and the scalars a ^ > 0, aT > 0, and ccq> 0 are unknown real multiplicative
perturbations that are collected in the uncertainty vector q= [ak,cct,(Xq] g 0c 9? ,
which belongs to an uncertainty domain Q composed of vectors with strictly positive
elements. The nominal process p0(s)\= /?(5;q0) is recovered from (7.2) after setting
q = qo;=[l 1 l]7' to yield
-Or.s
Po(s)= (7.4)
Tq5 + 1
Note that the real multiplicative parametric uncertainty description (7.3) has the
associated additive perturbations
AK:= K0(aK -1) (7.5a)
Ar:= r0(aT -1)
(7.5b)


APPENDIX C
PROOF OF THEOREM 7.1
Proof. By definition the closed-loop system is robustly stable with respect to
0max, and for any q = [aKaT ag]T ed£>max, the set £>max uq is not robustly stable. From
Lemma 7.2 this implies that q must satisfy the equality
A(co\OJ r0\cce)
a,
a,
= b()
(C.l)
for some co> 0. The solutions [a K aTf of (C.l) at each frequency depends on the
rank of the matrix A(co;&0 / ro;a0), and only solutions that satisfy aK >0, ax> 0, and
a e> 0 are considered as admissible because <2max is the space of strictly-positive real
ordered triplets. First, at those frequencies where matrix A is full rank, the solutions for
aK and az are given by (7.12a)-(7.12b), as claimed in the theorem. It now suffices to
exclude from the admissible set all the solutions corresponding to frequencies where A
is rank deficient.
Matrix A is not of full rank whendet(A) = 0, or equivalently when
K0KC
cocos{aeco)
T0 &o
T, T0
sin (a0a>)
V* V1
A To )
o)2 0
Since the values of K0KC and 60 / r0 are nonzero, A is not full rank when either co = 0
or at frequencies where
co cos(a 0co) -sin(agco) = 0
(C.2)
*/ To
161


63
previous results are based on earlier work that derive a necessary condition for the least
upper bound of a matrix to equal the modulus of an eigenvalue of the matrix, namely, that
the corresponding right and left eigenvector are dual (Bauer 1962). Unfortunately, for
the general case of complex matrices, there are no equivalent analytical results on optimal
scaling by positive diagonal matrices, although there exist several numerical algorithms.
From a robust control perspective, the structured singular value, fj. (defined as
supp(MU) where V\ = \diag(e'0' ,ejdl < 6¡ < In,i = 1,2,,} (Doyle, 1982)),
UeV L J
is a widely accepted tool in the robust analysis of linear systems. It considers the
problem of robust stability for a known plant subject to a block-diagonal uncertainty
structure under feedback. In general, any block-diagram interconnection of systems and
uncertainties can be rearranged into the block-diagonal standard form. Calculating ¡i is
not trivial; in fact the problem has been proven to be NP-hard (Braatz et al., 1994). The
difficulty is that the spectral radius is non-convex over the set of unitary matrix
transformations. One approach is to consider upper bounds for the spectral radius that
can be calculated easily, and ideally should be attainable to eliminate conservatism. The
maximum singular value is reasonable choice for an upper bound because it is invariant
under unitary matrix transformations. In addition, the maximum singular value upper
bound can be decreased by optimizing over similarity transformations because the
spectral radius is invariant under such transformations. Ultimately, the problem becomes
one of conditioning a matrix through optimal similarity and unitary transformations to
achieve equality between the spectral radius and the maximum singular value.
In addressing the existence of solutions to the proposed optimization,
Kouvaritakis and Latchman introduce the major principal direction alignment (MPDA)


114
illustrate our design procedure we will consider the proportional-integral (PI) controller,
which has the advantage of elimination of offset. The transfer function of a PI controller
is given by
c(s) = Kc
\
\ Tis J
where Kc and r, are the controller parameters.
The first step of the controller synthesis is to identify the regions in parameter
space of the controllers that result in nominal stability. For PI controllers this can be
achieved through a D-partition of the complex plane of the controller parameters
(Kiselev, 1997), or by checking the close-loop poles of the system for the representative
set of controller parameters. These regions can be plotted in the parameter space with the
x-axis being Mr, and the y-axis being Kc. The manifolds that separate these regions in
the parameter space will be called the nominal stability boundary.
Next, the Nyquist robust stability margin is calculated for each point in the
representative set of controller parameters that are nominally stabilizing. From these
values of the Nyquist robust stability margin it is possible to plot level sets for constant
values of the Nyquist robust stability margin; the most important being the level set
corresponding to a Nyquist robust stability margin value of 1. The regions in parameter
space with Nyquist robust stability margin less than one correspond to robustly stabilizing
controllers. While it is possible to determine the robust stability regions using a D-
partition on the uncertain system, there is no measure of robust stability for points inside
these regions (Kiselev, 1997). Furthermore, the nominal stability region and the robust
stability region need not be connected.


9
The matrices X(A) and Y(A) are of the form
X(A) = [x,(A) x2(A) xn(A)]
Y(A) = [y1(A) y2(A) y(A)]
where the set of normalized left singular-vectors (input principal directions) {x( (A)} and
normalized right singular-vectors (output principal directions) {y,.(A)} for i = 1,2,, ,
respectively constitute orthonormal eigenbasis of AA* and A* A. Furthermore, a pair of
singular vectors {x;.(A),yi(A)} is associated with each singular value cr.(A) through the
relationship
Ay, (A) = cr (A)x;.(A) (2.2)
The maximum singular value is denoted maximum singular value can be associated with a repeated singular value, i.e.
output/input principal direction pair) {x(A),y(A)} is any pair of left and right singular
vectors x,.(A) and y, (A) that correspond to the maximum singular value and satisfy (2.2
). Necessarily, a major output principal direction and major input principal direction
respectively must be elements of the orthonormal eigensubspace of AA* and A*A
associated with the maximum singular value.
In this chapter the following definitions are used in relation to the eigenvalue
decomposition (Golub & Van Loan, 1983; Isaacson & Keller, 1966; Stewart, 1970). Let
T((A) be an eigenvalue of A; then a right eigenvector v. of A is any non-zero vector
that satisfies
Av(. = Av(.


121
As for the ITAE performance criteria, the controllers designed using the two methods are
nearly identical given that the changes in Nyquist robust stability margin values between
Tables 6.1 and 6.2 are small. The ISE and LAE controllers designed using the second
method can be considered more conservative than the controllers design using the first
method, because the relative decrease in Kc (increasing conservatism) is larger than the
relative decrease in t¡ (decreasing conservatism). It is not practical to determine relative
conservatism between the ITAE controllers, considering the small difference between the
two.
6.4. Conclusion
The Nyquist robust stability margin k^(co) is an effective scalar measure of
robust stability. The main contribution of this paper is the introduction of a general
definition of the critical perturbation radius pc(co) that can account for non-convex
critical uncertainty value sets. This generalization of the critical direction theory is
illustrated for systems with affine parametric uncertainty for which the critical
perturbation radius can be calculated precisely and efficiently with no computational
issues. The computation of the Nyquist robust stability margin involves planar geometry
operations and solving linear equality/inequality feasibility problems. A parameter space
design method for robust performance using the Nyquist robust stability margin, is also
demonstrated.


170
4
fdiscXtX) = YjfdiscrA^l^x)
1=0
where
fdiscrA^x^x) = -16(l-cos(a>t -ffl,))2 ?]!(<, -f>,)
fdisc r,l ( 1 . J = -32(1 COS, f>, ))2 (, ffl, )'
fdiscr,2 (,) = 16((1 C0S( ? ,)) + O C0S(ffl, ) C0S(fflx )))
x(l cos( /dco(£yiJ):=-16(1-cos(;(
fdiscr,4 ( 1JC) = -4(1 cos(y,) cos(yx ))f 5 (x , )4
which is always non-positive, because 0, > cos(1)cos( = 1,2,- and m = l,2,---. Therefore, there is a real solution y to (E.5) only when
a>l=7m and cox = co{ + 27im where = 1,2, and m = l,2,---. But from (E.3), for
cox = m and cox cox + 2mn we have {g>x)2 ={cox+27onf which is only true when
m = 0 implying cox=cox, a contradiction. Therefore, there is no solution 0,
cox> cox, and y > 0 such to (E.la) and (E.lb). As such, the curves never intersect, and
therefore the first frequency interval gives the lowest curve.
Q.E.D.


66
may be used for conditioning of matrices. In addition, positive matrices remain positive
under transformation by non-negative diagonal matrices leading to connections with
Perron-roots /r(P) (positive eigenvalues of largest modulus) of positive matrices
P g /?"*" (note, R+ is the set of positive real numbers). From this perspective, Stoer and
Witzgall (1962) show that for the positive matrix P and non-negative diagonal
matrices D
;r(P) = min lub(D_1PD) (4.1)
DeB
where := {\zg(dx,d2, ,d n)\di >0,i = 1,2,,}, and
|ax||
lub(A) := max i,1 = max||Ax||
** ||x|| WH11 11
is the least upper bound norm of a matrix A g Cnxn subordinate to the vector norm |||. It
is noted that the least upper bound norm is equivalent to the induced matrix norm, and
that when the subordinating norm is the Euclidean norm then lub(A) = In developing the result it is necessary to make use of a result from Bauer (1962)
that states that if A is an eigenvalue of A, then
\A\ = lub(A) (4.2)
is only possible if a right and left A -eigenvector are dual with respect to the norm to
which the bound norm is subordinate, where by definition a left A -eigenvector w of A
satisfies the relation w*A = Aw" (Golub & Van Loan, 1983; Isaacson & Keller, 1966;
Stewart, 1970). The reader is cautioned that some authors use the term left eigenvector
for an eigenvector of AT.


CHAPTER 4
SPECTRAL RADIUS MAXIMUM SINGULAR VALUE EQUIVALENCE UNDER
OPTIMAL SIMILARITY SCALING
4.1. Introduction
It is well known that the maximum singular value of a matrix is an upper bound of
the spectral radius (i.e., where M eC"x"). Determining the conditions
under which the upper bound is attained is a significant issue in the field of robust
control. One approach is to seek properties of matrices that are necessary and sufficient
for equality of the spectral radius and the maximum singular value. Another approach
uses optimization to condition the matrix through similarity and unitary transformations
in order to increase the spectral radius and decrease the maximum singular value upper
bound so that equality is achieved.
Previous work deals with the optimal conditioning of matrices from a numerical
accuracy stand point (Bauer, 1963) and focuses on similarity transformations using
nonnegative diagonal matrices. The scaling problem for non-negative matrices yields a
very elegant and precise result. It provides a closed form expression for the optimal
similarity scaling matrix for which the Perron-root (largest positive eigenvalue of a
positive matrix) equals the least upper bound subordinate to an absolute norm. In
addition there are analytical expressions for the elements of the optimal diagonal matrix
that involve the Perron-eigenvectors of the given positive matrix (Stoer and Witzgall,
1962). The relationship to the present work is that the least upper bound of the matrix
subordinate to the Euclidean norm is the maximum singular value of a matrix. The
62


106
The elements of <3c(y) can be identified from Fusing the sequential method described in
Section 5.6 to yield
which contains three elements due to the nonconvexity of Vc( yields
£() = gmin |l + z| = |l 0.6512 y'0.3736| = 0.5111
and then invoking definition (5.3) it is readily determined that
pc{co) = |l 0.4140-0.6277j\ 0.5111 = 0.3475
at the frequency co = 0.95 .
Figure 5.6. Frame for the uncertainty value set for the system of
Example 5.6.2 at the frequency co = 0.9500. The critical point -1+/0 and
the nominal point g0(jo) are represented by the "x" markers, the
intersections of the arcs with the critical line are represented by the "+"
markers, and the intersection that defines the boundary point used in the
calculation of pc(co) is represented by the "*" marker. The critical value
set Fc() is nonconvex at this frequency, and it is represented by the
union of two disjoint a straight-line segments.


131
Hurwitz. Therefore, from the continuity of the roots of >(s;q) for q eQ, Lemma 1
follows from the application of the Boundary Crossing Theorem (Bhattacharyya, 1995)
for quasipolynomials, specialized to the Hurwitz case where the stability region of
interest is the open left-half plane. Q.E.D.
Applying Lemma 7.1 to the system being considered yields the following stability
result.
Lemma 7.2. The closed-loop system of Figure 7.1 with PI control parameters
adjusted using a tuning correlation from Table 7.1 is robustly stable with respect
to all parametric uncertainties q = [a % aT olq\ eQ if and only if the inequality
a
K
Larj
A {co,ae)
holds for all q eQ and for all a>> 0, where
(co)
(7.9a)
A (co,a0):=
K0KC
K0KC
0
Vro
cos (aeco) + co sin^)

yTi;
>
-l
CO
co cos(o; dco)
( \
\TI J
sin (aeco)
0
em2x2 (7.9b)
b(): =
(7.9c)
-CO
Proof First note that equations (7.9a)-(7.9c) are in terms of the dimensionless
frequency co defined as co\=co00 where co is the standard frequency with units of
reciprocal seconds and 0O is the nominal value of the time delay. Therefore, without loss
of generality, it suffices to apply Lemma 7.1 to the image 8{jcb\ q0) = S(jco/ #0;q0) of
the quasipolynomials (7.7) i.e.,


90
belongs to the uncertainty value set V{co). The more general problem of determining if
an arbitrary point weC belongs to V{co) is solved in this section, and the results are
then utilized to reformulate Theorem 5.1 in terms of computable quantities.
The affine uncertain system (5.9) can be written in the vector-matrix form
g(s,q) =
r
oo
io
20
o"
~<1\
\
[l s se 1
*']
01
+
>.
21
- WJ>1
\
0/
>/
2/
- v
3p.
)
Sn(n0+Np f
o
o
d\0
^20
ft.
O
1
Vi'
\
r 1 m-1 m 1
[1 s s s J
d +
du
d2l
<
_^0m_
<
d2m
d pm
3p.
)
K + Dpq)
(5.10)
where sn and sd are vectors of lengths i +1 and m +1, containing powers of the Laplace
variables, and where n0 eR/+1, d0 eRm+1, Np R(<+1) / and Dp eR('+1)'p are constant
vectors and matrices that represent the structure of the affine parametric uncertainty. The
value set at frequency co is obtained by evaluating (5.10) at s = jco for all q e Q to yield
s. sn,R(no,R Rq^J + jsni^nol + N ¡q^
g(j>,q) = ~rn - r),^? G Q>
sd,R\do,R + DpRq) + jsd j\dQ J + Dp fqj
(5.11)
where
<=[!
-CO2
CO4
] eRr -CO2
CO5
-7 -]
eR^+,)/21
sIr = ['
-CO2
4
IeR^1
V, = [a>
-CO2
]
G Rf(-D/2l


14
Proof. The proof consists of two cases, namely, when the maximum singular
value is distinct, and when it is repeated. The proof is taken directly from Kouvaritakis
and Latchman (1985) and is relegated to Appendix B. Q.E.D.
For the case of a distinct maximum singular value, Theorem 2.1 as stated is
entirely accurate and the proof rigorous. Unfortunately, when there is a repeated
maximum singular value, Theorem 2.1 as stated is not entirely accurate and the proof is
not rigorous. In the proof, Equation B.5 states that the variable z must assume a given
form (i.e., that z = Y*(A)w must be at least one element of the form). This does not
mean that every major input and output principal direction pair results from
z = Y*(A)w ; instead it should be interpreted as meaning that there is at least one such
pair that results from z = Y*(A)w. Hence, when the maximum singular value is
repeated, there may exist a major input and output principal direction pair that is not
aligned even when the spectral radius equals the maximum singular value.
Counterexamples are given in the examples section. A modified statement of MPDA
with a proof based on duality arguments is provided in the next section.
2.4. Modified Statement of the Major Principal Direction Alignment Principle
The following theorem is a modification of the MPDA Theorem 2.1 which
accurately takes into account the case of a repeated maximum singular value.
Theorem 2.2. The spectral radius of any matrix A e Cnxn is equal to the
maximum singular value of A if and only if there exists a major input and major
output principal direction pair of A that is aligned.
Proof. To prove sufficiency note that alignment of a major input and major
output principal direction pair of A implies


APPENDIX B
PROOF OF THEOREM 2.1
Proof. The following proof is taken verbatim from (Kouvaritakis, 1985) and
consists of two cases, namely, when the maximum singular value is distinct, and when it
is repeated.
(i) Distinct Maximum Singular Values:
The principal directions of A are unique (with respect to each other) to within a
scaling factor ejd, so that alignment of the major input and major output principal
directions of A, implies
Ta = eJd*x (B.l)
Pre-multiplication of equation (B.l) by A gives
Ayx = v(A)xx =e'*AxA
or
Axa =e~jea(A)xx
so that e~jea emerges as an eigenvalue A, of A. Noting that the moduli of the
eigenvalues of A are always bound from above by cf(A), it follows that
\A\ = p(A) = a(A).
To prove the converse, assume that p(A) = cf(A) so that there exists a vector w
such that
Aw = ejl//a(A)w (B.2)
and
158


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
ROBUST STABILITY ANALYSIS METHODS FOR SYSTEMS WITH
STRUCTURE AND PARAMETRIC UNCERTAINTIES
By
Charles Thomas Baab
December 2002
Chairman: Oscar D. Crisalle
Major Department: Chemical Engineering
The major principal direction alignment principle is investigated in detail for the
case when the maximum singular value is repeated. A first result is a new proof based on
duality theory for the necessary and sufficient conditions that ensure equality of the
spectral radius and maximum singular value of a matrix; namely, that there must exists at
least one aligned pair of major input-output principal-direction vectors. A second result
is the development of a novel numerical optimization algorithm to solve the optimal
similarity-scaling problem that yields an upper bound for the structured singular value.
The algorithm provides a systematic procedure for identifying the steepest-descent search
direction even for the case when the singular value is repeated and the underlying
optimization problem is locally nondifferentiable. The key theoretical element is the
characterization of the subdifferential at every point of nondifferentiability.
viii


43
and
g(DMD )
d,
for i = 1,2,, with
1
H
xl2
*
II
*21
,x2 =
*22
1
H
i
Xn2.
an orthonormal set of left singular vectors and
>11"
>12 "
y. =
y21
> y2 =
y22
yn i.
_y 2.
(3.24)
(3.25)
(3.26)
an orthonormal set of right singular vectors corresponding to the maximum singular value
(DMD 1) of multiplicity 2. Equation (3.20) is obtained from equation (3.17) when the
maximum singular value has multiplicity 2, i.e.
g(u) = V/(d;u) = Using (3.25) for {x,,x2} and (3.26) for {y,,y2}, and considering one element of g(u)
the law of cosines gives
-1 \ _
&() =
a(DMD )
dt
a(DMD ')p
|xnu] I' + 2|jc(.]m1|x(.2w2 |cos(Z(xnw,) Z(xj2u2)) + \xi2u2
d;
\y^\2 + 2\yixux||yf2M2|cos(Z(ynM1) Z(yi2u2)) + \yi2u2
Using trigonometric and complex number identities gives


135
Lemma 7.3. Consider all parameters aK and aT given by (7.12) of Theorem
7.1. Then aK > 0 and aT> 0 if and only if
where
ffleQ,uQ2 uQjU-
(O.ffl.1)
if
\ <
T, T0
otherwise
r
S^dl^n 3)
if
1 T, zo
otherwise
f
(rf4
if
<
(todS^ns)
otherwise
and where conl, con2 are the positive zeros of
/():=
cos(ccqco) + cosm(aQCo)
zl r0
(7.13)
arranged in increasing order, and codx, cod2, are the positive zeros of
t 6
f (&>): = cocos(agco) n(cQCo) (7.14)
TI T0
arranged in increasing order.
Proof. A comprehensive proof in given in Appendix D. Q.E.D.
Lemma 7.3 gives the frequency intervals for which the values of aK and aT
produced by the map (7.12) are simultaneously positive. For each interval, the parametric
plots of aK and aT trace a curve in the aT aK space. From Theorem 7.1, the points


12
The proof makes use of dual norm dual-vector theory presented earlier. Details are in
Appendix A. Q.E.D.
2.2.4. Eigenvector-Singular Vector Equivalence Result
The following Lemma is a consequence of Lemma 2.1.
Lemma 2.2. If the spectral radius of a matrix A e Cnxn is equal to the maximum
singular value of A, then each normalized right eigenvector v. of A associated
with an eigenvalue of maximum modulus A;( A) is also a right singular vector y¡
of A associated with the maximum singular value cr(A).
Proof It suffices to prove that \¡ is a right eigenvector of A*A associated with
an eigenvalue whose square root is (x(A), because by definition the rights singular
vectors yf are an orthonormal eigenbasis of A*A and the singular values are the square
roots of the eigenvalues of A*A. First, from Lemma 2.1, it follows that for each
normalized right eigenvector v. of A associated with an eigenvalue of maximum
modulus A, (A) there exists a normalized left eigenvector w(. = v. of A. For each such
eigenvector v(.
A*Av(. = A*vt.A(.(A)
=(V;a)*a,.(a)
= (w*A)*A.(A)
= (A,.(A)w*)*A,.(A)
= v**A*(A)A,.(A)
=|.(a)|2v,


119
Table 6.1. Controllers resulting from minimizing the three performance criteria with
stability results.
Performance
Criteria
Optimal Kc
Optimal Tj
Robustly
Stabilizing
ISE
1.8738
13.8489
1.3157
No
IAE
0.9326
5.4623
0.9081
Yes
ITAE
0.4642
3.7649
0.8306
Yes
For the case of the ISE performance measure, the resulting controller will be robustly
stabilizing, which was not the case for the previous method. As for, the IAE and ITAE
performance measures, it is necessary to optimize along robust Nyquist stability margin
contours less than 0.9081 and 0.8306, respectively, to achieve a greater degree of robust
stability.
Figure 6.3. IAE contours superimposed on the robust stability boundary.


140
PkM^o)'= min ¡[aK-l,aT-l,ag-\]T
(7.17)
where Q, is the interval of frequencies that gives the robust stability region as prescribed
by Theorem 7.2, and aK and aT are given by (7.12) in Theorem 7.1. Note that (7.17)
explicitly shows the dependence on the ratio 0Q/ r0, the only parameter that (7.12) is
truly dependent on when considering a specific tuning rule chosen from Table 7.1. This
optimization has no convexity guarantees, but because it only involves two parameters, it
can easily be solved by an exhaustive numerical search. In addition, the constraint that
ag> 0 can be further restricted. To show this, for ae = 1 (i.e., when there is no
perturbation in the time delay) let
PkA0o/toY= mm
coqi j
[aK-\,aT-\]J
be the parametric stability margin when only considering uncertainty in the process gain
and in the process time constant. This margin is easily calculated by performing a
frequency sweep over the specified range. The minimization (7.18) need only be
performed over the range \-pKT(0o / r0) < ag < \ + pKz(0Q / r0), because pKz{00 / r0) is
an upper bound on (7.18), and for every ag outside this range
[aK-\,az-\,ag-X\1
^PKAeJ*o)-
The parametric stability margin (7.17) considers simultaneous perturbations in all
the uncertain parameters. However it is also of some interest to use the result of Theorem
7.1 and Theorem 7.2 on the region of stable parameter perturbations to obtain other
measures of stability, such as the classical gain and phase margin (Seborg, 1989), as well
as a margin defined in terms of perturbations only in the time delay. While the
parametric stability margin (7.17) is clearly a superior measure of robustness in its


82
The critical-direction theory advanced in Latchman and Crisalle (1995) and in
Latchman et al. (1997) is based on the observation that the smallest destabilizing
perturbations occur along the critical direction
dc{jco)\=
1+ g0(./fi>)
| l + ^o O')|
which is interpreted as the unit vector with origin at the nominal point g0(jco) and
pointing towards the critical point -1 + j0 (cf. Figure 5.2). This direction in turn defines
the critical line r(co) := g0(jco) + adc(jco), aeR+, where R+ denotes the nonnegative
real numbers. The critical line r(co) is interpreted as a ray that originates at the nominal
point g0(jco) and passes through the critical point -1 + j0. The intersection of the
uncertainty value set with the critical line determines the critical uncertainty value set
Vc(co) : = V(co) n r(co) which may be (/) a single straight-line segment or a single isolated
point (in which case Vc(co) is a convex set) or (ii) a union of disjoint straight-line
segments and isolated points (in which case Vc(co) is a nonconvex set). Figure 5.2 shows
the case of a nonconvex critical uncertainty value set. Finally, the boundary of the
uncertainty value set is denoted dV(co), and the set of critical boundary- intersections
(B (co) is defined as
(Bc(co): = {dV(co) n r(co)} \ g0 (jco)
where \ is the set-difference operator. For the special case where d'V(co)nr(co)
contains g0(jco) as its only element, the following definition is applied:
c():=feo(i)}


The critical-direction theory is extended to include nonconvex critical uncertainty
value sets through the introduction of a general definition of the critical perturbation
radius. The Nyquist robust stability margin is calculated for systems with affine
parametric uncertainty using an explicit map from the parameter space to the Nyquist
plane. A practical design approach based on parameter space methods is introduced.
First the controller parameters that result in robustly stable closed-loop systems are
determined. Then, a performance objective is optimized over the set of robustly
stabilizing controller parameters, resulting in a robustly stabilizing controller with some
optimal performance characteristics.
A formal robustness analysis of popular proportional-integral controller tuning
rules for systems approximated by a first-order-plus-time-delay model is presented. The
uncertainty in the process model is represented by multiplicative parametric perturbations
in the process gain, process time constant, and process time-delay. The robustness of the
uncertain system is characterized in terms of the set of all perturbations that result in
stable closed-loops. This set is used to calculate the standard gain and phase margins,
and the parametric stability margin which is a metric of robustness to simultaneous
variations in all three system parameters. These margins are used to compare the relative
robustness properties of several disturbance-rejection and tracking tuning rules in
widespread use.
IX




86
(5.3) of pc(co) is utilized the proof is extended as follows. From Theorem 5.1 the
uncertain closed loop system is stable if and only if -1 + j0 V co Therefore, to
prove that (5.8) is sufficient for robust stability, we must show that if &N(£y) then -1 + yO g V(co) Vco. To prove by contradiction, assume that £N(y) that 3co such that -1 + yO eV(co). Then applying definitions (5.3) and (5.5) for a
frequency at which -1 + jO ^(co) gives
M) =
pA>)
i+goC/)
i + gp') | + g()
11 + gpO') I
= 1 +
£()
1 + ^0 O')
where £(&>) is the nonnegative real scalar given by (5.4). Hence, kN(co) > 1 for at least
one frequency, which contradicts the assumption. Therefore, if &N( follows that -1 + j0 &V(co) Vco To prove that (5.8) is necessary for robust stability,
one must show that if -1 + j0 £V(co) Vco then kN(co) <1 Vco To establish this, note
that if -1 + jO <£V(co) Vco then by definitions (5.3) and (5.5)
M) =
Pc()
1 + So O)
i + goO) |-£()
11 + goO) I
= l-
£()
1 + gpO)
Viy
where^(u) is given by (5.4). In this case, however, since -1 + y'O gU(&>) it follows that
-1 + y'O (Bc(co), and thus <^(co) must necessarily be a positive number. Using this fact in
the above equality leads to the conclusion that (co) <1 Vco
Q.E.D.
From Theorem 5.2 it follows that the scalar &N(y) serves to quantify the robust
stability of the closed-loop system. The computation of &N (co) requires knowledge of the
critical perturbation radiuspc(co)defined in (5.3). The challenging task in a given
problem is in fact the calculation of the critical perturbation radius.


53
conditions for which MPDA is attainable when the maximum singular value is repeated.
These conditions are important, because they result in a non-conservative upper bound
for ju.
A sufficient condition for attainability of MPDA is that there exist a major input
and major output principal direction pair with elementwise equal moduli. This is
equivalent to the existence of u such that V/u (d;u) = 0. In contrast, the less stringent
sufficient condition for a minimum is 0 e df (d), where as the condition V/tt (d; u) = 0 is
equivalent to 0 being an element of the surface of df(d). For the case when the
maximum singular value has multiplicity 2 this becomes the condition that 0 is on the
surface of the ellipsoid. In other words
ctBc = 1 (3.38a)
and
q = 0 (3.38b)
Equations (3.38a) and (3.38b) represent the sufficient conditions for attainability of
MPDA when the maximum singular value is repeated twice. When the maximum
singular value is repeated more than once the sufficient condition for attainability of
MPDA becomes
minV/u(d;u) = 0 (3.39)
with u*u = 1. Condition (3.39) is not as convenient as (3.38), but is still useful as a
method for determining attainability of MPDA and thus the conservatism of the upper
bound of ¡U.


115
The final step in the controller synthesis is to optimize some measure performance
over the robustly stabilizing controllers to determine the final controller. Three classical
measures of performance will be considered:
Integral of the squared error (ISE)
co
ISE = \[e(t)fdt
0
Integral of the absolute value of the error (IAE)
oo
IAE = J \e(t)\dt
o
Integral of the time-weighted absolute error (ITAE)
oo
ITAE = J t\e(t)\dt
0
The error signal e(t) is the difference between the set point and the measurement. In
addition, the measures are base on an error signal resulting from a servo test of a unit-step
set-point change (Seborg, 1989).
A search over the set of robustly stabilizing controllers is necessary to find a
controller that maximizes a desired performance measure. There are several ways of
performing this search. One approach is to calculate the performance measure at the
same time as the Nyquist robust stability margin is calculated and plot both the level sets
of constant stability margin and the constant performance measure. A controller that
maximizes both stability robustness and nominal performance can then be determined
from this plot. Another approach is to search along constant Nyquist robust stability
margin contours by computing the performance measure along a desired contour that is
robustly stabilizing (i.e., less than 1). This leads to a controller that maximizes nominal


94
solution to this set-membership problem directly defines the robust stability of the
system. Therefore, the robust stability of systems with real affine parametric uncertainty
can be efficiently determined by solving a feasibility problem by specializing Theorem
5.3 to the case w = -1 + j0. The result is given in the following theorem.
Theorem 5.4. Consider the real affine uncertain system given in (5.6a)-(5.6b)
with assumptions (Al) and (A2). Then the closed loop system is robustly stable
under unity feedback if and only the following linear equality/inequality problem
in q eRp is infeasible at all frequencies:
A(-\ + jO)q = b(-\ + jO)
subject to
where
1
0
0
-1
0
0
0
1
0
0
-1
0
0
0
0
0
0
0
o'
0
-q'x
0
0
-?2~
1
9p
-1
A(-\ + jO) =
s N +s D
*n,Rly p,R T ^d^P'R
sT N +sT D
gR
2xp
b(-\ + jO) =
Sn,Rn0,R Sd,R^O,R
sT n d
0,/ 3d,Ju0,l
R2
(5.16a)
(5.16b)
Proof From Theorem 5.1, the condition -1 + j0 <£V(co) for all frequencies co is
necessary and sufficient for ensuring robust stability. The result follows by applying
Theorem 5.3 to the special case of the point w = -1 + j0. Q.E.D.


89
where q¡ and q¡ i =1,2, ...,/? are finite real bounds. Equations (5.9a)-( 5.9b) define a
class of finite-dimensional, linear, time-invariant, real systems with affine parametric
uncertainties. For completeness, the perturbation family A is implicitly understood to be
the set A:= {£($,#) = g(s,q)~ g(s,q0) \ q eQ] for this class of uncertainties.
The value set co) at a given frequency co is defined as the set of the Nyquist-
plane points g(jco,q) obtained for all q uncertainty value set V(co) and E(Q) represent the 2p~l p edges of the bounding set Q.
Furthermore, let g(jco,E(Q)) represent the frame of the value set, namely, the image of
the edges of Q on the Nyquist plane under the mapping Two important
properties of the value sets generated by systems with affine uncertainty are the following
(Fu, 1990): (i) at each fixed frequency the boundary dV(co) of the uncertainty value set
V(co) is spanned by the image of the edges of Q, e.g., dV(co) is spanned by the frame of
the value set; (ii) the image of each edge of Q is either a circular arc or a line segment that
can be easily calculated analytically. The second property is a consequence of the affine
structure of the uncertainty which induces a linear fractional mapping. In the following
sections we exploit these properties to develop a computational approach to find the
Nyquist stability margin for affine uncertain systems. The results allow the efficient
verification of the set membership (5.7) invoked in Theorem 5.1 via a linear feasibility
problem, and permit the calculation of the robust stability margin invoked in Theorem 5.2
via a systematic algorithm.
5.4. Robust Stability and Uncertainty Value-Set Membership
The first step in the computation of the generalized critical perturbation radius for
the uncertain system (5.9a)-(5.9b) is to determine whether the critical point -1 + y'O


138
whose right-hand sides unambiguously define the mappings fK:(a>,ag)->aK and
fT:(o),a0) > aT. Note that for a given coordinate ae the mapping fx defines a one-to-
one correspondence between eQ, and aT< 0. This implies that the inverse map
fx':(aT,ae) > co exists. Next, set ag = ag and aT = aT in (7.12b) to define
aT = fT(co,ag) and solve the equation to obtain co = f~\aT,ag) with rueQ, as
prescribed by Theorem 7.2. Finally, substitute the frequency value into (7.12a) to obtain
aK = fK(f~\aT,a0),agy.= f(aT,a0). Hence, the map (7.15) is readily computable. It
may be of interest to remark that in general f(aT,ag) does not have a closed-form
expression because f~\ccT,ag) itself does not have a closed-form expression. Extensive
numerical studies have shown, however, that approximate close-form expressions for
/' can be obtained with high accuracy through least-squares fit to simple functional
forms. Nonetheless, this venue is not pursued further in this paper. Finally, the system is
stable with respect to the arbitrary multiplicative uncertainty q = [aK aT a0]T if the
uncertainty satisfies
aK Using this expression it can easily be determined if a given set of uncertainties
destabilizes the system for a specific tuning rule and its associated choice of tuning
parameter. Furthermore, it follows that a complete characterization of the region of
stabilizable multiplicative perturbations is compactly given by
0max:= {q = Ia K az ae]T :00,ag>0)


ACKNOWLEDGMENTS
I would like to express my sincere appreciation to my advisor, Oscar Crisalle,
without whose support this dissertation would not have been possible. Im also grateful
for the opportunity he has given me to obtain a masters degree in electrical engineering.
I wish to thank Professors Richard Dickinson, Dinesh Shah, Spyros Svoronos, and
Haniph Latchman for serving on my supervisory committee. It was reassuring knowing
that no matter how mathematically intense my research became Dr. Latchman always
understood where I was and where I needed to go.
I thank V. R. Basker, Jon Engelstad, Serkan Kincal, and H. Mike Mahon who
have led the way and Chris Meredith and Brian Remark who will follow. They all have
not only contributed greatly to my research but have been good friends.
Finally, I wish to thank my family. In particular, my loving wife, Holly, and my
wonderful son, C. J., for their unending support.
IV


19
singular vectors is aligned with the corresponding right singular vectors yj and y2. The
problem becomes finding a matrix U such that
[*; *2]=[xi x2]u (2.ii)
[y, yj] = [y, y2]u (2.12)
and
with unitary constraint
y
1
u*u = i
The solution can be found by solving the system of equations that equates the moduli of
the elements of x, and y, and that constrains the arguments of elements of x, and y to
differ by 0, where the unknowns are the elements of U and the variable 0. Although
this is a simple problem in complex algebra, the resulting set of equations have many
terms and are relatively cumbersome. Further theoretical work in this area is discussed in
Chapter 3. Therefore, an alternative method is used to solve the problem. First, from
Lemma 2.2 it follows that the right eigenvector v, is also right singular vector y,;
therefore, if U = [u, u2] then the first part of the problem becomes finding a
normalized such that
y. =vi=[y. y2],
The normalized least squares solution to (2.13) is
ui
[y. y2]+vi
bi y2]+v,
0.5548 + 0.8057/
0.2072 + 0.0126/
(2.13)


11
X >
y x
For a dual pair (x0,y0) under a homogenous norm it follows that
Rey*x0 = |y0||D||xo|| yoxo which implies that Rey*x0 = yoxo- Hence, for a strictly
homogenous norm every pair of dual vectors (x0,y0) is also strictly dual pair. In
addition, there exist a strict dual y0 for any x0 ^ 0 and a strict dual x0 for any y0 0.
Furthermore, the concept of approximately dual vectors is proposed such that a pair
(x0,y0) is approximately dual if ||y0||
xo =
y0xc
= i.
In general, the dual norm of a p-norm ||x|| /=(Z\ xi\p)Vp is the associated p-norm
, where X! p + \! q = \. So the infinity-norm and the 1-norm are duals, and the dual
norm of the 2 (Euclidean) norm is itself. For the 2-norm, a pair (x0,y0) is dual if
y0 = x0 /||x0||;, and approximately dual if y0 = ej0xo /IbcJr .
2.2.3. Dual Eigenvector Result
The basis of the following Lemma is a result of Bauer (1962) on the field of
values of a matrix.
Lemma 2.1. If the spectral radius of a matrix A e C"*" is equal to the maximum
singular value of A, then for each normalized right eigenvector \¡ associated
with an eigenvalue of maximum modulus A, (A) there exists a normalized left
eigenvector w(. = vf such that v(. and wf form a dual pair wf|| v,..
Proof Lemma 2.1 is a specialization of Bauers result to the case of the
Euclidean norm, and is therefore in terms of the maximum singular value of the matrix.


127
M-.= 0o(ae-\) (7.5c)
so that K = K0 + AK, z = z0 + Az, and 6 = 60 + AO. Also, a scaled additive perturbation
is defined as
Aq: =
such that q = q0 + Aq The strict positivity of the multiplicative parameter in (7.3a)
implies that the sign of the gain is invariant in Q, a constraint that is normally met in
applications since often the sign of the gain is known from the physics of the underlying
problem or from experience. Even though it is possible to extend the analysis to include
negative values of a^, that case is not of practical interest and is therefore not
considered further in this study.
7.2.2. Proportional-Integral Control and Controller Tuning Rules
The proportional-integral controller considered in Figure 7.1 is of the classical form
(
c(s) = Kc
1 + -
1
(7.6)
v T Is
where the controller gain Kc 0 and the integral time-constant z, > 0 are adjustable
parameters. The PI controller (7.6) ensures the offset-free behavior of the closed loop in
both the standard servo-control problem (tracking of step changes in the set point in the
absence of disturbances) and in the standard regulation problem (rejection of step
changes in the disturbance while the set point remains constant). Tuning consists of
prescribing values of the control parameters that ensure satisfying a specific performance
criterion of interest to the control designer.


133
characteristics of the system. Hence, the robust stability characteristics of the system are
only dependent on 00 / r0 and the particular tuning rule selected.
7.3.2. Parametric Boundaries for Robust Stability
The theoretical development of the previous section is valid for any arbitrary
. . 2
uncertainty domain Q that is a simply-connected open subspace of consisting of
strictly positive elements and that contains the nominal point q0 = [l, 1, l]. This section
seeks to characterize the largest uncertainty domain for which the closed-loop system is
robustly stable, thereby giving the region of all stabilizable multiplicative perturbations of
the nominal parameters. Such a region, denoted Qmax, is fully described by its boundary
dQmax. Obviously, any vector q for which aK- 0, aT= 0, or ae = 0 is a possible
element of the boundary of Qmax. The challenge is to find all the strictly positive vectors
q that are elements of dQmax. First note that for every strictly positive element
q = [ctKazae]T of the parametric robust-stability boundary, the characteristic quasi
polynomial £(s;q) is not Hurwitz and the family of quasipolynomials produced by
Q={ q + Aq, ||Aq|| < £} contains at least one element that is Hurwitz, where e is an
arbitrary positive real scalar and ¡| is a vector norm. This fact gives rise to the following
theorem.
Theorem 7.1. If the strictly positive uncertainty vector (\ = [aKax ag]T is an
element of the parametric robust-stability boundary dQmax for the closed-loop
system of Figure 7.1 with PI control parameters adjusted using a tuning
correlation from Table 7.1, then q must satisfy the parametrized map


44
Cr(DMD ') i i2 | |2n| 12
&() = (K.l -|tI )N
a
i
^(DMD'1) i, | /7
2 (|x.,||x.2|cos(Zx, -
dt
ct(DMD )
d
i
g(DMD ')
d
(k.K|sin(A. -
(k2|2-kl2)k
^xn) \y¡\lk/21 cos(Zy., Z.y¡2))cos(Zw, Zm2)|m,||m2|-
A-2) \y,. Ik/21 sin(A, y,2)) sm(zw, z2 )|, ||21 +
which is of the form (3.20) where the elements of H are defined by (3.21)-(3.24)
respectively.
There are now three cases to consider. The first case is when n = 2. This is a
trivial case, in that the optimal similarity scaling is given by the Perron scaling.
Therefore, there is no need to further investigate the properties of the subdifferential
when n = 2 The other two cases are when n- 3 and when n > 3 As will be discussed
shortly, the case when n = 3 is a degenerate case of the more general case n > 3.
Therefore the case when n > 3 will be discussed next followed by the case when n = 3.
The first result is that the subdifferential given by (3.19) is contained within an affine set
of dimension 3. The result is stated in the following theorem.
Theorem 3.7. For n> 3 and d such that the maximum singular value
f (d) = the affine set S = {z e Rn Pz = q} where elements of the matrix
P\, 1
Pl,l
Pi, 3
1
0
... 0
p =
Pl,\
Pl,2
P2,3
0
1
... 0
Pn-3.1
Pn-3.2
Pn-3,3
0
0
... 1


16
Therefore, for each orthonormalized right eigenvector v( there is a major input/output
principal direction pair that is aligned. Namely
y,(^) = v.
and
x,(A) = e"y,(A) (2.7)
where
6> = arg(A.(A)) (2.8)
Finally, there is always at least one right eigenvector v. of A associated with an
eigenvalue of maximum modulus T, (A); therefore, there must exist at least one major
input/output principal direction pair that is aligned, which completes the proof. Q.E.D.
Theorem 2.2 is a precise statement of the MPDA property. The theorem
eliminates any ambiguity that may result when applying the MPDA property as stated in
Theorem 2.1 to the case of repeated maximum singular values. In addition, the proof of
necessity makes well-designed use of the earlier work on dual vectors and dual norms,
and avoids the confusions associated with the earlier proof. This section is concluded
with a simple corollary that restates the MPDA property in the duality terminology,
namely
Corollary 2.1. The spectral radius of any matrix A e Cn'n is equal to the
maximum singular value of A if and only if there exists a major input and major
output principal direction pair of A that is approximately dual with respect to the
Euclidean norm.
Proof It suffices to show that approximate duality of a major input/output
principal direction pair with respect to the Euclidean norm is equivalent to alignment of


48
c.
+K*
c =
C2
= 0.5
^2,2
^2,4
_C3_
J13,2
+ b3 A _
(3.30)
and the matrix B which characterizes the length of the axes of E and its orientation has
the form
B =
K\
b\,2
*1.3
Ki
^2,2
^2,3
A 3
^2,3
b-¡,3 _
\2,b
,3 A,3
} can
(3.31)
where the 6 parameters {bu,b22,b33,bh2,bl3,b23} can also be expressed in terms of the
constants hi} s. These expression can be obtained by picking six different values of u
with iTu = 1, setting z = [g, (u) g2 (u) g3(u)]T and then solving the resulting system
of six linear equations in terms of {bu,b22,b33,bi2,bl3,b23} obtained from (3.29).
Unfortunately these expressions are vary cumbersome, and therefore in practice it is
easier to just solve the system of six linear equations resulting from the numerical data of
the particular problem.
The following theorem combines Theorem 3.7 and the above result that
{&i(uXg2(u)g3(u)} with u u = 1 is an ellipsoid to give a useful characterization of
df{ d).
Theorem 3.8. For n> 3 and d such that the maximum singular value
f (d) = j(DMD ') has multiplicity 2, the subdifferential df (d) is given by
d/(d) = {zei?nPz = q, ([z, z2 z3]-cT)B([z, z2 z3]t-c)<1}
where constants P and q are given by (3.27) and c and B by (3.30) and (3.31).


47
dimensional convex set in an n -dimensional space. This means that the first 3 terms of
g(u) (i.e., {g1(u),g2(u),g-3(u)}) describe df(d). Therefore, to complete the
characterization of df(d) it is only necessary to investigate conv{g](u),g2(u),g3(u)}
for u*u = 1, and then translate this 3-dimensional set to the Rn using the affine functions
given in Theorem 3.7.
The convex hull, conv{g1(u),g2(u),g3(u)|u*u = 1}, is now shown to be a 3-
dimensional ellipsoid, and thus df(d) is a 3-dimensional ellipsoid. Consider equation
(3.20), even though u = [|q |eyZu' |m2]T has 4 parameters (i.e., |m,|, \u2\, Zux, and
Zw2), the function g(u) with u*u = 1 is a function of only 2 parameters. One of the
parameters is xu = |m,| and the other is 0U = Zw, -Zm2. The reason \u2\ is not a third
parameter is that u*u = 1 necessarily requires \u2\ = -^1 -|ux\~ Now consider a fixed
value of xu, the terms {g1(u),g2(u),g-3(u)} are of the form
g, (u) = eu + el 2 cos(0u) + e1>3 sin(^)
g2 (u) = e2A + e2>2 cos(6>u) + e2>3 sin(6>)
S3 (U) = e3.1 + e3.2 COS(0u ) + g3,3 SnC61 )
which is obviously a parametric representation of a 2-dimensional ellipse in a 3-
dimensional space centered at [eu e2] e31]T To satisfy u*u = l, |m,| must be an
element of [0,1]. Therefore, varying xu over its admissible range of 0 to 1 generates a
set of ellipses which form the surface of an ellipsoid. This ellipsoid is given by
E = {z ei?3|(z-c)TB(z-c) = 1} (3.29)
The center c of the E is given by


105
boundary occurs, and it does not reenter the value set until it realizes the second
intersection with the boundary. As a consequence of this geometry, the critical
uncertainty value set ^c(co) is the union of two disjoint line segments, and therefore
Vc( When applying Theorem 5.4 to this system for a particular frequency, the only
modification to the feasibility problem presented for the previous example concerns the
form of the inequality constraints (5.16b) which now adopt the form
1
0
o'
"10"
-1
0
0
10
0
1
0
0.3
0
-1
0
0.3
0
0
1
0.3
0
0
-1
0.3
(5.26)
instead of the form (5.24). The vector and matrix definitions (5.21c)-(5.21f) remain
unchanged. As in the previous example, it can be readily verified using an active-set
method that the linear equality/inequality problem (5.16a)-(5.16b) with the data
corresponding to this example is infeasible. Invoking Theorem 5.4, it then follows that
-1 + j0 g V(cd) and hence it can be claimed that that at co = 0.95 the value set excludes
the critical point.
The Nyquist robust stability margin kK(co) and the critical perturbation radius
pc(co) can now be calculated at any frequency using the method described in Section 5.5.
At the frequency co = 0.95, the set of intersections of the critical line r(co) with the frame
g(jco,E(Q)) is
F = (-0.6510 y'0.3738, 0.4196 y 0.6217, 0.6349 j0.3911,
. 0.4403 j0.5995, 0.6512 j0.3736, 0.6498 y 0.3751}


174
a.-I
COt(*) =
T, To
CO
and
tan( Tj T0
CO*
*S/* / \
respectively. Hence, any positive scaled frequency co* at which f(co) and are
dco
simultaneously zero must satisfy
-co
Tp &o
Ti To
at
^ae-1
T, T Q
CO*
or
co
1
Tp &o
T, T0
at
l-Iaia-a,
K Ti T0 j
(F.3)
^/* / \
Also, any positive scaled frequency co* at which fd(co) and are simultaneously
dco
zero must satisfy
co
T o Op
T i T0
at
T l Tp
*
CO
which also reduces to (F.3). Finally, the frequencies given by (F.3) must satisfy (F.la)
for f(co) to be zero and (F.lb) for fd(co) to be zero. Substitution gives


7
In addressing the existence of solutions to the proposed optimization,
Kouvaritakis and Latchman introduce the major principal direction alignment (MPDA)
property (1985). The result states that the spectral radius of a matrix is equal to the
maximum singular value of the matrix if and only if the major input and the major output
principal direction of the matrix are aligned. MPDA is a strict condition for a matrix, but
can be used to determine the optimal positive diagonal matrix and unitary matrix that
result in equality between the aforementioned definition of ¡j. and the maximum singular
value upper bound. The proof of the MPDA principle is based on linear algebra
arguments, and considers separately the cases of a unique and a repeated maximum
singular value. For either case the proof of sufficiency is straightforward. The proof of
necessity for the case of a unique maximum singular value is precise but not as clear-cut.
On the other hand, the proof of necessity for the case of a repeated maximum singular
value is slightly ambiguous.
The inspiration for the MPDA principle is early work on determining when the
spectral radius equals the maximum singular value for positive matrices transformed by
non-negative diagonal matrices (Stoer and Witzgall, 1962). One motivation for the focus
on positive matrices is that they have good numerical properties (i. e., less round off error)
and therefore may be used for conditioning of matrices. In addition, positive matrices
remain positive under transformations by non-negative diagonal matrices leading to
connections to Perron-roots /r(M) (positive eigenvalues of largest modulus) of positive
matrices M (Ortega, 1987). These results on positive matrices are based on the
mathematical concepts of dual norms and dual vectors utilized by Bauer (1962) which
lead to elegant proofs for many of the results.


CHAPTER 2
A DUALITY PROOF FOR THE
MAJOR PRINCIPAL DIRECTION ALIGNMENT PRINCIPLE
2.1. Introduction
The structured singular value, /i(M), defined as the supremum of the spectral
radius of MU over diagonal unitary matrices U (Doyle, 1982), is a widely accepted tool
in the robust analysis of linear systems. It considers the problem of robust stability for a
known plant subject to a block-diagonal uncertainty structure under feedback. In general,
any block-diagram interconnection of systems and uncertainties can be rearranged into
the block-diagonal standard form. Calculating /u is not trivial; in fact the underlying
optimization problem has been proven to be NP-hard (Braatz et al., 1994). The difficulty
is that the spectral radius is non-convex over the set of unitary matrix transformations.
One approach is to consider upper bounds for the spectral radius that can be calculated
easily, and ideally should be attainable to eliminate conservatism. The maximum
singular value is a reasonable choice for an upper bound because it is invariant under
unitary matrix transformations. In addition the maximum singular value upper bound can
be decreased by optimizing over similarity transformations, because the spectral radius is
invariant under such transformations. Ultimately, the problem becomes one of
conditioning a matrix through optimal similarity and unitary transformations to achieve
equality between the spectral radius and maximum singular value. Therefore,
determining the conditions under which the upper bound is attained is a significant issue
in the field of robust control.
6


69
lub(Do'PD0)
Re{(w*D0)(D-1PD0)(D-1v)}
which from (4.7) equals /r(D¡'PD0). Therefore, (4.1) holds where the minimizing D is
given by (4.4).
The relationship of Stoer and WitzgalTs positive matrix result to the spectral-
radius/maximum-singular-value problem can be shown by specifying the least upper
bound norm to be subordinate to the Euclidean norm, i.e.
lub(A) = cr( A) (4.9)
where A e Cnxn. Combining (4.9) and the fact that the Perron-root of a positive matrix is
the spectral radius, (4.1) becomes
p(P) = min <7(DPD) (4.10)
for positive matrices P and positive diagonal matrices D. In addition, from (4.4), there
is an analytical expression for the optimizing D0 given by
D0 = diag
( ,4/2 ,4/2
4/2 4
Vw>
>/2 wl/2
w
1/2
n y
(4.11)
where v > 0 and w > 0 are right and left Perron-vectors of P. Clearly, (4.10) shows that
for positive matrices there is a simple similarity transformation for which the spectral
radius attains its the maximum singular value upper bound.
4.2.3. Major Principal Direction Alignment Property
In solving various robust control problems it is necessary to determine the
conditions under which the spectral radius of a matrix attains its maximum singular value
upper bound. The major principal direction alignment (MPDA) property addresses this


18
and the singular values are
{o-t, The spectral radius equals the maximum singular value, i.e.
N = P(A) = ^(A) = cr, =cr2
In this case the eigenvalue of maximum modulus is unique and non-repeated, and the
maximum singular value is repeated. An inspection of the left and right singular vectors
reveals that x, ej9yx and x2 e'dy2 which appears to contradict the MPDA Theorem
2 which states that there must exist at least one major input/output principal direction pair
that is aligned. This apparent contradiction can be resolved by realizing (2.10) is only
one possible orthonormal eigenbasis of A*A whose vectors are right singular vectors.
Different orthonormal eigenbasis of A*A are achieved through unitary transformations
of the orthonormal bases of the eigenspaces of A* A associated with each particular
singular value. The eigenspace of A* A associated with a non-repeating singular values
is rank one; therefore an orthonormal basis consists of only one vector and the only
unitary transformation of this basis is of the form ejd. On the other hand, the eigenspace
of A*A associated with a repeating singular value has rank greater than one, and
therefore an orthonormal basis consists of more than one vector and a unitary
transformation of this basis is a unitary matrix whose size is the rank of the
corresponding eigenspace.
Hence, for this example, there must exist a unitary matrix that transforms the left
singular vectors x, and x2 into x, and x2 such that at least one of the transformed left


157
combining the previous two equations gives
cr(A)<
A W;
(A.2)
The maximum singular values of A and A* coincide, therefore by definition
cr(A) = l|z||=l
A z
(A.3)
Inequality (A.2) implies that w(. maximizes definition (A.3) such that
o-(A) =
A w.
which from the assumption gives
A,.(A) =
A w.
or equivalently
A w.
A,.(A)
= 1
(A.4)
Combing (A.l) and (A.4) with vf = 1 gives
W; A
T,.(A) '
V; =
A w.
T,.(A)
v; =1
Meaning
A w.
A,. (A)
y¡
D
But the dual of vf is uniquely determined to be w(, therefore
A w,
= w;.
A,. (A) '
or
w*A = A,.(A)w*
Such that w(. is left eigenvector of A and is dual to v.
Q.E.D.


64
property (1985). The result states that the spectral radius of a matrix is equal to the
maximum singular value of the matrix if and only if a major input principle-direction and
a major output principal-direction of the matrix are aligned. MPDA is a strict condition
for a matrix, but can be used to determine the optimal positive diagonal matrix and
unitary matrix that results in equality between the afore mentioned definition of /u and
the maximum singular value upper bound for the case when the maximum singular value
is distinct.
It is the goal of this work to establish relationships between results obtained from
different perspectives of the same spectral-radius/maximum-singular-value equivalence
problem. To this end, the earlier work by Bauer (1963) on positive matrices is extended
to the class of general complex matrices. The results are necessary conditions for
equality that are used to improve the calculation of /u through its upper bound.
4.2. Mathematical Background
4.2.1. Dual Norms and Dual Vectors
In the theoretical development that follows the mathematical concepts of dual
norms and dual vectors are utilized. These concepts are explained in a paper by Bauer
(1962) and are reviewed here to facilitate the theoretical development. Given a vector
norm ||| its dual vector norm ||-|| is defined as
^ Reyx
y L := max Key x = max¡pr
1 llD IMH IMI* ||x||
For such dual norms the Holder inequality
||y|D||x|>Rey\


124
(ITAE). A summary of relevant correlations is given in Section 7.2 of this paper. All of
the tuning correlations mentioned have been carefully developed by their authors to
approximately satisfy specific performance criteria, and to ensure that the closed loop is
stable. More recent work on similar controller synthesis methods is given by Schei
(1994), Langer and Landau (1999), and Kristiansson and Lennartsson (1999), but these
design methods are not further explored in this paper, because the earlier works are more
often cited and are therefore more reasonable candidates for the robust analysis technique
presented.
Although in all cases the tuning correlations recognize that model (7.1) is an
approximation, they do not address the fact that the process of parameter identification is
inherently affected by uncertainties. More specifically, the gain, time-constant, and time-
delay parameters may be respectively affected by errors AK, At and A6. This paper
seeks to quantify the robust stability of classical tuning correlations for PI controllers with
respect to uncertainties in the gain, time-constant, and time-delay parameters of the
model.
A major objective of this paper is the calculation of a meaningful parametric
stability margin for this class of systems, which can then be used as a quantitative metric
to compare alternative PI controller tuning rules with respect to robust stability. While
standard gain and phase margins may also be used to get some idea of robustness, it is
well known that gain and phase margins may be fragile safeguards in the presence of
system uncertainties, as for example in case of the state feedback design problem. The
results of this paper provide mechanisms for rigorously resolving these robustness issues
for the case of PI tuning controllers for systems approximated by first-order-plus-time-
delay models.


109
with its nearest boundary point). This effect is realized at a frequency slightly higher than
co = 0.9417 for the example under consideration, and the value of pc(co) must be
calculated using the tangent point. As the frequency is reduced slightly below the value
co = 0.9417 the critical line is no longer tangent to the template, and its intersection with
the value set defines a single continuous segment, hence, Vc(a>)recovers its convexity.
The calculation of pc{co) is now made using the only existing boundary point, which is
now located closer to the nominal point. Hence, this local decrease in frequency in the
neighborhood of co 0.9417 requires that the boundary point selected for the calculation
of pc(co) be changed from the tangent point (located closer to -1 + j0) to a non
neighboring point that is located closer to the nominal point; this explains the
discontinuity pc{co) which in turn causes kN(co) to be discontinuous.
5.8. Conclusions
The main contribution of this paper is the generalization of the critical direction
theory proposed to analyze the robust stability of systems whose critical uncertainty value
sets are nonconvex. The generalization is obtained by introducing a new definition of the
critical perturbation radius, and the effectiveness of the generalized theory is validated by
its success in assigning a quantitative robust stability measure, namely, a computable
Nyquist robust stability margin, to problems that involve affine parametric uncertainties
characterized by real vectors that belong to a rectangular polytope. For the case
considered, the calculation of the Nyquist robust stability margin involves planar
geometry operations and solving a series of linear equality/inequality feasibility problems
that do not pose major computational challenges.


CHAPTER 7
ROBUSTNESS OF CLASSICAL PROPORTIONAL-INTEGRAL
CONTROLLER DESIGN METHODS
7.1. Introduction
The classical proportional-integral-derivative controllers in use since the early 40s
are now all-pervasive in industrial applications. In particular, the proportional-integral
(PI) version of this classical controller has found widespread application in processes
where the presence of measurement noise does not permit taking advantage of the
beneficial effects of the derivative action. PI control ensures offset-free performance, and
through adequate tuning choices is able to deliver aggressive or sluggish responses, as
desired by the control designer. Petroleum refining plants may include as many as two
thousand PI control loops, while other applications, such as the precise regulation of force
at the tip of the stylus of an atomic-force microscope, may involve a single such loop.
A PI controller that is poorly tuned may render the loop inherently unstable, and in
a practical application this instability often leads to a permanent saturation of the
manipulated actuator, thus rendering the controller completely ineffectual. Unfortunately,
when the process involves a large number of PI loops, instability-induced saturations due
to inadequate tuning may remain largely unnoticed as long as a subset of key control
loops remain saturation free. The net effect is a decrease in the overall performance of
the entire control system because the unstable loops are unable to contribute their share to
improving the dynamic quality of the variables they manipulate. Since PI controllers are
tuned using a number of classical correlations stemming back to the famous Ziegler-
122


50
df (d) is that given by Theorem 3.6. As is shown in the next section, this still has some
utility in determining a steepest descent direction.
3.4. Determining the Steepest Descent Direction and Conditions for a Minimum
When the maximum singular value is distinct the gradient exist and the steepest
descent direction is given by -V/(d) / ||V/(d)||. Furthermore, the necessary and
sufficient condition for a minimum is V/(d) = 0 When the maximum singular value is
repeated the results of the previous section and Theorem 3.4 and Theorem 3.5 can be
used in a steepest descent optimization algorithm. First the case when the when the
maximum singular is repeated once is considered, because the ellipsoidal characterization
of 5/"(d) results in a convex optimization problem for determining the steepest descent
direction. This is followed by the more general case when the maximum singular value is
repeated more than once.
Using Theorem 3.5 and the ellipsoidal characterization of (d) given by
Theorem 3.8, the subgradient that gives the steepest descent direction is now given by the
optimization
£sd(d) = argmin
(3.32)
such that
P^ = q
(3.33a)
and
([£ £2 £3]-ct)B([£. £2 ^]J-c)<\
(3.33b)
Optimizaiton (3.32) with constraints (3.33a) and (3.33b) represent the minimum distance
from the origin to the ellipsoid df (d). Obviously, the objective function of optimization


149
6>0 / r0 giving qualitatively similar results. It is also possible to easily compare
quantitatively the various tuning rules over a range of tuning parameters. The following
figures show the value for the gain margin, phase margin, and parametric stability margin
for all the tuning rules being considered, over the accepted range of the tuning-parameter
ratio, namely 0.1 < 0O / r0 < 1.0.
Controller manufactures recommend that a well-tuned controller have a gain
margin between 1.7 and 2.0 (Seborg, 1989). Figure 7.5 shows that only the ITAE
regulation (ITAE-reg.) tuning rule satisfies this recommendation over the recommended
range of the tuning parameter 60 / r0.
Figure 7.5. Gain margin of the tuning rules vs. the tuning parameter.
The Ziegler-Nichols (ZN), Cohen-Coon (CC), and IAE regulation (IAE-reg.) tuning rules
are the next closest to this recommendation having gain margins in the range 1.45 to 2.25.
The figure also shows that the ISE regulation (ISE-reg.) tuning rule has poor stability


178
Kouvaritakis, B. and Latchman, H. (1985). Necessary and sufficient stability criterion
for systems with structured uncertainties: the major principal direction alignment
principle, International Journal of Control, vol. 42, no. 3, pp. 575-598.
Kristiansson, B. and Lennartsson, B. (1999). Optimal PED controllers including roll off
and Schmidt predictor structure, IFAC99, 14th World Congress ofIFAC, vol. F, pp.
297-302, Beijing, P.R. China.
Lancaster, P. and Tismenetsky, M. (1985). The Theory of Matrices, 2nd Edition,
Academic Press, Inc., San Diego, CA.
Langer, J. and Landau, I. D. (1999). Combined pole placement/sensitivity function
shaping method using convex optimization criteria, Automtica, vol. 35, no. 6, pp.
1111-1120.
Latchman, H. A. (1986). Frequency Response Methods for Uncertain Multivariable
Systems, Doctor of Philosophy Dissertation, Oxford University.
Latchman, H. A. and Crisalle, O. D. (1995). Exact robustness analysis for highly
structured frequency-domain uncertainties, Proceedings of the American Control
Conference, Seattle, WA, pp. 3982-3987, IEEE, Piscataway, NJ.
Latchman, H. A., Crisalle, O. D., and Basker, V. R. (1997). The Nyquist robust stability
margin A new metric for robust stability, International Journal of Robust and
Nonlinear Control, vol. 7, pp. 211-226.
Lopez, A. M., Miller, J. A., Smith, C. L., and Murrill, P. W. (1967). Tuning controllers
with error-integral criteria, Instrumentation Technology, vol. 14, no. 11, pp. 57-62.
Luenberger, D. G. (1989). Linear and Nonlinear Programming, Addison-Wesley, Menlo
Park, CA.
MacFarlane, A. G. J., (1982). Complex Variable Methods for Linear Multivariable
Feedback Systems, Taylor and Francis, London.
Murrill, P. W. (1967). Automatic Control of Processes, International Textbook Co.,
Scranton, PA.
Nyquist, H. (1932). Regeneration theory, Bell Systems Technical Journal, vol. 11, pp.
126-147.
Ogata, K. (1990). Modern Control Engineering, Prentice Hall, Englewood Cliffs, NJ.
Ortega, J. M. (1987). Matrix Theory, Plenum Press, New York.


42
(3.18) degenerates to a point. This and other degenerate cases should be taken into
consideration when using Theorem 3.6.
3.3.2. Characterization of the Subdifferential as an Ellipsoid.
When the maximum singular value is distinct, the subdifferential is given by the
point df(d) = V/u(d;u) = V/(d), where V/(d) is given in Section 3.2.5. The next step
is to explore the case when the maximum singular value is repeated once. In this section,
it is shown that df (d) is an ellipsoid when the maximum singular value is repeated once
by examining the properties of the function V/U(d;u). To simplify the notation define
the vector valued function g:C2 -> Rn as g(u):= V/U(d;u) where d is a fixed point
such that maximum singular value /(d) = Theorem 3.6, this gives
df(d) = conv{g(u)|u*u = l,u eC2}
(3.19)
To analyze (3.19), g(u) is expressed in terms of u = [\u] \e^Ux \u2 \eJ:u2 ]T as
(3.20)
|2
W,
where the elements of the constant matrix H are given by
(3.21)
*f2 COSI
(Zxn Zxi2) -1ynIyi2\cos(Zyn Zyi2)) (3.22)
K s = -2
5^(DMD )
d,
(Ki lk/21 sin(Z* Z^,.2) \yn Iyi2\ sin(Zyn Zyi2)) (3.23)


58
3.7.2. Example 3.2.
The following example is taken from Daniel et al. (1986). Let M = AB, where
0.65012 + 0.00000/ 0.00000 + 0.00000/
0.45970 + 0.00000/ 0.45970 + 0.00000/
0.45970 + 0.00000/ 0.00000 + 0.45970/
-0.39322 + 0.00000/ -0.53729 + 0.53729/
and
B =
0.00000 + 0.00000/
0.45970 + 0.00000/
0.45970 + 0.00000/
0.53729-0.53729/
0.65012 + 0.00000/
-0.45970 + 0.00000/
0.00000-0.45970/
0.39332 + 0.00000/
Again, in performing the infimization inf cr(DMD ) the point d = [l 1 1 1]T
corresponding to D = I has a maximum singular value repeated (i.e., cr^M) = ct2(M) = 1, with objective function /(d) = a(DMD_1) is non-differentiable at d = [1 1 1 1]T and the
results of the this chapter are used to solve the optimization by either determining a
steepest descent direction from the point d = [1 1 1 1]T or by determining if the point
satisfies the optimality and MPDA conditions.
The ellipsoidal characterization of the subdifferential is given by
d/(d) = (z eRn Pz = q, ([z, z2 z3]-cT)B([z, z2 z3]t-c)<1}
P = [l 1 1 l]
q = 0
where


31
(\-A)x + AyeC Vx e C,y e C,A e(0,l)
All affine sets are convex, as are half-spaces. A vector sum
Alxl+A2x2+-"+Amxm
is called a convex combination of x,,x2,---,xm if the coefficients A¡ are all non-negative
and Aj + A2+---+Am =1. A subset of Rn is convex if and only if it contains all the
convex combinations of its elements. The intersection of all the convex sets containing a
given subset S of Rn is called the convex hull of S and is denoted convS\ Necessarily,
convS is the smallest convex set containing S. In addition, for any S c Rn, convS
consists of all the convex combinations of the elements of S. In general, by the
dimension of a convex set C one means the dimension of the affine hull of C.
A supporting half-space to a convex set C is a closed half-space which contains
C and has a point of C in its boundary. A supporting hyperplane to C, is a hyperplane
which is the boundary of a supporting half-space to C. As such, a supporting hyperplane
to C is associated with a linear function which achieves its maximum on C. The
supporting hyperplanes passing through a given point a e C correspond to vectors b
normal to C at a as defined by (3.8).
Let a function /(d) be de defined on a convex set S a Rn (note, for the MPDA
problem /(d):= cr(DMD) where D = diag(d) and S is the positive orthant such that
DeD). In what follows, it is assume that the function /(d) is finite on its domain of
definition. The function /(d) is said to be convex on S if
/(d,+(l-)d2) < c/(d,) + (l-a)/(d2) Vd,,d2 eS, a e[0,1]


51
(3.32) is convex in the n parameters (£,,£2,,£} and the constraints (3.33a) and
(3.33b) are convex sets. This n -dimensional optimization can be reduced to a 3-
dimensional optimization by incorporating the equality constraints (3.32a) into the
objective function (3.32), because by Theorem 3.7, (3.33a) implies that {£4,£5,,£}
are affine function of {£, ,£2 ,£3}. The optimization given by (3.32) and (3.33) becomes
^sd(d) = argmin||4||2 = argminfe + £ + £ + Zw(?/ ~ A,i£i "Pi,2^2 A,3^3)' (3-34)
such that
([£, £2 ^3]-ct)B([^ £2 <^3]T -c) < 1 (3.35)
where the terms {f4,£5,,£} of £sd(d) are obtain from the affine functions of
{jj,£2,£3}. The objective function of optimization (3.34) is a positive semi-definite
quadratic function and is therefore convex. In addition, the constraint (3.35) is a convex
set. Therefore, determining the steepest descent direction when the maximum singular
value is repeated once reduces to a simple 3-dimensional convex quadratic optimization
over a convex set. Finally, from Theorem 3.8 the necessary and sufficient condition for a
minimum, i.e. 0 e df(d), reduces to
ctBc<1, q = 0 (3.36)
because [z, z2 z3]T=0 must be an element of the ellipsoid and when
[z, z2 z3]T = 0, the terms {z4,z5,---,z} are zero only when q = 0 (i.e., the affine set
S = {z Pz = q) must pass through the origin).
Now for the case when the maximum singular value is repeated more than once.
From Theorem 3.5, the steepest descent direction is obtained from the smallest


68
Therefore, the matrix
D0 = diag
rV<7 A
v*' ¥' XU
(4.4)
makes D0'v and D0w a dual pair for any right and left Perron-eigenvectors v > 0 and
w > 0.
Duality is only a necessary condition for (4.2). Therefore, to show (4.1) holds it
suffices to show (4.3) holds for those matrices D~'PD0 whose right and left Perron-
vectors Dq'v and D0w are dual, where D0 is given by (4.4). Using the definitions of
eigenvalues and eigenvectors it can be shown that
Re{(w-D0)(D0-|PD0)(D0",v)} = *(DlPD0)Re{(w'D0)(Dv)} (4.5)
and from the definition of duality of vectors it is true that
Re{(w'D0)(D¡'v)}
PoHId
d;'v
= 1
(4.6)
Combining (4.5) and (4.6) gives
Re{(w*D0)(D0-1PD0)(D1v)}
PcHL
Do'v
= ;r(D0'PD0)
(4.7)
Using of the bilinear characterization of the least upper bound
lub(A) := max
x,y*o
Re{y*Ax}
D
(4.8)
Stoer and Witzgal (1962) show there is a maximizing pair for (4.8) in the positive
orthant, and that the only maximizing pair in the positive orthant for lub(Do'PD0) is the
pair D¡'v and D0w such that


41
maximum singular value is non-differential x(DMD ') and y(DMD ) are given by
(3.5) and (3.6) and there exists a sufficiently small perturbation of d such that there
exists sequences x^D^MDj1) - x(DMD ') and y^D^MD^1) -> y(DMD ) which
are uniquely determined up a multiple of ej9 such that the gradients V/(dk) exist. All
that is left is to define V/U(d;u) by (3.17), which represents the limit of V/(d) as
d* -> d for some dA eZ), where all u such that iTu = 1 represents all possible limit
sequences dk > d for d* eD. Q.E.D.
Theorem 3.6 is the natural extension of the gradient result given in Section 3.2.5.
For the case when the maximum singular value is distinct, V/U(d;u) = V/(d) for all
u*u = 1 (i.e., u = u = e'e) such that df(d) = conv{V/(d)} = (V/(d)} as expected. On
the other hand, when the maximum singular value is repeated V/(d) does not exists.
Instead one has V/U(d;u), which is an extension of equation (3.16) for V/(d), in that
V/U(d;u) represents the vector obtain when equation (3.16) is evaluated at one of the
possible major output and input principal directions given by (3.5) and (3.6). Obviously,
V/M(d;u) is a subgradent, since it is an element of df(d). In fact, V/M(d;u) represents a
subgradent that is arbitrarily close to some V/(d*) where dA d. That is V/U(d;u)
for iTu = 1 represent the boundary of df{d). Note, that a repeated maximum singular
value does not necessary guarantee a non-differentiability. Consider the matrix M = I
where /(d) = ^(DMD'1) = The function V/U(d;u) = 0 for all d and u, such that df(d) = {0} = (V/(d)} where the
gradient exists and is identically zero. This is an extreme case where the set given by


112
a measure of robust stability. The process consists of two stages. First, the domain of
controller parameters that result in robustly stable closed-loop systems is determined.
Then, in a second stage the set of optimal robustly stabilizing controller parameters is
obtained by optimizing a performance functional over the domain found in the previous
stage. The result is a robustly stabilizing controller with specific optimal performance
characteristics. The Section 6.2 introduces the design methodology. Then, Section 6.3
gives a design example based on this methodology. Concluding remarks are made in the
final section.
6.2. Design Methodology
The robust stability analysis technique of the previous Chapter based on the Nyquist
robust stability margin can now be utilized in the design of robustly stabilizing
controllers. A common objective in robust systems synthesis is to design a controller that
is robustly stable and satisfies a nominal performance criterion. This is usually called
robust performance in the control literature. The robust stability of a controller can be
determined, for example, from the Nyquist robust stability margin plot across frequency
for the system containing the designed controller and the uncertain plant. If the system is
robustly stable and nominal performance is not required, then no further synthesis is
necessary. However, if nominal performance is requested, further design work is
required. To design for robust performance it is necessary to characterize the set of
robustly stabilizing controllers and performance objectives in a compatible manner.
In this chapter a practical two step design approach based on parameter space
methods (Siljak, 1989) is proposed. First the controller parameters that result in robustly
stable closed-loop systems are determined. Then, a performance objective is optimized
over the set of robustly stabilizing controller parameters, resulting in a robustly stabilizing


22
and singular value decomposition A = XZY*, where
X = [x, x2 x3] =
-0.0381 + 0.0008/
0.9550-0.2457/
-0.1616-0.0001/
0.8954-0.4364/
0.0414-0.0198/
0.0613 + 0.0444/
0.0789 + 0.0124/
-0.0952-0.1281/
-0.3854-0.9053/
Y = [y, y2 y3]
0.0000 1.0000 0.0000
0.9340+ 0.3268 0.0000 -0.0243-0.1422
-0.1138- 0.0887 0.0000 0.1537- 0.9775
and the singular values are
{ The spectral radius equals the maximum singular value, i.e.
|^i | = |A21 = |A3| = p{ A) = cr(A) = cr, = ct2 = o-3
where there are two eigenvalues of maximum modulus with one being non-repeated and
the other having a multiplicity of two. The maximum singular value is associated with a
repeated singular value of multiplicity three. Again, inspection of the left and right
singular vectors reveals that x, + ejdy,, x2 + ejdy2, and x2 ^ ejey2. From Theorem 2.2,
it is known that there is at least one major input/output principal direction pair that is
aligned, but it is not apparent if there are more than one. The possibility exists that all
three can be aligned through a unitary transformation, because there are three
independent eigenvectors associated with the eigenvalue of maximum modulus. The
unitary matrix that transforms all three singular vectors such that all input/output
principal directions pairs are aligned is not found by solving the resulting system of
complex algebraic equations, because the equations are even more cumbersome than
would for the previous example. In fact, in this example, the existing singular value
decomposition as not transformed at all. Instead, an alternative singular value


130
S(s; q) = aKK0Kc(Tjs + \)e~a()9oS + cctt0tiS2 + Tjs (7.7)
where q e Q, and where the PI control parameters Kc and r¡ are selected using a tuning
correlation taken from Table 7.1 and are therefore functions of qg. It is known that the
nominal quasipolynomial 8{s\ qg) is Hurwitz, because the tuning correlations considered
in Table 7.1 yield controller settings that guarantee the stability of the nominal closed
loop. Let
(i):={ denote the family of quasipolynomials generated by (7.7) for all q eQ, where the
3
uncertainty domain is a simply-connected open subspace of 91 consisting of vectors
with strictly positive elements. The robust stability of the closed-loop with respect to an
uncertainty domain Q is ensured if and only if the entire family of quasipolynomials
,d(s) is Hurwitz. This in turn can be ensured through the zero-exclusion principle
enunciated in the following theorem.
Lemma 7.1. Given the parametric uncertainty q^Qand the family of
quasipolynomials A(s) of constant degree whose nominal quasipolynomial
>(s;qg) is Hurwitz, then every element of zl(s) is Hurwitz if and only if the
image set A(ja>) excludes the origin for all co> 0.
Proof First note that the symbol a> is used to denote the standard frequency
variable measured in reciprocal seconds (in the sequel we introduce the dimensionless
frequency co = (o60). Also note that the degree of (7.7) is equal to 2 for all q eQ because
the coefficient arrgr/ of the monomial cctTqT[S is always nonzero since aT, rg,and
fj = ti(tq,6q/tq) are strictly positive parameters. As discussed before, >(s;qg) is


CHAPTER 8
CONCLUSIONS AND FUTURE WORK
8.1. Conclusions
This dissertation presents an in depth exploration of robust stability analysis
methods for systems with structured and parametric uncertainties.
Chapter 2 through Chapter 4 investigates the Major Principal Direction Alignment
(MPDA) property. Chapter 2 gives a revised statement of the MPDA that fully considers
the case of repeated maximum singular values. In addition, a new proof is presented that
makes use of dual norm and dual vector theory. Chapter 3 studies the optimization
problem that is an upper bound on the structured singular value, ¡i. In particular, when
the maximum singular value is repeated the objective function is non-differentiable (i.e.,
the gradient does not exist). This work presents a characterization of the
subdifferentiable {i.e., the set of all sub-gradients or generalized gradients) that can be
used to obtain the steepest descent direction and necessary and sufficient conditions for a
minimum. Specifically, for the case of a twice repeated maximum singular value the
subdifferentiable is shown to be a 3-dimensional ellipsoid in an n -dimensional space.
Finally, attainability of MPDA (which eliminates conservatism in the upper bound of /j)
is shown to be equivalent to the optimal point lying on the surface of the ellipsoid.
Chapter 4 gives a necessary condition for and optimum that requires the optimal point to
be an element of the null space of a matrix formed from the elements of the left and right
eigenvectors of the system matrix.
154


141
consideration of simultaneous variations in all parameters, a comparison of the results
obtained with the gain and phase margins is very instructive.
First consider the parametric gain margin defined as
ro):=/(U; where the dependence on the ratio 00 / r0 is again shown explicitly. Since the parametric
gain margin is the smallest multiplicative perturbation in the gain that destabilizes the
system when there are no perturbations in the process time constant and in the process
time delay, it becomes readily apparent that this definition is equivalent to the standard
classical gain margin GM (Ogata, 1990), namely
GM(#0 / r0) = pK{0Q / r0) (7.18)
We can also obtain an expression for the classical phase margin by identifying the
gain crossover frequency. This is done by first defining the parametric time delay
margin as
PeWo'* o):=jra sj f(l,ag-,0o/ro) = \ (7.19)
OC Q^\J
which is the smallest multiplicative perturbation in the time delay that destabilizes the
system when there are no perturbations in the process gain and in the process time
constant. It then follows that there exists a frequency for which the Nyquist plot of the
open-loop response of the system with the perturbation q = [1,1, pe{00 / r0)] passes
through the critical point -1 + j0, because the closed loop characteristic quasipolynomial
(7.7) is identically zero. At this frequency the magnitude of the open-loop response of the
time-delay perturbed system is |-1 + y0| = 1. The magnitude of the open-loop response of
the nominal system is also 1 at this frequency, because the open-loop magnitude response
is independent of perturbations in the time-delay. Therefore, this frequency is the gain-


23
decomposition is constructed from the three eigenvectors associated with the eigenvalue
of maximum modulus. First, one right singular vector is obtained from the eigenvector
associated with the eigenvalue that is not repeated, i.e., y\ = v, as dictated by Lemma 2.2
and the corresponding left singular is given by
x; = y; = e--4oooyy;
according to (2.7) and (2.8) of Theorem 2.2. The remaining two right singular vectors
are obtained from v2 and v3, the eigenvectors associated with repeated eigenvalue of
maximum modulus. It can be shown that both of these eigenvectors are eigenvectors of
A*A, but that alone does not make them both right singular vectors, because singular
vectors are obtained from the orthonormal eigenbasis of A* A It is easy to show that v,
is normal and orthogonal to v2 and v3. Therefore, the remaining step is to
orthonormalize v2 and v3. One such orthonormalization is
y'2 = v2
V3-V2V3-V2
v3-v>3-v
The corresponding left singular vectors are then given by
x2 = ej^y\ = e--6000jy'2
x3 =
ej^)y\ =
-0.6000j
The alternative singular value decomposition is A = X 2Y where
X = [xi xi x3]
0.7878 0.3331/
-0.0910-0.2525/
0.2946-0.3310/
-0.2525-0.3294/
0.2895 + 0.3114/
0.7952 + 0.1210/
0.2144 + 0.2240/
-0.1311 + 0.8544/
-0.0666 0.3902/


88
that the uncertainty value set ^(co) itself does not have to be convex for the critical
uncertainty value set Vc(co) to be convex
5.3. Systems with Affine Uncertainty Structure
In this section the generalized critical direction theory is specialized to systems
with real parametric uncertainties that appear in an affine fashion, namely, an uncertain
rational function of the form
p
no (s) + ^q¡n¡(s)
g(s,q) = 7 qeQ (5.9a)
do( s) + ^iqidi(s)
i=i
where
e
*o(*):=2>oy
k=0
and
m
d0(sy-=T,doksl
k=o
are known nominal polynomials,
ni(s) = Yjniksk
k=0
and
=!<**'
k=0
are known perturbation polynomials, and q = [ql q2 ... q^ e Rp is a vector of real
perturbation parameters belonging to the bounded rectangular polytope
Q = [qeRp\ qT (5.9b)


103
The elements of (Bc {co) can be found using the systematic procedure discussed in
Section 5.6; however, it is also possible to identify the set from Figure 5.3, where it is
clear that only one element of F, namely -0.5660- j0.8584, lies on the boundary;
therefore (Bc{co) = {-0.5660 j0.8584}. As expected, <2c(n>) contains only one element
because Fc{co) is convex. Next, using definition (5.3) one finds that at co = 0.7 the
critical perturbation radius is pc(co) = 0.1694.
Figure 5.4. Critical perturbation radius pc{co) for the system considered
in Example 5.6.1.
Furthermore, since at this frequency g0(jco) = -0.4896-j1.0096, it follows that
&N( that at co = 0.7 the value set excludes the critical point.
Figures 5.4 and 5.5 respectively show the values of pc{co) and ku(co) calculated
for a grid of 100 nonnegative frequency points that are equally spaced in a logarithmic
scale in the range [0.001, 10]. From Figure 5.5 it is readily concluded that kN(co) < 1 for


To Holly and C.J.


177
Cohen, G. H. and Coon, G. A. (1953). Theoretical considerations of retarded control,
Transactions of ASME, vol. 75, pp. 827.
Cormen, T., Leiserson, C., and Rivest, R. (1990). Introduction to Algorithms, McGraw
Hill, New York.
Daniel, R. W., Kouvaritakis, B., and Latchman, H. A. (1986). Principal direction
alignment: a geometric framework for the complete solution to the p -problem, IEE
Proceedings Part D, vol. 133, no. 2, pp. 45-56.
Demyanov, V. F. and Vasilev, L. V. (1985). Nondifferentiable Optimization,
Optimization Software, Inc., Publications Division, New York.
Doyle, J. C. (1982). Analysis of feedback systems with structured uncertainty, IEE
Proceedings Part D, vol. 129, pp. 242-250.
Doyle, J. C. (1983). Synthesis of robust controllers and filters, Proceedings of the 22nd
IEEE Conference on Decision and Control, pp. 109-114.
Doyle, J. C., Glover, K., Khargonekar, P. P., and Francis, B. A. (1989). State space
solution to standard H2 and control problems, IEEE Transactions on Automatic
Control, vol. 34, no. 8, pp. 831-847.
Francis, B. A. and Zames, G. (1984). On Hm optimal sensitivity theory for SISO
feedback systems, IEEE Transactions on Automatic Control, vol. 29, no. 1, pp. 9-16.
Fu, M. (1990). Computing the frequency response of linear systems with parametric
perturbations, Systems and Control Letters, vol. 15, pp. 45-52.
Golub, G. H., and Van Loan, C. F. (1983). Matrix Computations, Johns Hopkins
University Press, Baltimore, MD.
Isaacson, E., and Keller, H. B. (1966). Analysis of Numerical Methods, John Wiley &
Sons, New York.
Kharitonov, V. L. (1978). Asymptotic stability of an equilibrium position of a family of
systems of linear differential equations, Differentsial nye Uravneniya, vol. 14, pp.
2086-2088.
Kharitonov, V. L. (1979). Asymptotic stability of an equilibrium position of a family of
systems of linear differential equations, Differential Equations, vol. 14, pp. 1483-
1485.
Kiselev, O. N., Lan, L. H., and Polyak, B. T. (1997). Frequency responses under
parametric uncertainty, Automation and Remote Control, vol. 58, no. 4, pp. 645-661.


99
Each of the four steps outlined above can be quantified in more precise terms. To
execute step (i), consider a ray r(p0,px) and an oriented arc a(a0falta2) whose
respective supporting line L(/?0,/?,) and supporting circle C, are given by (5.17) and
(5.18). The set of intersections 7(,) of the supporting line and supporting circle is
determined by the real solutions of the quadratic algebraic equation at2+bt + c = 0,
where a = \px- p0f b = 2 Re{(/?, p0)(p0 z,.)}, and c = |/?0|2 + |z,.|2 (r2 + 2 Re{/?0z,.}).
It readily follows that there are no intersections if the discriminant d: = b2 -4ac is
negative. Furthermore, if the discriminant is zero the supporting line is tangent to the
circle and there is only one intersection. Finally, there are two intersections if the
discriminant is positive. Now let {i1}f2}, denote the one or two real solutions to the
quadratic equation for the case where the discriminant is non-negative. The set of
intersection points 7(,) is composed of the points z in equation (5.17) obtained by setting
t = tk, k= 1, 2. To execute step (ii) and find 7r(,) a 7(0 it is sufficient to discard the
points of 7(,) that correspond to negative values of tk. To complete step (m), the set
7^? c 7r(,) can be determined using the cross product properties. Consider an arc
a ,(<20, ax,a2) and an intersection point y e 7r(,). Then y belongs to if and only if the
arcs a¡{a0,ax,a2) and ai(a0,y,a2) are both positive or both negative (i.e., both arcs turn
in the same direction). Finally, step (iv) is completed by setting the set F as just the union
of the sets obtained when considering all the arcs C, i = 1,2, , k .


59
B =
3.5575
1.0750
0.0000
1.0750
6.0773
1.0750
0.0000
1.0750
3.5576
and
c =
0.0548
-0.0749
0.0548
The point d = [1 1 1 1]T is optimal, because the necessary optimality
conditions q = 0 and cTBc = 0.0378 < 1 of (3.36) are satisfied implying 0 63/(d). This
means the upper bound infa(DMD_1) is 1.0000. On the other hand, the MPDA
attainability condition cTBc = 1 is not satisfied. Therefore, MPDA is not attainable and
the upper bound is conservative, i.e. /(M) < inf direction alignment (PDA) method proposed in Daniel et al. (1986) or a direct attempt at
solving the lower bound supp(MU) must by used to obtain an exact value of the
structured singular value.
3.7.3. Example 3.3.
This last example shows that even though the maximum singular value is repeated
at the optimum it may still be possible to attain MPDA and thus eliminate the
conservatism in the upper bound of ¡i. Consider the matrix
M =
-0.0274 + 0.2253/
0.2201 -0.2277/
-0.4758 + 0.2550/
0.1192 0.0574/
-0.0974 0.3482/
-0.0622 + 0.0571/
0.2355 0.0394/
-0.1977 0.1981/
-0.2418-0.0274/
0.1610 + 0.1308/
-0.0597 + 0.0705/
0.1303 + 0.0643/
-0.1025 + 0.0008/
0.1239 0.2037/
-0.1589 0.0976/
-0.0147 + 0.0149/
-0.0632 + 0.1792/
0.1533 + 0.1583/
0.3778 0.3278/
-0.4272-0.1706/
0.1624-0.1333/
-0.3688 0.1437/
0.2666 + 0.1838/
-0.0824 + 0.3762/
0.1610 + 0.0723/


ROBUST STABILITY ANALYSIS METHODS FOR SYSTEMS WITH
STRUCTURED AND PARAMETRIC UNCERTAINTIES
By
CHARLES THOMAS BAAB
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
2002


2
As a result, Hx optimal control (Zames, 1981; Francis and Zames, 1984; Doyle,
1983; Safonov and Verma, 1985; Doyle et al., 1989) was introduced as a framework to
effectively deal with robust stability and performance problems. The theory provides a
precise formulation and solution to the problem of synthesizing an output feedback
compensator that minimizes the Hx norm of a prescribed system transfer function. The
method considers unstructured uncertainties where the only information known about the
uncertainty is a norm bound. Typically, more information about the uncertainty is known
than a simple norm bound. As a consequence, several robust analysis methods have been
developed to consider these structure uncertainties.
Possibly the most effective and comprehensive result is the structured singular
value (/v) analysis method introduced by Doyle (1982), which considers the problem of
robust stability for a known plant subject to a block-diagonal uncertainty structure under
feedback. In general, any block-diagram interconnection of systems and uncertainties
can be rearranged into the block-diagonal standard form. The value of n corresponds to
the smallest uncertainty that will destabilize the system. Unfortunately, calculating ¡u is
not trivial; in fact the underlying optimization problem has been proven to be NP-hard
(Braatz et al., 1994). However, there is a convex optimization that gives a conservative
upper bound for ju. In addressing the existence of solutions to the proposed convex
optimization, Kouvaritakis and Latchman introduce the major principal direction
alignment (MPDA) property (1985), which gives necessary and sufficient conditions for
¡a to equal its upper bound, thus eliminating the conservatism.
Another robust stability analysis method for structure uncertainties is the critical-
direction theory developed by Latchman and Crisalle (1995) and Latchman et al. (1997)


84
between two distances, namely, the distance from the critical point -1 + j0 to the
nominal point g0(jco) (represented by | \ + gQ{jco) |) and the distance from the critical
point -1 + jO to the closest critical-boundary intersection (represented by On the
other hand, when -1 + j0 is an element of V(co), the lower statement in (5.3) states that
the critical perturbation radius is taken as the sum of the two distances in question.
Observe that when the critical uncertainty value set is convex, ) has only one
element (i.e. there is only one critical boundary intersection), and definition (5.3)
becomes equivalent to definition (5.2). Note also that to compute the critical perturbation
radius from (5.3) it is necessary to have full knowledge of the set of critical boundary
intersections f(£u) and to be able to evaluate whether the set membership condition
-1 + jO eV(co) holds; both of these issues are completely resolved in Section 5.3 and
Section 5.4 of this chapter for the case of systems with real affine parametric
uncertainties. For either definition it can be shown that pc(co) > 0 for all frequencies.
Finally, the Nyquist robust stability margin
M"):=
pA>)
i+s0O")
(5-5)(6)
is defined as the ratio of the critical perturbation radius to the distance between the
nominal point g0(ja>) and the critical point -1 + j0 measured along the critical
direction. Note that kN{co) > 0 for all frequencies.
5.2.2. Analysis of Robust Stability
The analysis of the robust stability of the uncertain feedback system being
considered can be resolved in terms of the following theorem.


125
The rest of the chapter is organized as follows. Section 7.2 introduces the closed
loop feedback system, including the uncertain first-order-plus-time-delay process and the
proportional-integral controller, and also provides a general discussion of tuning rules.
The theory necessary for analyzing robust stability is given in Section 7.3, where the zero-
exclusion principle is applied to the uncertain closed-loop quasi-polynomial, resulting in
a general characterization of the set of stable perturbations. Section 7.3 also introduces
measures of robust stability, including the gain margin, phase margin, and a novel
parametric stability margin. In Section 7.4 the results of Section 7.3 are applied to
determine the set of stabilizable perturbations for the ITAE regulation tuning rule and the
parametric stability margin results are given for a variety of tuning rules. Finally,
concluding remarks are made in Section 7.5.
7.2. Preliminaries
The uncertain process and the PI controller are arranged in the standard feedback
configuration shown in Figure 7.1, where the variable y(s) is the process output, r(s) is the
set point, d{s) is an additive disturbance, and e(s) = r(s) -y(s) is the feedback error. The
uncertainty in the model and the structure and tuning correlations for the controller are
discussed next.
d
Figure 7.1. Feedback control structure with proportional-integral
controller c(s) and uncertain process p(s\ q).


79
real intervals, a situation of relevance to many engineering problems. Early advances in
this field are due to Kharitonov (1978, 1979) who derived necessary and sufficient
conditions for the robust stability of interval polynomials, that is, polynomials with
independent coefficients that take values in closed real intervals. An extension of
Kharitonovs theorem to rational interval plants is proposed in Chapellat et al. (1989),
where the objective is to assess the stability of a family of plants by testing a subset of
extreme plants or extreme segments. The number of extreme plants required to determine
robust stability depends on the functional relationship between the uncertain parameters
and their bounding interval-sets. Comprehensive results based on extreme plants or
segments are known to exist only for a restricted set of uncertainty structures. A detailed
account of Karitonov-like methods can be found in Barmish (1994) and in the references
therein. For contextual value, it is worth mentioning that many of the methods proposed
are based on determining the stability of a set of Kharitonov plants (or extreme plants)
derived from the interval bounding-set description. For example, Chapellat et al. (1989)
and Bartlett et al. (1990) give conditions that use 32 Kharitonov segments or edges.
Barmish et al. (1992) prove that when using first-order compensators it is necessary and
sufficient that sixteen of the extreme plants be stable; furthermore, under certain
conditions only eight or twelve plants are necessary.
In this chapter the generalized critical direction theory is applied to systems with
affine parametric uncertainty and exploits earlier results of Fu (1990) regarding the
mapping of the uncertain parameters from their polytopic domain to the Nyquist plane to
develop a computationally tractable algorithm for calculating the Nyquist robust stability
margin. The chapter is organized as follows. Section 5.2 generalizes the critical direction
theory for systems with nonconvex critical value sets. Sections 5.3 through 5.8 are


APPENDIX F
SIGN CHANGES IN EQUATIONS (7.12A) AND (7.12B)
The set of frequencies that give aK> 0 and aT> 0 is identified by those
frequencies at which the sign of aK and aT change. In addition, only positive
frequencies are considered because for co = 0, it is known that aK = 0 and ar is
arbitrary. The sign of a K changes when the sign of the numerator or the denominator of
(7.12a) changes and the sign of aT changes when the sign of the numerator or the
denominator of (7.12b) changes. For co> 0 the sign of the numerator of aK never
changes, and because the numerator of a r is continuous and differentiable the sign of the
numerator of aT changes at those frequencies for which the function fn(co) given by
(7.13) of Theorem 7.2 satisfies
/.(*>) = o
provided
dfA) .= -l^adS\n(ae)) + svc\(aec) + caecos(aec) 0
dco t t0
For co> 0, the denominators of aK and aT change sign at those frequencies for which
fd(co) given by (7.14) of Theorem 7.2 satisfies
/() = o
provided
dfd(C0) ._ cos(a^)_^^sin(a?y)_£-.?.a|Cos(a(?£) ^ 0
dco T, T0
171


27
3.2. Mathematical Background
3.2.1. The Singular Value Decomposition
The following definitions are associated with the singular value decomposition
(Ortega, 1987). In this chapter only square matrices are being considered, therefore the
definitions are specialized for the case of square matrices, but it is noted that the singular
value decomposition theory is applicable to rectangular matrices.
The singular value decomposition of an arbitrary matrix A e Cx" is given by
A = X(A)Z(A)Y*(A) (3.2)
where £(A):= diag(crl(A),<72(A),---,<7n(A)) is the diagonal matrix of singular values
places in descending order, and X(A) and Y(A) are unitary matrices. The singular
values of square matrix A e Cx are given by
cr,.(A):= Ja^A'A), i' = l,2,,w
where A¡ (A* A) represent the i-th eigenvalues of the matrix A* A and where the singular
values are ordered such that
o-,(A)>o-2(A) >--->t(A)
The matrices X(A) and Y(A) are of the form
X(A) = [x,(A) x2(A) x(A)]
Y(A) = [y,(A) y2(A) -.. y(A)]
where the set of normalized left singular-vectors (input principal directions) (x(.(A)} and
the set of normalized right singular-vectors (output principal directions) (y,(A)} for
i = 1,2, respectively constitute orthonormal eigenbasis of AA* and A* A, such that
AA*xf(A) = cr] (A)x,.(A)


139
Clearly, the size of Qmax can be used to compare different tuning rules. Obviously, tuning
rules with larger regions of stable parameter perturbations are more robustly stable. To
better characterize the size of the stability region the next section introduces an
appropriate parametric stability margin along with the classical gain and phase margins,
and discusses their relationship to the robust stability boundary. The parametric stability
margin is then used to characterize the robustness of the controllers for quantitative
purposes when considering a given tuning rule, and for qualitative purposes when
comparing alternative tuning rules.
7.3.4. Stability Margins
The parametric stability margin is defined as the length (in a vector-norm sense)
of the smallest scaled additive perturbation Aq = q q0 that destabilizes the closed-loop
system of Figure 7.1. This margin serves as a quantitative measure of the robustness of
the closed system with respect to the parametric uncertainty referenced to the nominal
point q0 =[1,1,1]T, and is useful as means of comparing the performance of proposed
controller tuning rules. The value of the parametric stability margin depends on the norm
used to measure the length of the smallest destabilizing perturbation Aq In this paper
the lx norm is adopted because it represents the smallest box that destabilizes the system
and that contains the nominal point in the uncertain parameter space. Note, other norms
may certainly be considered (the l2 represents a sphere in the uncertain parameter space),
but the /M was chosen because the resulting box gives an easily understood bound in
terms of the physical parameters of the system.
The 4, parametric stability margin is calculated by solving the minimization
problem


CHAPTER 1
INTRODUCTION
1.1. Motivation
Uncertainty is a fact of any real-world system. This uncertainty inherently
translates to the model of the system used for control design, and is most often in the
form of neglected dynamics or variations in model parameters. An important
requirement of any control system is that it be robust (i.e., it functions satisfactorily under
these uncertainties), and the design of such control systems is known as Robust Control.
An important aspect of the robust control problem is the robust analysis problem which is
determining if a control system satisfies stability and performance requirements given an
admissible set of uncertainties.
Robust stability is obviously a necessary requirement for robustness, and has been
studied since the earliest days of feedback control, which originated to desensitize control
systems to changes in the process as well as stabilize unstable systems. The classical
design techniques focused on frequency domain methods such as those based on Bode
plots and Nyquist plots (Nyquist, 1932) and resulted in the gain and phase stability
margins. With the advent of the space race of the 1960s, the focus of control engineers
was shifted away from frequency domain robust stability methods to the field of optimal
control. In fact, the linear quadratic regulator (LQG) design appeared to give controllers
with good stability properties, but in the late 1970s it was found that LQG and other
prevailing methods of control design such as state feedback through observers lost their
stability guarantees under uncertainty.
1


71
A ediag(2,, This class of systems is especially amenable to spectral radius-preserving similarity
scaling, and through simple transformations is representative of the more general class of
full structured uncertainties.
Using Nyquist arguments in the complex plane, it can be shown that
sup p(MA) < 1
A
(4.12)
is a necessary and sufficient stability condition, where the complex matrix M is function
of the systems transfer function matrix evaluated a particular frequency. The
optimization problem (4.12) is non-convex, but it can be simplified by introducing the
following positive diagonal similarity scaling
p(MA) = p(D_1MDA) < ff(DMDA)
/TA-b
Furthermore, using geometric arguments based on the MPDA principle, it can be shown
that the supermizing diagonal-matrix Aopl has the form
a opt = QU
where Q = diag(g, ,q2,'",q) with q¡ e R+ and
U e'U:=^diag{eje' ,eJ02,---,eje")0<0i <2 n,i = 1,2, * *,
The optimization problem (4.12) becomes equivalent to
sup yo(MA) = supp(MQU) < inf cr(D 'MQD)
UeV
DeB
(4.13)
and the necessary and sufficient stability condition becomes
inf DeB
(4.14)


165
/
lim
0+
Tn 0,
co cos(a ) - sin(a 0)
v r/ ro y
0 w/ze > 1
*/ r0
0 when ag-1
*/ ro
0+ w/je ae<\
TI *0
T 0 T 0
Therefore lim a = +oo when 1 < ag < oo, and lim a = -oo when ag < 1
ro
a>-(r
*/ *0
The next step is to determine the positive frequencies for which the sign of aT
changes. As discussed in Appendix F, when considering only positive frequencies the
sign of the numerator of (7.12b) changes at the frequencies conl,con2, , and the sign of
the denominator of (7.12b) changes at the frequencies codx,cod2, . From equation (F.2)
of Appendix F, coni codj for all i = 1,2, , and j = 1,2, ; therefore, conl, con2, and
cod\,cOd2,"- are the frequencies at which the sign of aT changes. Starting at co = 0+,
Tn 0
lim ar = +oo when 1 < ag -> 0
TI To
codi represent a frequency at which the sign of ax changes. The relative ordering of coni
and codi is given by (F.2b). Therefore, when 1< ag ?i To
where the relative ordering of coni and codi is now given by (F.2a). For the case where
Tn 0,
ag< 1, lim aT --co, and again the sign changes at coni and eodi as the frequency
^ I *0
co->0
increases. Therefore, when ag <1, ar is positive for
*1 To
CO E Qr b. (cOdl,COni) LJ ((Od2 ^ (& d3t 3)^'


BIOGRAPHICAL SKETCH
Charles Thomas Baab was bom in Dayton, Ohio, on September 20, 1973. He
graduated with a B.S. in chemical engineering from the Georgia Institute of Technology
in June, 1996 with highest honors. Mr. Baab joined the graduate program at the
University of Florida in August of 1996. He graduated with a M.S. in electrical and
computer engineering in August of 2002. He received his Ph.D. in chemical engineering
in December 2002. He is married to Dr. Holly Loud Baab and has one wonderful son,
Carl John C. J.\
180


CHAPTER 5
GENERALIZATION OF THE NYQUIST ROBUST STABILITY MARGIN AND ITS
APPLICATION TO SYSTEMS WITH REAL AFFINE PARAMETRIC
UNCERTAINTIES
5.1. Introduction
The critical-direction theory developed by Latchman and Crisalle (1995) and
Latchman et al. (1997) addresses the problem of robust stability of systems affected by
uncertainties that can be characterized in terms of frequency-domain value sets. The
approach introduces the Nyquist robust stability margin kN(co) as a scalar measure of
robustness analogous to the structured singular value fa (Doyle, 1982) and the
multivariable stability margin km (Safonov, 1982) within the value-set paradigm. This
chapter extends the critical direction theory to the more general case where the critical
value-set may be nonconvex. The key to extending the theory is the introduction of a
generalized definition of the critical perturbation radius in a fashion that preserves all
previous results. The nonconvexity of the critical value set is observed in a number of
interesting problems, including the case studied by Fu (1990) consisting of rational
systems where the uncertainty appears affinely in the form of real parameters that belong
to a known rectangular polytope. The generalized critical direction theory is applied to
this particular class of uncertain systems, and is used to calculate the required Nyquist
robust stability margin with high precision and in the context of a computationally
manageable framework.
The robust stability problem studied by Fu is part of an extensive literature on
systems where the uncertainty appears in the form of parameters that vary in prescribed
78


8
It is the goal of this chapter to revisit the MPDA principle from the viewpoint of
duality theory. To facilitate the reading, the next section provides relevant mathematical
background including a discussion of the singular value decomposition, a summary of
dual-norm and dual-vector concepts, and a dual eigenvalue result. Section 2.3 introduces
the original MPDA theorem. Section 2.4 provides a modified statement of the MPDA
principle theorem that explicitly considers a repeated maximum singular value with a
proof based on the dual-norm and dual-vector theory. Several examples are given in
Section 2.5 and concluding remarks are made in Section 2.6.
2.2. Mathematical Background
2.2.1. The Singular Value Decomposition and Eigenvalue Decomposition
In this chapter only square matrices are being considered; therefore the definitions
are specialized for this type of matrices, but it is noted that the singular value
decomposition theory is applicable to generally rectangular matrices. The singular value
decomposition of an arbitrary matrix A e C*" is given by
A = X(A)£(A)Y*(A) (2.1)
where £(A):= J/ag(cr1(A),cr2(A),---,cr^(A)) is the diagonal matrix of singular values
organized in descending order, and X(A) and Y(A) are unitary matrices. The singular
values of square matrix A e C"*n are given by
/A|.(A*A), i = l,2,,n
where A,. (A* A) represent the z'-th eigenvalues of the matrix A* A and where the singular
values are ordered such that
cr,(A)>o-2(A)>--->crn(A)


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45
and the vector q = [q] q2 q_3]T satisfy
Pi,l
^2,1
^3,1
-f
-1
~^/+3,1
P',2
\,2
^2,2
^3,2
0
~A +3,2
Pi,3
\,3
^2,3
^3,3
0
~\+3,3
Vi .
Ka
^2,4
^3,4
-1
_~hj+3 4 ~
(3.27)
for i = 1,2, 3, and where hi j are given by (3.21)-(3.24). Additionally, the
dimension of S is 3.
Proof It is sufficient to show that every g(u) given by (3.20) with u*u = 1 is an
element of the affine set S = {z Rn Pz = q}, because if a set (i.e., {g(u)|u*u = 1}) is
contained within an affine set (i.e., S) then convex hull of the set (i.e.,
df (d) = conv{g(u)|iTu = 1}) is also contained within the affine set. Therefore, it must be
shown that for all u*u = 1, g(u) satisfies each of the n 3 linear equations that defines
the affine set. The first linear equation is
[Pi, i P\,2 Pi,3 1 0 " 0]g(u) = ?,
which must hold for all iTu = 1. This becomes
A.i£i (u) + Pi,282 () + Pi, 383 (u) + 84 (u) = 91
which from equation (3.20) is equivalent to
P\ 1 (K1 \ui |2 + A, 2 cos(Zw, Zu2 )|w, Iu21 + hx 3 sin(Zw, Zu2 )|m, ||u21 + A, 4 \u212) +
P\ 2 (^21 \u\ |2 + K 2 cos(Zw, Zu2 )\ux!u21 + h2 3 sin(Zw, Zu2 )|w, \\u21 + h2 4 \u212) +
P\ 3 (K, Iu{ I' + A3 2 cos(Zw, Zu2 )|w, \u21 + A3 3 sin(Zw, Zu2 )|w, ||w2 | + A3 4 \u2|) +
(h4a \ux I" + h4 2 cos(Zw, Zu2 )|w, \u2 | + h4 3 sin(Zw, Zu2 )|w, ||w21 + h4 4 \u212) = qx
or


129
behavior consistent with their explicit goal of attaining one-quarter decay ratios in the
response. The ITAE-prescribed settings, in contrast, tend to produce slower responses.
Simulation studies can be carried out to suggest that the ITAE settings are often more
robust than those proposed by the LAE and ISE methods, but no rigorous quantitative
evidence has been provided in the previous literature. This paper seeks to quantify the
relative stability robustness of all the tuning correlations represented in Table 1 by means
of a parametric stability margin whose computation is one of the main objectives of this
paper.
Table 7.1. Controller tuning correlations for proportional-integral controllers for
the servo and the regulation problems.
Method
Problem
KoK
tJt,
Reference
Ziegler-Nichols
Servo
control
0.9(0, / r,)'1
0.3(0, / r,)-'
(Ziegler, 1942)
Cohen-Coon
Servo
control
^ + 0.9 (0o/ro)-
20 + 9(0, /r,)"'
30 + 3(0,/r)
(Cohen, 1953)
IAE
Servo
control
0.758(0, / r0)"0 86'
l.O2-O.323(0o/ro)
(Rovira, 1969)
ITAE
Servo
control
O.586(0o / r0)"'916
1.03-0.165(0o/ro)
(Rovira, 1969)
ISE
Regulation
control
1.305(0, / r0)"*,6
0.492(0,/r,)-739
(Lopez, 1967)
IAE
Regulation
control
0.984(0,/r,)"0 986
O.6O8(0o/ro)707
(Lopez, 1967)
ITAE
Regulation
control
0.859(0,/r0)'9
0.674(0, /r)-68
(Lopez, 1967)
7.3. Analysis of Robust Stability
7.3.1. Conditions for Robust Stability
Straightforward block-diagram algebra operations show that the stability of the
closed-loop system of Figure 7.1 is determined by the properties of the quasipolynomial
of degree 2 (Bhattacharyya, 1995)


153
reg. parametric stability margin is pKx0 0.07. This implies that a 7% additive
perturbation of the nominal process parameters causes an unstable closed-loop system. In
fact, for any values of K0, r0, and a value 00 satisfying 60/ r0 = 0.1, a controller
designed using the ISE regulation tuning rules is destabilized by the process values
K = \.07K0, r = 0.93r0, and 0 = l.O7Oo. The ISE-reg. tuning rule is not very robust,
because the nominal values of K0, r0, and 00 are only estimates for which the
approximation error could easily be greater that 0.07. As for the robustness of the other
tuning rules, the control engineer could take advantage of available estimates of modeling
error bounds and use Figure 7.8 as a guide in choosing a sufficiently robust controller
tuning rule.
7.5. Conclusions
The robustness of a very popular class of PI controller tuning rules used in
conjunction with first-order plus time-delay models of industrial processes is of
substantial ongoing interest. This paper provides a mechanism for rigorously evaluating
the robustness of various tuning rules with respect to variations in the gain, delay, and
time-constant of the system model. An application of the zero-exclusion principle leads to
a characterization of the region of stable perturbations, which is then used to generate
analytical and graphical tests for robust stability. The parametric stability margin and
classical gain and phase margins were also computed based on the region of stable
perturbations and used to study the comparative robustness merits of various PI tuning
rules. This rigorous analysis serves to confirm some commonly held views on the merits
of particular tuning rules, but also highlights the pronounced lack of stability robustness
in rules such as the ISE regulation tuning rule.


57
'106.6882
-13.8175
-21.7949
B =
-13.8175
32.1680
17.6270
-21.7949
17.6270
15.3504
is obtained by the method mentioned after equation (3.31), and the vector
c =
-0.0231
0.1718
0.0092
is given by (3.30). The point d = [l 1 1 1 1]T is obviously not optimal, because the
necessary optimality condition q = 0 of (3.36) is not satisfied. Consequently, MPDA
does not hold either. Therefore, the next step is to find a steepest descent direction in
order to decrease the objective function in the next step of an iterative optimization
algorithm. The subgradient that gives the steepest descent direction is obtained by
solving the simple 3-parameter optimization given by (3.34) and (3.35) and is determined
to be
Ssd(d)
0.0264
0.1069
-0.0795
0.1338
-0.1877
Finally, the steepest descent direction is
Ld(d)
IMd)||
-0.0988
-0.3996
0.2970
-0.5002
0.7015


151
tuning rules satisfy this recommendation over the range of 6>0 / r0 considered. The CC,
ITAE-reg., and IAE-reg. can be considered the best tuned because the phase margin stays
within the range 20 to 65 degrees. For values of <90 / r0 below 0.5, the ISE-reg. tuning
rule lacks robustness because the phase margin is less the 20 degrees. Again, for the ZN,
CC, ITAE-reg., IAE-reg., and ISE-reg. tuning rules the general trend is that the phase
margin increases with increasing 0O/ r0, with the ZN tuning rule showing a dramatic
increase in phase margin. This again implies that the tuning rules result in inherently
more conservative controllers as the time-delay-to-time-constant ratio increases. For the
ITAE-servo and IAE-servo the phase margin remains relatively constant as the time-
delay-to-time-constant ratio increases, staying in the range 55 to 65 degrees. The
parametric time delay margin (7.19) is shown in Figure 7.7.
parameter.


168
aK()2) =
71
2 a
T, *O
*(3) =
;r
2 K0Kr ^ a
T, *O
such that ^(u,) < ^(^2) < ^(^3) <. Therefore, for ar =
/ v1
Tr
\TI J
, Q, gives the
lowest value of a
K
To show that the curves never intersect it must be shown that for every frequency
cox gQ, there does not exist a frequency cox eQr such that
aK(cox) = aK(cox)
(E.la)
and
az{cox) = aT{cox)
(E.lb)
co
To notationally simplify the problem consider the frequency scaling co , then from
a0
(7.12a) and (7.12b) equations (E.la) and (E.lb) become
co,
69.
cox cos(, ) y sin(£,) cox cos(x )-y sin( x)
(E.2a)
y cos(cox) + cox sin( 69,) y cos(cox) + cox sin(69t)
(cox cos(69,)-y sin(69,))69, (cox cos(cox)-y sin(69x))69,
(E.2b)
rn 6,
where the only free parameter is y = ag. By combining (E.2a) and (E.2b) the
Ti To
equations simplify to



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PAGE 191

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136
on these curves are candidate members for the robust stability boundary. Unfortunately,
there is an infinite number of frequency intervals in Q, and therefore also an infinite
number of curves. It is not possible to check all the frequency intervals and
corresponding curves to determine the robust stability boundary. Fortunately, the
following theorem states that the first frequency interval gives the curve that is the robust
stability boundary, and that all other frequency intervals need not be considered.
T
Theorem 7.2. The strictly positive uncertainty vector q = [a g aT olq\ is an
element of the parametric robust-stability boundary dQmiX for the closed-loop
system of Figure 7.1 with PI control parameters adjusted using a tuning
correlation from Table 7.1 if and only if q satisfies (7.12) for some co eQ,,
where
if
i U To
at
otherwise
< 00
Proof. First, Qmix is the region that contains the nominal point and that is
bounded by the curves given by all the frequency intervals and aK =0, ar =0, and
a0 = 0. Second, note that from (7.12) and (7.13)-(7.14) the endpoints of each frequency
interval are the zeros of the numerator and denominator of aT such that aT ranges from
zero to infinity over each frequency interval. Also, aT is monotonically decreasing over
each frequency segment. Therefore, there is a one-to-one relationship between positive
values of aT and frequency in each interval. That is, given a positive value of aT there
is one and only one corresponding frequency in each frequency interval. This in turn
implies that for any positive value of aT there is a unique value of a K for each frequency


60
The point d = [l 1 1 1 1]T corresponding to D = I has a maximum singular value
j(DMD"') = non-differentiable at d = [1 1 1 1 1]T and the results of the this chapter are used to
efficiently solve the optimization by either determining a steepest descent direction from
the point d = [1 1 1 1 1]T or by determining if the point satisfies the optimality and
MPDA conditions.
The ellipsoidal characterization of the subdifferential is given by
d/(d) = {zeZrPz = q, ([z, z2 z3]-cT)B([z1 z2 z3]t-c)<1}
where
P =
0.0828
0.9172
-0.9106 0.4221 1.0000 0.0000
1.9106 0.5779 0.0000 1.0000
0.0000
0.0000
B =
37.4471
-7.5773
25.1289
-7.5773
28.1213
-6.3079
25.1289
-6.3079
26.8994
and
c =
-0.2398
0.0632
0.2010
The point d = [1 1 1 1 1]T is optimal, because it satisfies the necessary
optimality conditions (3.36). Furthermore, the MPDA attainability conditions (3.38) are


123
Nichols methods first proposed in 1942 (Ziegler and Nichols, 1942), it is therefore
relevant to analyze the robust-stability of such tuning correlations. This robust-stability
problem has not been previously studied in the literature, but recent advances in the
theory of robust control analysis, in particular dealing with real parametric uncertainties,
now make it possible to address the issue in a quantitative fashion.
Since the appearance of Ziegler and Nichols' seminal work, a vast literature has
emerged on alternative approaches for tuning controllers of the PI type under the
assumption that the process being controlled is adequately described by the open-loop
stable first-order-plus-delay model
p(s) =
Ke
-Os
ZS + 1
(7.1)
where K is the process gain, r > Ois the time constant, and 6 > 0 is the time delay. The
three model parameters can be obtained from open-loop step-response tests, from
statistical parameter-estimation methods, model reduction techniques, etc. (Seborg et al.,
1989), (Wallen, 1999).
Among the large number of correlations in use today based on the first-order-plus-
dead-time process paradigm and found in most introductory texts on control engineering
are the widely-cited prescriptions proposed by Cohen and Coon (1953), who tuned
controllers using the criterion that good performance is attained when the response
realizes a one-quarter decay ratio. Also very often used are the correlations based on
error-integral criteria proposed and developed by Lopez et al. (1967), Murrill (1967),
Rovira et al. (1969), and Smith and Corripio (1985), among others, and that include
performance criteria such as the integral of the absolute value of the error (LAE), the
integral of the squared error (ISE), and the integral of the time-averaged absolute error


179
Rockafellar, R. T. (1970). Convex Analysis, Princeton University Press, Princeton, NJ.
Rovira, A. A., Murrill, P. W., and Smith, C. L., Tuning controllers for setpoint
changes ''Instrumentation Technology, vol. 16, no. 12, pp. 67- 69, 1969.
Safonov, M. G. (1982). Stability margins of diagonally perturbed multivariable feedback
systems, IEE Proceedings Part D, 129, pp. 251-256.
Safonov M. G., and Verma, M. S. (1985). / optimization and Hankel approximations,
IEEE Transactions on Automatic Control, vol. 30, no. 3, pp. 279-280.
Schei, T. S. (1994). Automatic tuning of PID controllers based on transfer function
estimation,"Automtica, vol. 30, no. 12, pp. 1983-1989.
Seborg, D. E., Edgar, T. F., and Mellichamp, D. A. (1989). Process Dynamics and
Control. John Wiley & Sons, New York.
Siljak, D. D. (1989). Parameter space methods for robust control design: a guided tour,
IEEE Transactions on Automatic Control, vol. 34, pp. 674-688.
Smith, C. A. and Corripio, A. B. (1985). Principles and Practice of Automatic Process
Control, John Wiley & Sons, New York.
Stewart, G. W. (1970). Introduction to Matrix Computations, Academic Press, Inc., New
York.
Stoer, J. and Witzgall, C. (1962). Transformations by diagonal matrices in a normed
space, Numerische Mathematik, vol. 4, pp. 158-171.
Tzafestas, S. G., (1984). Multivariable Control, New Concepts and Tools, Reidel,
Dordrecht.
Wallen, A. (1999). A tool for rapid system identification, Proceedings of the 1999
Conference on Control Applications, Kohala Coast, HI, pp. 1555-1560.
Zames, G. (1981). Feedback and optimal sensitivity: Model reference transformation,
multiplicative seminorms, and approximate inverses, IEEE Transactions on
Automatic Control, vol. 34, no. 2, pp. 301-320.
Ziegler, J. G. and Nichols, N. B. (1942). Optimum settings for automatic controllers,
Transactions of ASME, vol. 64, pp. 759.


28
and
A*Ay,.(A) = cr(2(A)y,.(A) (3.3)
Furthermore, a pair of singular vectors {x(.(A),y, (A)} is associated with each singular
value cr.(A) through the relationship
Ay,.(A) = c7,.(A)xf(A) (3.4)
The maximum singular value is defined as singular value can be associated with a repeated singular value, i.e. = vector pair (or major output/input principal direction pair) (x(A),y(A)} is any pair of
left and right singular vectors that corresponding to the maximum singular value and
satisfy (3.4). Necessarily, a major output principal direction and major input principal
direction respectively must respectively be normalized elements of the eigensubspaces of
AA* and A* A associated with the maximum singular value. If {x,.(A)} for i = 1,2,,?*
and {y,(A)} for i = l,2,---,r are orthonormal bases for these eigensubspaces that satisfy
(3.4), then any and all major output principal directions and major input principal
directions are respectively given by
x(A) = [x,(A) x2(A) xr(A)]u = x,(A)w,. (3.5)
i=i
and
y(A) = [y,(A) y2(A) yr(A)]u = y,.(A)w,. (3.6)
i=i
where u e C satisfies
u*u = 1


61
also satisfied, namely q = 0 and cTBc = l. Therefore, the upper bound
inf a(DMD 1 ) = 1.0000 is tight and the structured singular value is exactly
MM) = 1 .0000 even though the maximum singular value is repeated such that the
objective function is nondifferentiable.
3.8. Conclusions
The MPDA principle approach to solving the structured singular value problem is
investigated. In the infimization that gives an upper bound to mu, a repeated maximum
singular value results in a non-differentiablity of the objective function. Therefore,
efficient gradient descent optimization algorithms that use the analytical expression for
the gradient must be modified. The first result of this paper is characterization of the
subdifferential which represents the set of all sub-gradients or generalized gradients. In
addition, for the case of a once repeated maximum singular value it is shown that the
subdifferential is in fact a 3-dimensional ellipsoid in and -dimensional space. Using
results from non-differential optimization theory, the steepest descent direction is obtain
from this characterization of the subdifferential to facilitate the optimization.
Furthermore, conditions for optimality are presented which are based zero being an
element of the subdifferential. Finally, attainability of MPDA at the optimum is shown to
be equivalent to zero being on the boundary of the subdifferential enhancing the PDA
results when themaximum singular value is repeated.


Copyright 2002
by
Charles Thomas Baab


98
Po x Pi
The sign of the cross product determines the relative orientation of p0 and p] with
respect to the origin: px lies to the left (right) of p0 if p0 x px < 0 (>0), and pl and p0
are collinear if p0 x px = 0 (Cormen et al., 1990). The cross product can also be used to
determine the direction in which an oriented circular arc "turns". Let a(a0,ax,a2) be a
circular arc that originates at a0, passes through ax, and ends at a2. Then, the arc turns
left (right) if (a2 -a0) x (ax a0) > 0 (< 0). If (a2 a0) x (a] -a0) = 0 the three points are
collinear, and the arc degenerates into a line segment. The arc a(a0,£z,,zz2) is said to be
positive (negative) if it turns left (right).
The following four-step strategy to compute the intersection of a ray and a finite
number of arcs is proposed: (i) find I(,) = L(g0(ja>,-\ +jO))nC¡, the set of the
intersections (if any) of the supporting line (i.e., the line that contains the critical ray) and
the z-th supporting circle (i.e., the circle that contains the z'-th arc); (zz) find
7r(,) = v(g0(jco,-\ + y'0))n I{l), the set of intersections of the critical ray and the supporting
circle; (z'z'z) find = a,, n I(rl) c 7r(,), the intersections of the critical ray and the z'-th arc;
and (z'v) find the union over all arcs of all the possible ray-arc intersections.
l The set F corresponds to the desired set of intersections of the critical line r(co) with the
elements of the frame of g(ja>,YL(Q)) that are described by arcs of circles. The
remaining elements of the set F are of course the set of points the represent the
intersections of r(co) with the elements of the frame of g(jco,Yi(Q)) that are described by
straight-line segments.


40
df(d) = conv{z g /?" z = ^lim Vf(dk), d* > d, d^. g D}
where D is the set of points where /(d) = 5(DMD_l) is differentiable (i.e., the
maximum singular value is distinct). The gradients S7f(dk) are given by (3.16), and are
determined from the from the sequence of major output principal directions
x,(D,MD¡') and major input principal directions y^D^MD/), which are uniquely
determined up a multiple of e,e. From the perturbation theory of matrices (Lancaster
and Tismenetsky, 1985), analytic perturbations on normal matrices (i.e.,
(DMD )(DMD 1 )) have continuous eigenvalues and eigenvectors in a neighborhood
of the perturbation. Now, the right singular vectors {y;(DMD"')} are an orthonormal
eigenbasis of (DMD-1)*(DMD-1). Therefore, the right singular vectors are continuous
in D. This implies that for points where the maximum singular is non-differential each
major input principal direction y(DMD ) (all of which are given by (3.6)) is the limit of
a sequence of major input principal directions y^D^MDj1) that correspond to a
maximum singular value that is differentiable. In addition the converse is true; that a
sequence of major input principal directions y^D^MD^1) that corresponds to a
sequence of maximum singular values Jt(D,MDj') that are differential converges to
major input principal direction y(DMD ') given by (3.6) if the sequence at(DtMDj')
converge to the maximum singular value for the left singular vectors/major output principal directions. Therefore, for all dk g D
as dt->d then \k (D,MD¡) - x(DMD ) and y* (DjMDj1) - y(DMD 1) which
are given by (3.5) and (3.6) with u*u = l. In addition, for every point where the


54
3.6. Reconciling the Results with the PDA Results
The principal direction alignment (PDA) principle (Daniel et al., 1986) states the
infimum of (3.1) occurs at a stationary point of the largest singular value for which a
stationary point exists starting with the maximum singular value. If the maximum
singular value is repeated then there is no stationary point (the maximum singular value is
non-differentiable), and an attempt is made to find a stationary point of the second largest
singular value, and so on. This statement is not entirely accurate. Consider the case
when at the infimum, the singular value is repeated, and therefore the gradient does not
exist. As such the gradient can not be 0 and there is no stationary point, but it is possible
to have a repeated maximum singular value and still achieve MPDA as demonstrated by
Example 3.3. As such, the infimum occurs at a non-stationary point contradicting the
PDA theory.
The PDA theory can rectified as follows. First, a more accurate statement than
stating the infimum occurs at a stationary point (i.e. when all the partials are zero) of a
singular value is to state that the infimum occurs at a point where exist a left and right
singular vector pair that element wise equal moduli. The work of the previous section
gives the conditions for under which it is possible to equate the moduli when a singular
value is repeated. If the moduli can be equated, then MPDA achieved, otherwise it is
necessary to use the PDA algorithm by infimizing the next singular value.
3.7. Examples
The following three examples demonstrate the results of the previous sections.
The first example shows how to determine the steepest descent direction. The second
example demonstrates the conditions for a minimum. The third example illustrates the
conditions for which MPDA is attainable.


132
S(^-,q) = ccKK0Kc
jco ^
T j h 1
00 J
-A
j_u
(9n /7A2 j<*>
0 +aTT0Tj(-) +Tl~z~
VQ
or
S{^-; q) = akK0KcU^* +1) (cos(a0co) j sin(a0co)) -aT co2 +j~co (7.10)
00 00 00 00
Then from (7.10) and Lemma 7.1 it follows that the system is robustly stable with respect
to Q if and only if
(
a kKqKc
\
T T T
j~r~(o + \ (cos(a0co)-jsm(adco))-aT-^Lco2+j-^-co*O (7.11)
V 0Q y 0Q 0Q
for all q e Q and all co > 0, or similarly co > 0 because 0O is strictly positive. Separating
the real and imaginary parts and factoring out the parameter perturbations a K and ar
gives the following vector-matrix form of inequality (7.11):
f
K0KC
KqKc
*0 00
VTI To
COS(agco) + co sin(cCqCO)
cocos(agco)- sm(aQCo)
Tj Tq
( n }
0Q
^r0>
0
-1
CO
\_CCr J
0
-co
which is the same as (7.9) completing the proof.
Q.E.D.
The net effect of using multiplicative perturbations and the dimensionless
frequency co as opposed to the standard frequency co is that the expression (7.9)
explicitly shows the role of the ratio 60/ t0. Note that 60 / r0 is the only parameter
needed to extract the value of the factors KqKc and r0 / r, from the tuning correlations
of Table 7.1; hence, the product (r0 / r,)(0Q / r0) is not further simplified in (7.9b)-
(7.9c). Furthermore, through (7.9) Lemma 7.2 completely describes the robust stability


This dissertation was submitted to the Graduate Faculty of the College of Engineering
and to the Graduate School and was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
December 2002
Pramod Khargonekar
Dean, College of Engineering
Winfred M. Phillips
Dean, Graduate School


29
that is, u must be on the unit ball in C".
3.2.2. Statement of the Major Principal Direction Alignment Principle
The following theorem is a modification of the MPDA principle as proposed in
Kouvaritakis and Latchman (1985) which takes into account the case of a repeated
maximum singular value.
Theorem 3.1. The spectral radius of a matrix A e Cnxn is equal to the maximum
singular value of A if and only if there exists a major input and major output
principal direction pair of A that is aligned, i.e. there exists a pair (x(A),y(A)}
such that
y(A) = eJffx(A) (3.7)
for some 6 e [0,2/r).
Proof The proof is given in Kouvaritakis and Latchman (1985), and Chapter 2
offers an alternative proof based on the theory of dual vectors and dual norms. Q.E.D.
Given the optimal matrices U and D, Theorem 3.1 gives a necessary and
sufficient condition for the left hand side (spectral radius) and right hand side (maximum
singular value) of (3.1) to hold with equality. It is apparent that equation (3.7) requires
that the major input and major output principal directions have element-by-element equal
moduli and a constant element-by-element phase difference.
3.2.3. Affine Sets, Convex Sets, and Convex Functions
If x and y are different point in Rn, the set of points of the form
(1 -A)x + Ay = x + A(y-x), A eR
is called the line through x and y. A subset M of Rn is called an affine set if
{\-A)x + AyeM Vx e M,y e M,A e R


LIST OF REFERENCES
Baab, C. T., Cockbum, J. C., Latchman, H. A., and Crisalle, O. D. (2001). Extension of
the Nyquist robust stability margin to systems with nonconvex values sets,
Proceedings of the American Control Conference, Washington, D.C., pp. 1414-1419,
IEEE, Piscataway, NJ.
Baab, C. T., Cockbum, J. C., Latchman, H. A., and Crisalle, O. D. (2001).
Generalization of the Nyquist robust stability margin concept and its application to
systems with real affine parametric uncertainties, International Journal on Robust
and Nonlinear Control, vol. l,pp. 1415-1434.
Barmish, B. R. (1992). Extreme point results for robust stabilization of interval plants
with first order compensators, IEEE Transactions on Automatic Control, vol. 37, pp.
707-715.
Barmish, B. R. (1994). New Tools for Robustness ofLinear Systems, McMillan
Publishing Co., New York.
Bartlett, A. C., Tesi, A., and Vicino, A. (1990). Frequency response of uncertain
systems with interval plants, IEEE Transactions on Automatic Control, vol. 38, pp.
929-933.
Bauer, F. L. (1962). On the field of values subordinate to a norm, Numerische
Mathematik, vol. 4, pp. 103-113.
Bauer, F. L. (1963). Optimally scale matrices, Numerische Mathematik, vol. 5, pp. 73-
87.
Bauer, F. L., Stoer, J., and Witzgall, C. (1961). Absolute and monotonic norms,
Numerishce Mathematik, vol. 3, pp. 257-264.
Bhattacharyya, S. P., Chapellat, H., and Keel, L. H. (1995). Robust ControlThe
Parametric Approach, Prentice Hall, Upper Saddle River, NJ.
Braatz, R. D., Young, P. M., Doyle, J. C., and Morari, M. (1994). Computational
complexity of p calculation, IEEE Transactions on Automatic Control, vol. 39, pp.
1000-1002.
Chapellat, H. and Bhattacharyya, S. P. (1989). A generalization of Kharitonovs
theorem: Robust stability of interval plants, IEEE Transactions on Automatic
Control, vol. 34, pp. 1100-1108.
176


38
da2(M) ~ 5ct(M)
dd;
= 2ct(M)-
ddt
such that an expression for the partial derivative of (t(DMD ') with respect to the
diagonal element d¡ of matrix D is given by
aCT(DMD) ff(DMD')
dd:
d:
^.(DMD-1)!2 -|y.(DMD'')
(3.15)
When the maximum singular value principal direction x(DMD ') and the major input principal direction y(DMD "') are
determined by a scaling factor ej0 of the left singular vector x,(DMD) and right
singular vector y,(DMD 1), respectively. Therefore, x/DMD ') and y;(DMD ')
are
unique and the partial derivatives (3.15) exists for i = 1,2,, such that the gradient is
given by
V/(d) = Vct(DMD ) = k-l
x(DMD ') y(DMD ')
(3.16)
where the absolute value || is considered an element-by-element operator when applied
to a vector. As the preceding development has verified, when the maximum singular
value is distinct the gradient of the objective function /(d) = j(DMD ') exists and is
given by (3.16). In addition, the subdifferential is given by (3.11). When the maximum
singular value is repeated the gradient no longer exists, but it is possible to determine the
subdifferential and therefore a steepest descent direction. This is the main theoretical
result of this paper and is given in the next section.


83
Note that to determine (Bc(co) it is necessary to have knowledge of the uncertainty value
set boundary only along the critical line. Clearly, (Bc (co) contains a single element if
Vc(co) is a convex set, and contains at least two elements if Vc(co) is nonconvex.
When the critical value set Vc{co) is convex (as in the case of star-shaped value
sets with respect to the nominal point, for example), the critical perturbation radius is
defined as (Latchman and Crisalle, 1995; Latchman et al., 1997)
Pc(>): = max {a\z = g0(jco) + adc(jco) eV(a) } (5.2)
a&Rj L J
Definition (5.2) states that the critical perturbation radius for the case of a convex set
Vc(co) is simply the distance along the critical direction between the nominal point
g0{jco) and the uncertainty value set boundary dV(co). Note also that the perturbation
radius captures the "size" of the uncertainty that is relevant for stability analysis.
Definition (5.2) is not suitable, however, for the case of nonconvex critical value sets
Vc(a>). In this chapter the following generalization of the definition of the critical
perturbation radius is proposed, which is applicable to both the convex and nonconvex
cases:
i+goU)\-£() if-i+jO£V{(o)
1 + g0 (joj) | + ;(c) otherwise
(5.3)
where
%(co)= min |l + z| (5.4)
ze$c(a>) 1
represents the distance from -1 + j0 to the point in <3f(&>) that is closest to the critical
point -1 + j0. The upper statement in definition (5.3) states that when -1 + j0 is not an
element of V{co), the critical perturbation radius pc(co) is defined as the difference


(3.28)
46
(Pi,
1*1,1
+ A,2*2,l
Pi, 3*3,1
+ *4,l)|Wl| +
(a,
1*1,2
+ A,2*2,2
+ A,3*3,2
+ /j4 2)cos(Zw, -
-Z2),
(a,
1*1,3
+ A,2*2,3
+ A,3*3,3
+ /24 3)sin(Zw, -
z2)KJ
(a,
1*1,4
+ Pi, 2*2,4
+ Pi,3*3,4
+*4,4)h|2 =0i
Now, equation (3.27) gives
*2,1
*3,1
-fi
*/+3,1
*1,2
*2,2
*3,2
0
P/,2
_*/+3,2
*i ,3
*2,3
*3,3
0
P/,3
_*/+3,3
.*1,4
*2,4
*3,4
-1
_ 0/ .
__*/+3,4 _
such that (3.28) becomes
(0 i ~K\ +\i)h|2 +
(/z4 2 + h4 2) cos(ZW[ Zw2 )|i |m2 I +
(-*4,3 +\3)sin(Zwi Zu2)\u]\\u2\ +
(01 *4,4 *4,4 )|^2 | 01
or
|2 i
w, | + m2
which holds for all iTu = l. Hence, for all u *u = l, g(u) satisfies the first linear
equation that defines S. In fact, the preceding arguments hold for all n- 3 linear
equations that define S. Therefore, g(u) is contained within the affine set S for all
u*u = 1 implying that the subdifferential is contained within S. Finally, the n- 3
linearly independent rows of P are a basis for the orthogonal complement of the
subspace parallel to S such that the dimension of the affine set S is 3. Q.E.D.
Theorem 3.7 implies that the last n- 3 terms of g(u) can be expressed as an
affine functions of the first 3 terms of g(u) such that the subdifferential df(d) is a 3-


I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
CuctuxMJ)
Oscar D. Crisalle, Chairman
Professor of Chemical Engineering
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
iniph A. Latchman
Professor of Electrical and Computer
Engineering
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
Richard B. Dickinson
Professor of Chemical Engineering
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
- o .
Dinesh O. Shah
Professor of Chemical Engineering
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
dissertation for the degree of Doctor of Philosophy.
X c
v/>/*v\=jK3
Spyros A. Svoronos
Professor of Chemical Engineering


120
Table 2. Controllers resulting from minimizing the three performance criteria along the
0.8 robust Nyquist stability margin contour.
Performance
Criteria
Optimal Kc
Optimal z,
Robustly
Stabilizing
ISE
0.9609
8.5317
0.8
Yes
LAE
0.6164
4.5204
0.8
Yes
ITAE
0.4894
3.8476
0.8
Yes
To this end, the values of the performance measures along the Nyquist robust stability
margin contour of 0.8 are calculated and the optimizing controllers are shown in Table 2.
A comparison of Table 6.1 and Table 6.2 shows that to achieve the greater degree
of robust stability (i.e. a robust Nyquist stability margin value of 0.8) the ISE and LAE
controllers designed using the second method require smaller values of the proportional-
gain Kc and integral time constant z,.
Figure 6.4. ITAE contours superimposed on the robust stability boundary.


172
because the denominators of aK and ax are also continuous and differentiable. It can
/ \
now be shown that fn(co) and cannot simultaneously be zero, and likewise for
dco
/ \
fd(co) and First, the positive zeros of fn(co) given by conl, con2, and the
dco
positive zeros of fd{co) given by codl, cod2, are characterized. Consider the frequency
*
scaling co =, the positive zeros fn(co) and fd(co) are given by the positive solutions
aa
to
cot( -co
*0 @o
*1 To
(F.la)
a,
and
tan(ft)*) =
co
*0 @0
*0
(F.lb)
a.
respectively. These are simply the intersections of the cotangent curve with a negative-
slope line and the tangent curve with a positive-slope line. Let co*nl,co*nZ,--- be the
positive solutions to (F.la). For positive values of u* the cotangent curve is negative
TC 1)71
only when co* e (, 7z) u ( ,2/r)u- . Also, in each of these ranges the cotangent curve
goes from 0 to -oo, therefore the intersections with a negative sloped line must occur at
1 3 .
-7T

55
3.7.1. Example 3.1.
Let M = AB where
and
A =
-0.1582-0.3074/
0.4198-0.5890/
0.2182 + 0.0182/
0.1039 0.4891/
-0.0765 + 0.2315/
0.3252 + 0.3078/
0.0843 0.0067/
0.7031 + 0.1455/
-0.4090 + 0.0507/
-0.3256-0.0301/
B =
-0.3681-0.3181/
-0.2708 + 0.0371/
-0.4548 + 0.5280/
0.3127-0.1501/
0.2842 + 0.0444/
0.2366 + 0.2711/
0.0536 + 0.3304/
-0.0931-0.2255/
-0.0013-0.0917/
0.8244 + 0.1044/
In performing the infimization infcr(DMD !), consider the point d = [l 1 1 1 1]T
corresponding to D = I. The maximum singular value a(DMD') = (i.e., CTj(M) = objective function /(d) = a(DMD_1) is non-differentiable at d = [l 1 1 1 1]T and
the results of the this chapter are used to efficiently solve the optimization by either
determining a steepest descent direction from the point d = [1 1 1 1 1]T or by
determining if the point satisfies the optimality and MPDA conditions.
First, the ellipsoidal characterization of the subdifferential is obtained using the
method of Section 3.3.2. An orthonormal set of right singular vectors corresponding to
the repeated maximum singular value is


APPENDIX A
PROOF OF LEMMA 2.1
Proof. The following proof is modification of the proof given by Bauer (1962) in
that it is specialized to the case of the Euclidean norm, and is therefore in terms of the
maximum singular value of the matrix. Excluding the trivial case /t,(A) = 0, assume
without loss of generality that
0
Let v(. be any normalized right eigenvector of A with eigenvalue A¡(A), and choose w(
such that
which for the Euclidean norm implies uniquely that w(. = v,- / j|vf = vf. From the
definition of dual vectors
Multiplying by T(.(A) gives
w,.v,. = 1
T,.(A)w*v,. = A,.(A)
Using the definition of eigenvector gives
And from the assumption
The Holder inequality implies
w*Av. =A,.(A)
w* Av,. = wyAv(. <
A w.
(A.l)
156


162
0
a K
*, r0 .
-ar.
As expected, when co = 0 the equation associated with the imaginary part of the image of
the uncertainty drops out, because the image of the uncertainty is a real number. The
solution for the case co 0 is therefore obtained from the first equation of (C.l), namely,
= 0
and since K0KC, t0/t,, and 0o/to are non-zero, the solution set is given by setting
aK = 0 and aT equal to an arbitrary real number. This solution set is not admissible
because aK= 0. At the non-zero frequency solutions to (C.2) the second equation of
(C.l) becomes inconsistent, and therefore there is no solution aK and ar to (C.l) at
these frequencies. In terms of the solutions given by (7.12), these are frequencies at
which the expressions for aK and for ax tend to infinity. Therefore, all relevant
solutions to (C.l) are given by (7.12). Q.E.D.


134
-co
aK =
/
K0KC
Q
cocos(aeco)-
Tn
f \
\Zi )
sin (aea>)
(7.12a)
r =
Vro
\Ti)
cos(a0 On
(
co cos(a 0co) -
Tn
(7.12b)
\Ti J
sin (a0co)
co
for some co> 0.
Proof. By definition the closed-loop system is robustly stable with respect to
, and for any q = [aK aT ae]T e d0max, the set 0max u q is not robustly stable. From
Lemma 7.2 this implies that q must satisfy the equality A{co,ae)[aK aTf = b( some co >0. All relevant solutions to this vector-matrix equality are given by (7.12a)-
(7.12b). Further details of the proof are given in Appendix C. Q.E.D.
Note that for a given ratio 0q / Tq the candidate uncertainty-boundary coordinates
aK and ax in (7.12) are parametrized in terms of ae. Equations (7.12a)-(7.12b) trace
curves in the aT a ^ space as the frequency varies for a fixed value of 0q / Tq and for
an arbitrarily selected value ae> 0. A subset of these curves defines the boundary set
d(7max. In particular, note that if at a given frequency the values aK and aT obtained
from (7.12) are not simultaneously positive, then they are not elements of d(2max because
such pairs are not admissible. Therefore, it can be anticipated that dQnux can be
characterized by evaluating (7.12) within a set of selected frequency intervals since at
some frequencies the maps yield inadmissible solutions. The subintervals of frequency
that yield admissible solutions can be identified by finding the frequencies at which a K
and aT defined in (7.12) are simultaneously positive.


72
Furthermore, using MPDA arguments it can be shown that the optimizing D0 in (4.14)
results in the equality
supp(MQU) = ^(D-'MQDo)
U eU
when the maximum singular value is distinct at the infimum.
4.3. Main Result Extension of the Positive Matrix Result to
General Complex Matrices
The positive matrix result of Stoer and Witzgall as stated by (4.1) and specialized
to the Euclidean norm by (4.10) gives a positive diagonal similarity scaling (4.11) that
results in equality of the spectral radius and maximum singular value of a positive matrix.
When applied to robust control problems that involve complex matrices, the positive
matrix result is usually only sub-optimal. Therefore, it is necessary to extend the result to
the class of complex matrices. Unfortunately, much of the theoretical development is
dependent on the characteristic properties of positive matrices. Therefore, when
generalizing the result to complex matrices it is not possible to explicitly state that there
exists a similarity scaling that will result in equality of the spectral radius and maximum
singular value of a matrix. Nevertheless, it is possible determine the necessary conditions
for the existence of a positive diagonal similarity scaling that leads to equality. The result
is given in the following theorem.
Theorem 4.2. Let A e Cnxn have a right eigenvector v and a left eigenvector w
associated with an eigenvalue T(A) of maximum modulus such that
|A(A)| = p(A), and let D0 = diag(J0A,d02,---,d0n) be define as
D0:= argmino:(D1AD)
DeD
where := [diag(d],d2,---,dn)\ d¡ > 0,i = 1,2, ,/?}. Then if


160
where
9
y = JZaM
i=i
9
x=Y,aixi
i=1
i=i
The proof of sufficiency remains exactly the same. To prove necessity,
proceeding as before we obtain from equations (B.2) and (B.3)
z'X2(A)z 2
z z
= \A)
(BA)
for
z = Y\A)w (B.5)
Since the first q elements of the diagonal matrix Y(A) are all equal to cf(A), if follows
from equation (4) that for a normalized w, z must now assume the form
z = \ccx, a2, , aq, 0, , o]f
with
'Za*ai=l
i=i
Back substitution of w = Y(A)z into equation (B.2) reveals that
x = X aixi=eivy = ^ Z aiy¡
i=i i=i
and this completes the proof.
Q.E.D.


36
3.2.5. Expression for the gradient when the maximum singular value is distinct.
The mathematical background will now focus on the problem at hand, namely
performing the infimization
inf ct(DMD_1) (3.13)
DeB
The objective function /(d) = a(DMD ') (where D = diag(d) and the domain of the
objective function is the positive orthant such that D gD) is convex as was already
stated. Latchman (1986) has stated that when the maximum singular value is distinct, the
gradient exists and is given by a relatively simple expression. The following is the
derivation of this expression. After defining M:=DMD-1 to simplify the notation, the
singular value decomposition and equation (3.3) give
g2 (DMD-1) = g2 (M) = y* (M)M*My(M) (3.14)
If it exists, the partial derivative of (3.14) with respect to the diagonal element di of
matrix D is given by
8g2{ M) 5(y*(M)M*My(M))
dd:
dd:
which by the chain rule becomes
5ct2(M) dy*(M) ~ ~ 5y(M) d(M*M)_ ~
- -M My(M) + y (M)M ^- + y (M) -y(M)
dd:
dd;
dd:
dd:
Using (3.3) gives
da2(M) 2
dd:
= a2(M)
5y*(M) ~ _* ~ 5y(M)
JV J y(M) + y (M)-
dd:
dd:
~ 5(M*M)_ ~
+ y (M)y(M)
dd:
which simplifies to


143
the system. Consider for example a system that has large gain and large phase margins,
but becomes unstable when there are relatively small (compared to the gain and phase
margin) perturbations in both the process gain and time delay. On the other hand, a
system with a poor gain or phase margin can safely be anticipated to have poor robust
stability characteristics. Therefore, the gain and phase margin are useful in helping the
designer to avoid controllers with poor robust stability characteristics, but they may not
be sufficient to ensure stability robustness. The next section presents results that compare
the robustness properties of various classical PI controller tuning rules.
7.4. Results of Numerical Studies
7.4.1. Region of Stable Perturbations for the ITAE Regulation Tuning Rule
Using the theoretical development of the previous section it is possible to
determine the region of stable perturbations of the process parameters for the ITAE
regulation tuning rule. The other tuning rules have qualitatively similar results. From
Table 7.1, the ITAE regulation tuning rule gives
K<>KC = 0.859(£?0 / r0)"977
and
^ = 0.674
TI
For the purposes of illustrating the theory proposed, consider nominal process parameters
K0 = 1, t0 = 1, and 0O = 0.5. The tuning-parameter ratio is 0O / r0 = 0.5, giving
K0KC = 1.691 and r0/r7 = 1.080. First consider the case where there is no multiplicative
perturbation of the time delay, such that ae = 1. Standard calculations show that the gain
margin for the loop is GM = 1.816 and the phase margin for the loop is PM = 38.61
degrees. From the theoretical developments, the robust stability boundary is determined


92
Now consider an arbitrary Nyquist-plane point w = wR+ j w, g C with finite
magnitude. Clearly, w g V{co) if and only if there exists a vector q e Q such that
g(jco,q) = w. Using (5.11) this condition is equivalent to finding a vector q eQ that
solves the equation
-rj-' r )A(5.12)
sd,R\^o,R + Rq) + jsd j\d0 j +Dp Iqj
This problem can be characterized as a linear equality/inequality problem as shown
below.
Theorem 5.3. Let weC be an arbitrary point with finite magnitude on the
Nyquist plane. Then w e V(co) if and only if there exists a feasible solution
q e Rp to the linear equality
A(w) q = b(w)
subject to the linear-inequality constraint
1
0
0
o'
-1
0
0
0
-q~x
0
1
0
0
0
-1
0
0
- 0
0
0
1
0
0
0
-1
r*~p.
where
A(w): =
Sn,R^p,R WRSc,rD p Sn,I^p,I WRSd.lDpj WfSd RDp R
gR
2x p
b(w): =
n
O.R
n
0,1
+ WRSd,R^0,R W¡SdjdQj
WRSd,1^0,1 W/Sd Rd0 r
gR2
(5.13a)
(5.13b)
(5.14)
(5.15)


145
expected. For any positive values of the multiplicative perturbations a K and a r Figure
7.2 shows whether the ITAE regulation tuning rule with G0/ r0 = 0.5 and a0 = 1 is
stable; the general trend is that an increase in aT results in an increase in the range of
stable values of a K.
rule when 60 / r0 = 0.5 and ae = 1. The *+ marker represents the
nominal point aK = 1, aT = 1, and ae = 1. The ** marker denotes the
gain margin /^^(O.S) = GM(0.5) = 1.816.
To include perturbations in the time delay, the stability boundary is now obtained
for a range of values of a9 (with the tuning ratio parameter remaining a constant value of
00/ t0 = 0.5). Evaluating (7.12a) and (7.12b) over the appropriate frequency intervals as
determined in Theorem 7.2 for values of ae = 0.5, 1.0, 1.5, 1.781, and 2.5 yields the
parametric plot of Figure 7.3.


144
from the first range of frequencies for which a K and aT are simultaneously positive.
t 6
Because ae = 0.540 < 1, Theorem 7.2 states that this range is given by (cod],conX),
T, To
where cod] is the first positive zero of the function fd{co), and conX is the first positive
zero of the function fn(co) given in Lemma 7.2. From (F.2), codx is in the range (0,-^-;r)
and conl is in the range ( k, tt) A simple numerical search in these ranges gives
cod] = 1.122 and conl = 2.961. Now equations (12a) and (12b) can be evaluated over the
frequency interval co e (1.122,2.961) while keeping #0 / r0 = 0.5 and ae- 1. From
Theorem 7.2, the resulting perturbation pairs are all the boundary-point pairs (aK,ar)
with ag = 1 and are plotted in Figure 7.2.
The curve in Figure 7.2 represents the robust stability boundary. All points below
the curve define the region of stable perturbations. The *+ in Figure 7.2 represents the
nominal point aK = 1, aT = 1, and ae = 1. Obviously, the nominal point is part of the
region of stable perturbations. The figure shows that from the nominal point it is possible
to decrease towards zero the multiplicative perturbation in the process gain aK, or
arbitrarily increase the multiplicative perturbation in the process time constant aT, while
the system remains closed-loop stable. On the other hand, if a K is sufficiently increased,
or at sufficiently, decreased the closed-loop system becomes unstable. These points of
critical instability are given by the intersection of the stability boundary with a horizontal
or vertical line passing through the nominal point, respectively, with the former being the
standard gain margin (7.18), which from Figure 7.2 has a value of GM = 1.816 as


101
and
n0,R
1 0 -1
0.12 0.06 0
(5.21c)
n0J =
0.1
27
1
0.7 0.2
0 0
0 0 0"
2 1 0
0 0 0
-1 0
0.5 -0.5
(5.2 Id)
(5.21e)
(5.21 f)
A(-\+j0)
1.1572 -0.4527 -1.4624
-0.8534 -0.1930 0.0848
(5.22)
K-i + jO)
-0.8049
-2.5870
For this problem, the constraint (5.16b) is
1
0
0"
_3
-1
0
0
3
0
1
0
3
0
-1
0
q<
3
0
0
1
3
0
0
-1
_3_
(5.23)
(5.24)
It can be readily verified using an active-set method (Luenberger, 1984) that the linear
equality/inequality problem (5.16a)-(5.16b) with the data shown above is infeasible.
Invoking Theorem 5.4 it then follows that -1 + j0 g^(co) and hence it can be claimed
that that at co = 0.7 the value set V(a>) excludes the critical point.


24
Y = [y'i y2 y\]
0.8554-0.0000/
0.0145-0.2681/
0.4002-0.1901/
-0.0224-0.4144/
0.0631 + 0.4205/
0.5880 + 0.5489/
0.0505 + 0.3059/
-0.5907 + 0.6311/
0.1654-0.3597/
and the singular values are still
{O',.0-2.O' ,} = {2,2,2}
By construction, the input/output principal direction pairs are aligned as follows
X; = e-0-4000^;
x2 = e~0 6000jy2
x*3 = e-0-6000^;
This result shows that it is possible for several pairs of input/output principal directions to
be aligned when there is singular value multiplicity. Note that the alignment factors,
however, are not necessarily identical.
2.6. Conclusions
This chapter clarifies the implications of the MPDA principle by explicitly
considering the case of a repeated maximum singular value. An alternative proof of the
necessity of the MPDA property is presented that is based on dual norm and vector
theory. This proof shows the ties the MPDA property has to the earlier duality work
which partly inspired it. Examples show that the alignment properties of the input/output
principal direction pairs associated with maximum singular value are directly related to
the eigenvectors associated with eigenvalues of maximum modulus in terms of both the
multiplicity and the amount of alignment.


155
Chapter 5 gives an extension of the critical direction theory to systems with non-
convex value sets through the introduction a general definition of the Nyquist robust
stability margin the preserves the earlier results. Chapter 6 gives a parameter space
method for determined a Proportional-Integral (PI) controller that has guaranteed stability
properties based on the Nyquist robust stability margin while optimizing integral time
error performance objectives.
Lastly, Chapter 7 gives an extensive stability analysis of classical PI controller
tuning rules based approximate first-order-plus-time-delay models. The results give the
region of all stable perturbations in the model parameters as well as plots of the gain and
phase margin and parametric stability margin as a function of the tuning parameter.
8.2. Future Work
One obvious area of future work is applying the stability analysis results of
Chapter 7 to Proportional-Integral-Derivative (PID) controller tuning rules. Another
research interest of author is stability analysis of the predictive controllers, and work is
currently proceeding on designing robust predictive controllers by determining the
parametric stability margin as a function of the tuning parameters of the predictive
controller, namely the prediction horizon, control horizon, and weighing on the input.


117
space Kc and z,. For this example the D-partition method of (Kiselev, 1997) is used to
determine the nominal stability boundary. Then, the robust stability boundary is found
for a representative set of nominal stabilizing controller parameters, by determining the
unity Nyquist robust stability margin contour. The results are shown in Figure 6.1.
Figure 6.1. Stability regions with nominal (continuous line) and robust
(dashed line) stability boundaries. The dashed lines correspond to
contours where &N = 1.
Following the first design method proposed, the performance measure is calculated at
each value of the controller parameters. The resulting contours are shown in Figures 6.2,
6.3, and 6.4 for each of the performance objectives. Notice that the IAE and ITAE
minimum, Figures 6.3 and 6.4, occur within the robust stability region. This indicates
that the controllers designed for nominal IAE and ITAE performance are also robustly
stabilizing. In contrast, the controller suggested by the ISE criterion, Figure 6.2, is not


118
robustly stabilizing. Computation of the Nyquist stability margin verifies this as is shown
in Table 6.1. The ISE minimum, Figure 6.2, occurs outside the robust stability region,
and therefore is not robustly stabilizing. These results are as expected, because in general
the controllers designed using the ITAE performance criteria are more conservative, i.e.
less aggressive, than those designed using the LAE performance criteria, which in turn are
more conservative than controllers designed using the ISE criteria. Since the ISE design
is more aggressive, it is less likely to be robustly stabilizing, as is shown in the example.
Figure 6.2. ISE contours superimposed on the robust stability boundary.
The second proposed design method can now be used to design controllers that
have a greater degree of robust stability. This is done by minimizing the performance
measures along a constant robust Nyquist stability margin contour less than 1.


49
Proof. The subdifferential df (d) is just the convex hull of the 3-dimensional
ellipsoid E given by (3.29) translated to Rn by making it an element of the affine set
given by Theorem 3.7. The convex hull of E is the union of itself and its interior which
is given by conv£ = {z e R3 (z- c)TB(z- c) < 1}.
Q.E.D.
To complete the ellipsoidal characterization of df(d) when the maximum
singular value is repeated twice the case of n = 3 is now discussed. When n = 3, g3(u)
is an affine function of g, (u) and g2 (u), such that affdf (d) becomes a 2-dimensional
plane in 3-dimensional space. The effect is that the 3-dimensional ellipsoid E is
degenerate in that it has an axis of length zero, because it is required to be a subset of a 2-
dimensional plane. Consequently, convE = E such that df (d) becomes a 2-dimensional
ellipse including its interior in a 3-dimensional space. Also, df(d) has no relative
interior (i.e., there are no elements of df(d) that are not also on the boundary of df(d)).
Finally, note that degenerate cases are possible. Consider, the matrix
M = diag([ 1 1 0 0]T) such that the maximum singular value is repeated twice. The
above analysis gives H = 0 such that equation (3.27) is not meaningful. For this case
df{d) is no longer contained within a 3-dimensional affine set, but is actually
df (d) = {0} which is a special ellipsoid whose axis are all length zero.
Theorem 3.8 and the preceding paragraph concerning the case of n = 3 give the
desired ellipsoidal characterization of df(d) when the maximum singular value is
repeated once. The next logical set is to extend the results of this section to the case
when the maximum singular value is repeated more than once. Unfortunately, the
preceding ellipsoidal characterization no longer holds and the only characterization of


34
assumption made in the results section, namely, that the function being considered (i.e.,
/(d):= cr(DMD_1)) is nondifferentiable only at points in its domain. The second
theorem gives a characterization of the subdifferential that is used to construct an
expression for df (d) when the function is nondifferentiable.
Theorem 3.2. Let f be a convex function defined on a convex set S c R", and
let D be the set of points where f is differentiable. Then D is a dense subset
S, and its complement in S, given by D, is a set of measure zero. Furthermore,
the gradient mapping V/:d > V/(d) is continuous on D.
Proof. Two different proofs are given in Demyanov and VasiTev (1985) and
Rockefeller (1972). Both proofs are based on measure theory, and show that there are
countable number of sets where / is not differentiable. Q.E.D.
Theorem 3.2 essentially states that / is differentiable almost everywhere in S.
Theorem 3.3. Let f be a convex function defined on a convex set S a R", and
let D be the set of points where f is differentiable. Then
df(d) = conv{z eRn z = hm V/(d*), d, d, d, eD}
Proof. Again, two different proofs are given in Demyanov and VasiTev (1985)
and Rockafeller (1972). Both proofs use the continuity of the gradient on D given in
Theorem 3.2 to show that the limit sequences exist and that they converge to exposed
points of df (d). Therefore the df (d) is the convex hull of all such limit sequences. Q.E.D
As presented Theorem 3.3 seems of little practical value, because to construct
df (d) from it requires the construction of an infinite number of limit sequences. This
not the case as is shown in the results section. The next to theorems deal with solving the


73
/9(A) = min Dell
the following three conditions hold
i)
[ where
Whf-H
Hhf
hi
K)
1
N =
KIN2
KIH2-K|
hi
|2
i ii i2
2
|2 i i
L KIN
kKI
lvlr
1 Kl.
arg(v,.)
= arg(w,.), i = l,2,-
and either
iii-a)
hlhl=1
1=1
or
iii-b)
|w(.| = 0 for at least onei e1,2,,
Proof Assume (4.15) holds where D0 is an optimizing D such that
(4.15)
(4.16)
(4.17)
(4.18)
(4.19a)
(4.19b)
|^(A)| = where /1(A) is an eigenvalue of maximum modulus. Following the development of the
positive matrix result, a necessary condition for (4.20) to hold is that the corresponding
right and left eigenvectors of D¡AD0 be dual with respect to the Euclidean norm. Given


35
optimization problem of infimizing /(d). One gives conditions for an infimum, the
other gives an expression for the steepest desent direction.
Theorem 3.4. For the convex function f (d) to obtain its optimum value on S at
the point d0, it is necessary and sufficient that
Oe0/(do)
Proof A detailed proof is given in Demyanov and Vasilev (1985). Basically,
the condition is sufficient, because epif is entirely above a horizontal supporting
hyperplane at d0. The condition is necessary, because if 0 £df(d0) then it is possible to
find a direction that would decrease /(d0) such that /(d0) is not optimum. Q.E.D.
The steepest decreasing direction is given in the following theorem.
Theorem 3.5. If 0 g df (d), then the subgradient given by
£sd(d) = argmin||S|| (3-12)
d)
points in the opposite direction of the steepest descent direction. That is,
g(d) =
-5sd(d)
IMd)||
is the steepest descent direction of f at d.
Proof. A detailed proof is given in Demyanov and Vasilev (1985). The proof is
based on finding the direction that gives the smallest directional derivative as given by
(3.10). Q.E.D.
It is obvious that Theorem 3.4 and Theorem 3.5 are of great utility for any
steepest descent nondifferentiable optimization algorithm.


65
holds an is sharp, i.e., for any y0 there exists at least one x0, and for any x0 there exists
at least one y0 such that the equality holds (Bauer, 1962). If such a pair (x0,y0) with
||y0||D||xo|| = Rey¡x0 also satisfies the scaling condition
IWUM=i
it is called a dual pair. Note that the dual vector of x is often written (x)D. A pair
(x0,y0) is strictly dual and is written y0||Dx0 if ||y0||D||x0|| = y¡x0 = 1. For strictly
homogenous norms (i.e., those satisfying ||ox| = |or|||x|| for all complex scalars a) the
Holder inequality may be sharpened to (Bauer, 1962)
y x
For a dual pair (x0,y0) under a homogenous norm it follows that
Rey0x0 =||y0||D|x0||> y0x0 which implies that Rey0x0 = y0x0. Hence, for a strictly
homogenous norm every pair of dual vectors (x0,y0) is also strictly dual pair. In
addition, there exists a strict dual y0 for any x0 0 and a strict dual x0 for any y0 0.
In general, the dual norm of a p-norm ||x|| := (XIKT)'^ > assorted
p-norm where 1 / p +1 / q = 1. So the infinity-norm and the 1-norm are duals, and the
dual norm of the 2 (Euclidean) norm is itself. For the 2-norm, a pair (x0,y0) is dual if
y0 = xo/ F
112
4.2.2. Positive Matrix Result
Early work on determining when the spectral radius equals the maximum singular
value is concerned with positive matrices transformed by non-negative diagonal matrices,
because they have good numerical properties (i.e., less round off errors) and therefore


96
in F that lie on the boundary dF(co). In accordance with the general definition of
(Bc(co), the point g0(jco) must be excluded. Therefore, if g0(jco) g F, the set F must be
redefined by excluding from it the element g0(jco) through the assignment
F\g0(ja>) F. Without loss of generality, assume that the points in F are ordered in
increasing distance from the nominal point g0(jco), and that Pi and Pk are respectively
the closest and farthest points from g0(jco). Clearly, contains the point Pk.
When k > 1 the additional elements of (Bc (co) can be readily identified from F by
considering all the straight-line segments PnPn+1, n = 1, 2, ..., k-l, Pn eF. Clearly, if
one interior point of the segment PnPn+x lies outside the value set F(co), it can be
concluded that both Pn and Pn+] lie on the boundary of the value set, and hence, both
Pn and Pn+] are elements of Bc(co). It is easily seen that this is a necessary and
sufficient condition for the membership of Pn and Pn+l in (Bc (co). All the elements of
(Bc (co) can be systematically identified through such a sequential analysis of the points in
F. Note that to determine if the end points of a given segment Pn Pn+] belong to (Bc (co) it
suffices to test if any interior point, say the midpoint, of the segment lies outside of
Vc(co). This set membership condition can be readily determined by applying
Theorem 5.3 to the segment midpoint.
5.6. Intersection of a Ray and Arcs in the Complex Plane
As discussed above, the first step in the computation of the critical perturbation
radius consists of finding the set of points F = { P¡ (co), i = 1, 2, ..., k) that correspond to
intersections of the critical line r(co) with the frame g(jco,E(Q)). This is equivalent to
determining the intersection of a ray (the critical line) and arcs or straight-line segments


146
rule for 60 / r0 = 0.5 and selected values of ae.
The figure shows that as ae increases, the region of stable perturbations becomes smaller
(i.e., for a given value of the perturbation in the process time-constant ax the range of
stable perturbations in process gain aK decreases). The reverse is also true for
decreasing ag values. The inclusion of perturbations in the time delay is better
illustrated in Figure 7.4 where the results obtained for varying ae are plotted on a
contour plot where the x-axis is aT, the y-axis is a0, and the level curves are constant
values of the stability-boundary surface (7.15). For example, consider a multiplicative
perturbation in the process time constant of ar = 1.5 and a multiplicative perturbation in
the process time delay of a9 = 1.5. The contour plot shows that /(1.5,1.5) = 1.6 at this
point, implying that the system remains stable as long as the multiplicative perturbation in
the process gain aK is less than 1.6 as suggested by (7.16). The gain margin (7.18) is


87
When Vc{co) is convex, definition (5.2) indicates that pc{co) represents the
distance between the point g0(jco) and the (unique) point where the critical line
intersects the boundary of V(co). On the other hand, when Vc(co) is nonconvex there are
multiple points where the critical line intersects the boundary of V(co). In such cases,
definition (5.3) indicates that pc(co) is a function of the distance between g0(ja>) and
the boundary-intersection point that is closest to the critical point -1 + j0. Since in many
cases the convexity of V[{co) at any given frequencies may not be known a priori, the
generalized critical radius definition allows the application of the critical direction theory
without conservatism to a more general class of uncertain systems, including the case of
real affine uncertain systems discussed in ensuing sections. The Nyquist robust stability
margin k^{co) computed using the general definition (5.3) for pc{co) is attractive from
an analysis standpoint because through Theorem 5.2 it gives necessary and sufficient
conditions for robust stability. On the other hand, if kN (co) is computed using equation
(5.2) for pc{co), then the condition &N( when the set Vc{co) is nonconvex. From a control design point of view, however, it may
be advantageous to adopt the computationally simpler definition (5.2) even for the case
where Vc (co) is nonconvex, and accept the result as a suboptimal design, as is done in the
context of the structured singular value paradigm where control design is based on an
upper bound rather than on the exact value of the structured singular value. It must be
remarked, however, that when Vc(co) is in fact convex, using definition (5.2) for pc{co)
makes the resulting condition &N(u) stability; and in such cases the results are not conservative. It must also be emphasized