目錄 i Index A A/D converters 468 digitization 453 Abscissa of convergence 610 Absolute value complex number 622 acker.m 361, 366, 377 Ackermann s estimator formula 380 Ackermann s formula derivation 648 pole placement 360,645 647 SRL 368 undamped oscillator example 360 AC motor actuators 34 Actuators 3 control system design procedure546 sizing root locus extension methods 512 Actuator saturation feedback system 510 Adaptive control 530 Adams prize 10 Addition matrix laws for 632 Additional poles effect of 94 98 exercise problems related to 122 123 locus branches 196 Adjugate matrix 633 Agilent Technologies243 Aircraft lateral and longitudinal control control system design procedure 561 574 nonlinear equations 561 563 Aircraft coordinates definition 562 Aircraft response using MATLAB example 96 98 Airy, G. B. 9 Aizerman s conjecture 533 Algebraic manipulations 623 624 Aliasing example 469 z and s plane459 Altimeter 569 Altitude-hold autopilot control system design procedure 569 574 Amplitude armature voltage torque speed curves 35 frequency response design method 295 Amplitude ratio 63 Amplitude stabilization 295 Analog computer components 336 implementation example 337 338 Analog controller discrete equivalent example 148 149 Analog implementation 209 210 Analog prefilters aliasing reduction 469 Analog to digital (A/D) converters 468 digitization 453 Analysis tools discrete design 473 474 Analytic functions complex variables 626 Analyzing system response tools 2 Anti-alias prefilters 469 470 effect of sample rate selection 472 Anti-windup compensation PI controller example 512 513 Anti-windup methods 533 Arbitrary nonsingular matrix 635 Argument principle 258 259, 627 629 contour evaluation 629 stated 259 Armature 34 Armature voltage amplitudes torque speed curves 35
Index 669 Arrival angles rules for positive locus 185 187 Assigned zeros 399 Associative law for addition 631 Associative law for multiplication 632 Åström, K. J.15, 451 Asymptotes angles of positive locus 185 center of positive locus 185 frequency response 248 Asymptotically stable Lyapunov 526 Asynchronous sampling sample rate selection digital control 473 Attitude hold 571 Attitude rate control 572 Augmented state equations integral control 406 Automatic control 0 Automobile cruise control component block diagram 5 Automobile speed control 172 Automobile suspension 20 Automotive engine fuel air ratio control system design procedure 574 580 Autonomous estimator description 397 example 398 399 Autopilot design block diagram 225 root locus 215 time response plots 215 Auxiliary variable control law design 358 axis.m 264, 267 B Back emf laws and equations 34 Back emf voltage 33 35 Ball levitator linearization of motion example 501 502 scaling example 99 Band center 87 Band reject filter 554 Bandwidth frequency response 243 Barometric altimeter altitude hold autopilot 570 Bang-bang control 525 Bel 264 Bell, Alexander Graham 264 Bell Laboratories 264 Bellman, R. 11 Bergen, A. R. 875 Bertram, J. E. 530 Bilateral Laplace transforms 610 Bilinear approximation (see Tustin s method) 461 Black, H. S. 11 Block diagram 80 84 algebra examples 81 cruise control plant5 reduction using MATLAB 84 signal flow graph 109 simplification example 85 state equations 339 349 third order system 336 transfer function example 82 83 Blocking zeroes tracking properties 414 Block triangular matrix inverse 634 Bode, H. W.11 Bode gain phase relationship 272 275 crossover frequency example 273 exercise problems related to 320 Bode gain phase theorem stated 272 weighting function graphically illustrated 273 bode.m 63, 242, 254 Bode plot complex poles example 252 253 complex poles and zeros example 253 254 frequency response advantages 246 minimum and nonminimum phase systems 254 multiple crossover frequency system example 271 273
670 Index real poles and zeros example 250 252 RHP 267 rules 250 Bode plot technique frequency response 244 254 Boeing 264 aircraft coordinates 262 lateral and longitudinal control control system design procedure 561 574 Boyd, S. P. 423 Brake point transfer function 248 Breakaway points 180 Break-in point 181 Break point 248 Bremmer, H. 66 Bridge tee circuit 29 Bryson, A. E., Jr. 11 Bryson s rule 372 Butterworth configuration 421 C c2d.m 151,281,395 Callender, A. 11,152 Campbell, G. A.66 Cancellations transfer function 76 Canon, R. H., Jr. 37 Canonical forms control 339 modal 341 observer 378 state equations339 466 canon.m 348 Capacitor frequency response characteristics example 242 symbol and equation 28 Carburetor 574 Cart stick balancer 46 Cascaded tanks 448 Case studies airplane altitude 570 airplane yaw damper 566 automobile fuel air ratio 574 disk drive 580 rapid thermal processing 585 satellite attitude 550 Catalytic converter 575 Cauchy s Principle of the Argument 258 Cauchy s residue theorem 627 Cauchy s theorem627 Cayley Hamilton theorem characteristic equation 639 square matrix 635 Characteristic equation 351 Cayley Hamilton theorem 639 closed loop system 356 Characteristic polynomial matrix 635 Cheap control 374 Circle criterion 532,533, 536 relationship with describing functions537 sector conditions 532 sector for signum nonlinearity 533 Clark, R. N. 233 Classical control 11 design 331 Clock digitization 454 Closed loop bandwidth phase margin 276 sample rate selection 471 Closed loop control 6, 127 Closed loop estimator 375 Closed loop frequency response 275 276 exercise problems related to 315 robust servomechanism 415 Closed loop poles root locus 178 Truxal s formula 400 Closed loop stability frequency response determination 257 Closed loop step responses 154 Closed loop system characteristic equation356 LTR 423 transfer function reference input 404 Closed loop transfer function 245,260 261 Closed loop zeros Truxal s formula 400 Cofactor matrix632 Column space fundamental subspaces 637 Column vector matrix 631 Communication satellite graphical illustration 24 Commutative law for addition 631
Index 671 Companion form matrix 358 Comparator 4 Compensation control law and estimator 354 control system design procedure 547 exercise problems related to 319 324 frequency response designed method 276 295 root locus 203,508 Compensator design (state-space) conditionally stable 392 exercise problems related to 445 449 full order example 378 388 reduced order example 390 391 transfer function 421 Complementary sensitivity function 302 Complete transient response 65 Complex addition 623 Complex division624 Complex multiplication 623 Complex number definitions622 623 Complex numbers arithmetic of623 Complex poles Bode plot examples 252 254 s -plane plot 87 88 Complex roots partial fraction expansion inverse Laplace transform example 617 619 Complex variables 622 629 Complex zeros Bode plot example 252 254 Composite curve transfer function class 250 Composite plot frequency response 245 Computed torque497, 499 Computer digital control 470 Computer-aided Bode plot complex poles and zeros 254 Computer aids 13 Conditionally stable compensator design 392 Conditionally stable system block diagram 506 frequency response design method 270 root locus 270 root locus extension methods 506 stability properties example 271 Conformal mapping 630 Conjugate complex number 622 Constant closed loop magnitude Nichols chart contours 296 Continuation locus 189 Continuity relation incompressible fluid flow 38 Continuous and discrete systems Simulink simulation 394 step response comparison 395 Continuous implementation digital implementation comparison 461 Continuous signal Laplace transform 458 460 Continuous system block diagram 453 Contour Cauchy s theorem627 isolated singularity 627 s -plane 629 Contour evaluation 259 argument principle 628 illustrated 259 Control canonical form 339 block diagram339 controllable system 646 derivation 660 equations 340 third order system 357 transfer function to state space (tf2ss) 340 transformation to controllability matrix 343 Control characteristic equation 356 Controllability 639 643 definitions344, 639, 640 642 Controllability Gramian 642 Controllability matrix 644 diagonal system 640 equation 343 transformation to control canonical form 343 Controllable systems 343 control canonical form 646 Control law estimator combined 385 396 Control law design
672 Index full state feedback 354 365 Ackermann s formula 359 363 exercise problems related to 437 440 finding355 362 pendulum example 356 357 reference input 360 365 system diagram 356 zero location example 361 Controllers 3 continuous and discrete comparison plots 151,210, 282 Simulink diagram 151, 281, 395 digital implementation 145 152 dimension 425 equations 397, 399 polynomial form 424 transfer function example 196 197 Control responses analog and digital implementations 210 Control system design 545 599 Boeing 747 lateral and longitudinal control 561 574 case studies 550 599 automotive engine fuel air ratio 574 580 definition 257 exercise problems related to 599 609 perspective 545 principles 545 549 procedural steps 545 549 read/write head assembly control 580 585 RTP systems 585 597 satellite attitude control design549 562 Control system model ODE 11 12 Control theory 9 conv.m 71, 159, 370 Convolution Laplace transform 69 property 615 Convolution integral 60 61 example 60 Cosine signal derivative Laplace transform example 613 Cover up method computing residues627 inverse Laplace transforms 70 Crane with hanging load 45, 56 Crossover frequency 269 stability margins 269 Cruise control model simple system example 18 21 Cruise control plant block diagram5 Cruise control step response example 21, 332 334 Cruise control transfer function using MATLAB example 77 ctrb.m 642 Current source symbol and equation 28 D D/A converters 469 digitization 454 Dakota autopilot 211 Damped natural frequency 87 Damping locus z -plane 458 Damping ratio 87 vs. overshoot 92 vs. phase margin 268 Damping response digital vs. continuous design example478 479 damp.m 478 DC gain Final Value Theorem example 72 z -plane 460 DC motor lag compensation example 89 90 lead compensation example 279 283 modeling example33 34 reference input example 364 365 sketch 32 DC motor actuators 32 DC motor position control system type example 141 142 DC motor transfer function using MATLAB example 77 78 DC servo example 391 392 modified dominant second order pole locations redesign example 396
Index 673 reduced order estimator redesign example 392 393 root locus pole assignment 391 DC servo compensator SRL redesign example 393 395 Decay ratio tuning 152 Decibel246 Delay approximation contrasting methods 221 hold operation 255 Delayed sinusoidal signal Laplace transform example 613 Departure angles calculation root locus design 197 rules for positive locus 186 Derivative discrete control laws 475 Derivative control anticipatory nature 144 Derivative of the error (D) 142 Describing function 498, 512, 514 conditionally stable system 519 for saturation nonlinearity 514 for relay nonlinearity 516 for hysteresis nonlinearity 517 stability analysis 518 stability analysis for hysteresis nonlinearity 520 Describing functions root locus extension method 505 Design considerations frequency response design method 294 295 Design criterion spacecraft attitude control example 274 275 Design parameters lead networks 283 Design trade-off satellite 369 Design trade-off (continued) yaw damper 569 Desired gain graphical calculation 201 Determinant properties 632 633 square matrix631 632 Diagonal matrix 634 block diagram640 Diagonal system controllability matrix 640 Difference equations 479 state space design method 479 Differential equations exercise problems related to51 53 geometry 330 state-variable form dynamic models 332 Differentiation complex variables 625 Laplace transform property613 Laplace transforms 68 Digital control 452 486 digitization 453 454 discrete design474 479 discrete equivalent design461 468 dynamic analysis 455 460 emulation design 461 468 exercise problems related to 487 496 hardware characteristics468 470 perspective 452 sample rate selection 471 472 state space design method 497 484 Digital controller block diagram 146, 453 space station attitude example 465 466 state space design methods example 481 484 Tustin s method example 461 463 Digital design continuous design damping and step response example 478 479 Digital implementations 209 210 continuous implementation comparison 461 controllers 145 152 Digital signals 145 Digital to analog (D/A) converters 469 Digitization 453 454 Diophantine equation 425 Dirac, Paul 60 impulse 60 Direct design rational transfer functions 424 427 Direct digital design. (see Discrete design) 452 Direct transfer function formulation 424 Direct transmission term 331 Discrete control laws475
674 Index Discrete controller compensator design 394 Discrete design 474 479 analysis tools 474 474 digitization 454 example 474, 475 477 feedback properties 475 477 Discrete equivalent analog controller example 148 149 Discrete root locus example 474 Discrete signals 145 digitization 453 Discrete state space representation 1/s 2 plant example 481 Discrete systems and continuous Simulink simulation 395 step response comparison 395 dynamic analysis 454 460 Discrete time functions z-transform and Laplace transforms 454 457 Discrete transfer function 474 Disk drive servomechanism example 411 415 Disk read/write mechanism graphical illustration 24 schematic 25 Distinct complex roots partial fraction expansion inverse Laplace transform example 617 620 Distinct real roots example 70 71 Distributed parameter systems 47 49 Disturbance 3 Disturbance inputs system type 139 140 Disturbance rejection 127, 130 132 properties robust servomechanism 541 sample rate selection 472 473 steady-state tracking extended estimator 417 420 Divide and conquer state space design 331 Dominant second order poles 365 367 example 367 368 Double integrator plant 191 transfer function 191 Double pendulum 50 Double precision digital control470 Doyle, J. C.421 Drebbel, Cornelis 8 8 Drebbel s incubator drawing 8 feedback 8 8 Duality estimation and control 379 Dutch roll yaw damper 566 Dynamic analysis discrete systems 455 460 Final Value Theorem 460 s and z relationship 458 460 z -transform 455 456 z -transform inversion 456 458 Dynamic compensation 203 210 exercise problems related to 229 233 Dynamic models 17 49 electric circuits 28 31 electromechanical systems 31 35 exercise problems related to 50 56 fluid flow models 35 44 heat flow models 35 44 linearization 500 mechanical systems 18 20 perspective 17 room temperature 36 scaling 98 state-variable form differential equations 332 Dynamic response 58 113 exercise problems related to 114 126 feedback changes 134 perspective 58 state equations 349 353 Dynamic system saturation 505 E Eigenvalue 345 Eigenvalues matrix 635 Eigenvector 345 Eigenvectors matrix 635 eig.m 346, 348, 482 Electric circuits differential equation determination example 28 29
Index 675 dynamic models 27 31 elements 28 equations and transfer functions example 29 31 exercise problems related to 51 53 Electric power line conductor 125 Electromagnet 31 32 Electromechanical systems dynamic models 31 35 exercise problems related to 53 54 Elementary block diagrams examples 81 Elementary matrix operation 631 Emulation digitization 461 Emulation design461 468 applicability limits 468 damping and step response example 478 479 digital approximation methods comparison 467 MMPZ method 466 MPZ method 463 466 space station attitude digital controller example 465 466 stages 461 Tustin s method 461 462 example 462 463 EPROM 470 Equilibrium 500,562 Equivalent gain analysis frequency response 512 root locus 505 Error feedback control 132 Error constants 136 Error space approach robust tracking 490 417 definition 409 design example 415 416 robust control equations 410 Estimate error characteristic equation 375 Estimator 354 and controller mechanism 386 control law combined 385 395 extended 417 Estimator design 375 385 exercise problems related to 440 445 full order 375 338 pendulum example 376 378 pole selection 383 385 reduced order 338 383 pendulum example 382 383 SRL pendulum example 385 Estimator equations 418 Estimator error equation 389 Estimator modes uncontrollability 405 Estimator pole selection 383 385 design rules 383 Estimator SRL equation 384 Euler s relations complex variables 625 Evans, W. R. 11, 177 Evans form characteristic equation 179 Evans method 178 Exact discrete equivalent 473 Expensive control 373 374 Experimental data obtaining models 108 109 sources 108 Exponential envelope second-quarter system response 89 Exponentially decaying sinusoid Laplace transform example 613 Exponential order Laplace transforms 610 Extended estimator 416 420 motor speed system block diagram 420 steady-state tracking example 419 420 system for tracking block diagram 418 Extra pole effect of 97 F Factored zero-pole form 79 Fast poles 86 Feedback advantage 132 amplifier 10, 28 analysis 4 7 configuring rate 144 control block diagram 130 DC motor position control 141 142 design trade-off 7 Drebbel s incubator 7 8 equations 130 131
676 Index exercise problems related to 162 176 liquid-level control 7 mathematical model 4 properties 127 161 perspective 127 Feedback changes dynamic response 134 Feedback control 2 13 component block diagram 4 components 4 definition 0 error 130 exercise problems related to 14 16 illustrated 129 systematic study 10 Feedback law integral control 407 feedback.m 84, 211, 478 Feedback of state 355 Feedback output error state estimate equation 375 Feedback properties discrete design 475 discrete design example 476 477 Feedback scheme autopilot design 211 Feedback structure example 196 197 Feedback system 2 7, 81 actuator saturation 510 block diagram 137 drawing 300 root locus 178 182 Feedforward 1 integral model example 416 417 of disturbances 547 Fibonacci numbers 487 Final Value Theorem 72 73 DC gain example 73 example 72 incorrect use of example 72 stable system 72 Truxal s formula 156 Finite zeros 75 First order system SRL 369 First order term transfer function 248 Fixed point arithmetic 471 Flexibility flexible disk read/write mechanism example 24 25 Flexible disk drive state-variable form example 335 Flexible disk read/write mechanism flexibility example 24 25 Float valve feedback 7 Fluid flow models dynamic models 36 44 exercise problems related to 55 56 Flyball governor 8 operating parts 10 Folding z and s -plane 459 Forced differential equations solution Laplace transforms 74 zero initial conditions 74 75 Forcing function 350 Forward path 109 Fourth order system modal canonical form block diagram 341 Free body diagram disk read/write mechanism 25 Newton s law 18 suspension system 21 Free running digitization 454 Frequency control system design procedure 545 546 Frequency response 11, 62, 240 256 characteristics example 241 control system design procedure 546 example 63 64 exercise problems related to 309 313 irst-order system example 624 lag lead compensation 403 lead compensation 277 PD compensation 277 state space design methods 388 Frequency response characteristics lead compensator example 242 243 Frequency response data
Index 677 experimental data source 108 phase margin 268 Frequency response design method 239 309 exercise problems related to 309 328 lag compensation 268 290 perspective 309 PI compensation 285 Frequency response plots 240 LTR design 422 Frequency response resonant peak vs. phase margin 269 Frequency shift Laplace transform property 613 Laplace transforms 68 Fuel-air ratio control 598 Fuel injection 576 Full order compensator design satellite attitude control example 387 389 Full-order estimator 374 Fundamental subspaces 637 G Gain calculation reference input 362 Gained stabilization 208 Gain margin frequency response design method 267 LTR 420 magnitude and phase plot 268 Gain phase relationship demonstration 272 Gain selection reference input 406 Gain stabilization frequency response design method 294 Gantmacher, F. R. 631 General controller polynomial form 424 Global positioning system (GPS) 2 GM 267 Golden Nugget Airlines Problem 235,324 GP-B satellite 435 GPS 2 Gramian 647, 648 gram.m 647, 648 Graphical evaluation magnitude and phase 625 Graphical interpretation numerical integration 147 Gyroscope yaw damper 564 H Hanging crane rotational motion example 44 47 translational motion example 44 47 Hard disk control system design procedure 580 585 Hardware characteristics digital control 468 670 Heat exchanger illustrated 38 modeling equations example 37 39 pure time delay example 429 431 root locus 222 tuning example 154 155 Heat flow equations example 36 37 models dynamic models35 36 exercise problems related to55 56 Heat transfer RTP system 590 Heaviside, Oliver 28 Helicopter near hover 227, 445 Hessenberg Matrix 438 Hidden technology 2 High frequency plant uncertainty effect of 294 High order system transfer function process reaction curve 151 Hold operation delay 454 Homogeneous differential equations solution Laplace transforms example 72 73 Huygens, Christian 9 Hydraulic actuators 43 modeling example 43 44 Hydraulic piston modeling example 40
678 Index I Identity matrix 632 Illustrative root locus 191 200 exercise problems related to 227 228 Imaginary operator definition 622 Impulse function transform Laplace transforms example 66 67 Impulse in theory 144 impulse.m 86, 90, 96 Impulse response 58, 84, 88 using MATLAB 86 Incompressible fluidflow 39 44 Incubator, Drebbel 7 Inductor symbol and equation 28 Inertial acceleration 18 initial.m377, 378 Initial Value Theorem 619 620 example 619 Inner loop design altitude-hold autopilot 571 Input generator internal model principle 412 Input magnitude stability nonlinear example 499 Input matrix 331 Input shaping filter 4 Integral discrete control laws 476 Integral control 406 420 augmented state equations 406 block diagram 407 description 406 409 error space design example 415 416 exercise problems related to 449 451 feedback law 406 internal model approach 411 motor speed system example 408 409 polynomial solution 426 structure 407 Integral model feedforward example 416 416 Integral of the error (I) 142 Integration complex variables 621 Laplace transform property 613 Laplace transforms 68 Integrator 336 op-amp example 30 31 Integrator anti-windup509 512 techniques 511 Intense initial pulse 144 Internal model approach integral control 410 Internal model principle 412 Internal stability 101 Interrupt digitization 453 Inverse block triangular matrix 634 Inverse Laplace transform partial fraction expansion 617 619 Inverse Laplace transforms partial-fraction expansion 69 71 example 70 71 Inverse nonlinearity 500 Inverse Nyquist plot 299 300 Inverse transform 65 Inverse transformation state equation analysis 349 Inversion matrix 633 Inverted pendulum 46 SRL estimator design 385 example 370 372 step response 372 inv.m 347, 349 Isoclines 523 Isolated singularity complex variables 627 contour 627 K Kailath, T. 642 Kalman, R. E. 12, 530, 639 Khalil, H. 514, 530 Kinematics 45 Kirchhoff s current law (KCL) 28 Kirchhoff s voltage law (KVL) 28 Kuo, B. 35 L L Hôpital s rule Truxal s formula 156
Index 679 Lag compensation 207 208 circuit illustration 210 compensation characteristic 294 DC motor example 289 290 definition 203 design procedure 287 frequency response design method 286 290 state space method example 401 404 Lamp nonlinearity RTP system 594 Lancaster, F. W. 572 LAPACK 345 Laplace transforms 59 80, 349, 610 620 continuous signal 458 460 definition 66 67 differential equations solving advantages of 74 dynamic analysis 454 455 exercise problems related to 114 116 for problem solving 72 74 homogeneous differential equations solution example 72 73 impulse function transform example 66 67 linear systems transfer functions 65 66 properties 65,67 69, 610 616 simple discrete time functions 456 sinusoid transform example 67 step and ramp transforms example 66 table 611 LaSalle, L. P. 530 Laurent series expansion form 627 Law of generators 32 Law of motors 31 Lead compensation 277 278, 277 285 analog and digital implementations 209 circuit illustration 210 compensation characteristic 294 DC motor example 279 283 definition 203 design example 205 206 design procedure 282,547 discrete control laws 476 frequency response 277 maximum phase increase 278 primary design parameters 282 using MATLAB 213 Lead compensator 193 design type one servomechanism system 284 285 example 193 194 frequency response characteristics 242 243 state-space example 390 temperature control system example 283 284 Lead ratio frequency response 279 Least common multiple extended estimator 417 Left companion matrix 387 Left eigenvector 642 Left eigenvectors matrix 635 Left half plane (LHP) effect of zero 192 Final Value Theorem 460 LTR 420 SRL 368 time functions 86 zeros nonminimum-phase systems 253 253 LHP. (see Left half plane (LHP)) 573 Limit cycle root locus extension method 507 Linear differential equations solutions 500 501 standard form 331 Linearization 500 by feedback 503 by inverse nonlinearity 504 by small-signal analysis 500 control system design procedure 547 definition 500 dynamic models 500 nonlinear feedback 503 nonlinear pendulum example 501 rapid thermal processing example 504 water tank height and outfl ow example 41 43 Linearization of motion ball levitator example 501 502 Linear quadratic based compensator design 421 Linear quadratic regulator (LQR) (see also Symmetric root locus (SRL)) 367 368 gain and phase margins 374
680 Index Nyquist plots 370 regulator poles limiting behavior 373 374 regulators robustness properties 374 tape drive example 372 373 Linear systems analysis using MATLAB 76 80 transfer functions Laplace transforms 65 66 ways of representing 76 Linear time invariant equations 1 Linear time invariant systems stability 100 101 Liquid flow control 7 Liquid level control 7 Liquid-level control feedback 7 Lissajous pattern 64 Locus real-axis parts 183 Logarithmic decrement 121 loglog.m 60, 242, 253, 422 logspace.m 60 Long division z -transform inversion 457 Loop gain 109 Loop path 109 Loop transfer recovery (LTR) 420 424 design frequency response plots 422 example 421 424 Loudspeaker geometry 31 32 modeling example 31 32 with circuit example 32 33 Low sensitivity design criteria 294 lqe.m 422, 592 LQR. (see Linear quadratic regulator (LQR))367 lqr.m 372 372,591 lsim.m 79, 313 LTR 420 424 Lumped parameter model49 Lyapunov, A. M. 11, 498 Lyapunov direct method 527 position feedback system example 529 530 Lyapunov equation 528, 644 Lyapunov first method 529 Lyapunov function 499, 527 properties 527 Lyapunov indirect method 529 Lyapunov second method. (see Lyapunov direct method) 499 Lyapunov stability 526 532 definition 526 exercise problems related to 543 544 nonlinear system 529 second-order system example 529 Lypapunov stability analysis 526 asymptotically stable 526 first method 526, 529 second method 526, 527 redesign of adaptive control530 stable in the large 526 Lyapunov stability theorem 527 M Magnitude complex number 622 frequency response 241 graphical evaluation 625 transfer function class 247 Magnitude condition parameter value 228 229 Magnitude plot gain and phase margin 268 M and N circles 296 Manual control 0 margin.m 284, 315, 422 Mason, S. J. 109 Mason s rule complex systems example 111 signal-flow graph 109 112 simple system example 110 Matched pole-zero (MPZ) method emulation design 462 466 MMPZ comparison 466 Mathematical model feedback 4 MathWorks 13 MATLAB 2, 13
Index 681 linear system analysis 76 80 step response 19 MATLAB (continued) acker 360,379 pole location 367 axis 263, 365 bode 83, 242, 253 c2d 151, 281, 395 canon 348 commands function name and description 648 648 conv 71, 159, 370 ctrb 642 damp 478, 642 eig 345, 348, 482 feedback 83, 211, 478 gram 642, 644 impulse 86, 90, 96 initial 377, 383 inv 462, 464 loglog 63, 242, 253 logspace 63 lqe 422, 592 lqr 493 lsim 79, 313 margin 284, 315, 423 max 305 nichols 296 nyquist 261, 263, 267 obsv 643 ones 79 pade 220 parallel 83 place 360, 379 plot 26, 79 poly 74 printsys 77 pzmap 86 residue 71, 74 rlocfind 198, 315 rlocus 181, 315, 370 rltool 228 roots 353 semilogx 64, 242, 422 semilogy 305 series 83, 422, 478 sqrt 413 ss2tf 338, 350 ss2zp 338 ssdata 484 step 20, 26, 78 tf20, 26, 64, 83 tf2ss 340 tf2zp 79, 338 tzero 352, 354, 482 Matrix definitions 631 operations 631 special 634 Matrix exponential 636 Matrix identity 637 Matrix inverse 631 632 Matrix pencil 641, 643 644 Matrix theory 631 638 max.m 305 Maxwell, James Clerk 9 Mayr, O. 7 McLauren series 220 McRuer, D. 12 Mechanical systems dynamic models18 20 exercise problems related to 50 51 Method of computed torque 500 Microphone 54 Microprocessors control applications 470 Minimum phase systems Bode plot 253 Minor matrix 632 Mixed control system pure discrete equivalent comparison 474 MMPZ method emulation design 466 MPZ comparison 467 Modal canonical form block diagram 340 fourth-order system block diagram 341 state equations example 340 341 Modal form transformation 345 example 346 347 Model definition 17 Modern contro89 Modern control design 331 Modes of the system 75 Modified dominant second order pole locations DC servo
682 Index redesign example 397 Modified matched pole-zero (MMPZ) method emulation design 466 MPZ comparison 467 Modified PD control example 193 194 Modulus complex number 623 Moler, C. B. 480 Monic 179 Monic polynomials 425 Motion equation development rigid bodies 27 Motor AC 34 DC 323 Motor position control root locus example 179 180 Motor speed PID example 145 Motor speed control plant transfer function example 149 150 Motor speed system extended estimator block diagram 420 integral control example 408 409 MPZ method cemulation design 463 466 MMPZ comparison 467 Multiple crossover frequency system Bode plot example 271 272 Multiplication matrix laws for 632 Multiplication by time Laplace transform property 616 Laplace transforms 69 Multipoint injection automotive engine fuel air ratio 576 N Natural frequency 335 z -plane 458 Natural mode 335 Natural responses 84 stability 89 N circles 296 Negative feedback 81 Negative locus 179 definition 215 plotting rules 215 217 sketching example 217 Negative root locus (see Negative locus) 215 Neutrally stability frequency response design method 255 258 exercise problems related to 313 Neutral stable system 101 graphical illustration 153 Newton s law rotational motion 23 translational motion 18 Nichols, N. 151, 296 Nichols chart 296 constant closed loop magnitude contours 296 PID example 297 stability margins example 298 nichols.m 296 Nodes 109 Noncollocated actuator 197 198 Noncollocated case root locus design 197 198 Noncollocated sensor 197 198 Nonlinear differential equations solution 500 501 Nonlinearity complications automotive engine fuel air ratio 577 Nonlinear radiation RTP system 590 Nonlinear sensor automotive engine fuel air ratio 576 Nonlinear systems 497 Lyapunov stability 526 root locus extension methods505 510 Nonminimum phase systems Bode plot 253 frequency response 253 254 LTR 422 Nonminimum phase zero 96 example 96 Nonsingular matrix 632
Index 683 Norm matrix 638 Normal form 354 state space design 330 Normal modes state equations 340 Notch compensation 208 209 definition 203 root locus extension method 509 Notch filter block diagram 510 Null space fundamental subspaces 639 Numerical integration graphical interpretation 147 Nyquist, H. 11 Nyquist and Shannon sampling theorem 469 Nyquist criterion Lyapunov stability 536 Nyquist frequency s and z-plane 459 nyquist.m 262, 264, 267 Nyquist plot 260 defining gain and phase margin 268 evaluation 260 LQR design 294 multiple crossover frequency system example 271 272 open loop unstable system example 264 267 plotting procedure 261 pole locations 370 second order system example 261 262 third order system example 262 264 time delay system example 306 307 using MATLAB 263 264 vector margin 270 Nyquist stability criterion 257 267 exercise problems related to 313 314 O Observability 378, 639, 643 645 definitions 643 Observability Gramian 645 Observability matrix 378 379 equation 378 Observer 354 Observer canonical form 344, 377 379 block diagram 378 DC servo 389 equation 377 illustrated 345 third order system 378 obsv.m 643 ODE control system model 11 One-sided Laplace transforms 66, 610 ones.m 79 Op-amp integrator example 30 31 schematic symbol 29 simplified circuit 29 summer 30 Open loop 5 autopilot design 211 Open loop control system 0 illustrated 128 Open loop cruise control 5 Open loop estimator 375 Open loop transfer function 256, 523 Open loop unstable system Nyquist plot example 264 267 Optimal control 368 Optimal design control system design procedure 548 Optimal estimation 384 Ordinary differential equation (ODE) control system model 11 Orthogonal fundamental subspaces 637 Orthogonal matrix 637 Oscillator time scaling example 338 Oscillatory behavior example 154 155 Oscillatory systems block example 508 saturation nonlinear example 508 Oscillatory time response example 88 90 Output equation 349 state description 341 Output matrix 331
684 Index Output response analog and digital implementations 210 Overshoot definition 90 plot 95 time domain specification 90 92 vs. damping ratio 92 P Padé approximant 220 pade.m 220 Paper machine 14 parallel.m 84 Parameter consideration of two 218 219 Parameter changes output dependency block diagram 157 system gain sensitivity 131 132 time response sensitivity 157 159 Parameter range vs. stability example 104 107 Parameter selection satellite attitude control 552 Parameter value selection 201 203 exercise problems related to 228 229 Parseval s theorem 534 Partial fraction expansion distinct complex roots inverse Laplace transform example 617 619 frequency response 240 inverse Laplace transform 617 619 inverse Laplace transforms 69 71 example 70 71 repeated real roots inverse Laplace transform example 619 Partial state control canonical form 358 Path 109 Path gain 109 PD.(see Proportional derivative (PD)) 277 Peak amplitude second order transfer function 249 Peak times definition 90 time domain specification 91 92 Pendulum Example control law 356 357 estimator design 376 377 linear and nonlinear response 500 nonlinear equations 26 reduced order estimators 382 383 rotational motion 26 27 SRL estimator design 385 rotational motion step input 26 Performance bound function example 300 302 plot 301 Perturbations expressions for 157 robust 410 Perturbed equations 157 Phase 63 frequency response 241 graphical evaluation 625 Phase condition 182 Phase lag between output and input 65 vs. time delay 305 Phase margin closed loop bandwidth 276 frequency response data 269 frequency response design method 267 LTR 420 magnitude and phase plot 269 vs. damping ratio 269 vs. frequency response resonant peak 269 vs. transient response overshoot 269 Phase plane 522 state space design 330 Phase-plane plot with saturation 524 Phase plot 249 gain and phase margin 269 Phase stabilization 208 phase response design method 294 PHR test 641-643 Phugoid mode altitude-hold autopilot 572 PID. (see Proportional-integral-derivative (PID)) 142, 290 Piper Dakota autopilot 211 Pitch angle 562 place.m 362, 380, 396 Plant 4 1/s 2 discrete state space representation example 481 connected to estimator 377
Index 685 Plant changes control system design procedure 548 Plant evaluation/modification control system design procedure 547 Plant inversion LTR 421 Plant open loop pole root locus example 180-181 Plant transfer function motor speed control example 149-150 Plant uncertainty example 302 plot 301 plot.m 26, 79 PM 267 PM to damping ratio rule 269 Polar plot 260 Pole and zeros finding using MATLAB 347 349 Bode plot example 250 252 compensator design 445 correlation example 89 90 definition 75 inverse Laplace transforms 70 number of argument principle 627 630 of the system 75 partial fraction expansion inverse Laplace transform example 616 rational transfer function 74 76 response character indication 74 Pole assignment root locus DC servo 391 SRL 393 Pole location/placement 85 90 Ackermann s formula 359 645 647 corresponding impulse responses 88 exercise problems related to 118 119 polynomial transfer functions example 425 426 selection 366 375 dominant second order poles 366 368 example 377 368, 369 373 exercise problems related to 440 441 methods 366, 375 SRL 368 374 Pole selection SRL 396 Pole-zero cancellation 404, 624 Pole-zero patterns effects of on dynamic response 98 Pole-zero specifications control system design procedure 545 546 poly.m 74 Polynomial solution integral control 426 Polynomial transfer functions pole placement example 425 426 reduced order design example 426 428 Pontryagin, L. S. 11 Popov-Hautus-Rosebrock (PHR) test 641 642 Position error constant 255 Position feedback system Lyapunov direct method example 529 530 Positive definite matrix 638 determining methods 638 Positive feedback 81 Positive locus 179, 182 plotting rules 183 190 Positive root locus. (see Positive locus) 201 Positive semidefinite matrix 638 Prefilter RTP system 594 printsys.m 77 Problem solving using Laplace transforms 73 74 Process 4 Process noise estimator pole selection 381 Process reaction curve 152 graphical illustration 152, 154 Programmable read only memory (EPROM) 470 Proportional derivative (PD) compensation 277 compensation characteristic 294 frequency response 278 Proportional feedback control (P) 142 148 discrete control laws 475 Proportional-integral-derivative (PID) 10, 127 controller 142, 144 146 example Nichols chart 297
686 Index frequency response design method 290 294 motor speed example 145 regulators Ziegler-Nichols tuning 152 155 spacecraft attitude control example 290 294 state-space design 547 Proportional integral (PI) anti-windup compensation example 512 512 compensation frequency response design method 285 control system 105, 143 144 compensation characteristic 294 Prototype testing control system design procedure 548 Proximate time-optimal system (PTOS) 525, 526 Pseudorandom binary signal (PRBS) 109 Pseudorandom-noise data experimental data source 108 109 Pure discrete equivalent mixed control system comparison 474 Pure time delay design for system 428 431 exercise problems related to 496 heat exchanger example 429 431 pzmap.m 86 Q QR algorithm 272 Quality factor 87 Quantized signals 146 Quarter car-model 21 Quarter decay ratio example 154 155 graphic illustration 193 Quasidiagonal matrix 637 R Ragazzini, J. R. 27 RAM 470 Ramp response first order system Laplace transform example 615 Ramp signal robust tracking 409 Random access memory (RAM) 470 Range space fundamental subspaces 637 Rank matrix 635 Rapid thermal processing (RTP) system control system design procedure 585 597 laboratory model 589 linear model 590 Rational function complex variables 627 Rationalization algebraic manipulations 625 Rational transfer functions direct design 424-427 Reaction jets 551 Reaction wheels 551 Read/write head assembly control system design procedure 580 585 Read-only memory (ROM) 470 Rectangular matrix 631 Reduced order compensator design satellite attitude control example 390 390 Reduced order compensator transfer function 387 Reduced order design polynomial transfer function model example 426 428 Reduced order estimator 380 381 DC servo redesign example 392 393 pendulum example 382 383 structure 382 Reference input alternative structures 365 estimator 369 406 gain selection 406 general structure 397 406 example 363 364 full state feedback 362 365 block diagram 363 gain equation 362 oscillator step response 364 selection methods 397 type 1 system example 364 365 Reference spectrum plot 360
Index 687 Reference tracking 127 system type 135 137 Reference value 13 Region of convergence 610 Regulators 0 compensator design 445 Relay 505 Repeated poles partial fraction expansion inverse Laplace transform example 619 Repeated real roots partial fraction expansion inverse Laplace transform example 619 Reset control 510 Reset windup 510 residue.m 71, 74 Residues complex variables 626 627 Residue theorem 627 Resistor symbol and equation 28 Resonant frequency Nichols chart 296 Resonant peak frequency response 243 Response by convolution 750 60 to sinusoid 62 vs. pole locations and real roots example 85 86 Response sensitivity 168 RHP. (see Right half-plane (RHP)) 80 Right eigenvectors matrix 635 Right half-plane (RHP) 80 81 Bode plot 267 compensator transfer function 386 LTR 422 Lyapunov stability 528 root locus extension method 216 SRL 368 unstable closed loop system 186 187 zeros 96 example 97 nonminimum-phase systems 254 255 Rigid bodies motion equation development 27 Ring-lasers gyroscope yaw damper 564 Rise time definition 84 time domain specification 85 rlocfind.m 198, 315 rlocus.m 181, 315, 369 rltool 228 RMS 423 RMS value 423 424 Robust 410 Robust control definition 410 equations error space 410 Robust control (continued) sinusoid example 411 415 Robust properties 410 LQR regulators 374 system type 137 Robust servomechanism closed loop frequency response 415 disturbance rejection properties 414 Simulink block diagram 414 tracking properties 414 Robust tracking 406 420 error space approach 409 417 exercise problems related to 449 451 Rolling mill 117 Roll mode yaw damper 565 ROM 470 Room temperature control system component block diagram 3 dynamic model 36 Root locus 11, 177 0 degree locus. (see Negative locus) 182 180 degree locus. (see Positive locus) 182 autopilot design 213 closed loop poles 219 combine control and estimator 389 compensation 508 complex multiple roots example 198 199 conditionally stable system 270 DC servo pole assignment 390 reduced order estimator 392 definition 182 design example 210 215 exercise problems related to 224, 227 228, 233 237 feedback system 178 182
688 Index illustrative 191 200 lead design 205 lead lag 209 motor position control example 179 180 notch compensations 209 plant open-loop pole example 180 181 plotting rules 190 191 reduced order controller 390 rules application example 197 198 satellite attitude control PD control 191 192 satellite control collocated flexibility 196 197 lead compensator 192 193 small value for pole 293 294 transition value for pole 194 196 sketching guidelines 182 191 exercise problems related to 224 SRL pole assignment 393 stability examples 255 time delay 220 heat exchanger 221 Root locus design method 177 223 exercise problems related to 224 238 perspective 177 Root locus forms 179 Root locus method extensions 287 219 exercise problems related to 237 238 of Evans 178 Root mean square (RMS) value 423 424 roots.m 104, 193, 353 Rotational motion hanging crane example 44 47 Newton s law 23 pendulum example 25 27 satellite attitude control model example 23 24 Rotor free body diagram 33 Routh, E. J. 10 Routh s array 102 103 Routh s criterion Lyapunov stability 529 Routh s stability criterion 101 107, 111 112 Routh s test example 103 104, 107 108 Row vector matrix 631 RTP system control system design procedure 585 597 Rudder 562 Run-to-run control 608 S Saberi, A. 420 Sample digitization 453 Sampled data system 453 Sampled signals 146 Sample period digitization 453 Sample rate digitization 453 lower limit 471 Sample rate selection digital control 471 472 anti-alias prefilter 472 asynchronous sampling 473 disturbance rejection 472 472 tracking effectiveness 471 472 digitization 453 Sampling theorem Nyquist and Shannon 469 Sandberg, I. W. 536 Sastry, S. 530 Satellite flexible appendages example 252 254 Satellite attitude control application control system design procedure 549 561 example 140 141 full order compensator design 387 388 LTR 422 424 reduced order compensator design 388 390 rotational motion 23 24 SRL 369 370 state-variable form 332 model 141 Satellite control root locus collocated flexibility 196 197 lead compensator 193 194 small value for pole 194 195 transition value for pole 195 196 Satellite control schematic 24
Index 689 Satellite tracking signal 134 Satellite transfer function using MATLAB example 79 80 Saturation dynamic system 505 oscillatory systems nonlinear example 508 Scaling ball levitator example 99 dynamic models 500 time 68 Schmitt trigger circuit 517 Schmitz, E. 49 Second order equations external signals 416 Second order servomechanism example 401 404 Second order step responses transfer functions 96 Second order system block diagram 82 Lyapunov stability example 529 Nyquist plot example 262 263 responses 88 exponential envelope 89 step response plots 88 Second order term transfer function class 249 Semiconductor wafer manufacturing control system design procedure 585 597 semilogx.m 64, 242, 424 semilogy.m 305 Sensitivity 127 parameter changes 131 132 time response parameter change 157 159 Sensitivity function 130, 300 design limitations 303 305 example 303 304 plot and computation example 305 specifications 300 305 exercise problems related to 326 327 Sensor 4 control system design procedure 546 Sensor noise estimator pole selection 384 Separation principle compensator design 445 series.m 83, 422, 478 Servomechanisms 12 increasing velocity constant zero assignment 401 404 Servomechanism structure block diagram 218 Servomechanism system lead compensator design example 284 285 Servo motor torque-speed curves 35 Servo motor. (see also DC servo) 35 Servo speed control SRL example 369 Servo systems 0 Servo with tachometer feedback system type 138 139 Settling time definition 90 time domain specification 92 93 Shift in frequency Laplace transform property 613 Short-period mode altitude-hold autopilot 572 Signal decay rate of 86 Signal-flow graph block diagram 110 Mason s rule 109 112 Similarities complex variables 626 627 Similarity transformations matrix 635 Simple closed contour Cauchy s theorem 627 Simple design criterion spacecraft attitude control example 274 275 Simple discrete time functions z-transform and Laplace transforms 456 Simulation block diagram transfer function 336 338 control system design procedure 548 Simulink 2 block diagram robust servomechanism 413 LTR
690 Index block diagram 423 Simulink nonlinear simulation automotive engine fuel air ratio 579 RTP system 544 Simulink simulation continuous and discrete systems 395 Single point automotive engine fuel air ratio 576 Singularities transfer function 247 Singular value matrix 637 Singular value decomposition matrix 637 Singular vectors matrix 638 Sinusoid robust control example 412 415 with frequency Laplace transform example 612 Sinusoidal signal Laplace transform example 612 time integral 615 time product 617 Sinusoid of frequency compensator structure 412 Sinusoid transform Laplace transforms example 67 Sizing the actuator root locus extension methods 512 Slow poles 86 Small signal linearization 499 500 Smith compensator 428 Smith regulator time delay 428 Spacecraft attitude control example PID compensation 293 294 simple design criterion 274 275 Space station attitude emulation design digital controller example 465 466 Space station digital controller direct discrete design example 475 477 Specifications control system design procedure 544 546 Specific heat heat flow 37 Spectral analyzers 243 Speed control 2, 10 comparison plots 151 exercise problems related to 162 163 Speed controller example 143 Spiral mode yaw damper 566 S -plane and z -plane relationship digital control 458 460 Cauchy s theorem 627 complex poles 87 contour 621 RHP contour 259 specification transformation example 93 time domain specification 93 time functions 86 sqrt.m 413 Square matrix 622 Cayley-Hamilton theorem 628 determinant 623 trace 622 SRL. (see Symmetric root locus (SRL)) 367 ss2tf.m MATLAB function 338, 350 ss2zp.m 388 ssdata.m 484 ss.m 79, 348 Stability 84, 101 108 examples system definition and root locus 256 exercise problems related to 123 126 Stability (continued) feedback system 104 input magnitude nonlinear example 506 linear time invariant systems 100 101 Lyapunov 526 natural responses 89 necessary condition for 101 Routh s criterion 187 188 vs. parameter range example 104 105 vs. two parameter ranges example 105 107 Stability analysis 10
Index 691 Stability augmentation Boeing 562 Stability condition frequency response 258 Stability margins frequency response design method 267 272 exercise problems related to 315 319 Nichols chart example 298 Stability properties conditionally stable system example 271 Stability robustness performance bound function 301 Stabilization amplitude 294 phase 294 Stable compensator 392 Stable minimum phase system 272 Stable system block diagram 506 definition 101 Staircase algorithm 483 Star tracker satellite attitude control 550 State description equation 349 thermal systems zeros 352 thermal system transfer function example 350 350 State equations analysis 338 354 block diagrams 339 349 canonical forms 339 349 examples 340 341, 346 349, 353 354 exercise problems related to 435 436 State estimate equation feedback output error 375 State feedback control law state variable equation 645 State of the system 331 State space frequency response design methods 388 State space control design 330 State space design 329 432 advantages 330 331 compensator design 385 395 control law design full state feedback 354 365 estimator design 374 385 exercise problems related to 435 451 gain selection 406 integral control 406 420 LTR 420 424 Lyapunov stability 526 532 perspective 329 pole location selection 365 374 pure time delay 428 431 rational transfer functions 424 428 reference input with estimator 396 406 robust tracking 406 420 state equation analysis 338 354 State space method lag compensation example 401 404 State space pole placement method example 390 392 State space to transfer function (ss2tf) MATLAB function 353 States space design elements schematic diagram 355 States space design methods 528 529 example 527 digital controller 528 529 State transformation 342 State variable design 12 State variable equation state feedback control law 645 State variable form differential equations dynamic models 332 example flexible disk drive 335 satellite attitude control 332 Steady state error 253 255 command inputs and disturbances 294 determination example 254 Steady state information stochastic experimental data source 108 Steady state phase difference 64 Steady state tracking 134 157 disturbance rejection extended estimator 419 420 exercise problems related to 172 176 Steam engine 8 Index 907 Step and ramp transforms Laplace transforms example 66 step.m 20, 26, 78, 333