Đang tải... (xem toàn văn)
(BQ) Part 1 book Organic chemistry has contents: Structure determines properties, alcohols and alkyl halides, addition reactions of alkenes, stereochemistry, nucleophlic substitution, conlugation in alkadienes and allylic systems, arenes and aromaticity, spectroscopy,... and other contents.
CHAPTER The Root Locus Method 7.1 Introduction 7.2 The Root Locus Concept 408 7.3 The Root Locus Procedure 7.4 Parameter Design by the Root Locus Method 7.5 Sensitivity and the Root Locus 408 7.6 Three-Term (PID) Controllers 7.7 Design Examples 413 431 437 444 447 7.8 The Root Locus Using Control Design Software 7.9 Sequential Design Example: Disk Drive Read System 458 7.10 Summary 463 465 PREVIEW The performance of a feedback system can be described in terms of the location of the roots of the characteristic equation in the s-plane A graph showing how the roots of the characteristic equation move around the s-plane as a single parameter varies is known as a root locus plot The root locus is a powerful tool for designing and analyzing feedback control systems We will discuss practical techniques for obtaining a sketch of a root locus plot by hand We also consider computer-generated root locus plots and illustrate their effectiveness in the design process We will show that it is possible to use root locus methods for controller design when more than one parameter varies This is important because we know that the response of a closed-loop feedback system can be adjusted to achieve the desired performance by judicious selection of one or more controller parameters The popular PID controller is introduced as a practical controller structure with three adjustable parameters We will also define a measure of sensitivity of a specified root to a small incremental change in a system parameter The chapter concludes with a controller design based on root locus methods for the Sequential Design Example: Disk Drive Read System DESIRED OUTCOMES U p o n completion of Chapter 7, students should: Zi J J _l Understand the powerful concept of the root locus and its role in control system design Know how to sketch a root locus and also how to obtain a computer-generated root locus plot Be familiar with the PID controller as a key element of many feedback systems in use today Recognize the role of root locus plots in parameter design and system sensitivity analysis Be capable of designing a controller to meet desired specifications using root locus methods 407 408 7.1 Chapter The Root Locus Method INTRODUCTION lire relative stability and the transient performance of a closed-loop control system are directly related to the location of the closed-loop roots of the characteristic equation in the s-plane It is frequently necessary to adjust one or more system parameters in order to obtain suitable root locations Therefore, it is worthwhile to determine how the roots of the characteristic equation of a given system migrate about the s-plane as the parameters are varied; that is, it is useful to determine the locus of roots in the s-plane as a parameter is varied The root locus method was introduced by Evans in 1948 and has been developed and utilized extensively in control engineering practice [1-3] The root locus technique is a graphical method for sketching the locus of roots in the s-plane as a parameter is varied In fact, the root locus method provides the engineer with a measure of the sensitivity of the roots of the system to a variation in the parameter being considered The root locus technique may be used to great advantage in conjunction with the Routh-Hurwitz criterion The root locus method provides graphical information, and therefore an approximate sketch can be used to obtain qualitative information concerning the stability and performance of the system Furthermore, the locus of roots of the characteristic equation of a multiloop system may be investigated as readily as for a single-loop system If the root locations are not satisfactory, the necessary parameter adjustments often can be readily ascertained from the root locus [4] 7.2 THE R O O T L O C U S C O N C E P T The dynamic performance of a closed-loop control system is described by the closed-loop transfer function T(s) Y(s) p(s) R(s) q(sY (7.1) where p(s) and q(s) are polynomials in s The roots of the characteristic equation q(s) determine the modes of response of the system In the case of the simple singleloop system shown in Figure 7.1, we have the characteristic equation + KG(s) = 0, (7.2) where K is a variable parameter The characteristic roots of the system must satisfy Equation (7.2), where the roots lie in the s-plane Because is a complex variable, Equation (7.2) may be rewritten in polar form as \KG(s)\/KG(s) = - + /0, FIGURE 7.1 Closed-loop control system with a variable parameter K Ri.s) • Y{s> (7.3) Section 7.2 The Root Locus Concept 409 and therefore it is necessary that \KG(s)\ = and /KG(s) = 180° + /c360°, (7.4) where k ~ 0, ± , ±2, ± , The root locus is the path of the roots of the characteristic equation traced out in the s -plane as a system parameter is changed The simple second-order system considered in the previous chapters is shown in Figure 7.2 The characteristic equation representing this system is A(s) = + KG(s) = + K = 0, s(s - 2) or, alternatively, A(s) = s2 + 2s t K = s2 + 2£o)lts + w?, = (7.5) Trie locus of the roots as the gain K is varied is found by requiring that \KG(s)\ - K = s(s + 2) (7.6) and /KG(s) = ±180°, ±540°, (7.7) The gain K may be varied from zero to an infinitely large positive value For a second-order system, the roots are sus2 = ~C(»n ±o>„V£ - 1, (7.8) -1 and for £ < 1, we know that B = cos £ Graphically, for two open-loop poles as shown in Figure 7.3, the locus of roots is a vertical line for t, < in order to satisfy the angle requirement, Equation (7.7) For example, as shown in Figure 7.4, at a root Su the angles are K s(s + 2) FIGURE 7.2 Unity feedback control system The gain K is a variable parameter R(s) "> J = -/sx - /(s1 + 2) = - [(180° - B) + 0] = -180° s=$i fc K s(s + 2) (7.9) 410 Chapter The Root Locus Method /« 4L * i K increas rig- FIGURE 7.3 Root locus for a second-order system when Ke< K, < K2 The locus is shown as heavy lines, with arrows indicating the direction of increasing K Note that roots of the characteristic equation are denoted by " • " on the root locus i r t ! s s N "1 Ae /V [Ke K T - l = -t = 180°, or, alternatively, 180° Accounting for all possible root locus segments at remote locations in the s-plane, we obtain Equation (7.30) The center of the linear asymptotes, often called the asymptote centroid, is determined by considering the characteristic equation in Equation (7.24) For large values of s, only the higher-order terms need be considered, so that the characteristic equation reduces to However, this relation, which is an approximation, indicates that the centroid of n - M asymptotes is at the origin, s = A better approximation is obtained if we consider a characteristic equation of the form K with a centroid at crA The centroid is determined by considering the first two terms of Equation (7.24), which may be found from the relation M K t WS + Zi) llg»* +w-' + -+ft, From Chapter 6, especially Equation (6.5), we note that M i>M-\ = ¾ n and fl = «-i 1=1 Sty 7=1 Considering only the first two terms of this expansion, we have + M ,n-M t (n TTT = _ u \ n-M-\ Index Absolute stability, A system description that reveals whether a system is stable or not stable without consideration of other system attributes such as degree of stability, 356,406 Acceleration error constant, Ka The constant evaluated as lim[j ,2 G(5)l The sreadys-*0 state error for a parabolic input, r(t) = At1/!, is equal toA/Ka, 298 Acceleration input, steady-state error, 297-298 Accelerometer, 71,83 Ackermann's formula, 756, 767-768,772,777-778, 809-810,816 Across-variable, 43,45 Actuator, The device that causes the process to provide the output; the device that provides the motive power to the process, 62,142 Additive perturbation, A system perturbation model expressed in the additive form Ga{s) = G(s) + A(s) where G(s) is the nominal plant, A(s) is the perturbation that is bounded in magnitude, and Ca(s) is the family of perturbed plants, 834,899 Agricultural systems, 13 Aircraft, and computer-aided design, 19 unmanned, 15 Aircraft autopilot, 853 Aircraft attitude control, 319 Airplane control, 266,474-475, 482,747-748 All-pass network, A nonminimum phase system that passes all frequencies with equal gain, 513-514,566 Alternative signal flow graph, and block diagram models, 165-170 Amplidyne, 127 Amplifier, feedback, 219 Amplitude quantization error, 906-907, 950 Analogous variables, 47 Analog-to-digital converter, 902,906 Analysis of robustness, 834-836 Angle of departure, The angle at which a locus leaves a complex pole in the s-plane, 422-423,426,441^143,491 Angle of the asymptotes, The angle that the asymptote makes with respect to the real axis,4>A, 415,418,491 Armature-controlled motor, 64-65,69,81,94,117,127,137, 139 Array operations in MathScript, 979 Array operations in MATLAB, 959-960 Artificial hand, 11,14,36 Assumptions, Statements that reflect situations and conditions that are taken for granted and without proof In control systems, assumptions are often employed to simplify the physical dynamical models of systems under consideration to make the control design problem more tractable, 42,83-84,142 Asymptote, The path the root locus follows as the parameter becomes very large and approaches infinity, 415 of root locus, 415 Asymptote centroid, The center of the linear asymptotes, 416 aA, Asymptotic approximation for a Bode diagram, 502 Automatic control, history of, 4-8 Automatic fluid dispenser, 200,202 Automatic test system, 795-797 Automation, The control of an industrial process by automatic means, 6,39 Automobile steering control system, Automobiles, hybrid fuel vehicles, 21, 40 Auxiliary polynomial, The equation that immediately precedes the zero entry in the Routh array, 365,496 Avemar ferry hydrofoil, 736 Axis shift, 369 Backward difference rule, A computational method of approximating the time derivative of a function given by x(kT) x(kT) - x((k l)r) T where t = kT,T is the sample time, and k = 1,2, , 925,950 Bandwidth, The frequency at which the frequency response has declined dB from its low-frequency value, 520, 566,596,665 Bellman, R., Biological control system, 14 Black, H.S., 5-6,8,130,830 Block diagram, Unidirectional, operational block that represents the transfer 1007 1008 Index functions of the elements of the system, 71,72 Block diagram models, 71-76, 107-116 alternative signal-flow graphs, 165-170 signal-flow graphs, 154-165 Block diagram transformations, 73-74 Bobbin drive, 356 Bode,H.W., 500,830 Bode plot, The logarithm of magnitude of the transfer function is plotted versus the logarithm of o>, the frequency The phase, cj>, of the transfer function is separately plotted versus the logarithm of the frequency, 500-501, 541,567 asymptotic approximation, 502 Boring machine system, 232 Bounded response, 356 Branch on signal-flow graph, 76 Break frequency, The frequency at which the asymptotic approximation of the frequency response for a pole (or zero) changes slope, 502, 505,566 Breakaway point The point on the real axis where the locus departs from the real axis of the 5-plane, 418-420,491 Bridge,Tacoma Narrows, 357-359 Camera control, 308-312,341 Canonical form, A fundamental or basic form of the state variable model representation, including phase variable canonical form, input feedforward canonical form, diagonal canonical form, and Jordan canonical form, 211 Capek, Karel, 10 Cascade compensation network, A compensator network placed in cascade or series with the system process, 671-675,755 Cauchy's theorem If a contour encircles Z zeros and P poles of F( s), the corresponding contour encircles the origin of the F(s)-plane N = Z - P times clockwise, 568, 571-575,665 Characteristic equation, The relation formed by equating to zero the denominator of a transfer function, 52,142,387 Circles, constant, 596 Closed-loop feedback control system, A system that uses a measurement of the output and compares it with the desired output, 3,39,214 Closed-loop sampled-data system, 912 Closed-loop transfer function, A ratio of the output signal to the input signal for an interconnection of systems when all the feedback or feedfoward loops have been closed or otherwise accounted for Generally obtained by block diagram or signal flow graph reduction, 74,142, 387 388 Command following A n important aspect of control system design wherein a nonzero reference input is tracked, 779, 826 Compensation, The alteration or adjustment of a control system to provide a suitable performance, 668 using a phase-lag network on the Bode diagram, 691 using a phase-lag network on the s-plane, 692 using a phase-lead network on the Bode diagram, 675 using a phase-lead network on the s-plane, 681 using analytical methods, 700 using integration networks, 688 using state-variable feedback, 757 Compensator, An additional component or circuit that is inserted into the system to equalize or compensate for the performance deficiency, 477,668,755.757 Compensator design, full-state feedback and observer, 773 Complementary sensitivity function, The function C(s) = w „ , , ^ , that + Gc(s)G(s) satisfies the relationship S(s) + C(s) = 1, where 5(5) is the sensitivity function The function C(s) = T(s) is the closed-loop transfer function, 216,834,899 Complexity, A measure of the structure, intricateness, or behavior of a system that characterizes the relationships and interactions between various components, 16,39,276 in cost of feedback, 231 Complexity of design The intricate pattern of interwoven parts and knowledge required, 16 Components, The parts, subsystems, or subassemblies that comprise a total system, 276 in cost of feedback, 231 Computer control systems, 901,902 for electric power plant, 13 Computer-aided design, 19 Computer-aided engineering (CAE), 21 Conditionally stable system, 475 Conformal mapping, A contour mapping that retains the angles on the s-plane on the F(s)-plane, 570,655 Congress, 14 Constant M circles, 597 Constant N circles, 597 1009 Index Continuous design problem, 38, 139,208,270,349,402,485, 561,659,747,821,891,947 Contour map, A contour or trajectory in one plane is mapped into another plane by a relation F(s), 569 Contours in the s-plane, 569 Control engineering, 2,8-9 Control system An interconnection of components forming a system configuration that will provide a desired response, 2,39 characteristics using m-files, 246 design, 17 modern examples, 8-16 Controllability, 757-763 Controllability matrix, A linear system is (completely) controllable if and only if the controllability matrix P c = [B AB A B A"B] has full rank, where A is an nxn matrix; for single-input, single-output linear systems, the system is controllable if and only if the determinant of the nxn controllability matrix P c is nonzero, 758,826 Controllable system, A system with unconstrained control input u that transfers any initial state x(0) to any other state x(r), 758,826 conv function, 105,968 Convolution integral, 280 Corner frequency See Break frequency Cost of feedback, 231-232 Coulomb damper, 45 Critical damping, The case where damping is on the boundary between underdamped and overdamped, 54,142 Critically damped system, 103 Damped oscillation An oscillation in which the amplitude decreases with time, 56,142 Dampers, 45 Damping ratio, A measure of damping; a dimensionless number for the second-order characteristic equation, 54, 142,292 estimation of, 292 DC motor, An electric actuator that uses an input voltage as a control variable, 142 armature controlled, 64,81 field controlled, 63 Deadbeat response, 755 Decade, A factor of ten in frequency (e.g., the range of frequencies from rad/sec to 10 rad/sec is one decade), 502,566 of frequencies, 502 Decibel (dB), The units of the logarithmic gain, 566 Decoupled state variable format, 166 Design, The process of conceiving or inventing the forms, parts, and details of a system to achieve a reasoned purpose, 16-17,39 Design of a control system, The arrangement or the plan of the system structure and the selection of suitable components and parameters, 755 robot control, 396 in the time domain, 757 using a phase-lag network on the Bode diagram, 696 using a phase-lag network on the s-plane, 691 using a phase-lead network on the Bode diagram, 675 using a phase-lead network on the s-plane, 681 using integration networks, 688 using state-feedback, 756 Design specifications, 278 Detectable, A system in which the states that are unobservable are naturally stable., 761,826 Diagonal canonical form, A decoupled canonical form displaying the n distinct system poles on the diagonal of the state variable representation A matrix, 166,211 Differential equations, An equation including differentials of a function, 42,143 Differential operator, 50 Differentiating circuit, 68 Digital computer compensator, A system that uses a digital computer as the compensator element, 918-921 Digital control system, A control system using digital signals and a digital computer to control a process, 901-950 Digital control systems using control design software, 935 Digital controllers, implementation of, 925 Digital to analog converter, 905 Direct system, 213 See also Open-loop control system Discrete-time approximation, An approximation used to obtain the time response of a system based on the division of the time into small increments A ; , 211 Disk drive read system See Sequential design example Disturbance rejection property, 221-224 Disturbance signal An unwanted input signal that affects the system's output signal, 220-225,276 Dominant roots, The roots of the characteristic equation that represent or dominate the closed-loop transient response, 288,353, 427,491,521,566 Dynamics of physical systems, 41 Electric power industry, Electric traction motor, 13 93-95 • 1010 Index Electrohydraulic actuator, 66, 69,129 Engineering design, The process of designing a technical system, 16-17,40 English channel tunnel boring system, 232 Engraving machine 523-526, 537-538 Epidemic disease, model of, 167-168,372 Equilibrium state, 167 Error, steady-state, 228 Error constants, acceleration, 296 ramp, 297 step, 295 Error signal, The difference between the desired output, R{s), and the actual output Y(s); therefore E(s) = R(s) - Y(s), 110,143,276 Estimation error The difference between the actual state and the estimated state e(r) = x(t) - x(t), 769,826 Euler's method, A first-order explicit integration method utilized to obtain numerical solutions of differential equations, 211 Evans, W.R., 408 Examples of control systems, Exponential matrix function, 150 Extender, 135,206,742 Federal Reserve Board, 14 Feedback, amplifier, 219 control system, 3,9-11, 720-726 cost of, 231-232 full-state control design, 763 negative, 3,6 positive, 32 of state variables, 782,784 Feedback control system, and disturbance signals, 220-225 feedback function, 111-113,968 Feedback signal, A measure of the output of the system used as feedback to control the system, 3,40,110 Feedback systems, history of, Final value The value that the output achieves after all the transient constituents of the response have faded Also referred to as the steady-state value, 54 of response of y(t), 54 Final value theorem, The theorem that states that lim y(t) t—*co = lim 5Y(5), where Y(s) is the Laplace transform of y ( , 54 Flow graph See Signal-flow graph Flyball governor, A mechanical device for controlling the speed of a steam engine, 4-5,40 Forward rectangular integration, A computational method of approximating the integration of a function given by x(kT)**x({k-l)T) + Tx((k-1)T), where t = kT, T is the sample time, and k = 1,2, , 925,950 Fourier transform The transformation of a function of time, / ( f ) into the frequency domain, 496 Fourier transform pair, A pair of functions, one in the time domain, denoted by f(t), and the other in the frequency domain, denoted by F{j Yw) s(s + 2^,,) — CLOSED-LOOP MAGNITUDE PLOT UNIT STEP RESPONSE Overshoot -• Z)\ogMpM ^ '/; Rise time Pea t me ^ * Settling time (to within % of the final value) J Percent overshoot J + e -ivP/l-f J and = P J Maximum magnitude (f s 0.7) pto = io&r^' Vw2 Time-to-peak r C0n Settling time J M„ 0), Time J Resonant frequency {£ :£ 0.7) &), = w„Vl - 2£2 *„Vi - f2 Rise time (time to rise from 10% to 90% of final value) J Bandwidth (0.3 < f < 0.8) a>fi = (-1.19« + 1.85K ^ = 2,16f + 0.60 (0.3 < £ < 0.8) P1D Controller: < « * ) = = K r + KDs *i + s U + Zi)(S + 22) s TABLE PAGE 5.5 Summary of Steady-State Errors 298 5.6 The Optimum Coefficients of T(s) Based on the ITAE Criterion for a Step Input 308 5.7 The Optimum Coefficients of T(s) Based on the ITAE Criterion for a Ramp Input 312 10.2 Coefficients and Response Measures of a Deadbeat System 706 10.7 A Summary of the Characterislics of Phase-Lead and Phase-Lag Compensation Networks 729 ... 7 .1, as shown in Figure 7 .11 (b) Alternatively, we differentiate K(.s) K(s + 1) s(s + 2) His) FIGURE 7 .10 Closed-loop system 5+3 " • Yis) Chapter The Root Locus Method Table 7 .1 p(s) 0. 411 0. 419 ... magnitude criterion, Equation (7 .16 ) For example, the gain K at the root s — s- = - is found from (7 .16 ) as STEP gjjQlgi + 2| si\si + 4| = or K = 1- 111 -1 + 41 (7.28) | - + 2| This magnitude... 4)(52 + 8s + 32) + K = s4 + 12 s3 + 64s2 + 12 85 + K = (7.50) Therefore, the Routh array is sA 64 K 12 12 8 bx K , ci where bi — 12 (64) - 12 8 — = 53.33 and 12 53.33 (12 8) - YIK q = 53.33 Hence, the