2022年完整word版,《自动控制原理》试卷及答案 .pdf

AUTOMATIC CONTROL THEOREM (1) Derive the transfer function and the differential equation of the electric network shown in Fig.1. (12% ) Consider the system shown in Fig.2. Obtain the closed-loop transfer function)()(SRSC, )()(SRSE. （12%）Fig.2 The characteristic equation is given 010)6(5)(123KSKSSSGH. Discuss the distribution of the closed-loop poles. (16%) There are 3 roots on the LHP There are 2 roots on the LHP There are 1 roots on the LHP There are no roots on the LHP . K=? Consider a unity-feedback control system whose open-loop transfer function is )6.0(14 .0)(SSSSG. Obtain the response to a unit-step input. What is the rise time for this system? What is the maximum overshoot? （10%）C G1 G2 G3 G4 H2 H1 R E C2 R2 V2(S) R1 C1 V1(S) Fig.1 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 1 页，共 29 页5. Sketch the root-locus plot for the system )1()(SSKSGH. ( The gain K is assumed to be positive.) Determine the breakaway point and K value. Determine the value of K at which root loci cross the imaginary axis. Discuss the stability. (12%) 6. The system block diagram is shown Fig.3. Suppose )2(tr, 1n. Determine the value of K to ensure 1SSe. (12%) Fig.3 7. Consider the system with the following open-loop transfer function: )1)(1()(21STSTSKSGH. Draw Nyquist diagrams. Determine the stability of the system for two cases, the gain K is small, K is large. (12%) 8. Sketch the Bode diagram of the system shown in Fig.4. (14%) 2SK)3(4SSR E N C 32SS)10)(5()2(SSSR(S) C(S) Fig.4 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 2 页，共 29 页212121121212)()()(CCSCCRRCSCCRSVSV2423241321121413211)()(HGHGGGGGGGHGGGGGGGSRSC 0K6 K0 K6 no answer the breakaway point is 1 and 1/3; k=4/27 The imaginary axis S=j; K=25.75. 3K) 154.82)(181.34)(1481. 3)(1316.0() 11 .0(62.31)(SSSSSSGH精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 3 页，共 29 页AUTOMATIC CONTROL THEOREM (2) Derive the transfer function and the differential equation of the electric network shown in Fig.1. (12% ) Consider the equation group shown in Equation.1. Draw block diagram and obtain the closed-loop transfer function)()(SRSC. （16% ）Equation.1 )()()()()()()()()()()()()()()()()()()()(3435233612287111SXSGSCSGSGSCSXSXSXSGSXSGSXSCSGSGSGSRSGSX Use Routh s criterion to determine the number of roots in the right-half S plane for the equation 0400600226283)(12345SSSSSSGH. Analyze stability.（12% ） Determine the range of K value ,when )1 (2ttr, 5.0SSe. （12% ）Fig.2 C1 C2 V2(S) R1 R1 V1(S) Fig.1 R E ) 2(4322SSKSSC 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 4 页，共 29 页Fig.3 shows a unity-feedback control system. By sketching the Nyquist diagram of the system, determine the maximum value of K consistent with stability, and check the result using Routh s criterion. Sketch the root-locus for the system （20%）Fig.3 Sketch root-locus diagram. （18% ） Determine the transfer function. Assume a minimum-phase transfer function.（10% ）Im Re Im Re Im Re Im Re Im Re Im Re 0.1 1 2 3 4 L(dB) 40 30 20 5 0 20 0 20 40 -60 C R R E )54(2SSSK精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 5 页，共 29 页1)(1)()(2122112221112SCRCRCRSCRCRSVSV)(1)()(8743215436324321GGGGGGGGGGGGGGGGSRSC There are 4 roots in the left-half S plane, 2 roots on the imaginary axes, 0 root in the RSP. The system is unstable. 208KK=20 ) 154.82)(181.34)(1481. 3)(1316.0() 11 .0(62.31)(SSSSSSGH精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 6 页，共 29 页AUTOMATIC CONTROL THEOREM (3) List the major advantages and disadvantages of open-loop control systems. (12% )Derive the transfer function and the differential equation of the electric network shown in Fig.1.（16% ）Consider the system shown in Fig.2. Obtain the closed-loop transfer function)()(SRSC, )()(SRSE, )()(SPSC. （12%） The characteristic equation is given 02023)(123SSSSGH. Discuss the distribution of the closed-loop poles. (16%) 5. Sketch the root-locus plot for the system )1()(SSKSGH. (The gain K is assumed to be positive.) Determine the breakaway point and K value. Determine the value of K at which root loci cross the imaginary axis. Discuss the stability. (14%) Fig.1 R2 C2 R1 C1 U1 U2 P C G1 G2 G3 G5 H1 R G4 H2 H3 Fig.2 E 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 7 页，共 29 页6. The system block diagram is shown Fig.3. 21SKG, )3(42SSG. Suppose )2(tr, 1n. Determine the value of K to ensure 1SSe. (15%) 7. Consider the system with the following open-loop transfer function: )1)(1()(21STSTSKSGH. Draw Nyquist diagrams. Determine the stability of the system for two cases, the gain K is small, K is large. (15%) N R E C G1 G2 Fig.3 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 8 页，共 29 页Solution: The advantages of open-loop control systems are as follows: Simple construction and ease of maintenance Less expensive than a corresponding closed-loop system There is no stability problem Convenient when output is hard to measure or economically not feasible. (For example, it would be quite expensive to provide a device to measure the quality of the output of a toaster.) The disadvantages of open-loop control systems are as follows: Disturbances and changes in calibration cause errors, and the output may be different from what is desired. To maintain the required quality in the output, recalibration is necessary from time to time. 1)(1)()()(2122112221122112221112SCRCRCRSCRCRSCRCRSCRCRSUSU351343212321215143211)()(HGGHGGGGHGGHGGGGGGGGSRSC35134321232121253121431)1 ()()(HGGHGGGGHGGHGGHGGHGGGGSPSCR=2, L=1 S:the breakaway point is 1 and 1/3; k=4/27 The imaginary axis S=j; K=2 5.75. 3K精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 9 页，共 29 页AUTOMATIC CONTROL THEOREM (4) Find the poles of the following )(sF: sesF11)(12%) Consider the system shown in Fig.1,where6.0and 5nrad/sec. Obtain the rise timert, peak timept, maximum overshootPM, and settling timestwhen the system is subjected to a unit-step input. （10%）Consider the system shown in Fig.2. Obtain the closed-loop transfer function)()(SRSC, )()(SRSE, )()(SPSC. （12%） The characteristic equation is given 02023)(123SSSSGH. Discuss the distribution of the closed-loop poles. (16%) C(s) Fig.1 )2(2nnssR(s) P C G1 G2 G3 G5 H1 R G4 H2 H3 Fig.2 E 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 10 页，共 29 页5. Sketch the root-locus plot for the system )1()(SSKSGH. (The gain K is assumed to be positive.) Determine the breakaway point and K value. Determine the value of K at which root loci cross the imaginary axis. Discuss the stability. (12%) 6. The system block diagram is shown Fig.3. 21SKG, )3(42SSG. Suppose )2(tr, 1n. Determine the value of K to ensure 1SSe. (12%) 7. Consider the system with the following open-loop transfer function: )1)(1()(21STSTSKSGH. Draw Nyquist diagrams. Determine the stability of the system for two cases, the gain K is small, K is large. (12%) 8. Sketch the Bode diagram of the system shown in Fig.4. (14%) 32SS)10)(5()2(SSSR(S) C(S) Fig.4 N R E C G1 G2 Fig.3 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 11 页，共 29 页 Solution: The poles are found from 1seor 1)sin(cos)(jeejFrom this it follows that n2,0),2, 1 ,0(n. Thus, the poles are located at njs2Solution: rise timesec55.0rt, peak timesec785.0pt, maximum overshoot095.0PM, and settling timesec33.1stfor the%2criterion, settling timesec1stfor the %5criterion. 351343212321215143211)()(HGGHGGGGHGGHGGGGGGGGSRSC35134321232121253121431)1 ()()(HGGHGGGGHGGHGGHGGHGGGGSPSCR=2, L=1 5. S:the breakaway point is 1 and 1/3; k=4/27 The imaginary axis S=j; K=2 5.75. 3K精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 12 页，共 29 页AUTOMATIC CONTROL THEOREM (5) Consider the system shown in Fig.1. Obtain the closed-loop transfer function)()(SRSC, )()(SRSE. （18%） The characteristic equation is given 0483224123)(12345SSSSSSGH. Discuss the distribution of the closed-loop poles. (16%) Sketch the root-locus plot for the system )15.0)(1()(SSSKSGH. (The gain K is assumed to be positive.) Determine the breakaway point and K value. Determine the value of K at which root loci cross the imaginary axis. Discuss the stability. (18%) The system block diagram is shown Fig.2. 1111STKG, 1222STKG. Suppose 0r, 1n. Determine the value of SSe. Suppose 1r, 1n. Determine the value of SSe. (14%) C E R G1 G2 G3 H2 H1 H4 H3 Fig.1 N R E C G1 G2 Fig.2 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 13 页，共 29 页 Sketch the Bode diagram for the following transfer function. )1()(TssKsGH, 7K, 087.0T. (10%) A system with the open-loop transfer function )1()(2TSsKSGHis inherently unstable. This system can be stabilized by adding derivative control. Sketch the polar plots for the open-loop transfer function with and without derivative control. (14%) Draw the block diagram and determine the transfer function. (10%) R C U1(s) U2(s) 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 14 页，共 29 页321)()(GGGSRSCR=0, L=3,I=2 2121KKKess21211KKKess11)()(12RCssUsU精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 15 页，共 29 页AUTOMATIC CONTROL THEOREM (6) Consider the system shown in Fig.1. Obtain the closed-loop transfer function)()(SRSC, )()(SRSE. （18%）The characteristic equation is given 012012212010525)(12345SSSSSSGH. Discuss the distribution of the closed-loop poles. (12%) Sketch the root-locus plot for the system )3() 1()(SSSKSGH. (The gain K is assumed to be positive.) Determine the breakaway point and K value. Determine the value of K at which root loci cross the imaginary axis. Discuss the stability. (15%) The system block diagram is shown Fig.2. SG11, )125.0(102SSG. Suppose tr1, 1 .0n. Determine the value of SSe. (12%) Fig.1 C E R G1 G2 H2 H1 H3 N C R E G1 G2 0.5 Fig.2 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 16 页，共 29 页 Calculate the transfer function for the following Bode diagram of the minimum phase. (15%) For the system show as follows, )5(4)(sssG,1)(sH, (16%) Determine the system output )(tcto a unit step, ramp input. Determine the coefficient PK, VKand the steady state error to ttr2)(. Plot the Bode diagram of the system described by the open-loop transfer function elements )5 .01()1(10)(ssssG, 1)(sH. (12%) dB 0.1 1 4 8 16 -40 -20 0dB/dec 20 0 w 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 17 页，共 29 页32221212321221122211)1()()(HHGHHGGHHGGHGHGHGGGSRSCR=0, L=5 )1611()14)(1)(110(05.0)(2ssssssGtteetc431341)(tteettc41213445)(PK, 8 .0VK, 5.2sse精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 18 页，共 29 页AUTOMATIC CONTROL THEOREM (7) Consider the system shown in Fig.1. Obtain the closed-loop transfer function)()(SRSC, )()(SRSE. （16%） The characteristic equation is given 01087444)(123456SSSSSSSGH. Discuss the distribution of the closed-loop poles. (10%) Sketch the root-locus plot for the system 3) 1()(SSKSGH. (The gain K is assumed to be positive.) Determine the breakaway point and K value. Determine the value of K at which root loci cross the imaginary axis. Discuss the stability. (15%) Show that the steady-state error in the response to ramp inputs can be made zero, if the closed-loop transfer function is given by: nnnnnnasasasasasRsC1111)()(;1)(sH（12%） Calculate the transfer function for the following Bode diagram of the minimum phase. （15%）R E C G1 G2 G3 Fig.1 -20dB/dec -40 -40 w1 w2 w3 dB w 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 19 页，共 29 页 Sketch the Nyquist diagram (Polar plot) for the system described by the open-loop transfer function )12 .0(11.0)(sssSGH, and find the frequency and phase such that magnitude is unity. （16%） The stability of a closed-loop system with the following open-loop transfer function )1()1()(122sTssTKSGHdepends on the relative magnitudes of 1Tand 2T. Draw Nyquist diagram and determine the stability of the system. （16%）( 00021TTK) 3213221132112)()(GGGGGGGGGGGGSRSCR=2, I=2,L=2 )1() 1()(32122ssssGosrad5 .95/986.0精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 20 页，共 29 页AUTOMATIC CONTROL THEOREM (8) Consider the system shown in Fig.1. Obtain the closed-loop transfer function )()(SRSC, )()(SRSE. （16%） The characteristic equation is given 04)2(3)(123SKKSSSGH. Discuss the condition of stability. (12%) Draw the root-locus plot for the system 22)4()1()(SSKSGH;1)(sH. Observe that values of K the system is overdamped and values of K it is underdamped. (16%) The system transfer function is)1)(21()5.01()(ssssKsG,1)(sH. Determine the steady-state error SSewhen input is unit impulse)(t、unit step)( 1 t、unit ramp tand unit parabolic function 221t . (16%) Calculate the transfer function (minimum phase); Draw the phase-angle versus (12%) C R E G1 G2 G3 G4 Fig.1 -20dB/dec -40 -40 w1 w2 w3 dB w 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 21 页，共 29 页Draw the root locus for the system with open-loop transfer function. )3)(2()1 ()(ssssKsGH(14%) )1()(3TssKsGHDraw the polar plot and determine the stability of system. (14%) 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 22 页，共 29 页43214321432143211)()(GGGGGGGGGGGGGGGGSRSCK528. 0S:0K14 overdamped ;0.0718K14 underdamped S: )(t0sse; )(1 t0sse; tKess1; 221tsseS:21K; )1() 1()(32121ssssG精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 23 页，共 29 页AUTOMATIC CONTROL THEOREM (9) Consider the system shown in Fig.1. Obtain the closed-loop transfer function)()(SRSC, )()(SNSE. （12%） The characteristic equation is given 0750075005 .34)(123KSSSSGH. Discuss the condition of stability. (16%) Sketch the root-locus plot for the system )1(4)()(2ssasSGH. (The gain ais assumed to be positive.) Determine the breakaway point and avalue. Determine the value of aat which root loci cross the imaginary axis. Discuss the stability. (12%) Consider the system shown in Fig.2. 1)(1sKsGi, )1()(2TssKsG. Assume that the input is a ramp input, or attr)(where ais an arbitrary constant. Show that by properly adjusting the value of iK, the steady-state error SSein the response to ramp inputs can be made zero. (15%) E N C G1 G2 G3 G5 R G4 H3 Fig.1 H1 H2 C(s) E(s) R(s) G1(s) G2(s) Fig.2 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 24 页，共 29 页 Consider the closed-loop system having the following open-loop transfer function: ) 1()(TSSKSGH. Sketch the polar plot ( Nyquist diagram). Determine the stability of the closed-loop system. (12%) Sketch the root-locus plot. (18%) Obtain the closed-loop transfer function)()(SRSC. （15%）Im Re Im Re Im Re Im Re Im Re Im Re C G1 G2 G3 G4 H2 H1 R 精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 25 页，共 29 页354211335421243212321313542143211)1()()(HGGGGHGHGGGGHGGGGHGGHGHGGGGGGGGGSRSC354211335421243212321335422341)()(HGGGGHGHGGGGHGGGGHGGHGHGGGHHGSNSE45.30KS: N=1 P=1 Z=0; the closed-loop system is stable 2423241321121413211)()(HGHGGGGGGGHGGGGGGGSRSC精选学习资料 - - - - - - - - - 名师归纳总结 - - - - - - -第 26 页，共 29 页AUTOMATIC CONTROL THEOREM (10) Consider the system shown in Fig.1. Obtain the closed-loop transfer function )()(SRSC, )()(SNSC. （16%） The characteristic equation is given 01510520)(1234SSKSSSGH. Discuss the condition of stability. (14%) Consider a unity-feedback control system whose open-loop transfer function is )6.0(14 .0)(SSSSG. Obtain the response to a unit-step input. What is the rise time for this system? What is the maximum overshoot? （10%） Sketch the root-locus plot for the system )25.01 ()5.01()(sSsKSGH. (The gain K i