# A Linear Systems Primer

Springer Science & Business Media, Dec 3, 2007 - Technology & Engineering - 517 pages

Based on a streamlined presentation of the authors' successful work Linear Systems, this textbook provides an introduction to systems theory with an emphasis on control. The material presented is broad enough to give the reader a clear picture of the dynamical behavior of linear systems as well as their advantages and limitations. Fundamental results and topics essential to linear systems theory are emphasized. The emphasis is on time-invariant systems, both continuous- and discrete-time.

Key features and topics:

* Notes, references, exercises, and a summary and highlights section at the end of each chapter.

* Comprehensive index and answers to selected exercises at the end of the book.

* Necessary mathematical background material included in an appendix.

* Three core chapters guiding the reader to an excellent understanding of the dynamical behavior of systems.

* Detailed coverage of internal and external system descriptions, including state variable, impulse response and transfer function, polynomial matrix, and fractional representations.

* Explanation of stability, controllability, observability, and realizations with an emphasis on fundamental results.

* Detailed discussion of state-feedback, state-estimation, and eigenvalue assignment.

* Emphasis on time-invariant systems, both continuous- and discrete-time. For full coverage of time-variant systems, the reader is encouraged to refer to the companion book Linear Systems, which contains more detailed descriptions and additional material, including all the proofs of the results presented here.

* Solutions manual available to instructors upon adoption of the text.

A Linear Systems Primer is geared towards first-year graduate and senior undergraduate students in a typical one-semester introductory course on systems and control. It may also serve as an excellent reference or self-study guide for electrical, mechanical, chemical, and aerospace engineers, applied mathematicians, and researchers working in control, communications, and signal processing.

Also by the authors: Linear Systems, ISBN 978-0-8176-4434-5.

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### Contents

 System Models Differential Equations and InitialValue Problems 1 12 Preliminaries 6 121 Notation 7 13 InitialValue Problems 8 131 Systems of FirstOrder Ordinary Differential Equations 9 132 Classiﬁcation of Systems of FirstOrder Ordinary Differential Equations 10 133 nthOrder Ordinary Differential Equations 11 14 Examples of InitialValue Problems 13
 74 Poles and Zeros 282 741 Smith and SmithMcMillan Forms 283 742 Poles 284 743 Zeros 286 744 Relations Between Poles Zeros and Eigenvalues of A 290 75 Polynomial Matrix and Matrix Fractional Descriptions of Systems 292 751 A Brief Introduction to Polynomial and Fractional Descriptions 294 752 Coprimeness and Common Divisors 298

 Existence Continuation Uniqueness and Continuous Dependence on Parameters 17 16 Systems of Linear FirstOrder Ordinary Differential Equations 20 161 Linearization 21 162 Examples 24 17 Linear Systems Existence Uniqueness Continuation and Continuity with Respect to Parameters of Solutions 27 18 Solutions of Linear State Equations 28 19 Summary and Highlights 32 110 Notes 33 Exercises 34 An Introduction to StateSpace and InputOutput Descriptions of Systems 47 23 StateSpace Description of DiscreteTime Systems 50 24 InputOutput Description of Systems 56 242 Linear DiscreteTime Systems 60 243 The Dirac Delta Distribution 65 244 Linear ContinuousTime Systems 68 25 Summary and Highlights 71 26 Notes 73 Exercises 74 Response of Continuous and DiscreteTime Systems 76 The State Transition Matrix Φtt₀ 78 322 The State Transition Matrix 82 323 Nonhomogeneous Equations 84 33 The Matrix Exponential eAt Modes and Asymptotic Behavior of x Ax 85 332 How to Determine eAt 86 333 Modes Asymptotic Behavior and Stability 94 34 State Equation and InputOutput Description of ContinuousTime Systems 100 342 Transfer Functions 102 343 Equivalence of StateSpace Representations 105 35 State Equation and InputOutput Description of DiscreteTime Systems 108 352 The Transfer Function and the zTransform 112 353 Equivalence of StateSpace Representations 115 354 SampledData Systems 116 355 Modes Asymptotic Behavior and Stability 121 36 An Important Comment on Notation 126 37 Summary and Highlights 127 38 Notes 129 References 130 Exercises 131 Stability 141 42 The Concept of an Equilibrium 142 43 Qualitative Characterizations of an Equilibrium 144 44 Lyapunov Stability of Linear Systems 148 45 The Lyapunov Matrix Equation 153 46 Linearization 164 47 InputOutput Stability 170 48 DiscreteTime Systems 173 482 Linear Systems 176 483 The Lyapunov Matrix Equation 179 484 Linearization 185 485 InputOutput Stability 186 49 Summary and Highlights 188 410 Notes 189 References 190 Exercises 191 Controllability and Observability Fundamental Results 195 521 Reachability and Controllability 196 522 Observability and Constructibility 200 523 Dual Systems 203 53 Reachability and Controllability 204 531 ContinuousTime TimeInvariant Systems 205 532 DiscreteTime Systems 213 54 Observability and Constructibility 218 541 ContinuousTime TimeInvariant Systems 219 542 DiscreteTime TimeInvariant Systems 225 55 Summary and Highlights 230 56 Notes 232 Exercises 233 Controllability and Observability Special Forms 237 621 Standard Form for Uncontrollable Systems 238 622 Standard Form for Unobservable Systems 241 623 Kalmans Decomposition Theorem 244 63 EigenvalueEigenvector Tests for Controllability and Observability 248 64 Controller and Observer Forms 250 641 Controller Forms 251 642 Observer Forms 263 65 Summary and Highlights 269 66 Notes 271 References 272 Internal and External Descriptions Relations and Properties 277 73 Relations Between Lyapunov and InputOutput Stability 281
 753 Controllability Observability and Stability 303 754 Poles and Zeros 304 76 Summary and Highlights 306 77 Notes 308 Exercises 309 Realization Theory and Algorithms 313 821 ContinuousTime Systems 314 822 DiscreteTime Systems 315 83 Existence and Minimality of Realizations 316 832 Minimality of Realizations 317 833 The Order of Minimal Realizations 321 DiscreteTime Systems 323 84 Realization Algorithms 324 842 Realizations in ControllerObserver Form 326 843 Realizations with Matrix A Diagonal 339 844 Realizations Using SingularValue Decomposition 341 85 Polynomial Matrix Realizations 343 86 Summary and Highlights 345 87 Notes 346 State Feedback and State Observers 350 92 Linear State Feedback 352 922 Eigenvalue Assignment 355 ContinuousTime Case 369 924 InputOutput Relations 372 925 DiscreteTime Systems 376 DiscreteTime Case 377 93 Linear State Observers 378 ContinuousTime Systems 383 ContinuousTime Systems 385 DiscreteTime Systems 387 DiscreteTime Systems 391 94 ObserverBased Dynamic Controllers 392 941 StateSpace Analysis 393 942 Transfer Function Analysis 397 95 Summary and Highlights 400 96 Notes 403 References 404 Exercises 405 Feedback Control Systems 411 1022 Systems Connected in Feedback Conﬁguration 413 103 Parameterization of All Stabilizing Feedback Controllers 422 1031 Stabilizing Feedback Controllers Using Polynomial MFDs 423 1032 Stabilizing Feedback Controllers Using Proper and Stable MFDs 426 104 Two Degrees of Freedom Controllers 431 1041 Internal Stability 432 1042 Response Maps 435 1043 Controller Implementations 439 1044 Some Control Problems 445 105 Summary and Highlights 447 106 Notes 449 References 451 Exercises 452 Appendix 455 A12 Vector Spaces 456 A2 Linear Independence and Bases 460 A22 Linear Independence 461 A23 Linear Independence of Functions of Time 462 A24 Bases 463 A3 Linear Transformations 464 A31 Linear Equations 465 A32 Representation of Linear Transformations by Matrices 466 A33 Solving Linear Algebraic Equations 469 A4 Equivalence and Similarity 471 Matrix Case 472 A43 Equivalence and Similarity of Matrices 473 A5 Eigenvalues and Eigenvectors 474 A52 The Cayley Hamilton Theorem and Applications 475 A53 Minimal Polynomials 477 A6 Diagonal and Jordan Canonical Form of Matrices 478 A7 Normed Linear Spaces 483 A8 Some Facts from Matrix Algebra 486 A9 Numerical Considerations 487 A91 Solving Linear Algebraic Equations 488 A92 Singular Values and Singular Value Decomposition 491 A93 LeastSquares Problem 496 A10 Notes 497 Solutions to Selected Exercises 498 Index 505 Copyright