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Chapter 1 Introduction 1

1-1 The Study of Systems 1

1-2 The Scope of the Book 2

Chapter 2 Linear Spaces and Linear Operators 6

2-1 Introduction 6

2-2 Linear Spaces over a Field 7

2-3 Linear Independence, Bases, and Representations 12

Change of Basis 17

2-4 Linear Operators and Their Representations 19

Matrix Representations of a Linear Operator 21

2-5 Systems of Linear Algebraic Equations 26

2-6 Eigenvectors, Generalized Eigenvectors, and Jordan-Form Representations of a Linear Operator 33

Derivation of a Jordan-Form Representation 38

2-7 Functions of a Square Matrix 45

Polynomials of a Square Matrix 45

Functions of a Square Matrix 51

Functions of a Matrix Defined by Means of Power Series 54

2-8 Norms and Inner Product 57

2-9 Concluding Remarks 60

Problems 62

Chapter 3 Mathematical Descriptlons of Systems 70

3-1 Introduction 70

3-2 The Input-Output Description 72

Linearity 73

Causality 76

Relaxedness 77

Time Invariance 80

Transfer-Function Matrix 81

3-3 The State-Variable Description 83

The Concept of State 83

Dynamical Equations 86

Linearity 87

Time Invariance 89

Transfer-Function Matrix 90

Analog and Digital Computer Simulations of Linear Dyna-mical Equations 91

3-4 Examples 94

Dynamical Equations for RLC Networks 101

3-5 Comparisons of the Input-Output Description and the State-Variable Description 106

3-6 Mathematical Descriptions of Composite Systems 108

Time-Varying Case 108

Time-Invariant Case 111

Well-Posedness Problem 114

3-7 Discrete-Time Systems 121

3-8 Concluding Remarks 124

Problems 125

Chapter 4 Linear Dynamical Equations and Impulse-Response Matrices 133

4-1 Introduction 133

4-2 Solutions of a Dynamical Equation 134

Time-Varying Case 134

Solutions of x = A（t）x 134

Solutions of the Dynamical Equation E 139

Time-Invariant Case 141

4-3 Equivalent Dynamical Equations 146

Time-Invariant Case 146

Time-Varying Case 151

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

4-4 Impulse-Response Matrices and Dynamical Equations 154

Time-Varying Case 154

Time-Invariant Case 157

4-5 Concluding Remarks 161

Problems 162

Chapter 5 Controllability and Observability of Linear Dynamical Equatlons 168

5-1 Introduction 168

5-2 Linear Independence of Time Functions 170

5-3 Controllability of Linear Dynamical Equations 175

Time-Varying Case 175

Differential Controllability, Instantaneous Controllabil-ity, and Uniform Controllability 180

Time-Invariant Case 183

Controllability Indices 187

5-4 Observability of Linear Dynamical Equations 192

Time-Varying Case 192

Differential Observability, Instantaneous Observabil-ity, and Uniform Observability 196

Linear Time-Invariant Dynamical Equations 197

Observability Indices 198

5-5 Canonical Decomposition of a Linear Time-Invariant Dyna-mical Equation 199

Irreducible Dynamical Equations 206

5-6 Controllability and Observability of Jordan-Form Dynamical Equations 209

5-7 Output Controllability and Output Function Controllability 214

5-8 Computational Problems 217

5-9 Concluding Remarks 226

Problems 227

Chapter 6 Irreducible Realizations, Strict System Equivalence, and Identification 232

6-1 Introduction 232

6-2 The Characteristic Polynomial and the Degree of a Proper Rational Matrix 234

6-3 Irreducible Realizations of Proper Rational Functions 237

Irreducible Realization of β3/D（s） 237

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

Observable Canonical-Form Realization 240

Controllable Canonical-Form Realization 243

Realization from the Hankel Matrix 245

Jordan-Canonical-Form Realization 249

Realization of Linear Time-Varying Differential Equations 252

6-4 Realizations of Vector Proper Rational Transfer Functions 253

Realization from the Hankel Matrix 257

6-5 Irreducible Realizations of Proper Rational Matrices: Hankel Methods 265

Method Ⅰ.Singular Value Decomposition 268

Method Ⅱ.Row Searching Method 272

6-6 Irreducible Realizations of （s）: Coprime Fraction Method 276

Controllable-Form Realization 276

Realization of N（s）D-1（s）, Where D（s） and N（s） Are NotRight Coprime 282

Column Degrees and Controllability Indices 284

Observable-Form Realization 285

6-7 Polynomial Matrix Description 287

6-8 Strict System Equivalence 292

6-9 Identification of Discrete-Time Systems from Noise-Free Data 300

Persistently Exciting Input Sequences 307

Nonzero Initial Conditions 309

6-10 Concluding Remarks 313

Problems 317

Chapter 7 State Feedback and State Estimators 324

7-1 Introduction 324

7-2 Canonical-Form Dynamical Equations 325

Single-Variable Case 325

Multivariable Case 325

7-3 State Feedback 334

Single-Variable Case 334

Stabilization 339

Effect on the Numerator of g（s） 339

Asymptotic Tracking Problem—Nonzero SetPoint 340

Multivariable Case 341

Method Ⅰ 341

Method Ⅱ 345

Method Ⅲ 347

Nonuniqueness of Feedback Gain Matrix 348

Assignment of Eigenvalues and Eigenvectors 351

Effect on the Numerator Matrix of G（s） 352

Computational Problems 353

7-4 State Estimators 354

Full-Dimensional State Estimator 355

Method Ⅰ 357

Method Ⅱ 358

Reduced-Dimensional State Estimator 361

Method Ⅰ 361

Method Ⅱ 363

7-5 Connection of State Feedback and State Estimator 365

Functional Estimators 369

7-6 Decoupling by State Feedback 371

7-7 Concluding Remarks 377

Problems 378

Chapter 8 Stablllty of Llnear Systems 384

8-1 Introduction 384

8-2 Stability Criteria in Terms of the Input-Output Description 385

Tine-Varying Case 385

Time-Invariant Case 388

8-3 Routh-Hurwitz Criterion 395

8-4 Stability of Linear Dynamical Equations 400

Time-Varying Case 400

Time-Invariant Case 407

8-5 Lyapunov Theorem 412

A Proof of the Routh-Hurwitz Criterion 417

8-6 Discrete-Time Systems 419

8-7 Concluding Remarks 425

Problems 425

Chapter 9 Llnear Tlme-Invarlant Composlte Systems: Characterlza-tlon, Stablllty, and Deslgns 432

9-1 Introduction 432

9-2 Complete Characterization of Single-Variable Composite Systems 434

9-3 Controllability and Observability of Composite Systems 439

Parallel Connection 440

Tandem Connection 441

Feedback Connection 444

9-4 Stability of Feedback Systems 448

Single-Variable Feedback System 449

Multivariable Feedback System 451

9-5 Design of Compensators: Unity Feedback Systems 458

Single-Variable Case 458

Single-Input or Single-Output Case 464

Multivariable Case—Arbitrary Pole Assignment 468

Multivariable Case—Arbitrary Denominator-Matrix Assignment 478

Decoupling 486

9-6 Asymptotic Tracking and Disturbance Rejection 488

Single-Variable Case 488

Multivariable Case 495

Static Decoupling—Robust and NonrobustDesigns 501

State-Variable Approach 504

9-7 Design of Compensators: Input-Output FeedbackSytems 506

Single-Variable Case 506

Multivariable Case 511

Implementations of Open-Loop Compensators 517

Implementation Ⅰ 517

Implementation Ⅱ 519

Applications 523

Decoupling 523

Asymptotic Tracking, Disturbance Rejection, andDecoupling 526

9-8 Concluding Remarks 534

Problems 536

Appendix A Elementary Transformations 542

A-1 Gaussian Elimination 543

A-2 Householder Transformation 544

A-3 Row Searching Algorithm 546

A-4 Hessenberg Form 551

Problems 553

Appendix B Analytic Functions of a Real Variable 554

Appendix C Minimum Energy Control 556

Appendix D Controllability after the Introduction of Sampling 559

Problems 564

Appendix E Hermitian Forms and Singular Value Decomposition 565

Problems 570

Appendix F On the Matrix Equation AM + MB = N 572

Problems 576

Appendix G Polynomials and Polynomial Matrices 577

G-1 Coprimeness of Polynomials 577

G-2 Reduction of Reducible Rational Functions 584

G-3 Polynomial Matrices 587

G-4 Coprimeness of Polynomial Matrices 592

G-5 Column- and Row-Reduced Polynomial Matrices 599

G-6 Coprime Fractions of Proper Rational Matrices 605

Problems 618

Appendix H Poles and Zeros 623

Problems 635

References 636

Index 657