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LINEAR CONTROL SYSTEMS

LINEAR CONTROL SYSTEMS. Ali Karimpour Associate Professor Ferdowsi University of Mashhad. Topics to be covered include : Design of controller in time domain. Various controller configurations. Different kind of controllers. Controller realization .

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LINEAR CONTROL SYSTEMS

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

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

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

  4. Various controller configurations. 4 Feed forward compensation.(Two degree of freedom) 4 کنترلر پیش خور (دو درجه آزادی) G ( s ) p + e(t) r(t) c(t) u(t) + + - Controlled process Controller Controller ساختارهای کنترلی متفاوت 4

  5. Controller Series Compensation Structure جبران سازی سری In this Course we consider the series or cascade compensation. PID controllers. In This course Lead lag controllers Different kind of controllers In Graduate courses H2 Controllers H∞ Controllers Adaptive Controllers ………….. NN Controllers ………….. Intelligent Controllers 5

  6. PID Controllers کنترلر PID 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 Proportional only: Proportional plus Integral: Proportional plus derivative: Proportional, integral and derivative: 6

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

  8. PID and Operational Amplifiers کنترلر PID و تقویت کننده عملیاتی 8

  9. PID and Operational Amplifiers کنترلر PID و تقویت کننده عملیاتی 9

  10. PID and Operational Amplifiers کنترلر PID و تقویت کننده عملیاتی 10

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

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

  13. 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. Ziegler-Nichols Oscillation Method Ziegler-Nichols Reaction Curve Method Cohen-Coon Reaction Curve Method Time domain design Frequency domain design Tuning of PID Controllers تنظیم کنترلرهای PID

  14. Ziegler-Nichols Design طراحی زیگلر نیکولز This procedure is only valid for open loop stable plants. • Open-Loop Tuning • 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 TD is the system time delay and T1 is the time constant.

  15. 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. Ziegler-Nichols Reaction Curve Method(Open-Loop Case) طراحی زیگلر نیکولز حالت حلقه باز

  16. Kd Ki Kp PID PI Ziegler-Nichols Reaction Curve Method(Open-Loop Case) طراحی زیگلر نیکولز حالت حلقه باز P

  17. Consider step response of an open-loop system as: Numerical Example مثال عددی

  18. Kd Ki Kp PID PI Numerical Example مثال عددی P

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

  20. Kd Ki Kp P PID PI Ziegler-Nichols Oscillation Method(Closed-loop) طراحی زیگلر نیکولز بروش نوسانی(حلقه بسته)

  21. 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. Numerical Example مثال عددی

  22. Applying the procedure we find: Kc = 8 and ωc = 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. Solution حل

  23. Response to step reference پاسخ سیستم به پله

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

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

  26. Is it possible to set the value of k such that the damping ratio of complex poles be 0.707andramp error constant be 100? 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 ؟ + - Time domain design طراحی حوزه زمانی Clearly the design is not possible ???!!!??? Other controllers

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

  28. + - 48.4 -12.5 Tuning PD controller طراحی کنترلر PD Tuning kD by graphical method. -50 -35

  29. + - Tuning PD controller طراحی کنترلر PD Tuning kD by mathematical method

  30. + - Tuning PD controller طراحی کنترلر PD Why P.O. > 4.3%

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

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

  33. طراحی کنترلر PI (ادامه) We now need to set kP. حال نیاز داریم که kPرا تعیین کنیم. -11.5 -10.9 -4.6 -25 0 Tuning PI controller (Continue) Root loci with PI controller

  34. + - Tuning PI controller طراحی کنترلر PI Why P.O. > 4.3%

  35. + + With PI controller - - With PD controller Compare PI and PD controllers مقایسه کنترلرهای PI و PD

  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. + - + - 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. + - + - 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. Exercises تمرینها 7 Consider following structure: 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; b) Derive the parameters in the controller.

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