numerical algorithms for inverse eigenvalue problems...

NUMERICAL ALGORITHMS FOR

INVERSE EIGENVALUE PROBLEMS ARISING

IN CONTROL AND NONNEGATIVE MATRICES

By

Kaiyang Yang

SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

AT

THE AUSTRALIAN NATIONAL UNIVERSITY

CANBERRA, AUSTRALIA

OCTOBER 2006

c© Copyright by Kaiyang Yang, October 2006


DEPARTMENT OF

INFORMATION ENGINEERING, RSISE

The undersigned hereby certify that they have read and recommend

to the Research School of Information Sciences and Engineering for

acceptance a thesis entitled “Numerical Algorithms for Inverse

Eigenvalue Problems Arising in Control and Nonnegative

Matrices” by Kaiyang Yang in partial fulfillment of the requirements

for the degree of Doctor of Philosophy.

Dated: October 2006

Research Supervisors:Prof. John B. Moore

Dr. Robert Orsi

Examing Committee:Prof. Iven Mareels

Prof. Andrew Lim

ii


Date: October 2006

Author: Kaiyang Yang

Title: Numerical Algorithms for Inverse Eigenvalue

Problems Arising in Control and Nonnegative

Matrices

Department: Information Engineering, RSISE

Degree: Ph.D.

Permission is herewith granted to The Australian National Universityto circulate and to have copied for non-commercial purposes, at its discretion,the above title upon the request of individuals or institutions.

Signature of Author

THE AUTHOR RESERVES OTHER PUBLICATION RIGHTS, ANDNEITHER THE THESIS NOR EXTENSIVE EXTRACTS FROM IT MAYBE PRINTED OR OTHERWISE REPRODUCED WITHOUT THE AUTHOR’SWRITTEN PERMISSION.

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iii

To Bojiu Yang and Ping Sun, My Parents

and Hongsen Zhang, My Husband.

iv

Table of Contents

Table of Contents v

List of Figures ix

List of Symbols xi

Statement of Originality xii

Acknowledgements xiv

Abstract xvi

I Introduction 1

1 Introduction 2

1.1 Inverse Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Problems Arising in Control . . . . . . . . . . . . . . . . . . . 4

1.1.3 Problems Arising in Nonnegative Matrices . . . . . . . . . . . 4

1.2 Research Motivations and Contributions . . . . . . . . . . . . . . . . 5

1.2.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

II Background 11

2 Projections 13

v

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Alternating Projections . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Computational Complexity 20

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 What is NP-Hard? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Computational Complexity in Control . . . . . . . . . . . . . . . . . 22

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

III Problems Arising in Control 23

4 A Projective Methodology for Generalized Pole Placement 27

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2.1 The Symmetric Problem . . . . . . . . . . . . . . . . . . . . . 32

4.2.2 The General Nonsymmetric Problem . . . . . . . . . . . . . . 36

4.3 Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3.1 Classical Pole Placement: Random Problems . . . . . . . . . . 43

4.3.2 Classical Pole Placement: Particular Problem . . . . . . . . . 44

4.3.3 Continuous Time Stabilization: Random Problems . . . . . . 45

4.3.4 Continuous Time Stabilization: Particular Problem . . . . . . 47

4.3.5 Discrete Time Stabilization: Random Problems . . . . . . . . 49

4.3.6 Discrete Time Stabilization: Particular Problem . . . . . . . . 50

4.3.7 A Hybrid Problem . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 A Projective Methodology for Simultaneous Stabilization and De-

centralized Control 53

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2.1 Simultaneous Stabilization . . . . . . . . . . . . . . . . . . . . 56

5.2.2 Decentralized Control . . . . . . . . . . . . . . . . . . . . . . . 59


5.3.1 Simultaneous Stabilization: Random Problems . . . . . . . . . 61

5.3.2 Simultaneous Stabilization: Particular Problems . . . . . . . . 63

5.3.3 Decentralized Control: Random Problems . . . . . . . . . . . 63

vi

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6 Trust Region Methods for Classical Pole Placement 66

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2 Trust Region Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.2.1 Basic Methodology . . . . . . . . . . . . . . . . . . . . . . . . 69

6.2.2 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . 72

6.3 Derivative Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.4 Additional Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 75


6.5.1 Random Problems . . . . . . . . . . . . . . . . . . . . . . . . 77

6.5.2 Particular Problems . . . . . . . . . . . . . . . . . . . . . . . 79

6.5.3 Repeated Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 81

6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7 A Gauss-Newton Method for Classical Pole Placement 84

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.2 The Gauss-Newton Method . . . . . . . . . . . . . . . . . . . . . . . 87


7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

IV Problems Arising in Nonnegative Matrices 92

8 A Projective Methodology for Nonnegative Inverse Eigenvalue Prob-

lems 94

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

8.2 The Symmetric Problem . . . . . . . . . . . . . . . . . . . . . . . . . 98

8.3 The General Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 102


8.4.1 SNIEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.4.2 NIEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

9 Newton Type Methods for Nonnegative Inverse Eigenvalue Prob-

lems 116

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

9.2 Newton Type Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 118

9.3 Derivative Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 119

vii


9.4.1 SNIEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

9.4.2 NIEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

V Conclusion and Future Work 126

10 Conclusion and Future Work 127

10.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

10.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

A Results for Classical Pole Placement 132

B More Computational Results for Stabilization 134

Bibliography 137

viii

List of Figures

1.1 Flow chart of the thesis. . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Projection point onto convex set is unique. . . . . . . . . . . . . . . . 14

2.2 Projection points onto nonconvex set may be multiple. . . . . . . . . 15

2.3 Alternating projections between two intersecting convex sets. . . . . . 17

2.4 Alternating projections between a convex set and a nonconvex set in-

tersecting each other. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1 Illustration of simultaneous stabilization. . . . . . . . . . . . . . . . . 25

3.2 Illustration of decentralized control. . . . . . . . . . . . . . . . . . . . 25

4.1 Examples of generalized static output feedback pole placement problem. 30

4.2 Alternating projections for generalized static output feedback pole

placement problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Performance for classical pole placement using up to 10 initial conditions. 44

4.4 Performance for discrete time stabilization using up to 10 initial con-

ditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.5 The closed loop poles corresponding to a solution for the considered

hybrid problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.1 Quadratic convergence near solution of the Levenberg-Marquardt al-

gorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7.1 Quadratic convergence near solution of the Gauss-Newton algorithm. 91

ix

8.1 Illustration of the problem formulation for SNIEP. . . . . . . . . . . . 99

8.2 Illustration of the problem formulation for NIEP. . . . . . . . . . . . 104

8.3 Linear convergence of the SNIEP algorithm. . . . . . . . . . . . . . . 110

9.1 Quadratic convergence near solution of the SNIEP algorithm. . . . . 122

9.2 Quadratic convergence near solution of the NIEP algorithm. . . . . . 125

x

List of Symbols

R the set of real numbers.

C the set of complex numbers.

Rm×p the set of real m× p matrices.

Cm×p the set of complex m× p matrices.

On the set of orthogonal n× n matrices.

Sn the set of real symmetric n× n matrices.

Sn+ the set of real positive semidefinite n× n matrices.

AT the transpose of matrix A.

A∗ the complex conjugate transpose of matrix A.

tr(A) the sum of the diagonal elements of a square matrix A.

λ(A) the set of eigenvalues of matrix A ∈ Rn×n.

ρ(A) the maximum of the real parts of the eigenvalues of A ∈ Rn×n.

diag(v) the n× n diagonal matrix for v ∈ Cn whose i’th diagonal term is vi.

Re(a) the real part of a ∈ C.

Im(a) the imaginary part of a ∈ C.

vec(A) ∈ Cmp consists of the columns of A stacked below each other for A ∈ Cm×p.

A⊗B the Kronecker product of A and B.

‖ · ‖2 the vector 2-norm.

‖ · ‖F the Frobenius norm of a matrix. (In the cases of no confusion, F is omitted.)

xi

Statement of Originality

I hereby declare that this submission is my own work, in collaboration with others,

while enrolled as a PhD candidate at the Department of Information Engineering,

Research School of Information Sciences and Engineering, the Australian National

University. To the best of my knowledge and belief, it contains no material previously

published or written by another person nor material which to a substantial extent

has been accepted for the award of any other degree or diploma of the university or

other institute of higher learning, except where due acknowledgement has been made

in the text.

Most of the technical discussions in this thesis are based on the following publi-

cations:

• K. Yang, J. B. Moore, and Robert Orsi. Gauss-Newton Method for Solving

Static Output Feedback Pole Placement Problem. In preparation.

• K. Yang, R. Orsi, and J. B. Moore. Newton Type Methods for Solving Inverse

Eigenvalue Problems for Nonnegative Matrices. In preparation.

• K. Yang and R. Orsi. Simultaneous Stabilization and Decentralized Control: a

Projective Methodology. In preparation.

• K. Yang and R. Orsi. Static Output Feedback Pole Placement via a Trust

Region Approach. Submitted to IEEE Transactions on Automatic Control.

xii

• K. Yang and R. Orsi. Generalized Pole Placement via Static Output Feedback:

a Methodology Based on Projections. Automatica, 42 (12), 2006.

• R. Orsi and K. Yang. Numerical Methods for Solving Inverse Eigenvalue Prob-

lems for Nonnegative Matrices. In Proceedings of the 17th International Sympo-

sium on Mathematical Theory of Networks and Systems (MTNS), pp.2284∼2290,

Kyoto, Japan, 2006.

• K. Yang and R. Orsi. Pole Placement via Output Feedback: a Methodology

Based on Projections. In Proceedings of the 16th IFAC World Congress, 6 pages,

Prague, Czech Republic, 2005.

• K. Yang, R. Orsi, and J. B. Moore. A Projective Algorithm for Static Output

Feedback Stabilization. In Proceedings of the 2nd IFAC Symposium on System,

Structure and Control (SSSC), pp.263∼268, Oaxaca, Mexico, 2004.

xiii

Acknowledgements

I express my deepest gratitude to Professor John B. Moore and Dr. Robert Orsi for

being my supervisors and offering me so much guidance and help throughout this

research. John’s invaluable help, guidance and insights greatly influenced me. His

optimistic attitudes towards both research and life impressed me so much and will

always encourage me in my later life. Dr. Robert Orsi had been my supervisor and

the person I most closely worked with. His most careful scientific research attitude,

broad knowledge, invaluable day-to-day supervision and much patience are highly

appreciated. His encouragements will support me to go further in future research.

My special thanks go to Dr. Robert Mahony for being my advisor. He advised

me on some key issues in my research and offered me the chance to do tutoring which

was an important and cherished component of my PhD training.

Professor Uwe Helmke at the Wurzburg University of Germany brought the non-

negative inverse eigenvalue problems to our attention which led to a successful topic

in this research. I would like to thank his generous guidance.

Professor Iven Mareels at the University of Melbourne of Australia advised us on

various parts of the literature for our research. Only with that, could we have a deep

understanding of the previous research situation. I would like to thank his important

help.

I am grateful to Dr. Mei Kobayashi, Mr. Hiroyuki Okano and Mr. Toshinari Itoko

for their collaboration and help during my visit to IBM Tokyo Research Laboratory,

which made my three months’ stay very fruitful and enjoyable.

Department of Information Engineering of the Australian National University and

xiv

the SEACS group in National ICT Australia (NICTA) offered me warm environments

throughout my PhD. I especially thank Dr. Knut Huper, our program leader in

NICTA, for his help and guidance. I also especially thank Dr. Jochen Trumpf and

Dr. Alexander Lanzon for their much help and discussion. My special thanks go to

all our departmental staff and friends for their invaluable friendship.

Finally, the most importantly, I always thank my family for their endless love

and support. My parents dedicated all their love and effort to my twenty years’ long

education. They always support my decisions to pursue my dreams even I need to be

10, 000 kilometers away. They keep my heart warm and my belief firm.

My husband, Hongsen Zhang, is the most amazing person in my life. He gave

me all his love, support, encouragement, understanding and patience during my hard

work. Only with his unfailing love, could I possess the courage to go further in my

life and work.

xv

Abstract

An Inverse Eigenvalue Problem (IEP) is to construct a matrix which possesses both

proscribed eigenvalues and desired structure. Inverse eigenvalue problems arise in

broad application areas such as control design, system identification, principle com-

ponent analysis, structure analysis etc. There are many different types of inverse

eigenvalue problems and despite of a great deal of research effort being put into this

topic many of them are still open and are hard to be solved.

In this dissertation, we propose optimization algorithms for solving two types

of inverse eigenvalue problems, namely, the static output feedback problems and the

nonnegative inverse eigenvalue problems. Consequently, this dissertation is essentially

composed of two parts.

In the first part, three novel methodologies for solving various static output feed-

back pole placement problems are presented. The static output feedback pole place-

ment framework encompasses various pole placement problems. Some of them are

NP-hard, for example, classical pole placement [31]. That is, an efficient (i.e. poly-

nomial time) algorithm that is able to correctly solve all instances of the problem

cannot be expected. In this dissertation, a projective methodology, two trust region

methods and a Gauss-Newton method are proposed to solve various instances of the

pole placement problems.

In the second part, two novel methodologies for solving nonnegative/stochastic

inverse eigenvalue problems are presented. Nonnegative matrices arise in many ap-

plication areas and attract a lot of research in matrix analysis community. Stochastic

xvi

inverse eigenvalue problem has potential applications in Markov chains and the the-

ory of probability etc. In the small dimensional cases, i.e., the dimension of the

resulting matrix is less or equal to 5, there exists necessary and sufficient conditions

to fully characterize the problem. However when the dimension grows larger, the

problem becomes much harder to be solved. The existing necessary conditions are

too general and sufficient conditions are too specific. In general the proofs of the suf-

ficient conditions are nonconstructive. In this dissertation, a projective methodology

and two Newton type methods are proposed which are widely applicable to various

nonnegative inverse eigenvalue problems.

All of the problems considered are important and challenging in their area. The op-

timization methodologies are clearly stated and the algorithms are intensively tested.

More than the problems being solved in this thesis, the algorithms appear to be quite

useful in a lot more related problems, i.e., inverse eigenvalue problems subject to

different structural constraints.

xvii

Part I

Introduction

1

Chapter 1

Introduction

1.1 Inverse Eigenvalue Problems

1.1.1 Background

The spectral properties of a physical system govern its dynamical performance. Hence

the computation of eigenvalues enables the basic understanding of the underlying

physical system. In contrast, an inverse eigenvalue problem is to reconstruct a physical

system from desired dynamical behavior, namely, its eigenvalues. In this research, we

concentrate our attention on the problems whose systems can be somehow expressed

by matrices.

It is clear that an inverse eigenvalue problem is trivially solved if there is no

restriction on its structure. We simply construct a diagonal matrix with desired

eigenvalues as the diagonal entries. However in practice we usually require that the

resulting matrix from a specific inverse eigenvalue problem is physically realizable

and thus additional structural constraints are imposed. For example in static output

feedback problems, the closed loop systems form an affine subspace with the form

A + BKC. In general the solution to an inverse eigenvalue problem must satisfy two

2

3

constraints – the spectral constraint referring to the prescribed spectral data, and the

structural constraint referring to the desired structure.

There are various types of inverse eigenvalue problems and attract a lot of re-

search. For more details on different problems, existing theoretical results, numerical

algorithms, applications and open problems, refer to an excellent book [20].

Associated with any inverse eigenvalue problem are two fundamental questions:

the solvability and the computability. The solvability is to determine a necessary

and/or a sufficient condition under which an inverse eigenvalue problem is solvable.

The computability is to develop efficient and reliable algorithms to construct the

matrices with prescribed eigenvalues and desired structure, where the problem is

feasible. Both questions are difficult and challenging.

Inverse eigenvalue problems arise in a remarkable variety of applications, for ex-

ample, control design, system identification, seismic tomography, principle compo-

nent analysis, exploration and remote sensing, antenna array processing, geophysics,

molecular spectroscopy, particle physics, structure analysis, circuit theory, mechanical

system simulation [20].

In this dissertation, we propose optimization algorithms for the inverse eigenvalue

problems arising in control and nonnegative matrices. We use both classical methods,

i.e. Newton type methods, and novel methods, i.e. alternating projections and trust

region methods. What follows describe the problems being considered in greater

detail.

4

1.1.2 Problems Arising in Control

One of the most basic control tasks is the static output feedback pole placement.

That is, given system matrices A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and a list of desired

eigenvalues λD ∈ Cn, find a static output feedback controller K ∈ Rm×p such that

λ(A + BKC) = λD. Static output feedback control problems are a type of special

inverse eigenvalue problem as the structural constraint is the closed loop system has

the form A + BKC. These problems have simple expressions and wide applications.

They have been intensively researched in the past half a century. Despite this, many of

them are still open and some of them are proved to be NP-hard, for example, classical

pole placement and simultaneous stabilization. Development of novel, efficient and

reliable algorithms for these hard problems are certainly of great interest.

1.1.3 Problems Arising in Nonnegative Matrices

Nonnegative matrices are those whose entries are nonnegative. Stochastic matrices,

which are a type of special nonnegative matrices, are with each row sum to 1. Non-

negative/stochastic matrices are widely used in game theory, Markov chains, theory

of probability, probabilistic algorithms, discrete distributions, categorical data, group

theory, matrix scaling and economics [20]. Nonnegative inverse eigenvalue problem

is to construct a square nonnegative matrix with desired eigenvalues. When the di-

mension of the desired matrix is greater than 5, there is no necessary and sufficient

condition available. The existing necessary conditions are too general and sufficient

conditions are too specific. In general the proofs of sufficient conditions are noncon-

structive. To the best to our knowledge, there are only a few algorithms in literature

5

and they are not applicable to high dimensional problems. Development of novel al-

gorithms which are efficient and can be used to solve large scale problems is of great

interest.

1.2 Research Motivations and Contributions

1.2.1 Motivations

Identifying the open inverse eigenvalue problems arising in control and nonnegative

matrices and their hardness, our main interest here is to develop new optimization

algorithms. In the following, we stress several key points on why the considered

problems are interesting.

1. Pole placement in the generality (see for Chapter 4), which allows flexibility in

choosing the pole placement regions, has not previously been considered. Taking

the advantage that the pole placement regions need not to be convex or even

connected, there are broad choices of the pole placement regions for various

control tasks. Though many algorithms for different instances of generalized

pole placement have been proposed, none of them unifies the solution method

for various problems in one framework.

2. Simultaneous stabilization (see for Chapter 5) is an important problem in robust

control with broad applications. It is proved to be NP-hard [6]. Decentralized

control (see for Chapter 5) arises naturally from the need of controlling large

scale systems, where a centralized controller is not possible to apply. With a

bound on the norm of the controller, decentralized control problem is NP-hard

6

[6]. Despite the great deal of work that has been done on these problems, these

problems are not thoroughly solved and keep attracting a lot of research effort.

3. Classical pole placement (see for Chapter 6 and 7) is one of the most important

open problems in control design. It is shown to be NP-hard [31]. Though simply

expressed, the problem is hard to be solved.

4. Inverse eigenvalue problems for nonnegative/stochastic matrices attracted much

research in the past fifty years. However there are only one algorithm for sym-

metric nonnegative inverse eigenvalue problems and another for general non-

symmetric problems. Due to the nature of existing algorithms, they can hardly

solve high dimensional problems. Nonnegative/stochastic inverse eigenvalue

problems have broad potential applications.

1.2.2 Contributions

In regard to algorithm development, we mainly apply three methodologies, i.e. the

projective methodology, Newton type methods and trust region methods. Since each

of them has its own advantage, we highlight three key points on the contributions of

this work.

• In projective methodologies (see for Chapter 4, 5 and 8), we formulate the prob-

lems into feasibility problems involving two closed sets. In the case where one

of the sets is nonconvex, how to project onto the nonconvex set is an open hard

problem. We propose a substitute map which gives a reasonable estimation

of the true projection and provides good performance in computational exper-

iments. The idea of handling nonconvexity in such problem settings can be

7

used to many other related problems, especially those involving nonsymmetric

matrices.

• In trust region methods and Newton type methods (see for Chapter 6, 7 and

9), we formulate the problems as nonlinear least squares problems. The idea

of applying these methods to the specific problem settings is a perfect joint of

optimization techniques and practical problems. The proposed algorithms are

broadly tested to be efficient and reliable. Such ideas should be useful in many

more problems.

• Various instances of control tasks via static output feedback (see for Chapter

4) are unified into single framework, namely, generalized static output feedback

pole placement. The flexibility in choosing pole placement regions enables the

encompassment of a large scope of control tasks, especially less standard pole

placement problems. Hence the algorithms for solving these problems can be

applied to those which can be put in this framework.

1.3 Outline of the Thesis

This thesis consists of 5 parts including 10 chapters. Part I is the introduction and

Part V is the conclusion and future work. The main content is presented in 3 parts.

Figure 1.1 is a flow chart of the thesis, which clearly indicates on how the thesis is

organized. A more detailed introduction to the main content is as follows:

Part II : Background

In this work, optimization algorithms are developed to tackle inverse eigenvalue

problems arising in control and nonnegative matrices. All of them are hard to be

8

Introduction

Part II

Projections

Computational Complexity

Background

Part III

Problems Arising in Control

A Projective Methodology for Simultaneous Stabilization and

Decentralized Conrol

A Projective Methodology for Generalized Pole Placement

Trust Region Methods for Classical Pole Placement

Part IV

Problems Arising in Nonnegative Matrices

Newton Type Methods for Nonnegative Inverse

Eigenvalue Problems

A Projective Methodology for Nonnegative Inverse

Eigenvalue Problems

Conclusion and Future Work

A Gauss-Newton Method for Classical Pole Placement

Figure 1.1: Flow chart of the thesis.

solved. Different methodologies are employed to the algorithm developments,

i.e., alternating projections, Newton type methods and trust region methods.

Overviews of trust region methods and Newton type methods are given in the

chapters they are employed. Projections and alternating projections are used in

three chapters (see for Chapter 4, 5 and 8) and are introduced in the background

part. This part also includes a brief introduction to computational complexity

and the computational complexity in control.

Part III : Arising in Control

Control is one of the two main application areas considered in this work. Op-

timization algorithms for various static output feedback control problems are

9

presented in part III. First of all, generalized pole placement via static output

feedback problem is considered. They are formulated as feasibility problems

and solutions can be found via a projective methodology. Following the same

idea, simultaneous stabilization via static output feedback and stabilization via

decentralized static output feedback are considered. Alternating projection idea

is applied. Then trust regions methods are developed for solving classical pole

placement via static output feedback. As this problem is NP-hard, the trust

region methods appear to be performing extremely well. Lastly a novel Gauss-

Newton method is considered for the classical pole placement problem. The

problem is formulated as a constrained nonlinear optimization problem and

minimization is achieved via a Gauss-Newton method.

Part IV : Arising in Nonnegative Matrices

Problems arising in nonnegative and stochastic matrices are the other main

application area in this work. Two different methodologies are presented for

solving nonnegative/stochastive inverse eigenvalue problems, namely, a projec-

tive methodology and Newton type methods. The algorithms appear to be very

useful for various problems. The algorithms are also extendable to many other

inverse eigenvalue problems, especially those involving nonsymmetric matrices.

1.4 Summary

This chapter forms a base of this thesis. We first introduce the background of the

inverse eigenvalue problems. Identifying its broad applications and the existing diffi-

culties in various inverse eigenvalue problems, we are motivated to focus our attention

10

on the problems arising in control and nonnegative/stochastic matrices. Based on

this, we are ready to proceed to the following chapters with research details.

Part II

Background

11

12

This part consists of two chapters. They cover the key background information on

projections and computational complexity and are indispensable to understand the

main parts of this research.

Chapter 2 contains general properties of projections, alternating projections and

recalls how alternating projections can be used to find a point in the intersection of a

finite number of closed (convex) sets. It plays a key part in the algorithms in Chapter

4, 5 and 8. Basic knowledge is introduced and important properties are highlighted.

In Chapter 3, a brief introduction to computational complexity is presented. It is

essential to understand how hard the NP-hard problems are since several important

NP-hard problems are tackled in this work.

Chapter 2

Projections

2.1 Introduction

This chapter contains general properties of projections, alternating projections and

introduces the method of alternating projections. Since alternating projection idea,

especially its variation applied to the problems involving nonconvex set, is crucial

to the algorithms in Chapter 4, 5 and 8, we gather such materials in this separate

chapter. We first introduce the basic knowledge and then emphasize on the key

properties of the concepts introduced.

2.2 Projections

Projections play a key part in the algorithms. This section contains general properties

of projections.

Let x be an element in a Hilbert space H and let D be a closed (possibly noncon-

vex) subset of H. Any d0 ∈ D such that ‖x− d0‖ ≤ ‖x− d‖ for all d ∈ D will be

13

14

called a projection of x onto D. In the cases of interest here, namely where H is a

finite dimensional Hilbert space, there is always at least one such point for each x. A

function PD : H → H will be called a projection operator (for D) if for each x ∈ H,

PD(x) ∈ D and

‖x− PD(x)‖ ≤ ‖x− d‖ for all d ∈ D.

Where convenient, we will use y = PD(x) to denote that y is a projection of x

onto D. We emphasize that y = PD(x) only says y is a projection of x onto D and

does not make any statement regarding uniqueness.

If D is convex as well as closed then each x has exactly one projection point PD(x)

[51]. In Figure 2.1, D is a closed convex set and x is an arbitrary starting point. PD(x)

is a projection of x onto D and PD(x) is unique.

()

Figure 2.1: Projection point onto convex set is unique.

However if D is nonconvex but closed then each x may have multiple projection

points PD(x)’s. In Figure 2.2, D is a closed nonconvex set and x is an arbitrary

starting point. Due to the nonconvexity of D, the projections of x onto D may be

not unique. PD(x)1 and PD(x)2 are both of the smallest distance to x among all the

points in D and hence are both projections.

15

()2

()1

Figure 2.2: Projection points onto nonconvex set may be multiple.

2.3 Alternating Projections

In Chapter 4, 5 and 8, all problems of interest are feasibility problems of the following

abstract form.

Problem 2.3.1. Given closed sets D1, . . . ,DN in a finite dimensional Hilbert space

H, find a point in the intersectionN⋂

i=1

Di

(assuming the intersection is nonempty).

(In fact, we will solely be interested in the case N = 2.)

If all the Di’s in Problem 2.3.1 are convex, a classical method of solving Problem

2.3.1 is to alternatively project onto the Di’s. This method is often referred to as the

Method of Alternating Projections (MAP). If the Di’s have nonempty intersection, the

successive projections are guaranteed to asymptotically converge to an intersection

point [9].

Theorem 2.3.2 (MAP). Let D1, . . . ,DN be closed convex sets in a finite dimen-

sional Hilbert space H. Suppose⋂N

i=1Di is nonempty. Then starting from an arbitrary

initial value x0, the sequence

xi+1 = PDφ(i)(xi), where φ(i) = (i mod N) + 1,

16

converges to an element in⋂N

i=1Di.

We remark that the usefulness of MAP for finding a point in the intersection of

a number of sets is dependent on being able to compute projections onto each of the

Di’s.

The significance of Theorem 2.3.2 is that it defines a systematic numerical algo-

rithm for finding a point in the intersection of closed convex sets [67].

Theorem 2.3.3. Let D1, . . . ,DN be closed convex sets in a finite dimensional Hilbert

space H. Given constants t1, . . . , tN in the interval (0, 2), for i = 1, . . . , N , define the

operators

Ri = (1− ti)Id + tiPDi.

If⋂N

i=1Di is nonempty, then starting from an arbitrary initial value x0, the following

sequence

xi+1 = Ri(mod N)+1(xi)

converges to an element in⋂N

i=1Di.

Proof. See [36]. An alternate proof can also be found in [78].

Here Id denotes the identity operator on H; Id(x) = x for all x ∈ H. Each Ri of

the theorem is a projection operator onto Di if and only if ti = 1. If ti 6= 1 then Ri

is referred to as a relaxed projection. Theorem 2.3.3 is a generalization of Theorem

2.3.2 which corresponds to the case t1 = · · · = tN = 1. We will see later how the

freedom to choose the ti’s not equal to 1 can be useful. Figure 2.3 illustrates the

alternating projections between two intersecting closed convex sets. The pink region

and the green region represent two closed convex sets. The blue region is the feasible

17

set, i.e. the intersection of these two sets. Starting from an arbitrary point x, a point

in the feasible set can be found via alternating projections.

feasible set

starting point

Figure 2.3: Alternating projections between two intersecting convex sets.

When ti ∈ (0, 1), Ri is under projection where we step only the fraction ti from

the starting point to the target set. Hence Ri does not really map into the target

set, so we do not usually apply this technique to algorithms. On the contrary when

ti ∈ (1, 2), Ri is over projection where we step further ‘into’ the target set. So over

projection is sometimes used to find a point in the intersection in a finite number of

steps [8].

When one or more Di’s are nonconvex, Theorem 2.3.3 no longer applies and start-

ing the algorithm of Theorem 2.3.3 from certain initial values may result in a sequence

of points that does not converge to a solution of the problem [23]. Figure 2.4 illus-

trates the alternating projections between two intersecting sets involving nonconvex

set. The pink region represents a closed convex set. However the green region is a

closed nonconvex set. The blue region is the feasible set, i.e., the intersection of these

two sets. If we choose an arbitrary starting point x, alternating projection scheme

does not converge to a solution in the feasible set. If we choose another arbitrary

starting point y, alternating projection scheme is working and a point in the feasible

18

set can be found.

feasible set

starting point starting point

Figure 2.4: Alternating projections between a convex set and a nonconvex set inter-

secting each other.

If however there are only two sets, then the following distance reduction property

always holds.

Theorem 2.3.4. Let D1 and D2 be closed (nonempty) sets in a finite dimensional

Hilbert space H. For any initial value y0 ∈ D2, if

x1 = PD1(y0),

y1 = PD2(x1),

x2 = PD1(y1),

then

‖x2 − y1‖ ≤ ‖x1 − y1‖ ≤ ‖x1 − y0‖ .

Proof. The second inequality holds as y1 is a projection of x1 onto D2 and hence its

distance to x1 is less than or equal to the distance of x1 to any other point in D2 such

as y0. The first inequality holds by similar reasoning.

19

Corollary 2.3.5. If for i = 0, 1, . . .,

xi+1 = PD1(yi), yi+1 = PD2(xi+1),

that is, the xi’s and yi’s are successive projections between two closed sets, then

‖xi − yi‖ is a nonincreasing function of i.

Suppose one is interested in solving Problem 2.3.1 in the case of two sets, D1 and

D2, when one or both sets are nonconvex. If projections onto these sets are com-

putable, a solution method is to alternately project onto D1 and D2. Corollary 2.3.5

ensures that the distance ‖xi − yi‖ is nonincreasing with i. While this is promis-

ing, there is, however, no guarantee that this distance goes to zero and hence that a

solution to the problem will be found.

Most of the literature on alternating projection methods deals with the case of

convex subsets of a (possibly infinite dimensional) Hilbert space; a survey of these

results is contained in [2]. The text [27] is also recommended. There is much less

available for the case of one or more nonconvex sets; see in particular [23].

2.4 Summary

In this chapter, an overview of projections is given. We introduce the general prop-

erties of projections, alternating projections and recalls how alternating projections

can be useful in finding a common point in the intersection of finite number of closed

(convex) sets. Moreover the alternating projections involving nonconvex set are also

analyzed. Please refer to Chapter 4, Chapter 5 and Chapter 8 for the algorithms

using projective ideas.

Keywords: projections, alternating projections.

Chapter 3

Computational Complexity

3.1 Introduction

In this research, we tackle a few NP-hard problems which are long open problems

in control, for example, classical pole placement and simultaneous stabilization via

static output feedback. NP-hardness is directly related to computability and com-

plexity. In this chapter, we present a brief introduction to computational complexity

to give a basic idea on what are NP-hard problems. As large volume of literature

on computational complexity in its own right exists, especially, many on computa-

tional complexity analysis of control related problems, we just mention a few for more

details (see for [6] and [31] and a recent survey [5]).

3.2 What is NP-Hard?

A decision problem is one where the desired output is binary, and can be interpreted

as ‘yes’ or ‘no’.

20

21

A decidable problem is one for which there exists an algorithm that always halts

with the right answer. But there also exist undecidable problems, for which there is

no algorithm that always halts with the right answer.

If T (s) of an algorithm is defined as a function of size to be the worst case running

time over all instances of size s, we say that an algorithm runs in polynomial time if

there exists some integer k such that

T (s) = O(sk)

and we define P as the class of all (decision) problems that admit polynomial time

algorithms. Due to both practical and theoretical reasons, P is generically viewed as

the class of problems that are efficiently solvable.

There are many decidable problems of practical interest for which no polynomial

time algorithm is known. Many of these problems belong to a class known as NP

(nondeterministic polynomial time), that includes all of P. A decision problem is

said to belong to NP if every ‘yes’ instance has a ‘certificate’ of being a ‘yes’ instance

whose validity can be verified with a polynomial amount of computation.

If there exists a problem within the class NP that every problem in NP can be

reduced to it in polynomial time, then this problem is said to be NP-complete.

If a problem is at least as hard as some NP-complete problem, then it is said

to be NP-hard. NP-hardness of a problem means that it is about as difficult as a

NP-complete problem.

22

3.3 Computational Complexity in Control

Computational complexity analysis into various application areas is an active research

topic. It is helpful for the people to understand how valuable an algorithm is or how

much we can optimize the solution. In control, there are many problems proved to

be NP-hard or NP-complete. That is, a polynomial time algorithm that can correctly

solve all instances of such problems cannot be expected. For example, classical pole

placement via static output feedback and simultaneous stabilization via static output

feedback are NP-hard problems that are considered in this work. More than these,

there are many NP-hard problems in robust stability analysis, nonlinear control,

optimal control and Markov decision theory etc. For more details on these open hard

control problems, please refer to a recent survey [5].

3.4 Summary

In this chapter, a brief introduction to computational complexity is given. The main

purpose is to introduce NP-hardness and hence gets to know how hard the NP-hard

problems are. Note that computational complexity analysis is an independent and

abundant research area in its own right.

Keywords: computational complexity, NP-hard, undecidable.

Part III

Problems Arising in Control

23

24

This part consists of four chapters. They include optimization algorithms devel-

oped for solving inverse eigenvalue problems arising in control.

In Chapter 4, we present a projective methodology for solving static output feed-

back pole placement problems of the following rather general form: given n subsets

of the complex plane, find a static output feedback that places in each of these sub-

sets a pole of the closed loop system. The algorithm presented is iterative in nature

and is based on alternating projection ideas. Each iteration of the algorithm involves

a Schur matrix decomposition, a standard least squares problem and a combinato-

rial least squares problem. While the algorithm is not guaranteed to always find a

solution, computational results are presented demonstrating the effectiveness of the

algorithm.

In Chapter 5, we extend the projective methodology presented in Chapter 4 to

tackle some different control problems – simultaneous stabilization via static output

feedback and stabilization via decentralized static output feedback.

Unlike the static output feedback pole placement in the generality considered in

Chapter 4, which is handling one system each time, simultaneous stabilization is to

stabilize multiple systems simultaneously using one static output feedback controller.

In Figure 3.1, P1, . . . , Pn are n independent systems and we are supposed to find a

controller K such that all the closed loop systems are stable.

On the contrary, decentralized control is to split the control task into multiple

channels. In each channel, a controller is employed to stabilize the subsystem. De-

centralized control is generally arising in controlling large scale systems, where cen-

tralized control is not possible to apply. In Figure 3.2, P is a plant which is divided

into n subsystems P1, . . . , Pn. For each subsystem Pi (i = 1, . . . , n), a controller

25

Figure 3.1: Illustration of simultaneous stabilization.

Ki is employed to stabilize it. Again the algorithms are not guaranteed to always

find a solution. The effectiveness of the algorithms is demonstrated in computational

results.

Figure 3.2: Illustration of decentralized control.

In Chapter 6, we present two closely related algorithms for classical pole place-

ment via static output feedback. The pole placement problem is formulated as an un-

constrained nonlinear least squares optimization problem involving the desired poles

and the closed loop system poles. Minimization is achieved via two different trust

26

region approaches that utilize the derivatives of the closed loop poles. Extensive nu-

merical experiments show that the algorithms are very effective in practice though

convergence to a solution is not guaranteed for either algorithm. Near solutions,

both algorithms typically converge quadratically. While the algorithms require the

desired poles to be distinct, effective strategies for dealing with repeated poles are

also presented.

In Chapter 7, we present a Gauss-Newton method for the problem of classical pole

placement via static output feedback. The pole placement problem is formulated as an

constrained nonlinear least squares optimization problem involving the Schur form of

the closed loop system. Minimization is achieved via a Gauss-Newton method. Near

solutions, the algorithm is typically converge quadratically. As algorithm does not

require the desired poles to be distinct, there is no formal limitation on the proposed

algorithm. Further convergence analysis and more extensive experiments will be

carried out in future work.

Chapter 4

A Projective Methodology forGeneralized Pole Placement

4.1 Introduction

There has been a great deal of research done on the problems of pole placement and

stabilization via static output feedback. An overview of theoretical results, existing

algorithms and historical developments can be found in [10], [26], [29], [60], [61]

and [69]. (For the convenience of the reader the appendix A lists some of the main

theoretical pole placement results that have appeared in the literature to date.) Our

main interest here is in algorithms, and in this regard, for pole placement, the survey

paper [61] states that existing sufficiency conditions are mainly theoretical in nature

and that there are no good numerical algorithms available in many cases when a

problem is known to be solvable. Despite the great deal of work that has been done

in this area, new algorithms for these important problems are still of great interest.

In this chapter we will actually consider the following generalized static output

feedback pole placement problem.

Problem 4.1.1. Given A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and closed subsets C1, . . . , Cn

27

28

⊂ C, find K ∈ Rm×p such that

λi(A + BKC) ∈ Ci for i = 1, . . . , n.

Here λi(A + BKC) denotes the i’th eigenvalue of A + BKC.

Problem 4.1.1 encompasses many types of pole placement problems. Indeed by

varying the choice of Ci’s, Problem 4.1.1 can for example be specialized to the following

problems:

1. Classical pole placement :

Ci = ci, ci ∈ C.

Here each region Ci is an individual point on the complex plane, see for Figure

4.1(a).

2. Relaxed classical pole placement :

Ci = z ∈ C | |z − ci| ≤ ri.

Here each region Ci is a disk centered at ci ∈ C with radius ri ≥ 0, see for Figure

4.1(b).

3. Stabilization type problems for continuous time and discrete time systems :

C1 = . . . = Cn = z ∈ C | Re z ≤ −α, α > 0, (4.1.1)

and

C1 = . . . = Cn = z ∈ C | |z| ≤ α, 0 < α < 1,

respectively, see for Figure 4.1(c) and (d).

29

4. Hybrid problems : for example, problems of the type shown in Figure 4.1(e).

Here c ∈ C and the aim is to place a pair of poles at c and c, and to place the

remaining poles in the truncated cone C:

C1 = c, C2 = c, C3 = . . . = Cn = C.

As far as we are aware, pole placement in the generality presented in Problem

4.1.1 has not previously been considered. The most closely related results from the

existing literature are as follows. Early work in [37] considers pole placement in a

single region specified by polynomials. While a Lyapunov type necessary and sufficient

condition is given for a matrix to have its eigenvalues in such a region, this condition

is polynomial in the matrix in question and hence not readily amenable to controller

design. In [16], LMI conditions are presented that are sufficient though not necessary

for pole placement in various convex regions. These results cover state feedback and

full-order dynamic output feedback but not static output feedback. Another LMI

based approach, again for a single, though this time, possibly disconnected region, is

considered in [7]. A method for placing poles in distinct convex regions (each region

is specified using linear programming constraints or second order cone constraints)

is given in [38] however the method is based on eigenvalue perturbation results and

hence appears largely limited to cases where the open loop poles are already quite

close to the desired poles. In [63], pole placement in distinct convex regions (each

region is a disk or a half plane) is achieved via a rank constrained LMI approach

though the results are only for state feedback.

This chapter presents an algorithm for Problem 4.1.1. The approach employed

here is quite different to each of the approaches mentioned above. Problem 4.1.1 is

shown to be equivalent to finding a point in the intersection of two particular sets,

30

(a) Classical pole placement. (b) Relaxed classical pole placement.

(c) Continuous time stabilization. (d) Discrete time stabilization.

(e) A hybrid problem.

Figure 4.1: Examples of generalized static output feedback pole placement problem.

31

one of which is a simple convex set, the other a rather complicated nonconvex set.

The algorithm is iterative in nature and is based on an alternating projection like

scheme between these two sets. Each iteration of the algorithm involves a Schur

matrix decomposition and a standard least squares problem. If the Ci’s are not all

equal, each iteration also requires a combinatorial least squares matching step.

Alternating projection type ideas have been employed previously for output feed-

back stabilization, see in particular [33], [35] and [57]1. A distinguishing feature of

our algorithm is that, unlike these methods, our algorithm does not involve LMIs. (A

further technical difference is that rather than solving feasibility problems that in-

volve symmetric matrices, the problem solved by the algorithm is a feasibility problem

involving nonsymmetric matrices.)

The algorithm can be applied to problems with rather general choices for the Ci

regions. In fact the only formal requirement is the following, which we state in the

form of an assumption.

Assumption 1. It is possible to calculate projections onto each of the Ci’s: given

z ∈ C it is possible to find zi ∈ Ci such that |z − zi| ≤ |z − c| for all c ∈ Ci.

In particular, the Ci’s must be closed sets though they need not be convex or even

connected.

For given A, B, C and Ci’s, Problem 4.1.1 may or may not have a solution. Indeed,

one would expect that determining whether a particular instance of Problem 4.1.1 is

solvable is in general difficult. For example, the problem of determining whether the

classical pole placement problem is solvable for particular A, B, C and desired poles

has recently been shown to be NP-hard [31]. Given the difficulty of Problem 4.1.1, an

1They were first used in control design to solve certain convex problems, see [34].

32

efficient (i.e., polynomial time) algorithm that is able to correctly solve all instances

of the problem cannot be expected, and while the algorithm presented here is often

quite effective in practice, it is not guaranteed to find a solution even if a solution

exists.

This chapter is structured as follows 2. Section 4.2 presents the solution method-

ology. To motivate the solution methodology we first restrict our attention to systems

that have a symmetric state space representation (A = AT , C = BT ) and present

an algorithm for this easier class of problems. The general problem is then consid-

ered. Section 4.3 contains computational results of applying the algorithm to various

instances of Problem 4.1.1.

4.2 Methodology

4.2.1 The Symmetric Problem

Systems with a symmetric state space realization, that is, systems with state space

matrices satisfying A = AT , C = BT , occur in various contexts, for example RC-

networks. In order to motivate our solution methodology, we first consider the fol-

lowing special case of Problem 4.1.1 for symmetric systems3.

Problem 4.2.1. Given A ∈ Sn, B ∈ Rn×m, and closed subsets C1, . . . , Cn ⊂ R, find

K ∈ Sm such that

λi(A + BKBT ) ∈ Ci for i = 1, . . . , n.

2Please refer to Chapter 2 for the background of projections.3See [52] for more regarding the classical pole placement problem for symmetric systems and a

rather surprising result regarding arbitrary pole placement for such systems.

33

Note that in Problem 4.2.1 the static output feedback matrix K is required to

be symmetric and that the Ci’s are assumed to be subsets of the real numbers. This

latter assumption is natural as given any symmetric K, A+BKBT is also symmetric

and hence has real eigenvalues.

We assume A, B and C1, . . . , Cn are given and fixed. Let

L = Z ∈ Sn | Z = A + BKBT for some K ∈ Sm

and let M denote the set of symmetric matrices with eigenvalues in the specified

regions C1, . . . , Cn,

M = Z ∈ Sn | λi(Z) ∈ Ci, i = 1, . . . , n.

The symmetric problem can be stated as follows:

Find X ∈ L ∩M.

We now show that projections onto both sets L and M can be calculated and

hence that alternating projections can be employed as a solution method.4

For now on, Sn will be regarded as a Hilbert space with inner product

〈Y, Z〉 = tr(Y Z) =∑i,j

yijzij,

and associated norm ‖Z‖ = 〈Z, Z〉 12 (the Frobenius norm). For i = 1, . . . , n, PCi

will

denote a projection operator for Ci.

The set L is an affine subspace and hence is convex. Projection of X ∈ Sn onto

L involves solving a standard least squares problem.

4As the system is symmetric and we restrict K to be symmetric, the symmetric static outputfeedback continuous time stabilization problem is equivalent to an LMI problem. Hence, if theproblem is solvable, a numerical solution to the problem can be readily found using existing LMIalgorithms, see for example [71]. The L and M for continuous time stabilization problem are bothconvex and projections onto these sets are readily calculated.

34

Lemma 4.2.2. The projection of X ∈ Sn onto L is given by PL(X) = A + BKBT

where K is a solution of the least squares problem

arg minK∈Sm

‖(B ⊗B) vec(K)− vec(X − A)‖2 .

A proof of a more general version of this result is given in the next subsection.

While the set M is not convex, it is still possible to calculate projections onto M.

How to do this is shown in Theorem 4.2.4 below. The result is based on the following

result of Hoffman and Wielandt.

Lemma 4.2.3. Suppose Y, Z ∈ Sn have eigenvalue-eigenvector decompositions

Y = V DV T , D = diag(λY1 , . . . , λY

n ),

Z = WEW T , E = diag(λZ1 , . . . , λZ

n ),

where V,W ∈ Rn×n are orthogonal and λY1 ≥ . . . ≥ λY

n and λZ1 ≥ . . . ≥ λZ

n . Then

‖D − E‖ ≤ ‖Y − Z‖ , (4.2.1)

where ‖ · ‖ denotes the Frobenius norm.

Proof. See for example [40, Corollary 6.3.8].

Theorem 4.2.4. Given Y ∈ Sn, let Y = V DV T be an eigenvalue-eigenvector de-

composition of Y with D = diag(λ1, . . . , λn). Let σ be a permutation of 1, . . . , nsuch that amongst all possible permutations, it minimizes

n∑

k=1

|λk − PCσ(k)(λk)|2.

Define

PM(V, D) = V DV T

35

where

D = diag(PCσ(1)(λ1), . . . , PCσ(n)

(λn)).

Then PM(V, D) is a best approximant in M to Y in the Frobenius norm.

Proof. Let Y be as in the theorem statement. As PM(V,D) ∈M, it remains to show

‖Y − PM(V,D)‖ ≤ ‖Y − Z‖ for all Z ∈M. (4.2.2)

Without loss of generality, suppose the eigenvalues of Y are ordered, i.e., λ1 ≥ . . . ≥λn. Similarly, for Z ∈ M, let Z = WEW T be an eigenvalue-eigenvector decomposi-

tion with E = diag(λZ1 , . . . , λZ

n ) and λZ1 ≥ . . . ≥ λZ

n .

As the Frobenius norm is orthogonally invariant,

‖Y − PM(V,D)‖ = ‖D − D‖. (4.2.3)

Let π be a permutation of 1, . . . , n such that λZk ∈ Cπ(k), k = 1, . . . , n. (Such a

permutation exists as Z ∈M and hence must have an eigenvalue in each of the Ci’s.)

Define E = diag(PCπ(1)(λ1), . . . , PCπ(n)

(λn)). It follows from the definition of D that

‖D − D‖ ≤ ‖D − E‖. (4.2.4)

As for each k, |λk − PCπ(k)(λk)| ≤ |λk − λZ

k |, it also follows that

‖D − E‖ ≤ ‖D − E‖. (4.2.5)

Combining (4.2.3), (4.2.4), (4.2.5) and inequality (4.2.1) from Lemma 4.2.3 gives

the inequality in (4.2.2) and the proof is complete.

Note that to calculate PM(V, D) we keep the original orthogonal matrix V and

simply modify the diagonal matrix D to D. The fact that V remains unchanged

motivates our solution method for the general nonsymmetric case.

36

4.2.2 The General Nonsymmetric Problem

Consider again Problem 4.1.1. Throughout this subsection it is assumed A, B, C and

C1, . . . , Cn are given and fixed.

From now on Cn×n will be regarded as a Hilbert space with inner product

〈Y, Z〉 = tr(Y Z∗) =∑i,j

yij zij.

The associated norm is the Frobenius norm ‖Z‖ = 〈Z, Z〉 12 .

In this subsection we redefine L to be the set of all possible closed loop system

matrices,

L = Z ∈ Rn×n | Z = A + BKC for some K ∈ Rm×p,

and redefine M to be the set of complex matrices with eigenvalues in the specified

regions C1, . . . , Cn,

M = Z ∈ Cn×n | λi(Z) ∈ Ci, i = 1, . . . , n.

Problem 4.1.1 can now be stated as:

Find X ∈ L ∩M.

A solution strategy to solve Problem 4.1.1 would be to employ an alternating pro-

jection scheme, alternatively projecting between L and M. A difficulty occurs in

trying to do this. While L is convex and M is nonconvex, just as for the symmetric

problem, M in this case is a substantially more complicated set and how to calculate

projections onto this set is a difficult unsolved problem. That is, given a point Z, it

is not clear how to find a point in M of minimal distance to Z.

37

As will be verified by the experiments, an alternating projection like scheme can

still be quite successful if for M a suitable substitute mapping is used in place of an

actual projection operator.

Figure 4.2 illustrates the problem formulation for Problem 4.1.1 and the alternat-

ing projection scheme. The pink line represents the set L as it is an affine subspace.

The green region represents the set M which is a rather complicated closed noncon-

vex set. The dotted parts are the feasible set, i.e. the intersection of the two sets.

Starting from an arbitrary point x, we know exactly how to do projection onto L.

However instead of the actual projection onto M which is a pink point with dotted

arrow, our proposed substitute mapping gives a reasonable estimation at the black

point in M.

feasible set

starting point

Figure 4.2: Alternating projections for generalized static output feedback pole place-

ment problems.

Before proceeding, recall Schur’s result [40, Th 2.3.1].

Theorem 4.2.5. Given Z ∈ Cn×n with eigenvalues λ1, . . . , λn in any prescribed order,

there is a unitary matrix V ∈ Cn×n and an upper triangular matrix T ∈ Cn×n such

38

that

Z = V TV ∗,

and Tkk = λk, k = 1, . . . , n.

The following map is proposed as a substitute for a projection map onto M.

Though it is not a true projection map, the notation PM will still be used. Our choice

for PM is motivated by the projection operator for M for the symmetric problem.

We stress that as PM is not a true projection map, unlike for the symmetric problem,

the distance reduction property of Theorem 2.3.4 may not hold. While PM has

various desirable properties (see below), the theoretical convergence properties of the

algorithm are currently unclear, providing interesting questions for future research.

In what follows, Assumption 1 is assumed to hold.

Definition 4.2.6. Suppose V ∈ Cn×n is unitary and T ∈ Cn×n is upper triangular.

Let σ be a permutation of 1, . . . , n such that amongst all possible permutations, it

minimizesn∑

k=1

|Tkk − PCσ(k)(Tkk)|2. (4.2.6)

Define

PM(V, T ) = V TV ∗

where T is upper triangular and given by

Tkl =

PCσ(k)(Tkk), if k = l,

Tkl, otherwise.

Note that PM maps into the set M. Note also that, just as for the symmetric

problem, finding σ involves solving a combinatorial least squares problem. (This will

be discussed further later in the section.)

39

In order to apply PM to Z ∈ Cn×n, a Schur decomposition of Z must first be found.

A given Z may have a nonunique Schur decomposition and Z = V1T1V∗1 = V2T2V

∗2

does not necessarily imply PM(V1, T1) = PM(V2, T2). Hence, PM may give different

points for different Schur decompositions of the same matrix. (Though it was not

discussed at the time, similar comments apply to the PM mapping for the symmetric

case.) This is not so important as different Schur decompositions lead to points in

M of equal distance to the original matrix, as is now shown.

Theorem 4.2.7. Suppose Z = V1T1V∗1 = V2T2V

∗2 where V1, V2 ∈ Cn×n are unitary

and T1, T2 ∈ Cn×n are upper triangular. Then

‖PM(V1, T1)− Z‖ = ‖PM(V2, T2)− Z‖ .

Proof. Suppose Z = V TV ∗ where V is unitary and T is upper triangular. If T is

the matrix given in Definition 4.2.6, then by the unitary invariance of the Frobenius

norm,

‖PM(V, T )− Z‖ = ‖T − T‖.

As ‖T−T‖2 equals the quantity in (4.2.6), ‖PM(V, T )−Z‖ depends only on T11, . . . , Tnn

(and the sets C1, . . . , Cn). The Tkk’s are the eigenvalues of Z and hence, aside from

ordering, are not decomposition dependent. The result now follows by noting that

(4.2.6) does not depend on the ordering of the Tkk’s.

PM(V, T ) keeps V fixed and modifies T . Theorem 4.2.8 below shows that of all

the points in M of the form V TV ∗, T ∈ M upper triangular, i.e., of all the points

in M that have a Schur decomposition with the same V matrix, PM(V, T ) is closest

(or at least equal closest) to the original point Z = V TV ∗.

40

Theorem 4.2.8. Suppose Z = V TV ∗ ∈ Cn×n with V unitary and T upper triangular.

Then PM(V, T ) satisfies

‖PM(V, T )− Z‖ ≤ ‖V TV ∗ − Z‖ for all upper triangular T ∈M.

Proof. Let T be an upper triangular matrix in M. The unitary invariance of the

Frobenius norm implies the result will be established if it can be shown

‖T − T‖ ≤ ‖T − T‖,

where T is the matrix given in Definition 4.2.6. As both T and T are upper triangular

and T ∈M, it follows that

‖T − T‖2 =n∑

k=1

|Tkk − Tkk|2 +∑

k<l

|Tkl − Tkl|2 (4.2.7)

and that Tkk ∈ Cσ(k), k = 1, . . . , n, for some permutation σ.

The result now follows by noting that ‖T −T‖2 equals the quantity in (4.2.6) and

that this value must be less than or equal to the first summation on the right hand

side of the equality in (4.2.7).

As for the symmetric problem, projection of X ∈ Cn×n onto L involves solving a

standard least squares problem.

Lemma 4.2.9. The projection of X ∈ Cn×n onto L is given by PL(X) = A + BKC

where K is a solution of the least squares problem

arg minK∈Rm×p

∥∥(CT ⊗B) vec(K)− vec[Re(X)− A]∥∥

2.

Proof. We would like to find K ∈ Rm×p that minimizes

‖X − (A + BKC)‖2 . (4.2.8)

41

As A,B and C are real matrices, it follows that (4.2.8) equals

‖Re(X)− (A + BKC)‖2 + ‖Im(X)‖2 , (4.2.9)

and hence that the problem is equivalent to minimizing the first term in (4.2.9).

The result now follows by noting that for any Z ∈ Cn×n, ‖Z‖ = ‖vec(Z)‖2, and

that for any (appropriately sized) matrices P, Q and R, vec(PQR) = (RT ⊗P ) vec(Q)

[41].

Here is our algorithm for Problem 4.1.1.

Algorithm:

Problem Data. A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and C1, . . . , Cn ⊂ C.

Initialization. Choose a randomly generated matrix Y ∈ Rn×n.

repeat

1. X := PL(Y ).

2. Calculate a Schur decomposition of X: X = V TV ∗.

3. Y := PM(V, T ).

until ‖X − Y ‖ < ε.

Note that as Y = PM(V, T ) = V TV ∗, see Definition 4.2.6, ‖X − Y ‖= ‖V TV ∗ − V TV ∗‖= ‖T − T‖ = (

∑ |Tkk − Tkk|2) 12 . As the Tkk’s are the eigenvalues of X ∈ L and the

Tkk’s are the eigenvalues of Y ∈ M, the algorithm stops when X (which equals

A + BKC for some K) has eigenvalues sufficiently close to those of a matrix that

satisfies the pole placement constraints. In particular, each eigenvalue of such an X

cannot violate the pole placement constraints by more than ε.

42

As mentioned previously, PM involves finding a permutation σ that minimizes

(4.2.6). The first step in solving this combinatorial least squares problem is calculating

the squared distance of each Tkk to each subset Cl and placing these values in a n×n

cost matrix D:

Dkl := |Tkk − PCl(Tkk)|2. (4.2.10)

The problem is now equivalent to finding a permutation σ such that∑

Dkσ(k) is

minimal.

The problem of finding a minimizing σ given a cost matrix D is a linear assignment

problem which can be solved in O(n3) time using the so called Hungarian method, see

[53] for details.

Note that if the Ci’s are not all distinct, for example this occurs for stabilization

problems and what here have been called hybrid problems, the complexity of the

matching problem is reduced. In fact for stabilization problems, all the Ci’s are the

same and no matching step is required. For the hybrid problem shown in Figure 4.1

(e), it is only necessary to check n(n − 1) possibilities corresponding to which two

Tkk’s are matched to c and c. Hence for this hybrid problem, the direct approach is

faster than using the Hungarian method.

Also note that, given a cost matrix D, an alternative to the Hungarian method

is the following faster suboptimal matching strategy. Find the (or a) smallest entry

in D, say Dkl. Match Tkk with Cl and cross out row k and column l of D. Now

only consider the uncrossed out entries in D and repeat, until all n matches have

been made. This method does not always find the optimal matching though it can

often be an quite effective substitute for the Hungarian method. It will be termed

suboptimal matching. Surprisingly, as will be shown in the next section, by using

43

suboptimal matching in the algorithm it was possible to solve a particular problem

which was not solvable using the Hungarian method.

4.3 Computational Results

This section contains computational results of applying the algorithm to various in-

stances of Problem 4.1.1. We include results for classical pole placement, continuous

time and discrete time stabilization, and a hybrid problem.

The algorithms for each problem were implemented in Matlab 6.5 and all results

were obtained using a 3.06 GHz Pentium 4 machine.

Throughout this section a randomly generated matrix will be a matrix whose

entries are drawn from a normal distribution of zero mean and variance 1.

4.3.1 Classical Pole Placement: Random Problems

This subsection contains results for some randomly generated classical pole placement

problems. A 1000 problems with n = 6, m = 4 and p = 3 were created. Each problem

was created as follows. A,B, C and K were generated randomly. A scalar multiple of

the identity was added to A to ensure the stability degree of A + BKC was equal to

α = 0.1. The desired poles were taken to be the poles of A+BKC. Initial conditions

were chosen randomly.

An attempt was made to solve each problem using up to 10 different initial condi-

tions and a maximum of a 1000 iterations per initial condition. With the termination

parameter ε set to ε = 10−3, the overall success rate was 91%. (The success rate

increases to 95% if up to 20 initial conditions are used.) For the problems that were

44

solved, the average number of iterations taken was 1.8 × 103 and the average time

taken was 1.5 CPU seconds.

Figure 4.3 shows the performance of the algorithm for classical pole placement

problem if we try to solve each problem using up to 10 initial conditions.

1 2 3 4 5 6 7 8 9 1045

50

55

60

65

70

75

80

85

90

95

No. of initial conditions

Suc

cess

rat

e (%

)

Figure 4.3: Performance for classical pole placement using up to 10 initial conditions.

Note that for classical pole placement problems, in (4.2.10), if Cl = cl then

PCl(Tkk) = cl.

4.3.2 Classical Pole Placement: Particular Problem

The following problem is taken from [66] and is of interest as the set of desired poles

overlaps with the set of open loop poles. The system matrices are the following:

45

A =

1 0 0 0

0 2 0 0

0 0 −3 0

0 0 0 −4

, B =

1 0

0 1

1 0

1 1

, C =

1 0

1 0

0 1

0 1

T

.

In this problem the set of open loop poles is 1, 2,−3,−4 and the aim is to

place the closed-loop poles at −1,−2,−3,−5. While initial attempts to solve this

problem failed, the problem was solved by using suboptimal matching and replacing

Step 3 of the algorithm with the relaxed projection ‘Y := (1 − t)X + tPM(V, T )’,

t ∈ (0, 2) constant. Strictly speaking this is not a relaxed projection in the true sense

as PM(V, T ) may not be a projection of X = V TV ∗, however, the idea is clearly the

same. Solutions were successfully found by taking t close to 0; t = 0.1, 0.2 and 0.3 can

all be used to successfully find a solution. Likelihood of success increases and speed

of convergence decreases the closer t is to 0. With t = 0.3 and ε = 10−3, solutions can

typically be found in about 1.2 × 104 iterations and about 4.7 CPU seconds.(With

ε reduced greatly to ε = 10−14 a solution was found in about 106 iterations and 275

CPU seconds.)

Note: when employing relaxed projections, the loop termination criterion of the

algorithm is different. It should be replaced by ‘until ‖X − PM(V, T )‖ < ε’ as now

‖X − Y ‖ = ‖X − [(1− t)X + tPM(V, T )]‖ = t ‖X − PM(V, T )‖.

4.3.3 Continuous Time Stabilization: Random Problems

In the comparison paper [26], a number of methods for continuous time stabilization

via static output feedback are compared. In this subsection we repeat the same

numerical experiments of [26] using our algorithm, to see how our algorithm compares.

46

The experiments involve randomly generated problems. A 1000 problems were

generated for each of a number of different choices of system dimensions n, m and

p. In each case, A, B, C and K were generated randomly. A scalar multiple of

the identity was added to A to ensure it had a stability degree of one. A was then

replaced by A−BKC. All stable A’s were discarded.

In applying our algorithm, the Ci’s were chosen the same as in (4.1.1) with α = 0.1.

As we only require the poles of the closed loop system to be stable, we terminated

the algorithm as soon as all poles had real part less than zero. An attempt was made

to solve each problem using up to 10 different initial conditions and a maximum of a

100 iterations per initial condition.

Results are given in Table 4.1. As can be seen, all problems were solved. Most

problems were solved in 10 iterations or less and average solution times were very

small. These results are as good as the results for the best two algorithms tested in

[26].

Other values for α, such as 0.2, 0.5, 1, and 2, were also tried and produced just

as good results, indicating a certain robustness of the algorithm with respect to this

parameter.

The number of iterations required for the stabilization problems presented in this

subsection is much less than the number required for the classical pole placement

problems of the prior subsections. One would expect the stabilization problems to be

easier to solve and hence this result is not unexpected. This probably indicates that

the pole placement problems are harder problems, as one would expect. The other

possibility is that the algorithm is some how better suited to solving stabilization

problems rather than pole placement problems.

47

(n,m, p) (3, 1, 1) (6, 1, 1) (9, 1, 1) (3, 2, 1) (6, 4, 3) (9, 5, 4)

1 ≤ i ≤ 10 996 990 991 993 989 98210 < i ≤ 100 3 7 8 3 11 17

100 < i ≤ 1000 1 3 1 4 0 1NC 0 0 0 0 0 0T 0.0007 0.0007 0.001 0.0006 0.001 0.002

Table 4.1: A comparison of performance for different n, m and p. i denotes the number

of iterations and listed are the number of problems that converged within different

iteration ranges. ‘NC’ denotes the number of problems that did not converge within

1000 iterations. T denotes the average convergence time in CPU seconds for the

problems that converge within 1000 iterations.

Finally we note that being able to solve the continuous time stabilization problem

enables one to solve other related problems. For example the reduced order dynamic

output feedback stabilization problem can also be solved via s system augmentation

technique; see for example [69].

4.3.4 Continuous Time Stabilization: Particular Problem

The following problem taken from [44] appears frequently in the literature. The sys-

tem considered is the nominal linearized model of a helicopter:

A =

−0.0366 0.0271 0.0188 −0.4555

0.0482 −1.0100 0.0024 −4.0208

0.1002 0.3681 −0.7070 1.4200

0.0000 0.0000 1.0000 0.0000

,

48

B =

0.4422 0.1761

3.5446 −7.5922

−5.5200 4.4900

0.0000 0.0000

, C =

0

1

0

0

T

.

In this problem we wish to place the closed loop eigenvalues in the set z ∈C |Re(z) ≤ −α with α = 0.1. To achieve this, we apply the algorithm with A

replaced by A+αI. Before proceeding, we define P γM(V, T ) as follows. It is a modified

version of Definite 4.2.6 specifically for the continuous time stabilization problems.

Definition 4.3.1. Let γ ∈ R be nonpositive. For any V ∈ Cn×n unitary and any

T ∈ Cn×n upper triangular, define

P γM(V, T ) = V TV ∗,

where

Tkl =

minγ, Re(Tkk)+ i Im(Tkk), if k = l and Re(Tkk) ≥ 0,

Tkl, otherwise.

Numerical experiments show that the performance of the algorithm can actually

be improved by using P γM which depends on the parameter γ ∈ R. P γ

M shifts the

real parts of the unstable eigenvalues to γ (≤ 0) rather than to 0. (If γ = 0 then P 0M

is just PM.) As we will see in this subsection, choosing certain value of γ increases

the likelihood of finding solutions. An intuitive justification why γ < 0 can improve

convergence is the following. During the iteration process, applying PL will tend to

shift the eigenvalues back towards the right side of the complex plane. If we replace

the real parts of the unstable eigenvalues with γ < 0, even though PL may shift the

eigenvalues a little bit to the right, they may still end up in the left half plane as

desired.

49

Regarding the choice of the parameter γ, a number of different values were se-

lected for this problem. When γ ≥ −17, the algorithm was not always convergent.

However, when γ ≤ −18, for example −18,−19,−20, . . . , the algorithm appeared to

be always convergent. Typically the algorithm converged within 1000 iterations, with

computational time under 0.7 CPU seconds. A particular solution is

K =[

0.0939 1.1127]T

for which A + BKC has eigenvalues −0.1440,−9.3716,−0.1765± i0.7909.

4.3.5 Discrete Time Stabilization: Random Problems

This subsection contains results for some randomly generated discrete time sta-

bilization problems. For each problem, the aim is to place all poles in the set

C = z | |z| ≤ α, α = 0.9. A 1000 (A,B,C) triples with n = 6, m = 4 and

p = 3 were randomly generated. Triples with A stable were discarded and replaced.

As in Section 4.3.1, an attempt was made to solve each problem using up to 10

different initial conditions and a maximum of a 1000 iterations per initial condition.

With ε = 10−3, the success rate for 1 initial condition was 61% and the overall success

rate was 80%. A plot of success rate versus number of initial conditions is shown in

Figure 4.4. For the problems that were solved, the average number of iterations taken

was 3.3× 103 and the average time taken was 0.37 CPU seconds.

Note: in (4.2.10), PCl(Tkk) equals αTkk

|Tkk| if |Tkk| ≥ α and Tkk otherwise.

50

1 2 3 4 5 6 7 8 9 1060

65

70

75

80

85

No. of initial conditions

Suc

cess

rat

e (%

)

Figure 4.4: Performance for discrete time stabilization using up to 10 initial condi-

tions.

4.3.6 Discrete Time Stabilization: Particular Problem

The following example is taken from [32]. We wish to stabilize the following system:

A =

0.5 0 0.2 1.0

0 −0.3 0 0.1

0.01 0.1 −0.5 0

0.1 0 −0.1 −1.0

, B =

1 0

0 1

0 0

−1 0

, C =

1 0

0 0

0 1

1 1

T

.

Note A is not stable. We take C as in Section 4.3.5 and ε = 10−3. This problem

was easily solved for all initial conditions tried, a stabilizing K could be found on

average in 3 iterations or less.

51

4.3.7 A Hybrid Problem

So far we have presented results for three traditional classes of problems. In this sub-

section we demonstrate the generality of the algorithm by considering a less standard

problem, namely a hybrid problem of the type shown in Figure 4.1(e).

The problem parameters are c = −0.5+ i3, C = z ∈ C | Re z ≤ −2 and | Im z| ≤|Re z|, n = 13, m = 3 and p = 5. To ensure solvability, B, C and K were randomly

generated and A set to A = V T V T −BKC with V a random orthogonal matrix and

T a real block upper triangular matrix with a spectrum satisfying the constraints. (T

was assigned the spectrum −0.5±i3,−2,−2±i,−2.3,−2.5,−3±i3,−3.5±i3.1,−4±i4 and was created by choosing appropriate 1× 1 and 2× 2 blocks for its diagonal.

The remaining upper triangular entries of T were chosen randomly.)

With ε = 10−3, 64% of initial conditions tested resulted in a solution of this

problem within 5000 iterations. For the initial conditions that lead to convergence,

the average number of iterations taken was 8.6× 102 and the average time taken was

3.1 CPU seconds. The closed loop poles corresponding to a particular solution are

shown in Figure 4.5.

Note: the least squares matching steps were done directly rather than using the

Hungarian method.

4.4 Summary

In this chapter a new methodology for solving a broad class of output feedback pole

placement problems is presented. While the methodology is not guaranteed to find a

solution, numerical experiments presented demonstrate that it can be quite effective

52

−5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−5

−4

−3

−2

−1

0

1

2

3

4

5

Figure 4.5: The closed loop poles corresponding to a solution for the considered hybrid

problem.

in practice. A particular strength of the algorithm is that, in addition to being able

to solve classical pole placement and stabilization problems, it can also be used to

solve less standard pole placement problems.

Keywords: static output feedback, pole placement, stabilization, alternating projec-

tions.

Chapter 5

A Projective Methodology for

Simultaneous Stabilization and

Decentralized Control

5.1 Introduction

In this chapter, a projective methodology is presented for solving static output feed-

back simultaneous stabilization and stabilization by decentralized static output feed-

back problems. This chapter contains an overview of the algorithms and some of the

numerical results. An analysis of convergence properties and extensive computational

experiments are needed and will be done in the future work.

In this chapter, we actually consider the following two problems.

Problem 5.1.1. : Static Output Feedback Simultaneous Stabilization.

Given Ai ∈ Rni×ni , Bi ∈ Rni×m, and Ci ∈ Rp×ni for each i = 1, . . . , k , find K ∈ Rm×p

53

54

such that all the closed loop systems Ai + BiKCi for each i = 1, . . . , k are stable.

Simultaneous stabilization problem was first introduced in [62] and [73]. This

problem arises frequently in practice, due to plant uncertainty, plant variation, failure

modes, plants with several modes of operation, or nonlinear plants linearized at several

different equilibria [76].

Static output feedback simultaneous stabilization problem has been shown to be

NP-hard [6]. That is, an efficient (i.e., polynomial time) algorithm that is able to

correctly solve all instances of the problem cannot be expected.

Even for three single input single output systems, no tractable simultaneous sta-

bilization design procedure has been proposed [15]. An effective approach to tackle a

variety of simultaneous stabilization problems is through numerical means.

Note that if Ci’s are all identity, this problem reduces to state feedback simultane-

ous stabilization problem. If k = 1, it reduces to the standard static output feedback

stabilization problem.

Various numerical algorithms have been proposed for solving Problem 5.1.1, for

example, [11] and [12]. Unlike these methods, our algorithm does not involve LMIs.

Note that Problem 5.1.1 is different from what is usually referred to as the ‘si-

multaneous stabilization problem’ in [4] and [72]. Problem 5.1.1 is to find a static

output feedback controller simultaneously stabilizing k > 1 systems. While the other

problem is to seek a dynamic compensator which stabilizes the given system and the

compensator is stable itself, which is also referred to as ‘strong stabilization problem’.

The other problem considered in this chapter is the stabilization by decentralized

static output feedback.

Problem 5.1.2. : Stabilization by Decentralized Static Output Feedback.

55

Given A ∈ Rn×n, Bi ∈ Rn×mi, and Ci ∈ Rpi×n for each i = 1, . . . , k, find Ki ∈Rmi×pi’s such that

A +k∑

i=1

BiKiCi

is stable.

For many large systems like electric power systems, transportation systems and

whole economic systems, it is desirable to decentralize the control task. On the one

hand, using decentralized control will reduce the design complexity greatly; on the

other hand, decentralized control is especially preferable if the measurements have

been taken on local channels and the controls can be applied on local channels only.

Hence decentralized control has drawn a great deal of research into large scale control

problems.

Problem 5.1.2 has been extensively researched and a lot of numerical algorithms

have been proposed, for example, [13] and [47].

Stabilization by decentralized static output feedback is NP-hard if one imposes a

bound on the norm of the controller or if the blocks are constrained to be identical,

that is, all Ki’s are identical [6].

Note that Problem 5.1.2 can be regarded as a generalization of the centralized

problem to include a block diagonal structure constraint on K.

This chapter presents a projective methodology for both Problem 5.1.1 and 5.1.2.

The problems are shown to be equivalent to finding a point in the intersection of

two particular sets. Alternating projection ideas are employed to find a point in the

intersection. One of the sets is an affine subspace and hence convex. We know exactly

how to do the projection onto it. However the other set is closed but nonconvex and

moreover it is a rather complicated set. To the best of our knowledge, how to project

56

onto this set is an open difficult problem. We use a substitute mapping that maps

onto the set, which gives a reasonable estimation of the actual projection operator.

The effectiveness of the algorithms is demonstrated in computational results.

The structure of the chapter is as follows 1. The methodology for both problems

is presented in Section 5.2. Section 5.3 contains computational results demonstrating

the effectiveness of the algorithms.

5.2 Methodology

This section presents the solution algorithms for Problem 5.1.1 and 5.1.2. Throughout

this section Cn×n will be regarded as a Hilbert space with inner product 〈Y, Z〉 =

tr(Y Z∗) =∑

i,j yij zij and the associated norm is the Frobenius norm ‖Z‖ = 〈Z, Z〉 12 .

5.2.1 Simultaneous Stabilization

Throughout this subsection it is assumed that (Ai, Bi, Ci)’s for i = 1, . . . , k are given

and fixed.

In this subsection, we define LS to be the set of all possible closed loop system

matrices,

LS = Z1 ∈ Rn1×n1 , . . . , Zk ∈ Rnk×nk | Z1 = A1 + B1KC1, . . . , Zk = Ak + BkKCk

for some K ∈ Rm×p ,

and define MS to be the set of matrices with eigenvalues in the left half complex

plane,

MS = Z1 ∈ Cn1×n1 , . . . , Zk ∈ Cnk×nk | ρ(Z1) ≤ 0, . . . , ρ(Zk) ≤ 01Please refer to Chapter 2 for background of projections.

57


Find X1, . . . , Xk ∈ LS ∩MS.

A solution strategy to solve Problem 5.1.1 would be to employ an alternating

projection scheme, alternatively projecting between LS and MS. While LS is an

affine subspace and hence convex, MS is in general a rather complicated nonconvex

set. Alternating projections between LS and MS are not guaranteed to converge.

More importantly, how to calculate projections onto MS is an open hard problem.

As will be shown in the experiments, an alternating projection like scheme can

still be quite successful if instead of using a true projection map for MS, a suitable

substitute is employed.

The Schur’s result in Theorem 4.2.5 shows that a complex square matrix is uni-

tarily equivalent to an upper triangular complex matrix.

The following mapping is used as a substitute for the true projection map into

MS. Though it is not a true projection map, the notation PMSwill still be used.

The choice for PMSis motivated by the solution for symmetric static output feedback

stabilization problem, where a true projection can be found.

Definition 5.2.1. For any Vi ∈ Cn×n unitary and any Ti ∈ Cn×n upper triangular,

define

PMS(Vi, Ti) = ViTiV

∗i ,

where

Tikl=

min0,Re(Tikk)+ iIm(Tikk

), if k = l,

Tikl, otherwise.

Given starting points Z1 ∈ Rn1×n1 , . . . , Zk ∈ Rnk×nk , we do PMS(Vi, Ti) to Zi for

each i = 1, . . . , k. Note that PMS(Zi) maps into MS. In order to apply PMS

to Zi,

58

a Schur decomposition of Zi must first be found. A given Zi may have nonunique

Schur decompositions and they may give different projection points. This fact is not

so important as different Schur decompositions lead to points in MS of equal distance

from the original matrix. The proof of this fact is similar to the proof for Theorem

4.2.7 and hence is omitted here. PMS(Vi, Ti) keeps Vi fixed and modifies Ti. Refer

to the proof of Theorem 4.2.8 for a similar idea which shows that of all the points

in MS that have a Schur decomposition with the same Vi matrix, PMS(Vi, Ti) is the

closest (or at least equal closest) to the original point Zi = ViTiV∗i .

The projection of Xi ∈ Rni×ni for each i = 1, . . . , k onto LS involves solving a

standard least squares problem.

Lemma 5.2.2. The projection of Xi ∈ Rni×ni onto LS is given by PLS(Xi) = Ai +

BiKCi where K is a solution of the least squares problem

arg minK∈Rm×p

‖ (k∑

i=1

CTi ⊗Bi ) vec (K)−

k∑i=1

vec [Re (Xi)− Ai ] ‖2

Proof. This proof is similar to the proof for Lemma 4.2.9 and hence is omitted here.


Algorithm:

Problem Data. Ai ∈ Rni×ni , Bi ∈ Rni×m, Ci ∈ Rp×ni for each i = 1, . . . , k.

Initialization. Choose randomly generated matrices Yi ∈ Rni×ni for each i = 1, . . . , k.

repeat

1. Xi := PLS(Yi).

59

2. Calculate a Schur decomposition of each Xi: Xi = ViTiV∗i .

3. Yi := PMS(Vi, Ti).

until ρ(Xi) ≤ 0 for all i = 1, . . . , k.

As we only require the eigenvalues of all the closed loop systems to be stable, we

terminate the algorithm as soon as all the eigenvalues have real parts less or equal to

zero.

5.2.2 Decentralized Control

Throughout this subsection it is assumed that A, Bi and Ci for each i = 1, . . . , k are

given and fixed.

In this subsection we define LD to be the set of all possible closed loop system

matrices,

LD = Z ∈ Rn×n ‖ Z = A +k∑

i=1

BiKiCi for some K1 ∈ Rm1×p1 , . . . , Kk ∈ Rmk×pk,

and define MD to be the set of matrices with eigenvalues in the left half complex

plane,

MD = Z ∈ Cn×n ‖ ρ(Z) ≤ 0


Find X ∈ LD ∩MD.

Again alternating projection scheme is employed to solve Problem 5.1.2. As the

method of projecting onto MD is an unsolved problem, the following PMD(V, T ) is

employed as a substitute of the actual projection operator. Computational experi-

ments show that this scheme is quite successful in practice.

60

The following map PMD(V, T ) is used as a substitute for the projection map onto

MD.

Definition 5.2.3. For any V ∈ Cn×n unitary and any T ∈ Cn×n upper triangular,

define

PMD(V, T ) = V TV ∗,

where

Tkl =

min0,Re(Tkk)+ iIm(Tkk), if k = l,

Tkl, otherwise.

The projection of X ∈ Rn×n onto LD involves solving a standard least squares

problems.

Lemma 5.2.4. The projection of X ∈ Rn×n onto LD is given by PLD(X) = A +

∑ki=1 BiKiCi where vec(K) is a solution of the least squares problem and vec(K)

stacks vec(Ki) in order

arg minK∈R

Pimi×pi

‖ [ CT1 ⊗B1 . . . CT

k ⊗Bk ] vec (K)− vec [Re (X)− A ] ‖22

Proof. This proof is similar to the proof of Lemma 4.2.9 and hence is omitted here.


Algorithm:

Problem Data. A ∈ Rn×n, Bi ∈ Rn×mi , Ci ∈ Rpi×n for each i = 1, . . . , k.

Initialization. Choose a randomly generated matrix Y ∈ Rn×n.

repeat

1. X := PLD(Y ).

61

2. Calculate a Schur decomposition of X: X = V TV ∗.

3. Y := PMD(V, T ).

until ρ(X) ≤ 0.


This section contains computational results of applying the algorithms to both Prob-

lem 5.1.1 and 5.1.2. We present results for both randomly generated problems and

particular problems from literature. As mentioned in the introduction, extensive

numerical experiments will be done and it is a part of the future work of this thesis.

The algorithms for each problem were implemented in Matlab 6.5 and all results


Throughout this section a randomly generated matrix will be a matrix whose

entries are drawn from a normal distribution of zero mean and variance 1.

5.3.1 Simultaneous Stabilization: Random Problems

This subsection contains results for some randomly generated static output feedback

simultaneous stabilization problems. A 1000 problems were created for each of a

number of different choices for the system dimensions (n,m, p). Each problem was

created as follows. System matrices Ai, Bi, Ci for each i = 1, . . . , k and K were

generated randomly. A scalar multiple of the identity was added to Ai to ensure it

had a stability degree of 0.1. Ai was then replaced by Ai − BiKCi. All stable Ai’s

were discarded.

Numerical experiments show that the performance of the algorithm can actually be

62

improved by using a slightly modified version of PMSwhich depends on a parameter

γ ∈ R. A formal definition of similar projection map can be found in Definition 4.3.1,

where P γMS

can be obtained using the similar idea. P γMS

shifts the real parts of the

unstable eigenvalues to γ (≤ 0) rather than to 0. In these experiments, we chose

γ = −1. An attempt was made to solve each problem using up to 10 different initial

conditions and a maximum of 1000 iterations per initial condition.

(n,m, p) (3, 1, 1) (6, 4, 3) (9, 5, 4)k 3 6 3 6 3 6

S.R. 97% 98% 100% 100% 100% 99%i 138 251 38 161 49 222T 0.11 0.41 0.05 0.42 0.11 0.87

Table 5.1: A comparison of performance for different n,m, p and the number of

systems k. S. R. denotes the success rate, T denotes the average convergence time

in CPU seconds, and i the average number of iterations. T and i are based only on

those problems that were successfully solved.

As can be seen from Table 5.1, the results are quite good. The algorithm solved

most of the problems. The average solution times were very small. Note that the

different choices of dimensions have their own difficulties. Kimura’s stabilization

condition is m + p > n (see Theorem A.0.3). Only (n,m, p) = (6, 4, 3) meets this

condition. In the case of (n,m, p) = (9, 5, 4), m + p = n. While in the case of (3, 1, 1)

problems m + p < n, which are quite hard problems.

63

5.3.2 Simultaneous Stabilization: Particular Problems

This subsection contains results for some particular problems of simultaneous stabi-

lization from literature. In order to present a greater number of results, rather than

presenting the details of each problem, only references are given. For each problem,

100 random initial conditions were tested and the maximum number of iterations

per initial condition was set to 1000. As soon as all the systems were stable, the

experiments were terminated.

No. references (n,m, p) k S.R. (%) T i

1 [11, Ex. 1] (2, 1, 1) 3 98% 0.13 282 [11, Ex. 2], [12, Ex. 2], [75, Ex. 1] (3, 1, 2) 4 96% 0.14 313 [11, Ex. 3] (3, 1, 3) 3 96% 0.18 414 [11, Ex. 4], [12, Ex. 1] (2, 1, 1) 3 95% 0.18 40

Table 5.2: Results for particular examples from literature. T and i are based only on

those problems that were successfully solved.

As can be seen from Table 5.2, performance was very good. Solutions to each

problem could be found. The success rates are all greater or equal to 95% and both

the average iterations and average times taken for successfully solved problems are

very small.

5.3.3 Decentralized Control: Random Problems

This subsection contains results for some randomly generated problems of stabiliza-

tion by decentralized static output feedback. A 1000 problems were created for each

of a number of different choices for the system dimensions n and controller dimen-

sions (mi, pi) for i = 1, . . . , k. Each problem was created as follows. System matrices

64

A,Bi, Ci and Ki for i = 1, . . . , k were generated randomly. A scalar multiple of iden-

tity was added to A to ensure it had a stability degree of 0.1. A was then replaced

by A−∑ki=1 BiKiCi. All stable A’s were discarded.

Numerical experiments show that the performance of the algorithm can actually

be improved by using P γMD

with similar idea as from Definition 4.3.1. Here we choose

the value of γ to be −1.

n 10 10 20 20 50 50k 2 3 2 3 2 3

(m1, p1) (2, 2) (2, 2) (3, 4) (3, 4) (5, 7) (5, 7)(m2, p2) (2, 2) (1, 3) (4, 3) (4, 3) (7, 5) (7, 5)(m3, p3) - (3, 1) - (4, 4) - (6, 6)

S.R. 100% 100% 100% 100% 100% 100%i 39 9.7 17 3.8 22 6.5T 0.04 0.01 0.10 0.05 3.3 2.2

Table 5.3: A comparison of performance for randomly generated problems with dif-

ferent n,mi, pi and the number of subsystems k. T and i are based only on those

problems that were successfully solved.

As can be seen from Table 5.3, the algorithm is performing very well in solving

every problems. Not surprisingly that for the same n-dimensional problems, the more

channels the control task is divided in the faster the solution would be found and the

less iterations were needed.

5.4 Summary

In this chapter we present a novel methodology for solving static output feedback

simultaneous stabilization and stabilization by decentralized static output feedback

65

problems. Numerical experiments show that the algorithms are very effective in

practice, though the methodology is not guaranteed to find a solution.

Keywords: static output feedback, simultaneous stabilization, decentralized control,

alternating projections.

Chapter 6

Trust Region Methods for ClassicalPole Placement

6.1 Introduction

Pole placement via static output feedback is a classical problem in systems and con-

trol theory, and there exists a great deal of research into this topic. Unfortunately, the

problem of determining solvability of static output feedback pole placement problems

has recently been shown to be NP-hard [31]. Though sufficient conditions for solv-

ability exist, the survey paper [61] states that these conditions are mainly theoretical

in nature and that there are no good numerical algorithms available in many cases

when a problem is known to be solvable. New algorithms for this important problem

are certainly of interest.

In this chapter we present two related numerical algorithms for solving static

output feedback pole placement problems.

Problem 6.1.1. Given system matrices A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and desired

eigenvalues λD ∈ Cn, find K ∈ Rm×p such that

λ(A + BKC) = λD.

66

67

Two trust region approaches are considered for solving the following unconstrained

nonlinear least squares problem

minK∈Rm×p

f(K) :=1

2

∥∥λ(A + BKC)− λD∥∥2

2. (6.1.1)

Here λ(A+BKC) denotes the vector of eigenvalues of A+BKC, with entries sorted

to give the minimum norm. Trust region methods, which are well known in the

optimization community, are a type of iterative method for minimizing nonconvex

functions. The specific trust region methods we use are the trust region Newton

method and the Levenberg-Marquardt method. In order to employ the trust region

Newton method, at each iteration, the first and second derivatives of the eigenvalues

of A+BKC must be calculated. The Levenberg-Marquardt based algorithm has the

advantage of only requiring the first derivatives of the eigenvalues. Both resulting

algorithms have the desirable property that, near solutions, they typically converge

quadratically.

A technicality that arises in the proposed approaches is that the eigenvalues of

A + BKC may not be differentiable everywhere. They will however be differentiable

at all points at which A + BKC has distinct eigenvalues. A consequence of this is

that the algorithms are only appropriate for problems for which the desired poles are

distinct. It turns out that the algorithms can still be used to solve problems whose

desired poles are distinct but whose separation is quite small and hence this does not

appear to be a serious limitation.

The idea of solving pole placement type problems by utilizing eigenvalue deriva-

tives is not completely new. Related ideas have been used to solve a noncontrol

related inverse eigenvalue problem involving symmetric matrices [30]. However, the

68

methods presented in [30] are all local in nature and hence require one to start suffi-

ciently close to a solution for them to converge. An important distinguishing feature

of our algorithms is that the use of a trust region methodology means that this is

not the case for our algorithms. First derivatives of eigenvalues (though not second

derivatives) have also been used to solve various control problems. For example, in

[38] they are used to try to achieve pole placement in certain convex regions. The

methodology that is used there is quite different to the one used here and is based on

convex programming techniques. As it requires that the open loop poles are already

quite close to the desired poles, this method also only works locally. Along the same

lines as [38], first derivatives of eigenvalues have also been used for robustness analysis

and stabilization, see [59].

This chapter is structured as follows. Section 6.2 contains an overview of trust

region methods. In order to use trust region methods to solve the pole placement

problem, the first and second derivatives of the f are required. Details of these calcu-

lations, including how to calculate derivatives of the eigenvalues, are given in Section

6.3. Trust region methods require the function to be minimized to be differentiable.

While f will typically only be differentiable on an open dense set, this turns out to

be sufficient. Such issues are addressed in Section 6.4. Section 6.5 contains compu-

tational results, including results for a number of problems from the literature.

6.2 Trust Region Methods

This section gives an overview of trust region methods. It is assumed that the function

f : RN → R to be minimized is (sufficiently) smooth. The actual f we wish to

minimize is given in (6.1.1) and it may not satisfy this assumption. Issues related to

69

this fact are addressed in Section 6.4. Additional information on trust region methods

can be found in [24] and [56] .

6.2.1 Basic Methodology

Trust region methods can be used to minimize smooth nonconvex functions and are

iterative in nature. It is assumed that the function f : RN → R to be minimized

is (sufficiently) smooth. The actual f we wish to minimize in problem formulations

may not satisfy this assumption. Issues related to this fact are addressed in Section

6.4. Given a current iterate xk, they construct a possibly nonconvex, quadratic ap-

proximation of the objective function about xk. This model is only assumed to be a

good approximation in a certain ball centered about xk. This is the so-called ‘trust

region’. It turns out that, numerically, it is possible to readily minimize a quadratic

function over a ball. Doing so gives a candidate step pk. The step pk is only accepted

if the difference in the objective function, f(xk) − f(xk + pk), is sufficiently close to

the difference predicted by the model. If pk is not acceptable, the trust region radius

is decreased and the process repeated. On the other hand, if the model gives a good

prediction, the radius of the trust region may be increased to allow a larger step in

the next iteration.

What follows describes the trust region method in greater detail. At each iteration,

the quadratic approximation is assumed to be of the form

mk(p) = f(xk) +∇f(xk)T p +

1

2pT Bkp.

Here Bk is typically either the Hessian of f at xk or some approximation of this

Hessian. If Bk is the Hessian, then mk is simply the 2nd order Taylor approximation

70

of f at xk. As will be discussed below, it may also be useful to consider other choices

for Bk.

Each constrained minimization problem is of the form

minp∈RN

mk(p) s.t. ‖p‖2 ≤ ∆k, (6.2.1)

where ∆k > 0 is the current trust region radius. The solution pk of (6.2.1) gives a

potential step. Whether or not it is a suitable step is assessed by considering the

ratio of actual reduction of the objective to the predicted reduction:

ρk =f(xk)− f(xk + pk)

mk(0)−mk(pk). (6.2.2)

The overall trust region method is as follows.

Trust Region Method, Generic Algorithm ([56])

Given ∆ > 0, ∆0 ∈ (0, ∆), and η ∈ [0, 14):

for k= 0, 1, 2, . . .

Obtain pk by (approximately) solving (6.2.1);

Evaluate ρk from (6.2.2);

if ρk < 14

∆k+1 = 14∆k

else

if ρk > 34

and ‖pk‖2 = ∆k

∆k+1 = min2∆k, ∆else

∆k+1 = ∆k;

if ρk > η

xk+1 = xk + pk

71

else

xk+1 = xk;

end(for).

Approximate solutions of the constrained quadratic minimization problem (6.2.1)

can be obtained in a number of ways. One way is the nearly exact solution method

described in [56, Section 4.2]: it can be shown that problem (6.2.1) is equivalent to

finding a p and a scalar γ ≥ 0 such that the following conditions hold,

‖p‖2 ≤ ∆k,

(Bk + γI)p = −∇f(xk),

γ(∆k − ‖p‖2) = 0,

(Bk + γI) is positive semidefinite.

Without going into the details we mention that finding a p and γ that satisfy these

conditions is equivalent to solving a one dimensional root finding problem in γ which

can be solved using a Newton method.

Regarding the choice of Bk’s, the Hessian of f at xk is a natural choice. In this case,

the method is called the trust region Newton method. When the objective function

f is a least squares cost, say f(x) = 12

∑Mi=1 r2

i (x) for some functions ri : RN → R,

there is another suitable choice. In this case, if we define r(x) = (r1(x), . . . , rM(x))T

and let J(x) denote the Jacobian of r(x), then

∇f(x) = J(x)T r(x) and ∇2f(x) = J(x)T J(x) +M∑i=1

ri(x)∇2ri(x),

and a good choice for Bk is J(xk)T J(xk). The advantages of this choice for Bk include

the fact that it does not require the calculation of the second derivatives of the ri’s and

72

that it gives a good approximation of ∇2f(xk) when f(xk) is small, that is, when each

ri(xk) is small. For this choice of Bk’s, the method is called the Levenberg-Marquardt

method.

6.2.2 Convergence Results

This section contains some general convergence results. These results have been

specialized for our purposes and the given references usually refer to more general

results. The results in this section all assume the nearly exact solution method is

used for the subproblems (6.2.1) and that the algorithm parameter η is nonzero, that

is, η ∈ (0, 14).

A simple but important property of trust region methods is that the cost is non-

increasing from one iteration to the next: for all k ≥ 0, f(xk) ≥ f(xk+1).

Here are some less simple properties. The following result concerns global conver-

gence to stationary points.

Theorem 6.2.1 ([56, Th 4.8]). Suppose that on the sublevel set

x | f(x) ≤ f(x0), (6.2.3)

f is twice continuously differentiable and bounded below, and that ‖Bk‖ ≤ β for some

constant β. Then

limk→∞

∇f(xk) = 0.

Theorem 6.2.1 holds for both the trust region Newton method and the Levenberg-

Marquardt method. For the former method, the following result also holds.

Theorem 6.2.2 ([56, Th 4.9]). Suppose the set (6.2.3) is compact, that f is twice

continuously differentiable on this set, and that Bk = ∇2f(xk). Then the xk’s have a

73

limit point x∗ that satisfies the following first and second order necessary conditions

for a local minima,

∇f(x∗) = 0, ∇2f(x∗) is positive semidefinite.

The following local convergence result for the trust region Newton method implies

that near a (strict) local minimum the method reduces to a pure Newton method.

Theorem 6.2.3 ([56, Th 6.4]). Suppose that Bk = ∇2f(xk). Further, suppose the

sequence of xk’s converges to a point x∗ that satisfies the following first and second

order sufficient conditions for a (strict) local minima,

∇f(x∗) = 0, ∇2f(x∗) is positive definite,

and that f is three continuously differentiable in a neighborhood of x∗. Then the trust

region bound ∆k becomes inactive for all k sufficiently large.

A consequence of Theorem 6.2.3 is that local convergence for the trust region

Newton method is usually quadratic, just as for the pure Newton method.

The final result in this section shows that the Levenberg-Marquardt method is

often locally quadratically convergent to global minima. (Note that this may not be

the case for local minima that are not global minima.)

Theorem 6.2.4 ([56, Section 10.2]). Suppose the ri’s that determine f are three

continuously differentiable in a neighborhood of a global minima x∗. Suppose further

that J(x∗)T J(x∗) is positive definite. Then the Levenberg-Marquardt method is locally

quadratically convergent to x∗.

74

6.3 Derivative Calculations

In order to apply the trust region methods, we need to calculate the appropriate first

and second derivatives. As already mentioned, the eigenvalues of A + BKC may not

be differentiable everywhere. For example, the eigenvalues of

[4 0

0 0

]+

[k

l

] [1 1

](6.3.1)

are 2±√4 + 4k when l = −k and hence they are not differentiable at (k, l) = (−1, 1).

The next result, which follows from a result in [43, Section 2.5.7], shows that lack

of differentiability cannot occur at points at which the eigenvalues are distinct.

Theorem 6.3.1. Consider a matrix valued function A : RN → Rn×n. Suppose A(x)

is k-times continuously differentiable in x in an open neighborhood Ω. Furthermore

suppose that at each point in Ω, A(x) has distinct eigenvalues. Then the eigenvalues

of A(x) are k-times continuously differentiable in Ω.

Suppose the conditions of Theorem 6.3.1 are satisfied with k ≥ 2. Then we can

write down explicit expressions for the first and second derivatives of the eigenvalues.

(Part of the following results appear in [48]. The rest we have proved using similar

techniques to those appearing in that paper.) If λi denotes the ith eigenvalue of

A(x), suppose D = diag(λ1, . . . , λn) and let X ∈ Cn×n and Y ∈ Cn×n be such that

A(x)X = XD and Y ∗X = I. Then

∂λi

∂xk

=

(Y ∗∂A(x)

∂xk

X

)

ii

, (6.3.2)

and if we define

P = Y ∗∂A(x)

∂xk

X and Q = Y ∗∂A(x)

∂xl

X, (6.3.3)

75

then

∂2λi

∂xk∂xl

=

(Y ∗∂

2A(x)

∂xk∂xl

X

)

ii

+n∑

j=1j 6=i

PijQji + PjiQij

λi − λj

. (6.3.4)

These results can be used to calculate the derivatives of our objective function

f(K) = 12

∑ni=1(λi − λD

i )∗(λi − λDi ). Differentiating we have

∂f(K)

∂Kkl

= Re

n∑

i=1

(λi − λDi )∗

∂λi

∂Kkl

(6.3.5)

and

∂2f(K)

∂Kkl∂Kpq

= Re

n∑

i=1

(∂λi

∂Kkl

)∗ (∂λi

∂Kpq

)+

n∑i=1

(λi − λDi )∗

∂2λi

∂Kkl∂Kpq

. (6.3.6)

Note that ‘∂A(x)∂xk

’ is given by

∂(A + BKC)

∂Kkl

= BkCl, (6.3.7)

where Bk is the kth column of B and Cl is the lth row of C. Identity (6.3.7) implies

that the first term appearing in (6.3.4) is always zero. Combining (6.3.2)–(6.3.7) we

now have a complete characterization of the first and second derivatives of our cost

(at points where A + BKC has distinct eigenvalues).

Note that when applying the Levenberg-Marquardt method, the approximate sec-

ond derivatives are given by the first term in (6.3.6),

∂2f(K)

∂Kkl∂Kpq

≈ Re

n∑

i=1

(∂λi

∂Kkl

)∗ (∂λi

∂Kpq

). (6.3.8)

6.4 Additional Comments

When evaluating the cost f(K), the eigenvalues of A + BKC must be matched

with the desired eigenvalues in a least squares sense. Suppose a given problem has

76

distinct desired eigenvalues and that it is solvable. Then, sufficiently near a solution

of the problem, the eigenvalues of A + BKC will be distinct, which eigenvalues of

A + BKC match to which desired eigenvalues will not change, and the eigenvalues

of A + BKC will depend smoothly on K. As a result, for problems that are solvable

and have distinct desired eigenvalues, our objective function f will be smooth in

a neighborhood of solutions. An important consequence of this is that the results

from Section 6.2.2 regarding local convergence to solutions still apply. In particular,

near solutions of problems with distinct eigenvalues, both our algorithms will often

converge quadratically.

The comments above address the behavior of the algorithms in a neighborhood

of a solution. What about behavior far away from a solution? Are the algorithms

even defined in such regions? Considering the steps involved, all that is required for

the algorithms to be well defined is that, for each iterate, A + BKC have distinct

eigenvalues. If the desired eigenvalues are distinct and a generic initial condition is

used, it is unlikely that for either algorithm, that for any iterate, A + BKC has

repeated eigenvalues. Hence, under these mild assumptions, the algorithms should

be well defined and this is indeed what is observed in practice.

If the desired eigenvalues are not distinct, the cost may not be differentiable at

a solution. This indicates that the requirement of distinct desired eigenvalues is also

necessary. This does not limit the usefulness of the algorithms too much however as

desired eigenvalues can always be perturbed slightly so that they are distinct. While

having distinct but close eigenvalues does lead to a degree of ill-conditioning in our

algorithms, the algorithms can still be effectively utilized in such cases, as will be

shown in the numerical results section.

77

We also mention that, if the desired eigenvalues are distinct, it is our belief that,

modulo small changes, the global results of Section 6.2.2 should still hold, at least

generically.


This section contains computational results of applying the algorithms to various

problems. The algorithms were implemented in Matlab 6.5 and all results were ob-

tained using a 3.19 GHz Pentium 4 machine.

6.5.1 Random Problems

A 1000 random problems were created for each of a number of different choices for the

system dimensions (n,m, p). Each problem was created as follows. System matrices

A, B, and C were generated randomly; their entries were drawn from a normal

distribution of zero mean and variance 1. λD was taken to be the spectrum of a

randomly generated matrix and a scalar was added to λD to ensure maxi Re λDi =

−0.1. Each triple (n,m, p) was chosen to satisfy

mp > n. (6.5.1)

As the problems are randomly generated and satisfy condition (6.5.1), Wang’s suffi-

cient condition ensures each problem is solvable, [74].

An attempt was made to solve each problem using up to 5 different initial condi-

tions and a maximum of a 2000 iterations per initial condition. Initial conditions were

chosen randomly. The convergence condition used was∥∥λ(A + BKC)− λD

∥∥2

< ε

with ε = 10−3. Results for both algorithms are given in Table 6.1.

78

(n,m,p) (3,2,2) (6,4,3) (9,5,5)

Trust S.R. 100% 100% 91%Region T 0.10 4.2 125Newton i 31 247 1037

Levenberg- S.R. 100% 100% 99%Marquardt T 0.03 0.12 0.73

i 15 104 293

Table 6.1: A comparison of performance for different n, m and p. S.R. denotes the

success rate, T denotes the average convergence time in CPU seconds, and i the

average number of iterations. T and i are based only on those problems that were

successfully solved. ε = 10−3.

As can be seen, the results for both algorithms are quite good. Not surpris-

ingly, given the reduced computation required for its implementation, the Levenberg-

Marquardt based algorithm is faster than the trust region Newton based algorithm;

notice in particular the large difference in T for the (9, 5, 5) problem. What is perhaps

surprising is that the Levenberg-Marquardt based algorithm is more likely to find a

solution (at least within the number of iterations that were allowed). This suggests

that the Levenberg-Marquardt based algorithm is superior to the trust region New-

ton based algorithm. Note however that we have observed instances where, using the

same initial condition, the former algorithm converges to a local minima while the

latter algorithm converges to a solution.

The problems in Table 6.1 are ‘easy’ in the sense that they actually satisfy

Kimura’s condition (see Theorem A.0.3), m + p > n, and in most cases the num-

ber of variables mp is significantly larger than n. Results for some harder problems

are presented in Table 6.2. For these problems, mp−n = 1. An attempt was made to

solve each problem using up to 5 different initial conditions and a maximum of 5000

79

iterations per initial condition. Initial conditions were chosen randomly.

(n,m,p) (5,3,2) (7,2,4) (9, 2, 5)

Trust S.R. 100% 89% 65%Region T 1.9 21 100Newton i 539 2319 4747

Levenberg- S.R. 99% 96% 81%Marquardt T 0.29 2.0 6.4

i 308 1502 3163

Table 6.2: Some harder random problems. T and i are based only on those problems

that were successfully solved. ε = 10−3.

As can be seen, for these problems, success rates are less, even though we have

allowed a greater number of iterations per initial condition. Overall though the results

for these difficult problems are still quite good.

Figure 6.1 shows a typical plot of ‖λ(A+BKC)−λD‖2 versus i. It has the desired

property that near solution the convergence was quadratic.

6.5.2 Particular Problems

This subsection contains results for particular problems from the literature. In order

to present a greater number of results, rather than presenting the details of each

problem, only references are given. Results are presented only for the Levenberg-

Marquardt based algorithm.

For each problem, 100 random initial conditions were tested and the maximum

number of iterations per initial condition was set to 1000 (expect for Problem 6 for

which the maximum number of iterations per initial condition was set to 1500). The

termination parameter ε was reduced to ε = 10−6. Results are given in Table 6.3.

80

0 50 100 150 200 250

0.001

0.01

0

0.1

1

Iteration No. i

‖λ(A

+BKC

)−λD‖ 2

Figure 6.1: Quadratic convergence near solution of the Levenberg-Marquardt algo-

rithm.

As can be seen, performance was again very good. Solutions to each problem

could be found. Aside from Problem 6, for those initial conditions that lead to a

solution, average convergence times were less than 0.5 CPU seconds and solutions

could be found from many different initial conditions.

Problem 6 was the most sensitive to initial conditions. In fact, the results for

Problem 6 in the table are based on choosing the entries of initial K’s from a normal

distribution of zero mean and variance 100. (Choosing initial conditions for this

problem in the same manner as for all the other problem lead to a rather low success

rate of 5%.)

81

No. references (n, m, p) T i S.R. (%)1 [1, ex 1, case 1] (4, 2, 2) 0.19 103 842 [1, ex 1, case 2] (4, 2, 2) 0.16 77 923 [1, ex 2], [70] (5, 2, 4) 0.21 88 964 [64] (4, 3, 2) 0.07 26 1005 [49, ex 1] (4, 2, 2) 0.10 40 986 [49, ex 2] (6, 3, 2) 2.0 923 447 [49, ex 3] (5, 3, 2) 0.29 132 1008 [66, ex 1] (4, 3, 2) 0.06 21 1009 [66, ex 2] (3, 1, 2) 0.01 4.1 10010 [66, ex 3] (4, 2, 2) 0.03 8.9 10011 [46] (8, 4, 3) 0.46 151 9212 [25] (3, 2, 2) 0.05 18 100

Table 6.3: Particular problems. T and i are based only on those problems that were


6.5.3 Repeated Eigenvalues

Each of the problems considered in the prior subsection (as well as all the random

problems) had distinct desired eigenvalues. In this subsection we consider what can

be achieved if the desired eigenvalues are not distinct.

Consider the following problem from [14],

A =

−1 0.5 −0.2 0.85 0.45 0.9

0 −0.5 −2 0.9 0.4 0.1

0.15 0 −2 −0.2 0.1 0.8

0 0.1 −0.25 −0.8 0 0.2

−0.2 0.4 0 −0.5 −2 0.1

0.6 −0.7 0 0.2 0 −2.5

, B =

1 0

1 0

1 0

1 1

0.5 1

0 1

, C =

0 0 1

0 0 1

0 1 0

0 1 0

1 0 0

1 0 0

T

,

with desired eigenvalues λD = −3,−3,−2,−2,−1,−1T . Notice that this problem

has three pairs of repeated eigenvalues.

The algorithms do not provide a way to exactly solve this problem. However, a

82

fairly good approximate solution can be found by considering a slightly perturbed

desired spectrum with distinct eigenvalues. For example, suppose λD is replaced with

λDδ = −3 − δ,−3,−2 − δ,−2,−1 − δ,−1T with δ = 10−4. Then this perturbed

problem can often be solved.

An alternative strategy is to solve a series of perturbed problems with decreasing

δ’s. First solve a perturbed problem with δ = 10−1. Then, setting δ = 10−2 and using

the solution of the prior problem as an initial condition, solve this new perturbed

problem. Continue this process with δ = 10−3 and δ = 10−4.

Using the Levenberg-Marquardt based algorithm, the first strategy lead to a so-

lution for 55% of initial conditions tried, with average convergence time of 0.72 CPU

seconds. The second strategy was successful in 59% of cases, with average convergence

time of 0.35 CPU seconds.

We note that the main problem we encountered in solving these problems was not

convergence to local minima, though this can occur, but rather that near solutions

the Hessian of the cost can have very large eigenvalues. This leads to numerical

issues when trying to solve the constrained quadratic subproblems (6.2.1). The code

we have implemented for these subproblems works very well in the vast majority of

cases though we expect it could still be improved further and hence that even better

results may be achievable.

6.6 Summary

In this chapter two related numerical methods for the static output feedback pole

placement problem have been presented. Both algorithms are well behaved globally

and have the property that local convergence to solutions often occurs quadratically.

83

Extensive computational results presented indicate that the algorithms can be highly

effective in practice. While it is required that the desired poles are distinct, the

algorithms can still be successfully utilized for problems with repeated poles if small

perturbations to the desired poles are allowed.

Keywords: pole placement, static output feedback, trust region method, Newton’s

method, Levenberg-Marquardt method, eigenvalue derivatives.

Chapter 7

A Gauss-Newton Method for

Classical Pole Placement

7.1 Introduction

In this chapter we present a Gauss-Newton algorithm for solving static output feed-

back pole placement problems see Problem 6.1.1.

The problem is formulated as a constrained nonlinear least squares problem and

the minimization is achieved via a Gauss-Newton algorithm. This chapter contains

an overview of the algorithm and some of the numerical results. As part of the future

work, a convergence analysis and more numerical experiments will be carried out.

Consider Problem 6.1.1 again. Define

f(Q,G, K) = ‖Q(D + G)QT − (A + BKC)‖2. (7.1.1)

Here A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n are given system matrices. D is a block

diagonal matrix with some 1× 1 and 2× 2 blocks placed properly on the diagonal. D

84

85

is constructed in the following way to possess the desired eigenvalues λD ∈ Cn. The

1 × 1 blocks contain the real eigenvalues in λD. Each 2 × 2 block contains the real

parts of a pair of complex conjugate eigenvalues on the diagonal and the values of the

imaginary parts on the subdiagonal. Note that the eigenvalues are sorted to give the

minimal norm, which is a combinatorial least squares problem (this problem and its

solution methods are explained in Section 4.2.2).

A Gauss-Newton method is considered for solving the following constrained non-

linear least squares problem

Problem 7.1.1.

min f(Q,G, K)

s.t Q ∈ On;

G block super triangular;

K ∈ Rm×p.

For clarity on the form of G, look at the following example. Suppose λD =

−1,−2± i,−3, then

D =

−1 0 0 0

0 −2 1 0

0 −1 −2 0

0 0 0 −3

and G =

0 g1 g2 g3

0 0 0 g4

0 0 0 g5

0 0 0 0

. (7.1.2)

In Section 4.2.2, we recall a Schur’s result that any matrix Z ∈ Cn×n is unitarily

equivalent to an upper triangular matrix [40, Th 2.3.1]. Here we recall another Schur’s

result that for any matrix Z ∈ Rn×n, there is a real orthogonal matrix V ∈ On such

that Z = V TV T where T is a block upper triangular matrix [40, Th 2.3.4].

86

Theorem 7.1.2. Given Z ∈ Rn×n with eigenvalues λ1, . . . , λn in any prescribed order,

there is an orthogonal matrix V ∈ On and a block upper triangular matrix T ∈ Rn×n

such that

Z = V TV T (7.1.3)

and Tii = Re(λi), i = 1, . . . , n. Complex conjugate eigenvalues form 2×2 blocks along

the diagonal.

Proof. See, for example, [40, Th 2.3.4].

Note the problem formulation of Problem 7.1.1 is motivated from the Schur’s form

of real matrices in Theorem 7.1.2.

Gauss-Newton method is a variation of standard Newton’s method and is itera-

tive in nature. Gauss-Newton method has the advantage of just requiring the first

derivative of the cost function. In order to employ Gauss-Newton method to solve

Problem 7.1.1, we need to calculate the first derivative at the each iteration. As we

have three variables in f(Q,G, K) and these variables have dependent relationships,

we first use elimination technique to simply the cost function. Then we do the first

order Taylor approximation to derive the first derivatives.

This chapter is organized as follows 1. The main algorithm is presented in Section

7.2. Since the calculation is intensive and is hard to present, Section 7.2 contains an

overview of the algorithm. Section 7.3 presents some results of applying the algorithm

to some randomly generated problems. Further analysis and experiments will be done

in the future work.

1For more information on Gauss-Newton method, please refer to Section 9.2

87

7.2 The Gauss-Newton Method

This section presents the solution algorithm.

In cost function f(Q,G, K), we have three variables. Assume K and Q are given

and fixed, the calculation of G reduces to a standard least squares problem. The

solution of G is a formula in variables K and Q. We put the solution back into

(7.1.1) and get a new cost function in K and Q. Assume again K is given and fixed,

the calculation of Q in the new cost function reduces to a standard least squares

problem in Q. The solution of K is a formula in Q. We put this solution back into

the cost function. After these two steps, we obtain a new cost function in variable

Q only, from which we can do the first order Taylor approximation and hence apply

Gauss-Newton method. What follows describes this procedure in greater detail.

Throughout this section, we assume A,B and C are given and fixed. The capital

letters are used to represent matrices and small bold letters represent vectors.

Stage I : eliminate G

f(Q,G, K) = ‖Q(D + G)QT − (A + BKC)‖2

= ‖vec(G)− vec[QT (A + BKC)Q−D]‖22

= ‖Qgg− vec(V )‖22

= 0.

gopt = (QTg Qg)

−1QTg v. (7.2.1)

where V := QT (A + BKC)Q−D. So the new cost function is:

f(Q, K) = ‖ [ Qg(QTg Qg)

−1QTg − I ] vec ( QT (A + BKC)Q−D ) ‖2

2

88

Look at example (7.1.2) again. Note that g = [ g1 g2 g3 g4 g5 ]T and Qg has the

form

Qg =

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

T

,

and to a given problem, Qg is a constant matrix.

Stage II : eliminate K

Define Q := Qg(QTg Qg)

−1QTg − I. We have

f(Q, K) = ‖ Q vec ( QT (A + BKC)Q−D ) ‖22

= ‖ Q vec ( QT AQ−D ) + Q ( QT CT ⊗QT B )k ‖22

= 0.

so kopt = −(Y T Y )−1Y Tw. (7.2.2)

where Y := Q ( QT CT ⊗QT B ) and w := Q vec ( QT AQ−D ).

Define Q := I − Y (Y T Y )−1Y T , we have a new cost function

f(Q) = ‖Qw‖22

Note that Q is a constant matrix since Qg is fixed once the problem is given.

The cost function we are going to proceed with Gauss-Newton method is the

following:

f(Q) = ‖Qw‖22

= ‖I − Q ( QT CT ⊗QT B ) [ ( QC ⊗QBT ) QT Q ( QT CT ⊗QT B ) ]−1

( QC ⊗QBT )QT Q vec ( QT AQ−D ) ‖22

89

So far the cost function f(Q,G, K) has been reduced to a new cost function f(Q).

The first order Taylor approximation of f(Q) is based on the fact that the first order

Taylor approximation of an orthogonal matrix Q = Q0(I + Ω + O(Ω2)) where Ω is

a skew-symmetric matrix. The calculation of the Jacobian of f(Q) is quite lengthy

and hence we omit here.


Algorithm:

Problem Data. A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and λD ∈ Cn.

Initialization. Choose a randomly generated orthogonal matrix Y ∈ On , a randomly

generated block super triangular matrix G (the form is determined by λD) and

a randomly generated matrix K ∈ Rm×p.

repeat

1. Calculate gopt using (7.2.1).

2. Calculate kopt using (7.2.2).

3. Do first order Taylor approximation of f(Q).

4. Derive ∆Q to update Q.

until f(Q,G, K) < ε.


This section contains some numerical results of applying the algorithm to randomly

generated problems. The algorithm was implemented in Matlab 6.5 and all results


90

A 1000 random problems were created for each of a number of different choices

for the system dimensions (n, m, p). Each problem was created as follows. A,B and

C were randomly generated, i.e., whose entries are drawn from a normal distribution

of zero mean and variance 1. To ensure the problem is feasible, the desired eigen-

values λD are taken from a random matrix and a scalar was added to λD to ensure

maxi Re λDi = −0.1. Initial conditions were chosen randomly. Each triple (n,m, p)

was chosen to satisfy m+p > n. As the problems are randomly generated and satisfy

condition m + p > n, Kimura’s condition (see Theorem A.0.3) ensures each problem

is solvable.

(n,m,p) (6, 4, 3) (9, 5, 5)

Gauss- S.R. 98% 99%Newton T 0.57 0.92Method i 79 31

Table 7.1: A comparison of performance for different n, m and p. S.R. denotes the


average number of iterations. T and i are based only on those problems that were


As can be seen from Table 7.1, the results for the algorithm are very good. It

solves most of the randomly generated problems. Figure 7.1 shows a typical plot of

f(Q,G,K) versus i. It has the desired property that near solution, the convergence

was quadratic. As mentioned in the introduction, a more comprehensive analysis and

numerical experiments will be carried out in near future.

91

0 10 20 30 40 5010

−8

10−6

10−4

10−2

100

102

Iteration No. i

f(Q,G,K

)

Figure 7.1: Quadratic convergence near solution of the Gauss-Newton algorithm.

7.4 Summary

In this chapter, a Gauss-Newton algorithm is proposed for solving static output feed-

back pole placement problem. The problem is formulated as a constrained nonlinear

least squares problem and minimization is achieved via a Gauss-Newton method. As

this chapter gives an overview of the algorithm and contains some of the numerical

results, convergence analysis and numerical experiments will be done in near future,

which we believe will be performing well.

Keywords: pole placement, static output feedback, Gauss-Newton method.

Part IV

Problems Arising in Nonnegative

Matrices

92

93

In this part, numerical algorithms are presented for solving two inverse eigenvalue

problems, namely, the inverse eigenvalue problem for nonnegative matrices (NIEP)

or stochastic matrices (StIEP) and the inverse eigenvalue problem for symmetric

nonnegative matrices (SNIEP).

In Chapter 8, we present two related numerical methods, one for NIEP/StIEP and

another for SNIEP. The methods are iterative in nature and utilize alternating pro-

jection ideas. For the algorithm for the symmetric problem, the main computational

component of each iteration is an eigenvalue-eigenvector decomposition, while for the

other problem, it is a Schur matrix decomposition. Numerical results are presented

demonstrating that the algorithms are very effective in solving various problems in-

cluding high dimensional problems.

In Chapter 9, two related numerical algorithms are presented for NIEP/StIEP and

SNIEP. Both algorithms are iterative in nature. One is based on Newton’s method

and the other one is based on Gauss-Newton method. The main computational com-

ponents of each iteration are the first and second order derivatives of the eigenvalues.

Extensive numerical experiments show that the algorithms are very effective in prac-

tice though convergence to a solution is not guaranteed for either algorithm.

Chapter 8

A Projective Methodology for

Nonnegative Inverse Eigenvalue

Problems

8.1 Introduction

A real n× n matrix is said to be nonnegative if each of its entries is nonnegative.

The Nonnegative Inverse Eigenvalue Problem (NIEP) is the following: given a

list of n complex numbers λ = λ1, . . . , λn, find a nonnegative n × n matrix with

eigenvalues λ (if such a matrix exists).

A related problem is the Symmetric Nonnegative Inverse Eigenvalue Problem

(SNIEP): given a list of n real numbers λ = λ1, . . . , λn, find a symmetric non-

negative n× n matrix with eigenvalues λ (if such a matrix exists).

The NIEP and SNIEP are different problems even if λ is restricted to contain only

94

95

real entries; there exist lists of n real numbers λ for which the NIEP is solvable but

the SNIEP is not [42].

Finding necessary and sufficient conditions for a list λ to be realizable as the

eigenvalues of a nonnegative matrix has been a challenging area of research for over

fifty years and this problem is still unsolved; see the recent survey paper [28]. As

noted in [19, Section 6], while various necessary or sufficient conditions exist, the

necessary conditions are usually too general while the sufficient conditions are too

specific. Under a few special sufficient conditions, a nonnegative matrix with the

desired spectrum can be constructed; however, in general, proofs of sufficient con-

ditions are nonconstructive. Two sufficient conditions that are constructive and not

restricted to small n are, respectively, given in [65], for the SNIEP, and [68], for the

NIEP with real λ. (See also [58] for an extension of the results of the latter paper.)

A good overview of known results relating to necessary or sufficient conditions can be

found in the recent survey paper [28] and general background material on nonnegative

matrices, including inverse eigenvalue problems and applications, can be found in the

texts [3] and [54]. We also mention the recent paper [22], which can be used to help

determine whether a give list λ may be realizable as the eigenvalues of a nonnegative

matrix.

In this chapter we are interested in generally applicable numerical methods for

solving NIEPs and SNIEPs. To the best of our knowledge, the only algorithms that

have appeared up to now in the literature consist of [18] for the SNIEP and [21]

for the NIEP. In the case of [18], the following constrained optimization problem is

considered:

minQT Q=I,R=RT

1

2

∥∥QT ΛQ−R R∥∥2

. (8.1.1)

96

Here Λ is a constant diagonal matrix with the desired spectrum and stands for the

Hadamard product, i.e., componentwise product. Note that the symmetric matrices

with the desired spectrum are exactly the elements of QT ΛQ | Q ∈ On and that

the symmetric nonnegative matrices are exactly the elements of R R | R ∈ Sn.In [18], a gradient flow based on (8.1.1) is constructed. A solution to the SNIEP is

found if the gradient flow converges to a Q and an R that zero the objective function.

The approach taken in [21] for the NIEP is similar but is complicated by the fact that

the set of all matrices, both symmetric and nonsymmetric, with a particular desired

spectrum is not nicely parameterizable. In particular, these matrices can no longer

be parameterized by the orthogonal matrices.

In this chapter we present a numerical algorithm for the NIEP and another for

the SNIEP. In both cases, the problems are posed as problems of finding a point in

the intersection of two particular sets. Unlike the approaches in [18] and [21] which

are based on gradient flows, our algorithms are iterative in nature. For the SNIEP,

the solution methodology is based on an alternating projection scheme between the

two sets in question. The solution methodology for the NIEP is also based on an

alternating projection like scheme but is more involved, as we will shortly explain.

While alternating projections can often be a very effective means of finding a

point in the intersection of two or more convex sets, for both the SNIEP and NIEP

formulations, one set is nonconvex. Nonconvexity of one of the sets means that

alternating projections may not converge to a solution. This is in contrast to the case

where all sets are convex and convergence to a solution is guaranteed.

As mentioned above, for each problem, one set in the problem formulation is non-

convex. For the NIEP, this set is particularly complicated; it consists of all matrices

97

with the desired spectrum. At least some of the members of this set will be non-

symmetric matrices and it is this that causes complications. In particular, though

the set is closed and hence projections are well defined theoretically, how to calculate

projections onto such sets is an unsolved difficult problem. An alternate method for

mapping onto this set is used, which is motivated by the control counterpart in Part

II. Though the resulting points are not necessarily projected points, they are members

of the set and share a number of other desirable properties. As will be shown, this

alternate ‘projection’ is very effective in our context. Furthermore, we believe that

it may also be quite effective for other inverse eigenvalue problems involving non-

symmetric matrices. For more on other inverse eigenvalue problems, see the survey

papers [17] and [19], and the recent text [20].

Before concluding this introductory section we would like to point out how the

NIEP is related to another problem involving stochastic matrices. A n× n matrix is

said to be stochastic if it is nonnegative and the sum of the entries in each row equals

one. Another variation of the NIEP is the STochastic Inverse Eigenvalue Problem

(StIEP): given a list of n complex numbers λ = λ1, . . . , λn, find a stochastic n× n

matrix with eigenvalues λ (if such a matrix exists). It turns out that the NIEP and

the StIEP are almost exactly the same problem, as we now show. (See also [21].)

The vector of all 1’s is always an eigenvector for a stochastic matrix, implying

each stochastic matrix must have 1 as an eigenvalue. Also, the maximum row sum

matrix norm of a stochastic matrix equals 1 and hence the spectral radius cannot be

greater than 1, and as a result, must actually equal 1. Suppose λ satisfies the above

mentioned necessary conditions to be the spectrum of a stochastic matrix and that

a nonnegative matrix A with this spectrum can be found. Then if an eigenvector x

98

of A corresponding to the eigenvalue 1 can be chosen to have positive entries (by the

Perron-Frobenius theorem this is certainly possible if A is irreducible), then, if we

define D = diag(x), it is straightforward to verify that

D−1AD

is a stochastic matrix with the desired spectrum. (In fact it can be shown that

if λ satisfies the above mentioned necessary conditions, then it is the spectrum of a

stochastic matrix if and only if it is the spectrum of a nonnegative matrix [77, Lemma

5.3.2].)

This chapter is structured as follows 1. The SNIEP algorithm is presented first,

in Section 8.2, and then insights from this algorithm are used to address the more

difficult NIEP in Section 8.3. Numerical results for both algorithms are presented in

Section 8.4.

8.2 The Symmetric Problem

Our algorithm for solving the SNIEP consists of alternately projecting onto two par-

ticular sets. The details are given in this section.

Given a list of real eigenvalues λ = λ1, . . . , λn, renumbering if necessary, suppose

λ1 ≥ . . . ≥ λn. Let

Λ = diag(λ1, . . . , λn), (8.2.1)

and let M denote the set of all real symmetric matrices with eigenvalues λ,

M = A ∈ Sn |A = V ΛV T for some orthogonal V . (8.2.2)

1Please refer to Chapter 2 for background information on projections.

99

Let N denote the set of symmetric nonnegative matrices,

N = A ∈ Sn |Aij ≥ 0 for all i, j. (8.2.3)

The SNIEP can now be stated as the following particular case of Problem 2.3.1:

Find X ∈M∩N . (8.2.4)

Our solution approach is to alternatively project between M and N , and we next

show that it is indeed possible to calculate projections onto these sets.

Figure 8.1 illustrates the problem formulation and the alternating projections for

SNIEP. For visualization convenience, the figure is demonstrated in R3. However it

should be clear that the real problem can be any dimensional. The pink ball represents

the set M (actually nonconvex) and the box region represents N . The shaded pink

region is the intersection of the two sets. We know exactly how to do projections onto

both N and M and hence both projections are well established.

starting point

feasible set

Figure 8.1: Illustration of the problem formulation for SNIEP.

100

First, in order for the term ‘projection’ to make sense, we need to define an

appropriate Hilbert space and associated norm. From now on, Sn will be viewed as

a Hilbert space with inner product 〈A,B〉 = tr(AB) =∑

i,j aijbij. The associated

norm is the Frobenius norm ‖A‖ = 〈A, A〉 12 .

The projection of A ∈ Sn ontoM is given by Theorem 8.2.1 below. More precisely,

it gives a projection of A onto M. The reason for this is that the set M is nonconvex

and hence projections onto this set are not guaranteed to be unique.

M is nonconvex if its defining λ contains a pair of nonequal eigenvalues. For

example, if n = 2, consider

A =

[λ1 0

0 λ2

]and B =

[λ2 0

0 λ1

].

If λ1 6= λ2, then the convex combination (A+B)/2 does not have the same spectrum

as A and B.

Theorem 8.2.1 is based on the result of Hoffman and Wielandt, see Lemma 4.2.3

(see for example [40, Corollary 6.3.8]).

Theorem 8.2.1. Given A ∈ Sn, let A = V diag(µ1, . . . , µn)V T with V a real or-

thogonal matrix and µ1 ≥ . . . ≥ µn. If Λ is given by (8.2.1), then V ΛV T is a best

approximant in M to A in the Frobenius norm.

Proof. A matrix B ∈ M is a projection of A onto M if ‖A−B‖ ≤ ‖A−M‖ for all

M ∈ M. M ∈ M if and only if M = WΛW T for some real orthogonal matrix W .

Hence Lemma 4.2.3 implies

‖diag(µ1, . . . , µn)− Λ‖ ≤ ‖A−M‖ for all M ∈M. (8.2.5)

If B = V ΛV T then, using the fact that the Frobenius norm is orthogonally invariant,

101

it follows that

‖A−B‖ =∥∥V (diag(µ1, . . . , µn)− Λ)V T

∥∥

= ‖diag(µ1, . . . , µn)− Λ‖ .

(8.2.6)

The result follows by combining (8.2.5) and (8.2.6).

Projection onto N is straightforward and is given by Theorem 8.2.2 below.

Theorem 8.2.2. Given A ∈ Sn, define A+ ∈ Sn by

(A+)ij = max Aij, 0 for all 1 ≤ i, j ≤ n. (8.2.7)

A+ is the best approximant in N to A in the Frobenius norm.

Proof. The projection of x ∈ R onto the nonnegative real numbers equals max x, 0.The general result follows by noting that if B ∈ Sn, and in particular if B ∈ N , then

‖A−B‖ =

(∑i,j

|Aij −Bij|2) 1

2

and hence that the problem reduces to n2 decoupled scalar problems.

Our proposed algorithm for solving the SNIEP is the following.

SNIEP algorithm:

Problem Data. List of desired real eigenvalues λ = λ1, . . . , λn, λ1 ≥ . . . ≥ λn.

Initialization. Choose a randomly generated symmetric nonnegative matrix Y ∈ Sn+.

repeat

1. Calculate an eigenvalue-eigenvector decomposition of Y :

Y = V diag(µ1, . . . , µn)V T , µ1 ≥ · · · ≥ µn.

2. X := V diag(λ1, . . . , λn)V T .

102

3. X := (X + XT )/2.

4. Y := X+.


In the above algorithm, X+ is given by (8.2.7).

Note that at each iteration of the algorithm, X has the desired spectrum λ and Y

is nonnegative. If ε is small, say ε = 10−14, termination of the loop ensures X equals

Y (approximately) and hence that Y solves the SNIEP.

Due to small numerical inaccuracy, X from Step 2 of the algorithm may not be

perfectly symmetric. Step 3 makes it so.

Of course, while Corollary 2.3.5 ensures ‖X − Y ‖ is nonincreasing from one iter-

ation to the next, the set M is nonconvex and hence there is no guarantee that the

algorithm will terminate.

8.3 The General Problem

In this section we consider the NIEP. As we will see, the main difference between the

SNIEP and the NIEP is that the set of real, not necessarily symmetric, matrices with

a given (possibly complex) spectrum no longer has as nice a characterization as the

set of real symmetric matrices with a given spectrum.

Throughout this section, Cn×n will be regarded as a Hilbert space with inner

product 〈A,B〉 = tr(AB∗) =∑

i,j aij bij and the associated norm is the Frobenius

norm ‖A‖ = 〈A,A〉 12 .

The Schur’s result in Theorem 4.2.5 shows that a complex square matrix is uni-

tarily equivalent to an upper triangular matrix.

103

We now redefine some terms from the prior section.

Let λ = λ1, . . . , λn be a given list of complex eigenvalues. Define

T = T ∈ Cn×n |T is upper triangular with spectrum λ. (8.3.1)

Theorem 4.2.5 implies that the set of all complex matrices with spectrum λ is

given by the following set:

M = A ∈ Cn×n |A = UTU∗ for some unitary U and some T ∈ T . (8.3.2)

Let N denote the set of (not necessarily symmetric) nonnegative matrices,

N = A ∈ Rn×n |Aij ≥ 0 for all i, j. (8.3.3)

Having redefined M and N , the NIEP can now be stated as the following partic-

ular case of Problem 2.3.1:

Find X ∈M∩N . (8.3.4)

Figure 8.2 illustrates the problem formulation and the alternating projections for

NIEP. The pink region represents the set M which is a rather complicated nonconvex

set and the box region represents the set N . Again we illustrate in R3 for visual-

ization convenience. The shaded pink region is the intersection of the two sets. The

projections onto N is well established. While the calculation of the projection onto

M denoted by the pink dot with dotted arrow is an unsolved problem, a substitute

mapping is applied denoted by the black dot.

We would like to use alternating projections to solve the NIEP, alternatively pro-

jecting between M and N . However, to the best of our knowledge, the way to

calculate projections onto M is an unsolved problem. Suppose instead we could find

104

starting point

feasible set

Figure 8.2: Illustration of the problem formulation for NIEP.

a mapping that was in some sense a reasonable substitute for a projection map for

M. Using this substitute mapping and the projection map for N in an alternating

projection like scheme may still produce a viable algorithm. The following function

PM is used as a substitute for a true projection map onto M.

Definition 8.3.1. Suppose U ∈ Cn×n is unitary and T ∈ Cn×n is upper triangular.

Let λi, i = 1, . . . , n, be a permutation of the list of eigenvalues λ such that, among all

possible permutations, it minimizes

n∑i=1

|λi − Tii|2. (8.3.5)

Define

PM(U, T ) = UTU∗, (8.3.6)

where T ∈ T is given by

Tij =

λi, if i = j,

Tij, otherwise.

105

Note that PM maps into the set M.

A given matrix A ∈ Cn×n may have a nonunique Schur decomposition and A =

U1T1U∗1 = U2T2U

∗2 does not imply PM(U1, T1) = PM(U2, T2). The nonuniqueness of

Schur’s decomposition is demonstrated by the following example [40]:

T1 =

1 1 4

0 2 2

0 0 3

and T2 =

2 −1 3√

2

0 1√

2

0 0 3

are unitarily equivalent via

U =1√2

1 1 0

1 −1 0

0 0√

2

as UT1U∗ = T2. If λ = 0, 0, 0, it is readily verified that PM(U, T1) 6= PM(I, T2).

Hence PM is decomposition dependent.

It turns out that the fact that PM may give different points for different Schur

decompositions of the same matrix is not particularly important. Because different

Schur decompositions lead to points in M of equal distance from the original matrix.

The proof of this fact is similar to the proof for Theorem 4.2.7.

The next theorem shows that given A = UTU∗, if we restrict attention to matrices

of the form UTU∗, T ∈ T , then PM(U, T ) is a point in M closest to A.

Theorem 8.3.2. Suppose A = UTU∗ ∈ Cn×n with U a unitary matrix and T upper

triangular. Then PM(U, T ) satisfies

‖PM(U, T )− A‖ ≤ ‖UTU∗ − A‖ for all T ∈ T .

Proof. This proof is similar to the proof for Theorem 4.2.8 and it omitted here.

106

For completeness, we note that, given A = UTU∗, PM(U, T ) may not satisfy

‖PM(U, T )− A‖ ≤ ‖M − A‖ for all M ∈M.

For example, if

U =1

5

[−3 4

4 3

]and T =

[1 −3

0 2

],

and

U =1

5

[−4 3

3 4

]and T =

[0 −3

0 0

],

then, if λ = 0, 0, one can readily verify that

‖PM(U, T )− UTU∗‖ ‖U T U∗ − UTU∗‖.

As for the symmetric case, projection onto N is straightforward.

Theorem 8.3.3. Given A ∈ Cn×n, define A+ ∈ Rn×n by

(A+)ij = maxRe(Aij), 0 for all 1 ≤ i, j ≤ n. (8.3.7)

A+ is the best approximant in N to A in the Frobenius norm.

Proof. The projection of z ∈ C onto the nonnegative real numbers is given by

maxRe(z), 0. The remainder of the proof follows by exactly the same reasoning

used in the proof of Theorem 8.2.2.

Our proposed algorithm for solving the NIEP is the following.

NIEP algorithm:

Problem Data. List of desired complex eigenvalues λ = λ1, . . . , λn.

Initialization. Choose a randomly generated nonnegative matrix Y ∈ Rn×n.

107

repeat

1. Calculate a Schur decomposition of Y : Y = UTU∗.

2. X := PM(U, T ).

3. Y := X+.


In the above algorithm, PM(U, T ) is given by Definition 8.3.1 and X+ is given by

(8.3.7).

As for the SNIEP algorithm, at each iteration of the NIEP algorithm, X has the

desired spectrum λ and Y is nonnegative. If ε is small, say ε = 10−14, termination of

the loop ensures X equals Y (approximately) and hence that Y solves the NIEP.

Remark 8.3.1. If each of the members of λ are real and we seek a symmetric non-

negative matrix with spectrum λ, then the NIEP algorithm reduces to the SNIEP

algorithm. More precisely, this is true if the members of λ are real, if the initial

condition Y is a symmetric nonnegative matrix, and, for Schur decompositions used

in the NIEP algorithm, U is restricted to be real. This can be shown by comparing

the steps in each algorithm.

Indeed, suppose the current Y is symmetric and nonnegative. For any Schur

decomposition of Y , T must be real diagonal matrix. As we restrict the U matrix

to be real, such a decompositioin is nothing but a standard eigenvalue-eigenvector

decomposition for a symmetric matrix (though the eigenvalues are not necessarily

ordered along the diagonal of T ).

As both the elements of λ and the diagonal entries of T are real, the permutation

that minimizes (8.3.5) can be easily characterized. Indeed, in this case (8.3.5) is

minimized if and only if

108

n∑i=1

λiTii (8.3.8)

is maximized. From Lemma 4.2.3, (8.3.8) is maximized if the λi’s are ordered in the

same way as the Tii’s. This implies that if Y is symmetric, the step of producing a

X from Y is the same in both algorithms.

Lastly, projection of a symmetric matrix onto (8.3.3) gives the same matrix as

projection onto (8.2.3) and hence this step in both algorithms is also the same. This

established our claim.

We close this section by noting that unlike the SNIEP algorithm, for the NIEP

algorithm there is no guarantee that ‖X − Y ‖ is nonincreasing from one iteration to

the next.


This section contains some numerical results for both the SNIEP and NIEP algo-

rithms.

All computational results were obtained using a 3 GHz Pentium 4 machine. The

algorithms were coded using Matlab 7.0.

Throughout this section, when we say a matrix is ‘randomly generated’ we mean

each entry of that matrix is randomly drawn from the uniform distribution on the

interval [0, 1]. When dealing with the SNIEP algorithm, all randomly generated

matrices are chosen symmetric.

For both algorithms, the initial starting Y is always randomly generated. For

both algorithms, the convergence tolerance ε is set to 10−14.

109

8.4.1 SNIEP

This subsection starts with some results for randomly generated SNIEPs. To ensure

each problem is feasible, each desired spectrum is taken from a randomly generated

matrix.

Results for various problem sizes n are given in Table 8.1. For each value of n,

1000 problems were considered. The table contains the average number of iterations

required to find a solution, the average time required to find a solution, and the

success rate. As can be seen, the algorithm performed extremely well and was able

to solve every problem. In all cases, both the average number of iterations and the

average solution time was very small.

n i T % solved

5 19 0.0016 10010 18 0.0030 10020 17 0.0075 100100 12 0.15 100

Table 8.1: SNIEP: A comparison of performance for different problem sizes n. i

denotes the average number of iterations and T denotes the average convergence time

in CPU seconds.

Remark 8.4.1. It is interesting to note that T increases with n, as would be expected,

while i decreases. A reason for this could be the following. For any choice of desired

eigenvalues, M is a smooth manifold. In addition, if the eigenvalues defining M are

distinct, as they will be if they were taken from a randomly generated matrix, then

the dimension of M is n(n−1)2

; see [39, Chapter 2]. The dimension of Sn is n(n+1)2

.

110

Hence,

dim Mdim Sn

=n− 1

n + 1,

which is an increasing function of n. For larger n, M is ‘thicker’ relative to the

ambient space and hence, intuitively, the corresponding SNIEP is easier to solve.

0 2 4 6 8 10 12 1410

−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Iteration No. i

‖Xi−X‖

Figure 8.3: Linear convergence of the SNIEP algorithm.

Suppose X1, X2, . . . , is a sequence of X’s produced by the SNIEP algorithm

and that these points converge to a solution X. Figure 8.3 shows a typical plot

of∥∥Xi − X

∥∥ versus i. Convergence is clearly linear. This is to be expected: Suppose

X is a point on the boundary of N and that the (Xi)+’s lie in a particular face of N .

As M is a manifold, near X it looks locally like an affine subspace of Sn. As the face

of N also looks locally like an affine subspace, we could expect local linear conver-

gence as alternating projections between two intersecting affine subspaces converge

linearly [27].

111

Randomly generated problems have properties that are not shared by all SNIEPs.

For example, as already mentioned, randomly generated problems have distinct eigen-

values. We next consider a problem with repeated eigenvalues, namely λ = 3− t, 1+

t,−1,−1,−1,−1 for 0 < t < 1. The t = 1/2 version of the problem is also consid-

ered in [18], where a numerical solution is sought via the gradient flow approach of

that paper. An analytic solution to this problem is given in [65].

Notice that for any value of t the desired eigenvalues sum to zero and hence

there exist arbitrarily small perturbations of the spectrum which lead to an infeasible

SNIEP. In particular this problem cannot have any solutions in the interior of N .

We have tried the SNIEP algorithm on a number of other problems with repeated

eigenvalues with excellent results. This is the hardest problem we have encountered

so far.

The results of applying the algorithm to the problem for various values of t are

given in Table 8.2. They are based on running the algorithm a 100 times for each

value of t.

t i T % solved

0.25 480 0.061 1000.5 470 0.061 970.75 340 0.050 650.95 310 0.046 59

Table 8.2: SNIEP: A problem with repeated eigenvalues, λ = 3 − t, 1 +

t,−1,−1,−1,−1. i and T do not include the attempts that had not converged

after 5000 iterations.

First, the results indicate that the SNIEP algorithm is not always successful in

112

finding a solution. However, they also show that the algorithm can still be quite

successful if a number of initial conditions are tried. It is interesting to note that the

algorithm becomes more sensitive to the choice of the initial condition the larger t is.

Notice that as t → 1, the eigenvalues 3− t and 1+ t both converge to the same value,

and the dimension of the manifold M (which depends solely on the multiplicities of

the eigenvalues) goes from 9 when 0 < t < 1 to 8 when t = 1 [39, Chapter 2].

Aside: Regarding initial conditions, as noted before, both the SNIEP and NIEP

algorithms use a nonnegative initial starting point. This is important and, in fact,

the performance of neither algorithm is as good if non-nonnegative initial condition

are used.

Here is a solution that was found to the t = 12

problem:

0 0 0√

32

0 1

0 0 1 12

1 0

0 1 0 12

1 0√32

12

12

0 12

√32

0 1 1 12

0 0

1 0 0√

32

0 0

.

This solution is different to both the solution in [18] and the solution in [65]. A

number of other solutions were also found.

Here is a X+ corresponding to an infeasible X (again for the t = 12

problem):

12

0 0 0 0√

32

0 0 78

78

78

0

0 78

0 78

78

0

0 78

78

0 78

0

0 78

78

78

0 0√32

0 0 0 0 0

.

113

The eigenvalues of this matrix are λ = 258, 11

2,−7

8,−7

8,−7

8,−1.

8.4.2 NIEP

This subsection starts with some results for randomly generated NIEPs. Again, to

ensure each problem is feasible, each desired spectrum is taken from a randomly

generated matrix. Results are given in Table 8.3.

n i T % solved

5 26 0.011 99.710 44 0.045 99.820 48 0.12 99.8100 200 12 96.6

Table 8.3: NIEP: A comparison of performance for different problem sizes n. i and

T do not include the problems that had not converged after 5000 iterations.

As can be seen, the results are again very good with almost all problems solved.

The results indicate that NIEPs are harder to solve than SNIEPs. Also, the

number of iterations, time, and time per iteration are greater. Part of the reason for

an increase in time per iteration will be the extra computation required to calculate

the least squares matching component of each PM(U, T ) calculation; see (8.3.5). (For

SNIEPs, the corresponding step is easy: the eigenvalues are real and just need to be

sorted in decreasing order.)

For the NIEPs, both i and T increased with n.

The final problem we consider is taken from [21]. It is to find a stochastic matrix

with (presumably randomly generated) spectrum λ = 1.0000,−0.2608, 0.5046, 0.6438,

114

−0.4483. Furthermore the problem requires the matrix to have zeros in certain po-

sitions. In the context of Markov chains, we require the states to form a ring and

that each state be linked to at most two immediate neighbors. The zero pattern is

given by the zeros of the following matrix:

Z =

1 1 0 0 1

1 1 1 0 0

0 1 1 1 0

0 0 1 1 1

1 0 0 1 1

. (8.4.1)

Our algorithm as it stands is not able to solve this problem though it is able to do so

if a simple modification is made. Using Z from (8.4.1), define

N = A ∈ Rn×n | Aij ≥ 0 and Aij = 0 if Zij = 0.

N is still a convex set. In the NIEP algorithm, replacing projection onto N by

projection onto N gives solutions (nonnegative matrices) with zeros in the desired

places. Using the transformation discussed in the introduction of the paper, solutions

found by the algorithm can be converted into stochastic matrices with the same

spectrum. Note that this transformation preserves zeros.

Using this methodology readily produced many solutions. An example is

X =

0.6931 0.2887 0 0 0.0182

0.1849 0.2422 0.5729 0 0

0 0.5476 0.3622 0.0902 0

0 0 0.5437 0.1233 0.3330

0.3712 0 0 0.6103 0.0185

.

115

Another solution is

X =

0.8634 0.0431 0 0 0.0936

0.6224 0 0.3776 0 0

0 0.4935 0.1564 0.3501 0

0 0 0.1107 0.0115 0.8778

0.3452 0 0 0.2467 0.4080

.

Notice that this latter solution has an extra zero. While this X still solves the

problem, by further modifying N it is possible to ensure zeros appear only in the

places specified by (8.4.1) and nowhere else.

For example, using

N = A ∈ Rn×n | Aij = 0 if Zij = 0 and Aij ≥ δ otherwise,

with δ > 0 a small constant, does the trick. Note that the stochastic matrix trans-

formation leaves positive entries positive.

8.5 Summary

In the chapter we presented two related numerical methods, one for the NIEP, which

can also be used to solve the inverse eigenvalue problem for stochastic matrices, and

another for the SNIEP. While this chapter deals with two specific types of inverse

eigenvalue problems, the ideas presented should also be applicable to many other

inverse eigenvalue problems, including those involving nonsymmetric matrices.

Keywords: inverse eigenvalue problem, nonnegative matrices, stochastic matrices,

alternating projections, Schur’s decomposition.

Chapter 9

Newton Type Methods for

Nonnegative Inverse Eigenvalue

Problems

9.1 Introduction

In this chapter, we present two related Newton methods for solving SNIEP and

NIEP/StIEP. Details of the considered problems can be found in Section 8.1.

For NIEP, given a list of complex eigenvalues λD ∈ Cn, two approaches are con-

sidered for solving the following unconstrained nonlinear least squares problem

minA∈Rn×n

f(A) :=1

2‖λ(A A)− λD‖2

2. (9.1.1)

Here AA denotes the Hadamard product of A, i.e. componentwise product. Note

that any nonnegative matrix is exactly an element of A A | A ∈ Rn×n. λ(A A)

denotes the vector of eigenvalues of A A, with entries sorted to give the minimum

116

117

norm.

Along the same line for SNIEP, given a list of real eigenvalues λD ∈ Rn, two

approaches are considered for solving the following problem

minA∈Sn

f(A) :=1

2‖λ(A A)− λD‖2

2. (9.1.2)

Note that any symmetric nonnegative matrix is exactly an element of AA | A ∈ Sn.

Newton type methods, which are well known in the optimization community, are a

type of iterative methods. The specific Newton type methods we use are the standard

Newton’s method and the Gauss-Newton method. In order to employ the Newton

type methods, at each iteration, the first and second derivatives of the eigenvalues

of A A must be calculated. The Gauss-Newton method has the advantage of only

requiring the first derivatives of the eigenvalues. Both algorithms presented have the

desired property that, near solutions, they often converge quadratically.

The eigenvalues of A A may not be differentiable everywhere. However they

will be differentiable at all points at which A A has distinct eigenvalues. Hence the

algorithms can still be used to solve problems whose desired eigenvalues are distinct

but whose separation is quite small and hence this does not appear to be a serious

limitation.

This chapter is structured as follows. Section 9.2 contains a brief overview of

Newton type methods, as Newton’s method is well known and is a standard and

classical method, only basic formulas are given. In order to use these methods to

solve the SNIEP and NIEP/StIEP, the first and second derivatives of (9.1.1) and

(9.1.2) are required. Details of these calculations are given in Section 9.3. Section 9.4

contains computational results.

118

9.2 Newton Type Methods

This section gives an overview of Newton type methods.

Newton’s method plays a central role in the development of numerical techniques

for optimization. It arises naturally from the Taylor approximation to a function.

Newton’s method is an very important tool in solving many optimization problems.

Most of the existing practical methods for optimization (e.g. Gauss-Newton method

and Quasi-Newton method) are variations of Newton’s method [55]. Additional in-

formation on Newton type methods can be found in [55] and [56].

It is assumed that the function f : RN → R to be minimized is (sufficiently)

smooth. Let f be twice continuously differentiable. Given a current iterate xk and

an increment step pk, the quadratic approximation is assumed to be of the form

f(xk + p) = f(xk) +∇f(xk)T p +

1

2pT Bkp.

Here Bk is typically either the Hessian of f at xk or some approximation of this Hes-

sian. If Bk is the Hessian, then f(xk+p) is simply the 2nd order Taylor approximation

of f at xk. In the variations of standard Newton’s method, other choices of Bk may

be employed.

Newton iterate takes the form

xk+1 = xk −H−1k ∇f(xk), k ≥ 0. (9.2.1)

Note in Newton iterate, the choice of Bk is the Hessian of f at xk. When the

objective function f is a least squares cost, say f(x) = 12

∑Mi=1 r2

i (x) for some functions

ri : Rn → R. If we define r(x) = (r1(x), . . . , rM(x))T and let J(x) denote the Jacobian

119

of r(x), then

∇f(x) = J(x)T r(x) and ∇2f(x) = J(x)T J(x) +M∑i=1

ri(x)∇2ri(x),

and a good choice for Bk is J(xk)T (xk). For this choice of Bk’s, the method is called

the Gauss-Newton method.

Note that a rather surprising result of the proposed algorithms is that usually

the convergence properties of both Newton’s method and Gauss-Newton method can

only be analyzed from a local point of view. That is, we have to start sufficiently

close to a solution for the algorithm to converge and only in this case local quadratic

convergence can be expected. This does not appear to be the case to our algorithms.

The computational results show that we can choose random starting points for the

algorithms to converge. Typically the algorithms converge quadratically near solu-

tions.

9.3 Derivative Calculations

In order to apply the Newton type methods from the prior section, we need to calculate

the appropriate first and second derivatives. Theorem 6.3.1 shows that if A A in

(9.1.1) and (9.1.2) have distinct eigenvalues, the eigenvalues of AA are differentiable.

Suppose the conditions in Theorem 6.3.1 are satisfied with k ≥ 2. Then we can

write down explicit expressions for the first and second derivatives of the eigenvalues.

The calculation of the first derivative of eigenvalues is given in (6.3.2) and the second

derivative in (6.3.4).

These results can be used to calculate the derivatives of our objective function

120

f(A) = 12

∑ni=1(λi − λD

i )∗(λi − λDi ). Differentiating we have

∂f(A)

∂Akl

= Re

n∑

i=1

(λi − λDi )∗

∂λi

∂Akl

(9.3.1)

and

∂2f(A)

∂Akl∂Apq

= Re

n∑

i=1

(∂λi

∂Akl

)∗ (∂λi

∂Apq

)+

n∑i=1

(λi − λDi )∗

∂2λi

∂Akl∂Apq

. (9.3.2)

Note that ‘∂A(x)∂xk

’ is given by

∂(A A)

∂Akl

= 2AkleTk el, (9.3.3)

where ek is a zero vector with only a 1 on the k-th position. Identity (9.3.3) implies

that the first term appearing in (6.3.4) is always zero. Combining (6.3.2), (6.3.4),

(9.3.1), (9.3.2) and (9.3.3) we now have a complete characterization of the first and

second derivatives of our cost (at points where A A has distinct eigenvalues).

Note that when applying the Gauss-Newton method, the approximate second

derivatives are given by the first term in (9.3.2),

∂2f(A)

∂Akl∂Apq

≈ Re

n∑

i=1

(∂λi

∂Akl

)∗ (∂λi

∂Apq

). (9.3.4)

Remark 9.3.1. The calculations of the first and second derivatives for SNIEP and

NIEP/StIEP are roughly the same. The only difference would be that the first deriva-

tive needs less calculation for SNIEP as (9.3.1) for SNIEP is symmetric.


This section contains some numerical results for both SNIEP and NIEP problems.

All computational results were obtained using a 3.19 GHz Pentium 4 machine.

The algorithms were coded using Matlab 6.5.

121

n 5 10 20

Newton’s S.R. 100% 100% 100%method T 0.45 13 595

i 5.9 5.7 5.9

Gauss- S.R. 100% 100% 100%Newton T 0.03 0.16 4.8method i 5.9 5.7 5.8

Table 9.1: SNIEP: A comparison of performance for different n. S.R. denotes the


average number of iterations.

Throughout this section, when we say a matrix is ‘randomly generated’ we mean

each entry of that matrix is randomly drawn from the uniform distribution on the

interval [0, 1]. When dealing with the SNIEP algorithm, all randomly generated

matrices are chosen symmetric.

9.4.1 SNIEP

This subsection presents some results for SNIEPs.

First we present some results for randomly generated problems. To ensure each

problem was feasible, each desired spectrum was taken from a randomly generated

matrix. Results for various problem sizes n are given in Table 9.1. For each value of n,

1000 problems were created (with exception to n = 20 of Newton’s method where 100

problems were created). Initial conditions were chosen randomly. The convergence

condition used was ‖λ(A A)− λD‖2 < ε with ε = 10−14.

As can be seen, the algorithms perform extremely well and they solve every prob-

lem. As expected, the average solution time for the Gauss-Newton method is much

122

1 2 3 4 5 6

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Iteration No. i

||λ(AA

)−λD|| 2

Figure 9.1: Quadratic convergence near solution of the SNIEP algorithm.

less than that for the Newton’s method. The average numbers of iterations are

roughly the same for both algorithms. (The algorithms were able to solve problems

up to n = 48. Due to the computational constraint of the computer, we did not get

results for problems larger than this size. As far as we tested, the success rate for

n = 48 problems was also 100%. However the computation was extensive and each

iteration consumed a lot of time.)

Figure 9.1 shows a typical plot of ‖λ(A A) − λD‖2 versus i. As expected near

solution, the convergence was quadratic.

Randomly generated problems have properties that are not shared by all SNIEPs,

for example, they have distinct eigenvalues. We next show results for a problem with

repeated eigenvalues, namely λD = 3− t, 1 + t,−1,−1,−1,−1 for 0 < t < 1. Refer

to Section 8.4 for details on the literature of this particular problem and the existing

solution methods.

Table 9.2 gives the results of applying the Gauss-Newton algorithm to the problem

123

t i T % solved

0.25 67 0.35 96%0.5 123 0.64 91%0.75 141 0.76 83%0.95 206 0.99 89%

Table 9.2: SNIEP: A particular problem with repeated eigenvalues. i and T are only

based on the problems that had converged within 1000 iterations. ε = 10−3.

for various values of t. They are based on running the algorithm for 100 times for

each value of t.

Though the algorithm did not successfully find a solution in very trial, the per-

formance was again very good. As mentioned in the introduction, the eigenvalues of

AA may not be differentiable where it has repeated eigenvalues. As we can see from

Table 9.2, it is not a serious limitation on applying our algorithm to problems with

repeated eigenvalue.

9.4.2 NIEP

This subsection presents some results for NIEPs.

First we present the results for some randomly generated problems. Again to

ensure each problem is feasible, each desired spectrum is taken from a randomly

generated matrix.

Results for various problem sizes n are given in Table 9.3. For each value of n,

1000 problems were created (with exception to n = 20 of Newton’s method where 100

problems were created.). The table contains the average number of iterations required

to find a solution, the average time required to find a solution and the success rate.

As can be seen, the algorithms perform very good in solving random problems.

124

n 5 10 20

Newton’s S.R. 98.7% 98.0% 91%method T 0.78 32 1913

i 8.0 9.3 12

Gauss- S.R. 99.1% 98.3% 98.2%Newton T 0.04 0.62 45method i 8.1 9.6 13

Table 9.3: NIEP: A comparison of performance for different n. T and i are based only

on those problems that were successfully solved within 1000 iterations. ε = 10−14.

Figure 9.2 shows a typical plot of ‖λ(A A) − λD‖2 versus i. As expected near

solution the convergence was quadratic.

As introduced in Section 8.1, under rather mild assumption on the desired spec-

trum, a nonnegative matrix is readily transformed to a stochastic matrix. The final

problem we consider is from the counterpart in Section 8.4. It is to find a stochastic

matrix with spectrum λ = 1.0000,−0.2608, 0.5046, 0.6438,−0.4483. Furthermore

the problem requires the matrix to have zeros in certain positions as the pattern given

in (8.4.1). Using either Newton’s method or Gauss-Newton method produced many

solutions. A special solution is

X =

0 0.9667 0 0 0.0333

0.1866 0 0.8133 0 0

0 0.0828 0.8581 0.059 0

0 0 0.0004 0.5811 0.4185

0.6083 0 0 0.3916 0

.

Notice that this solution has three extra zeros, which shows the flexibility of the

proposed algorithms.

125

1 2 3 4 5 6 7 8 9 10 11 1210

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

102

Iteration No. i

‖λ(AA

)−λD‖ 2

Figure 9.2: Quadratic convergence near solution of the NIEP algorithm.

9.5 Summary

In the chapter two Newton algorithms are presented for solving NIEP/StIEP and

SNIEP problems. As demonstrated by experiments, they perform very good. While

they deal with two specific types of inverse eigenvalue problems, the ideas are ap-

plicable to many other inverse eigenvalue problems subject to different structural

constraint, especially those involving nonsymmetric matrices.

Keywords: inverse eigenvalue problem, nonnegative matrices, Newton’s method,

Gauss-Newton method.

Part V


126

Chapter 10


10.1 Conclusion

The main goal of this research is to develop optimization algorithms to tackle some

interesting, important and challenging inverse eigenvalue problems arising in con-

trol and nonnegative matrices. A common feature shared by all the problems being

solved is that they have simple problem expressions and/or formulations, but they

are all very hard to be solved. Hence novel, efficient, reliable and general applicable

algorithms are of great interest.

On the control part, we concentrate on static output feedback related problems.

The control tasks range from the most basic stabilization to much more sophisticated

hybrid pole placement and simultaneous stabilization problem etc. The fact that the

desired pole placement regions need not to be convex or even connected allows much

flexibility in choosing pole placement regions to meet various control design tasks.

The three methodologies have their own features.

The projective methodology is generally applicable to various control problems.

127

128

The problem formulation is indeed a new perspective and is one of the main contribu-

tions. The computation of ‘projection’ onto nonconvex set is an open hard problem.

We propose reasonable substitute to make the alternating projection scheme working.

Such ideas should be widely applicable to control and noncontrol related problems,

especially those involving nonsymmetric matrices. The algorithms are iterative in

nature and the computation of each iteration is very fast, which makes it possible

to solve high dimensional problems. As there is no guarantee the algorithms will

always converge, the freedom to choose multiple starting points greatly increase the

likelihood that solutions could be found.

The trust region methods are proposed for solving classical pole placement prob-

lem and will be extended to pole placement problems in generality. The problem

formulation is naturally arisen from the control goals and has a clear geometric inter-

pretation. As the algorithms are iterative in nature, the computation of each iteration

is comparatively expensive. However in most cases near solutions, the convergence is

quadratic, which ends up with a low overall computational cost.

The Gauss-Newton method is built on a novel problem formulation and sophisti-

cated computations. The problem formulation is motivated by the real Schur’s form

of the closed loop system matrices. The algorithm is generally applicable to various

problems, including those with repeated eigenvalues. Though its computation is ex-

tensive and choice of starting point is important, the algorithm performs very well in

experiments.

On nonnegative matrices part, we tackle nonnegative/symmetric nonnegative/stochastic

inverse eigenvalue problems respectively. To the best of our knowledge, there are only

129

few algorithms exiting in literature dealing with these problems. As the wide applica-

tion of nonnegative matrices, these problems have broad potential applications. The

features of the methodologies are as follows.

The projective methodology is built on a simply problem formulation with clear

geometric meaning. The projective algorithms are very fast and can be applied to

high dimensional problems. The way of handling nonconvexity in the problem setting

should be applicable to related problems, especially those involving nonsymmetric

matrices.

The Newton type methods are proposed for constrained nonlinear optimization

problem formulations. They utilize the first and second derivatives of eigenvalues and

are iterative in nature. The algorithms have the desired property that near solutions

the convergence is typically quadratic.

A lot of experiments were carried out for all the algorithms proposed and the

results were compared with other existing methods. As observed from the computa-

tional results and some preliminary future work results, we believe the algorithms in

this thesis are new, interesting and promising.

10.2 Future Work

Inverse eigenvalue problem is a big family and encompasses a large scope of problems.

No matter the theoretical analysis on necessary and/or sufficient conditions under

which a problem is solvable or the practical issue on developing efficient and reliable

algorithms, a lot more work need to be done. In this section, we mention a few future

work directions. We believe these problems should be solvable based on either the

methodologies proposed in this thesis or some variations.

130

1. Chapter 5 presents an overview of a projective methodology and some of the

numerical results of static output feedback simultaneous stabilization and decen-

tralized control problems. In the same framework more related control problems

could be solved, for example, reduced order output feedback control, regional

pole placement, H2/H∞ control etc. As simultaneous stabilization via static

output feedback is NP-hard, it is interesting to see the performance of the pro-

posed algorithm in more examples.

2. In Chapter 6, trust region methods are introduced to solve classical pole place-

ment problems. With little modification, these methods can be used to solve all

other problems in generalized pole placement framework. As the performance

for classical pole placement is extremely good, we expect the performance for

other problems especially those less standard problems will be as good.

3. Chapter 7 introduces an interesting problem formulation for classical pole place-

ment via static output feedback and proposes a Gauss-Newton method for solv-

ing it. A detailed analysis of convergence properties and related issues is needed.

In addition, as we believe this method is able to handle problems with repeated

eigenvalues, it does not have any formal constraints. However this guess needs

to be proved in numerical experiments.

4. COMPleib is a problem database containing many milestone static output feed-

back control problems. As we tested the projective methodology for solving

static output feedback stabilization problems (results are shown in Appendix

B), it is interesting to see the performance using trust region methods and

Gauss-Newton method, especially for those problems where mp < n.

131

5. As mentioned previously, the projective methodology, Newton type methods,

and trust region methods can all be potentially applied to other inverse eigen-

value problems subject to different structural and spectral constraints. For

example [20, Problem 4.15], given L = (it, jt)lt=1 of double subscripts and a

set of values a1, . . . , al over a field F, and a set of values λD = λ1, . . . , λn,find a matrix X ∈ Fn×n such that

λ(X) = λD

Xit,jt = at, for t = 1, . . . l.

This is the most general setting for inverse eigenvalue problems with prescribed

entries. The choice of the field F can be real or complex. The methodologies

employed in this thesis are readily applied to solve such problems.

Appendix A

Results for Classical Pole

Placement

This appendix lists some of the main theoretical pole placement results that have

appeared in the literature to date. References to the literature have already been

given in the introduction of Chapter 4.

When referring to ‘arbitrary pole placement’ it is with the understanding that

complex poles are placed in complex conjugate pairs.

Theorem A.0.1. Arbitrary pole placement implies the system is controllable and

observable, and that mp ≥ n.

The necessity of controllability and observability can be seen by considering the

controllability and observability canonical forms. The necessity of the dimension

constraint follows from fact that if one wants to arbitrarily assign n values, at least

as many degrees of freedom are required of the input variable.

For following result was shown by Wang in [74].

132

133

Theorem A.0.2. Consider fixed system dimension n, m and p, and suppose mp > n.

Arbitrary pole placement is possible for generic systems with these dimension.

The sufficient condition just mentioned is for generic systems. For particular

systems we have the following sufficient condition by Kimura [45].

Theorem A.0.3. Consider a controllable and observable system satisfying m+p > n.

Given any choice of desired poles λ1, . . . , λn (with complex poles in complex conjugate

pairs) and arbitrary ε > 0, there exists K such that |λi(A + BKC) − λi| < ε, i =

1, . . . , n.

Appendix B

More Computational Results for

Stabilization

COMPLeib is a control problem database containing many milestone control prob-

lems.

We ran the projective algorithm presented in Chapter 4 on the static output

feedback stabilization problems given in the COMPLeib, which were not open loop

stable. Details of the problems can be found in [50].

For each problem, 100 randomly initial conditions are tested and the maximum

number of iterations per initial condition was set to 1000. Once a stabilizing solution

was found, the algorithm was terminated.

134

135

Problem (n,m, p) S.R. i T ρ(A + BKC)AC1 (5, 3, 3) 97% 15 0.005 -0.47AC2 (5, 3, 3) 100% 12 0.003 -0.52AC4 (4, 1, 2) 100% 18 0.003 -0.05AC5 (4, 2, 2) 100% 517 0.11 -0.0013AC7 (9, 1, 2) 55% 148 0.065 -6.9e-4AC8 (9, 1, 5) 43% 929 0.32 -0.10AC9 (10, 4, 5) 98% 129 0.088 -0.099AC10 (55, 2, 2) - - - -AC11 (5, 2, 4) 3% 199 0.00048 -5.1e-4AC12 (4, 3, 4) 100% 2.7 0.0011 -0.72AC13 (28, 3, 4) - - - -AC14 (40, 3, 4) - - - -AC18 (10, 2, 2) 100% 612 0.31 -2.9e-4HE1 (4, 2, 1) - - - -HE3 (8, 4, 6) 100% 5.6 0.0034 -0.20HE4 (8, 4, 6) 100% 7.1 0.0055 -0.16HE5 (8, 4, 2) 3% 527 0.24 -6.8e-5HE6 (20, 4, 6) 100% 2.2 0.005 -0.005HE7 (20, 4, 6) 100% 1.9 0.0032 -0.005JE2 (21, 3, 3) 54% 131 0.25 -1.28JE3 (24, 3, 6) 58% 155 1.3 -1.1581REA1 (4, 2, 3) 51% 62 0.018 -0.54REA2 (4, 2, 2) 71% 23 0.0057 -0.071REA3 (12, 1, 3) 100% 22 0.018 -0.02REA4 (8, 1, 1) - - - -DIS2 (3, 2, 2) 100% 2.6 0.0005 -0.94DIS4 (6, 4, 6) 100% 2.7 0.016 -0.72DIS5 (4, 2, 2) 100% 523 0.13 -0.002WEC1 (10, 3, 4) 100% 508 0.26 -0.0036BDT2 (82, 4, 4) 100% 13 0.43 -0.075IH (21, 11, 10) 100% 10 0.095 -0.5PAS (5, 1, 3) - - - -TF1 (7, 2, 4) 100% 80 0.033 -0.08TF2 (7, 2, 3) 100% 4.8 0.002 -1e-5TF3 (7, 2, 3) - - - -NN1 (3, 1, 2) - - - -NN2 (2, 1, 1) 100% 1.5 0 -0.98NN3 (4, 1, 1) - - - -NN5 (7, 1, 2) - - - -

136

Problem (n,m, p) S.R. i T ρ(A + BKC)NN6 (9, 1, 4) - - - -NN7 (9, 1, 4) - - - -NN9 (5, 3, 2) - - - -NN10 (8, 3, 3) - - - -NN12 (6, 2, 2) - - - -NN13 (6, 2, 2) 100% 15 0.0045 -0.086NN14 (6, 2, 2) 100% 15 0.005 -0.092NN15 (3, 2, 2) 100% 2.4 0.0006 -0.31NN16 (8, 4, 4) 100% 2.2 0.0018 -0.30NN17 (3, 2, 1) 1% 1 0 -0.55

Table B.1: S.R. denotes the success rate, T denotes the average convergence time

in CPU seconds, and i the average number of iterations. T and i are based only on

those problems that were successfully solved. ‘−′ indicates a stabilizing solution was

not found.

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