线性系统的根轨迹分析法 (22).pdf

1 Chapter 6 Compensation of Automatic Control Systems 6.1 Basic Concepts and Forms of Compensation 6.2 Phase Lead Compensation 6.3 Phase Lag Compensation 6.4 Lag-lead Compensation 6.5 PID Controller 2 1-ImReMaintain the magnitude-frequency characteristics of the low-frequency band of the system and attenuate the magnitude of the high-frequency band in the system.1 1 Why do we need phase lag compensation Phase lag compensation Low-frequency band:High open-loop gain,steady-state error meets the requirements.The phase lag compensation attenuates the system magnitude in the intermediate and high-frequency band,making the magnitude crossover frequency of the system decrease.Sufficient phase margin can be obtained by using the phase angle of the original system at the new magnitude crossover frequency.Meanwhile it improves the system capability of resisting the high-frequency noise interference.Increase open-loop gain K K1 1 K K2 2 Low-frequency band:Small open-loop gain,steady-state error is to big.Intermediate-frequency band:The phase margin is small,and the transient performance get worse.wc2 ,2 2 wc1 ,1 1 w0+w+wc1 wc2 1 1 2 2 If the original system has a fast phase attenuation at the magnitude crossover frequency,the lead compensation cannot be used.3 Two important parameters and T T 2 2 Phase lag compensator and its transfer function Phase lag compensation)s(oU2R)s(iUC1R+U sTsTsUsio()11()A passive lag compensation network.()+TRR C12+RRR12201The transfer function of phase lag compensator +TsG sTs1(),011c+4 3 3 Zeros and poles of phase lag compensator Phase lag compensation 1TT1Re Im 0 In the complex plane,the pole of phase lag compensator is always on the right side of the zero point.+TsG sTs1(),011c5 w ww w w w 0954054-09-w w L04 4 Bode plot of the lag compensator Phase lag compensation w wwTT()arctan()arctan()+w wwLTT()20lg 1()20lg 1()22Logarithmic frequency characteristic+TsG sTs1(),011c Magnitude-frequency characteristic Phase-frequency characteristic Phase angle is negative.-20dB/dec 20lg 1T1T1T1Tm1wTm10lg20lg6 101001000101010010001 /)(Bd/)(L09-081-0402002-04-5 5 Lag compensation based on Bode plot Phase lag compensation)s(Y)s(U)s(H)s(G)s(cG+TsG sTs1(),011c 1 1 wc1 Phase angle is negative L(wc2)wc2 20lg T1T1Before compensation 7 101001000101010010001 /)(Bd/)(L09-081-0402002-04-5 5 Lag compensation based on Bode plot Phase lag compensation 1 1 2 2 Find the open-loop gain that meets the steady-state performance index.Determine the magnitude crossover frequency before compensation wc1 and phase margin 1 Choose a new magnitude crossover frequency wc2 Expected phase margin 0 0()wc180+2Calculate the parameters of the phase lag compensator +wLc20lg()02Select the zero point of the lag compensator and calculate the parameter T wc1 wTc10112The designed phase lag corrector is+TsG sTsc1()1Verify that the performance indicators of the system after compensation whether meet the requirements Phase margin before compensation Attenuate the magnitude of the intermediate and high-frequency bands 2 2 Find the frequency point that meets the phase margin requirement on the phase-frequency characteristic curve of the original system Phase angle is negative 0+=()512L(wc2)wc2 20lg wLc1020()2T1T1wTc1012Compensated phase margin Before compensation 8 Example Example The open-loop transfer function of the unit feedback system is .Try to design a phase lag compensator to meet:gG sKs s+()2500(25)Consider the systems requirements for steady-state error.The system is an type I system,ess=R/Kv,input R=1+s sKsG ssKKssvgg(25)lim()lim100250000+wwwwws sjjG jG ssjsj(25)(1 0.04)()()2500100eKssg11000.01If ,then ,make Kg1Kg1Solve(1)Phase margin is greater than 45;(2)For unit speed function input,the steady-state error of output is less than or equal to 0.01rad.9 dar/1010010001Bd/)(L /)(dar/101001000151051-03-54-06-57-09-501-021-531-051-561-081-0402002-04-Example Draw the Bode plot of the system before compensation when Kg=1,and determine the magnitude crossover frequencywc1and phase margin 1 w wc1=47=47 1 1=2828 L+www()20lg10020lg20lg 1(0.04)2Let L(w w0 0,ww+221(0.04)10000+ww625-6250000=042().wwcc90arctan0 0415211wc471().+wwcc18018090arctan0 0428111Then Thus The expression of the phase-frequency characteristic of the system before compensation is().ww90arctan0 04The expression of the magnitude-frequency characteristic of the system before compensation is So w wc1=？1 1=？10 Example Choose a new magnitude crossover frequency wc2 Calculate parameter wLc100.2742620()2 Select the zero of the compensator and calculate the parameter T wLc20lg()2+wc()180+(512)180+45513020 wwc()90arctan(0.04)1302 wtgc25 40212wc2=21 The amplitude of the original system is+wwwwLc()20lg100lglg 1(0.04)11.23682212 wTc102.112T1.7363 The designed phase lag compensator is+sTssGsTs11 1.7363762110.4c11 Example+ssss ssG sss(10.04)(1 1.7363)(25)(0.575946)()100(10.4762)685.65(2.1).(.)(.)(.+wwwccc47625n 045 83418090arctan 0 04arctan 1 7363arcta2222 Draw the Bode plot of the compensated system,and verify the magnitude crossover frequency of the system wc2 and phase margin 2 whether meet the performance index requirements The open-loop transfer function of the system after compensation is Original system Compensator The phase frequency characteristic of the system after compensation is(.)(.)(.)+wwww()=90arctan 0 04arctan 1 7363arctan 0 4762The phase margin of the system after compensation is 12 Example 12 1.01010010001da r/。/)(da r/Bd/)(L1.01010010001003-06-09-021-051-081-060402002-04-w wc14747 1 1 =28=28 w wc22121 2 2=45.83=45.83 13 From:U(1)1.51.41.31.21.110.90.80.70.60.50.40.30.20.100.50.450.40.350.30.250.20.150.10.050Step response of closed-loop system before and after compensation Output of the closed-loop system Time/s Example Before Lag compensation Lead compensation 14 6 6 Influence of the lag compensator on the system and its limitation Phase lag compensation Influence of the lag compensator on the system Limitation Relative stability of the system Overshoot Phase margin Gain margin Kg Magnitude crossover frequency wc System response speed Since the lag compensator has an attenuation effect on the magnitude of the intermediate and high-frequency bands,and has no attenuation for the magnitude of the low-frequency band,the steady-state accuracy of the system can be improved by increasing the system gain.When the original system cannot find a phase margin point that meets the requirements in the low-frequency range of the phase-frequency characteristics,phase lag compensation cannot be used.Closed-loop system bandwidth wb 15 Understand the basic concepts of phase lag compensation.Master the design ideas and methods of phase lag compensation.