Finite Zeros in Discrete Time Control Systems

✍ Scribed by Jerzy Tokarzewski

• Publisher: Springer
• Year: 2006
• Tongue: English
• Leaves: 332
• Series: Lecture Notes in Control and Information Sciences
• Edition: 1
• Category: Library

✦ Synopsis

This book presents a state space approach to the analysis of zeros of MIMO LTI discrete-time systems, using the Moore-Penrose pseudoinverse and singular value decomposition of the first nonzero Markov parameter of a system. The book begins with definition of invariant zeros and goes as far as a general characterization of output-zeroing inputs and the corresponding solutions, explicit formulas for maximal output-nulling invariant subspaces and for the zero dynamics.

✦ Table of Contents

Contents......Page 7
1 Introduction......Page 10
1.1 Smith Zeros and Decoupling Zeros......Page 11
1.2 Scope of the Book......Page 14
1.3 Glossary of Symbols......Page 16
2.1 Definitions of Zeros......Page 18
2.2 Decoupling Zeros......Page 22
2.3 Invariant and Transmission Zeros......Page 27
2.4 The Output-Zeroing Problem......Page 30
2.5 Invariant Zeros and Output-Zeroing Inputs......Page 34
2.6 Relationships Between Smith and Invariant Zeros......Page 45
2.7 Algebraic Criteria of Degeneracy......Page 54
2.8 Exercises......Page 62
3.1 Preliminary Characterization of Invariant Zeros......Page 74
3.1.1 Invariant Zeros in Proper Systems......Page 75
3.1.2 Invariant Zeros in Strictly Proper Systems......Page 76
3.2 Output-Zeroing Inputs......Page 80
3.2.1 Output-Zeroing Inputs for Proper Systems......Page 82
3.2.2 Output-Zeroing Inputs for Strictly Proper Systems......Page 89
3.3 Exercises......Page 100
4.1 System Zeros in Strictly Proper Systems......Page 115
4. 1. 1 Decoupl i ng and Transmi ssi on Zeros......Page 117
4.1.2 First Markov Parameter of Full Rank......Page 120
4.2 System Zeros in Proper Systems......Page 128
4.2.1 Decoupling and Transmission Zeros......Page 129
4.2.2 Proper Systems with Matrix D of Full Rank......Page 130
4.3 Systems with Vector Relative Degree......Page 136
4.4 Exercises......Page 145
5 Singular Value Decomposition of the First
Markov Parameter......Page 157
5.1 Invariant and Smith Zeros in Strictly Proper Systems......Page 159
5.2 A Procedure for Computing Zeros of Strictly Proper Systems......Page 173
5.3 Invariant and Smith Zeros in Proper Systems......Page 181
5.4 A Procedure for Computing Zeros of Proper Systems......Page 194
5.5 Exercises......Page 200
6 Output-Nulling Subspaces in
Strictly Proper Systems......Page 205
6.1 Systems with the Identically Zero Transfer-Function Matrix......Page 206
6.2 First Markov Parameter of Full Column Rank......Page 212
6.3 Invariant Subspaces......Page 216
6.4 SVD and the Output-Zeroing Problem......Page 230
6.5 Zeros, Output-Nulling Subspaces and Zero Dynamics......Page 239
6.6 Exercises......Page 245
7.1 Matrix D of Full Column Rank......Page 253
7.2 Invariant Subspaces......Page 257
7.3 SVD and the Output-Zeroing Problem......Page 268
7.4 Zeros, Output-Nulling Subspaces and Zero Dynamics......Page 275
7.5 Exercises......Page 277
8.1 Definitions of Zeros......Page 279
8.3 Sufficient and Necessary Condition for Degeneracy......Page 282
8.4 Zeros and SVD of Matrix E......Page 283
8.4.1 Invariant Zeros......Page 284
8.4.2 Output Decoupling Zeros......Page 287
8.4.3 Input Decoupling Zeros......Page 288
8.5 Markov Parameters and the Weierstrass Canonical Form......Page 289
8.6 Invariant Zeros and the First Markov Parameter......Page 291
8.6.1 First Markov Parameter of Full Column Rank......Page 295
8.6.2 SVD of the First Markov Parameter......Page 298
8.7 Exercises......Page 303
Epilogue......Page 305
A.1 Reachability and Observability......Page 307
A.2 Canonical Decomposition of the State Space......Page 310
B.1 The Moore-Penrose Inverse of a Matrix......Page 313
B.2 Singular Value Decomposition of a Matrix......Page 316
B.3 Endomorphisms of a Linear Space over C......Page 317
C.1 Polynomial and Rational Matrices......Page 320
References......Page 324