PID Controllers

1. PID Controllers Jack Stankovic University of Virginia Spring 2015

2. I Control: Integral Control PI Control: Proportional-Integral Control D Control: Derivative Control PD Control: Proportional-Derivative Control PID Control: Proportional-Integral-Derivative Control Summary Outline

3. P Control • What is wrong with P control? • Non-zero steady-state error • Why? • When (current, instantaneous) error becomes zero then there is no longer a control signal

4. I Control • What is I control? • The controller output is proportional to the integral of all past errors U(z) E(z) Integral Controller

5. Integral Control Adds Pole D(z) N(z) + + Y(z) U(z) + E(z) + + T(z) R(z) KI z/ z-1 G(z) -

6. PI Control

7. D Control • D Control: the control output is proportional to the rate of change of the error • D control is able to make an adjustment prior to the appearance of even larger errors. • D control is never used alone, because of its zero output when the error remains constant.

8. PD Control

9. PID Control

10. I Control • Mostly used with P control • Here, by itself to demonstrate its main effects • Zero steady state error • Slow response

11. Introducing I Control • What is I control? • The controller output is proportional to the integral of all past errors U(z) E(z) Integral Controller

12. I Control D(z) N(z) + + Y(z) U(z) + E(z) + + T(z) R(z) KI z/ z-1 G(z) -

13. Steady-State Error with I Control • Start with Example 9.1: the IBM Lotus Domino Server Recall Here H(z) = 1

14. Example (continued) Multiply top and bottom by (z-1) then set z = 1

15. Why is the steady state error zero

16. General Case for G(z) • The steady-state error of a system with I control is 0, as long as the close-loop system is stable.

17. The steady-state error due to disturbance

18. Example 9.3 • Disturbance Rejection in the IBM Lotus Domino Server

19. Transient Response with I Control • I Control eliminates the steady-state error, but it slows the system down • The reason is the integrator adds an open-loop pole at 1, which generates a closed-loop pole that is usually close to 1. • Example 9.2: Closed-loop poles of the IBM Lotus Domino Server

20. Example 9.2 (continued) • Observe the root locus • The largest closed-loop pole is always closer to the unit circle than the open-loop pole 0.43. Here there is only P control Here there is I control

21. PI Control • Common controller • Fast response by P control • Accurate response by I control • Good combination • Another example of Pole Placement Design • Previously we did pole placement for P controller

22. PI Control Design by Pole Placement • Slightly different than for P control (see p 305 in text/handout) • Design Goals: • Assumption: G(z) is a first-order system • A higher-order system is approximated by a first-order system (chapter 3)

23. PI Control Design by Pole Placement • Example 9.5: Consider the IBM Lotus Domino server • Determine transfer function of system and SASO requirements Stable – poles within unit circle Accuracy – zero steady state error Note: 2 poles (one from G(z) and one from I control)

24. PI Control Design by Pole Placement • Compute the desired closed loop poles • Construct and expand the desired characteristic polynomial )

25. Construct the modeled characteristic polynomial

26. Example 9.5 (continued) • Expand the modeled characteristic polynomial Set the desired equal to the modeled

27. Example 9.5 (continued) • Solve for

28. Example 9.5 (continued) • Then check • Poles are ( .5 + .33) and (.5 - .33) so the system is stable • = 1 so there is no steady state error

29. Example 9.5 (continued) P control leads to quicker response I control leads to 0 steady-state error

30. PI Control Design Using Root Locus • The new issue: • The root locus allows only one parameter to be varied • A PI controller has two parameters: • The P control gain, and the I control gain • Solution to this issue: • Determine possible locations of the PI controller’s zero, relative to other poles and zeros • For each relative location of the zero, draw the root locus • For the most promising relative locations, try a few possible exact locations • Simulate to verify the result

31. Summary • I control adjust the control input based on the sum of the control errors • Eliminate steady-state error • Increase the settling time • D control adjust the control input based on the change in control error • Decrease settling time • Sensitive to noise • P, I and D can be used in combination • PI control, PD control, PID control

32. Summary (continued) • Pole placement design • Find the values of control parameters based on a specification of desired closed-loop properties. • Root locus design • Observe how closed-loop poles change as controller parameters are adjusted

33. Relationship to WSN and RTS • Consider PRR problem in WSN • Consider Miss Ratio Problem in RTS

34. Extra Slides

35. Example 9.4 • Moving-average filter plus I control • A moving average slows down the system responses • An I control also slows down the system response • So the combination leads to undesirable slow behavior • Example: IBM Lotus Domino server + I control + Moving-average filer

36. Moving-average filter vs. I controller • An I controller works like a moving-average filter: • More response to sustained change in the output than a short transient disturbance • An I controller drives the steady-state error to 0, but the moving-average filter does not.

37. PI Control

38. PI Control Design by Pole Placement • Approaches: • Step 1: compute the desired closed-loop poles • Step 2,3,4: find the P control gain and I control gain • Step 5: Verify the result • Check that the closed-loop poles lie within the unit circle • Simulate transient response to assess if the design goals are met

39. PI Control

40. Steady-state Error with PI Control • PI has a zero steady-state error, in response to a step change in the reference input • It also holds for the disturbance input

41. PI Control Design Using Root Locus • Example 9.6: PI control using root locus

42. Example 9.6 (continue)

43. Example 9.6 (continue) P control leads to quicker response I control leads to zero steady-state error

44. CHR Controller Design Method

45. CHR Controller Design Method

46. CHR Controller Design Method • Example 9.7:

47. Example 9.7 (continue) • No simulation is needed to verify it. Why? - Only one option in the table

48. D Control

49. D Control • A Real CS Example: • An IBM Lotus Domino server is used for healthy consulting. (MaxUsr, RIS) • Bird flu happens in this area. More and more people request for the service. • More and more hardware is added to the server. • So the reference point keeps increasing. • To deal with the increasing reference point, do we have better choices than P/I control? • How about setting the control output proportional to the rate of error change?

50. D Control • D Control: the control output is proportional to the rate of change of the error • D control is able to make an adjustment prior to the appearance of even larger errors. • D control is never used alone, because of its zero output when the error remains constant. • The steady-state gain of a D control is 0.