See discreteconvolution costate, 433434 cost function, 418, 429 terms in, 422 Cramers rule, 172, 176 critically damped, 324, 328332 crossover point, 247, 268 current estimator, 442 current observers, 369374 current phasor, expression for, 25 D damping factor for linear friction, 147 data hold, 102 data reconstruction, 113121 first-order hold, 118119 fractional-order hold, 119121 reconstructed version of e(t), 113 using polynomial extrapolation, 113 zero-order hold, 114118 data-reconstruction device, 101 DC gain, 131, 203 dc motor system, 18 decimal-to-binary conversion algorithms, 293 delayed z-transform, 136137 derivation procedure, 171172 derivative of a matrix, 505506 diagonal matrix, 502 differentiator transfer function, 309 digital computer, 3536 digital controllers, 12 with nonzero computation time, 141 nth-order linear, 140 digital control system, 1215 digital filter, 37 differentiation of a function, 308 in U.S. Navy aircraft carriers, 37 digital-to-analog (D/A) converter, 35, 100, 134 discrete convolution technique, 5758 discrete Riccati equation, 434, 439440 discrete state equations of a sampleddata system, 150154 526 Index discrete state matrices, 180181 discrete state models for digital control systems, 183188 discrete state-space model for the closed loop, 187 discrete-time systems, 12, 3537 with time delays, 139142 discrete unit impulse function, 43, 256 discrete unit step function, 41 disturbances, 13 double-sided z-transform, 38 dynamic systems, identifying, 394 batch least squares, 409 black-box identification, 394401 choice of input, 412413 least-squares system identification, 401407 practical factors for identification, 412414 recursive least-squares system identification, 409412 sampling frequency, 413 signal scaling, 413414 forward path, 492 Fourier transform of e(t), 111, 115 results from, 108110 fractional-order hold, 119121 frequency response magnitudes for, 121 impulse response of, 120 transfer function of, 119 frequency aliasing, 116 frequency foldover, 116 frequency response, interpretation of, 110, 259261 frequency spectrum of e(t), 109 full-order current observer, 369370 fundamental matrix, 85 E H eigenvalues, 74, 435438, 502 eigenvectors, 435438, 503 electric circuit law, 25 electric power, 26 electric power system, 484 electric power system models, topology identification in, 484488 environmental chamber control system, 461466 error-control condition, 382 error signal, 15, 18, 21 E*(s), 169, 208 amplitude of the discontinuity of e(t), 500 for e(t) = - t, 105 for e(t) = u(t), 104105 evaluation of, 105108, 496500 Laplace transform of, 105 properties of, 110113 relationship between E(z) and, 126127 theorem of residues, 498499 zeros of, 110 Euler method, 219 Eulers identity, 511 Eulers relation, 54, 107, 110 F feedback, parallel, or minor-loop compensation, 286 feedback path, 103 filter transfer function, 298, 302, 310 final-value theorem, 204, 470, 515 first-order hold, 118119 frequency response of, 119120 first-order linear differential equation, 23, 36 flow graphs, 5962 G gain margin, 255 general rational function, 510 generating function, defined, 38 grey-box identification, 391 G(z), 198 Hankel matrix, 396, 399400 hardware configuration of system, 349 high frequency gain, 287 high-order systems, computations for, 154155 I ideal filter, 112 ideal sampler, 102104 defined, 104 ideal time delays, systems with, 139142 identity matrix, 501 IEEE 39-bus power system, 486, 487 impulse functions, 134 impulse modulator, 103 inertia, 485 infinite bus, 24 infinite-horizon linear-quadraticGaussian (IH-LQG) design, 442, 444446 initial-condition (zero-input) response, 518 input space of system, 63 integral of a matrix, 506 integrator transfer function, 308 inverse Laplace transform, 103, 149 inverse z-transform, 200, 206207 discrete convolution technique, 5758 inversion-formula method, 56 partial-fraction expansion method, 52 power series method, 51 inversion-formula method, 56 K Kalman filters, 374, 440444 Kirchoffs law, 25 Kronecker delta function, 440 L Laplace transform, 17, 24, 37, 52, 57, 102103, 508519 of constant-coefficient linear differential equations into algebraic equations, 516519 convolution property of, 133 definition of, 508 of exponential function, 508 inverse, 508, 511513 of linear time-invariant continuoustime systems, 3738 properties, 513514 for system response, 218 transfer function, 169 lateral control system, 13 least squares estimation, 392, 401, 486 least-squares minimization, 446 least-squares system identification, 401407 linear quadratic (LQ) optimal control, 424428 linear time-invariant difference equations, solving, 4851, 59 linear time-invariant (LTI) discrete-time systems, 12 bilinear transformation, 234238 characteristic equation of, 234 Jury stability test, 239243, 245246 Nyquist criterion for, 248256 root locus for, 244247 RouthHurwitz criterion, 236241, 246 stability, 230233 linear time-invariant (LTI) systems, 167, 391 linear time-varying discrete system, 8990 loop, 492 loop gain, 492 low-order single-input single-output systems, 378380 M marginally stable, 231 Marine Air Traffic Control and Landing System (MATCALS), 466, 468 Masons gain formula, 6162, 68, 80, 150, 172, 174, 176177, 491493 MATLAB pidtool, 319321, 477484 MATLAB sisotool, 332333 matrix, 18, 501507 adjoint of, 503 algebra of, 505507 cofactor of, 503 derivative of a, 505506 determinant of, 504 diagonal, 502 identity, 501 inverse of, 504 inversion lemma, 504 Index minor of, 503 multiplication of a, 502, 505 partitioned, 502 symmetric, 502 trace of a, 502 transpose of, 502 McDonnell-Douglas Corporation F4 aircraft, 14 mechanical power, 26 memory locations (shift registers), 60 minimum-cost function, 424 minimum principle, 433434 modal matrix, 7778 modified z-transform, 136139, 499 properties of, 137 Moore-Penrose pseudo-inverse of , 392 motor back emf, 18 multiplication of matrices, 505 multiplication of scalars, 505 multiplication of vectors, 505 N neonatal fractional inspired oxygen, PID feedback controllers for MATLAB pidtool PIDF controllers, 477484 plant transfer function, 474476 Taubes PID controller, 476477 Newtons laws, 390, 516 second law of motion, 25 Nichols chart, 264266 ninth-order ordinary nonlinear differential equation, 1415 nonsynchronous sampling, 142145 nontouching loops, 492 nth-order continuous-time system, 37 nth-order differential equation, 518 nth-order linear difference equation,37 nth-order linear digital controller, 140 numerical integration algorithm, 219222 Nyquist criterion for discrete-time systems, 248256, 311 characteristic equation, 249 frequency response for G(z), 253 gain and phase margins, 255 MATLAB program to plot Nyquist diagram, 254255 Nyquist diagram, 250251 Nyquist path, 249250 pulse transfer functions, 255256 s-plane Nyquist diagram, 249250 theorem, 249 transfer function, 248 z-plane Nyquist diagram, 251252 O observability, concepts of, 374378 observer-based control systems, 369374 observer canonical form, 68 open-loop dc gain, 203 open-loop sampled-data systems, 168 open-loop systems containing digital filters, 133134 model, 134 open-loop transfer function, 234 optimal control law for system, 428429 optimality, principle of, 421424 original signal flow graph, 171, 173174, 176177 output-feedback controller, 26 overshoot, 206 P parameter Estimation, 391 partial-fraction expansion method, 52, 510, 512 path, 492 path gain, 492 peak overshoot, 280282 percent peak overshoot, 26 performance index, 418 persistency of excitation, 412 phase-lag compensator, 287294 advantages, 303 phase-lead compensation, 294295 advantages of, 303 closed-loop frequency responses, 299, 301 design procedure, 295298 disadvantages of, 303 MATLAB program, 291292, 299 open-loop frequency responses, 299300, 302303 step responses, 300 phase-lead filter, 299 phase margin, 255 phase margin of the compensated system, 289 phase variable canonical form, 68 physical sampler, 103 pitch angle, 21 plant defined, 11 dynamics of, 12 pole assignment/pole placement, 343346 polezero cancellations, 391 polezero locations, 110 positive definite quadratic form, 507 positive semidefinite quadratic form, 507 power amplifier, 101 power series method, 51 prediction errors, covariance of, 442 prediction observer, 353 predictor-corrector algorithm, 221222 primary strip, 110111 proportional-integral (PI) compensator, 35 proportional-plus-derivative (PD) controller, 247 proportional-plus-integral (PI) compensator, 243 527 proportional-plus-integral-plusderivative (PID) controller, 37, 279, 309313, 463 analog version of, 463 block diagram, 464 design process, 313315 frequency response for, 310311 MATLAB program, 316318 step response behavior, 464465 transfer function, 309310, 312 pseudo inverse, 392. Control Systems study material includes control systems notes, control systems books, control systems syllabus, control systems question paper, control systems case study, control systems questions and answers, control systems courses in control systems pdf form. 0000004031 00000 n 9 July 2021. The initial value of f(t) is f(0) = lim sF(s) = lim c sS sS s2 d = 1 s2 + a2 which, of course, is correct. Solutions Manual Digital Control System Analysis & Design 4/E, Charles L. Phillips, Troy Nagle, Aranya Chakrabortty - Free download as PDF File (.pdf), Text File (.txt) or read online for free. Three examples of finding the inverse Laplace transform are given next. Control Systems Engineering S. Palani, TMH. ScienceDirect is a registered trademark of Elsevier B.V. ScienceDirect is a registered trademark of Elsevier B.V. Reference Data for Engineers (Ninth Edition), Fundamental approaches to control system analysis, A Generalized Framework of Linear Multivariable Control, Linear and Non-Linear Stability Analysis in Boiling Water Reactors, Advanced Control Design with Application to Electromechanical Systems, The main objective of this chapter is to present a broad range of well-worked out and recent theoretical tools in the field of advanced, General Aviation Aircraft Design (Second Edition), and when designing the control system. Academia.edu no longer supports Internet Explorer. (a) Find [ f1(t)][ f2(t)]. The damped sinusoid has an amplitude of 2R and a phase angle of w, where R and w are defined in (A5-15). Hence a method is required for converting from a general rational function to the forms that appear in the tables. 7En@. A5-17. Thus for any function f(t), [ f(t)] = [ f(t)u(t)] = F(s) (A5-7) ExAMplE A5.1 The Laplace transform of the time function f(t) = 5u(t) + 3-2t will now be found. AlAin University Of Science And Technology, Australian Academy of Higher Learning Pty Ltd T A Technical Institute of Victoria. Computer examples using MATLABnd Simulinkhave been introduced throughout the book to supplement and enhance standard hand-solved examples. In the last few years, it has been incorporated in Mathematica (via MathModelica) and Maple (via MapleSim). A block diagram cannot be constructed until inputs and outputs of each component have been defined. Di erential Control Now suppose we furthermore have a performance speci cation: Overshoot Rise Time Settling Time T G(s) D s +-u(s) y(s) Problem: Both of these environments can be used to create dynamic simulations of chemical systems reasonably easily, and EMSO features a large library of chemical engineering unit operations. In the above article, a student can download control systems notes for B Tech EEE 2nd year and control systems notes for B Tech ECE 2nd year. From (A5-11), k1 = (s - a + jb)F(s) s = a - jb = R jw k2 = (s - a - jb)F(s) s = a + jb = R-jw = k *1 (A5-15) where the asterisk indicates the conjugate of the complex number. See also, Moore Penrose pseudo inverse pulse transfer function, 127133 Q quadratic cost function, 419421 quadratic form, 419, 506 R radar-noise disturbances, 14 radar unit, 13 rectangular rule for numerical integration, 36, 219 recursive least-squares system identification, 409412 reduced-order observer, 364369 regulator control system, 378 repeated-root terms, 510 residue of a function, 106 residue of F(s), 510 resonance, 267268 response, defined, 15 resultant system stability margins, 293 Riccati equation, 434, 437 rise time, 280282 robotic control system, 2122 root locus for a system, 321334 characteristic equation, 321322 filter dc gain equal unity, 322 MATLAB program for, 324326, 330333 phase-lag design, 322324 phase-lead design, 326328 round-off errors in computer, 220 Runge-Kutta rule, 223 S sampled-data control systems, 100103, 113 sampled signal flow graph, 172 sample period, 13 sampler-data hold device, 101103 satellite model, 1618 second-order differential equation, 17 second-order system, 441 second-order transfer function, 17 series compensation, 285 sensitivity, 283 servomechanism, 2021 servomotor system, 1822, 429, 454461 computer data format, 456 phase-lag filter, 460461 528 Index servomotor system (continued ) phase lead design, 459461 system frequency response, 458460 system hardware, 455 system model, 456459 settling time, 280282 Shannons sampling theorem, 112113 shifting theorem, 138 signal flow graph, 60 corresponding state equations of, 6869 similarity transformations, 73 properties, 74 simulation diagrams of analog plant, 154 for discrete-time systems, 5962 single-machine infinite bus (SMIB) power system, 2426 continuous-time state-variable model for, 26 current phasor, expression for, 25 set of symbols, 25 single-sided z-transform, 38 single-valued function relationship, 64 singular value decomposition (SVD), 396 sink node, 492 SNR (signal to noise ratio), 413 source node, 492 specific heat of liquid, 23 stabilizable, 439 starred transform, 102, 111 state equations numerical method via digital computer, 87 recursive solution, 8486 z-transform of, 8184, 8687 state estimator closed-loop state equations, 363364 closed-loop system characteristic equation, 362363 controller transfer function, 359362 error dynamics, 354356 errors in, 354 example, 356359 observer model, 352353 plantobserver system, 355 state transition matrix, 85, 149 computer method for finding, 87 properties of, 88 state-variable formulations, 71 converting continuous state equations, 178183 for diagonal elements, 7678 for digital controller, 183188 examples, 7180 for mechanical system, 147150 of open-loop sampled-data systems, 145146 using linear-transformation matrix, 7576 using partial-fraction expansion, 7273 using similarity transformations, 7374 state-variable model of a system, 6364 example, 6468 multivariable discrete system, state equations for, 7071 transfer function, state equations for, 6970 static systems, 391393 steady-state accuracy, 215218 steady-state optimal control, 434438 stiffness factor, 147 summing junction, 491 symmetric matrix, 502 synchronous generator, 484 system characteristic equation, 207208 system identification, 390 system time response, 198200 for all instants of time of sampleddata system, 201 analog system unit-step response, 200 effects of sampling, 200202 mapping s-plane into z-plane, 208215 simulation of, 218222 T Taubes PID controller, 476477 temperature control system, 2224, 463466 Texas Instruments TI9900 microprocessor system, 455 thermal capacity of liquid, 23 thermal system. DYNAMICS & CONTROL 3 CONTROL Section 2: Basics of Control System Analysis Dr. Study Resources. Concordia University MECH 371 ANALYSIS AND DESIGN OF CONTROL SYSTEMS Fall 2022 Assignment #2 E 2.23 Determine the closed-loop transfer function? # (c) Find x(t) for the case that x(0) = 1 and x(0) = 1. Included are a study of non-minimum phase systems and the concepts of gain margin, phase margin, delay margin, bandwidth, stability, and sensitivity. Your email address will not be published. (c) Write the characteristic equations for the systems of Problem A5-13. Procedural variables are those that can affect performance via the procedures imposed on, or employed in, the task environment in which that performance is observed (e.g., the nature of instructions, the training schedule, etc.). (b) Take the inverse transform of each F(s) in part (a) to verify the results. See the end of the References section for further information on software packages. Take the Laplace transform of f(t) in part (c) to verify your result. Snorri Gudmundsson BScAE, MScAE, Ph.D., FAA DER (ret. 0000003249 00000 n This part of the solution is therefore the complementary solution of the differential equation. If we let p1 = a - jb and p2 = a + jb, (A5-9) can be written as F(s) = k3 k1 k2 kn + + + g + s - p3 s - pn s - a + jb s - a - jb (A5-14) The coefficients k1 and k2 can be evaluated using (A5-11) as before. Course Hero uses AI to attempt to automatically extract content from documents to surface to you and others so you can study better, e.g., in search results, to enrich docs, and more. 0000004336 00000 n The supertwisting algorithm is adopted to control the rotor side converter (RSC) and the grid side converter (GSC) at a fixed switching frequency. 0000003805 00000 n We must always keep in mind that deriving reasonable mathe-matical models is the most important part of the entire analysis of control systems. Sorry, preview is currently unavailable. Modern Control Systems Richard C. Dorf 2011 Modern Control Systems, 12e, is ideal for an introductory undergraduate course in control systems for engineering students. We usually denote the transfer function by G(s). The ai coefficients in (A5-30) are parameters of the physical system described by the differential equation, such as mass, friction coefficient, spring constant, inductance, and resistance. 0000002311 00000 n Digital Control Systems Analysis and Design is appropriate for a one semester/two-quarter senior-level course in digital or discrete-time controls. We apprehend how the Bode magnitude and phase diagrams are constructed, with emphasis on the details for second-order systems, and how they are beneficial to control systems analysis and synthesis.