1. LINEAR CONTROL SYSTEMS Ali Karimpour Associate Professor Ferdowsi University of Mashhad

2. Lecture 16 Dr. Ali Karimpour May 2013 2 Lecture 16 Topics to be covered include: v Design of controller in time domain. Various controller configurations. Different kind of controllers. Controller realization. v Time domain design of the PID controllers. Design of PID controllers. Design of PD controllers. Design of PI controllers. Time domain design of control systems

3. Lecture 16 Dr. Ali Karimpour May 2013 3 Various controller configurations. ساختارهای کنترلی متفاوت + - r(t) e(t) u(t) c(t) 1 Series or cascade compensation. 1 ساختار کنترلی سری 2 State-feedback control. 2 کنترل فیدبک حالت + - r(t) u(t) x(t) )(sG p c(t) 3 Forward compensation with series compensation. (Two degree of freedom) 3 جبران سازی پیش رو با جبران سازی سری ( دو درجه آزادی ) + - r(t) e(t) u(t) c(t) )(sG p

4. Lecture 16 Dr. Ali Karimpour May 2013 44 Various controller configurations. ساختارهای کنترلی متفاوت 4 Feed forward compensation. (Two degree of freedom) 4 کنترلر پیش خور ( دو درجه آزادی ) + r(t) e(t) u(t) c(t) )(sG p + + - Controller Controlled process

5. Lecture 16 Dr. Ali Karimpour May 2013 5 Series Compensation Structure In this Course we consider the series or cascade compensation. جبران سازی سری Controller In This course Different kind of controllers PID controllers. Lead lag controllers H 2 Controllers H ∞ Controllers Intelligent Controllers ………….. Adaptive Controllers NN Controllers ………….. In Graduate courses

6. Lecture 16 Dr. Ali Karimpour May 2013 66 PID Controllers PID has become almost universally used in industrial control. These controllers have proven to be robust and extremely beneficial in the control of many important applications. PID stands for:P (Proportional) I (Integral) D (Derivative) The standard form PID are: An Alternative form for PID کنترلر PID Proportional only: Proportional plus Integral: Proportional plus derivative: Proportional, integral and derivative:

7. Lecture 16 Dr. Ali Karimpour May 2013 77 Lead-lag Compensators Closely related to PID control is the idea of lead-lag compensation. The transfer function of these compensators is of the form: If a p) in other form. If a>1 this is a lead network. Or (z<p) in other form. کنترلر پیش فاز پس فاز

8. Lecture 16 Dr. Ali Karimpour May 2013 PID and Operational Amplifiers کنترلر PID و تقویت کننده عملیاتی 88

9. Lecture 16 Dr. Ali Karimpour May 2013 99 PID and Operational Amplifiers کنترلر PID و تقویت کننده عملیاتی

10. Lecture 16 Dr. Ali Karimpour May 2013 10 PID and Operational Amplifiers کنترلر PID و تقویت کننده عملیاتی

11. Lecture 16 Dr. Ali Karimpour May 2013 11 Effects of the PD control on the time response. تاثیر کنترلر PD بر پاسخ زمانی + + + - PD controller Derivative part can improve the oscillation. جمله مشتق می تواند رفتار گذرا را بهبود بخشد.

12. Lecture 16 Dr. Ali Karimpour May 2013 12 Effects of the PI control on the time response. تاثیر کنترلر PI بر پاسخ زمانی + + +- PI controller Loop transfer function without controller Loop transfer function with controller PI controller can improve error by increases the type of system by one

13. Lecture 16 Dr. Ali Karimpour May 2013 13 Tuning of PID Controllers Because of their widespread use in practice, we present below several methods for tuning PID controllers. Actually these methods are quite old and date back to the 1950’s. Nonetheless, they remain in widespread use today. In particular, we will study. u Ziegler-Nichols Oscillation Method u Ziegler-Nichols Reaction Curve Method u Cohen-Coon Reaction Curve Method u Time domain design u Frequency domain design تنظیم کنترلرهای PID

14. Lecture 16 Dr. Ali Karimpour May 2013 14 Ziegler-Nichols Design طراحی زیگلر نیکولز This procedure is only valid for open loop stable plants. u Open-Loop Tuning u Closed-Loop Tuning According to Ziegler and Nichols, the open-loop transfer function of a system can be approximated with time delay and single-order system, i.e. where T D is the system time delay and T 1 is the time constant.

15. Lecture 16 Dr. Ali Karimpour May 2013 15 Ziegler-Nichols Reaction Curve Method(Open-Loop Case) For open-loop tuning, we first find the plant parameters by applying a step input to the open-loop system. The plant parameters K, TD and T1 are then found from the result of the step test as shown in Figure. طراحی زیگلر نیکولز حالت حلقه باز

16. Lecture 16 Dr. Ali Karimpour May 2013 16 PI KpKp KiKi KdKd PID P طراحی زیگلر نیکولز حالت حلقه باز Ziegler-Nichols Reaction Curve Method(Open-Loop Case)

17. Lecture 16 Dr. Ali Karimpour May 2013 17 Numerical Example Consider step response of an open-loop system as: مثال عددی

18. Lecture 16 Dr. Ali Karimpour May 2013 18 PI KpKp KiKi KdKd PID P Numerical Example مثال عددی

19. Lecture 16 Dr. Ali Karimpour May 2013 19 Ziegler-Nichols Oscillation Method(Closed-loop) This procedure is only valid for open loop stable plants and it is carried out through the following steps u Set the true plant under proportional control, with a very small gain. u Increase the gain until the loop starts oscillating. Note that linear oscillation is required and that it should be detected at the controller output. u Record the controller critical gain K c and the oscillation period of the controller output, T. u Adjust the controller parameters according to Table طراحی زیگلر نیکولز بروش نوسانی ( حلقه بسته )

20. Lecture 16 Dr. Ali Karimpour May 2013 20 PI KpKp KiKi KdKd PIDP Ziegler-Nichols Oscillation Method(Closed-loop) طراحی زیگلر نیکولز بروش نوسانی ( حلقه بسته )

21. Lecture 16 Dr. Ali Karimpour May 2013 Consider a plant with a model given by Find the parameters of a PID controller using the Z-N oscillation method. Obtain a graph of the response to a unit step input reference. 21 Numerical Example مثال عددی

22. Lecture 16 Dr. Ali Karimpour May 2013 22 Solution Applying the procedure we find: K c = 8 and ω c =  3. T=3.62 Hence, from Table, we have The closed loop response to a unit step in the reference at t = 0 is shown in the next figure. حل

23. Lecture 16 Dr. Ali Karimpour May 2013 23 Response to step reference پاسخ سیستم به پله

24. Lecture 16 Dr. Ali Karimpour May 2013 24 Time domain design طراحی حوزه زمانی + - در سیستم زیر آیا می توان k را بگونه ای تنظیم کرد که نسبت میرائی قطبهای مختلط سیستم 0.707 گردد؟ Is it possible to set the value of k such that the damping ratio of complex poles be 0.707? -25 0 Yes بله

25. Lecture 16 Dr. Ali Karimpour May 2013 25 Time domain design طراحی حوزه زمانی + - در سیستم زیر آیا می توان k را بگونه ای تنظیم کرد که ثابت خطای شیب معادل 100 گردد؟ Is it possible to set the value of k such that ramp error constant be 100? Yes بله

26. Lecture 16 Dr. Ali Karimpour May 2013 26 Time domain design طراحی حوزه زمانی + - در سیستم زیر آیا می توان k را بگونه ای تنظیم کرد که نسبت میرائی قطبهای مختلط سیستم 0.707 و ثابت خطای شیب معادل 100 گردد ؟ Is it possible to set the value of k such that the damping ratio of complex poles be 0.707 and ramp error constant be 100 ? Clearly the design is not possible ???!!!??? Other controllers

27. Lecture 16 Dr. Ali Karimpour May 2013 27 در سیستم ضرایب کنترلر را بگونه ای تنظیم کنید که نسبت میرائی قطبهای مختلط سیستم 0.707 و ثابت خطای شیب معادل 100 گردد ؟ Determine the controller coefficient such that the damping ratio of complex poles be 0.707 and ramp error constant be 100 ? + - Tuning PD controller طراحی کنترلر PD

28. Lecture 16 Dr. Ali Karimpour May 2013 28 + - Tuning k D by graphical method. Tuning PD controller طراحی کنترلر PD -12.5 48.4 -50-35

29. Lecture 16 Dr. Ali Karimpour May 2013 29 + - Tuning k D by mathematical method Tuning PD controller طراحی کنترلر PD

30. Lecture 16 Dr. Ali Karimpour May 2013 30 + - Tuning PD controller طراحی کنترلر PD Why P.O. > 4.3%

31. Lecture 16 Dr. Ali Karimpour May 2013 31 Tuning PI controller طراحی کنترلر PI + + + - + - Clearly type of system is 2 so: در سیستم ضرایب کنترلر را بگونه ای تنظیم کنید که نسبت میرائی قطبهای مختلط سیستم 0.707 و ثابت خطای شیب معادل 100 گردد ؟ Determine the controller coefficient such that the damping ratio of complex poles be 0.707 and ramp error constant be 100 ?

32. Lecture 16 Dr. Ali Karimpour May 2013 32 Tuning PI controller (Continue) طراحی کنترلر PI ( ادامه ) + - حال نیاز داریم که نسبت میرائی قطبهای مختلط سیستم 0.707 گردد. We now need damping ratio of complex poles be 0.707. Root loci with proportional controller -25 0

33. Lecture 16 Dr. Ali Karimpour May 2013 33 Tuning PI controller (Continue) طراحی کنترلر PI ( ادامه ) حال نیاز داریم که k P را تعیین کنیم. We now need to set k P. Root loci with PI controller -25 0 -10.9-4.6 -11.5

34. Lecture 16 Dr. Ali Karimpour May 2013 34 + - Tuning PI controller طراحی کنترلر PI Why P.O. > 4.3%

35. Lecture 16 Dr. Ali Karimpour May 2013 35 Compare PI and PD controllers مقایسه کنترلرهای PI و PD + - + - With PI controller With PD controller

36. Lecture 16 Dr. Ali Karimpour May 2013 36 Exercises تمرینها 1- In the following system design a PID controller with Ziegler-Nichols Oscillation Method + - 2- In the following system design a PID controller with Ziegler-Nichols Oscillation Method + -

37. Lecture 16 Dr. Ali Karimpour May 2013 37 Exercises تمرینها 3 In the following system design a PD controller such that that the damping ratio of complex poles be 0.6 and ramp error constant be 80. + - + - 4 In the following system design a PD controller such that that the damping ratio of complex poles be 0.6 and ramp error constant be 80.

38. Lecture 16 Dr. Ali Karimpour May 2013 38 Exercises تمرینها 5 In the following system design a PI controller such that that the damping ratio of complex poles be 0.6 and ramp error constant be 80. + - + - 6 In the following system design a PI controller such that that the damping ratio of complex poles be 0.6 and ramp error constant be 80.

39. Lecture 16 Dr. Ali Karimpour May 2013 39 Exercises تمرینها Let the input impedance be generated by a resistor R2 be in series with a resistor R1 and a capacitor C1) that are in parallel, and let the feedback impedance be generated by a resistor Rf and a capacitor C f. a) Show that this choices lead to form a PID controller with high frequency gain limit as; 7 Consider following structure: b) Derive the parameters in the controller.