INDEX
Acceleration constant:
definition of, 577
determination from Bode diagram, 62
relationship to poles and zeros, 584–585
Ackermann's formula for design using pole placement, 137–143
application example of, 139–143
Adaptive control systems, pitch flight, 111–117
Advanced z transform:
table of, 235
Aircraft: adaptive pitch flight control of, 111–117
Analog-to-digital converters, 209–221
Antennas of tracking radars, mechanical resonance of, 508–510
APOLLO 11 mission, description of, 488–492
Aquaculture control system, optimal control application to, 499–500
Asymptotic stability, definition of, 408–414
Autopilots, control of airplane's roll using, 544–546
Backlash, definition of, 341–342
describing function of, 342–344
back_lsh subroutine, Toolbox, 356
Bandwidth:
effect of compensation on, 58, 93–96
effect of noise on, 58
BASIC computer program:
describing function from, 363–369
Bilinear transformation, 252
modified, 257
Bode-diagram method, 16–21, 24–25
approximate closed-loop response, 24–25
case study design examples using, 507–516
digital system application of, 256–262, 528–537
Bode's theorems:
Break frequency, definition of, 24–25
Calculus of variations, 462–467
Bolza problem, 463
Euler-Lagrange equation for, 465
Lagrange problem, 462
Cascade compensation, methods of, 36–44
Cauchy's theorem:
application to digital systems, 247
Center, definition of, 374
Characteristic response, second-order system, 566–573
Chemical control systems, 546
Circle criterion, See Generalized circle criterion
Closed-loop frequency response, 21–24
Compensation:
cascade-compensation techniques, 36–44
digital control application of, 257–261, 263–271
linear continuous system use of, 33–36
minor-loop feedback method of, 44–46
Nichols chart method of, 21–24, 74–77
root locus method of, 25–29, 74–93
tradeoffs among methods of, 93–96
Complimentary sensitivity, definition of, 164
Computer programs:
back_lsh, 356
dead_zn, 356
describing function using, 363–369
margins, 16
MATLAB, 16–28, 272–273, 361–363, 549–565
phase-plane obtained using, 367–401
relays, 356
rlpoba, 25
root locus obtained using, 25
rootangl, 25
rootmag, 25
Controllability, definition of, 131–134
example of, 134
test for, 134
Damped frequency oscillation, definition of, 570–571
Damping factor, 571
critical, 568
definition of, 571
overdamped, 570
underdamped, 571
Dead zone, describing function of, 337–338
Dead zone nonlinearity, definition of, 337
dcad_zn subroutine, Toolbox, 356
case study design example using, 507–516
Dead zone, 337
guidelines for use of, 425–427
MATLAB application to, 361–363
prediction of oscillation using, 352–357
software programs for obtaining, 361–369
Design:
Bode-diagram method of, 51–74, 256–262, 507–516, 528–537
cascade compensation techniques, 36–44
general procedures for, 505–507
linear continuous system, 33–177
minor-loop feedback method of, 44–46
Nichols chart method of, 21–24, 74–77
robot's angular joint control, 516–522
root-locus method of, 25–29, 74–93
space vehicle attitude control, 477–492
temperature control of liquid, 528–537
tracking radar position loop, 507–516
Design procedure, general procedure for, 505–507
Differential equations, 326–327
linear, 326
Digital control systems, 209–324, 528–537
block-diagram algebra, 236–242
Bode diagram applied to, 256–262, 528–537
Case study of temperature control, 528–537
digital filler, design of, 285–289
examples of, 248–252, 256–285, 528–537
Jury's criterion applied to, 255–256
MATLAB applied to, 21–24, 74–77
Nyquist diagram application to, 245–252
Ragazzini's method applied to, 273–285
root-locus diagram applied to, 263–271
Routh–Hurwitz applied to, modified, 252–253
sampling characteristics in, 211–215
Schur–Cohn criterion applied to, 253–255
signal-flow graphs applied to, 241–242
stability analysis of, 245–271, 528–537
Digital-to-analog converters, 209–211
Digitization process, design of digital fillers for, 285–289
Discrete-data control systems, See Digital control systems
Disturbance,
effect on hydrofoil of sea wave, 537–542
effect on tracking radar of wind, 507–516, 543–544
Hamilton-Jacobi equation for, 469
relation to maximum principle, 477, 494–496
Ecological control systems, 500–501
Energy consumption minimization, use of optimal control for, 485–488
Error constants, definition of, 574–585
Estimator design:
attitude system case study design, 477–492
linear-state variable-feedback, 119–131, 143–156, 522–528
pole placement, 119–131, 137–156
Estimator gain matrix M, definition of, 143
Euler-Lagrange equation, 465
Feedback compensation, 33–36, 44–45, 54–62, 64–67, 74–208
Final-value theorem: definition of z transform, 232–233
Finite stability, definition of, 409
First method of Liapunov, 405–406
amplitude characteristics of, 220
phase characteristics of, 220
transfer function of, 219
FORTRAN Programs:
phase-plane obtained using, 397–404
Fourier series:
applied to describing function, 336–351
applied to discrete systems, 213–215
Friction:
describing function of, 347–351
Fuel consumption minimization, use of optimal control for, 482–485
Gain margin, definition of, 18
multiple, 19
Gain-phase diagram, See Nichols chart
Generalized circle criterion, 422–425
guidelines for use of, 425–427
Global stability, definition of, 409
examples illustrating, 170–171, 174–175
theoretical concepts of, 162–167
Hamilton-Jacobi equation, derivation of, 469
Hard self-excitation, derivation of, 328–329
Helicopter control system, 114–116
Human control systems, 114–116
Hydrofoils, effect of torque disturbances on, 537–541
pitch-control system for, 104–105
Hysteresis:
definition of, 345
describing function of, 345–347
Ideal relay, characteristics of, 437
Initial-value theorem:
definition of z transform, 232
Internal combustion engine:
optimization of, 457
computer computational method, 229–230
partial fraction expansion for, 228
power-series expansion for, 228–229
Isocline method, phase-plane construction using, 372, 378–381
Jacobian matrices, definition of, 333
Jitter, due to noise, 5
Jump resonance, definition of, 328–329
Jury's stability criterion, 255–256
Bode-diagram compensation using, 54, 56–58
Nichols chart compensation using, 74–75
root-locus compensation using, 83–91
Bode diagram compensation using, 54, 56–5 8
describing function compensation, 78–81
digital control compensation, 256–261
linear-state variable feedback, 130
root locus compensation using, 78–81
Liapunov's stability criteria, 399, 405–415
design examples using, 414–415
guidelines for use of, 425–427
scalar function definiteness definition used for, 410–412
Limit cycles:
definition of, 328
phase-plane illustration of, 385–386
Linear algebraic techniques, 175–177
Linear differential equation, 326
constant coefficient, 326
homogeneous solution, 3
particular solution, 3
time-variable representation of, 326–327
stability of, 3
Linearizing approximations, 330–334
Linear-state variable feedback, 119–131
controller and estimator design, 143–154
controller design using, 124–131
estimator design using, 143–146
general design procedure for, 119–124
regulator design using, 146–154
Liquid-level control systems, 528–537
Local stability, definition of, 409
Logspace MATLAB command, definition of, 563
Lunar Excursion Module, equations for, 107–108, 488–492
Maintainability considerations, 5
margins subroutine, Toolbox, 16
Mason's theorem:
digital system application of, 241–242
Bode diagram obtained using, 16–21
conversion between continuous and discrete time, 15–16
data representation of, 559–562
Demonstration m-file, 550
describing function analysis using, 361–362
fundamental concepts, 556
Help, 550
Modern Control System Theory and Design Toolbox, 549
Nichols chart obtained using, 21–24
Nonlinear Toolbox, 551
Nyquist diagram obtained using, 12–16
root-locus diagram obtained from, 25–28
Simulink Toolbox, 551
Student Edition of MATLAB, 549
Symbolic Toolbox, 551
Synopsis File, 550
transformation from state-space to transfer function, 564
transformation from transfer function to state-space form, 15
retrieving from anonymous FTP server, 549
tutorial, 550
Tutorial File, 550
Matrix:
closed-loop resolvent, 122
controller gain K, 120
estimator gain M, 143
Jacobian, 333
Maximum percent overshoot:
definition of, 573
root-locus determination of, 81–83
attitude control design using, 477–488
co-state for, 474
Hamiltonian function for, 474
Lunar soft-landing system design, 488–492
minimum-energy problem solution, 485–488
minimum-fuel problem solution, 482–485
minimum-time problem solution, 480–482
Maximum value of peaking, Nichols chart method for finding, 21–24
Method of isoclines, phase-plane construction using, 372, 378–381
Modern Control System Theory and Design Toolbox, 58
retrieving from anonymous FTP server, 58
table of commands used by, 74
Modified bilinear transformation, 257
Multivariable systems, 167–169
NASA programs:
Lunar Excursion Module, 488–492
Orbiting Astronomical Observer, 442–443
Ranger unmanned space vehicle, 446–447
Negative-definite function, definition of, 410
Networks,
Nichols chart:
compensation and design using, 74–77
MATLAB applied to obtain, 21–24
maximum value of peaking from, 22–23
Node, phase-plane, 375
Noise:
effect of bandwidth on, 58
effect of compensation on, 58, 93–96
Nonlinear control systems, 325–456
methods available for analyzing, 329–330, 425–427
Nonlinear differential equations, 326–327
Nonlinear system stability:
describing functions for finding, 324–369
generalized circle criterion for, 422–425
guidelines for selecting method, 425–427
Liapunov's criteria for finding, 399, 405–415
phase-plane method for finding, 371–415
piecewise-linear method for, 369–371
Popov's method for finding, 415–422
Nonlinearities:
Nuclear control systems, temperature-control system of, 103
Numerical methods, use of digital computers for, 332
Nyquist's stability criterion:
digital control system application of, 245–252
obtaining Nyquist diagram using MATLAB, 10–16
On-off nonlinearity, describing function of, 345–347
Optimal control theory, 457–504
aquaculture application of, 499–500
attitude control application of, 477–488
ecological system application of, 500–501
lunar soft-landing problem, 459, 488–492
minimum energy-problem, 459, 485–488
minimum fuel-consumption problem, 459, 482–485
minimum-time problem, 458–459, 480–482
quadratic optimal control problem, 460, 501–502
Orbiting Astronomical Observatory, 442–443
Overdamped control system, definition of, 570
Overshoot:
curves of, 574
definition of, 572
Partial fraction expansion, z-transform inversion using, 228
Peak time, 572
Performance indices, quadratic, 460, 501–502
Phase portrait, definition of, 371–372
Phase trajectory, definition of, 372
Phase-lag network:
Bode-diagram compensation using, 54, 56–58
Nichols chart compensation using, 74–75
robot joint case study use of, 516–521
root-locus compensation using, 83–91
Phase-lag–lead network:
Phase-lead network:
Bode-diagram compensation using, 54, 56–58
describing function compensation, 78–81
digital control compensation using, 256–261
linear-state variable feedback using, 130
root-locus compensation using, 78–81
design examples using the, 392–397
digital computer program for obtaining, 397–399
energy determination from, 407–410
guidelines for use of, 425–427
representative phase portraits, 388–390
software programs for obtaining, 397–399
space-attitude control example, 481–485
time determination from, 386–388
with external forcing functions, 389–392
Phase-shift approximation, straight-line, 70–74
PD compensators, 47
Piecewise-linear approximations, application to nonlinear systems, 330–334
Pole-placement design:
using Ackermann's formula, 137–143
using linear-state-variable feedback, 119–131
Pontryagin's maximum principle, 473–492
attitude control design using, 477–492
co-state for, 474
Hamiltonian function for, 474
Lunar soft-landing system design, 488–492
minimum-energy problem solution, 48–488
minimum-fuel problem solution, 482–485
minimum-time problem solution, 480–482
guidelines for use of, 425–427
Position constant:
definition of, 576
relationship to poles and zeros, 581–582
Position limiting, phase-plane illustration of, 389–390
Positive-definite function, definition of, 410
Power consumption, considerations of, 5
Power-series expansion, discrete system application of, 128–129
Programs:
back_lsh, 356
dead_zn, 356
describing function using, 363–369
discrete Bode and root locus, 272–273
inverse z transform obtained from, 229–231
margins, 16
MATLAB, 15–28, 361–362, 549–565
Nichols chart obtained from, 21–24
phase-plane obtained using, 397–404
relays, 356
rlaxis, 25
rlpoba, 25
root locus obtained using, 25–28
rootangl, 25
rootmag, 25
Quadratic form of scalar function used for Liapunov's method, definition of, 410–411
Quadratic optimal control problem, 460, 501–502
Quasilinear control systems, definition of, 330
use of staleness factor with, 282–285
Ranger unmanned space vehicle, 446–447
Rate feedback:
describing function compensation, 360–361
linear system compensation using, 44–45
Rate limiting, phase-plane illustration of, 388–389
Regulator design, linear-state variable-feedback, 146–154
Relay servo:
ideal, 444
phase-plane analysis of, 392–397
relays subroutine, Toolbox, 356
Reliability considerations, 5
rlaxis subroutine, Toolbox, 25
rlpoba subroutine, Toolbox, 25
Robotics:
case study design of joint for, 516–521
Ford Motor Company, 517
welding of automobiles using, 544
Robust control systems, 156–162
case study for controlling a hydrofoil, 537–541
tracking radar positioning system robustness example, 547–548
Root-locus method, 409–454, 549–567
case study design application of, 522–528
compensation and design using, 75–93
digital system application of, 528–537
MATLAB applied to obtain, 25–28
rootangl subroutine, Toolbox, 25
rootmag subroutine, Toolbox, 25
Routh–Hurwitz stability criterion:
application to root locus method, 87
modified form for digital systems, 252–253
Saddle point, definition of, 383–385
Sampled-data control systems: 209–324
case study of temperature control, 528–537
definition of, 209
Sampled-data control system, See Digital control systems
Sampling process:
frequency-domain characteristics, 213–216
ideal, 212
definition of, 339
describing function of, 340–341
Schedule considerations, 5
Schur–Cohn stability criterion, 253–255
Second-method of Liapunov, 406–415
Second-order control systems, characteristics of, 566–574
Sensitivity, definition of, 3–4
complementary, 164
Shannon's sampling theorem, 214–215
Signal-flow graphs:
case study design example using, 507–516
digital system application of, 241–242
Simulation, nonlinear control system, 426–27
Singular points, definition of, 382–385
Size of equipment, considerations of, 5
Soft self-excitation, definition of, 328–329
Space vehicles,
case study design examples, 522–528
Lunar Excursion Module, 107–108, 488–492
navigation equations for, 331–333
Orbiting Astronomical Observer, 442–443
Ranger Unmanned Space Vehicle, 446–447
Stability:
Bode method for determining, 16–21, 24–25
bounded input-bounded output, 3
comparison of various methods, 93–96
finite, 409
global, 409
guidelines for use of methods, 93–96
linear continuous system, 9–208
local, 409
Nichols method for determining, 21–24, 74–77
Nyquist method for determining, 10–16, 245–252
Routh–Hurwitz method, 87, 252–253
summary of linear system methods, 93–96
definition of, 347
describing function of, 347–351
phase-plane application problem, 376–377
Subharmonic generation, definition of, 329
Subroutines from Advanced Modern Control System Theory and Design Toolbox:
back_lsh, 356
dead_zn, 356
margins, 16
relays, 356
rlaxis, 25
rlpoba, 25
rootangl, 25
rootmag, 25
Tables:
acceleration constant and error, 578
advanced z transform, 235
compensation methods comparison, 94–95
MATLAB commands and functions, 563–564
MATLAB program listings for:
Bode diagram obtained using, 18, 21
Nichols chart obtained using, 24
Nyquist diagram obtained using, 13, 16
root locus method obtained using, 27–28
position constant and error, 576
steady-state constants summary, 578
velocity constant and error, 577
z transform, 227
Tank-level control system, design of, 528–537
Taylor series expansion, 332, 405, 469
Temperature control system:
nuclear power plant, 103
Thermal control systems, case study design of, 528–537
Time minimization, use of optimal control for, 458–459, 480–482
Toolbox subroutines, Advanced Modem Control System Theory and Design, See Subroutines for Advanced Modem Control System Theory and Design Toolbox
Tracking radar:
design of positioning system for, 507–516
Transfer-function concept:
Transformation between:
state-space to transfer function form, 564
transfer function to state-space form, 15
Transient response, 4
determination from root locus, 75–93
Two-phase ac servomotor, nonlinear
characteristics of, 330, 435–436
Type of control system, definition of, 575
Undamped natural frequency, definition of, 571
Underdamped control system, definition of, 571
Van der Pol equation, Liapunov's method applied to, 405
Velocity constant:
definition of, 578
determination from Bode diagram, 55
relationship to poles and zeros, 583–584
w transforms:
case study design example using, 528–537
modified, 257
Ward-Leonard system, case study example using, 513
Weight considerations, 5
z plane, relationship to s plane, 223–225
block diagram algebra, 236–242
exponential decay, 226
initial-value theorem, 232–233
table of, 227
transfer function using, 236–242
unit step, 225
amplitude characteristics of, 218–219
phase characteristics of, 218–219
transfer function of, 218