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1、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 1 Basic principle Lag-lead compensation The lag-lead compensation uses its lead part to increase
2、the phase margin of the system,and use its lag part to improve the steady-state performance of the system.This compensation method takes into account the advantages of phase lag compensation and phase lead compensation.It can not only accelerate the response speed and reduce the overshoot,but also s
3、uppress the high-frequency noise and improve the steady-state accuracy.3 2 2 Lag-lead compensator and its transfer function 1C)s(oU2R)s(iU2C1RTTssTTG skT sT s1)1,(1)(),(1)(1)1221c12The passive lag-lead compensation network.is the lag compensation part.T sT s1122T sT s1111is the lead compensation par
4、t.Lag-lead compensation 4 3 3 Distribution of zeros and poles of the lag-lead compensator T12T12Re Im 0 T sT sG skTTT sT s(1)(1)(),1,(1)(1)21c1212T11T1Lead compensation part.Lag compensation part.Lag-lead compensation 5 4 4 Bode plot of the lag-lead compensator T sTsG skTsT s(1)(1)(),1(1)(1)21c12Bod
5、e Diagram0-5-10-15-20)s/dar(/Bd/LT0.011T0.11T11T10190450-45-90-20T0.011)s/dar(/ged/T0.11T11T101TTk1010121Lag-lead compensation Assuming 6 5 5 Lag-lead compensation based on Bode plot e.g.The open-loop transfer function of the known unit feedback system is .Try to design a lag-lead compensator to mee
6、t the following requirements.s ssG sK(1)(2)()According to the requirement of static speed error coefficient,the following formula can be listed as s ssKsG ssKKssv(1)(2)2lim()lim1000K20Solution(1)the phase margin is 50and the gain margin is greater than or equal to10dB.(2)Static speed error coefficie
7、nt is 10/s.Lag-lead compensation Example 7 Draw the Bode plot of the system before compensation when K=20.10.00164216.02.01.040.020.0)s/dar(/)()s/dar(/Bd/)(L0164216.02.01.040.020.010.0072-081-09009-04-02-0020406g g=-32 Lg=-13dB The system is unstable before compensation.Example 8 10.00164216.02.01.0
8、40.020.0)s/dar(/)()s/dar(/Bd/)(L0164216.02.01.040.020.010.0072-081-09009-04-02-0020406 Select the magnitude crossover frequency of the compensated system.=1.5 We choose the compensated magnitude crossover frequency as 1.5rad/s,according to the requirement,the phase lead required here is 50.Example 9
9、 Determine the break frequency of the lag part in the lag-lead compensator.The parameter is determined by the maximum phase lead.T100.15rad/s11.52The larger break frequency is chosen as one tenth of the system magnitude crossover frequency after compensation,the parameter T2 can be obtained as 1sin1
10、mssss0.0151166.710.1516.6711When the parameter is equal to 10,m=54.9,according to the requirement,the phase lead required here is 50,so choose=10.Thus,another break frequency of the phase lag part is .T0.015rad/s12The transfer function of the lag part of the lag-lead compensator is .Example 10 Deter
11、mine the transfer function of the lead part of the lag-lead compensator.0164216.02.01.040.020.010.0)s/dar(/)()s/dar(/Bd/)(L络网正校0164216.02.01.040.020.010.0072-081-09009-04-02-0020406 =1.5 13dB-13dB（1.5rad/s,-13dB）0.7 7 20dB/dec-20dB line 0dB line ssss711 0.14310.71 1.4311The transfer function of the
12、phase lead part in the lag-lead compensator is as follows.Example 11 sssssssG sG s G sc(1 0.143)(1 66.7)(1)(1 0.5)10(1 1.43)(1 6.67)()()()k Draw the Bode plot of the compensated system to verify whether the phase margin and gain margin meet the design requirements.The open-loop transfer function of
13、the compensated system is as follows.ssG sss1 0.1431 66.7()1 1.431 6.67cThe transfer function of the lag-lead compensator is as follows.after compensation.before compensationcompensatorlead-lag the dec/dB40-dec/dB20-dec/dB20-dec/dB40-dec/dB60-dec/dB40-dec/dB20-)s/rad/()/()s/rad/(dB)/(L1001.05.16216.
14、02.01.004.002.05.1compensatorlead-lag the.after compensation.before compensation1064216.02.01.004.002.001.0270-180-90090-40-20-0204060-13dB-32 16dB 50 The designed lag-lead compensator can meet the performance requirements.Example 12 6 6 Comparison of three ccompensators Phase lag compensation.Phase
15、 Lead Compensation Phase Lag Compensation Lag-lead Compensation Transfer function Principle Impact on the system The phase lead compensation is usually used to improve the system phase margin,so as to improve the transient performance.Phase lag compensation suppresses high-frequency interference and
16、 noise.The magnitude crossover frequency and the bandwidth of the closed-loop system decrease,and the response speed of the system slows down.Phase lag compensation is usually used to improve the steady-state performance of the system and suppress the influence of noise.Application Providing a phase
17、 lead near the magnitude crossover frequency to enable the phase margin of the system to meet the requirements.The lead compensator increases the system magnitude crossover frequency and closed-loop system bandwidth,accelerates the response speed and decreases the overshoot.，TsG sTs1()11c，TsG sTs1()
18、11cT sT sG skT sT s(1)(1)(),1(1)(1)21c12The lag-lead compensation increases the low-frequency gain of the system;the system bandwidth and stability margin increase;the system response speed is accelerated;the steady-state accuracy is improved.It is usually applied to the case that the system is unst
19、able before compensation,and the system after compensation is required to have faster response speed,larger phase margin and higher steady-state accuracy.It attenuates the high-frequency magnitude of the original system,and makes the magnitude crossover frequency move to the low-frequency direction.
20、The phase characteristics of intermediate and high-frequency are basically unchanged,so that the system phase margin can meet the requirements.The lag-lead compensation makes use of its lead part to increase the phase margin of the system.At the same time,the lag part is used to improve the steady-state performance of the system,taking into account the advantages of phase lag compensation and phase lead compensation.13 Understand the basic concept of lag-lead compensation.Master its design principle and methods.