Introduction to linear control systems

✍ Scribed by Bavafa-Toosi, Yazdan

• Publisher: Elsevier Academic Press
• Year: 2017
• Tongue: English
• Leaves: 929
• Category: Library

✦ Synopsis

Introduction to Linear Control Systems is designed as a standard introduction to linear control systems for all those who one way or another deal with control systems. It can be used as a comprehensive up-to-date textbook for a one-semester 3-credit undergraduate course on linear control systems as the first course on this topic at university. This includes the faculties of electrical engineering, mechanical engineering, aerospace engineering, chemical and petroleum engineering, industrial engineering, civil engineering, bio-engineering, economics, mathematics, physics, management and social sciences, etc. The book covers foundations of linear control systems, their raison detre, different types, modelling, representations, computations, stability concepts, tools for time-domain and frequency-domain analysis and synthesis, and fundamental limitations, with an emphasis on frequency-domain methods. Every chapter includes a part on further readings where more advanced topics and pertinent references are introduced for further studies. The presentation is theoretically firm, contemporary, and self-contained. Appendices cover Laplace transform and differential equations, dynamics, MATLAB and SIMULINK, treatise on stability concepts and tools, treatise on Routh-Hurwitz method, random optimization techniques as well as convex and non-convex problems, and sample midterm and endterm exams. The book is divided to the sequel 3 parts plus appendices. PART I: In this part of the book, chapters 1-5, we present foundations of linear control systems. This includes: the introduction to control systems, their raison detre, their different types, modelling of control systems, different methods for their representation and fundamental computations, basic stability concepts and tools for both analysis and design, basic time domain analysis and design details, and the root locus as a stability analysis and synthesis tool. PART II: In this part of the book, Chapters 6-9, we present what is generally referred to as the frequency domain methods. This refers to the experiment of applying a sinusoidal input to the system and studying its output. There are basically three different methods for representation and studying of the data of the aforementioned frequency response experiment: these are the Nyquist plot, the Bode diagram, and the Krohn-Manger-Nichols chart. We study these methods in details. We learn that the output is also a sinusoid with the same frequency but generally with different phase and magnitude. By dividing the output by the input we obtain the so-called sinusoidal or frequency transfer function of the system which is the same as the transfer function when the Laplace variable s is substituted with . Finally we use the Bode diagram for the design process. PART III: In this part, Chapter 10, we introduce some miscellaneous advanced topics under the theme fundamental limitations which should be included in this undergraduate course at least in an introductory level. We make bridges between some seemingly disparate aspects of a control system and theoretically complement the previously studied subjects. Appendices: The book contains seven appendices. Appendix A is on the Laplace transform and differential equations. Appendix B is an introduction to dynamics. Appendix C is an introduction to MATLAB, including SIMULINK. Appendix D is a survey on stability concepts and tools. A glossary and road map of the available stability concepts and tests is provided which is missing even in the research literature. Appendix E is a survey on the Routh-Hurwitz method, also missing in the literature. Appendix F is an introduction to random optimization techniques and convex and non-convex problems. Finally, appendix G presents sample midterm and endterm exams, which are class-tested several times.

✦ Table of Contents

Cover......Page 1
Introduction to Linear Control Systems......Page 3
Copyright......Page 4
Dedication......Page 5
Preface......Page 6
Acknowledgments......Page 13
Part I: Foundations......Page 14
1.1 Introduction......Page 15
1.2 Why control?......Page 16
1.3 History of control......Page 17
1.4 Why feedback?......Page 19
1.5 Magic of feedback......Page 20
1.6 Physical elements of a control system......Page 21
1.7 Abstract elements of a control system......Page 22
1.8 Design process......Page 23
1.9 Types of control systems......Page 25
1.10.1 Stability and performance......Page 26
1.10.2 Sensitivity and robustness......Page 28
1.10.3 Disturbance......Page 29
1.11.1 Stability and performance......Page 30
1.11.2 Sensitivity and robustness......Page 31
1.11.3 Disturbance and noise......Page 32
1.11.4 Reliability, economics, and linearity......Page 33
1.12 The 2-DOF control structure......Page 39
1.13 The Smith predictor......Page 44
1.15 Modern representation—Generalized model......Page 45
1.16 Status quo......Page 47
1.16.1 Overview......Page 48
1.16.1.1 Summary......Page 52
1.16.1.2 The forgotten......Page 56
1.16.2 Relation with other disciplines......Page 57
1.16.3 Challenges......Page 58
1.16.4 Outlook......Page 60
1.18 Notes and further readings......Page 63
1.19 Worked-out problems......Page 68
1.20 Exercises......Page 77
References......Page 86
Further Reading......Page 91
2.1 Introduction......Page 96
2.2 System modeling......Page 97
2.2.1 State-space......Page 98
2.2.1.1 Linearization......Page 100
2.2.1.2 Number of inputs and outputs......Page 105
2.2.2 Frequency domain......Page 106
2.2.2.1 Finding the output......Page 107
2.2.3 Zero, pole, and minimality......Page 110
2.3 Basic examples of modeling......Page 111
2.3.1 Electrical system as the plant......Page 112
2.3.2 Mechanical system as the plant......Page 113
2.3.3 Liquid system as the plant......Page 114
2.3.4 Thermal system as the plant......Page 115
2.3.5 Hydraulic system as the plant......Page 116
2.3.6 Chemical system as the plant......Page 117
2.3.7 Structural system as the plant......Page 119
2.3.9 Economics system as the plant......Page 121
2.3.10 Ecological system as the plant......Page 122
2.3.11 Societal system as the plant......Page 123
2.3.12 Physics system as the plant......Page 125
2.3.13.1 Exact modeling of delay......Page 126
2.3.13.2 Approximate modeling of delay......Page 131
2.3.14.2 Amplifiers......Page 132
2.4 Block diagram......Page 134
2.5 Signal flow graph......Page 139
2.5.1 Basic terminology of graph theory......Page 140
2.5.2 Equivalence of BD and SFG methods......Page 142
2.5.3 Computing the transmittance of an SFG......Page 143
2.7 Notes and further readings......Page 146
2.8 Worked-out problems......Page 151
2.9 Exercises......Page 183
References......Page 203
3.1 Introduction......Page 211
3.2 Lyapunov and BIBO stability......Page 212
3.3 Stability tests......Page 216
3.4 Routh’s test......Page 218
3.4.1 Special cases......Page 220
3.5 Hurwitz’ test......Page 223
3.6 Lienard and Chipart test......Page 224
3.7 Relative stability......Page 225
3.8 D-stability......Page 227
3.9 Particular relation with control systems design......Page 229
3.10 The Kharitonov theory......Page 230
3.11 Internal stability......Page 231
3.12 Strong stabilization......Page 234
3.13 Stability of LTV Systems......Page 235
3.14 Summary......Page 238
3.15 Notes and further readings......Page 239
3.16 Worked-out problems......Page 242
3.17 Exercises......Page 252
References......Page 262
4.1 Introduction......Page 267
4.2 System type and system inputs......Page 268
4.3 Steady-state error......Page 269
4.4 First-order systems......Page 273
4.5 Second-order systems......Page 274
4.5.1 System representation......Page 275
4.5.2 Impulse response......Page 276
4.5.3 Step response......Page 277
4.5.3.1 Time response characteristics......Page 278
4.6 Bandwidth of the system......Page 283
4.6.1 First-order systems......Page 284
4.6.2 Second-order systems......Page 286
4.6.3 Alternative derivation......Page 289
4.6.4 Higher-order systems......Page 291
4.7 Higher-order systems......Page 293
4.8 Model reduction......Page 295
4.9 Effect of addition of pole and zero......Page 299
4.10 Performance region......Page 302
4.11 Inverse response......Page 303
4.12 Analysis of the actual system......Page 305
4.12.1 Sensor dynamics......Page 306
4.12.2 Delay dynamics......Page 314
4.13 Introduction to robust stabilization and performance......Page 317
4.13.1 Open-loop control......Page 319
Design for disturbance and noise rejection......Page 320
Design for sinusoidal reference tracking......Page 323
4.15 Notes and further readings......Page 326
4.16 Worked-out problems......Page 328
4.17 Exercises......Page 350
References......Page 358
5.1 Introduction......Page 361
5.2 The root locus method......Page 364
5.3 The root contour......Page 382
5.4 Finding the value of gain from the root locus......Page 383
5.5.1.2 Systems with NMP zeros......Page 386
5.5.1.3 Examples of systems without NMP zeros......Page 387
5.5.1.4 Examples of system with NMP zeros......Page 396
5.5.2 Simple systems......Page 402
5.6 Summary......Page 409
5.7 Notes and further readings......Page 410
5.8 Worked-out problems......Page 411
5.9 Exercises......Page 440
References......Page 450
Part II: Frequency domain analysis & synthesis......Page 452
6.1 Introduction......Page 453
6.2 Nyquist plot......Page 455
6.2.1 Principle of argument......Page 456
6.2.2 Nyquist stability criterion......Page 458
6.2.3 Drawing of the Nyquist plot......Page 459
6.2.4 The high- and low-frequency ends of the plot......Page 462
6.2.5 Cusp points of the plot......Page 465
6.2.6 How to handle the proportional gain/uncertain parameter......Page 466
6.2.7 The case of j-axis zeros and poles......Page 467
6.2.8 Relation with root locus......Page 478
6.3 Gain, phase, and delay margins......Page 480
6.3.1 The GM concept......Page 481
6.3.1.1 Definition of GM in the Nyquist plot context......Page 490
6.3.2 The PM and DM concepts......Page 492
6.3.3 Stability in terms of the GM and PM signs......Page 507
6.3.4 The high sensitivity region......Page 509
6.5 Notes and further readings......Page 511
6.6 Worked-out problems......Page 513
6.7 Exercises......Page 531
References......Page 537
7.1 Introduction......Page 539
7.2.3 Log magnitude......Page 540
7.2.5 Octave and decade......Page 541
7.2.7.1 Gain K......Page 542
7.2.7.2 Zeros at origin (jω)+m......Page 543
7.2.7.4 Real zeros not at origin (1+jωT)+m......Page 544
7.2.7.6 Error in Lm......Page 545
7.2.7.9 Double poles [1+2ζωnjω+1ωn2(jω)2]−m......Page 546
7.2.8 How to draw the Bode diagram with hand......Page 547
7.3 Bode diagram and the steady-state error......Page 552
7.4 Minimum phase and nonminimum phase systems......Page 555
7.4.4 NMP pole with negative gain: −1/(p−s)=1/(s−p), p%3e0......Page 556
7.4.5 Determination of NMP systems from the Bode diagram......Page 561
7.5 Gain, phase, and delay margins......Page 562
7.6 Stability in the Bode diagram context......Page 567
7.8 Relation with Nyquist plot and root locus......Page 568
7.9 Standard second-order systems......Page 569
7.10 Bandwidth......Page 570
7.11 Summary......Page 573
7.13 Worked-out problems......Page 574
7.14 Exercises......Page 589
References......Page 596
8.1 Introduction......Page 597
8.2 S-Circles......Page 599
8.3 M-Circles......Page 600
8.4 N-circles......Page 602
8.5 M- and N-Contours......Page 603
8.6 KMN chart......Page 604
8.7.1 Gain, phase, and delay margins......Page 607
8.7.2 Stability......Page 608
8.7.3 Bandwidth......Page 611
8.8 The high sensitivity region......Page 613
8.9 Relation with Bode diagram, Nyquist plot, and root locus......Page 614
8.11 Notes and further readings......Page 615
8.12 Worked-out problems......Page 616
8.13 Exercises......Page 630
References......Page 635
9.1 Introduction......Page 637
9.2 Basic controllers: proportional, lead, lag, and lead-lag......Page 639
9.3 Controller simplifications: PI, PD, and PID......Page 657
9.4 Controller structures in the Nyquist plot context......Page 660
9.5 Effect of the controllers on the root locus......Page 663
9.6 Design procedure......Page 666
9.7.1 Heuristic rules......Page 668
9.7.2.1 Pole placement method......Page 669
9.7.2.2 Direct synthesis......Page 671
9.7.2.3 Skogestad tuning rules......Page 672
9.7.3 Optimization-based rules......Page 676
9.8 Internal model control......Page 678
9.9 The Smith predictor......Page 679
9.10.2 Integral control—I-term......Page 680
9.10.6 Series proportional-integral-derivative—Series PID......Page 681
9.10.8 Lag......Page 682
9.11 Summary......Page 683
9.12 Notes and further readings......Page 684
9.13 Worked-out problems......Page 685
9.14 Exercises......Page 722
References......Page 727
Part III: Advanced Issues......Page 730
10.1 Introduction......Page 731
10.2 Relation between time and frequency domain specifications......Page 732
10.3 The ideal transfer function......Page 734
10.4 Controller design via the TS method......Page 741
10.5 Interpolation conditions......Page 742
10.6 Integral and Poisson integral constraints......Page 745
10.7.1 Implications of open-loop integrators......Page 752
10.7.2 MP and NMP poles and zeros......Page 754
10.7.3 Imaginary-axis poles and zeros......Page 758
10.8.1 Maximal actuator movement......Page 760
10.8.2 Minimal actuator movement......Page 762
10.8.4 Sensor speed......Page 763
10.9 Delay......Page 764
10.10 Eigenstructure assignment by output feedback......Page 765
10.10.1 Regulation......Page 768
10.10.2 Tracking......Page 772
10.11 Noninteractive performance......Page 773
10.12 Minimal closed-loop pole sensitivity......Page 777
10.13.1 Structured perturbations......Page 781
10.13.2 Unstructured perturbations......Page 782
10.14 Special results for positive systems......Page 783
10.15 Generic design procedure......Page 784
10.16 Summary......Page 788
10.17 Notes and further readings......Page 789
10.18 Worked-out problems......Page 792
10.19 Exercises......Page 805
References......Page 815
Appendices A–G......Page 823
A.1 Introduction......Page 824
A.2 Basic properties and pairs......Page 826
A.2.2 Table of some Laplace transform pairs......Page 827
A.3 Differentiation and integration in time domain and frequency domain......Page 829
A.3.2 Differentiation formula in time domain......Page 831
A.3.4 Frequency domain formulae......Page 832
A.3.5 Some consequences......Page 833
A.4 Existence and uniqueness of solutions to differential equations......Page 834
References......Page 835
B.1.1 Electrical systems......Page 836
B.1.2 Mechanical systems......Page 837
B.1.3 Chemical systems......Page 838
B.2 Equivalent systems......Page 839
B.3 Worked-out problems......Page 840
References......Page 845
C.1 Introduction......Page 847
C.2 MATLAB®......Page 848
C.2.1.1 Script file......Page 850
C.2.1.2 Function file......Page 851
C.2.2.1 LTI models......Page 853
C.2.2.6 Model dynamics......Page 854
C.2.2.11 Pole placement......Page 855
C.3 Simulink......Page 856
C.4 Worked-out problems......Page 858
References......Page 874
D.1 Introduction......Page 875
D.2.1 Deterministic systems......Page 876
D.2.2 Stochastic systems......Page 878
D.2.3 Miscellaneous......Page 879
D.3 Lipschitz stability......Page 880
D.4 Lagrange, Poisson, and Lyapunov stability......Page 881
D.5 Finite-time and fixed-time stability......Page 883
D.5.1 Fixed-time decentralized stability of large-scale systems......Page 884
D.5.1.1 Large-scale system description......Page 885
D.6 Summary......Page 886
References......Page 887
E.2 Applications of the Routh’s array......Page 890
References......Page 893
F.2 Convex optimization......Page 897
F.3 Nonconvex optimization......Page 898
F.5 Genetic algorithms......Page 899
References......Page 911
G.1 Sample Midterm Exam (ME) (4h, closed book/notes)......Page 913
G.2 Sample Endterm Exam (EE) (4h, closed book/notes)......Page 914
Index......Page 918
Back Cover......Page 929

✦ Subjects

Linear control systems