Solution Manual for Digital Control Engineering - Analysis and Design

Front Cover......Page 1
Digital Control Engineering: Analysis and Design......Page 2
Copyright page......Page 3
Contents......Page 4
Numerous examples......Page 12
Standard mathematics prerequisites......Page 13
New to this edition......Page 14
Organization of text......Page 15
Supporting material......Page 18
1 Introduction to Digital Control......Page 20
1.2 The structure of a digital control system......Page 21
1.3.1 Closed-loop drug delivery system......Page 22
1.3.3 Control of a robotic manipulator......Page 23
Resources......Page 25
2.1 Analog systems with piecewise constant inputs......Page 28
2.2 Difference equations......Page 30
2.3 The z-transform......Page 31
2.3.1 z-Transforms of standard discrete-time signals......Page 32
2.3.2 Properties of the z-transform......Page 34
Time advance......Page 35
Multiplication by exponential......Page 36
Complex differentiation......Page 37
Long division......Page 38
Partial fraction expansion......Page 39
2.3.4 The final value theorem......Page 47
2.4 Computer-aided design......Page 48
2.5 z-Transform solution of difference equations......Page 50
2.6.1 Convolution summation......Page 51
2.6.2 The convolution theorem......Page 53
2.7 The modified z-transform......Page 56
2.8 Frequency response of discrete-time systems......Page 58
2.8.1 Properties of the frequency response of discrete-time systems......Page 61
2.8.2 MATLAB commands for the discrete-time frequency response......Page 63
2.9 The sampling theorem......Page 64
2.9.1 Selection of the sampling frequency......Page 65
Problems......Page 68
Computer exercises......Page 71
3.1 ADC model......Page 74
3.2 DAC model......Page 75
3.3 The transfer function of the ZOH......Page 76
3.4 Effect of the sampler on the transfer function of a cascade......Page 77
3.5 DAC, analog subsystem, and ADC combination transfer function......Page 80
3.6 Systems with transport lag......Page 88
3.7 The closed-loop transfer function......Page 90
3.8 Analog disturbances in a digital system......Page 93
3.9 Steady-state error and error constants......Page 94
3.9.2 Sampled ramp input......Page 96
3.10.1 MATLAB......Page 98
3.10.2 Simulink......Page 99
Problems......Page 104
Computer exercises......Page 108
4.1 Definitions of stability......Page 110
4.2 Stable z-domain pole locations......Page 112
4.3.1 Asymptotic stability......Page 113
4.3.2 BIBO stability......Page 114
4.3.3 Internal stability......Page 117
4.4.1 MATLAB......Page 120
4.4.2 Routh-Hurwitz criterion......Page 121
4.5 Jury test......Page 123
4.6 Nyquist criterion......Page 128
4.6.1 Phase margin and gain margin......Page 133
Problems......Page 142
Computer exercises......Page 144
5.1 Root locus......Page 146
5.3 Design specifications and the effect of gain variation......Page 151
5.4 Root locus design......Page 154
5.4.1 Proportional control......Page 156
5.4.2 PD control......Page 157
5.4.3 PI control......Page 166
5.4.4 PID control......Page 172
5.5 Empirical tuning of PID controllers......Page 175
Problems......Page 180
Computer exercises......Page 182
6.1 z-Domain root locus......Page 184
6.2 z-Domain digital control system design......Page 187
Observation......Page 189
6.2.1 z-Domain contours......Page 190
6.2.2 Proportional control design in the z-domain......Page 194
6.3 Digital implementation of analog controller design......Page 199
Forward differencing......Page 200
Backward differencing......Page 201
6.3.2 Pole-zero matching......Page 202
6.3.3 Bilinear transformation......Page 205
6.3.4 Empirical digital PID controller tuning......Page 218
6.4 Direct z-domain digital controller design......Page 219
6.5 Frequency response design......Page 224
6.6 Direct control design......Page 232
6.7 Finite settling time design......Page 237
Problems......Page 249
Computer exercises......Page 252
7.1 State variables......Page 254
7.2 State–space representation......Page 257
7.2.2 Linear versus nonlinear state–space equations......Page 259
7.3 Linearization of nonlinear state equations......Page 262
7.4 The solution of linear state–space equations......Page 265
7.4.1 The Leverrier algorithm......Page 270
Leverrier algorithm......Page 271
7.4.2 Sylvester’s expansion......Page 274
7.4.3 The state-transition matrix for a diagonal state matrix......Page 276
Properties of constituent matrices......Page 279
7.4.4 Real form for complex conjugate eigenvalues......Page 281
7.5 The transfer function matrix......Page 283
7.5.1 MATLAB commands......Page 284
7.6 Discrete-time state–space equations......Page 285
7.6.2 Complex conjugate eigenvalues......Page 288
7.7 Solution of discrete-time state–space equations......Page 290
7.7.1 z-Transform solution of discrete-time state equations......Page 291
7.8 z-Transfer function from state–space equations......Page 296
7.9 Similarity transformation......Page 298
7.9.1 Invariance of transfer functions and characteristic equations......Page 301
Problems......Page 302
Computer exercises......Page 308
8 Properties of State–Space Models......Page 312
8.1.1 Asymptotic stability......Page 313
8.1.2 BIBO stability......Page 316
8.2 Controllability and stabilizability......Page 320
8.2.1 MATLAB commands for controllability testing......Page 326
8.2.2 Controllability of systems in normal form......Page 327
8.2.3 Stabilizability......Page 328
8.3 Observability and detectability......Page 332
8.3.1 MATLAB commands......Page 335
8.3.3 Detectability......Page 336
8.4 Poles and zeros of multivariable systems......Page 338
8.4.1 Poles and zeros from the transfer function matrix......Page 339
8.4.2 Zeros from state–space models......Page 342
8.5 State–space realizations......Page 344
Systems with no input differencing......Page 345
Systems with input differencing......Page 347
8.5.2 Controllable form in MATLAB......Page 349
8.5.3 Parallel realization......Page 350
Parallel realization for MIMO systems......Page 353
8.5.4 Observable form......Page 355
8.6 Duality......Page 357
8.7 Hankel realization......Page 358
Resources......Page 362
Problems......Page 363
Computer exercises......Page 368
9.1 State and output feedback......Page 370
9.2 Pole placement......Page 372
9.2.1 Pole placement by transformation to controllable form......Page 375
9.2.2 Pole placement using a matrix polynomial......Page 376
9.2.3 Choice of the closed-loop eigenvalues......Page 378
9.2.5 Pole placement for multi-input systems......Page 383
9.3 Servo problem......Page 386
9.4 Invariance of system zeros......Page 391
9.5.1 Full-order observer......Page 393
9.5.2 Reduced-order observer......Page 396
9.6 Observer state feedback......Page 399
9.6.1 Choice of observer eigenvalues......Page 402
9.7 Pole assignment using transfer functions......Page 408
Problems......Page 412
Computer exercises......Page 416
10.1 Optimization......Page 418
10.1.1 Unconstrained optimization......Page 419
10.1.2 Constrained optimization......Page 421
10.2 Optimal control......Page 423
10.3 The linear quadratic regulator......Page 428
10.3.1 Free final state......Page 429
10.4 Steady-state quadratic regulator......Page 438
10.4.1 Output quadratic regulator......Page 439
10.4.2 MATLAB solution of the steady-state regulator problem......Page 440
10.4.3 Linear quadratic tracking controller......Page 442
10.5 Hamiltonian system......Page 445
10.5.1 Eigenstructure of the Hamiltonian matrix......Page 448
Problems......Page 452
Computer exercises......Page 455
11.1 Discretization of nonlinear systems......Page 458
11.1.1 Extended linearization by input redefinition......Page 459
11.1.2 Extended linearization by input and state redefinition......Page 461
11.1.3 Extended linearization by output differentiation......Page 462
11.1.4 Extended linearization using matching conditions......Page 464
11.2 Nonlinear difference equations......Page 466
11.3 Equilibrium of nonlinear discrete-time systems......Page 467
11.4.1 Lyapunov functions......Page 469
11.4.2 Stability theorems......Page 471
11.4.4 Lyapunov stability of linear systems......Page 473
11.4.5 MATLAB......Page 476
11.4.6 Lyapunov’s linearization method......Page 477
11.4.7 Instability theorems......Page 478
11.4.8 Estimation of the domain of attraction......Page 480
11.5 Stability of analog systems with digital control......Page 482
11.6 State plane analysis......Page 484
11.7.1 Controller design using extended linearization......Page 489
11.7.2 Controller design based on Lyapunov stability theory......Page 492
11.8 Input-output stability and the small gain theorem......Page 493
11.8.1 Absolute stability......Page 500
Problems......Page 504
Computer exercises......Page 508
12.1.1 Software requirements......Page 510
12.1.2 Selection of ADC and DAC......Page 513
12.2.1 Antialiasing filters......Page 514
12.2.2 Effects of quantization errors......Page 517
12.2.3 Phase delay introduced by the ZOH......Page 522
12.3 Controller structure......Page 523
12.4.1 Filtering the derivative action......Page 526
12.4.2 Integrator windup......Page 528
12.4.3 Bumpless transfer between manual and automatic mode......Page 531
12.4.4 Incremental form......Page 534
12.5 Sampling period switching......Page 535
12.5.1 MATLAB commands......Page 538
12.5.2 Dual-rate control......Page 545
Resources......Page 547
Problems......Page 548
Computer exercises......Page 549
APPENDIX I: Table of Laplace and z-Transforms......Page 552
APPENDIX II: Properties of the z-Transform......Page 554
A.1 Matrices......Page 556
A.3.2 Transposition......Page 557
Multiplication by a matrix......Page 559
Properties of determinants......Page 563
A.5 Inverse of a matrix......Page 564
A.6 Trace of a matrix......Page 565
Linearly independent vectors......Page 566
A.8 Eigenvalues and eigenvectors......Page 567
Lower triangular matrix......Page 568
A.9 Partitioned matrix......Page 570
Equivalent norms......Page 572
Frobenius norm......Page 573
A.12 Quadratic forms......Page 574
A.13 Singular value decomposition and pseudoinverses......Page 576
A.14 Matrix differentiation/integration......Page 580
A.15 Kronecker product......Page 582
Resources......Page 583
Index......Page 584