A Linear Systems PrimerBased 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 timeinvariant systems, both continuous and discretetime.
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. * Helpful guidelines for the reader in the preface. * 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 statefeedback, stateestimation, and eigenvalue assignment. * Emphasis on timeinvariant systems, both continuous and discretetime. For full coverage of timevariant 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 firstyear graduate and senior undergraduate students in a typical onesemester introductory course on systems and control. It may also serve as an excellent reference or selfstudy guide for electrical, mechanical, chemical, and aerospace engineers, applied mathematicians, and researchers working in control, communications, and signal processing.

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Contents
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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 
505  