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Multidimensional Systems Theory

and Applications

Second Edition

by

N.K. Bose

HRB SystemsProfessor and University Endowed Fetter Fellow

Department

of Electrical Engineering

The Pennsylvania State University UniversityPark Pennsylvania

U.S.A.

With contribution by

N.K. Bose

B. Buchberger

and

J.P.

Guiver

KLUWER ACADEMIC PUBLISHERS

DORDRECHT / BOSTON / LONDON

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Table of ontents

List of Acronyms xi

List of Notations xii

Preface xiii

Acknowledgments xvii

Introduction xix

1.

TRENDS IN MU LTIDIMEN SIONAL SYSTEMS THEORY 1

1 Introduction 1

2 M ultidimensional Systems Stability 2

2.1 M ultidimensiona l Digital Filters 3

2.2 M ultivariate Netw orks 9

2.3 Delay-Differential System s 12

2.4 Stiff Differential Systems 13

2.5 M ultipass Processes 15

3 Multivariate Realization Theory 15

3.1 State-space Realization Theo ry 16

3.2 Transform Dom ain Realization Theory 18

3.2.1 Integral Representation of Positive Real Functions 20

3.2.2 Sum of Squares Representation of a Form in Netw ork Synthesis 22

4 TC-D Problem of M oments and

Its

Ap plications

in

Multidimensional Systems

Theory 23

5 Role of Irreducible Polynom ials in M ultidimen sional Systems Theory 26

6 Hilbert Transform and Spec tral Fac torization 27

7 Conclusions 28

8 Updates 30

V ll

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viii

MULTIDIMENSIONAL SYSTEMS THEORY AND APPLICATIONS

2.

CAUSAL

ND

WEAKLY CAU SAL 2-D FILTERS WITH APPLICATIONS

IN STABILIZATION 35

1 Scalar 2-D Inpu t / Output Systems 35

2 Stability 38

3 Structural Stability 43

4 M ulti-Input M ulti-Outpu t Systems 43

5 Stabilization of Scalar System s 44

6 Characterization of Stabilizers for Scalar Systems 47

7 Stabilization of Strictly Causal Transfer M atrices 53

7.1 MTMO Feedb ack System s and The ir Stabilization 55

8 Characterization of Stabilizers for

MOVIO

Systems 59

9 Stabilization of Weakly Causal Systems 63

10 Stabilization of MEMO Weakly Causal Systems 73

11 Conclusions 77

12 Updates 77

3. THE EQUATION A x = b OVER THE RING

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Tableof Contents

ix

5. MULTIVARIATE POLY NOM IALS, MA TRICES, AN D MATRIX-FRACTION

DESCRIPTIONS 129

1 Introduction 129

2 Relative Primen ess and GCD Extraction from M ultivariate Polyn om ials 131

2.1 Tests for Relative Prime ness 131

2.2 Primitive factorization algorithms for GC D extraction 133

2.2.1 M ultivariate GC D extraction from the Sylvester matrix 133

2.3 Bezout M atrix Based M ethod 135

2.3.1 M ultivariate Polyn om ial GC D Extraction from Be zout M atrix 136

3 Polyn om ial M atrix Primitive Factorization in the Bivariate Case 138

3.1 Primitive Factorization of Bivariate Polynom ial M atrices 140

3.2 General Factorization of Bivariate Polyn om ial M atrices 147

4 M ultivariate Polynom ial M atrix Factorization 151

4.1 Various Types of M ultivariate Polyn om ial M atrix Factorizations 151

4.2 M odules 151

4.3 Various Deg rees of Prime ness 155

4.4 i2-M odu les and Vector Spaces 159

5 Computations for Cop rimeness Using Grobner Bases 161

5.1 Algo rithm for Comp uting the Right Inverse of ZL C M atrices 162

5.2 M inor coprim eness and GrObner Bases 163

5.3 Problems in Factor Coprimeness and GCLD GC RD ) Extraction 168

6 Generalization of the Serre Conjecture and its Consequences 170

6.1 Normal Full Rank M atrix Case 170

6.1.1 Factorization Problem in No rmal Full Rank Case 174

6.2 De generate Ran k Case 174

7 Factorization as a Product of Elem entary M atrix Factors 176

8 App lications in M ultidimensional System s Stabilization 178

9 Behavioral Approach 184

10 Conclusions 187

6. RECENT IMPACTS OF MU LTIDIMENSIONAL SYSTEM S RESEA RCH 191

1 Introduction 191

2 Inference of Stabilty of Sets of M ultidimension al Systems From Subsets

of Low Cardinality 192

2.1 Un ivariate Com plex Hurwitz Polyn om ial Case 192

2.2 M ultivariate Scattering Hu rwitz Polyn om ial Case 193

2.3 Robust W avenumber Respon se 198

2.4 Stability Of Interval M atrix 199

3 M ultiple Deconvolution Operators for Rob ust Superresolution 200

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MU LTIDIMENSIONAL SYSTEMS THEORY AND APPLICATIONS

3.1 Recursive Least Squares RLS) Algorithm for Superresolution 202

3.2 Backprojection Algorithm for Superresolution 206

4 M ultisensor Array-Based Superresolution 207

4.1 M ultisensor High Resolution Ima ge Reconstruction M odel 209

4.2 Regularization M ethods 211

5 Wavelets for Superresolution 213

6 Other Recent Applications 216

7 Conclusions 218

7. MULTIVARIATE RATIONAL APPRO XIMAN TS OF TH E PADE-TY PE 221

1 Introduction and M otivation 22 1

2 M ultivariate Pad6-Type Approx imants Scalar Case) 222

3 Pade-Type Matrix Appro ximants 226

4 Conclusions 227

5 Updates 228

8 OLD PROBLEM SOLUTIONS AND SO ME N E W PROBLEMS 231

1 Progress Towards Solution of Open Problems since 1985 231

1.1 Nonessential Singularities of the Second Kind NSSK ) 231

1.2 Rational Spectral Factoriza tion 232

1.3 Feedback Stabilization 233

1.4 Robust Continuous System Stability 235

1.5 Stability of Disc rete Systems Un der Coefficient Perturb ation 236

1.6 Rob ust Stability of M ultivariable System s 237

2 New Open Problems 238

2.1 Greatest Com mon Left Right) Divisor Com putation 238

2.2 Determinantal Left Right Joint) Factorization 239

2.3 Unim odular Com pletion Over Ring of Proper Stable M ultivariate

Rational Functions 240

2.4 M inimax Con troller and GrObner Bases 242

References 243

Index 265