Acceleration constant:
definition of, 280
determination from Bode diagram, 536
relationship to poles and zeros, 286–287
ACET program, description of, 463
Ackermann’s formula for design using pole placement, 611–617
application example of, 613–617
Adaptive control systems, 252–254
human, 6
Adjoint matrix, definition of, 104
Aeronautical control systems, 1
Aircraft:
adaptive pitch flight control of, 589–591
attitude control system for, 16
Bell Boeing V-22 Osprey military tilt rotor, 16–17
Bell Boeing 609 commercial tilt rotor, 17
equations for landing phase, 172–173
instrument landing system for, 172–173
nose-wheel steering system for, 500
UH-60 Black Hawk combat assault helicopter, 17–18
wind-shear detection for, 14
Airline reservation system, control of, 33
Amplidyne, 202
Analog computer, 94–101, 116–118, 133–134
Analogy circuit concept, 221–224
Analog-to-digital converters, 683–695
Analytic functions:
definition of, 41
derivative of, 40
Antennas of tracking radars, mechanical resonance of, 981–983
Antiskid control system, 7
Antisubmarine warfare system, control of, 33
Aquatic control system, 26
modeling equations for, 177–178
Armature-controlled dc servomotor, 198–202
decreasing time constant of, 230
elimination of disturbances with, 231–232
with back emf for stabilization, 231
Asymptotic stability, definition of, 236
Attitude control:
rocket, 501
Automatic floor scrubbers, control of, 34
Automatic inventory control, 314–315
Automatic warehousing, 7, 314–315
Automation, 7
Automobile, model of leasing agency of, 35
control of airplane’s roll using, 1018–1020
Band-pass network, circuit for, 79–81
Band-rejection network, circuit for, 79–81
Bandwidth:
effect of compensation on, 532, 567–579
effect of noise on, 532
BASIC computer program, Bode diagram from, 386–389
Bioengineering control systems, 1
Biomedical control systems, 23–25
Bode-diagram method, 362–398, 525–548
approximate closed-loop response, 541–548
approximate phase-shift from, 544–548
break frequency of, 368
computer programs for obtaining, 380–385
MATLAB used to obtain, 380–385
relationship to Nyquist diagram, 374–375
Bode’s theorems:
Branches, 86
Break frequency, definition of, 368
Break-in point, definition of, 415–418
Bulk modulus of oil, definition of, 215
Cameras:
automatic focus control system for, 25–26
Nikon N6006 35-mm camera, 25–26
Capacitance:
of mechanical system, 223
of pressure system, 223
thermal, 223
Cascade compensation, methods of, 510–518
Cascaded elements, transfer function of, 82
Cauchy-Reimann conditions, 41
Cauchy’s theorem:
application to digital systems, 721
principle of the argument, 350
Central Japan Railways Company, 11–14
Characteristic equation, 83, 333–334
obtaining from classical methods, 334–335
obtaining from state variables, 335–340
Characteristic response, second-order system, 240–247
Chemical control systems, 1, 7, 32, 1020
pH factor control of, 32
Circle criterion, see Generalized circle criterion
Circuit concept and analogy, 222–224
Classical approach, second-order system analysis, 240–247
Climate control system, 7
Closed-loop control system, definition of, 4–6
Closed-loop frequency response, relationship to time-domain, 405–408
Closed-loop resolvent matrix, definition of, 596
Closed-loop system matrix, definition of, 596
Closed-loop systems, transfer function, 82–95
CODAS-II program, description of, 462
Cofactors of matrices, definition of, 104
Command, control and communication system, 33
Command/control/communication:
airline reservation system, 33
antisubmarine warfare system, 33
control of, 33
ground traffic control system, 33
police command system, 33
satellite communication system, 33
strategic system, 33
Commercially available software:
ACET, 463
CODAS-II, 462
MATLAB, 461
Companion matrix, definition of, 110–111
Compensation:
Bode-diagram method of, 525–541
cascade-compensation techniques, 507–508, 510–518
definition of, 507
linear continuous system use of, 507–510
minor-loop feedback method of, 518–520
Nichols chart method of, 548–551
root locus method of, 549, 552–567
tradeoffs among methods of, 567–570
definition of, 39
multivalued function of, 42–43
ordinary points of, 40
singled-valued function of, 42–43
singularities of, 42
zeros of, 43
Complimentary sensitivity, definition of, 638
Computer programs:
ACET, 463
Bode diagram obtained using, 380–385
Codas II, 462
commercially available software, 456, 460–463
margins, 375
MATLAB, 58–75, 118–121, 380–385, 437–442, 460–461
Nichols chart obtained using, 403–405
rlpoba, 429
rootangl, 429
root locus obtained using, 437–442
rootmag, 429
time response obtained using, 121–125, 393–398
wpmp, 402
Conditionally stable systems, 357–359
Constant magnitude loci, 400–402
Constant phase-angle loci, 400–402
Control-canonical form, 100
Controllability, definition of, 605–608
example of, 608
test for, 607
Control stroke, hydraulic system, 214
Control winding of ac servomotor, 206–207
Critical damping, definition of, 242
Cruise control system, 7
Ctrl-C program, description of, 461–462
Damped frequency oscillation, definition of, 241–247
Damping factor, 245
Damping ratio:
best damping ratio to use, 252–255
critical, 242
relation to the phase margin, 357
dc armature-controlled motor, 198–202
dc field-controlled servomotor, 203–206
dc Ward-Leonard system, 202–203
Dead-reckoning system, application of, 34
Decibel, definition of, 364
Delay time factor, Bode diagram of, 371–372
Depth control, submarine, 22, 28, 176
Design:
Bode-diagram method of, 528–541
cascade compensaton techniques, 510–518
linear continuous system, 507–591
minor-loop feedback method of, 518–520
Nichols chart method of, 548–551
Determinant, 104
Determinant of signal-flow graph, definition of, 88
Diagonal matrix, definition of, 102
Differentiation:
Laplace transformation of, 54
Differentiation network, 79–81
Digital computer, data storage and retrieval for, 324–326
Disturbance:
Dominant poles, 555
Ecological control systems, 26, 177–178, 181–182, 497
modeling equations for, 177–178, 497
waste control system example of, 181–182
Economic considerations, 296, 482–483
Economic systems, control of, 1, 26, 31–32, 170
Eigenvalues, definition of, 338
Electro-mechanical devices:
armature-controlled dc motor, 198–202
field-controlled dc servomotor, 203–206
state-variable representation of, 201, 205–206, 209
transfer functions of, 195–213
two-phase ac servomotor, 206–213
Elevator control system, 29, 31
English set of units, 194
conversion from International set of units, 194
Error constants, definition of, 279–280
Estimator design:
linear-state variable-feedback, 593–605, 617–630
Estimator gain matrix M, definition of, 617
Euler-Lagrange equation, 939
Feedback compensation, 507–510, 515–520, 522–525, 528–541, 548–657
Feedforward compensation, 164
Field-controlled dc motor, 203–206
automatic speed control, 5
Final-value theorem:
definition of Laplace transform, 55
Fluidic control systems, 478–479
Ford Motor Company, 8
FORTRAN programs:
Bode diagram obtained using, 386–398
root locus obtained using, 443–454
time-response obtained using, 122–124
Forward path:
definition of, 87
gain of, 87
definition of, 50
Fourier series:
Fourier transform, definition of, 49–50
Frequency for maximum peaking, Nichols chart method for finding, 399
Frequency spectrum, 159
Gain margin, definition of, 357
Gain-crossover frequency, definition of, 374
Gain-phase diagram, see Nichols chart
illustrative problem of, 226–227
Generator constant, definition of, 197
Gimbal, gyroscope use of, 229
Grant’s rule, 421
Gyroscope, 229
examples illustrating, 642–645, 648–649
theoretical concepts of, 637–641
Helicopter control system, 589–591
Help-Mate robot, 10
Homogeneous solution, differential equations, 130
Human control systems, 6, 23–25, 30–31, 485–486, 492, 588–590
Human transfer function, 492–494, 500–501
state-variable representation of, 216–217
transfer function of, 216, 218
valve-controlled motor, 217–219
Hydraulic-electrical analogies, 223
Hydraulic motor and pump, 214–217
state-variable representation, 216–217
transfer function of, 216, 218
Hydraulic valve-controlled motor, 217–219
transfer function of, 218
pitch-control system for, 578–579
IAE performance criterion, definition of, 291
Identity matrix, definition of, 102
Inductance:
of hydraulic systems, 223
of mechanical system, 223
Information theory, 38
Initial-value theorem:
definition of Laplace transform, 55
Input vector, definition of, 102
Instrument landing system, description of, 172–173
Integration:
Interest accrual, control system for, 170–171
Internal combustion engine:
control of, 31
optimization of, 931
International set of units, 194
conversion to English set of units, 194
Inventory control systems, 7
Inverse Laplace transform, approach of, 56–58
Inverse of a square matrix, 106–107
ISE performance criterion, definition of, 291
ISTAE performance criterion, definition of, 291
ISTSE performance criterion, definition of, 291
ITAE performance criterion, 291, 294–295
definition of, 291
ITSE performance criterion, definition of, 291
Kirchoff’s equations, 186–188, 223
Bode-diagram compensatioin using, 510–511
Nichols chart compensation using, 548–551
root-locus compensation using, 557–567
definition of, 51
differential equation solution, 54
final-value theorem, 55
initial-value theorem, 55
inversion of using MATLAB, 71–75
shifting theorem, 55
unit ramp, 52
Bode diagram compensation using, 530–541
circuit for, 80
linear-state variable feedback, 603–604
root locus compensation using, 552–555
Liberty Mutual Insurance Group, 24
Linear algebraic techniques, 649–651
Linear differential equation, 75–77, 130, 190
constant coefficient, 75–77, 190
homogeneous solution, 130
particular solution, 130
Linear-state variable feedback, 593–630
controller and estimator design, 620–630
controller design using, 598–605
estimator design using, 617–620
general design procedure for, 593–598
regulator design using, 620–628
Linear system, 54–55, 507–508, 510–518, 525–541, 549–567
compensation of, 507–508, 510–518, 525–541, 549–567
Liquid-level control systems, 2–4, 488–489
Logarithm, definition of, 364
Log-magnitude versus phase plot, 363–371
Logspace MATLAB command, definition of, 381–382
Loop gain, definition of, 87
Lunar Excursion Module, equations for, 171–172
M contours, definition of, 400–401
Machine-tool control, computer-controlled, 309–310, 322–323
Management control systems, 1
Man-machine control systems, 6, 23–25, 492–494, 500–501
Manual control systems, transfer function of, 237–239, 500–501
Map drive control system, 263–265
Mapping contours from s-plane to F(s)-plane, 344–349
Margins subroutine, Toolbox, 375
Marketing program, equations for, 180–181
Mason’s theorem:
application problems for, 90–93
definition of, 88
Mass-capacitance analogy, 223
Mass-inductance analogy, 223
Bode diagram obtained using, 380–385
Control System Toolbox, 71, 73–74
Demonstration m-file, 59
fundamental concepts of, 66–68
Laplace transform inversion using, 71, 74–75
Modern Control System Theory and Design Toolbox, 58
Nichols chart obtained using, 402–405
Nonlinear Toolbox, 60
Nyquist diagram obtained using, 359–362
rise time curve obtained using, 289–290
root-locus diagram obtained from, 437–442
Simulink Toolbox, 60
Student Edition of MATLAB, 58, 61–62, 64
Symbolic Toolbox, 60
Synopsis File, 59
transformation from state-space to transfer function, 120–121
transformation from transfer function to state-space form, 119–120
transient response of systems using, 124–129
retrieving from anonymous FTP server, 58
tutorial, 59
Tutorial File, 59
Matrix:
addition/subtraction of matrices, 104–105
adjoint matrix, 104
closed-loop resolvent, 596
cofactor of, 104
diagonal matrix definition, 102
differentiation of a matrix, 107
estimator gain M, 617
identity matrix definition, 102
integration of a matrix, 108
inverse of a square matrix, 106–107
multiplication by a scalar, 105
multiplication of two matrices, 106
skew-symmetric matrix, 103
symmetric matrix, 103
transpose of a matrix, 103
zero matrix, 104
Matrix operations, rules of, 104–108
Maximization of K(σ) method, use of for root-locus method, 415–418
Maximum percent overshoot:
definition of, 247
root-locus determination of, 556–557
Maximum value of peaking, Nichols chart method for finding, 399–404
McDonneell Douglas Corporation, 14–16
Mechanical control systems, 189–95
Mechanical devices:
state-variable representation of, 190–193, 195
transfer functions of, 191
Mechanical-electrical analogies, 221–224
Metadyne, 202
Military control systems:
strategic missile submarine, 20–22
Minimization nof K(σ) method, 415–418
Minimum-phase systems, definition of, 364
Missile systems:
acceleration-control steering, 262–263
launcher control system for, 3–5, 274–276
roll attitude control of, 501
Modeling:
circuit concept and analogy, 221–224
transfer functions of control systems, 255–260
using principle of conservation, 221–222
Models of:
acquaculture control systems, 177–178
armature-controlled dc motor, 198–202
automobile leasing agency, 35
command, control, and communication system, 33
ecological control systems, 177–178, 497
economic systems, 26
field-controlled dc servomotor, 203–206
human control systems, 23–25, 30–31, 485–486, 492, 588–590
hydraulic motor and pump, 214–217
hydraulic valve-controlled motor, 217–219
inventory control system, 7
mass-marketing campaign, 180–181
organization of an engineering corporation, 35
thermal control systems, 219–221
Modern Control System Theory and Design Toolbox, 58
retrieving from anonymous FTP server, 58
table of commands used by, 74
Moment of inertia, determination of, 211–213
Motor constant, definition of, 208
Motor time constant, 208
Multivariable systems, definition of, 109
Myoelectric control signals, 24
NASA programs:
first untehered walk in, 237–239
International Space Station, 8–11
Lunar Excursion Module, 171–172
Shuttle Mission Simulator Motion-based trainer, 9, 12
Navigation systems, accuracy of, 254, 296
Networks:
table of transfer functions for, 80–81
Newton’s law of motion, 191–192
compensation and design using, 548–551
MATLAB applied to obtain, 403–405
maximum value of peaking from, 402–404
Node, signal-flow graph, 86
Noise:
effect of bandwidth on, 532
effect of compensation on, 532, 567–569
Nonminimum-phase systems, definition of, 364–365
Nuclear control systems, 1
nuclear reactor control, 27–28
temperature-control sysem of, 577
Numerical methods, Runge-Kutta, 394
Nyquist’s stability criterion, 349–362
Nyquist’s theorem:
obtaining Nyquist diagram using MATLAB, 359–362
relationship to Bode diagram, 355, 357–359
Observer-canonical form, definition of, 100
illustrative example of, 98–101
table summarizing computing devices, 99
Ordinary points, definition of, 39–40
Output equation, definition of, 111
Output node, definition of, 86–87
Output vector, definition of, 109
Overdamped control system, definition of, 244
Overshoot:
definition of, 247
Pacemaker, electronic, 315
Paper color control system, 232–234
Parabolic input, 280
Partial fraction expansion, Laplace transform inversion, 56-58
Particular solution, differential equation, 130
Path:
definition of, 87
gain of, 87
Peak time, 246
Performance indices, 291, 294–295
IAE, 291
ISE, 291
ISTAE, 291
ISTSE, 291
ITSE, 291
Phase-crossover frequency, definition of, 374
Bode-diagram compensation using, 510–512
characteristics of, 511
Nichols chart compensation using, 548–551
root-locus compensation using, 557–567
Phase-lag-lead network:
Phase-lead network:
Bode-diagram compensation using, 530–532
linear-state variable feedback using, 601–604
root-locus compensation using, 552–555
Phase margin:
definition of, 355
relation to damping ratio, 357
Phase-shift approximation, straight-line, 544–548
Phase-variable canonical form, 110–112
definition of, 111
PD compensators, 515–516, 520–521
Pole-placement design:
using Ackermann’s formula, 611–617
using linear-state-variable feedback, 593–605
Poles:
definition of, 42
of order n, 43
second-order, 241
Police command systems, control of, 33
Position constant:
definition of, 279
relationship to poles and zeros, 283–285
Positive feedback systems, root locus of, 434–437
Power consumption, considerations of, 295–296, 532
Principle of conservation, definition of, 221–222
Program CC, description of, 462–463
Programs:
Bode diagram obtained from, 380–385
CODAS II, 462
commercially available software, 456, 460–463
margins, 375
MATLAB, 58–75, 118–126, 380–385, 437–440, 460–461
Nichols chart obtained from, 403–405
rlaxis, 428
rlpoba, 429
rootangl, 429
root locus obtained using, 437–442
rootmag, 429
time response obtained using, 121–125, 393–398
Quadratic-phase lag network, Bode diagram of, 369–371
Quasilinear control systems, definition of, 206–207
Railroad control systems, 10–14
automatic control of, 14
Linear Express, 12
magnetic levitation used with, 10–13
Rate feedback, linear system compensation using, 518–520
Reduction of multiloop systems:
Regulator design, linear-state variable-feedback, 620–628
Regulex, 202
Resistance:
of hydraulic system, 223
of mechanical system, 223
of thermal system, 223
Rise time, definition of, 289
rlaxis subroutine, Toolbox, 429
rlpoba subroutine, Toolbox, 429
RoboKent, 34
Robotics:
automobile assembly using, 8, 266–267, 327–328
Ford Motor Company, 8
GMFanuc Robotics Corporation, 266–267, 327–328
material transport using, 8, 10
NASA space shuttle using, 9
painting automobiles using, 266–267
RoboKent, 34
welding of automobiles using, 327–328
Robust control systems, 630–636
Rocket flight, state and output equations for, 114–125
Root-locus method, 409–454, 549–567
compensation and design using, 549–567
computer program for obtaining, 443–454
dipole, 559
MATLAB applid to obtain, 437–443
maximization of K(σ) method, 415–417
negative feedback construction, 409–434
positive feedback construction, 434–437
Routh-Hurwitz method applied to, 418
rootangl subroutine, Toolbox, 429
rootmag subroutine, Toolbox, 429
Rotorol, 202
Routh-Hurwitz stability criterion, 340–344, 418
application to root locus method, 418
Runge-Kutta, numerical integration using, 394
Satellite, equations for a spherical, 170
Satellite communications, control of, 33
Second-order control systems, 240–261
best damping ratio to use, 252–255
state-variable signal-flow graph, 248–252
transient response of, 241–247
complementary, 638
definition of, 274
Servomechanism, 5
Settling time, deifnition of, 289
Shifting theorem, definition of, 55
Shinkansen train system, 13–14
Ship control systems, 1
Ship stabilization system, 323–324
Ship’s steering system, 29–30, 490
illustrative problem on, 304–305
application problems using, 89–95
branches of, 86
determinant of the graph for, 88
feedback loops for, 87
loop gain for, 87
Mason’s theorem to reduce, 88
nodes of, 86
path definition of, 87
path gain of, 87
reduction of, 88
relationship to state variables, 116–119
sink, definition of, 87
source, definition of, 86
MV-22A tilt rotor, Hughes Operational Flight Trainer of, 196
NASA Shuttle Mission Simulator Motion-based trainer, 8–9, 12
UH-60 Black Hawk Flight simulator, 17–19
Singularities, definition of, 42
Sink, defintiion of, 87
Size of equipment, considerations of, 295–296
Skew-symmetric matrix, definition of, 103
Sociological system model, equations for, 179–180
Source, definition of, 86
Space station, International, NASA, 8–11
Space vehicles:
International Space Station, 8–11
Lunar Excursion Module, 171–172
reentering earth’s atmosphere, 580–581
Specific heat of water, definition of, 220
field-controlled dc motor with, 5
turbine, 588
Spring-mass-damper system, 189–195
Stability:
Bode method for determining, 362–398, 525–548
bounded input-bounded output, 332
comparison of various methods, 463–464
guidelines for use of methods, 463–464
Liapunov, 873, 879–896
linear continuous system, 331–591
local, 883
Nichols method for determining, 398–405
Nyquist method for determining, 349–362
Routh-Hurwitz method, 340–344, 418, 726–727
stabilization definition, 331–335
summary of linear system methods, 463–464
State equations, 110–112, 121–124, 186–189
State space, definition of, 108
State of a system, definition of, 108
State transition equation:
definition of, 110
State transition matrix, 129–135
exponential series form for, 137–141
properties of, 130
State-variable concept:
advantages of using, 109
companion matrix, definition of, 110–111
definition of, 108
phase-variable canonical form, 110–112
state of a system, 108
state space definition, 108
state vector definition, 108
State-variable diagram, 115–118
State-variable signal-flow graph, 116–118
State vector:
definition of, 102
norm of, 336
Strategic missile submarines, USS Tennessee, SSBN-734, 20–22
Strategic systems, command/control/communication, 33
Submarines:
depth control system for, 22, 176, 480, 493
diving and control function for, 254
Subroutines from Modern Control System Theory and Design Toolbox:
margins, 375
rlaxis, 428
rlpoba, 429
rootangl, 429
rootmag, 429
wpmp, 402
Superposition, definition of, 801
Symmetric matrix, definition of, 103
Tables:
aceleration constant and error, 280
Bode-diagram program listings, 382–383, 385, 386–393
compensation methods comparison, 568–569
conversion factors in going from the International System to the English set of units, 194
ITAE form for zero ramp error, 295
ITAE form for zero step error, 295
Laplace transform, 53, 1023–1024
MATLAB commands and functions, 72–73
MATLAB program listings for:
Bode diagram obtained using, 382–383, 385
describing function, 836–837
Laplace transform inversion, 73–75
Nichols chart obtained using, 406
Nyquist diagram obtained using, 350, 362
root locus method obtained using, 439, 441
transformation from state-space to transfer function form, 121
transformation from transfer function to state-space form, 120
transient response of systems, 127
Modern Control System Theory and Design Toolbox, commands used by, 72–73
mechanical rotation symbols, 194
mechanical translation symbols, 190
network transfer functions, 80–81
operational amplifier computing devices, 99
position constant and error, 279
root-locus program listing, 450–453
steady-state constants summary, 281
time-response program listing, 124
transfer function of networks, 80–81
velocity constant and error, 280
Tank-level control system, 2–4, 32, 488–489
Taylor series expansion, 207
Television camera control system, 320–321
Temperature control system, 7, 27, 30
nuclear power plant, 577
xylene chemical process, 587
Thermal capacitance, 223
Thermal control systems, 7, 27, 30
Thermal-electrical analogies, 224
Thermal resistance, 223
Bell Boeing V-22 Osprey military tilt rotor, 16–17
Bell Boeing 609 commercial tilt rotor, 17
Time delay factor, 371–372, 431–434
Time-domain response:
program for obtaining, 121–124, 393–398
relationship to frequency domain, 405–408
Time to the first peak:
definition of, 246
root-locus deermination of, 556
Time-invariant state equation, solution of, 135–137
Time response:
recurrence equation for, 121–124
recurrence equation sampling for, 122
Time-varying state equation, solution of, 111–112
Toolbox subroutines, Modern Control System Theory and Design, see Subroutines for Modern Control system Theory and Design Toolbox
Torpedo, control system for, 174–175
Torque constant, definition of, 200
Torque-speed curves, 207
Traffic control system, ground, 33
Train control system, high speed, 10–14
Trainers, control system for, 310–311
NASA Shuttle Mission Simulator Motion-based trainer, 8–9, 12
UH-60 Black Hawk Flight Simulator, 17–19
Transfer-function concept:
armature-controlled dc motor, 198–202
block diagram transformations, 83–84
field-controlled dc servomotor, 203–206
hydraulic motor and pump, 214–217
hydraulic valve-controlled motor, 217–219
table of transfer functions, 80–81
two-phase ac servomotor, 206–213
Transformation between:
state-space to transfer function form, 120–121
transfer function to state-space form, 119–120
Transient response, 240–247, 288–290, 555–557
determination from root locus, 555–557
exponential series form for, 137–141
properties of, 130
Transition from real axis method, root locus, 417–418
method use of, 424
Transitions Research Corporation, 10, 34
Transportation systems:
air, 14–20, 172–173, 500, 589–591
automobile, 35
space, 8–11, 171–172, 237–239, 271–272, 580–581
underwater, 22, 176, 254, 176, 480, 493
Transpose of a matrix, definition of, 103
Turbine speed-control system, 588
Two-phase ac servomotor, 206–213
illustrative problem of, 224–226
nonlinear characteristics of, 206–207
state-variable representation of, 209
transfer function of, 208
Undamped natural frequency, definition of, 241
Underdamped control system, definition of, 245
Van der Pol equation:
definition of, 171
Vector, state, 102
Velocity constant:
definition of, 279
determination from Bode diagram, 534–536
relationship to poles and zeros, 283–285
Velocity error, 279
with armature-current feedback, 230
Warehousing, automatic, 314
Waste-treatment control system, equations for, 181–182
Weight considerations, 295–296
Wind-shear detection system, 14–15
wpmp subroutine, Toolbox, 402
Zero-error systems, definition of, 292–293
Zero-input stable, definition of, 332
Zero matrix, definition of, 104
Zeros, definition of, 43
Zero steady-state ramp error, definition of, 279
Zero steady-state step error, definition of, 279