线性系统的根轨迹分析法 (24).pdf

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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 structure PID controller PID is a controller which applies proportional,integral and deri

2、vative control laws according to system deviation.It is characterized by mature technology,flexible application,and independence of accurate system model.TsG sKT s()11icpd)s(EobjectControlled )s(Y)s(UThe transfer function of PID is PID controller structure Proportional part Integral part Derivative

3、part pKipTsKpdK T s3 2 2 PI,PD and PID control PI control PD control PID control Transfer functions It has infinite gain at zero frequency,which improves the steady-state performance of the system.It has little effect on intermediate and high-frequency characteristics,and maintains the response spee

4、d and stability margin of the system.Using the phase lead compensator provided by PD controller,the phase margin of the system is increased.After compensation,the system magnitude crossover frequency increases and the system response speed is accelerated.Bode plot Compensation form Functions Lag com

5、pensation TsG sK()11icpG sKT s()1cpdTsG sKT s()11icpdIn the low-frequency region,PI control is mainly used to improve the system type and eliminate the steady-state error.In the high-frequency region,PD control is mainly used to increase the phase margin and improve the response speed.PID controller

6、 can improve the performance of the system.lgK20PPID controller Lead compensation Lag-lead compensation 4 3 3 Relationship between PID controller parameters and time domain performance index of the system Lag-lead compesation Parameter Rising Time Overshoot Adjustment Time Steady-state Error decreas

7、e increase minor change decrease decrease increase increase eliminate minor change decrease decrease KpTK1ipK TpdRelated to Kp PID controller parameter selection sequence.proportional factorintegral factorderivative factor.5 4 4 Example to illustrate the effect of PID controller e.g.Mass spring damp

8、ing system is shown in the figure.The external force F is taken as the input and the displacement x as the output.M=1kg,b=10Ns/m,k=20N/m,F=1(t).Design the P,PI,PD and PID controllers respectively to make the output response curve of the system meet the given performance requirements.（1）The steady-st

9、ate value of displacement is 1.（2）Faster rising time and transition process time.（3）Small overshoot.（4）Steady-state error is zero.xFMbkExample Lag-lead compesation 6 4 4 The transfer function of the system is as follows.F sMsbskX s()()12Solution AmplitudeTime(sec)Step Response0.050.0450.040.0350.030

10、.0250.020.0150.010.00502.521.510.50Step response curve of the system without any compensator ss102012Lag-lead compesation Example to illustrate the effect of PID controller Example 7 Compensator Control system block diagram Step response curve P PD PI PID)s(X)s(F)s(X)s(F)s(X)s(F)s(X)s(FAmplitudeTime

11、(sec)Step Response 1.510.501.41.210.80.60.40.20AmplitudeTime(sec)Step Response1.41.210.80.60.40.201.41.210.80.60.40.20AmplitudeTime(sec)Step Response1.41.210.80.60.40.200.30.250.20.150.10.050AmplitudeTime(sec)Step Response1.41.210.80.60.40.20121086420Example pK211020sspdsKK211020ssipsKK211020ssdipK

12、ssKK211020ss8 4 4 PID controller parameter tuning Critical proportioning method（1）First,only use the proportional control and select the smaller proportional gain Kp.（2）Kp is changed from being small to large until constant magnitude oscillation occurs.（3）Assume that the period of equal magnitude os

13、cillation is Tr and the proportional gain is Kpr,then PID parameters can be selected according to the table below.Lag-lead compesation rT9 4 4 controller Kp Ti Td P 0.5Kpr PI 0.45Kpr 0.83Tr PID 0.6Kpr 0.5Tr 0.12Tr Critical proportioning method Lag-lead compesation PID controller parameter tuning 10

14、e.g.Suppose the transfer function of the controlled object is as follows.The PID controller is used for series compensator,try to determine its parameters Kp,Ti,Td.s ssG s(1)(5)()10s ssKsK(1)(5)()pp Solution Only the proportional control function is put into operation,and the transfer function of th

15、e closed-loop system can be written.According to the Routh array,when Kpr is equal to 30,the closed-loop system has a pair of pure imaginary roots as s=1,21,25jThe corresponding equal magnitude oscillation period Tr can be calculated as=r rT52.812Example 11 controller Kp Ti Td Closed loop transfer f

16、unction P PI PID 15ssss6515()1532ssssss2.3313.9811.6531.4613.5()31.4613.5432sssssss1.418.4315.5525.2918()8.5325.2918432213.52.33181.4050.337PIDPIPAmplitudeTime(sec)Step Response 2.521.510.50400350300250200150100500Example 12 4 4 The limitations of the critical proportioning method is used to adjust

17、the controller parameters.The ideal controller parameters may not be obtained directly,fine tuning is also needed.The actual control object may not allow equal magnitude oscillation,or it cannot produce the same magnitude oscillation within the normal operation range.Lag-lead compesation Adjust PID controller parameter 13 Understand the basic structure of PID controller and the function of various parameters.Master the parameter tuning method of PID controller.