CHAPTER 19: Single-Loop IMC Control

Transcription

1 When I coplete this chapter, I want to be able to do the following. Recognize that other feedback algoriths are possible Understand the IMC structure and how it provides the essential control features Tune an IMC controller Correctly select between PID and IMC

2 Outline of the lesson. Thought exercise for odel-based control IMC structure Desired control features IMC algorith and tuning Application guidelines

3 Let s quickly review the PID algorith E ( t ) SP ( t ) CV ( t ) 1 d CV MV ( t ) K c E ( t ) E ( t' ) dt ' Td T I 0 dt This is the de facto standard! I PID was developed in 1940 s PID is not the only feedback algorith PID gives good balance of perforance and robustness PID does not always give the best perforance Multiple PIDs are used for ultivariable systes

4 Let s look ahead to the IMC structure and algorith We will have another algorith to learn!!!! IMC was developed forally in 1980 s, but the ideas began in 1950 s IMC uses a process odel explicitly IMC involves a different structure and controller IMC could replace PID, but we chose to retain PID unless an advantage exists A single IMC can be used for ultivariable systes

5 Let s do a thought experient: 1. We want to control the concentration in the tank. 2. Initially, A 6 wgt% We want A 7 wgt% 3. Fro data, we know that A/ v 0.5 wgt%/% open Solvent Reactant A A wgt% 6% What do we do? Valve % open tie

6 Let s do a thought experient: 1. We want to control the concentration in the tank. 2. Initially, A 6 wgt% We want A 7 wgt% 3. Fro data, we know that A/ v 0.5 wgt%/% open Solvent Reactant A A wgt% 6% What do we expect to happen? Valve % open 2% tie

7 Let s do a thought experient: 1. We want to control the concentration in the tank. 2. Initially, A 6 wgt% We want A 7 wgt% 3. Fro data, we know that A/ v 0.5 wgt%/% open 7.3% A wgt% 6% Solvent Reactant A Should we be surprised? What do we do now? Devise 2 different responses Valve % open 2% tie Hint: We used a odel for the first calculation. What do we do if the odel is in error?

8 Let s do a thought experient: 1. We want to control the concentration in the tank. 2. Initially, A 6 wgt% We want A 7 wgt% 3. Fro data, we know that A/ v 0.5 wgt%/% open 7.3% A wgt% 6% 6.91% Solvent Reactant A We apply the feedback until we have converged. Valve % open 2% -0.6% tie

9 7.3% A wgt% 6% 5.91% Solvent Valve % open 2% -0.6% A tie Reactant D(s) d (s) SP(s) MV(s) C (s) v (s) P (s) - CV(s) Discuss this diagra and coplete the closed-loop block diagra for the IMC concept.

10 7.3% A wgt% 6% 5.91% Solvent Valve % open 2% -0.6% A tie Reactant D(s) d (s) SP(s) MV(s) C (s) v (s) P (s) - CV(s) This is a odel of the feedback process. (s) - Model error E (s)

11 D(s) d (s) SP(s) T p (s) MV(s) CP (s) v (s) P (s) - CV(s) What is CP?? E (s) (s) - Transfer functions CP (s) controller v (s) valve P (s) feedback process (s) odel d (s) disturbance process Variables CV(s) controlled variable CV (s) easured value of CV(s) D(s) disturbance E (s) odel error MV(s) anipulated variable SP(s) set point T p (s) set point corrected for odel error

12 What controller calculation gives good perforance? It is NOT a PID algorith Let s set soe key features and deterine what cp will achieve these features 1. Zero steady-state offset for step-like inputs 2. Perfect control (CVSP for all tie) 3. Moderate anipulated variable adjustents 4. Robustness to odel isatch 5. Anti-reset-windup

13 What controller calculation gives good perforance? SP(s) - T p (s) MV(s) CP (s) What is CP?? D(s) v (s) d (s) P (s) (s) - CV(s) E (s) 1. Zero steady-state offset: CV D( s) 1 CP (1 cp ) ( ) v p S d What condition for cp (s) ensures zero steady-state offset for a step disturbance? Hint: Steady-state is at t.

14 What controller calculation gives good perforance? SP(s) - T p (s) MV(s) CP (s) What is CP?? D(s) v (s) d (s) P (s) (s) - CV(s) E (s) 1. Zero steady-state offset: CV D( s) li CV ( t) t 1 CP li s 0 (1 cp ( ) v D (1 KcpK s s 1 K ( K p cp ) p ) K K Oh yeah, the final value theore! S d d 0 ) CONCLUSION K cp (K ) -1 Easily achieved because both are in coputer!

15 What controller calculation gives good perforance? SP(s) - T p (s) MV(s) CP (s) What is CP?? D(s) v (s) d (s) P (s) (s) - CV(s) E (s) 2. Perfect dynaic control: CV D( s) 1 CP (1 cp ) ( ) v p S d What is required for perfect dynaic control? Do you expect that we will achieve this goal?

16 What controller calculation gives good perforance? SP(s) - T p (s) MV(s) CP (s) What is CP?? D(s) v (s) d (s) P (s) (s) - CV(s) E (s) 2. Perfect dynaic control: CV D( s) 1 (1 cp ( s)) d CV D( s) 0 1 CP CP (1 ( ) v cp ( ) v p S p ) S d CONCLUSION cp (s) ( (s)) -1 Controller is inverse of odel! Can we achieve this?

17 2. Perfect dynaic control: 3. Moderate anipulated variable adjustents θs Ke ( τs 1) then cp ( τs 1) e K θs CONCLUSION cp (s) ( (s)) -1 Controller is inverse of odel! Can we achieve this? This is a pure derivative, which will lead to excessive anipulated variable oves This is a prediction into the future, which is not possible Conclusion: Perfect control is not possible! (See other exaples in the workshops.)

18 2. Perfect dynaic control: 3. Moderate anipulated variable adjustents CONCLUSION cp (s) ( (s)) -1 K( τ1s 1) ( τ s 1)( τ s 1) 2 3 then cp ( τ 2s 1)( τ 3s 1) K ( τ s 1) 1 Controller is inverse of odel! Can we achieve this? This has a second derivative, which will lead to excessive anipulated variable oves This could have an unstable controller, if τ 1 is negative. This is unacceptable. Conclusion: Perfect control is not possible! (See other exaples in the workshops.)

19 Let s begin our IMC design with the results so far. CONCLUSION K cp (K ) -1 Easily achieved because both are in coputer! CONCLUSION cp (s) ( (s)) -1 Controller is inverse of odel! We have loosened the restriction to a condition that can be achieved. Now, what does approxiate ean? How can we define the eaning of approxiate so that we have a useful design approach?

20 Separate the odel into two factors, one invertible and the other with all non-invertible ters. The invertible factor has an inverse that is causal and stable, which results in an acceptable controller. The gain is the odel gain, K. The non-invertible factor has an inverse that is noncausal or unstable. The factor contains odels eleents with dead ties and positive nuerator zeros. The gain is the 1.0. What do we use for the controller?

21 Separate the odel into two factors, one invertible and the other with all non-invertible ters. The IMC controller eliinates all non-invertible eleents in the feedback process odel by inverting - (s). cp [ ] -1 Looks easy, but I need soe practice.

22 F S solvent F A pure A AC Class exercise: We have two odels for the feedback dynaics for the 3-tank ixer. Deterine cp (s) for each. Epirical odel 5.5s 0.039e (1 10.5s) Fundaental odel (1 5s) 3

23 Epirical odel 5.5s 0.039e (1 10.5s) Fundaental odel (1 5s) 3 (1 10.5s) cp ( 1 5s ) cp ( s ) Discuss these results. Do they ake sense? Are there any shortcoings? (Hint: Look at other desirable features.)

24 (1 10.5s) cp ( 1 5s ) cp ( s ) First derivative Third derivative! 3. Moderate anipulated variable adjustents 4. Robustness to odel isatch To achieve these features, we ust be able to slow down the controller. We chose to include a filter in the feedback path.

25 D(s) d (s) SP(s) T p (s) MV(s) CP (s) v (s) P (s) - CV(s) (s) - f (s) E (s) f τ f 1 s 1 N f The value of N f is selected so that the product of the controller and filter has a polynoial in s with a denoinator order the nuerator order. The filter is designed to prevent pure derivatives and the filter constant is tuned to achieve robust perforance.

26 Class exercise: Deterine the structure of the filter for the two possible controllers we just designed for the three tank ixing process. Epirical odel 5.5s 0.039e (1 10.5s) Fundaental odel (1 5s) 3 (1 10.5s) cp ( 1 5s ) cp ( s )

27 3 ) 5 ( ) ( s s ) 10.5 ( ) ( 5.5 s e s s Epirical odel Fundaental odel ) 10.5 (1 ) ( s s cp ) ( ) ( s s cp Class exercise: Deterine the structure of the filter for the two possible controllers we just designed for the three tank ixing process ) ( s s f f τ ) ( s s f f τ

28 Now, we ust deterine the proper value for the filter tie constant - this is controller tuning again! We know how to set the goals fro experience with PID. CV Dynaic Behavior: Stable, zero offset, iniu IAE MV Dynaic Behavior: daped oscillations and sall fluctuations due to noise. MV can be ore aggressive in early part of transient

29 We can tune using a siulation and optiization or Process reaction curve Solve the tuning proble. Requires a coputer progra. Apply, and fine tune v 1 TC v 2 COMBINED DEFINITION OF TUNIN First order with dead tie process odel Noisy easureent signal ± 25% paraeters errors between odel/plant IMC controller: deterine K f Miniize IAE with MV inside bound v 1 TC v 2 K 1 θ 5 τ 5 τ f??? We can develop tuning correlations. These are available for a lead-lag controller, cp (s) f (s).

30 0.55 IMC Tuning for a typical set of conditions and goals Scaled filter tie constant, τ f /(θτ) Fraction dead tie, θ/(θτ)

31 Class exercise: Deterine the filter tie constant for the IMC controller that we just designed for the three tank ixing process. Epirical odel 5.5s 0.039e (1 10.5s) (1 10.5s) cp f 1 τ f s 1 1

32 Class exercise: Deterine the filter tie constant for the IMC controller that we just designed for the three tank ixing process. Epirical odel 5.5s 0.039e (1 10.5s) (1 10.5s) cp f τ f 1 s The controller is lead-lag, so we can use the correlation. 2. θ/(θτ) 5.5/ τ f /(θτ) 0.38 ; τ f (0.38)(15.5) 6.1 inutes 4. See the next slide for the dynaic response. 5. We can fine tune to achieve the desired CV and MV behavior.

33 F S solvent F A pure A AC 3.5 concentration 3 AC Discuss the control perforance tie valve opening, % v A tie

34 Sith Predictor: A sart fellow naed Sith saw the benefit for an explicit odel in the late 1950 s. He invented the Sith Predictor, using the PI controller. The eleents in the box calculate an approxiate inverse D(s) d (s) SP(s) MV(s) C (s) v (s) P (s) - - CV(s) - (s) (s) - E (s) Notes: c (s) is a PI controller; - (s) is the invertible factor.

35 When do we select an IMC over a PID? 1. Very high fraction dead tie, θ/(θτ) > Very strong inverse responses. 3. Cascade priary controller with slow secondary (inner) dynaics 4. Feedforward with disturbance dead tie less than feedback dead tie. See the textbook for further discussion.

36 CHAPTER 19: IMC CONTROL WORKSHOP 1 Trouble shooting: You have deterined the odel below epirically and have tuned the IMC controller using the correlations. The closedloop perforance is not acceptable. What do you do? 1 Epirical odel K 1 θ 5 τ 5 IMC filter τ f 3.5 concentration anipulated flow tie tie

37 CHAPTER 19: IMC CONTROL WORKSHOP 2 Design features: We want to avoid integral windup. 1. Describe integral windup and why it is undesirable. 2. Modify the structure to provide anti-reset windup. D(s) d (s) SP(s) T p (s) MV(s) CP (s) v (s) P (s) - CV(s) (s) - f (s) E (s)

38 CHAPTER 19: IMC CONTROL WORKSHOP 3 Design features: We want to obtain good perforance for set point changes and disturbances. The filter below affects disturbances. Introduce a odification that enables us to influence set point responses independently. D(s) d (s) SP(s) T p (s) MV(s) CP (s) v (s) P (s) - CV(s) (s) - f (s) E (s)

39 CHAPTER 19: IMC CONTROL WORKSHOP 4 Design features: We discussed two reasons why we cannot achieve perfect feedback control. Identify other reasons and explain how they affect the IMC structure design. D(s) d (s) SP(s) T p (s) MV(s) CP (s) v (s) P (s) - CV(s) (s) - f (s) E (s)

40 When I coplete this chapter, I want to be able to do the following. Recognize that other feedback algoriths are possible Understand the IMC structure and how it provides the essential control features Tune an IMC controller Correctly select between PID and IMC Lot s of iproveent, but we need soe ore study! Read the textbook Review the notes, especially learning goals and workshop Try out the self-study suggestions Naturally, we ll have an assignent!

41 CHAPTER 19: LEARNIN RESOURCES SITE PC-EDUCATION WEB - Tutorials (Chapter 19) The Textbook, naturally, for any ore exaples. Addition inforation on IMC control is given in the following reference. Brosilow, C. and B. Joseph, Techniques of Model- Based Control, Prentice-Hall, Upper Saddle River, 2002

42 CHAPTER 19: SUESTIONS FOR SELF-STUDY 1. Discuss the siilarities and differences between IMC and Sith Predictor algoriths 2. Develop the equations that would be solved for a digital ipleentation of the IMC controller for the three-tank ixer. 3. Select a feedback control exaple in the textbook and deterine the IMC tuning for this process. 4. Find a feedback control exaple in the textbook for which IMC is a better choice than PID. 5. Explain how you would ipleent a digital algorith to provide good control for the process in Exaple 19.8 experiencing the flow variation described.