1. University of Virginia
PID Controllers
Gang Zhou
University of Virginia
February 2006

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

3. Introducing I Control What is wrong with P control?
Non-zero steady-state error
Prove it
A real CS case
See p259

4. Introducing I Control What is I control?
The controller output is proportional to the integral of all past errors

5. Steady-State Error with I Control
Start from Example 9.1: the IBM Lotus Domino Server

6. More General
The steady-state error of a system with I control is 0, as long as the close-loop system is stable.

7. The steady-state error due to disturbance

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

9. Example 9.2 (continue) Observe the root locus
The largest closed-loop pole is always closer to the unit circle than the open-loop pole 0.43.
Ask Ting why Fig 9.2 (a) has root locus, since it has only one pole and has no K. --- It is a P controller by default.

10. Example 9.3
Disturbance Rejection in the IBM Lotus Domino Server

11. 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
The settling time will be very long, here greater than 78.

12. 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.

13. PI Control

14. 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

15. PI Control Design by Pole Placement
Design Goals:
Assumption: G(z) is a first-order system
A higher-order system is approximated by a first-order system (chapter 3)
Equation 3.30 in Chapter 3

16. 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

17. PI Control Design by Pole Placement
Example 9.5: Consider the IBM Lotus Domino server

18. Example 9.5 (continue)

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

20. 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

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

22. Example 9.6 (continue)

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

24. CHR Controller Design Method
The bump test is in page 310

25. CHR Controller Design Method

26. CHR Controller Design Method
Example 9.7:

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

28. 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?

29. 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.

30. PD Control
The 0 pole is at page 317

31. PD Control
PD controllers are not appropriate for first-order systems because pole placement is quite limited
PD controllers can be used to reduce the overshoot for a system that exhibits a significant amount of oscillation with P control
Example: consider a second-order system

32. Example (continue)

33. Example (continue)
It is real good
It sounds good

34. Example (continue)

35. Example (continue)

36. PID Control

37. PID Control PI controllers are preferred over PID controller
D control is sensitive to the stochastic variations
A low-pass filter can be applied to smooth the system output. In that case, the D control only responds to large changes
But the filter slows down the system response
PID Control Design by Pole placement
Compute the dominant poles based on the design goals
Compute the desired characteristic polynomial
Compute the modeled characteristic polynomial
Solve for the gains of the P, I and D control by coefficient matching
Verify the result
PI controllers are preferred over PID controllers --- in the second paragraph in page 323

38. PID control design by pole placement
Example 9.8:
Consider the IBM Lotus Domino Server

39. Example 9.8 (continue)

40. Example 9.8 (continue)

41. 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, OD control, PID control

42. Summary (continue) Pole placement design Root locus 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
Empirical method:
The values of control parameters can be determined by empirical methods based on the step response of the open-loop system