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| موضوع: حل كتاب Modern Control Engineering Solution Manual الإثنين 30 أبريل 2012, 6:25 pm | |
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أخواني في الله أحضرت لكم حل كتاب Modern Control Engineering 5th ed Solution Manual Fifth Edition Katsuhiko Ogata
و المحتوى كما يلي :
Contents Preface ix Chapter 1 Introduction to Control Systems 1 1–1 Introduction 1 1–2 Examples of Control Systems 4 1–3 Closed-Loop Control Versus Open-Loop Control 7 1–4 Design and Compensation of Control Systems 9 1–5 Outline of the Book 10 Chapter 2 Mathematical Modeling of Control Systems 13 2–1 Introduction 13 2–2 Transfer Function and Impulse-Response Function 15 2–3 Automatic Control Systems 17 2–4 Modeling in State Space 29 2–5 State-Space Representation of Scalar Differential Equation Systems 35 2–6 Transformation of Mathematical Models with MATLAB 392–7 Linearization of Nonlinear Mathematical Models 43 Example Problems and Solutions 46 Problems 60 Chapter 3 Mathematical Modeling of Mechanical Systems and Electrical Systems 63 3–1 Introduction 63 3–2 Mathematical Modeling of Mechanical Systems 63 3–3 Mathematical Modeling of Electrical Systems 72 Example Problems and Solutions 86 Problems 97 Chapter 4 Mathematical Modeling of Fluid Systems and Thermal Systems 100 4–1 Introduction 100 4–2 Liquid-Level Systems 101 4–3 Pneumatic Systems 106 4–4 Hydraulic Systems 123 4–5 Thermal Systems 136 Example Problems and Solutions 140 Problems 152 Chapter 5 Transient and Steady-State Response Analyses 159 5–1 Introduction 159 5–2 First-Order Systems 161 5–3 Second-Order Systems 164 5–4 Higher-Order Systems 179 5–5 Transient-Response Analysis with MATLAB 183 5–6 Routh’s Stability Criterion 212 5–7 Effects of Integral and Derivative Control Actions on System Performance 218 5–8 Steady-State Errors in Unity-Feedback Control Systems 225 Example Problems and Solutions 231 Problems 263 iv ContentsChapter 6 Control Systems Analysis and Design by the Root-Locus Method 269 6–1 Introduction 269 6–2 Root-Locus Plots 270 6–3 Plotting Root Loci with MATLAB 290 6–4 Root-Locus Plots of Positive Feedback Systems 303 6–5 Root-Locus Approach to Control-Systems Design 308 6–6 Lead Compensation 311 6–7 Lag Compensation 321 6–8 Lag–Lead Compensation 330 6–9 Parallel Compensation 342 Example Problems and Solutions 347 Problems 394 Chapter 7 Control Systems Analysis and Design by the Frequency-Response Method 398 7–1 Introduction 398 7–2 Bode Diagrams 403 7–3 Polar Plots 427 7–4 Log-Magnitude-versus-Phase Plots 443 7–5 Nyquist Stability Criterion 445 7–6 Stability Analysis 454 7–7 Relative Stability Analysis 462 7–8 Closed-Loop Frequency Response of Unity-Feedback Systems 477 7–9 Experimental Determination of Transfer Functions 486 7–10 Control Systems Design by Frequency-Response Approach 491 7–11 Lead Compensation 493 7–12 Lag Compensation 502 7–13 Lag–Lead Compensation 511 Example Problems and Solutions 521 Problems 561 Chapter 8 PID Controllers and Modified PID Controllers 567 8–1 Introduction 567 8–2 Ziegler–Nichols Rules for Tuning PID Controllers 568 Contents v8–3 Design of PID Controllers with Frequency-Response Approach 577 8–4 Design of PID Controllers with Computational Optimization Approach 583 8–5 Modifications of PID Control Schemes 590 8–6 Two-Degrees-of-Freedom Control 592 8–7 Zero-Placement Approach to Improve Response Characteristics 595 Example Problems and Solutions 614 Problems 641 Chapter 9 Control Systems Analysis in State Space 648 9–1 Introduction 648 9–2 State-Space Representations of Transfer-Function Systems 649 9–3 Transformation of System Models with MATLAB 656 9–4 Solving the Time-Invariant State Equation 660 9–5 Some Useful Results in Vector-Matrix Analysis 668 9–6 Controllability 675 9–7 Observability 682 Example Problems and Solutions 688 Problems 720 Chapter 10 Control Systems Design in State Space 722 10–1 Introduction 722 10–2 Pole Placement 723 10–3 Solving Pole-Placement Problems with MATLAB 735 10–4 Design of Servo Systems 739 10–5 State Observers 751 10–6 Design of Regulator Systems with Observers 778 10–7 Design of Control Systems with Observers 786 10–8 Quadratic Optimal Regulator Systems 793 10–9 Robust Control Systems 806 Example Problems and Solutions 817 Problems 855 vi ContentsAppendix A Laplace Transform Tables 859 Appendix B Partial-Fraction Expansion 867 Appendix C Vector-Matrix Algebra 874 References 882 Index 88 Index A Absolute stability, 160 Ackermann’s formula: for observer gain matrix, 756–57 for pole placement, 730–31 Actuating error, 8 Actuator, 21–22 Adjoint matrix, 876 Air heating system, 150 Aircraft elevator control system, 156 Analytic function, 860 Angle: of arrival, 286 of departure, 280, 286 Angle condition, 271 Asymptotes: Bode diagram, 406–07 root loci, 274–75, 284–85 Attenuation, 165 Attitude-rate control system, 386 Automatic controller, 21 Automobile suspension system, 86 Auxiliary polynomial, 216 B Back emf, 95 constant, 95 Bandwidth, 474, 539 Basic control actions: integral, 24 on-off, 22 proportional, 24 proportional-plus-derivative, 25 proportional-plus-integral, 24 proportional-plus-integral-plusderivative, 35 two-position, 22–23 Bleed-type relay, 111 Block, 17 Block diagram, 17–18 reduction, 27–28, 48 Bode diagram, 403 error in asymptotic expression of, 403 of first-order factors, 406–07, 409 general procedure for plotting, 413 plotting with MATLAB, 422–25 of quadratic factors, 410–12 of system defined in state space, 426–27 Branch point, 18 Break frequency, 406 Breakaway point, 275–76, 285–86, 351 Break-in point, 276, 281, 285–86, 351 Bridged-T networks, 90, 520 Business system, 5Index 887 C Canonical forms: controllable, 649 diagonal, 650 Jordan, 651, 653 observable, 650 Capacitance: of pressure system, 107–09 of thermal system, 137 of water tank, 103 Cancellation of poles and zeros, 288 Cascaded system, 20 Cascaded transfer function, 20 Cauchy–Riemann conditions, 860–61 Cauchy’s theorem, 526 Cayley–Hamilton theorem, 668, 701 Characteristic equation, 652 Characteristic polynomial, 34 Characteristic roots, 652 Circular root locus, 282 Classical control theory, 2 Classification of control systems, 225 Closed-loop control system, 8 Closed-loop system, 20 Closed-loop frequency response, 477 Closed-loop frequency response curves: desirable shapes of, 492 undesirable shapes of, 492 Closed-loop transfer function, 19–20 Cofactor, 876 Command compensation, 630 Compensation: feedback, 308 parallel, 308 series, 308 Compensator: lag, 323, 503–04 lag–lead, 332–34, 511–13 lead, 312–13, 495–96 Complete observability, 683–84 conditions for, 684–85 in the s plane, 684 Complete output controllablility, 714 Complete state controllability, 676–81 in the s plane, 680–81 Complex-conjugate poles: cancellation of undesirable, 520 Complex function, 859 Complex impedence, 75 Complex variable, 859 Computational optimization approach to design PID controller, 583–89 Conditional stability, 299–300, 510–11 Conditionally stable system, 299–300, 458, 510–11 Conduction heat transfer, 137 Conformal mapping, 447, 462–64 Conical water tank system, 152 Constant-gain loci, 302–03 Constant-magnitude loci (M circles), 478–79 Constant phase-angle loci (N circles), 480–81 Constant v n loci, 296 Constant z lines, 298 Constant z loci, 296 Control actions, 21 Control signal, 3 Controllability, 675–81 matrix, 677 output, 681 Controllable canonical form, 649, 688 Controlled variable, 3 Controller, 22 Convection heat transfer, 137 Conventional control theory, 29 Convolution, integral, 16 Corner frequency, 406 Critically damped system, 167 Cutoff frequency, 474 Cutoff rate, 475 D Damped natural frequency, 167 Damper, 64, 132 Damping ratio, 165 lines of constant, 296 Dashpot, 64, 132–33 Dead space, 43 Decade, 405 Decibel, 403 Delay time, 169–70 Derivative control action, 118–20, 222 Derivative gain, 84 Derivative time, 25, 61 Detectability, 688 Determinant, 874 Diagonal canonical form, 694 Diagonalization of n*n matrix, 652 Differential amplifier, 78 Differential gap, 23, 24 Differentiating system, 231 Differentiation: of inverse matrix, 881 of matrix, 880 of product of two matrices, 880 Differentiator: approximate, 617 Direct transmission matrix, 31 Disturbance, 3, 26 Dominant closed-loop poles, 182 Duality, 754EAe t : computation of, 670–71 Eigenvalue, 652 invariance of, 655 Electromagnetic valve, 23 Electronic controller, 77, 83 Engineering organizational system, 5–6 Equivalent moment of inertia, 234 Equivalent spring constant, 64 Equivalent viscous-friction coefficient, 65, 234 Evans, W. R., 2, 11, 269 Exponential response curve, 162 F Feedback compensation, 308–09, 342, 519 Feedback control, 3 Feedback control system, 7 Feedback system, 20 Feedforward transfer function, 19 Final value theorem, 866 First-order lag circuit, 80 First-order system, 161–64 unit-impulse response of, 163 unit-ramp response of, 162–63 unit-step response of, 161–62 Flapper, 110 valve, 156 Fluid systems: mathematical modeling of, 100 Free-body diagram, 69–70 Frequency response, 398 correlation between step response and, 471–74 lag compensation based on, 502–11 lag–lead compensation based on, 511–17 lead compensation based on, 493–502 Full-order state observer, 752–53 Functional block, 17 G Gain crossover frequency, 467–69 Gain margin, 464–67 Gas constant, 108 for air, 142 universal, 108 Gear train, 232 system, 232–34 Generalized plant, 813, 815–17 diagram, 810–16, 853–54 H H infinity control problem, 816 H infinity norm, 6, 808 888 Index Hazen, 2, 11 High-pass filter, 495 Higher-order systems, 179 transient response of, 180–81 Hurwitz determinants, 252–58 Hurwitz stability criterion, 252–53, 255–58 equivalence of Routh’s stability criterion and, 255–57 Hydraulic controller: integral, 130 jet-pipe, 147 proportional, 131 proportional-plus-derivative, 134–35 proportional-plus-integral, 133–34 proportional-plus-integral-plusderivative, 135–36 Hydraulic servo system, 124–25 Hydraulic servomotor, 128, 130, 156 Hydraulic system, 106, 123–39, 149 advantages and disadvantages of, 124 compared with pneumatic system, 106 I Ideal gas law, 108 Impedance: approach to obtain transfer function, 75–76 Impulse function, 866 Impulse response, 163, 178–79, 195–97 function, 16–17 Industrial controllers, 22 Initial condition: response to, 203–11 Initial value theorem, 866 Input filter, 261, 630 Input matrix, 31 Integral control, 220 Integral control action, 24–25, 218 Integral controller, 22 Integral gain, 61 Integral time, 25, 61 Integration of matrix, 880 Inverse Laplace transform: partial-fraction expansion method for obtaining, 867–73 Inverse Laplace transformation, 862 Inverse of a matrix: MATLAB approach to obtain, 879 Inverse polar plot, 461–62, 537–38 Inverted-pendulum system, 68–72, 98 Inverted-pendulum control system, 746–51 Inverting amplifier, 78 I-PD control, 591–92 I-PD-controlled system, 592, 628–29, 643 with feedforward control, 642Index 889 J Jet-pipe controller, 146–47 Jordan blocks, 679 Jordan canonical form, 651, 695, 706–07 K Kalman, R. E., 12, 675 Kirchhoff’s current law, 72 Kirchhoff’s loop law, 72 Kirchhoff’s node law, 72 Kirchhoff’s voltage law, 72 L Lag compensation, 321 Lag compensator, 311, 321, 502 Bode diagram of, 503 design by frequency-response method, 502–11 design by root-locus method, 321, 323 polar plot of, 503 Lag network, 82, 542 Lag–lead compensation, 330, 335, 338, 377, 511–18 Lag–lead compensator: Bode diagram of, 558 design by frequency-response method, 513–17 design by root-locus method, 331–32, 380–82 electronic, 330–32 polar plot of, 512 Lag–lead network: electronic, 330–32 mechanical, 366 Lagrange polynomial, 708 Lagrange’s interpolation formula, 708 Laminar-flow resistance, 102 Laplace transform, 862 properties of, 865 table of, 863–64 Lead compensator, 311, 493 Bode diagram of, 494 design by frequency-response method, 493–502 design by root-locus method, 311–18 polar plot of, 494 Lead, lag, and lag–lead compensators: comparison of, 517–18 Lead network, 542 electronic, 82 mechanical, 365 Lead time, 5 Linear approximation: of nonlinear mathematical models, 43 Linear system, 14 constant coefficient, 14 Linear time-invariant system, 14, 164 Linear time-varying system, 14 Linearization: of nonlinear systems, 43 Liquid-level control system, 157 Liquid-level systems, 101, 103–04, 140–41 Log-magnitude curves of quadratic transfer function, 411 Logarithmic decrement, 237 Logarithmic plot, 403 Log-magnitude versus phase plot, 403, 443–44 LRC circuit, 72–73 M M circles, 478–79 a family of constant, 479 Magnitude condition, 271 Manipulated variable, 3 Mapping theorem, 448–49 Mathematical model, 13 MATLAB commands: MATLAB: obtaining maximum overshoot with, 194 obtaining peak time with, 194 obtaining response to initial condition with, 266 partial-fraction expansion with, 871–73 plotting Bode diagram with, 422–23 plotting root loci with, 290–91 writing text in diagrams with, 188–89 [A,B,C,D] = tf2ss(num,den), 40, 656, 698 bode(A,B,C,D), 422, 426 bode(A,B,C,D,iu), 426–27 bode(A,B,C,D,iu,w), 422 bode(A,B,C,D,w), 422 bode(num,den), 422 bode(num,den,w), 422, 425, 551 bode(sys), 422 bode(sys,w), 552 c = step(num,den,t), 190 for loop, 243, 249, 584 [Gm,pm,wcp,wcg,] = margin(sys), 468–69 gtext ('text'), 189 impulse(A,B,C,D), 195 impulse(num, den), 195 initial(A,B,C,D,[initial condition],t), 209 inv(A), 879 K = acker(A,B,J), 736 K = lqr(A,B,Q,R), 798 K = place(A,B,J), 736MATLAB commands (Cont.) K e = acker(A',C',L)', 773 K e = acker(Abb,Aab,L)', 773 K e = place(A',C',L)', 773 K e = place(Abb',Aab',L)', 773 [K,P,E] = lqr(A,B,Q,R), 798 [K,r] = rlocfind(num,den), 303 logspace(d1,d2), 422 logspace(d1,d2,n), 422–23 lqr(A,B,Q,R), 797 lsim(A,B,C,D,u,t), 201 lsim(num,den,r,t), 201 magdB = 20*log10(mag), 422 [mag,phase,w] = bode(A,B,C,D), 422 [mag,phase,w] = bode(A,B,C,D,iu,w), 422 [mag,phase,w] = bode(A,B,C,D,w), 422 [mag,phase,w] = bode(num,den), 422 [mag,phase,w] = bode(num,den,w), 422, 476 [mag,phase,w] = bode(sys), 422 [mag,phase,w] = bode(sys,w), 476 mesh, 192 mesh(y), 192, 249 mesh(y'), 192, 249 [Mp,k] = max(mag), 476 NaN, 799 [num,den] = feedback(num1,den1, num2,den2), 20–21 [num,den] = parallel(num1,den1, num2,den2), 20–21 [num,den] = series(num1,den1, num2,den2), 20–21 [num,den] = ss2tf(A,B,C,D), 41, 657 [num,den] = ss2tf(A,B,C,D,iu), 41–42, 58, 657 [NUM,den] = ss2tf(A,B,C,D,iu), 59, 659 nyquist(A,B,C,D), 436, 441–42 nyquist(A,B,C,D,iu), 441 nyquist(A,B,C,D,iu,w), 436, 441 nyquist(A,B,C,D,w), 436 nyquist(num,den), 436 nyquist(num, den,w), 436 nyquist(sys), 436 polar(theta,r), 545 printsys(num,den), 20–21, 189 printsys(num,den,'s'), 189 r = abs(z), 544 [r,p,k] = residue(num,den), 239, 871–72 [re,im,w] = nyquist(A,B,C,D), 436 [re,im,w] = nyquist(A,B,C,D,iu,w), 436 [re,im,w] = nyquist(A,B,C,D,w), 436 [re,im,w] = nyquist(num,den), 436 [re,im,w] = nyquist(num,den,w), 436 890 Index [re,im,w] = nyquist(sys), 436 residue, 867 resonant_frequency = w(k), 476 resonant_peak = 20*log10(Mp), 476 rlocfind, 303 rlocus(A,B,C,D), 295 rlocus(A,B,C,D,K), 290, 295 rlocus(num,den), 290–91 rlocus(num,den,K), 290 sgrid, 297 sortsolution, 584 step(A,B,C,D), 184, 186 step(A,B,C,D,iu), 184 step(num,den), 184 step(num,den,t), 184 step(sys), 184 sys = ss(A,B,C,D), 184 sys = tf(num,den), 184 text, 188 theta = angle(z), 544 w = logspace(d2,d3,100), 425 y = lsim(A,B,C,D,u,t), 201 y = lsim(num,den,r,t), 201 [y, x, t] = impulse(A,B,C,D), 195 [y, x, t] = impulse(A,B,C,D,iu), 195 [y, x, t] = impulse(A,B,C,D,iu,t), 195 [y, x, t] = impulse(num,den), 195 [y, x, t] = impulse(num,den,t), 195 [y, x, t] = step(A,B,C,D,iu), 184 [y, x, t] = step(A,B,C,D,iu,t), 184 [y, x, t] = step(num,den,t), 184, 190 z = re+j*im, 544 End of MATLAB commands Matrix exponential, 661, 669–674 closed solution for, 663 Matrix Riccati equation, 798, 800 Maximum overshoot: in unit-impulse response, 179 in unit-step response, 170, 172 versus z curve, 174 Maximum percent overshoot, 170 Maximum phase lead angle, 494, 498 Measuring element, 21 Mechanical lag–lead system, 366 Mechanical lead system, 365 Mechanical vibratory system, 236 Mercury thermometer system, 151 Minimal polynomial, 669, 704–06 Minimum-order observer, 767–77 based controller, 777 Minimum-order state observer, 752 Minimum-phase system, 415–16 Minimum-phase transfer function, 415 Minor, 876 Modern control theory, 7, 29 versus conventional control theory, 29Index 891 Motor torque constant, 95 Motorcycle suspension system, 87 Multiple-loop system, 458–59 N N circles, 480–81 a family of constant, 481 Newton’s second law, 66 Nichols, 2, 11, 398 Nichols chart, 482–85 Nichols plots, 403 Nonbleed-type relay, 111 Nonhomogeneous state equation: solution of, 666–67 Noninverting amplifier, 79 Nonlinear mathematical models: linear approximation of, 43–45 Nonlinear system, 43 Nonminimum-phase systems, 300–01, 415, 417 Nonminimum-phase transfer function, 415, 488 Nonuniqueness: of a set of state variables, 655 Nozzle-flapper amplifier, 110 Number-decibel conversion line, 404 Nyquist, H., 2, 11, 398 Nyquist path, 545 Nyquist plot, 403, 439–40, 443 of positive-feedback system, 535–37 of system defined in state space, 440–43 Nyquist stability analysis, 454–62 Nyquist stability criterion, 445–54 applied to inverse polar plots, 461–62 O Observability, 675, 682–88 complete, 683–85 matrix, 653 Observable canonical form, 650, 692 Observation, 752 Observed-state feedback control system, 761 Observer, 753 design of control system with, 786–93 full-order, 753 mathematical model of, 752 minimum-order, 767–73 Observer-based controller: transfer function of, 761 Observer controller: in the feedback path of control system, 787, 790–93 in the feedforward path of control system, 787–90 Observer-controller matrix, 762 Observer-controller transfer function, 761–62 Observer error equation, 753 Observer gain matrix, 755 MATLAB determination of, 773 Octave, 405 Offset, 258 On-off control action, 22–23 On-off controller, 22 One-degree-of-freedom control system, 593 op amps, 78 Open-loop control system, 8 advantages of, 9 disadvantages of, 9 Open-loop frequency response curves: reshaping of, 493 Open-loop transfer function, 19 Operational amplifier, 78 Operational amplifier circuits, 93–94 for lead or lag compensator: table of, 85 Optimal regulator problem, 806 Ordinary point, 861 Orthogonality: of root loci and constant gain loci, 301–02 Output controllability, 681 Output equation, 31 Output matrix, 31 Overdamped system, 168–69 Overlapped spool valve, 146 Overlapped valve, 130 P Parallel compensation, 308–09, 342–43 Partial-fraction expansion, 867–73 with MATLAB, 871–73 PD control, 373 PD controller, 614–15 Peak time, 170, 172, 193 Performance index, 793 Performance specifications, 9 Phase crossover frequency, 467–69 Phase margin, 464–67 versus z curve, 472 PI controller, 2, 614–15 PI-D control, 590–92 PID control system, 572–77, 583, 587, 617–21, 628–29, 642–43 basic, 590 with input filter, 629 two-degrees-of-freedom, 592–95 PID controller, 567, 577, 614–16, 620, 632 modified, 616 using operational amplifiers, 83–84Pilot valve, 124, 130 PI-PD control, 592 PID-PD control, 592 Plant, 3 Pneumatic actuating valve, 117–18 Pneumatic controllers, 144–45, 154–55 Pneumatic nozzle-flapper amplifier, 110 Pneumatic on-off controller, 115 Pneumatic pressure system, 142 Pneumatic proportional controller, 112–16 force-balance type, 115–16 force-distance type, 112–15 Pneumatic proportional-plus-derivative controller, 119–20 Pneumatic proportional-plus-integral control action, 120–22 Pneumatic proportional-plus-integralplus-derivative control action, 122–23 Pneumatic relay, 111 bleed type, 111 nonbleed type, 111 reverse acting, 112 Pneumatic systems, 106–23, 153 compared with hydraulic system, 106 Pneumatic two-position controller, 115 Polar grids, 297 Polar plot, 403, 427–28, 430, 432 Pole: 861 of order n, 861 simple, 861 Pole assignment technique, 723 Pole-placement: necessary and sufficient conditions for arbitrary, 725 Pole placement problem, 723–35 solving with MATLAB, 735–36 Positive-feedback system: Nyquist plot for, 536–37 root loci for, 303–07 Positional servo system, 95–97 Pressure system, 107, 109 Principle of duality, 687 Principle of superposition, 43 Process, 3 Proportional control, 219 Proportional control action, 24 Proportional controller, 22 Proportional gain, 25, 61 Proportional-plus-derivative control: of second-order system, 224 of system with inertia load, 223 Proportional-plus-derivative control action, 25 Proportional-plus-derivative controller, 22, 542 892 Index Proportional-plus-integral control action, 24 Proportional-plus-integral controller, 22, 121, 542 Proportional-plus-integral-plusderivative control action, 25 Proportional-plus-integral-plusderivative controller, 22 Pulse function, 866 Q Quadratic factor, 410 log-magnitude curves of, 411 phase-angle curves of, 411 Quadratic optimal control problem: MATLAB solution of, 804 Quadratic optimal regulator system, 793–95 MATLAB design of, 797 R Ramp response, 197 Rank of matrix, 875 Reduced-matrix Riccati equation, 795–97 Reduced-order observer, 752 Reduced-order state observer, 752 Reference input, 21 Regulator system with observer controller, 778–86, 789 Relative stability, 160, 217, 462 Residue, 867 Residue theorem, 527 Resistance: gas-flow, 107 laminar-flow, 101–02 of pressure system, 107, 109 of thermal system, 137 turbulent-flow, 102 Resonant frequency, 430, 470 Resonant peak, 413, 430, 470 versus z curve, 413 Resonant peak magnitude, 413, 470 Response: to arbitrary input, 201 to initial condition, 203–11 to torque disturbance, 221 Reverse-acting relay, 112 Riccati equation, 795 Rise time, 169–171 obtaining with MATLAB, 193–94 Robust control: system, 16, 806–17 theory, 2, 7 Robust performance, 7, 807, 812 Robust pole placement, 735 Robust stability, 7, 807, 809Index 893 Root loci: general rules for constructing, 283–87 for positive-feedback system, 303–07 Root locus, 271 method, 269–70 Routh’s stability criterion, 212–18 S Schwarz matrix, 268 Second-order system, 164 impulse response of, 178–79 standard form of, 166 step response of, 165–75 transient-response specification of, 171 unit-step response curves of, 169 Sensor, 21 Series compensation, 308–09, 342 Servo system, 95, 164–65 design of, 739–51 with tachometer feedback, 268 with velocity feedback, 175–77 Servomechanism, 2 Set point, 21 Set-point kick, 590 Settling time, 170, 172–73 obtaining with MATLAB, 194 versus z curve, 174 Sign inverter, 79 Simple pole, 861 Singular points, 861 Sinusoidal signal generator, 486 Sinusoidal transfer function, 401 Small gain theorem, 809 Space vehicle control system, 367, 538–39 Speed control system, 4, 148 Spool valve: linealized mathematical model of, 127 Spring-loaded pendulum system, 98 Spring-mass-dashpot system, 66 Square-law nonlinearity, 43 S-shaped curve, 569 Stability analysis, 454–62 in the complex plane, 182 Stabilizability, 688 Stack controller, 115 Standard second-order system, 189 State, 29 State controllability: complete, 676, 678, 680 State equation, 31 solution of homogeneous, 660 solution of nonhomogeneous, 666–67 Laplace transform solution of, 663 State-feedback gain matrix, 724 MATLAB approach to determine, 735–36 State matrix, 31 State observation: necessary and sufficient conditions for, 754–55 State observer, 751–77 design with MATLAB, 773 type 1 servo system with, 746 State observer gain matrix: 755 Ackermann’s formula to obtain, 756–57 direct substitution approach to obtain, 756 transformation approach to obtain, 755 State space, 30 State-space equation, 30 correlation between transfer function and, 649, 656 solution of, 660 State-space representation: in canonical forms, 649 of nth order system, 36–39 State-transition matrix, 664 properties of, 665 State variable, 29 State vector, 30 Static acceleration error constant, 228, 421 determination of, 421–22 Static position error constant, 226, 419 Static velocity error constant, 227, 420 Steady-state error, 160, 226 for unit parabolic input, 229 for unit ramp input, 228 in terms of gain K, 230 Steady-state response, 160 Step response, 699–700 of second-order system, 165–69 Summing point, 18 Suspension system: automobile, 86–87 motorcycle, 87 Sylvester’s interpolation formula, 673, 709–713 System, 3 Sytem types, 419 type 0, 225, 230, 419, 433, 487–88 type 1, 225, 230, 420, 433, 487–88 type 2, 225, 230, 421, 433, 487–88 System response to initial condition: MATLAB approach to obtain, 203–11 T Tachometer, 176 feedback, 343 Taylor series expansion, 43–45Temperature control systems, 4–5 Test signals, 159 Text: writing on the graphic screen, 188 Thermal capacitance, 137 Thermal resistance, 137 Thermal systems, 100,136–39 Thermometer system, 151–52 Three-degrees-of-freedom system, 645 Three-dimensional plot, 192 of unit-step response curves with MATLAB, 191–93 Traffic control system, 8 Transfer function, 15 of cascaded elements, 73–74 of cascaded systems, 20 closed-loop, 20 of closed-loop system, 20 experimental determination of, 489–90 expression in terms of A, B, C, and D, 34 of feedback system, 19 feedforward, 19 of minimum-order observer-based controller, 777 of nonloading cascaded elements, 77 observer-controller, 762, 780–82 open-loop, 19 of parallel systems, 20 sinusoidal, 401 Transfer matrix, 35 Transformation: from state space to transfer function, 41–42, 657 from transfer function to state space, 40–41, 656 Transient response, 160 analysis with MATLAB, 183–211 of higher-order system, 180 specifications, 169, 171 Transport lag, 417 phase angle characteristics of, 417 Turbulent-flow resistance, 102 Two-degrees-of-freedom control system, 593–95, 599–614, 636–41, 646–47 Two-position control action, 22–23 Two-position controller, 22 Type 0 system, 225, 230, 488 log-magnitude curve for, 419, 488 polar plot of, 433 Type 1 servo system: design of, 743–51 pole-placement design of, 739–46 Type 1 system, 420 log-magnitude curve for, 420, 488 polar plot of, 433 894 Index Type 2 system, 421 log-magnitude curve for, 421, 488 polar plot of, 433 U Uncontrollable system, 681 Undamped natural frequency, 165 Underdamped system, 166–67 Underlapped spool valve, 146 Unit acceleration input, 247 Unit-impulse response: of first-order system, 163 of second-order system, 178 Unit-impulse response curves: a family of, 178 obtained by use of MATLAB, 196–97 Unit-ramp response: of first-order system, 162–63 of second-order system, 197–200 of system defined in state space, 199–200 Unit-step response: of first-order system, 161 of second-order system, 163, 167, 169 Universal gas constant, 108 Unstructured uncertainty: additive, 852–53 multiplicative, 809 system with, 809 V Valve: overlapped, 130 underlapped, 130 zero-lapped, 130 Valve coefficient, 127 Vectors: linear dependence of, 674 linear independence of, 674 Velocity error, 227 Velocity feedback, 176, 343, 519 W Watt’s speed governor, 4 Weighting function, 17 Z Zero, 861 of order m, 862 Zero-lapped valve, 130 Zero placement, 595, 597, 612 approach to improve response characteristics, 595–97 Ziegler–Nichols tuning rules, 11, 568–77 first method, 569–70 second method, 570–71
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