Digital Control Engineering: Analysis and Design

601 Pages·2012·12.068 MB·English

by  M. Sami Fadali, Antonio Visioli

Digital Control
Engineering
Analysis and Design
Second Edition
M. Sami Fadali
Antonio Visioli
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Fadali,M.Sami.
Digitalcontrolengineering:analysisanddesign/M.SamiFadali,AntonioVisioli.(cid:1)
Secondedition.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-0-12-394391-0(hardback)
1.Digitalcontrolsystems. I.Visioli,Antonio.II.Title.
TJ223.M53F332013
629.809(cid:1)dc23
2012021488
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Contents
Preface......................................................................................................................xi
CHAPTER 1 Introduction to Digital Control.............................................1
1.1 Whydigitalcontrol?......................................................................2
1.2 The structure of adigitalcontrol system......................................2
1.3 Examplesof digital control systems.............................................3
1.3.1 Closed-loopdrug delivery system.......................................3
1.3.2 Computer control ofan aircraft turbojetengine.................4
1.3.3 Control ofa robotic manipulator.........................................4
Resources...............................................................................................6
Problems................................................................................................7
CHAPTER 2 Discrete-Time Systems..........................................................9
2.1 Analog systems with piecewise constant inputs...........................9
2.2 Difference equations...................................................................11
2.3 The z-transform...........................................................................12
2.3.1 z-Transformsof standarddiscrete-time signals.................13
2.3.2 Properties ofthe z-transform.............................................15
2.3.3 Inversion of the z-transform...............................................19
2.3.4 The final value theorem.....................................................28
2.4 Computer-aided design...............................................................29
2.5 z-Transform solution ofdifference equations.............................31
2.6 The time response ofa discrete-time system.............................32
2.6.1 Convolution summation.....................................................32
2.6.2 The convolutiontheorem...................................................34
2.7 The modified z-transform............................................................37
2.8 Frequency response ofdiscrete-time systems.............................39
2.8.1 Properties ofthe frequency response of
discrete-time systems.........................................................42
2.8.2 MATLABcommands for the discrete-time
frequency response.............................................................44
2.9 The sampling theorem.................................................................45
2.9.1 Selection ofthe sampling frequency.................................46
Resources.............................................................................................49
Problems..............................................................................................49
Computerexercises.............................................................................52
iii
iv Contents
CHAPTER 3 Modeling of Digital Control Systems..............................55
3.1 ADC model.................................................................................55
3.2 DAC model.................................................................................56
3.3 Thetransferfunction ofthe ZOH..............................................57
3.4 Effect of the sampler on the transfer function of acascade......58
3.5 DAC, analog subsystem,and ADC combination
transfer function..........................................................................61
3.6 Systems with transport lag..........................................................69
3.7 Theclosed-loop transfer function...............................................71
3.8 Analog disturbancesin adigitalsystem.....................................74
3.9 Steady-state error anderrorconstants........................................75
3.9.1 Sampled step input............................................................77
3.9.2 Sampled ramp input..........................................................77
3.10 MATLABcommands.................................................................79
3.10.1 MATLAB........................................................................79
3.10.2 Simulink..........................................................................80
Resources.............................................................................................85
Problems..............................................................................................85
Computer exercises.............................................................................89
CHAPTER 4 Stability of Digital Control Systems................................91
4.1 Definitionsof stability................................................................91
4.2 Stable z-domain pole locations...................................................93
4.3 Stability conditions.....................................................................94
4.3.1 Asymptotic stability..........................................................94
4.3.2 BIBOstability...................................................................95
4.3.3 Internalstability.................................................................98
4.4 Stability determination..............................................................101
4.4.1 MATLAB........................................................................101
4.4.2 Routh-Hurwitzcriterion..................................................102
4.5 Jury test.....................................................................................104
4.6 Nyquist criterion.......................................................................109
4.6.1 Phase margin and gain margin........................................114
Resources...........................................................................................123
Problems............................................................................................123
Computer exercises...........................................................................125
CHAPTER 5 Analog Control System Design........................................127
5.1 Rootlocus.................................................................................127
5.2 Rootlocususing MATLAB.....................................................132
5.3 Design specifications and the effect ofgain variation.............132
Contents v
5.4 Rootlocusdesign......................................................................135
5.4.1 Proportional control.........................................................137
5.4.2 PDcontrol........................................................................138
5.4.3 PIcontrol..........................................................................147
5.4.4 PID control.......................................................................153
5.5 Empirical tuning of PID controllers.........................................156
Resources...........................................................................................161
Problems............................................................................................161
Computerexercises...........................................................................163
CHAPTER 6 Digital Control System Design.........................................165
6.1 z-Domain root locus..................................................................165
6.2 z-Domain digital control system design...................................168
6.2.1 z-Domain contours...........................................................171
6.2.2 Proportional control design inthe z-domain...................175
6.3 Digital implementation of analog controllerdesign.................180
6.3.1 Differencing methods.......................................................181
6.3.2 Pole-zero matching..........................................................183
6.3.3 Bilinear transformation....................................................186
6.3.4 Empirical digitalPID controller tuning...........................199
6.4 Direct z-domain digital controller design.................................200
6.5 Frequency response design.......................................................205
6.6 Direct control design.................................................................213
6.7 Finite settling time design.........................................................218
Resources...........................................................................................230
Problems............................................................................................230
Computerexercises...........................................................................233
CHAPTER 7 State(cid:1)Space Representation..........................................235
7.1 State variables...........................................................................235
7.2 State(cid:1)space representation.......................................................238
7.2.1 State(cid:1)space representation inMATLAB........................240
7.2.2 Linear versus nonlinearstate(cid:1)space equations...............240
7.3 Linearizationofnonlinear state equations................................243
7.4 The solution oflinear state(cid:1)space equations...........................246
7.4.1 The Leverrier algorithm...................................................251
7.4.2 Sylvester’s expansion.......................................................255
7.4.3 The state-transition matrix for a diagonal state matrix...257
7.4.4 Real form for complexconjugate eigenvalues................262
7.5 The transfer functionmatrix.....................................................264
7.5.1 MATLABcommands.......................................................265
vi Contents
7.6 Discrete-time state(cid:1)space equations........................................266
7.6.1 MATLABcommands for discrete-time
state(cid:1)space equations......................................................269
7.6.2 Complex conjugate eigenvalues....................................269
7.7 Solution ofdiscrete-time state(cid:1)space equations......................271
7.7.1 z-Transformsolutionofdiscrete-timestateequations......272
7.8 z-Transferfunction fromstate(cid:1)space equations......................277
7.8.1 z-Transferfunction inMATLAB.....................................279
7.9 Similarity Transformation.........................................................279
7.9.1 Invariance of transfer functions and
characteristic equations....................................................282
Resources...........................................................................................283
Problems............................................................................................283
Computer exercises...........................................................................289
CHAPTER 8 Properties of State(cid:1)Space Models...............................293
8.1 Stabilityofstate(cid:1)space realizations.........................................294
8.1.1 Asymptotic stability.......................................................294
8.1.2 BIBO stability..................................................................297
8.2 Controllability and stabilizability............................................301
8.2.1 MATLABcommands for controllability testing.............307
8.2.2 Controllabilityof systemsinnormal form......................308
8.2.3 Stabilizability.................................................................309
8.3 Observability anddetectability.................................................313
8.3.1 MATLABcommands.......................................................316
8.3.2 Observability ofsystemsin normal form........................317
8.3.3 Detectability...................................................................317
8.4 Poles and zeros ofmultivariable systems.................................319
8.4.1 Poles and zeros from the transfer function matrix..........320
8.4.2 Zeros from state(cid:1)space models....................................323
8.5 State(cid:1)space realizations...........................................................325
8.5.1 Controllable canonicalrealization...................................326
8.5.2 Controllable form in MATLAB.....................................330
8.5.3 Parallel realization.........................................................331
8.5.4 Observable form...............................................................336
8.6 Duality...........................................................................................338
8.7 Hankel realization..................................................................339
Resources...........................................................................................343
Problems............................................................................................344
Computer exercises...........................................................................349
Contents vii
CHAPTER 9 State Feedback Control.....................................................351
9.1 State andoutputfeedback.........................................................351
9.2 Pole placement...........................................................................353
9.2.1 Pole placement by transformationto
controllable form..............................................................356
9.2.2 Pole placement using a matrix polynomial.....................357
9.2.3 Choice of the closed-loop eigenvalues............................359
9.2.4 MATLABcommands for pole placement.......................364
9.2.5 Pole placement for multi-input systems..........................364
9.2.6 Pole placement by output feedback.................................367
9.3 Servo problem............................................................................367
9.4 Invariance of system zeros........................................................372
9.5 State estimation..........................................................................374
9.5.1 Full-order observer...........................................................374
9.5.2 Reduced-order observer...................................................377
9.6 Observer state feedback.............................................................380
9.6.1 Choice of observer eigenvalues.......................................383
9.7 Pole assignment using transfer functions..................................389
Resources...........................................................................................393
Problems............................................................................................393
Computerexercises...........................................................................397
CHAPTER 10 Optimal Control.....................................................................399
10.1 Optimization..............................................................................399
10.1.1 Unconstrained optimization...........................................400
10.1.2 Constrained optimization...............................................402
10.2 Optimal control..........................................................................404
10.3 The linear quadratic regulator...................................................409
10.3.1 Free final state...............................................................410
10.4 Steady-state quadratic regulator................................................419
10.4.1 Output quadratic regulator.............................................420
10.4.2 MATLAB solution of the steady-state
regulator problem.........................................................421
10.4.3 Linear quadratic tracking controller..............................423
10.5 Hamiltonian system...................................................................426
10.5.1 Eigenstructure ofthe Hamiltonian matrix.....................429
Resources.............................................................................................433
Problems..............................................................................................433
Computer exercises..............................................................................436
viii Contents
CHAPTER 11 Elements of Nonlinear Digital Control Systems........439
11.1 Discretizationof nonlinearsystems..........................................439
11.1.1 Extended linearizationbyinput redefinition................440
11.1.2 Extended linearizationbyinput and
state redefinition............................................................442
11.1.3 Extended linearizationbyoutputdifferentiation..........443
11.1.4 Extended linearizationusing matching conditions.......445
11.2 Nonlinear difference equations.................................................447
11.2.1 Logarithmictransformation...........................................448
11.3 Equilibriumofnonlinear discrete-time systems.......................448
11.4 Lyapunov stability theory..........................................................450
11.4.1 Lyapunov functions.......................................................450
11.4.2 Stability theorems..........................................................452
11.4.3 Rate ofconvergence......................................................454
11.4.4 Lyapunov stability oflinear systems............................454
11.4.5 MATLAB.......................................................................457
11.4.6 Lyapunov’slinearization method..................................458
11.4.7 Instabilitytheorems.......................................................459
11.4.8 Estimation ofthe domain of attraction.........................461
11.5 Stability ofanalogsystems with digitalcontrol.......................463
11.6 State planeanalysis...................................................................465
11.7 Discrete-timenonlinearcontroller design.................................470
11.7.1 Controller design using extended linearization.............470
11.7.2 Controller design based on Lyapunov
stability theory...............................................................473
11.8 Input-outputstability andthe smallgain theorem....................474
11.8.1 Absolute stability...........................................................481
Resources.............................................................................................485
Problems..............................................................................................485
Computerexercises..............................................................................489
CHAPTER 12 Practical Issues...................................................................491
12.1 Design of the hardware and software architecture...................491
12.1.1 Software requirements...................................................491
12.1.2 Selection ofADC and DAC..........................................494
12.2 Choice ofthe sampling period..................................................495
12.2.1 Antialiasing filters.........................................................495
12.2.2 Effects ofquantization errors........................................498
12.2.3 Phase delay introduced bythe ZOH.............................503
Contents ix
12.3 Controller structure....................................................................504
12.4 PID control.................................................................................507
12.4.1 Filteringthe derivative action........................................507
12.4.2 Integrator windup...........................................................509
12.4.3 Bumpless transfer between manual
andautomatic mode.......................................................512
12.4.4 Incremental form............................................................515
12.5 Sampling period switching........................................................516
12.5.1 MATLABcommands.....................................................519
12.5.2 Dual-rate control............................................................526
Resources.............................................................................................528
Problems..............................................................................................529
Computer exercises..............................................................................530
APPENDIX I Table of Laplace and z-transforms...............................533
APPENDIX II Properties of the z-transform..........................................535
APPENDIX III Review of Linear Algebra.................................................537
Index......................................................................................................................565