1 / 43

PID Controllers

PID Controllers. Gang Zhou. University of Virginia February 2006. I Control: Integral Control PI Control: Proportional-Integral Control PD Control: Proportional-Derivative Control PID Control: Proportional-Integral-Derivative Control Summary. Outline. Introducing I Control.

gashton
Télécharger la présentation

PID Controllers

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. PID Controllers Gang Zhou University of Virginia February 2006

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

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

  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.

  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

  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)

  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

  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

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

  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

  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 • 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

More Related