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ps3fpe - Problems 167 em 3.42 Suppose that unity feedback...

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Unformatted text preview: Problems 167 em. 3.42 Suppose that unity feedback is to be applied around the listed open-loop systems. Use Routh’s stability criterion to determine whether the resulting closed-loop systems will of an aircraft :; be stable. 40+” and the pitch (33 KG“) = s(s§+2.t2+33+4) .5 . _ 2 5+4) (b) [(60) _ s2 (H1) _ Ma (Q) KGB) — 82(33+232—.\‘—1) 3.43 Use Routh’s stability criterion to determine how many roots with positive real parts the following equations have: (a) s4 + 853 + 32x2 4- so: + 100 = 0. ' (b) s5 + 10:4 + 3053 + 8032 + 344.: + 480 = 0. (c) :4 + 2:3 + 7.92 m 2.3 + 8 = 0. (t1)r3+r2 +205+7S =0. (e) s4 + 652 + 25 = 0. degrees. The :ording to the than 10% and ning the step lty assOciated r_4._..__.,____,__ _,_.,. Mvhcw :ms. 3.44 Find the range of K for which all the roots of the following polynomial are in the LHP: :5 + 534 +1053 +1059- + is +K : 0. Use MATLAB to verify your answer by plotting the roots of the polynomial in the s-plane for various values of K. 3.45 The transfer function of a typical tape-drive system is given by Ks+4 Go) : c_u_w slts + 0.5)(s + 1) (S" + 0.45 + 4)] where time is measured in milliseconds. Using Routh’s stability criterion, determine the amount of t the range of K for which this system is stable when the characteristic equation is 3), given some 'i 1 —l» ((5) = O. ' 3.46 Consider the closed-loop magnetic levitation system shown in Fig. 3.67. Determine the conditions on the system parameters (a, K, 2, p. K0) to guarantee closed-loop system i stability. Figure 3.67 + e K(s+z) it XD . . . R ”r 1’ Magnetic lewtation (5+p) (Em?) 3W that system _ 1 3.47 Consider the system shown in Fig. 3.68. (3) Compute the closad-loop characteristic equation. (b) For what values of (72A) is the system stable? Hint: An approximate answer may be found using l 9‘“ s 1 4 Ta Figure 3.68 Control system for Problem 3.47 tte errors but high )1 provides robust ystem less stable. These three kinds and describes the tables 4.2 and 4.3. to replace a given : transfer 'fu nction ids to uIIkTS) then id C2d. tracking input and : rate of 40°C/ sec. error constant (KP 3 ts? system? : rather than in the roller. )r at if the approx- height e(kT_;) and Figure 4.23 Three-amplifier topologies For Problem 4.2 Problems 207 PROBLEMS Probiemsfor Section 4.] .' The Basic Equations of Control 4.1 4.2 4.3 if S is the sensitivity of the unity feedback system to changes in the plant transfer function and '1‘ is the transfer function from reference to output, show that S + T = 1. We define the sensitivity of a transfer function G to one of its parameters k as the ratio of percent change in G to percent change in k. 8‘3 _ dG/G A dlnG _ it: .1156 k ‘ dk/k — dlnt: ‘ Gttk' The purpose of this prohlem is to examine the effect of feedback on sensitivity. In particular, we would like to compare the topologies shown in Fig. 4.23 for connecting three amplifier stages with a gain of —K into a single amplifier with a gain of — 10. (a) For each topology in Fig. 4.23, compute fi; so that ifK : 10, Y : —lOR. (b) For each topology, compute .3? when G = Y/R. [Use the respeCLiVe ,6,- valnes found in part (a).'_[ Which case is the least sensitive? (c) Compute the sensitivities of the systems in Fig. 4.23(h,c) to ,5; and 63. Using your results, comment on the relative need for precision in sensors and actuators. :32 (b) Compare the two structures shown in Fig. 4.24 wilh respect to sensitivity to changes in the overall gain due to changes in the amplifier gain. Use the relation idlnF_KctF S _ # _ _ a’ In K F dK as the measure. Select H1 and H2 so that the nominal system outputs satisfy Fl 2 F2, and assume KH1 > O. 208 Chapter 4 A First Analysis of Feedback Figure 4.24 Block diagrams for Problem 4.3 4.4 A unity feedback control system has the open-loop transfer function A (a) Compute the sensitivity of the closcdeloop transfer function to changes in the parameter A. lb) Compute the sensitiVity of the closed-100p transfer function to changes in the parameter a. (c) Ifthe unity gain in the feedhack changes to a value of t5 94 l. compute the sensitivity of the closed-loop transfer function with respect to ,8. 4.5 Compute the equation for the system error for the filtered feedback system shown in Fig. 4.4. 4.6 if S is the sensitivity of the filtered feedback system to changes in the plant transfer function and T is the transfer function from reference [0 output, compute the sum of S+T.ShowthatS+T= i iszH. (:1) Compute the sensitivity of the filtered feedback system shown in Fig. 4.4 with respect to changes in the plant transfer function, G. (b) Compute the sensitivity of the filtered feedback system shown in Fig. 4.4 with respect to changes in the controller transfer function, Dd (c) Compute the sensitivity of the filtered feedback system shown in Fig. 4.4 with respect to changes in the filter transfer function, F, (d) Compute the sensitivity of the filtered feedback system shown in Fig. 4.4 with respect to changes in the sensor transfer function. H. Problemsfor Section 4.2: Control of Steady-Stare Error 4.7 Consider the DC—motor c0ntrol system with rate (tachometer) feedback shown in Fig. 425(3). (a) Find values for K’ and it; so that the system of Fig. 4.25(b) has the same transfer function as the system of Fig. 4.25(a). (1)) Determine the system type with reSpect to tracking 6,- and compute the system K1; in terms of parameters K’ and id . (c) Does the addition of tachometer feedback with positive it, increase or decrease K”? 4.8 Consider the system shown in Fig. 4.26, where (3+4)2 0(3) = Km- Figure 4.25 Control system fl Figure 4.26 Control system f Problem 4.8 Figure 4.2? Control system Problem 4.9 :lt‘Ol cm are shown in cgleetcd arid the to a step distur— rder to make the .ccurately sketch gain K computed :s corresponding shootMp 5 5%. specifications. the new control bance torque be 3 tracking? ice rejection? i Figure 4.41 DC Motor speed-control block diagram for Problems 4.29 and 4.30 4.29 Problems 217 (3) Determine the system type and error constant with respect to tracking polynomial reference inputs of the system for P [D = kp], PD [D = kp + [(03], and PID [D = kp + kI/t- + [cps] controllers. Let kg = 19, it; 2 0.5, and kD = 4/19. (b) Determine the system type and error constant of the system with respect to distur- bance inputs for each of the three regulators in part (a) with respect to rejecting polynomial disturbances WU) at the input to the plant. (c) Is this system better at tracking references or rejecting disturbances? Explain your response briefly. (d) Verify your results for parts (a) and (b) using MATLAB by plotting unit step and ramp responses for both tracking and disturbance rejection. The DCsmotor speed control shown in Fig. 4.41 is described by the differential equation it + 6sz = 600%; - lSOOw, where y is the motor speed, va is the armature voltage, and w is the load torque. Assume the armature voltage is computed using the PI control law t vfl = — (kpe + [:1]; edit), where e = r — y. (a) Compute the transfer function from W to 1’ as a function of fig; and Jr}. (b) Compute values for kp and it; so that the characteristic equation of the closedsloop system will have roots at —60 :l: 60}. 4.30 4.31 For the sysrern in Problem 4.29. compute the following steady-state errors; (a) to a unit—step reference input; (h) to a unit-ramp reference input; (c) to a unit—step disturbance input; (d) for a unit-ramp disturbance input. (e) Verify your answers to (a) and (d) using MATLAB. Note that a ramp response can be generated as a step response of a system modified by an added integrator at the reference input. COnsider the satellite-attitude centrol problem shown in Fig. 4.42 where the normalized parameters are I : 10 spacecraft. inertia, N-lTl-SBC2/I'Etd , : reference satellite attitude, rad. ...
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