Control Theory: A guided tour

Control Theory: A guided tour, 3rd Edition......Page 4
Contents......Page 8
Acknowledgements......Page 20
Foreword......Page 22
The structure, content and purpose of the book......Page 24
Some historical threads in the development of control systems technology......Page 26
1.1 General systems ideas......Page 28
1.2 What is control theory? An initial discussion......Page 29
1.3 What is automatic control?......Page 33
1.4 Some examples of control systems......Page 34
2.1 Initial discussion......Page 38
2.3 Requirements for an automatic control system......Page 39
2.5 Diagrams illustrating and amplifying some of the concepts described so far and showing relationships to a software engineering context......Page 41
3.1 Feedback loops: further discussion......Page 46
3.2 What sorts of control laws are there?......Page 48
3.3 How feedback control works: a practical view......Page 49
3.4 General conditions for the success of feedback control strategies......Page 54
3.5 Alternatives to feedback control......Page 55
4.2 Definition of the Laplace transform......Page 56
4.3 Use of the Laplace transform in control theory......Page 58
4.5 System simplification through block manipulation......Page 59
4.7 Poles and zeros of a transfer function......Page 60
4.8.1 Meaning of pole locations......Page 61
4.9 Pole placement: synthesis of a controller to place the closed loop poles in desirable positions......Page 62
4.10 Moving the poles of a closed loop system to desirable locations: the root locus technique......Page 65
4.11.1 Obtaining a transfer function from a given frequency response curve......Page 66
4.11.2 Obtaining a transfer function from a transient response curve......Page 67
4.12 Determination of transfer functions by cross-correlation......Page 68
5.3 Frequency response of a linear system......Page 74
5.5 Frequency response and stability: an important idea......Page 75
5.7 Practical point: the need for stability margins......Page 77
5.9 Obtaining the frequency response of a system experimentally......Page 78
5.9.1 Obtaining the frequency response of a system experimentally: some practical difficulties......Page 79
5.10 Design based on knowledge of the response of a system to a unit step input......Page 80
5.11 How frequency response is obtained by calculation from a differential equation......Page 81
5.12 Frequency response testing can give a good estimate of a system’s transfer function......Page 83
5.13 Frequency response of a second-order system......Page 84
5.14 Nyquist diagram and Nichols chart......Page 87
6.1 Approaches to mathematical modelling......Page 90
6.3 Modelling a system that exists, based on data obtained by experimentation......Page 91
6.5 Methods/approaches/techniques for parameter estimation......Page 93
6.6 Why modelling is difficult: an important discussion......Page 95
6.9 Regression analysis......Page 96
6.10 Analysis of residuals......Page 97
7.1 What is meant by non-linearity......Page 120
7.2.3 Describing function method (described later in this chapter)......Page 122
7.2.4 Sector bound methods......Page 123
7.3 The describing function method for analysis of control loops containing non-linearities......Page 124
7.4 Linear second-order systems in the state plane......Page 126
7.5 Non-linear second-order systems in the state plane......Page 127
7.6 Process non-linearity: large signal problems......Page 128
7.7 Process non-linearity: small signal problems......Page 129
7.8.1 The motivation for linearisation......Page 130
7.9.1 An initial trivial example......Page 131
7.9.2 Comments......Page 132
7.10 Linearisation about a nominal trajectory: illustration......Page 133
7.11 The derivative as best linear approximation......Page 134
8.1 Stability: initial discussion......Page 138
8.2 Stability for control systems: how it is quantified......Page 140
8.4 Stability margin......Page 143
8.5 Stability tests for non-linear systems......Page 144
8.6 Local and global stability......Page 145
8.7 Lyapunov’s second (direct) method for stability determination......Page 146
8.8 What sets the limits on the control performance?......Page 148
8.9 How robust against changes in the process is a moderately ambitious control loop?......Page 151
8.11 Systems that are difficult to control: unstable systems......Page 153
8.11.1 Cancellation of an unstable pole by a matching zero in the controller......Page 154
8.11.2 Shifting an unstable pole by feedback......Page 155
8.12 Systems that are difficult to control: non-minimum phase systems......Page 162
8.13 Process dead time: a difficult dynamic element in the control loop......Page 166
8.13.1 How to control processes that have significant dead time......Page 167
8.14.1 Sensitivity functions and their interrelation......Page 168
8.14.2 Integral constraints in the time domain......Page 170
8.14.3 Design constraints caused by Bode’s theorem......Page 171
9.2.1 Rudimentary on–off control......Page 176
9.2.2 Introduction to variable structure systems and sliding mode control......Page 178
9.3.1 The three-term controller......Page 182
9.3.3 Illustration of the value of a derivative term to control the degree of damping......Page 183
9.3.4.1 To apply a step to the process that is to be controlled and use the response to calculate the coefficients......Page 185
9.3.4.2 To fit the controller into a closed loop with the process to be controlled and go through a tuning procedure online......Page 191
9.3.4.3 To fit a so-called self-tuning controller into closed loop with the process. After a learning period, the controller will hopefully have chosen its own coefficients......Page 192
9.4 Control systems for batch process......Page 193
9.5 Input shaping......Page 197
9.6 Gain scheduling (to allow a control system to operate successfully when the process to be controlled changes its characteristics over so wide a range that no constant controller can be found that performs adequately)......Page 199
9.6.2 LPV gain scheduling as a step forward from traditional gain scheduling......Page 200
9.7 Converting a user’s requirements into a control specification......Page 202
9.8.1 Methodologies and illustrations......Page 204
References on economic justification of investment in automation......Page 210
10.2 Computers as system components: devices that can change their state only at discrete times......Page 212
10.3 Discrete time algorithms......Page 214
10.4.1 Direct controller synthesis......Page 215
10.4.2 Gain plus compensation approach......Page 217
10.5 Overview: concluding comments, guidelines for algorithm choice and some comments on procedure......Page 220
11.1.2 The state vector......Page 226
11.2 The concept of state......Page 228
11.3 Alternative system descriptions......Page 229
11.4 The mapping representation of Σ......Page 230
11.5.2 Linearisation of the model......Page 232
11.6.1.2 The transition matrix......Page 234
11.6.1.4 Diagonalisation approach......Page 235
11.6.2 The time-varying case......Page 238
11.6.3 The periodically time-varying case......Page 241
11.8 Relation between the transfer-matrix description and the vector–matrix description......Page 242
11.10 System realisation......Page 243
11.11 Stability......Page 244
11.12 Reachability, controllability, observability and reconstructibility for continuous time systems......Page 245
11.13 The unforced state equation in discrete time......Page 246
11.15 Obtaining the L transform equivalent of the state equation......Page 248
11.17 Reachability, controllability, observability and reconstructibility for discrete time systems......Page 249
11.18.1 Introduction......Page 251
11.18.2 The reachability canonical form......Page 253
11.18.3 The controllability canonical form (phase-variable form) (Figure 11.9)......Page 255
11.18.4 The observability canonical form......Page 258
11.18.5 The reconstructibility canonical form......Page 259
11.18.6 State equations for multi-input, multi-output processes......Page 260
11.18.7 The Jordan canonical form......Page 261
11.19 The state-variable approach to control system design......Page 263
11.20.1 Control design based on state-variable feedback......Page 264
11.20.2 Modal control by state feedback......Page 266
12.1 Introduction......Page 276
12.2 A state space view of cascade control......Page 277
12.2.1 Establishing the state space equations by inspection from the block diagram......Page 278
12.2.3 Looking at the performance of the jacketed reactor under control, first with the single loop and then, for comparison purposes, with the inner loop operational......Page 280
12.3 An inverse Nyquist view of the entries in the A matrix of a system representation......Page 281
12.4 Illustration of modes and modal analysis......Page 283
12.5 Moving between different system representations: the relationship between state space and transfer function representations......Page 291
12.5.1 Poles and zeros of state space systems......Page 292
12.6.1 The RGA: in its simplest form, a forecaster of steady state interaction......Page 293
12.6.2 Illustrative example......Page 294
12.6.3 Singular value decomposition......Page 295
13.2 Optimisation: a few ideas that can form building blocks......Page 298
13.2.1 Discussion......Page 304
13.3 Time-optimal control......Page 305
13.4.1 LQR problems with infinite time horizon......Page 313
13.4.1.1 Solution of the LQR problem for three different choices of cost function: open loop stable process (see Figure 13.20, where the dynamic performances are compared)......Page 314
13.4.1.2 Solution of the Riccati equation for three different choices of cost function: open loop unstable process......Page 316
13.4.2 LQR problems with finite time horizon......Page 319
14.2 The separation principle......Page 326
14.4 How a state estimator works: the Kalman filter......Page 327
14.5 The Kalman filter: more detail......Page 328
14.6 Obtaining the optimal gain matrix......Page 330
14.7 Prerequisites for successful application of the Kalman filter in the form shown in Figure 14.4......Page 331
14.8 Discussion of practical points arising......Page 332
14.8.1 Use of the innovation sequence to modify R and Q......Page 333
14.9 Prediction and predictive control......Page 334
15.1 Motivation and introduction......Page 336
15.2.2 Elementary illustration: the effect of choice of p on the nature of the norm......Page 337
15.2.3 Non-elementary aside......Page 338
15.3.1 Guaranteed stability of a feedback loop......Page 339
15.3.2 Robust stability of a closed loop......Page 340
15.4.1 Setting the scene......Page 341
15.4.2 Robust stability......Page 342
15.4.3 Disturbance rejection......Page 344
15.5 Robust control design using a mixed sensitivity H∞ loop shaping approach: worked example......Page 345
15.7.1 Singular values and eigenvalues......Page 349
15.7.2 Eigenvalues of a rectangular matrix A......Page 351
15.7.4 Relations between frequency and time domains......Page 352
15.8.2 Simple illustration of the use of the ν gap metric......Page 353
15.8.3 More about the two metrics δν and bG,D......Page 354
15.8.4 The insight provided by the ν gap metric......Page 355
15.8.6 A discussion on the two metrics δν and bG,D......Page 356
15.9 Using LMI methods in control systems analysis and design......Page 357
15.9.2 Solving an LMI: the feasibility stage: worked example......Page 359
15.9.3 LMI applications to control: simple examples......Page 360
15.9.4 Motivating example: robust pole placement using LMIs......Page 361
15.10 An outline of how H∞ design works and how it spracticality can be usefully extended through μ synthesis......Page 362
15.10.1 The μ-synthesis method......Page 363
15.11 Robustness or adaptivity?......Page 364
16.1 Introduction......Page 370
16.2.1 Motivation......Page 371
16.2.3 Simple properties of a neuron demonstrated in the two-dimensional real plane......Page 372
16.2.5 Neural network training......Page 374
16.2.6 Neural network architectures to represent dynamic processes......Page 376
16.2.6.1 Three ways to make neural networks dynamic......Page 377
16.2.9 Neural nets: summary......Page 379
16.3.1 Introduction and motivation......Page 380
16.3.1.1 A simple illustration of how a crude rule of thumb can be encoded to produce an easily implementable control algorithm......Page 381
16.3.2 Some characteristics of fuzzy logic......Page 382
16.4.2 Artificial GAs......Page 383
16.4.4 GA summary......Page 385
16.4.5 References......Page 386
16.5.2 Controller switching......Page 387
16.6.1 Basic ideas......Page 388
16.6.3 Structural characteristics of an abstract learning system......Page 389
16.7.1 The properties that an intelligent system ought to possess......Page 391
16.8 Where next for AI techniques?......Page 392
17.1 A rapid review of how control technology developed......Page 394
17.2 The development of the control systems discipline: a structure......Page 397
17.3 The mathematical roots of control theory......Page 398
18.1 General remarks on the control literature and on the following references and recommended further reading......Page 400
18.3 Control-oriented software......Page 401
18.6 Methodologies for economic justification of investment in automation......Page 402
18.10 Robust control......Page 403
18.13 Genetic algorithm, genetic programming and other parallel evolutionary search methods......Page 404
18.16 Stochastic aspects of control......Page 405
18.19 Ordinary differential equations......Page 406
18.20 Differential topology/differential geometry/differential algebra......Page 407
18.22 Operator theory and functional analysis applied to linear control......Page 408
18.24 Miscellany......Page 409
18.26 Alphabetical list of references and suggestions for further reading......Page 410
A1 Control of product thickness in a strip rolling mill: from a control point of view this is predominantly a dead time problem......Page 434
A2 The cut-up problem......Page 438
A3 Control of the pressure inside small fuel-fired furnaces......Page 441
A4 Batch control: a brief case history of one process: Oxygen Steelmaking......Page 444
A5 A note on the introduction of novel measurement sensors into control systems......Page 446
Notation......Page 448
Afterword: Visualisation of the evolution of control design approaches: an overview in a single diagram......Page 450
Index......Page 452