Essentials of Control Techniques and Theory

340 Pages·2009·4.315 MB·English

by  John Billingsley

Essentials of
Control Techniques
and Theory
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Essentials of
Control Techniques
and Theory
John Billingsley
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CRC Press
Taylor & Francis Group
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Library of Congress Cataloging-in-Publication Data
Billingsley, J. (John)
Essentials of control techniques and theory / John Billingsley.
p. cm.
Includes index.
ISBN 978-1-4200-9123-6 (hardcover : alk. paper)
1. Automatic control. 2. Control theory. I. Title.
TJ223.M53B544 2010
629.8--dc22 2009034834
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Contents
Preface ............................................................................................................xi
Author .........................................................................................................xiii
SeCtIon I eSSentIalS of Control teChnIqueS—
What You need to KnoW
1 Introduction: Control in a nutshell; history, theory, art, and
Practice ............................................................................................3
1.1 The Origins of Control ......................................................................3
1.2 Early Days of Feedback .....................................................................5
1.3 The Origins of Simulation .................................................................6
1.4 Discrete Time ....................................................................................7
2 Modeling time. .................................................................................9
2.1 Introduction ......................................................................................9
2.2 A Simple System ................................................................................9
2.3 Simulation .......................................................................................11
2.4 Choosing a Computing Platform .....................................................12
2.5 An Alternative Platform ...................................................................15
2.6 Solving the First Order Equation .....................................................16
2.7 A Second Order Problem .................................................................19
2.8 Matrix State Equations ....................................................................23
2.9 Analog Simulation ...........................................................................24
2.10 Closed Loop Equations ...................................................................26
3 Simulation with JollIeS: JavaScript on-line learning
Interactive environment for Simulation .........................................29
3.1 Introduction ....................................................................................29
3.2 How a JOLLIES Simulation Is Made Up ........................................31
3.3 Moving Images without an Applet ..................................................35
3.4 A Generic Simulation ......................................................................38
v
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vi ◾ Contents
4 Practical Control Systems ...............................................................41
4.1 Introduction ....................................................................................41
4.2 The Nature of Sensors .....................................................................42
4.3 Velocity and Acceleration ...............................................................44
4.4 Output Transducers ........................................................................44
4.5 A Control Experiment ....................................................................46
5 adding Control ...............................................................................49
5.1 Introduction ....................................................................................49
5.2 Vector State Equations .....................................................................49
5.3 Feedback..........................................................................................52
5.4 Another Approach ...........................................................................53
5.5 A Change of Variables .....................................................................55
5.6 Systems with Time Delay and the PID Controller ...........................57
5.7 Simulating the Water Heater Experiment ........................................60
6 Systems with real Components and Saturating
Signals—use of the Phase Plane ....................................................63
6.1 An Early Glimpse of Pole Assignment .............................................63
6.2 The Effect of Saturation ...................................................................65
6.3 Meet the Phase Plane .......................................................................65
6.4 Phase Plane for Saturating Drive .....................................................70
6.5 Bang–Bang Control and Sliding Mode ...........................................74
7 frequency domain Methods ..........................................................77
7.1 Introduction ...................................................................................77
7.2 Sine-Wave Fundamentals .................................................................78
7.3 Complex Amplitudes .......................................................................79
7.4 More Complex Still-Complex Frequencies ......................................81
7.5 Eigenfunctions and Gain .................................................................81
7.6 A Surfeit of Feedback .......................................................................83
7.7 Poles and Polynomials .....................................................................85
7.8 Complex Manipulations ..................................................................87
7.9 Decibels and Octaves .......................................................................88
7.10 Frequency Plots and Compensators .................................................89
7.11 Second Order Responses..................................................................92
7.12 Excited Poles ....................................................................................93
8 discrete time Systems and Computer Control ...............................97
8.1 Introduction ....................................................................................97
8.2 State Transition ...............................................................................98
8.3 Discrete Time State Equations and Feedback ................................101
8.4 Solving Discrete Time Equations ..................................................102
8.5 Matrices and Eigenvectors .............................................................103
8.6 Eigenvalues and Continuous Time Equations ...............................104
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Contents ◾ vii
8.7 Simulation of a Discrete Time System ...........................................105
8.8 A Practical Example of Discrete Time Control ..............................107
8.9 And There’s More ..........................................................................110
8.10 Controllers with Added Dynamics ................................................112
9 Controlling an Inverted Pendulum ................................................115
9.1 Deriving the State Equations .........................................................115
9.2 Simulating the Pendulum ..............................................................119
9.3 Adding Reality ..............................................................................122
9.4 A Better Choice of Poles ................................................................123
9.5 Increasing the Realism ...................................................................124
9.6 Tuning the Feedback Pragmatically ...............................................126
9.7 Constrained Demand ....................................................................127
9.8 In Conclusion ................................................................................129
SeCtIon II eSSentIalS of Control theorY—What
You ought to KnoW
10 More frequency domain Background theory ..............................133
10.1 Introduction ..................................................................................133
10.2 Complex Planes and Mappings ......................................................134
10.3 The Cauchy–Riemann Equations ..................................................135
10.4 Complex Integration ......................................................................138
10.5 Differential Equations and the Laplace Transform ........................140
10.6 The Fourier Transform ..................................................................144
11 More frequency domain Methods ...............................................147
11.1 Introduction ..................................................................................147
11.2 The Nyquist Plot ............................................................................148
11.3 Nyquist with M-Circles .................................................................151
11.4 Software for Computing the Diagrams .........................................153
11.5 The “Curly Squares” Plot ...............................................................154
11.6 Completing the Mapping ..............................................................155
11.7 Nyquist Summary .........................................................................156
11.8 The Nichols Chart .........................................................................156
11.9 The Inverse-Nyquist Diagram ........................................................158
11.10 Summary of Experimental Methods ..............................................162
12 the root locus .............................................................................165
12.1 Introduction ..................................................................................165
12.2 Root Locus and Mappings .............................................................165
12.3 A Root Locus Plot .........................................................................169
12.4 Plotting with Poles and Zeroes ......................................................172
12.5 Poles and Polynomials ...................................................................173
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viii ◾ Contents
12.6 Compensators and Other Examples...............................................176
12.7 Conclusions ...................................................................................178
13 fashionable topics in Control .......................................................181
13.1 Introduction ..................................................................................181
13.2 Adaptive Control ...........................................................................182
13.3 Optimal Control ...........................................................................182
13.4 Bang–Bang, Variable Structure, and Fuzzy Control ......................182
13.5 Neural Nets ...................................................................................184
13.6 Heuristic and Genetic Algorithms .................................................184
13.7 Robust Control and H-infinity ......................................................185
13.8 The Describing Function ...............................................................185
13.9 Lyapunov Methods ........................................................................186
13.10 Conclusion ....................................................................................187
14 linking the time and frequency domains ....................................189
14.1 Introduction ..................................................................................189
14.2 State-Space and Transfer Functions ...............................................189
14.3 Deriving the Transfer Function Matrix .........................................190
14.4 Transfer Functions and Time Responses .......................................193
14.5 Filters in Software..........................................................................197
14.6 Software Filters for Data ................................................................199
14.7 State Equations in the Companion Form ......................................201
15 time, frequency, and Convolution ...............................................205
15.1 Delays and the Unit Impulse .........................................................205
15.2 The Convolution Integral ..............................................................207
15.3 Finite Impulse Response (FIR) Filters............................................209
15.4 Correlation ....................................................................................211
15.5 Conclusion ....................................................................................215
16 More about time and State equations ..........................................217
16.1 Introduction ..................................................................................217
16.2 Juggling the Matrices ....................................................................217
16.3 Eigenvectors and Eigenvalues Revisited .........................................218
16.4 Splitting a System into Independent Subsystems ...........................221
16.5 Repeated Roots ..............................................................................225
16.6 Controllability and Observability ..................................................227
17 Practical observers, feedback with dynamics .............................233
17.1 Introduction ..................................................................................233
17.2 The Kalman Filter .........................................................................233
17.3 Reduced-State Observers ...............................................................237
17.4 Control with Added Dynamics .....................................................242
17.5 Conclusion ...................................................................................246
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Contents ◾ ix
18 digital Control in More detail ......................................................247
18.1 Introduction ..................................................................................247
18.2 Finite Differences—The Beta-Operator .........................................247
18.3 Meet the z-Transform ....................................................................251
18.4 Trains of Impulses .........................................................................252
18.5 Some Properties of the z-Transform ...............................................254
18.6 Initial and Final Value Theorems ...................................................256
18.7 Dead-Beat Response ......................................................................257
18.8 Discrete Time Observers ...............................................................259
19 relationship between z- and other transforms ............................267
19.1 Introduction ..................................................................................267
19.2 The Impulse Modulator .................................................................267
19.3 Cascading Transforms ..................................................................268
19.4 Tables of Transforms .....................................................................271
19.5 The Beta and w-Transforms. ..........................................................272
20 design Methods for Computer Control ........................................277
20.1 Introduction ..................................................................................277
20.2 The Digital-to-Analog Convertor (DAC) as Zero Order Hold .......277
20.3 Quantization .................................................................................279
20.4 A Position Control Example, Discrete Time Root Locus ..............280
20.5 Discrete Time Dynamic Control–Assessing Performance .............282
21 errors and noise ...........................................................................289
21.1 Disturbances .................................................................................289
21.2 Practical Design Considerations ....................................................292
21.3 Delays and Sample Rates ...............................................................296
21.4 Conclusion ....................................................................................297
22 optimal Control—nothing but the Best .......................................299
22.1 Introduction: The End Point Problem ............................................299
22.2 Dynamic Programming ................................................................300
22.3 Optimal Control of a Linear System ..............................................305
22.4 Time Optimal Control of a Second Order System ........................306
22.5 Optimal or Suboptimal? ...............................................................308
22.6 Quadratic Cost Functions .............................................................309
22.7 In Conclusion ................................................................................315
Index ...........................................................................................................317
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