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Chapter 1 Introduction1

1-1 The Study of Systems1

1-2 The Scope of the Book2

Chapter 2 Linear Spaces and Linear Operators6

2-1 Introduction6

2-2 Linear Spaces over a Field7

2-3 Linear Independence, Bases, and Representations12

Change of Basis17

2-4 Linear Operators and Their Representations19

Matrix Representations of a Linear Operator21

2-5 Systems of Linear Algebraic Equations26

2-6 Eigenvectors, Generalized Eigenvectors, and Jordan-Form Representations of a Linear Operator33

Derivation of a Jordan-Form Representation38

2-7 Functions of a Square Matrix45

Polynomials of a Square Matrix45

Functions of a Square Matrix51

Functions of a Matrix Defined by Means of Power Series54

2-8 Norms and Inner Product57

2-9 Concluding Remarks60

Problems62

Chapter 3 Mathematical Descriptlons of Systems70

3-1 Introduction70

3-2 The Input-Output Description72

Linearity73

Causality76

Relaxedness77

Time Invariance80

Transfer-Function Matrix81

3-3 The State-Variable Description83

The Concept of State83

Dynamical Equations86

Linearity87

Time Invariance89

Transfer-Function Matrix90

Analog and Digital Computer Simulations of Linear Dyna-mical Equations91

3-4 Examples94

Dynamical Equations for RLC Networks101

3-5 Comparisons of the Input-Output Description and the State-Variable Description106

3-6 Mathematical Descriptions of Composite Systems108

Time-Varying Case108

Time-Invariant Case111

Well-Posedness Problem114

3-7 Discrete-Time Systems121

3-8 Concluding Remarks124

Problems125

Chapter 4 Linear Dynamical Equations and Impulse-Response Matrices133

4-1 Introduction133

4-2 Solutions of a Dynamical Equation134

Time-Varying Case134

Solutions of x = A（t）x134

Solutions of the Dynamical Equation E139

Time-Invariant Case141

4-3 Equivalent Dynamical Equations146

Time-Invariant Case146

Time-Varying Case151

Linear Time-Varying Dynamical Equation withPeriodic A（·）153

4-4 Impulse-Response Matrices and Dynamical Equations154

Time-Varying Case154

Time-Invariant Case157

4-5 Concluding Remarks161

Problems162

Chapter 5 Controllability and Observability of Linear Dynamical Equatlons168

5-1 Introduction168

5-2 Linear Independence of Time Functions170

5-3 Controllability of Linear Dynamical Equations175

Time-Varying Case175

Differential Controllability, Instantaneous Controllabil-ity, and Uniform Controllability180

Time-Invariant Case183

Controllability Indices187

5-4 Observability of Linear Dynamical Equations192

Time-Varying Case192

Differential Observability, Instantaneous Observabil-ity, and Uniform Observability196

Linear Time-Invariant Dynamical Equations197

Observability Indices198

5-5 Canonical Decomposition of a Linear Time-Invariant Dyna-mical Equation199

Irreducible Dynamical Equations206

5-6 Controllability and Observability of Jordan-Form Dynamical Equations209

5-7 Output Controllability and Output Function Controllability214

5-8 Computational Problems217

5-9 Concluding Remarks226

Problems227

Chapter 6 Irreducible Realizations, Strict System Equivalence, and Identification232

6-1 Introduction232

6-2 The Characteristic Polynomial and the Degree of a Proper Rational Matrix234

6-3 Irreducible Realizations of Proper Rational Functions237

Irreducible Realization of β3/D（s）237

Irreducible Realizations of g（s） = N（s）/D（s）240

Observable Canonical-Form Realization240

Controllable Canonical-Form Realization243

Realization from the Hankel Matrix245

Jordan-Canonical-Form Realization249

Realization of Linear Time-Varying Differential Equations252

6-4 Realizations of Vector Proper Rational Transfer Functions253

Realization from the Hankel Matrix257

6-5 Irreducible Realizations of Proper Rational Matrices: Hankel Methods265

Method Ⅰ.Singular Value Decomposition268

Method Ⅱ.Row Searching Method272

6-6 Irreducible Realizations of （s）: Coprime Fraction Method276

Controllable-Form Realization276

Realization of N（s）D-1（s）, Where D（s） and N（s） Are NotRight Coprime282

Column Degrees and Controllability Indices284

Observable-Form Realization285

6-7 Polynomial Matrix Description287

6-8 Strict System Equivalence292

6-9 Identification of Discrete-Time Systems from Noise-Free Data300

Persistently Exciting Input Sequences307

Nonzero Initial Conditions309

6-10 Concluding Remarks313

Problems317

Chapter 7 State Feedback and State Estimators324

7-1 Introduction324

7-2 Canonical-Form Dynamical Equations325

Single-Variable Case325

Multivariable Case325

7-3 State Feedback334

Single-Variable Case334

Stabilization339

Effect on the Numerator of g（s）339

Asymptotic Tracking Problem—Nonzero SetPoint340

Multivariable Case341

Method Ⅰ341

Method Ⅱ345

Method Ⅲ347

Nonuniqueness of Feedback Gain Matrix348

Assignment of Eigenvalues and Eigenvectors351

Effect on the Numerator Matrix of G（s）352

Computational Problems353

7-4 State Estimators354

Full-Dimensional State Estimator355

Method Ⅰ357

Method Ⅱ358

Reduced-Dimensional State Estimator361

Method Ⅰ361

Method Ⅱ363

7-5 Connection of State Feedback and State Estimator365

Functional Estimators369

7-6 Decoupling by State Feedback371

7-7 Concluding Remarks377

Problems378

Chapter 8 Stablllty of Llnear Systems384

8-1 Introduction384

8-2 Stability Criteria in Terms of the Input-Output Description385

Tine-Varying Case385

Time-Invariant Case388

8-3 Routh-Hurwitz Criterion395

8-4 Stability of Linear Dynamical Equations400

Time-Varying Case400

Time-Invariant Case407

8-5 Lyapunov Theorem412

A Proof of the Routh-Hurwitz Criterion417

8-6 Discrete-Time Systems419

8-7 Concluding Remarks425

Problems425

Chapter 9 Llnear Tlme-Invarlant Composlte Systems: Characterlza-tlon, Stablllty, and Deslgns432

9-1 Introduction432

9-2 Complete Characterization of Single-Variable Composite Systems434

9-3 Controllability and Observability of Composite Systems439

Parallel Connection440

Tandem Connection441

Feedback Connection444

9-4 Stability of Feedback Systems448

Single-Variable Feedback System449

Multivariable Feedback System451

9-5 Design of Compensators: Unity Feedback Systems458

Single-Variable Case458

Single-Input or Single-Output Case464

Multivariable Case—Arbitrary Pole Assignment468

Multivariable Case—Arbitrary Denominator-Matrix Assignment478

Decoupling486

9-6 Asymptotic Tracking and Disturbance Rejection488

Single-Variable Case488

Multivariable Case495

Static Decoupling—Robust and NonrobustDesigns501

State-Variable Approach504

9-7 Design of Compensators: Input-Output FeedbackSytems506

Single-Variable Case506

Multivariable Case511

Implementations of Open-Loop Compensators517

Implementation Ⅰ517

Implementation Ⅱ519

Applications523

Decoupling523

Asymptotic Tracking, Disturbance Rejection, andDecoupling526

9-8 Concluding Remarks534

Problems536

Appendix A Elementary Transformations542

A-1 Gaussian Elimination543

A-2 Householder Transformation544

A-3 Row Searching Algorithm546

A-4 Hessenberg Form551

Problems553

Appendix B Analytic Functions of a Real Variable554

Appendix C Minimum Energy Control556

Appendix D Controllability after the Introduction of Sampling559

Problems564

Appendix E Hermitian Forms and Singular Value Decomposition565

Problems570

Appendix F On the Matrix Equation AM + MB = N572

Problems576

Appendix G Polynomials and Polynomial Matrices577

G-1 Coprimeness of Polynomials577

G-2 Reduction of Reducible Rational Functions584

G-3 Polynomial Matrices587

G-4 Coprimeness of Polynomial Matrices592

G-5 Column- and Row-Reduced Polynomial Matrices599

G-6 Coprime Fractions of Proper Rational Matrices605

Problems618

Appendix H Poles and Zeros623

Problems635

References636

Index657