كتاب Modern Control Engineering

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Modern Control Engineering
Fifth Edition
Katsuhiko Ogata

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Contents
Preface ix
Chapter 1 Introduction to Control Systems 1
1–1 Introduction 1
1–2 Examples of Control Systems 4
1–3 Closed-Loop Control Versus Open-Loop Control 7
1–4 Design and Compensation of Control Systems 9
1–5 Outline of the Book 10
Chapter 2 Mathematical Modeling of Control Systems 13
2–1 Introduction 13
2–2 Transfer Function and Impulse-Response Function 15
2–3 Automatic Control Systems 17
2–4 Modeling in State Space 29
2–5 State-Space Representation of Scalar Differential
Equation Systems 35
2–6 Transformation of Mathematical Models with MATLAB 392–7 Linearization of Nonlinear Mathematical Models 43
Example Problems and Solutions 46
Problems 60
Chapter 3 Mathematical Modeling of Mechanical Systems
and Electrical Systems 63
3–1 Introduction 63
3–2 Mathematical Modeling of Mechanical Systems 63
3–3 Mathematical Modeling of Electrical Systems 72
Example Problems and Solutions 86
Problems 97
Chapter 4 Mathematical Modeling of Fluid Systems
and Thermal Systems 100
4–1 Introduction 100
4–2 Liquid-Level Systems 101
4–3 Pneumatic Systems 106
4–4 Hydraulic Systems 123
4–5 Thermal Systems 136
Example Problems and Solutions 140
Problems 152
Chapter 5 Transient and Steady-State Response Analyses 159
5–1 Introduction 159
5–2 First-Order Systems 161
5–3 Second-Order Systems 164
5–4 Higher-Order Systems 179
5–5 Transient-Response Analysis with MATLAB 183
5–6 Routh’s Stability Criterion 212
5–7 Effects of Integral and Derivative Control Actions
on System Performance 218
5–8 Steady-State Errors in Unity-Feedback Control Systems 225
Example Problems and Solutions 231
Problems 263
iv ContentsChapter 6 Control Systems Analysis and Design
by the Root-Locus Method 269
6–1 Introduction 269
6–2 Root-Locus Plots 270
6–3 Plotting Root Loci with MATLAB 290
6–4 Root-Locus Plots of Positive Feedback Systems 303
6–5 Root-Locus Approach to Control-Systems Design 308
6–6 Lead Compensation 311
6–7 Lag Compensation 321
6–8 Lag–Lead Compensation 330
6–9 Parallel Compensation 342
Example Problems and Solutions 347
Problems 394
Chapter 7 Control Systems Analysis and Design by the
Frequency-Response Method 398
7–1 Introduction 398
7–2 Bode Diagrams 403
7–3 Polar Plots 427
7–4 Log-Magnitude-versus-Phase Plots 443
7–5 Nyquist Stability Criterion 445
7–6 Stability Analysis 454
7–7 Relative Stability Analysis 462
7–8 Closed-Loop Frequency Response of Unity-Feedback
Systems 477
7–9 Experimental Determination of Transfer Functions 486
7–10 Control Systems Design by Frequency-Response Approach 491
7–11 Lead Compensation 493
7–12 Lag Compensation 502
7–13 Lag–Lead Compensation 511
Example Problems and Solutions 521
Problems 561
Chapter 8 PID Controllers and Modified PID Controllers 567
8–1 Introduction 567
8–2 Ziegler–Nichols Rules for Tuning PID Controllers 568
Contents v8–3 Design of PID Controllers with Frequency-Response
Approach 577
8–4 Design of PID Controllers with Computational Optimization
Approach 583
8–5 Modifications of PID Control Schemes 590
8–6 Two-Degrees-of-Freedom Control 592
8–7 Zero-Placement Approach to Improve Response
Characteristics 595
Example Problems and Solutions 614
Problems 641
Chapter 9 Control Systems Analysis in State Space 648
9–1 Introduction 648
9–2 State-Space Representations of Transfer-Function
Systems 649
9–3 Transformation of System Models with MATLAB 656
9–4 Solving the Time-Invariant State Equation 660
9–5 Some Useful Results in Vector-Matrix Analysis 668
9–6 Controllability 675
9–7 Observability 682
Example Problems and Solutions 688
Problems 720
Chapter 10 Control Systems Design in State Space 722
10–1 Introduction 722
10–2 Pole Placement 723
10–3 Solving Pole-Placement Problems with MATLAB 735
10–4 Design of Servo Systems 739
10–5 State Observers 751
10–6 Design of Regulator Systems with Observers 778
10–7 Design of Control Systems with Observers 786
10–8 Quadratic Optimal Regulator Systems 793
10–9 Robust Control Systems 806
Example Problems and Solutions 817
Problems 855
vi ContentsAppendix A Laplace Transform Tables 859
Appendix B Partial-Fraction Expansion 867
Appendix C Vector-Matrix Algebra 874
References 882
Index 88
Index
A
Absolute stability, 160
Ackermann’s formula:
for observer gain matrix, 756–57
for pole placement, 730–31
Actuating error, 8
Actuator, 21–22
Adjoint matrix, 876
Air heating system, 150
Aircraft elevator control system, 156
Analytic function, 860
Angle:
of arrival, 286
of departure, 280, 286
Angle condition, 271
Asymptotes:
Bode diagram, 406–07
root loci, 274–75, 284–85
Attenuation, 165
Attitude-rate control system, 386
Automatic controller, 21
Automobile suspension system, 86
Auxiliary polynomial, 216
B
Back emf, 95
constant, 95
Bandwidth, 474, 539
Basic control actions:
integral, 24
on-off, 22
proportional, 24
proportional-plus-derivative, 25
proportional-plus-integral, 24
proportional-plus-integral-plusderivative, 35
two-position, 22–23
Bleed-type relay, 111
Block, 17
Block diagram, 17–18
reduction, 27–28, 48
Bode diagram, 403
error in asymptotic expression of, 403
of first-order factors, 406–07, 409
general procedure for plotting, 413
plotting with MATLAB, 422–25
of quadratic factors, 410–12
of system defined in state space,
426–27
Branch point, 18
Break frequency, 406
Breakaway point, 275–76, 285–86, 351
Break-in point, 276, 281, 285–86, 351
Bridged-T networks, 90, 520
Business system, 5Index 887
C
Canonical forms:
controllable, 649
diagonal, 650
Jordan, 651, 653
observable, 650
Capacitance:
of pressure system, 107–09
of thermal system, 137
of water tank, 103
Cancellation of poles and zeros, 288
Cascaded system, 20
Cascaded transfer function, 20
Cauchy–Riemann conditions, 860–61
Cauchy’s theorem, 526
Cayley–Hamilton theorem, 668, 701
Characteristic equation, 652
Characteristic polynomial, 34
Characteristic roots, 652
Circular root locus, 282
Classical control theory, 2
Classification of control systems, 225
Closed-loop control system, 8
Closed-loop system, 20
Closed-loop frequency response, 477
Closed-loop frequency response curves:
desirable shapes of, 492
undesirable shapes of, 492
Closed-loop transfer function, 19–20
Cofactor, 876
Command compensation, 630
Compensation:
feedback, 308
parallel, 308
series, 308
Compensator:
lag, 323, 503–04
lag–lead, 332–34, 511–13
lead, 312–13, 495–96
Complete observability, 683–84
conditions for, 684–85
in the s plane, 684
Complete output controllablility, 714
Complete state controllability, 676–81
in the s plane, 680–81
Complex-conjugate poles:
cancellation of undesirable, 520
Complex function, 859
Complex impedence, 75
Complex variable, 859
Computational optimization approach to
design PID controller, 583–89
Conditional stability, 299–300, 510–11
Conditionally stable system, 299–300,
458, 510–11
Conduction heat transfer, 137
Conformal mapping, 447, 462–64
Conical water tank system, 152
Constant-gain loci, 302–03
Constant-magnitude loci (M circles),
478–79
Constant phase-angle loci (N circles),
480–81
Constant v
n loci, 296
Constant z lines, 298
Constant z loci, 296
Control actions, 21
Control signal, 3
Controllability, 675–81
matrix, 677
output, 681
Controllable canonical form, 649, 688
Controlled variable, 3
Controller, 22
Convection heat transfer, 137
Conventional control theory, 29
Convolution, integral, 16
Corner frequency, 406
Critically damped system, 167
Cutoff frequency, 474
Cutoff rate, 475
D
Damped natural frequency, 167
Damper, 64, 132
Damping ratio, 165
lines of constant, 296
Dashpot, 64, 132–33
Dead space, 43
Decade, 405
Decibel, 403
Delay time, 169–70
Derivative control action, 118–20, 222
Derivative gain, 84
Derivative time, 25, 61
Detectability, 688
Determinant, 874
Diagonal canonical form, 694
Diagonalization of n*n matrix, 652
Differential amplifier, 78
Differential gap, 23, 24
Differentiating system, 231
Differentiation:
of inverse matrix, 881
of matrix, 880
of product of two matrices, 880
Differentiator:
approximate, 617
Direct transmission matrix, 31
Disturbance, 3, 26
Dominant closed-loop poles, 182
Duality, 754EAe
t
:
computation of, 670–71
Eigenvalue, 652
invariance of, 655
Electromagnetic valve, 23
Electronic controller, 77, 83
Engineering organizational system, 5–6
Equivalent moment of inertia, 234
Equivalent spring constant, 64
Equivalent viscous-friction coefficient,
65, 234
Evans, W. R., 2, 11, 269
Exponential response curve, 162
F
Feedback compensation, 308–09, 342, 519
Feedback control, 3
Feedback control system, 7
Feedback system, 20
Feedforward transfer function, 19
Final value theorem, 866
First-order lag circuit, 80
First-order system, 161–64
unit-impulse response of, 163
unit-ramp response of, 162–63
unit-step response of, 161–62
Flapper, 110
valve, 156
Fluid systems:
mathematical modeling of, 100
Free-body diagram, 69–70
Frequency response, 398
correlation between step response
and, 471–74
lag compensation based on, 502–11
lag–lead compensation based on,
511–17
lead compensation based on, 493–502
Full-order state observer, 752–53
Functional block, 17
G
Gain crossover frequency, 467–69
Gain margin, 464–67
Gas constant, 108
for air, 142
universal, 108
Gear train, 232
system, 232–34
Generalized plant, 813, 815–17
diagram, 810–16, 853–54
H
H infinity control problem, 816
H infinity norm, 6, 808
888 Index
Hazen, 2, 11
High-pass filter, 495
Higher-order systems, 179
transient response of, 180–81
Hurwitz determinants, 252–58
Hurwitz stability criterion, 252–53, 255–58
equivalence of Routh’s stability
criterion and, 255–57
Hydraulic controller:
integral, 130
jet-pipe, 147
proportional, 131
proportional-plus-derivative, 134–35
proportional-plus-integral, 133–34
proportional-plus-integral-plusderivative, 135–36
Hydraulic servo system, 124–25
Hydraulic servomotor, 128, 130, 156
Hydraulic system, 106, 123–39, 149
advantages and disadvantages of, 124
compared with pneumatic system, 106
I
Ideal gas law, 108
Impedance:
approach to obtain transfer function,
75–76
Impulse function, 866
Impulse response, 163, 178–79, 195–97
function, 16–17
Industrial controllers, 22
Initial condition:
response to, 203–11
Initial value theorem, 866
Input filter, 261, 630
Input matrix, 31
Integral control, 220
Integral control action, 24–25, 218
Integral controller, 22
Integral gain, 61
Integral time, 25, 61
Integration of matrix, 880
Inverse Laplace transform:
partial-fraction expansion method for
obtaining, 867–73
Inverse Laplace transformation, 862
Inverse of a matrix:
MATLAB approach to obtain, 879
Inverse polar plot, 461–62, 537–38
Inverted-pendulum system, 68–72, 98
Inverted-pendulum control system,
746–51
Inverting amplifier, 78
I-PD control, 591–92
I-PD-controlled system, 592, 628–29, 643
with feedforward control, 642Index 889
J
Jet-pipe controller, 146–47
Jordan blocks, 679
Jordan canonical form, 651, 695, 706–07
K
Kalman, R. E., 12, 675
Kirchhoff’s current law, 72
Kirchhoff’s loop law, 72
Kirchhoff’s node law, 72
Kirchhoff’s voltage law, 72
L
Lag compensation, 321
Lag compensator, 311, 321, 502
Bode diagram of, 503
design by frequency-response method,
502–11
design by root-locus method, 321, 323
polar plot of, 503
Lag network, 82, 542
Lag–lead compensation, 330, 335, 338,
377, 511–18
Lag–lead compensator:
Bode diagram of, 558
design by frequency-response method,
513–17
design by root-locus method, 331–32,
380–82
electronic, 330–32
polar plot of, 512
Lag–lead network:
electronic, 330–32
mechanical, 366
Lagrange polynomial, 708
Lagrange’s interpolation formula, 708
Laminar-flow resistance, 102
Laplace transform, 862
properties of, 865
table of, 863–64
Lead compensator, 311, 493
Bode diagram of, 494
design by frequency-response method,
493–502
design by root-locus method, 311–18
polar plot of, 494
Lead, lag, and lag–lead compensators:
comparison of, 517–18
Lead network, 542
electronic, 82
mechanical, 365
Lead time, 5
Linear approximation:
of nonlinear mathematical models, 43
Linear system, 14
constant coefficient, 14
Linear time-invariant system, 14, 164
Linear time-varying system, 14
Linearization:
of nonlinear systems, 43
Liquid-level control system, 157
Liquid-level systems, 101, 103–04, 140–41
Log-magnitude curves of quadratic
transfer function, 411
Logarithmic decrement, 237
Logarithmic plot, 403
Log-magnitude versus phase plot, 403,
443–44
LRC circuit, 72–73
M
M circles, 478–79
a family of constant, 479
Magnitude condition, 271
Manipulated variable, 3
Mapping theorem, 448–49
Mathematical model, 13
MATLAB commands:
MATLAB:
obtaining maximum overshoot with,
194
obtaining peak time with, 194
obtaining response to initial condition
with, 266
partial-fraction expansion with,
871–73
plotting Bode diagram with, 422–23
plotting root loci with, 290–91
writing text in diagrams with, 188–89
[A,B,C,D] = tf2ss(num,den), 40, 656,
698
bode(A,B,C,D), 422, 426
bode(A,B,C,D,iu), 426–27
bode(A,B,C,D,iu,w), 422
bode(A,B,C,D,w), 422
bode(num,den), 422
bode(num,den,w), 422, 425, 551
bode(sys), 422
bode(sys,w), 552
c = step(num,den,t), 190
for loop, 243, 249, 584
[Gm,pm,wcp,wcg,] = margin(sys),
468–69
gtext ('text'), 189
impulse(A,B,C,D), 195
impulse(num, den), 195
initial(A,B,C,D,[initial condition],t), 209
inv(A), 879
K = acker(A,B,J), 736
K = lqr(A,B,Q,R), 798
K = place(A,B,J), 736MATLAB commands (Cont.)
K
e = acker(A',C',L)', 773
K
e = acker(Abb,Aab,L)', 773
K
e = place(A',C',L)', 773
K
e = place(Abb',Aab',L)', 773
[K,P,E] = lqr(A,B,Q,R), 798
[K,r] = rlocfind(num,den), 303
logspace(d1,d2), 422
logspace(d1,d2,n), 422–23
lqr(A,B,Q,R), 797
lsim(A,B,C,D,u,t), 201
lsim(num,den,r,t), 201
magdB = 20*log10(mag), 422
[mag,phase,w] = bode(A,B,C,D), 422
[mag,phase,w] = bode(A,B,C,D,iu,w),
422
[mag,phase,w] = bode(A,B,C,D,w),
422
[mag,phase,w] = bode(num,den), 422
[mag,phase,w] = bode(num,den,w),
422, 476
[mag,phase,w] = bode(sys), 422
[mag,phase,w] = bode(sys,w), 476
mesh, 192
mesh(y), 192, 249
mesh(y'), 192, 249
[Mp,k] = max(mag), 476
NaN, 799
[num,den] = feedback(num1,den1,
num2,den2), 20–21
[num,den] = parallel(num1,den1,
num2,den2), 20–21
[num,den] = series(num1,den1,
num2,den2), 20–21
[num,den] = ss2tf(A,B,C,D), 41, 657
[num,den] = ss2tf(A,B,C,D,iu), 41–42,
58, 657
[NUM,den] = ss2tf(A,B,C,D,iu), 59,
659
nyquist(A,B,C,D), 436, 441–42
nyquist(A,B,C,D,iu), 441
nyquist(A,B,C,D,iu,w), 436, 441
nyquist(A,B,C,D,w), 436
nyquist(num,den), 436
nyquist(num, den,w), 436
nyquist(sys), 436
polar(theta,r), 545
printsys(num,den), 20–21, 189
printsys(num,den,'s'), 189
r = abs(z), 544
[r,p,k] = residue(num,den), 239, 871–72
[re,im,w] = nyquist(A,B,C,D), 436
[re,im,w] = nyquist(A,B,C,D,iu,w), 436
[re,im,w] = nyquist(A,B,C,D,w), 436
[re,im,w] = nyquist(num,den), 436
[re,im,w] = nyquist(num,den,w), 436
890 Index
[re,im,w] = nyquist(sys), 436
residue, 867
resonant_frequency = w(k), 476
resonant_peak = 20*log10(Mp), 476
rlocfind, 303
rlocus(A,B,C,D), 295
rlocus(A,B,C,D,K), 290, 295
rlocus(num,den), 290–91
rlocus(num,den,K), 290
sgrid, 297
sortsolution, 584
step(A,B,C,D), 184, 186
step(A,B,C,D,iu), 184
step(num,den), 184
step(num,den,t), 184
step(sys), 184
sys = ss(A,B,C,D), 184
sys = tf(num,den), 184
text, 188
theta = angle(z), 544
w = logspace(d2,d3,100), 425
y = lsim(A,B,C,D,u,t), 201
y = lsim(num,den,r,t), 201
[y, x, t] = impulse(A,B,C,D), 195
[y, x, t] = impulse(A,B,C,D,iu), 195
[y, x, t] = impulse(A,B,C,D,iu,t), 195
[y, x, t] = impulse(num,den), 195
[y, x, t] = impulse(num,den,t), 195
[y, x, t] = step(A,B,C,D,iu), 184
[y, x, t] = step(A,B,C,D,iu,t), 184
[y, x, t] = step(num,den,t), 184, 190
z = re+j*im, 544
End of MATLAB commands
Matrix exponential, 661, 669–674
closed solution for, 663
Matrix Riccati equation, 798, 800
Maximum overshoot:
in unit-impulse response, 179
in unit-step response, 170, 172
versus z curve, 174
Maximum percent overshoot, 170
Maximum phase lead angle, 494, 498
Measuring element, 21
Mechanical lag–lead system, 366
Mechanical lead system, 365
Mechanical vibratory system, 236
Mercury thermometer system, 151
Minimal polynomial, 669, 704–06
Minimum-order observer, 767–77
based controller, 777
Minimum-order state observer, 752
Minimum-phase system, 415–16
Minimum-phase transfer function, 415
Minor, 876
Modern control theory, 7, 29
versus conventional control theory, 29Index 891
Motor torque constant, 95
Motorcycle suspension system, 87
Multiple-loop system, 458–59
N
N circles, 480–81
a family of constant, 481
Newton’s second law, 66
Nichols, 2, 11, 398
Nichols chart, 482–85
Nichols plots, 403
Nonbleed-type relay, 111
Nonhomogeneous state equation:
solution of, 666–67
Noninverting amplifier, 79
Nonlinear mathematical models:
linear approximation of, 43–45
Nonlinear system, 43
Nonminimum-phase systems, 300–01,
415, 417
Nonminimum-phase transfer function,
415, 488
Nonuniqueness:
of a set of state variables, 655
Nozzle-flapper amplifier, 110
Number-decibel conversion line, 404
Nyquist, H., 2, 11, 398
Nyquist path, 545
Nyquist plot, 403, 439–40, 443
of positive-feedback system, 535–37
of system defined in state space, 440–43
Nyquist stability analysis, 454–62
Nyquist stability criterion, 445–54
applied to inverse polar plots, 461–62
O
Observability, 675, 682–88
complete, 683–85
matrix, 653
Observable canonical form, 650, 692
Observation, 752
Observed-state feedback control system,
761
Observer, 753
design of control system with, 786–93
full-order, 753
mathematical model of, 752
minimum-order, 767–73
Observer-based controller:
transfer function of, 761
Observer controller:
in the feedback path of control system,
787, 790–93
in the feedforward path of control
system, 787–90
Observer-controller matrix, 762
Observer-controller transfer function,
761–62
Observer error equation, 753
Observer gain matrix, 755
MATLAB determination of, 773
Octave, 405
Offset, 258
On-off control action, 22–23
On-off controller, 22
One-degree-of-freedom control system,
593
op amps, 78
Open-loop control system, 8
advantages of, 9
disadvantages of, 9
Open-loop frequency response curves:
reshaping of, 493
Open-loop transfer function, 19
Operational amplifier, 78
Operational amplifier circuits, 93–94
for lead or lag compensator:
table of, 85
Optimal regulator problem, 806
Ordinary point, 861
Orthogonality:
of root loci and constant gain loci,
301–02
Output controllability, 681
Output equation, 31
Output matrix, 31
Overdamped system, 168–69
Overlapped spool valve, 146
Overlapped valve, 130
P
Parallel compensation, 308–09, 342–43
Partial-fraction expansion, 867–73
with MATLAB, 871–73
PD control, 373
PD controller, 614–15
Peak time, 170, 172, 193
Performance index, 793
Performance specifications, 9
Phase crossover frequency, 467–69
Phase margin, 464–67
versus z curve, 472
PI controller, 2, 614–15
PI-D control, 590–92
PID control system, 572–77, 583, 587,
617–21, 628–29, 642–43
basic, 590
with input filter, 629
two-degrees-of-freedom, 592–95
PID controller, 567, 577, 614–16, 620, 632
modified, 616
using operational amplifiers, 83–84Pilot valve, 124, 130
PI-PD control, 592
PID-PD control, 592
Plant, 3
Pneumatic actuating valve, 117–18
Pneumatic controllers, 144–45, 154–55
Pneumatic nozzle-flapper amplifier, 110
Pneumatic on-off controller, 115
Pneumatic pressure system, 142
Pneumatic proportional controller, 112–16
force-balance type, 115–16
force-distance type, 112–15
Pneumatic proportional-plus-derivative
controller, 119–20
Pneumatic proportional-plus-integral
control action, 120–22
Pneumatic proportional-plus-integralplus-derivative control action,
122–23
Pneumatic relay, 111
bleed type, 111
nonbleed type, 111
reverse acting, 112
Pneumatic systems, 106–23, 153
compared with hydraulic system, 106
Pneumatic two-position controller, 115
Polar grids, 297
Polar plot, 403, 427–28, 430, 432
Pole: 861
of order n, 861
simple, 861
Pole assignment technique, 723
Pole-placement:
necessary and sufficient conditions for
arbitrary, 725
Pole placement problem, 723–35
solving with MATLAB, 735–36
Positive-feedback system:
Nyquist plot for, 536–37
root loci for, 303–07
Positional servo system, 95–97
Pressure system, 107, 109
Principle of duality, 687
Principle of superposition, 43
Process, 3
Proportional control, 219
Proportional control action, 24
Proportional controller, 22
Proportional gain, 25, 61
Proportional-plus-derivative control:
of second-order system, 224
of system with inertia load, 223
Proportional-plus-derivative control
action, 25
Proportional-plus-derivative controller,
22, 542
892 Index
Proportional-plus-integral control action,
24
Proportional-plus-integral controller, 22,
121, 542
Proportional-plus-integral-plusderivative control action, 25
Proportional-plus-integral-plusderivative controller, 22
Pulse function, 866
Q
Quadratic factor, 410
log-magnitude curves of, 411
phase-angle curves of, 411
Quadratic optimal control problem:
MATLAB solution of, 804
Quadratic optimal regulator system,
793–95
MATLAB design of, 797
R
Ramp response, 197
Rank of matrix, 875
Reduced-matrix Riccati equation, 795–97
Reduced-order observer, 752
Reduced-order state observer, 752
Reference input, 21
Regulator system with observer
controller, 778–86, 789
Relative stability, 160, 217, 462
Residue, 867
Residue theorem, 527
Resistance:
gas-flow, 107
laminar-flow, 101–02
of pressure system, 107, 109
of thermal system, 137
turbulent-flow, 102
Resonant frequency, 430, 470
Resonant peak, 413, 430, 470
versus z curve, 413
Resonant peak magnitude, 413, 470
Response:
to arbitrary input, 201
to initial condition, 203–11
to torque disturbance, 221
Reverse-acting relay, 112
Riccati equation, 795
Rise time, 169–171
obtaining with MATLAB, 193–94
Robust control:
system, 16, 806–17
theory, 2, 7
Robust performance, 7, 807, 812
Robust pole placement, 735
Robust stability, 7, 807, 809Index 893
Root loci:
general rules for constructing, 283–87
for positive-feedback system, 303–07
Root locus, 271
method, 269–70
Routh’s stability criterion, 212–18
S
Schwarz matrix, 268
Second-order system, 164
impulse response of, 178–79
standard form of, 166
step response of, 165–75
transient-response specification of, 171
unit-step response curves of, 169
Sensor, 21
Series compensation, 308–09, 342
Servo system, 95, 164–65
design of, 739–51
with tachometer feedback, 268
with velocity feedback, 175–77
Servomechanism, 2
Set point, 21
Set-point kick, 590
Settling time, 170, 172–73
obtaining with MATLAB, 194
versus z curve, 174
Sign inverter, 79
Simple pole, 861
Singular points, 861
Sinusoidal signal generator, 486
Sinusoidal transfer function, 401
Small gain theorem, 809
Space vehicle control system, 367, 538–39
Speed control system, 4, 148
Spool valve:
linealized mathematical model of, 127
Spring-loaded pendulum system, 98
Spring-mass-dashpot system, 66
Square-law nonlinearity, 43
S-shaped curve, 569
Stability analysis, 454–62
in the complex plane, 182
Stabilizability, 688
Stack controller, 115
Standard second-order system, 189
State, 29
State controllability:
complete, 676, 678, 680
State equation, 31
solution of homogeneous, 660
solution of nonhomogeneous, 666–67
Laplace transform solution of, 663
State-feedback gain matrix, 724
MATLAB approach to determine,
735–36
State matrix, 31
State observation:
necessary and sufficient conditions for,
754–55
State observer, 751–77
design with MATLAB, 773
type 1 servo system with, 746
State observer gain matrix: 755
Ackermann’s formula to obtain, 756–57
direct substitution approach to obtain,
756
transformation approach to obtain, 755
State space, 30
State-space equation, 30
correlation between transfer function
and, 649, 656
solution of, 660
State-space representation:
in canonical forms, 649
of nth order system, 36–39
State-transition matrix, 664
properties of, 665
State variable, 29
State vector, 30
Static acceleration error constant,
228, 421
determination of, 421–22
Static position error constant,
226, 419
Static velocity error constant,
227, 420
Steady-state error, 160, 226
for unit parabolic input, 229
for unit ramp input, 228
in terms of gain K, 230
Steady-state response, 160
Step response, 699–700
of second-order system, 165–69
Summing point, 18
Suspension system:
automobile, 86–87
motorcycle, 87
Sylvester’s interpolation formula, 673,
709–713
System, 3
Sytem types, 419
type 0, 225, 230, 419, 433, 487–88
type 1, 225, 230, 420, 433, 487–88
type 2, 225, 230, 421, 433, 487–88
System response to initial condition:
MATLAB approach to obtain, 203–11
T
Tachometer, 176
feedback, 343
Taylor series expansion, 43–45Temperature control systems, 4–5
Test signals, 159
Text:
writing on the graphic screen, 188
Thermal capacitance, 137
Thermal resistance, 137
Thermal systems, 100,136–39
Thermometer system, 151–52
Three-degrees-of-freedom system, 645
Three-dimensional plot, 192
of unit-step response curves with
MATLAB, 191–93
Traffic control system, 8
Transfer function, 15
of cascaded elements, 73–74
of cascaded systems, 20
closed-loop, 20
of closed-loop system, 20
experimental determination of, 489–90
expression in terms of A, B, C, and D, 34
of feedback system, 19
feedforward, 19
of minimum-order observer-based
controller, 777
of nonloading cascaded elements,
77
observer-controller, 762, 780–82
open-loop, 19
of parallel systems, 20
sinusoidal, 401
Transfer matrix, 35
Transformation:
from state space to transfer function,
41–42, 657
from transfer function to state space,
40–41, 656
Transient response, 160
analysis with MATLAB, 183–211
of higher-order system, 180
specifications, 169, 171
Transport lag, 417
phase angle characteristics of, 417
Turbulent-flow resistance, 102
Two-degrees-of-freedom control system,
593–95, 599–614, 636–41, 646–47
Two-position control action, 22–23
Two-position controller, 22
Type 0 system, 225, 230, 488
log-magnitude curve for, 419, 488
polar plot of, 433
Type 1 servo system:
design of, 743–51
pole-placement design of, 739–46
Type 1 system, 420
log-magnitude curve for, 420, 488
polar plot of, 433
894 Index
Type 2 system, 421
log-magnitude curve for, 421, 488
polar plot of, 433
U
Uncontrollable system, 681
Undamped natural frequency, 165
Underdamped system, 166–67
Underlapped spool valve, 146
Unit acceleration input, 247
Unit-impulse response:
of first-order system, 163
of second-order system, 178
Unit-impulse response curves:
a family of, 178
obtained by use of MATLAB, 196–97
Unit-ramp response:
of first-order system, 162–63
of second-order system, 197–200
of system defined in state space,
199–200
Unit-step response:
of first-order system, 161
of second-order system, 163, 167, 169
Universal gas constant, 108
Unstructured uncertainty:
additive, 852–53
multiplicative, 809
system with, 809
V
Valve:
overlapped, 130
underlapped, 130
zero-lapped, 130
Valve coefficient, 127
Vectors:
linear dependence of, 674
linear independence of, 674
Velocity error, 227
Velocity feedback, 176, 343, 519
W
Watt’s speed governor, 4
Weighting function, 17
Z
Zero, 861
of order m, 862
Zero-lapped valve, 130
Zero placement, 595, 597, 612
approach to improve response characteristics, 595–97
Ziegler–Nichols tuning rules, 11, 568–77
first method, 569–70
second method, 570–71

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كتاب Modern Control Engineering - صفحة 2 761752

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Modern Control Engineering
Fifth Edition
Katsuhiko Ogata

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Contents
Preface ix
Chapter 1 Introduction to Control Systems 1
1–1 Introduction 1
1–2 Examples of Control Systems 4
1–3 Closed-Loop Control Versus Open-Loop Control 7
1–4 Design and Compensation of Control Systems 9
1–5 Outline of the Book 10
Chapter 2 Mathematical Modeling of Control Systems 13
2–1 Introduction 13
2–2 Transfer Function and Impulse-Response Function 15
2–3 Automatic Control Systems 17
2–4 Modeling in State Space 29
2–5 State-Space Representation of Scalar Differential
Equation Systems 35
2–6 Transformation of Mathematical Models with MATLAB 392–7 Linearization of Nonlinear Mathematical Models 43
Example Problems and Solutions 46
Problems 60
Chapter 3 Mathematical Modeling of Mechanical Systems
and Electrical Systems 63
3–1 Introduction 63
3–2 Mathematical Modeling of Mechanical Systems 63
3–3 Mathematical Modeling of Electrical Systems 72
Example Problems and Solutions 86
Problems 97
Chapter 4 Mathematical Modeling of Fluid Systems
and Thermal Systems 100
4–1 Introduction 100
4–2 Liquid-Level Systems 101
4–3 Pneumatic Systems 106
4–4 Hydraulic Systems 123
4–5 Thermal Systems 136
Example Problems and Solutions 140
Problems 152
Chapter 5 Transient and Steady-State Response Analyses 159
5–1 Introduction 159
5–2 First-Order Systems 161
5–3 Second-Order Systems 164
5–4 Higher-Order Systems 179
5–5 Transient-Response Analysis with MATLAB 183
5–6 Routh’s Stability Criterion 212
5–7 Effects of Integral and Derivative Control Actions
on System Performance 218
5–8 Steady-State Errors in Unity-Feedback Control Systems 225
Example Problems and Solutions 231
Problems 263
iv ContentsChapter 6 Control Systems Analysis and Design
by the Root-Locus Method 269
6–1 Introduction 269
6–2 Root-Locus Plots 270
6–3 Plotting Root Loci with MATLAB 290
6–4 Root-Locus Plots of Positive Feedback Systems 303
6–5 Root-Locus Approach to Control-Systems Design 308
6–6 Lead Compensation 311
6–7 Lag Compensation 321
6–8 Lag–Lead Compensation 330
6–9 Parallel Compensation 342
Example Problems and Solutions 347
Problems 394
Chapter 7 Control Systems Analysis and Design by the
Frequency-Response Method 398
7–1 Introduction 398
7–2 Bode Diagrams 403
7–3 Polar Plots 427
7–4 Log-Magnitude-versus-Phase Plots 443
7–5 Nyquist Stability Criterion 445
7–6 Stability Analysis 454
7–7 Relative Stability Analysis 462
7–8 Closed-Loop Frequency Response of Unity-Feedback
Systems 477
7–9 Experimental Determination of Transfer Functions 486
7–10 Control Systems Design by Frequency-Response Approach 491
7–11 Lead Compensation 493
7–12 Lag Compensation 502
7–13 Lag–Lead Compensation 511
Example Problems and Solutions 521
Problems 561
Chapter 8 PID Controllers and Modified PID Controllers 567
8–1 Introduction 567
8–2 Ziegler–Nichols Rules for Tuning PID Controllers 568
Contents v8–3 Design of PID Controllers with Frequency-Response
Approach 577
8–4 Design of PID Controllers with Computational Optimization
Approach 583
8–5 Modifications of PID Control Schemes 590
8–6 Two-Degrees-of-Freedom Control 592
8–7 Zero-Placement Approach to Improve Response
Characteristics 595
Example Problems and Solutions 614
Problems 641
Chapter 9 Control Systems Analysis in State Space 648
9–1 Introduction 648
9–2 State-Space Representations of Transfer-Function
Systems 649
9–3 Transformation of System Models with MATLAB 656
9–4 Solving the Time-Invariant State Equation 660
9–5 Some Useful Results in Vector-Matrix Analysis 668
9–6 Controllability 675
9–7 Observability 682
Example Problems and Solutions 688
Problems 720
Chapter 10 Control Systems Design in State Space 722
10–1 Introduction 722
10–2 Pole Placement 723
10–3 Solving Pole-Placement Problems with MATLAB 735
10–4 Design of Servo Systems 739
10–5 State Observers 751
10–6 Design of Regulator Systems with Observers 778
10–7 Design of Control Systems with Observers 786
10–8 Quadratic Optimal Regulator Systems 793
10–9 Robust Control Systems 806
Example Problems and Solutions 817
Problems 855
vi ContentsAppendix A Laplace Transform Tables 859
Appendix B Partial-Fraction Expansion 867
Appendix C Vector-Matrix Algebra 874
References 882
Index 88
Index
A
Absolute stability, 160
Ackermann’s formula:
for observer gain matrix, 756–57
for pole placement, 730–31
Actuating error, 8
Actuator, 21–22
Adjoint matrix, 876
Air heating system, 150
Aircraft elevator control system, 156
Analytic function, 860
Angle:
of arrival, 286
of departure, 280, 286
Angle condition, 271
Asymptotes:
Bode diagram, 406–07
root loci, 274–75, 284–85
Attenuation, 165
Attitude-rate control system, 386
Automatic controller, 21
Automobile suspension system, 86
Auxiliary polynomial, 216
B
Back emf, 95
constant, 95
Bandwidth, 474, 539
Basic control actions:
integral, 24
on-off, 22
proportional, 24
proportional-plus-derivative, 25
proportional-plus-integral, 24
proportional-plus-integral-plusderivative, 35
two-position, 22–23
Bleed-type relay, 111
Block, 17
Block diagram, 17–18
reduction, 27–28, 48
Bode diagram, 403
error in asymptotic expression of, 403
of first-order factors, 406–07, 409
general procedure for plotting, 413
plotting with MATLAB, 422–25
of quadratic factors, 410–12
of system defined in state space,
426–27
Branch point, 18
Break frequency, 406
Breakaway point, 275–76, 285–86, 351
Break-in point, 276, 281, 285–86, 351
Bridged-T networks, 90, 520
Business system, 5Index 887
C
Canonical forms:
controllable, 649
diagonal, 650
Jordan, 651, 653
observable, 650
Capacitance:
of pressure system, 107–09
of thermal system, 137
of water tank, 103
Cancellation of poles and zeros, 288
Cascaded system, 20
Cascaded transfer function, 20
Cauchy–Riemann conditions, 860–61
Cauchy’s theorem, 526
Cayley–Hamilton theorem, 668, 701
Characteristic equation, 652
Characteristic polynomial, 34
Characteristic roots, 652
Circular root locus, 282
Classical control theory, 2
Classification of control systems, 225
Closed-loop control system, 8
Closed-loop system, 20
Closed-loop frequency response, 477
Closed-loop frequency response curves:
desirable shapes of, 492
undesirable shapes of, 492
Closed-loop transfer function, 19–20
Cofactor, 876
Command compensation, 630
Compensation:
feedback, 308
parallel, 308
series, 308
Compensator:
lag, 323, 503–04
lag–lead, 332–34, 511–13
lead, 312–13, 495–96
Complete observability, 683–84
conditions for, 684–85
in the s plane, 684
Complete output controllablility, 714
Complete state controllability, 676–81
in the s plane, 680–81
Complex-conjugate poles:
cancellation of undesirable, 520
Complex function, 859
Complex impedence, 75
Complex variable, 859
Computational optimization approach to
design PID controller, 583–89
Conditional stability, 299–300, 510–11
Conditionally stable system, 299–300,
458, 510–11
Conduction heat transfer, 137
Conformal mapping, 447, 462–64
Conical water tank system, 152
Constant-gain loci, 302–03
Constant-magnitude loci (M circles),
478–79
Constant phase-angle loci (N circles),
480–81
Constant v
n loci, 296
Constant z lines, 298
Constant z loci, 296
Control actions, 21
Control signal, 3
Controllability, 675–81
matrix, 677
output, 681
Controllable canonical form, 649, 688
Controlled variable, 3
Controller, 22
Convection heat transfer, 137
Conventional control theory, 29
Convolution, integral, 16
Corner frequency, 406
Critically damped system, 167
Cutoff frequency, 474
Cutoff rate, 475
D
Damped natural frequency, 167
Damper, 64, 132
Damping ratio, 165
lines of constant, 296
Dashpot, 64, 132–33
Dead space, 43
Decade, 405
Decibel, 403
Delay time, 169–70
Derivative control action, 118–20, 222
Derivative gain, 84
Derivative time, 25, 61
Detectability, 688
Determinant, 874
Diagonal canonical form, 694
Diagonalization of n*n matrix, 652
Differential amplifier, 78
Differential gap, 23, 24
Differentiating system, 231
Differentiation:
of inverse matrix, 881
of matrix, 880
of product of two matrices, 880
Differentiator:
approximate, 617
Direct transmission matrix, 31
Disturbance, 3, 26
Dominant closed-loop poles, 182
Duality, 754EAe
t
:
computation of, 670–71
Eigenvalue, 652
invariance of, 655
Electromagnetic valve, 23
Electronic controller, 77, 83
Engineering organizational system, 5–6
Equivalent moment of inertia, 234
Equivalent spring constant, 64
Equivalent viscous-friction coefficient,
65, 234
Evans, W. R., 2, 11, 269
Exponential response curve, 162
F
Feedback compensation, 308–09, 342, 519
Feedback control, 3
Feedback control system, 7
Feedback system, 20
Feedforward transfer function, 19
Final value theorem, 866
First-order lag circuit, 80
First-order system, 161–64
unit-impulse response of, 163
unit-ramp response of, 162–63
unit-step response of, 161–62
Flapper, 110
valve, 156
Fluid systems:
mathematical modeling of, 100
Free-body diagram, 69–70
Frequency response, 398
correlation between step response
and, 471–74
lag compensation based on, 502–11
lag–lead compensation based on,
511–17
lead compensation based on, 493–502
Full-order state observer, 752–53
Functional block, 17
G
Gain crossover frequency, 467–69
Gain margin, 464–67
Gas constant, 108
for air, 142
universal, 108
Gear train, 232
system, 232–34
Generalized plant, 813, 815–17
diagram, 810–16, 853–54
H
H infinity control problem, 816
H infinity norm, 6, 808
888 Index
Hazen, 2, 11
High-pass filter, 495
Higher-order systems, 179
transient response of, 180–81
Hurwitz determinants, 252–58
Hurwitz stability criterion, 252–53, 255–58
equivalence of Routh’s stability
criterion and, 255–57
Hydraulic controller:
integral, 130
jet-pipe, 147
proportional, 131
proportional-plus-derivative, 134–35
proportional-plus-integral, 133–34
proportional-plus-integral-plusderivative, 135–36
Hydraulic servo system, 124–25
Hydraulic servomotor, 128, 130, 156
Hydraulic system, 106, 123–39, 149
advantages and disadvantages of, 124
compared with pneumatic system, 106
I
Ideal gas law, 108
Impedance:
approach to obtain transfer function,
75–76
Impulse function, 866
Impulse response, 163, 178–79, 195–97
function, 16–17
Industrial controllers, 22
Initial condition:
response to, 203–11
Initial value theorem, 866
Input filter, 261, 630
Input matrix, 31
Integral control, 220
Integral control action, 24–25, 218
Integral controller, 22
Integral gain, 61
Integral time, 25, 61
Integration of matrix, 880
Inverse Laplace transform:
partial-fraction expansion method for
obtaining, 867–73
Inverse Laplace transformation, 862
Inverse of a matrix:
MATLAB approach to obtain, 879
Inverse polar plot, 461–62, 537–38
Inverted-pendulum system, 68–72, 98
Inverted-pendulum control system,
746–51
Inverting amplifier, 78
I-PD control, 591–92
I-PD-controlled system, 592, 628–29, 643
with feedforward control, 642Index 889
J
Jet-pipe controller, 146–47
Jordan blocks, 679
Jordan canonical form, 651, 695, 706–07
K
Kalman, R. E., 12, 675
Kirchhoff’s current law, 72
Kirchhoff’s loop law, 72
Kirchhoff’s node law, 72
Kirchhoff’s voltage law, 72
L
Lag compensation, 321
Lag compensator, 311, 321, 502
Bode diagram of, 503
design by frequency-response method,
502–11
design by root-locus method, 321, 323
polar plot of, 503
Lag network, 82, 542
Lag–lead compensation, 330, 335, 338,
377, 511–18
Lag–lead compensator:
Bode diagram of, 558
design by frequency-response method,
513–17
design by root-locus method, 331–32,
380–82
electronic, 330–32
polar plot of, 512
Lag–lead network:
electronic, 330–32
mechanical, 366
Lagrange polynomial, 708
Lagrange’s interpolation formula, 708
Laminar-flow resistance, 102
Laplace transform, 862
properties of, 865
table of, 863–64
Lead compensator, 311, 493
Bode diagram of, 494
design by frequency-response method,
493–502
design by root-locus method, 311–18
polar plot of, 494
Lead, lag, and lag–lead compensators:
comparison of, 517–18
Lead network, 542
electronic, 82
mechanical, 365
Lead time, 5
Linear approximation:
of nonlinear mathematical models, 43
Linear system, 14
constant coefficient, 14
Linear time-invariant system, 14, 164
Linear time-varying system, 14
Linearization:
of nonlinear systems, 43
Liquid-level control system, 157
Liquid-level systems, 101, 103–04, 140–41
Log-magnitude curves of quadratic
transfer function, 411
Logarithmic decrement, 237
Logarithmic plot, 403
Log-magnitude versus phase plot, 403,
443–44
LRC circuit, 72–73
M
M circles, 478–79
a family of constant, 479
Magnitude condition, 271
Manipulated variable, 3
Mapping theorem, 448–49
Mathematical model, 13
MATLAB commands:
MATLAB:
obtaining maximum overshoot with,
194
obtaining peak time with, 194
obtaining response to initial condition
with, 266
partial-fraction expansion with,
871–73
plotting Bode diagram with, 422–23
plotting root loci with, 290–91
writing text in diagrams with, 188–89
[A,B,C,D] = tf2ss(num,den), 40, 656,
698
bode(A,B,C,D), 422, 426
bode(A,B,C,D,iu), 426–27
bode(A,B,C,D,iu,w), 422
bode(A,B,C,D,w), 422
bode(num,den), 422
bode(num,den,w), 422, 425, 551
bode(sys), 422
bode(sys,w), 552
c = step(num,den,t), 190
for loop, 243, 249, 584
[Gm,pm,wcp,wcg,] = margin(sys),
468–69
gtext ('text'), 189
impulse(A,B,C,D), 195
impulse(num, den), 195
initial(A,B,C,D,[initial condition],t), 209
inv(A), 879
K = acker(A,B,J), 736
K = lqr(A,B,Q,R), 798
K = place(A,B,J), 736MATLAB commands (Cont.)
K
e = acker(A',C',L)', 773
K
e = acker(Abb,Aab,L)', 773
K
e = place(A',C',L)', 773
K
e = place(Abb',Aab',L)', 773
[K,P,E] = lqr(A,B,Q,R), 798
[K,r] = rlocfind(num,den), 303
logspace(d1,d2), 422
logspace(d1,d2,n), 422–23
lqr(A,B,Q,R), 797
lsim(A,B,C,D,u,t), 201
lsim(num,den,r,t), 201
magdB = 20*log10(mag), 422
[mag,phase,w] = bode(A,B,C,D), 422
[mag,phase,w] = bode(A,B,C,D,iu,w),
422
[mag,phase,w] = bode(A,B,C,D,w),
422
[mag,phase,w] = bode(num,den), 422
[mag,phase,w] = bode(num,den,w),
422, 476
[mag,phase,w] = bode(sys), 422
[mag,phase,w] = bode(sys,w), 476
mesh, 192
mesh(y), 192, 249
mesh(y'), 192, 249
[Mp,k] = max(mag), 476
NaN, 799
[num,den] = feedback(num1,den1,
num2,den2), 20–21
[num,den] = parallel(num1,den1,
num2,den2), 20–21
[num,den] = series(num1,den1,
num2,den2), 20–21
[num,den] = ss2tf(A,B,C,D), 41, 657
[num,den] = ss2tf(A,B,C,D,iu), 41–42,
58, 657
[NUM,den] = ss2tf(A,B,C,D,iu), 59,
659
nyquist(A,B,C,D), 436, 441–42
nyquist(A,B,C,D,iu), 441
nyquist(A,B,C,D,iu,w), 436, 441
nyquist(A,B,C,D,w), 436
nyquist(num,den), 436
nyquist(num, den,w), 436
nyquist(sys), 436
polar(theta,r), 545
printsys(num,den), 20–21, 189
printsys(num,den,'s'), 189
r = abs(z), 544
[r,p,k] = residue(num,den), 239, 871–72
[re,im,w] = nyquist(A,B,C,D), 436
[re,im,w] = nyquist(A,B,C,D,iu,w), 436
[re,im,w] = nyquist(A,B,C,D,w), 436
[re,im,w] = nyquist(num,den), 436
[re,im,w] = nyquist(num,den,w), 436
890 Index
[re,im,w] = nyquist(sys), 436
residue, 867
resonant_frequency = w(k), 476
resonant_peak = 20*log10(Mp), 476
rlocfind, 303
rlocus(A,B,C,D), 295
rlocus(A,B,C,D,K), 290, 295
rlocus(num,den), 290–91
rlocus(num,den,K), 290
sgrid, 297
sortsolution, 584
step(A,B,C,D), 184, 186
step(A,B,C,D,iu), 184
step(num,den), 184
step(num,den,t), 184
step(sys), 184
sys = ss(A,B,C,D), 184
sys = tf(num,den), 184
text, 188
theta = angle(z), 544
w = logspace(d2,d3,100), 425
y = lsim(A,B,C,D,u,t), 201
y = lsim(num,den,r,t), 201
[y, x, t] = impulse(A,B,C,D), 195
[y, x, t] = impulse(A,B,C,D,iu), 195
[y, x, t] = impulse(A,B,C,D,iu,t), 195
[y, x, t] = impulse(num,den), 195
[y, x, t] = impulse(num,den,t), 195
[y, x, t] = step(A,B,C,D,iu), 184
[y, x, t] = step(A,B,C,D,iu,t), 184
[y, x, t] = step(num,den,t), 184, 190
z = re+j*im, 544
End of MATLAB commands
Matrix exponential, 661, 669–674
closed solution for, 663
Matrix Riccati equation, 798, 800
Maximum overshoot:
in unit-impulse response, 179
in unit-step response, 170, 172
versus z curve, 174
Maximum percent overshoot, 170
Maximum phase lead angle, 494, 498
Measuring element, 21
Mechanical lag–lead system, 366
Mechanical lead system, 365
Mechanical vibratory system, 236
Mercury thermometer system, 151
Minimal polynomial, 669, 704–06
Minimum-order observer, 767–77
based controller, 777
Minimum-order state observer, 752
Minimum-phase system, 415–16
Minimum-phase transfer function, 415
Minor, 876
Modern control theory, 7, 29
versus conventional control theory, 29Index 891
Motor torque constant, 95
Motorcycle suspension system, 87
Multiple-loop system, 458–59
N
N circles, 480–81
a family of constant, 481
Newton’s second law, 66
Nichols, 2, 11, 398
Nichols chart, 482–85
Nichols plots, 403
Nonbleed-type relay, 111
Nonhomogeneous state equation:
solution of, 666–67
Noninverting amplifier, 79
Nonlinear mathematical models:
linear approximation of, 43–45
Nonlinear system, 43
Nonminimum-phase systems, 300–01,
415, 417
Nonminimum-phase transfer function,
415, 488
Nonuniqueness:
of a set of state variables, 655
Nozzle-flapper amplifier, 110
Number-decibel conversion line, 404
Nyquist, H., 2, 11, 398
Nyquist path, 545
Nyquist plot, 403, 439–40, 443
of positive-feedback system, 535–37
of system defined in state space, 440–43
Nyquist stability analysis, 454–62
Nyquist stability criterion, 445–54
applied to inverse polar plots, 461–62
O
Observability, 675, 682–88
complete, 683–85
matrix, 653
Observable canonical form, 650, 692
Observation, 752
Observed-state feedback control system,
761
Observer, 753
design of control system with, 786–93
full-order, 753
mathematical model of, 752
minimum-order, 767–73
Observer-based controller:
transfer function of, 761
Observer controller:
in the feedback path of control system,
787, 790–93
in the feedforward path of control
system, 787–90
Observer-controller matrix, 762
Observer-controller transfer function,
761–62
Observer error equation, 753
Observer gain matrix, 755
MATLAB determination of, 773
Octave, 405
Offset, 258
On-off control action, 22–23
On-off controller, 22
One-degree-of-freedom control system,
593
op amps, 78
Open-loop control system, 8
advantages of, 9
disadvantages of, 9
Open-loop frequency response curves:
reshaping of, 493
Open-loop transfer function, 19
Operational amplifier, 78
Operational amplifier circuits, 93–94
for lead or lag compensator:
table of, 85
Optimal regulator problem, 806
Ordinary point, 861
Orthogonality:
of root loci and constant gain loci,
301–02
Output controllability, 681
Output equation, 31
Output matrix, 31
Overdamped system, 168–69
Overlapped spool valve, 146
Overlapped valve, 130
P
Parallel compensation, 308–09, 342–43
Partial-fraction expansion, 867–73
with MATLAB, 871–73
PD control, 373
PD controller, 614–15
Peak time, 170, 172, 193
Performance index, 793
Performance specifications, 9
Phase crossover frequency, 467–69
Phase margin, 464–67
versus z curve, 472
PI controller, 2, 614–15
PI-D control, 590–92
PID control system, 572–77, 583, 587,
617–21, 628–29, 642–43
basic, 590
with input filter, 629
two-degrees-of-freedom, 592–95
PID controller, 567, 577, 614–16, 620, 632
modified, 616
using operational amplifiers, 83–84Pilot valve, 124, 130
PI-PD control, 592
PID-PD control, 592
Plant, 3
Pneumatic actuating valve, 117–18
Pneumatic controllers, 144–45, 154–55
Pneumatic nozzle-flapper amplifier, 110
Pneumatic on-off controller, 115
Pneumatic pressure system, 142
Pneumatic proportional controller, 112–16
force-balance type, 115–16
force-distance type, 112–15
Pneumatic proportional-plus-derivative
controller, 119–20
Pneumatic proportional-plus-integral
control action, 120–22
Pneumatic proportional-plus-integralplus-derivative control action,
122–23
Pneumatic relay, 111
bleed type, 111
nonbleed type, 111
reverse acting, 112
Pneumatic systems, 106–23, 153
compared with hydraulic system, 106
Pneumatic two-position controller, 115
Polar grids, 297
Polar plot, 403, 427–28, 430, 432
Pole: 861
of order n, 861
simple, 861
Pole assignment technique, 723
Pole-placement:
necessary and sufficient conditions for
arbitrary, 725
Pole placement problem, 723–35
solving with MATLAB, 735–36
Positive-feedback system:
Nyquist plot for, 536–37
root loci for, 303–07
Positional servo system, 95–97
Pressure system, 107, 109
Principle of duality, 687
Principle of superposition, 43
Process, 3
Proportional control, 219
Proportional control action, 24
Proportional controller, 22
Proportional gain, 25, 61
Proportional-plus-derivative control:
of second-order system, 224
of system with inertia load, 223
Proportional-plus-derivative control
action, 25
Proportional-plus-derivative controller,
22, 542
892 Index
Proportional-plus-integral control action,
24
Proportional-plus-integral controller, 22,
121, 542
Proportional-plus-integral-plusderivative control action, 25
Proportional-plus-integral-plusderivative controller, 22
Pulse function, 866
Q
Quadratic factor, 410
log-magnitude curves of, 411
phase-angle curves of, 411
Quadratic optimal control problem:
MATLAB solution of, 804
Quadratic optimal regulator system,
793–95
MATLAB design of, 797
R
Ramp response, 197
Rank of matrix, 875
Reduced-matrix Riccati equation, 795–97
Reduced-order observer, 752
Reduced-order state observer, 752
Reference input, 21
Regulator system with observer
controller, 778–86, 789
Relative stability, 160, 217, 462
Residue, 867
Residue theorem, 527
Resistance:
gas-flow, 107
laminar-flow, 101–02
of pressure system, 107, 109
of thermal system, 137
turbulent-flow, 102
Resonant frequency, 430, 470
Resonant peak, 413, 430, 470
versus z curve, 413
Resonant peak magnitude, 413, 470
Response:
to arbitrary input, 201
to initial condition, 203–11
to torque disturbance, 221
Reverse-acting relay, 112
Riccati equation, 795
Rise time, 169–171
obtaining with MATLAB, 193–94
Robust control:
system, 16, 806–17
theory, 2, 7
Robust performance, 7, 807, 812
Robust pole placement, 735
Robust stability, 7, 807, 809Index 893
Root loci:
general rules for constructing, 283–87
for positive-feedback system, 303–07
Root locus, 271
method, 269–70
Routh’s stability criterion, 212–18
S
Schwarz matrix, 268
Second-order system, 164
impulse response of, 178–79
standard form of, 166
step response of, 165–75
transient-response specification of, 171
unit-step response curves of, 169
Sensor, 21
Series compensation, 308–09, 342
Servo system, 95, 164–65
design of, 739–51
with tachometer feedback, 268
with velocity feedback, 175–77
Servomechanism, 2
Set point, 21
Set-point kick, 590
Settling time, 170, 172–73
obtaining with MATLAB, 194
versus z curve, 174
Sign inverter, 79
Simple pole, 861
Singular points, 861
Sinusoidal signal generator, 486
Sinusoidal transfer function, 401
Small gain theorem, 809
Space vehicle control system, 367, 538–39
Speed control system, 4, 148
Spool valve:
linealized mathematical model of, 127
Spring-loaded pendulum system, 98
Spring-mass-dashpot system, 66
Square-law nonlinearity, 43
S-shaped curve, 569
Stability analysis, 454–62
in the complex plane, 182
Stabilizability, 688
Stack controller, 115
Standard second-order system, 189
State, 29
State controllability:
complete, 676, 678, 680
State equation, 31
solution of homogeneous, 660
solution of nonhomogeneous, 666–67
Laplace transform solution of, 663
State-feedback gain matrix, 724
MATLAB approach to determine,
735–36
State matrix, 31
State observation:
necessary and sufficient conditions for,
754–55
State observer, 751–77
design with MATLAB, 773
type 1 servo system with, 746
State observer gain matrix: 755
Ackermann’s formula to obtain, 756–57
direct substitution approach to obtain,
756
transformation approach to obtain, 755
State space, 30
State-space equation, 30
correlation between transfer function
and, 649, 656
solution of, 660
State-space representation:
in canonical forms, 649
of nth order system, 36–39
State-transition matrix, 664
properties of, 665
State variable, 29
State vector, 30
Static acceleration error constant,
228, 421
determination of, 421–22
Static position error constant,
226, 419
Static velocity error constant,
227, 420
Steady-state error, 160, 226
for unit parabolic input, 229
for unit ramp input, 228
in terms of gain K, 230
Steady-state response, 160
Step response, 699–700
of second-order system, 165–69
Summing point, 18
Suspension system:
automobile, 86–87
motorcycle, 87
Sylvester’s interpolation formula, 673,
709–713
System, 3
Sytem types, 419
type 0, 225, 230, 419, 433, 487–88
type 1, 225, 230, 420, 433, 487–88
type 2, 225, 230, 421, 433, 487–88
System response to initial condition:
MATLAB approach to obtain, 203–11
T
Tachometer, 176
feedback, 343
Taylor series expansion, 43–45Temperature control systems, 4–5
Test signals, 159
Text:
writing on the graphic screen, 188
Thermal capacitance, 137
Thermal resistance, 137
Thermal systems, 100,136–39
Thermometer system, 151–52
Three-degrees-of-freedom system, 645
Three-dimensional plot, 192
of unit-step response curves with
MATLAB, 191–93
Traffic control system, 8
Transfer function, 15
of cascaded elements, 73–74
of cascaded systems, 20
closed-loop, 20
of closed-loop system, 20
experimental determination of, 489–90
expression in terms of A, B, C, and D, 34
of feedback system, 19
feedforward, 19
of minimum-order observer-based
controller, 777
of nonloading cascaded elements,
77
observer-controller, 762, 780–82
open-loop, 19
of parallel systems, 20
sinusoidal, 401
Transfer matrix, 35
Transformation:
from state space to transfer function,
41–42, 657
from transfer function to state space,
40–41, 656
Transient response, 160
analysis with MATLAB, 183–211
of higher-order system, 180
specifications, 169, 171
Transport lag, 417
phase angle characteristics of, 417
Turbulent-flow resistance, 102
Two-degrees-of-freedom control system,
593–95, 599–614, 636–41, 646–47
Two-position control action, 22–23
Two-position controller, 22
Type 0 system, 225, 230, 488
log-magnitude curve for, 419, 488
polar plot of, 433
Type 1 servo system:
design of, 743–51
pole-placement design of, 739–46
Type 1 system, 420
log-magnitude curve for, 420, 488
polar plot of, 433
894 Index
Type 2 system, 421
log-magnitude curve for, 421, 488
polar plot of, 433
U
Uncontrollable system, 681
Undamped natural frequency, 165
Underdamped system, 166–67
Underlapped spool valve, 146
Unit acceleration input, 247
Unit-impulse response:
of first-order system, 163
of second-order system, 178
Unit-impulse response curves:
a family of, 178
obtained by use of MATLAB, 196–97
Unit-ramp response:
of first-order system, 162–63
of second-order system, 197–200
of system defined in state space,
199–200
Unit-step response:
of first-order system, 161
of second-order system, 163, 167, 169
Universal gas constant, 108
Unstructured uncertainty:
additive, 852–53
multiplicative, 809
system with, 809
V
Valve:
overlapped, 130
underlapped, 130
zero-lapped, 130
Valve coefficient, 127
Vectors:
linear dependence of, 674
linear independence of, 674
Velocity error, 227
Velocity feedback, 176, 343, 519
W
Watt’s speed governor, 4
Weighting function, 17
Z
Zero, 861
of order m, 862
Zero-lapped valve, 130
Zero placement, 595, 597, 612
approach to improve response characteristics, 595–97
Ziegler–Nichols tuning rules, 11, 568–77
first method, 569–70
second method, 570–71

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