Linear Systems Control: Deterministic and Stochastic Methods: Linear Systems Control Deterministic and Stochastic Methods.part2.rar

Linear Systems Control: Deterministic and Stochastic Methods
By Elbert Hendricks, Ole Jannerup, Paul Haase Sørensen


Publisher: Springer
Number Of Pages: 556
Publication Date: 2008-10-01
ISBN-10 / ASIN: 3540784853
ISBN-13 / EAN: 9783540784852
Binding: Hardcover




Modern control theory and in particular state space or state variable methods can be adapted to the description of many different systems because it depends strongly on physical modeling and physical intuition. The laws of physics are in the form of differential equations and for this reason, this book concentrates on system descriptions in this form. This means coupled systems of linear or nonlinear differential equations. The physical approach is emphasized in this book because it is most natural for complex systems. It also makes what would ordinarily be a difficult mathematical subject into one which can straightforwardly be understood intuitively and which deals with concepts which engineering and science students are already familiar. In this way it is easy to immediately apply the theory to the understanding and control of ordinary systems. Application engineers, working in industry, will also find this book interesting and useful for this reason.



In line with the approach set forth above, the book first deals with the modeling of systems in state space form. Both transfer function and differential equation modeling methods are treated with many examples. Linearization is treated and explained first for very simple nonlinear systems and then more complex systems. Because computer control is so fundamental to modern applications, discrete time modeling of systems as difference equations is introduced immediately after the more intuitive differential equation models. The conversion of differential equation models to difference equations is also discussed at length, including transfer function formulations.

A vital problem in modern control is how to treat noise in control systems. Nevertheless this question is rarely treated in many control system textbooks because it is considered to be too mathematical and too difficult in a second course on controls. In this textbook a simple physical approach is made to the description of noise and stochastic disturbances which is easy to understand and apply to common systems. This requires only a few fundamental statistical concepts which are given in a simple introduction which lead naturally to the fundamental noise propagation equation for dynamic systems, the Lyapunov equation. This equation is given and exemplified both in its continuous and discrete time versions.

With the Lyapunov equation available to describe state noise propagation, it is a very small step to add the effect of measurements and measurement noise. This gives immediately the Riccati equation for optimal state estimators or Kalman filters. These important observers are derived and illustrated using simulations in terms which make them easy to understand and easy to apply to real systems. The use of LQR regulators with Kalman filters give LQG (Linear Quadratic Gaussian) regulators which are introduced at the end of the book. Another important subject which is introduced is the use of Kalman filters as parameter estimations for unknown parameters.

The textbook is divided into 7 chapters, 5 appendices, a table of contents, a table of examples, extensive index and extensive list of references. Each chapter is provided with a summary of the main points covered and a set of problems relevant to the material in that chapter. Moreover each of the more advanced chapters (3 - 7) are provided with notes describing the history of the mathematical and technical problems which lead to the control theory presented in that chapter. Continuous time methods are the main focus in the book because these provide the most direct connection to physics. This physical foundation allows a logical presentation and gives a good intuitive feel for control system construction. Nevertheless strong attention is also given to discrete time systems.

Very few proofs are included in the book but most of the important results are derived. This method of presentation makes the text very readable and gives a good foundation for reading more rigorous texts.

A complete set of solutions is available for all of the problems in the text. In addition a set of longer exercises is available for use as Matlab/Simulink

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Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Invisible Thread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Classical Control Systems and their Background. . . . . . . . . . . . . . 2
1.2.1 Primitive Period Developments . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Pre-Classical Period Developments . . . . . . . . . . . . . . . . . . . 4
1.2.3 Classical Control Period . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4 Modern Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 State Space Modelling of Physical Systems . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Modelling of Physical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Linear System Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 State Space Models from Transfer Functions . . . . . . . . . . . . . . . 21
2.3.1 Companion Form 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.2 Companion Form 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Discrete Time Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Analysis of State Space Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1 Solution of the Linear State Equation . . . . . . . . . . . . . . . . . . . . . 59
3.1.1 The Time Varying System . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1.2 The Time Invariant System . . . . . . . . . . . . . . . . . . . . . . . . 64
3.2 Transfer Functions from State Space Models . . . . . . . . . . . . . . . 72
3.2.1 Natural Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3 Discrete Time Models of Continuous Systems . . . . . . . . . . . . . . . 76
3.4 Solution of the Discrete Time State Equation . . . . . . . . . . . . . . . 82
3.4.1 The Time Invariant Discrete Time System . . . . . . . . . . . . 83
3.5 Discrete Time Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . 87
3.6 Similarity Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.7 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.7.1 Stability Criteria for Linear Systems . . . . . . . . . . . . . . . . 104
3.7.2 Time Invariant Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 106
xi
3.7.3 BIBO Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.7.4 Internal and External Stability . . . . . . . . . . . . . . . . . . . . 115
3.7.5 Lyapunov’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.8 Controllability and Observability . . . . . . . . . . . . . . . . . . . . . . . . 121
3.8.1 Controllability (Continuous Time Systems) . . . . . . . . . . 124
3.8.2 Controllability and Similarity Transformations . . . . . . . 132
3.8.3 Reachability (Continuous Time Systems) . . . . . . . . . . . . 132
3.8.4 Controllability (Discrete Time Systems) . . . . . . . . . . . . . 137
3.8.5 Reachability (Discrete Time Systems) . . . . . . . . . . . . . . . 138
3.8.6 Observability (Continuous Time Systems) . . . . . . . . . . . 142
3.8.7 Observability and Similarity Transformations . . . . . . . . 146
3.8.8 Observability (Discrete Time Systems) . . . . . . . . . . . . . . 148
3.8.9 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.8.10 Modal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.8.11 Controllable/Reachable Subspace Decomposition . . . . . 154
3.8.12 Observable Subspace Decomposition . . . . . . . . . . . . . . . 157
3.9 Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
3.9.1 Controller Canonical Form. . . . . . . . . . . . . . . . . . . . . . . 159
3.9.2 Observer Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . 164
3.9.3 Duality for Canonical Forms . . . . . . . . . . . . . . . . . . . . . 167
3.9.4 Pole-zero Cancellation in SISO Systems . . . . . . . . . . . . . 168
3.10 Realizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
3.10.1 Minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
3.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
3.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
3.12.1 Linear Systems Theory . . . . . . . . . . . . . . . . . . . . . . . . . 183
3.13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
4 Linear Control System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
4.1 Control System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
4.1.1 Controller Operating Modes . . . . . . . . . . . . . . . . . . . . . . 196
4.2 Full State Feedback for Linear Systems . . . . . . . . . . . . . . . . . . . 199
4.3 State Feedback for SISO Systems . . . . . . . . . . . . . . . . . . . . . . . . 208
4.3.1 Controller Design Based on the Controller Canonical
Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
4.3.2 Ackermann’s Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . 210
4.3.3 Conditions for Eigenvalue Assignment . . . . . . . . . . . . . . 212
4.4 State Feedback for MIMO Systems . . . . . . . . . . . . . . . . . . . . . . 226
4.4.1 Eigenstructure Assignment
for MIMO Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
4.4.2 Dead Beat Regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
4.5 Integral Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
4.6 Deterministic Observers and State Estimation . . . . . . . . . . . . . . 251
4.6.1 Continuous Time Full Order Observers . . . . . . . . . . . . . 252
4.6.2 Discrete Time Full Order Observers . . . . . . . . . . . . . . . . 255
xii Contents
4.7 Observer Design for SISO Systems . . . . . . . . . . . . . . . . . . . . . . . 256
4.7.1 Observer Design Based on the Observer Canonical Form 256
4.7.2 Ackermann’s Formula for the Observer . . . . . . . . . . . . . 259
4.7.3 Conditions for Eigenvalue Assignment . . . . . . . . . . . . . . 264
4.8 Observer Design for MIMO Systems . . . . . . . . . . . . . . . . . . . . . 265
4.8.1 Eigenstructure Assignment for MIMO Observers . . . . . 266
4.8.2 Dead Beat Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
4.9 Reduced Order Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
4.10 State Feedback with Observers . . . . . . . . . . . . . . . . . . . . . . . . . . 272
4.10.1 Combining Observers and State Feedback . . . . . . . . . . 273
4.10.2 State Feedback with Integral Controller and Observer 277
4.10.3 State Feedback with Reduced Order Observer . . . . . . . 284
4.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
4.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
4.12.1 Background for Observers . . . . . . . . . . . . . . . . . . . . . . . 287
4.13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
5 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
5.1 Introduction to Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . 293
5.2 The General Optimal Control Problem . . . . . . . . . . . . . . . . . . . 294
5.3 The Basis of Optimal Control – Calculus of Variations . . . . . . . 296
5.4 The Linear Quadratic Regulator . . . . . . . . . . . . . . . . . . . . . . . . 304
5.4.1 The Quadratic Cost Function . . . . . . . . . . . . . . . . . . . . . 305
5.4.2 Linear Quadratic Control . . . . . . . . . . . . . . . . . . . . . . . . 307
5.5 Steady State Linear Quadratic Regulator . . . . . . . . . . . . . . . . . . 316
5.5.1 Robustness of LQR Control . . . . . . . . . . . . . . . . . . . . . . 324
5.5.2 LQR Design: Eigenstructure Assignment Approach . . . 325
5.6 Discrete Time Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . 328
5.6.1 Discretization of the Performance Index. . . . . . . . . . . . . 329
5.6.2 Discrete Time State Feedback . . . . . . . . . . . . . . . . . . . . . 330
5.6.3 Steady State Discrete Optimal Control . . . . . . . . . . . . . . 332
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
5.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
5.8.1 The Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . 340
5.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
6 Noise in Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
6.1.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
6.2 Expectation (Average) Values of a Random Variable . . . . . . . . 357
6.2.1 Average Value of Discrete Random Variables . . . . . . . . 361
6.2.2 Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 362
6.2.3 Joint Probability Distribution and Density Functions . . 366
6.3 Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
6.3.1 Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Contents xiii
6.3.2 Moments of a Stochastic Process . . . . . . . . . . . . . . . . . . 375
6.3.3 Stationary Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
6.3.4 Ergodic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
6.3.5 Independent Increment Stochastic Processes . . . . . . . . . 382
6.4 Noise Propagation: Frequency and Time Domains . . . . . . . . . . 392
6.4.1 Continuous Random Processes: Time Domain . . . . . . . 394
6.4.2 Continuous Random Processes: Frequency Domain . . . 397
6.4.3 Continuous Random Processes: Time Domain . . . . . . . 402
6.4.4 Inserting Noise into Simulation Systems . . . . . . . . . . . . . 408
6.4.5 Discrete Time Stochastic Processes . . . . . . . . . . . . . . . . . 412
6.4.6 Translating Continuous Noise into Discrete Time
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
6.4.7 Discrete Random Processes: Frequency Domain . . . . . . 416
6.4.8 Discrete Random Processes: Running in Time . . . . . . . . 420
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
6.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
6.6.1 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . 423
6.6.2 The Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
6.6.3 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . 424
6.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
7 Optimal Observers: Kalman Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
7.2 Continuous Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
7.2.1 Block Diagram of a CKF . . . . . . . . . . . . . . . . . . . . . . . . 436
7.3 Innovation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
7.4 Discrete Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
7.4.1 A Real Time Discrete Kalman Filter (Open Form) . . . . 449
7.4.2 Block Diagram of an Open Form DKF . . . . . . . . . . . . . 453
7.4.3 Closed Form of a DKF . . . . . . . . . . . . . . . . . . . . . . . . . . 456
7.4.4 Discrete and Continuous Kalman Filter Equivalence . . 461
7.5 Stochastic Integral Quadratic Forms . . . . . . . . . . . . . . . . . . . . . 464
7.6 Separation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
7.6.1 Evaluation of the Continuous LQG Index . . . . . . . . . . . 469
7.6.2 Evaluation of the Discrete LQG Index . . . . . . . . . . . . . . 475
7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
7.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
7.8.1 Background for Kalman Filtering . . . . . . . . . . . . . . . . . . 478
7.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
Appendix A Static Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
A.1 Optimization Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
A.1.1 Constrained Static Optimization . . . . . . . . . . . . . . . . . . 496
A.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
xiv Contents
Appendix B Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
B.1 Matrix Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
B.2 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . 503
B.3 Partitioned Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
B.4 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
B.5 Matrix Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
Appendix C Continuous Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . 511
C.1 Estimator Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
C.1.1 Time Axis Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
C.1.2 Using the LQR Solution . . . . . . . . . . . . . . . . . . . . . . . . 512
Appendix D Discrete Time SISO Systems . . . . . . . . . . . . . . . . . . . . . . . 515
D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
D.2 The Sampling Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
D.3 The Z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
D.4 Inverse Z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
D.5 Discrete Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
D.6 Discrete Systems and Difference Equations . . . . . . . . . . . . . . . 528
D.7 Discrete Time Systems with Zero-Order-Hold . . . . . . . . . . . . . 528
D.8 Transient Response, Poles and Stability . . . . . . . . . . . . . . . . . . 529
D.9 Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
D.10 Discrete Approximations to Continuous Transfer Functions. . 534
D.10.1 Tustin Approximation . . . . . . . . . . . . . . . . . . . . . . . . 535
D.10.2 Matched-Pole-Zero Approximation (MPZ) . . . . . . . . 536
D.11 Discrete Equivalents to Continuous Controllers . . . . . . . . . . . . 539
D.11.1 Choice of Sampling Period . . . . . . . . . . . . . . . . . . . . . 545
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

Linear Systems Control Deterministic and Stochastic Methods.part1

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Linear Systems Control Deterministic and Stochastic Methods.part2-4

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感谢楼主分享
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谢谢楼主！！！！！！！！！！！！！！！！！！！！！！！！！！！！

谢谢楼主！！！！！！！！！！！！！！！！！！！！！！！！！！！！

haohaohaohaohaohaohaohaohaohaohaohao

哇~~新書耶
好久沒見到系統控制有新書了呢
感謝樓主分享

未来空军揭秘,thank you for the job

很好很强大。。。。。。。。。。。。。

:27bb :27bb :27bb :27bb

谢谢啊。。。。。。。。。。。。。。。。

好书真多 :11bb :11bb

好书，谢谢楼主分享。。。。。。。。。。。。

：23de ：23de ：23de

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：cacakiki15de：8de

:13bb:27bb:21bb

非常棒的一本書籍
謝意樓主的分享

占位等好书！！

谢谢楼主！！！！！！！！！！！！！！！！！！！！！！！！！！！！

这本书我喜欢，好好学习

Advanced Signal Processing

nice. thanks

thanks

感谢楼主分享

多谢楼主的分享了

谢谢楼主
学习了

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