Index 669 Arrival angles rules for positive locus Assigned zeros 399 Associative law for addition 631 Associative law for multiplication 632 Å

Transcription

1 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, 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 exercise problems related to locus branches 196 Adjugate matrix 633 Agilent Technologies243 Aircraft lateral and longitudinal control control system design procedure nonlinear equations Aircraft coordinates definition 562 Aircraft response using MATLAB example Airy, G. B. 9 Aizerman s conjecture 533 Algebraic manipulations Aliasing example 469 z and s plane459 Altimeter 569 Altitude-hold autopilot control system design procedure Amplitude armature voltage torque speed curves 35 frequency response design method 295 Amplitude ratio 63 Amplitude stabilization 295 Analog computer components 336 implementation example Analog controller discrete equivalent example Analog implementation Analog prefilters aliasing reduction 469 Analog to digital (A/D) converters 468 digitization 453 Analysis tools discrete design Analytic functions complex variables 626 Analyzing system response tools 2 Anti-alias prefilters effect of sample rate selection 472 Anti-windup compensation PI controller example Anti-windup methods 533 Arbitrary nonsingular matrix 635 Argument principle , contour evaluation 629 stated 259 Armature 34 Armature voltage amplitudes torque speed curves 35

2 Index 669 Arrival angles rules for positive locus 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 Autonomous estimator description 397 example 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 Ball levitator linearization of motion example 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 algebra examples 81 cruise control plant5 reduction using MATLAB 84 signal flow graph 109 simplification example 85 state equations third order system 336 transfer function example Blocking zeroes tracking properties 414 Block triangular matrix inverse 634 Bode, H. W.11 Bode gain phase relationship 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 complex poles and zeros example frequency response advantages 246 minimum and nonminimum phase systems 254 multiple crossover frequency system example

3 670 Index real poles and zeros example RHP 267 rules 250 Bode plot technique frequency response Boeing 264 aircraft coordinates 262 lateral and longitudinal control control system design procedure 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 equations 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 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, 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

4 Index 671 Companion form matrix 358 Comparator 4 Compensation control law and estimator 354 control system design procedure 547 exercise problems related to frequency response designed method root locus 203,508 Compensator design (state-space) conditionally stable 392 exercise problems related to full order example reduced order example transfer function 421 Complementary sensitivity function 302 Complete transient response 65 Complex addition 623 Complex division624 Complex multiplication 623 Complex number definitions Complex numbers arithmetic of623 Complex poles Bode plot examples s -plane plot Complex roots partial fraction expansion inverse Laplace transform example Complex variables Complex zeros Bode plot example 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 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 definitions344, 639, 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 Control law design

5 672 Index full state feedback Ackermann s formula exercise problems related to finding pendulum example reference input system diagram 356 zero location example 361 Controllers 3 continuous and discrete comparison plots 151,210, 282 Simulink diagram 151, 281, 395 digital implementation dimension 425 equations 397, 399 polynomial form 424 transfer function example Control responses analog and digital implementations 210 Control system design Boeing 747 lateral and longitudinal control case studies automotive engine fuel air ratio definition 257 exercise problems related to perspective 545 principles procedural steps read/write head assembly control RTP systems satellite attitude control design Control system model ODE Control theory 9 conv.m 71, 159, 370 Convolution Laplace transform 69 property 615 Convolution integral 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 Cruise control plant block diagram5 Cruise control step response example 21, 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 example damp.m 478 DC gain Final Value Theorem example 72 z -plane 460 DC motor lag compensation example lead compensation example modeling example33 34 reference input example sketch 32 DC motor actuators 32 DC motor position control system type example DC motor transfer function using MATLAB example DC servo example modified dominant second order pole locations redesign example 396

6 Index 673 reduced order estimator redesign example root locus pole assignment 391 DC servo compensator SRL redesign example 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 Design criterion spacecraft attitude control example Design parameters lead networks 283 Design trade-off satellite 369 Design trade-off (continued) yaw damper 569 Desired gain graphical calculation 201 Determinant properties square matrix 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 digitization discrete design discrete equivalent design dynamic analysis emulation design exercise problems related to hardware characteristics perspective 452 sample rate selection state space design method Digital controller block diagram 146, 453 space station attitude example state space design methods example Tustin s method example Digital design continuous design damping and step response example Digital implementations continuous implementation comparison 461 controllers Digital signals 145 Digital to analog (D/A) converters 469 Digitization Diophantine equation 425 Dirac, Paul 60 impulse 60 Direct design rational transfer functions Direct digital design. (see Discrete design) 452 Direct transfer function formulation 424 Direct transmission term 331 Discrete control laws475

7 674 Index Discrete controller compensator design 394 Discrete design analysis tools digitization 454 example 474, feedback properties Discrete equivalent analog controller example 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 Discrete time functions z-transform and Laplace transforms Discrete transfer function 474 Disk drive servomechanism example Disk read/write mechanism graphical illustration 24 schematic 25 Distinct complex roots partial fraction expansion inverse Laplace transform example Distinct real roots example Distributed parameter systems Disturbance 3 Disturbance inputs system type Disturbance rejection 127, properties robust servomechanism 541 sample rate selection steady-state tracking extended estimator Divide and conquer state space design 331 Dominant second order poles example 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 Final Value Theorem 460 s and z relationship z -transform z -transform inversion Dynamic compensation exercise problems related to Dynamic models electric circuits electromechanical systems exercise problems related to fluid flow models heat flow models linearization 500 mechanical systems perspective 17 room temperature 36 scaling 98 state-variable form differential equations 332 Dynamic response exercise problems related to feedback changes 134 perspective 58 state equations 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

8 Index 675 dynamic models elements 28 equations and transfer functions example exercise problems related to Electric power line conductor 125 Electromagnet Electromechanical systems dynamic models exercise problems related to Elementary block diagrams examples 81 Elementary matrix operation 631 Emulation digitization 461 Emulation design applicability limits 468 damping and step response example digital approximation methods comparison 467 MMPZ method 466 MPZ method space station attitude digital controller example stages 461 Tustin s method example 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 definition 409 design example robust control equations 410 Estimate error characteristic equation 375 Estimator 354 and controller mechanism 386 control law combined extended 417 Estimator design exercise problems related to full order pendulum example pole selection reduced order pendulum example SRL pendulum example 385 Estimator equations 418 Estimator error equation 389 Estimator modes uncontrollability 405 Estimator pole selection 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 Experimental data obtaining models sources 108 Exponential envelope second-quarter system response 89 Exponentially decaying sinusoid Laplace transform example 613 Exponential order Laplace transforms 610 Extended estimator motor speed system block diagram 420 steady-state tracking example 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 design trade-off 7 Drebbel s incubator 7 8 equations

9 676 Index exercise problems related to liquid-level control 7 mathematical model 4 properties 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 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 Feedback scheme autopilot design 211 Feedback structure example Feedback system 2 7, 81 actuator saturation 510 block diagram 137 drawing 300 root locus Feedforward 1 integral model example of disturbances 547 Fibonacci numbers 487 Final Value Theorem 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 Flexible disk drive state-variable form example 335 Flexible disk read/write mechanism flexibility example Float valve feedback 7 Fluid flow models dynamic models exercise problems related to Flyball governor 8 operating parts 10 Folding z and s -plane 459 Forced differential equations solution Laplace transforms 74 zero initial conditions 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 Frequency response 11, 62, characteristics example 241 control system design procedure 546 example exercise problems related to 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 Frequency response data

10 Index 677 experimental data source 108 phase margin 268 Frequency response design method exercise problems related to lag compensation 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 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 translational motion example Hard disk control system design procedure Hardware characteristics digital control Heat exchanger illustrated 38 modeling equations example pure time delay example root locus 222 tuning example Heat flow equations example 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 Huygens, Christian 9 Hydraulic actuators 43 modeling example Hydraulic piston modeling example 40

11 678 Index I Identity matrix 632 Illustrative root locus exercise problems related to Imaginary operator definition 622 Impulse function transform Laplace transforms example Impulse in theory 144 impulse.m 86, 90, 96 Impulse response 58, 84, 88 using MATLAB 86 Incompressible fluidflow Incubator, Drebbel 7 Inductor symbol and equation 28 Inertial acceleration 18 initial.m377, 378 Initial Value Theorem 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 augmented state equations 406 block diagram 407 description error space design example exercise problems related to feedback law 406 internal model approach 411 motor speed system example polynomial solution 426 structure 407 Integral model feedforward example Integral of the error (I) 142 Integration complex variables 621 Laplace transform property 613 Laplace transforms 68 Integrator 336 op-amp example Integrator anti-windup 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 Inverse Laplace transforms partial-fraction expansion example Inverse nonlinearity 500 Inverse Nyquist plot Inverse transform 65 Inverse transformation state equation analysis 349 Inversion matrix 633 Inverted pendulum 46 SRL estimator design 385 example 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

12 Index 679 Lag compensation circuit illustration 210 compensation characteristic 294 DC motor example definition 203 design procedure 287 frequency response design method state space method example Lamp nonlinearity RTP system 594 Lancaster, F. W. 572 LAPACK 345 Laplace transforms 59 80, 349, continuous signal definition differential equations solving advantages of 74 dynamic analysis exercise problems related to for problem solving homogeneous differential equations solution example impulse function transform example linear systems transfer functions properties 65,67 69, 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 , analog and digital implementations 209 circuit illustration 210 compensation characteristic 294 DC motor example definition 203 design example 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 example frequency response characteristics state-space example 390 temperature control system example 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 LHP. (see Left half plane (LHP)) 573 Limit cycle root locus extension method 507 Linear differential equations solutions 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 Linearization of motion ball levitator example Linear quadratic based compensator design 421 Linear quadratic regulator (LQR) (see also Symmetric root locus (SRL)) gain and phase margins 374

13 680 Index Nyquist plots 370 regulator poles limiting behavior regulators robustness properties 374 tape drive example Linear systems analysis using MATLAB transfer functions Laplace transforms ways of representing 76 Linear time invariant equations 1 Linear time invariant systems stability 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) design frequency response plots 422 example Loudspeaker geometry modeling example with circuit example Low sensitivity design criteria 294 lqe.m 422, 592 LQR. (see Linear quadratic regulator (LQR))367 lqr.m ,591 lsim.m 79, 313 LTR Lumped parameter model49 Lyapunov, A. M. 11, 498 Lyapunov direct method 527 position feedback system example 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 definition 526 exercise problems related to 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 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 simple system example 110 Matched pole-zero (MPZ) method emulation design MMPZ comparison 466 Mathematical model feedback 4 MathWorks 13 MATLAB 2, 13

14 Index 681 linear system analysis 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 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 Matrix pencil 641, Matrix theory 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 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 Modal form transformation 345 example Model definition 17 Modern contro89 Modern control design 331 Modes of the system 75 Modified dominant second order pole locations DC servo

15 682 Index redesign example 397 Modified matched pole-zero (MMPZ) method emulation design 466 MPZ comparison 467 Modified PD control example 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 Motor speed PID example 145 Motor speed control plant transfer function example Motor speed system extended estimator block diagram 420 integral control example MPZ method cemulation design MMPZ comparison 467 Multiple crossover frequency system Bode plot example 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 sketching example 217 Negative root locus (see Negative locus) 215 Neutrally stability frequency response design method 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 Noncollocated case root locus design Noncollocated sensor Nonlinear differential equations solution 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 methods Nonminimum phase systems Bode plot 253 frequency response LTR 422 Nonminimum phase zero 96 example 96 Nonsingular matrix 632

16 Index 683 Norm matrix 638 Normal form 354 state space design 330 Normal modes state equations 340 Notch compensation 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 open loop unstable system example plotting procedure 261 pole locations 370 second order system example third order system example time delay system example using MATLAB vector margin 270 Nyquist stability criterion exercise problems related to O Observability 378, 639, definitions 643 Observability Gramian 645 Observability matrix equation 378 Observer 354 Observer canonical form 344, 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 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 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 Oscillatory systems block example 508 saturation nonlinear example 508 Oscillatory time response example Output equation 349 state description 341 Output matrix 331

17 684 Index Output response analog and digital implementations 210 Overshoot definition 90 plot 95 time domain specification vs. damping ratio 92 P Padé approximant 220 pade.m 220 Paper machine 14 parallel.m 84 Parameter consideration of two Parameter changes output dependency block diagram 157 system gain sensitivity time response sensitivity Parameter range vs. stability example Parameter selection satellite attitude control 552 Parameter value selection exercise problems related to Parseval s theorem 534 Partial fraction expansion distinct complex roots inverse Laplace transform example frequency response 240 inverse Laplace transform inverse Laplace transforms example 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 Pendulum Example control law estimator design linear and nonlinear response 500 nonlinear equations 26 reduced order estimators rotational motion SRL estimator design 385 rotational motion step input 26 Performance bound function example 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 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

18 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 Plant transfer function motor speed control example 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 Bode plot example compensator design 445 correlation example definition 75 inverse Laplace transforms 70 number of argument principle of the system 75 partial fraction expansion inverse Laplace transform example 616 rational transfer function response character indication 74 Pole assignment root locus DC servo 391 SRL 393 Pole location/placement Ackermann s formula corresponding impulse responses 88 exercise problems related to polynomial transfer functions example selection dominant second order poles example , exercise problems related to methods 366, 375 SRL 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 poly.m 74 Polynomial solution integral control 426 Polynomial transfer functions pole placement example reduced order design example Pontryagin, L. S. 11 Popov-Hautus-Rosebrock (PHR) test Position error constant 255 Position feedback system Lyapunov direct method example Positive definite matrix 638 determining methods 638 Positive feedback 81 Positive locus 179, 182 plotting rules Positive root locus. (see Positive locus) 201 Positive semidefinite matrix 638 Prefilter RTP system 594 printsys.m 77 Problem solving using Laplace transforms 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) discrete control laws 475 Proportional-integral-derivative (PID) 10, 127 controller 142, example Nichols chart 297

19 686 Index frequency response design method motor speed example 145 regulators Ziegler-Nichols tuning spacecraft attitude control example state-space design 547 Proportional integral (PI) anti-windup compensation example compensation frequency response design method 285 control system 105, 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 Pure discrete equivalent mixed control system comparison 474 Pure time delay design for system exercise problems related to 496 heat exchanger example pzmap.m 86 Q QR algorithm 272 Quality factor 87 Quantized signals 146 Quarter car-model 21 Quarter decay ratio example 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 laboratory model 589 linear model 590 Rational function complex variables 627 Rationalization algebraic manipulations 625 Rational transfer functions direct design Reaction jets 551 Reaction wheels 551 Read/write head assembly control system design procedure Read-only memory (ROM) 470 Rectangular matrix 631 Reduced order compensator design satellite attitude control example Reduced order compensator transfer function 387 Reduced order design polynomial transfer function model example Reduced order estimator DC servo redesign example pendulum example structure 382 Reference input alternative structures 365 estimator gain selection 406 general structure example full state feedback block diagram 363 gain equation 362 oscillator step response 364 selection methods 397 type 1 system example Reference spectrum plot 360

20 Index 687 Reference tracking 127 system type 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 Residue theorem 627 Resistor symbol and equation 28 Resonant frequency Nichols chart 296 Resonant peak frequency response 243 Response by convolution to sinusoid 62 vs. pole locations and real roots example Response sensitivity 168 RHP. (see Right half-plane (RHP)) 80 Right eigenvectors matrix 635 Right half-plane (RHP) Bode plot 267 compensator transfer function 386 LTR 422 Lyapunov stability 528 root locus extension method 216 SRL 368 unstable closed loop system zeros 96 example 97 nonminimum-phase systems 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 Robust 410 Robust control definition 410 equations error space 410 Robust control (continued) sinusoid example 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 error space approach exercise problems related to Rolling mill 117 Roll mode yaw damper 565 ROM 470 Room temperature control system component block diagram 3 dynamic model 36 Root locus 11, degree locus. (see Negative locus) degree locus. (see Positive locus) 182 autopilot design 213 closed loop poles 219 combine control and estimator 389 compensation 508 complex multiple roots example conditionally stable system 270 DC servo pole assignment 390 reduced order estimator 392 definition 182 design example exercise problems related to 224, , feedback system

21 688 Index illustrative lead design 205 lead lag 209 motor position control example notch compensations 209 plant open-loop pole example plotting rules reduced order controller 390 rules application example satellite attitude control PD control satellite control collocated flexibility lead compensator small value for pole transition value for pole sketching guidelines exercise problems related to 224 SRL pole assignment 393 stability examples 255 time delay 220 heat exchanger 221 Root locus design method exercise problems related to perspective 177 Root locus forms 179 Root locus method extensions exercise problems related to of Evans 178 Root mean square (RMS) value roots.m 104, 193, 353 Rotational motion hanging crane example Newton s law 23 pendulum example satellite attitude control model example Rotor free body diagram 33 Routh, E. J. 10 Routh s array Routh s criterion Lyapunov stability 529 Routh s stability criterion , Routh s test example , Row vector matrix 631 RTP system control system design procedure 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 anti-alias prefilter 472 asynchronous sampling 473 disturbance rejection tracking effectiveness digitization 453 Sampling theorem Nyquist and Shannon 469 Sandberg, I. W. 536 Sastry, S. 530 Satellite flexible appendages example Satellite attitude control application control system design procedure example full order compensator design LTR reduced order compensator design rotational motion SRL state-variable form 332 model 141 Satellite control root locus collocated flexibility lead compensator small value for pole transition value for pole Satellite control schematic 24

22 Index 689 Satellite tracking signal 134 Satellite transfer function using MATLAB example 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 Second order step responses transfer functions 96 Second order system block diagram 82 Lyapunov stability example 529 Nyquist plot example responses 88 exponential envelope 89 step response plots 88 Second order term transfer function class 249 Semiconductor wafer manufacturing control system design procedure semilogx.m 64, 242, 424 semilogy.m 305 Sensitivity 127 parameter changes time response parameter change Sensitivity function 130, 300 design limitations example plot and computation example 305 specifications exercise problems related to 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 Servomechanism structure block diagram 218 Servomechanism system lead compensator design example 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 Settling time definition 90 time domain specification 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 Similarities complex variables Similarity transformations matrix 635 Simple closed contour Cauchy s theorem 627 Simple design criterion spacecraft attitude control example Simple discrete time functions z-transform and Laplace transforms 456 Simulation block diagram transfer function control system design procedure 548 Simulink 2 block diagram robust servomechanism 413 LTR

23 690 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 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 Smith compensator 428 Smith regulator time delay 428 Spacecraft attitude control example PID compensation simple design criterion Space station attitude emulation design digital controller example Space station digital controller direct discrete design example Specifications control system design procedure Specific heat heat flow 37 Spectral analyzers 243 Speed control 2, 10 comparison plots 151 exercise problems related to Speed controller example 143 Spiral mode yaw damper 566 S -plane and z -plane relationship digital control 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, examples system definition and root locus 256 exercise problems related to Stability (continued) feedback system 104 input magnitude nonlinear example 506 linear time invariant systems Lyapunov 526 natural responses 89 necessary condition for 101 Routh s criterion vs. parameter range example vs. two parameter ranges example Stability analysis 10

24 Index 691 Stability augmentation Boeing 562 Stability condition frequency response 258 Stability margins frequency response design method exercise problems related to 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 State equations analysis block diagrams canonical forms examples , , exercise problems related to 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 advantages compensator design control law design full state feedback estimator design exercise problems related to gain selection 406 integral control LTR Lyapunov stability perspective 329 pole location selection pure time delay rational transfer functions reference input with estimator robust tracking state equation analysis State space method lag compensation example State space pole placement method example State space to transfer function (ss2tf) MATLAB function 353 States space design elements schematic diagram 355 States space design methods example 527 digital controller 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 command inputs and disturbances 294 determination example 254 Steady state information stochastic experimental data source 108 Steady state phase difference 64 Steady state tracking disturbance rejection extended estimator exercise problems related to Steam engine 8 Index 907 Step and ramp transforms Laplace transforms example 66 step.m 20, 26, 78, 333