A Linear Systems Primer

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

* 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 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

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
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