自动控制原理(中英文对照李道根)习题2题解.pdf

1、Solutions1SolutionsP2.1 The following differential equations represent linear time-invariant systems,where)(trdenotes the input and)(tcdenotes the output.Find the transfer function of each of the systems.(a)()(6d)(d11d)(d6d)(d2233trtcttcttcttc(b)(d)(d2)(d)(d4d)(d3d)(d2233trttrtcttcttcttcSolution:(a)

2、Taking Laplace transform,assuming zero initial conditions,we have)()(6)(11)(6)(23sRsCssCsCssCsHence,the transfer function is61161)()(23ssssRsC(b)In a same way,we have)()(2)(6)(4)(3)(23sRssRsCssCsCssCs14312)()(23sssssRsCP2.2A mass-spring-damper system is shown in Fig.P2.2.Determine the differential e

3、quation between the input force fand the output displacement x.Solution:Using Newtons second law yields22d)(dd)(d)()(ttxMttxBtKxtfRearranging in a normalized form gets)(1)(d)(dd)(d22tfMtxMKttxMBttxP2.3 Determine the transfer function)()(2sFsXfor the system shown in Fig.P2.3.Both masses slide on a fr

4、ictionless surface,and MNK/1.Solution:Using Newtons second law for both masses yieldsfxxKtxxM)(d)(d2122121222221dd)(txMxxKTaking Laplace transform yields)()()()()(212121sFsXsXKsXsXsMSolutions2)()()(22221sXsMsXsXKEliminating the intermediate variables results in)2(1)()(22sssFsXP2.4A thermistor has a

5、response to temperature represented byTeRR1.00where Ris resistance,Tis temperature in degrees Celsius,and 100000R.Find the linearized model for the thermistor operating at CT20and for a small range of variation of temperature.Solution:Expanding the expression of thermistor into Taylors series in the

6、 neighborhood of specified temperature,we haveTTtRTRtRTT0d)(d)()(0,CT200where 3.13510001.0d)(d21.0000eeRTtRTTTTT.Hence,the linearized model for the thermistor,at CT20,isT.R3135P2.5 Obtain the transfer functions,)()(sVsVio,for the passive networks shown in Fig.P2.5.Solution:Using Kirchhoffs laws,we h

7、ave(a)1)(111)()(212212CsRRCsRCsRRCsRsVsVio(b)212112122112111)()(RRCsRRCsRRRRRCsRCsRRsVsVio(c)()()1()()()()()2()()(1)(1)(1)(1)(2112112ssVRCsVsRCsVsVsVsVsRCssVCsVsCRsVRsVRsVRsVsCRioAioAioAioAovCiv1R2R(a)ovCiv1R2R(b)ov1CivRR(c)2CFigure P2.5.Solutions31)2(12)()(21221212212sCCRsCCRsRCsCCRsVsVioP2.6 Obtai

8、n the transfer functions,)()(sVsVio,for the active networks shown in Fig.P2.6.Solution:Using Kirchhoffs laws,we have(a)00100001)1()1()1()()(RCsRRsCRsCRRsVsVio(b)sCRsCRRsCRsCRsCRsVsVio10001000011)1()1()1()1()()(c)11)()()(1)()(1)(2)(1)(211221221A222111sCRsCRRRsVsVsVRRsVsVRsVsCRsVRsVsCRioBoBiA(d)303132

9、21232110)()()(1)(111)(1)(1ZZZZZZZZsVsVsVZsVZZZsVZsVZiooAAiKK0R1R0CovovivivKovivK0R1R0Coviv1C(a)(b)(c)(d)0C0R0R0R1R1Z1R2Z3Z1CFigure P2.6.Solutions4P2.7 Draw a block diagram showing all the variables for the system described by the following set of differential equations.)()()()()()()()()()()()()()()(

10、)(5022453452311211txKtctctnKtxtxtxxTtxtxtxtxKtxtntctrtx where)(tris the input,)(tcis the output,)(1tnand)(2tnare two disturbances,0K,1K,2K,and Tare constants.Solution:Taking Laplace transform with zero initial conditions and writing the result in a cause-and-effect form yields)()1()()()()()(1)()()()

11、()()()()()()()()()()()()()()()()()()()()()()()(502245345231121150222453452311211sxssKsCsNKsXsXsXTssXsXsXsXsXKsXsNsCsRsXsxKssCsCssNKsXsXsXsTsXsXsXsXsXKsXsNsCsRsXRearranging the variables)(sR,)(1sX,)(2sX,)(3sX,)(4sX,)(5sXand)(sCfrom left to right in order,we have the block diagram as shown.P2.8 A syst

12、em is described by the following set of equations:)()()()()()()()()()()()()()()()()()()()(3452333612287111sXsGsCsCsGsXsGsXsXsGsXsGsXsCsGsGsGsRsGsXDraw the block diagram showing all the variables for this system and find the transfer function for)()(sRsC.Solution:Taking Laplace transform with zero in

13、itial conditions and writing the result in a cause-and-effect form,it is not difficult to get the block diagram as shown.R1GTs1)1(0ssK2KC2N1N1X2X3X4X5XR1GC1X2X3X2G6G3G4G5G7G8GSolutions51G2G3G4G)(2sR)(1sC)(1sR)(2sCFigure P 2.9.P2.9 A system with two inputs and two outputs is shown in Fig.P2.9.Determi

14、ne the transfer functions)()(11sRsC,)()(21sRsC,)()(12sRsC,and)()(22sRsC.Write the expressions of)(1sCand)(2sC.Solution:The transfer function is a description of one input to one output.Letting 0)(2sR,we get)()()()(1)()()(4321111sGsGsGsGsGsRsC)()()()(1)()()()()(432132112sGsGsGsGsGsGsGsRsCLetting 0)(1

15、sR,we get)()()()(1)()()()()(432114321sGsGsGsGsGsGsGsRsC)()()()(1)()()(4321322sGsGsGsGsGsRsCThe expressions of)(1sCand)(2sCare given by)()()()()(1)()()()()()()()(1)()(243214311432111sRsGsGsGsGsGsGsGsRsGsGsGsGsGsC)()()()(1)()()()()()(4321243111sGsGsGsGsRsGsGsGsRsG)()()()(1)()()()()()()(43212313212sGsG

16、sGsGsRsGsRsGsGsGsCrespectively.P2.10 By block diagram deduction find the transfer functions of following systems.RGHCR1G2GHR1G2G3G3H2H1HC(d)(b)(a)CR1G2GHC(c)FigureP2.10.Solutions6Solution:(a)HGHHGHGHGsRsC1)1(11)()(b)HGGGsRsC22111)()()(HGGG2211(c)HGGGsRsC12111)()()(HGGG1211(d)223311321312333211133321

17、11)1)(1(11111)()(HGHGHGGGGGGHHGGGHGGHGGGHGGsRsC31313322113211HHGGHGHGHGGGGP2.11 For the following systems:Derive the transfer functions for)()(sRsCand)()(sNsCby block diagram deduction.)(sR1G2G3GH)(sC)(sN1G2G3G2H1H(b)(sN)(sC)(sR(a)FigureP2.11.RGHCRGHH1CR1G2GHCR1G2GHG2CR1G2GHCR1G2GHG1CR1G2G3G3H2H1HCR

18、1G2G3G3H2H1HC31 G11 GSolutions7Solution:(a)HGGGGsRsC11)()(2121HGGGGsRsC111)()(2132(b)221212112212211111)()(HGHGGGGHHGGGHGGGsRsC22121312212122311)1(111)1()()(HGHGGGGGHGHGGHGGGGsRsC)(sR1G2G3GH)(sC)(sN)(sR1G2GH1)(sC)(sR1G2G3GH)(sC)(sN1G2G3GH1)(sC)(sN32GG)1(21HGG)(sN)(sC1G2G3G2H1H)(sN)(sC)(sR1G2G2H1H)(s

19、C)(sR1G2G3G2H1H)(sN)(sC)(sR3G2G2H1H)(sC)(sN1G13GG2221HGG11HG)(sC)(sNSolutions8P2.12By block diagram deduction find the transfer functions of the following systems.Solution:(a)22112312212211221131)(111)1()()(HGGHGGGGHHGGGHGGGGGsRsC(b)1432123213324321142433232143323211111)()(HGGGGHGGGHGGGGGGhGHGHGGGGG

20、GHGGGGGsRsC)(sR1G2G3G1H)(sC(b)(a)2H2H1H)(sR1G2G3H)(sC3G4G2H1H)(sR1G2G3H)(sC(c)(d)2H1H)(sR1G2G3H)(sC3G4GFigure P2.12.)(sR1G2G3G1H)(sC2H)(sR1G2G13GG1H)(sC2H2H1H)(sR1G2G3H)(sC3G4G42GH1H)(sR1G32GG3H)(sC4GSolutions9)(sRK)(sH)(sC)(sE)5(2ssFigureP2.13.(c)2111211211111)1(11)(HHHGHHGHHHGGsGe32211121122111211

21、3221)1(11)1(1)()(HGHHHGHHGGHHHGHHGHGGGGsRsCee32121321211121211)1(HHHGGHGGHHHGHHGG(d)41241134343213434321111)()(GGHGGHHGGGGGGHGGGGGGsRsC1432123234343211HGGGGHGGHGGGGGGP2.13 For the system shown in the figure,determine K,and)(sHif it is required that the feedforward transfer function of the system)20)

22、(5()10(100)()(sssssEsCSolution:There is a relation for the system,mathematically,)()()()()()(sEsRsRsCsEsCCalculatingKsHssKssKsssHssKsRsC2)(2)5(2)5(2)5()(21)5(2)()(2H1H)(sR1G2G3H)(sC2111HHH)(sR1G2G3H)(sCeG2H1H)(sR1G2G3H)(sC3G4G2H1H)(sR11 G21GG3H)(sC43GG41 G4121GGHH)(sR21GG)(sC343431HGGGGSolutions10Ks

23、HsssHssssKsssHsssHsRsE2)(2)5()(2)5()5(2)5()(21)5()(21)()(we get)20)(5()10(100)(2)5(2)()(sssssHssKsEsCHence,the final result is50K,10)5(5)(ssssHP2.14 Are the two systems shown in Fig.?P2.14(a)and(b)equivalent?Explain.Solution:The two systems are not equivalent,because the three loops in(a)are nontouc

24、hing,however,the loops adand be,beand cfin(b)are touching.In fact,using Masons formula,we haveadbecfbecfadcfadbecfbeadabcsXsY1)()(11adcfcfbeadabcsXsY1)()(22P2.15 Find the overall transfer functions for the following systems.Solution:Using masons formula,we have(a)cgbchcgbchfabcdesRsC1)1()()(1x1yabcd

25、e11fabcfde112x2yFigureP2.14.)(sR)(sCabcde11(a)fgh)(sR)(sCabcde1(b)fgh)(sR)(sCabcde11(c)fgh)(sR)(sCabcde1(d)fgh1Figure P2.15.Solutions11(b)afchefghchbgafbgedabcdsRsC1)1()()(c)afchchbgafchbgeabcdsRsC1)1()()(d)aedfchfehgbgaedabcdfaeabcsRsC1)1()()(P2.16 Find the transfer function for the multi-loop cros

26、sing system shown in Fig.P.2.16.Solution:There are five loops and four forward paths.Hence we have212121211GGGGGGGG11Gp,1122Gp,12213GGp,13213GGp,1321212121312)()(GGGGGGGGsRsCP2.17 Draw equivalent signal flow graphs for the block diagrams in Fig.P.2.12.Find the transfer functions by use of Masons for

27、mula.Solution:(a)2211222211)()(HGGHGGGGGsRsC(b)2211222211)()(HGGHGGGGGsRsC)(sR1G2G)(sCFigure P2.16.)(sR1G2G3G1H)(sC2H1RC3G1G2G112H1H1R3G1G2G12H1H2H1H)(sR1G2G3H)(sC3G4GC14G3HSolutions12(c)32121321211121211)1()()(HHHGGHGGHHHGHHGGsRsC(d)1432134323243211)()(HGGGGHGGHGGGGGGsRsC11R1G2G12H1HC13H2H1H)(sR1G2G3H)(sC13G4G1R1G2G2H1HC3H12H1H)(sR1G2G3H)(sC3G4G