Robust stability analysis methods for systems with structured and parametric uncertainties

MISSING IMAGE

Material Information

Title:
Robust stability analysis methods for systems with structured and parametric uncertainties
Physical Description:
ix, 180 leaves : ill. ; 29 cm.
Language:
English
Creator:
Baab, Charles Thomas
Publication Date:

Subjects

Subjects / Keywords:
Chemical Engineering thesis, Ph.D   ( lcsh )
Dissertations, Academic -- Chemical Engineering -- UF   ( lcsh )
Genre:
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph.D.)--University of Florida, 2002.
Bibliography:
Includes bibliographical references.
Statement of Responsibility:
by Charles Thomas Baab.
General Note:
Printout.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 029636611
oclc - 51947152
System ID:
AA00013545:00001


This item is only available as the following downloads:


Full Text









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
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EQAIA2H5Z_J13C7Y INGEST_TIME 2013-02-14T13:33:41Z PACKAGE AA00013545_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES