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Absorption, dynamics of, 328-333 ACSL simulation software, 540 Adjoint of matrix, 443 Alarm, process and diagnostic, 549 AmpIitude ratio, 202 Analog computer, 517 Analog-to-digital converter, 35 1 Attenuation, 59 Autonomous system, definition, 485 Bandwidth, 129 Batch control, 546 Bilinear transformation, 378 Block diagram, 53, 112-113 chemical reactor, 135-141 standard symbols, 143-144 Bode diagram: asymptotic approximations, 210-212 controllers, 217-218 definition, 209 first-order system, 209-211 graphical rules, 213 second-order system, 213-216 systems in series, 211-212 transportation lag, 205 Bode stability criterion, 227-228 Bumpless transfer, 288, 551
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C, for valve, 305 Cascade control, 249-256 in valve positioner, 315 CC simulation software, 540 Characteristic equation, 167-168 roots of, 33-35, 167 sampled-data system, 377, 395 Chattering, in on-off control, 497 Chemical reactor, 135-141, 235 phase plane of, 479-483, 500-504 Clamping, 35 1 Closed-loop system, 112 CIosed-loop transfer functions, 143-149 Cofactor matrix, 442 Cohen-Coon tuning, 288-289 Comparator, 112 Computer control, 405-427, 543-557 Control system response, 151-159 Control valve (see Valve, control) Controller, 128-133, 545-556 calibration of, 13 1 cascade, 249-256 direct digital, 405 feedforward, 257-265 internal model, 272-279 microprocessor-based, 543-556 pneumatic versus electronic, 543-544 ratio, 265-267 sampled-data, 405-427
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Controller mechanism, 124-126 Controller modes, choice of, 283-284 motivation for, 132-133 Controller tuning, 282-295 Cohen-Coon process reaction curve, 288-290 comparison of methods, 291-295 Ziegler-Nichols, 233-234, 286-287 Comer frequency, 210 Criteria of control quality, 284-285 Critical damping, 95 Critical points, analysis of, 487492 definition of, 485 Cross-controller, 455 Crossover frequency, 227 Damped oscillator (see Spring-massdamper system) Damping, viscous, 90-92 Dead time (see Transportation lag) Dead zone, in on-off control, 498-500 Decay ratio, 97 Decibel, 211 Derivative action in control, 131 Derivatives, Laplace transform of, 16-19 Describing function, 506-5 11 of actual relay, 510 definition of, 510 Determinant of matrix, 442 Deviation variables, 51-52, 115 in distributed parameter systems, 342 Differential equations, 20-21 computer solution, 517-532 Digital-to-analog converter, 35 1 Displays, 548-549 Distance-velocity lag (see Transportation lag) Distributed control, 555-556 Distributed-parameter systems, 333-344 Error, 111 EXACT self-tuner, 548 Exponential stage (see First-order system) External feedback for anti-reset windup, 553-554
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Feedback: negative, 113 positive, 113 Feedforward control, 257-265 Foxboro tuning rules, 262-265 Fieldbus module, 555-556 Filter in internal model control, 275 Final-value theorem, 37-38 First-order lag, 52 (See also First-order system) First-order system, 49-53 computer simulation, 533-534 impulse response, 57-58 interacting, 83-86 . noninteracting, 80-82 in series arrangement, 80-86 sinusoidal response, 58-61 step response, 55-57 transfer function, 5 i-52 Flow control, 547-548 Focus, 487-490 Forcing function, 22, 52-55 Fourier series, 358-359, 507-509 Frequency response, 20 1-240 Bode diagram, 209-220 (See also Bode diagram) Bode stability criterion, 227-228 comparison with root locus, 174 in control system design, 224-240 of controllers, 2 17-2 18, 22 1, 236-238 definition, 203 of distributed-parameter systems, 337-338, 343-344 from elliptical phase diagram, 221 experimental determination of, 299-300 gain and phase margins, 228-233 heuristic stability arguments, 208-209, 224-227 Nyquist stability criterion, 227 from pulse test, 300-301 substitution rule, 201-202 of systems, 209-216 in series, 207, 211-213 Ziegler-Nichols settings, 233-234, 286-287 Frequency testing, 299
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Gain margin, 228-233 design specifications, 229 Gain-phase plot, 5 11 Gas absorber, dynamics of, 328-333 Harmonic analysis, 506-5 10 Heat conduction, dynamics of, 333-338 Heat exchanger: dynamics, of counterflow, 339-344 resonance in, 343-344 steam-jacketed kettle, 318-324 Heater, stirred-tank (see Stirred-tank heater) Hold first-order, 375 zero-order, 352-353 Hysteresis in valves, 315 Impulse function, 42, 54 Impulse modulated function, 353 Impulse modulation, 352 Initial-value theorem, 39 Integral, Laplace transform of, 434 Integral action in control, 130-131 Integral of error criteria: absolute value of error (WE), 285 square of error (ISE), 285 time-weighted absolute error (I IAE), 285 Integrator, 5 18, 532-533 Interacting systems, 80-86 Interaction: in control system, 453-454 in mercury thermometer, 87 Internal model control, 272-278 Inverse of matrix, 442-443 Inversion of Laplace transforms, 22-33 Isoclines, method of, 485-487 Lag, 61 Laplace transform, 1344 of integral, 43-44 inversion of, 22-33 table, 17-18 use in partial differential equations, 335-336
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Lead-lag transfer function, 547 Liapunov, method of, 491-492 Limit cycle, 315, 492-493 in exothermic chemical reactor, 502 in on-off control, 498, 506-511 Limiting in controller and valve, 550, 551-553 Linearization, 72-75, 321-324 in analysis of critical points, 490 Liquid level, 64-70 computer simulation, 520-525 Load change, 112 Loading, in liquid-level process, 85 Long division, BASIC program, 372-374 Lumped-parameter model, of distancevelocity lag, 338-339 for mercury thermometer, 50 Manometer, mercury, 105 Matrix, 441-442 Matrix differential equation, 432-433 Minimal prototype response, 408 Minor of matrix, 442 Mixing process, 70-7 1 Modeling, 344 Modified Z-transform, 384-392 table, 356-357 Multiloop system, block diagram reduction, 148-149 Multiple input-multiple output system (MIMO), 453 Multivariable control, 453466 decoupling, 46 1 interaction, 453454 stability, 464-466 Natural frequency, 97 Natural period, 97 Negative feedback, 113-114 overall transfer function, 144-148 Node, 488490 Nodebus in distributed control, 555 Noninteracting control, 458463 Nonlinear systems, 469-505 definition of, 47 1
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INOEX 275 Proportional-integral-derivative control, ideal transfer function, I32 Pulse function: as approximation to unit impulse, 67-69 response of liquid-level system to, 67-69 P&e Festing, 300 Pulse transfer function, 360-361 Quadratic lag (see Second-order system)
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Nonminimum phase characFeristics, Nomninimum phase lag, 338 NyquisF stability criterion, 227
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Qffset, definition, 153 On-off control, 130 of stirred-tank heater, 493-500 Open-loop transfer function, 167 Overall transfer function, from block diagram, 144-149 for positive feedback system, I47 Overdamped response, 95 Overshoot, 96 Pade approximation to transport lag, 103 Partial fractions, 22-33 Pendulum, 476-478 phase plane of, 490-492 Per&k of oscillation, 97 ultimate, 233 Phase angle, 59, 202 Phase lag, 59 Phase lead, 59 Phase margin, 228-233 design specification, 229 Phase plane, 471-493 graphical methods in, 48M93 Phase space, 471-483 PoIes and zeros, 178, 182 Positive feedback, 1 l3-114 overah transfer function, 147 Process dynamics, experimental, 296-301 theoretical, 3 18-345 Process identification, 296-301 semi-log method, 297-298 Process reaction curve, 288 Proportional band, 129 Proportional control, 111 Proportional controller, idea1 transfer function, 128-129 Proportional-derivative control, ideal transfer function, 131-132 Proportional-integral control, ideal transfer function, 130-131
Ramp function, 15 Ratio control, 265-267 RC circuit, 71-72 Regulator problem, 114 Relay: actual, harrnonic~anaIysis of, 506-510 electronic, ideaI, in on-off control, 493-494 Reset windup, 551-554 Resistance, 64 linear, 64 Resonance, 216 in heat exchanger, 344 Resonant peak, 215-216 Response time, 97 Rise time, 97 Root locus, 177-193 comparison with frequency response, 174 concept, 177-182 plotting of diagrams, 182-184 rules for plotting, for negative feedback, 184-186 sampled-data system, 380-382 Roots of equation, BASIC program, 193-195 Routh test for stability, 169-171 extensions, 175 in sampled-data systems, 378-379 Saddle point, 487-491 Sampled-data control, 347-428 closed-loop response, 364-366
design methods, 410-415 for first-order with transport lag, 39WQ3 open-loop response, 360-363 performance specifications, 408-409 pulse transfer function, 361 stability, 376-378 Sampling, 349-35 1 fast and slow, 410 Second-order system, 90-101 computer simulation, 520-521 dynamic parameters r and &. 91-92 impulse respor~se, 98-99 overdamped, graphical calculation of time constants, 105-106 overdamped, semi-log graphical method, 297-298 sinusoidal response, 99-101 step response, 92-98 transfer function, 92% Self-tuner, 547-548 Sensitivity, controller, 128 Servomechanism problem, 114 Set point, definition, 112 Settling time, 415-416 Simnon simulation software, 540 Simulation, computer, 517-539 Simulation software, 532-540 Single input-single output system (SISO), 453 Spit-tile, 189 Spring-mass-damper system, 90-92 phase plane of, 472-476 Stability, 164-174 Bode criterion, 227-228 conditional, 193 definition, 166 in nonlinear systems, 491-493 in multivariable system, 464-466 Routh test, 169-171 sampled-data control, 376-382 Started function, 353. State of system, definition, 484-485 State variable, 432-433 selection and types, 436-437 State-space methods, 429-468 transfer function matrix, 449 transition matrix, 447
Steady-state gain, 66 Step function, 15, 53-54 Step testing, 296-299 Stirred-tank heater: block diagram for contol of, 111-120 closed-loop response of, 151-159 on-off control, 493-494 Substitution rule in frequency response, 201-204 Summer, 532-533 Superposition, 52-53, 47 1 Sutro weir, 64 Taylor-series expansion, 73, 32 1 Thermometer dynamics, 49-52 Time constant, 5 1, 72 Tracking in controller and valve, 550-55 1 Trajectory, definition of, 484 Transducer, 139 Tr;msfer function, 49, 52-53 for distributed-parameter systems, 335-337, 343 simulation by computer, 532-534 Transfer function matrix, 446-452 Transfer lag, 83 Transform (see Laplace transform and Z-transform) Transition matrix, 447 Translation: of function, 40 of transform, 39 Transportation lag: computer simulation, 525-528 as a distributed parameter system, 338-339 Pade approximation, 102-103 transfer function, 101-102 Tuning rules, 282-295 TUTSIM simulation software, 532-540 Ultimate periodic response, 59 Underdamped response, 93-94 Unity feedback, 152
566 INDEX Valve, control, 303-316 C , 305 characteristics, 306-309 construction, 303-305 equal percentage, 308 hysteresis, 315 linear, 308 linearization of, 324-326 logarithmic, 308 positioner, 3 14-3 16 sizing, 305-306 transfer function, 127-128 Vector, column and row, 441 Weir, 64-65 Zero-order hold, 352-353 Zeros and poles, 178-182 Ziegler-Nichols settings, 233-234, 239-240, 286-287 z-transfoml, 354-355 inversion by long division, 362 inversion by partial fractions, 363-364 table, 356-357
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