Chapter 6 Model Predictive Control

1. Chapter 6Model Predictive Control Prof. Shi-Shang Jang National Tsing-Hua University Chemical Engineering Department

2. Historical Development • Questions: • Given a system what are the absolute limitations to output control? • How can one make use of a process model and account for model error? • Smith Predictor(1955) • Feedforward Control • Inferential Control (1975, 1979) • Dynamic Matrix Control (Shell, 1979) • Model Algorithmic Control (France, 1979) • Internal Model Control (Garcia and Morari, 1982)

3. Criteria for Controller Quality • Regulatory Behavior – Compensation for (unmeasured disturbances) • Servo Behavior- Follow set point changes (fast, smooth, no offset) • Robustness-Controller should be effective when there are modeling errors (both structure and parameters)

4. Criteria for Controller Quality-Continued • Constraints- Ability to deal with constraints on inputs and states (no windup) • Remark: 90% of loops can be handled by PID type controllers.

5. d + e u + + ys y GI GP - ym + - Gm Internal Model Control (IMC) Structure e=ys-y+ym

6. Analysis of Internal Model Structure • From the block diagram:

7. Properties of IMC (Principles of Internal Model Control) 1. (Dual Stability) If the model is perfect, stability of controller and plant is sufficient for overall system stability Proof: If Gp=Gm, then y=GpGI(ys-d)+d u=(ys-d) GI • Use IMC only on stable systems,unstable systems can be stabilized by feedback control • Constraints on inputs has no effect on stability

8. Properties of IMC (Principles of Internal Model Control)- continued 2. (Prefect Control) If model is perfect and invertible, and GI=Gp-1, then y=ys for any d • Notes: (1) This is “optimal control”. (2) Suppose Gp-1 is not realizable, then it is recommended to factor this transfer function into two terms: Gp(s)=G+(s)G-(s) where G+(s) is not realizable contains all time delays and RHP zeros. In this case the “best” controller possible is: GI(s)=G--1(s), this controller minimizes sum of the square of the errors in output. (3) This suggests the design of F(s) as GI=Gp-1 F(s) such that GI(s) realizable.

9. Example Which is realizable if we choose: Where n=degree of D(s)-degree of N(s)>0,  is chosen

10. Properties of IMC (Principles of Internal Model Control)- continued 3. Zero Offset There is no offset if we choose: GI(0)=1/Gm(0) Pf: • This means the output will attain the set point exactly in presence of persistent disturbances and set-point changes - integral feedback • b. Note that this is true even if the model is imperfect.

11. e(s) Gc(S) m(s) + - Gm(S) Properties of IMC (Principles of Internal Model Control)- continued 4. Comparison with Feedback Controller If we choose Then we get classical output feedback control. Note that GI(s) in the closed loop transfer function of the following:

12. d e(s) m(s) Gc(S) ys + Gc(S) y(s) + - - Gp(S) Gp(S) + - Properties of IMC (Principles of Internal Model Control)- continued Joining this with the IMC block diagram we get which reduced to a classical feedback control: d ys(s) + Gc Gp -

13. Properties of IMC (Principles of Internal Model Control)- continued Notes: (a) may be looked upon as a “poor” approximation to Gm-1(s). For large Kc, this approximation gets better. (b) Note the similarity of this structure to Smith Predictor also approximates the invertible part of Gm-1(s). (c) This feedback structure implies use of a filter:

14. On-Line Tuning of IMC • Choose a process model through plant tests • Choose a filter to make GI(s) realizable • Decrease  untill system becomes oscillatory. Brosilow recommends the following time constant: (where u=min. filter constant; Pu=period of oscillation) • If /D<1, then IMC will yield better performance than a PID controller. If 1< /D<2; then IMC and PID are competitive.

15. Computation of Approximate Inverses • In practice, it is easier to find an approximate inverse of the process transfer function in the time domain (using discrete models). • Time Domain View: • Given the past history of inputs to the process and current estimate of the disturbance, compute the current and future inputs which will make the output follow the desired set point. • Limitation in Practice: • The future will be limited to a finite time horizon (3-4 times time-constant of system. • Attention must be limited to values of output at discrete times.

16. Summary of MPC • MPC consists of three blocks: • Process model • A controller (approximate inverse) • A filter • Advantages: • Quality of response depends on controller design • Robustness depends on filter • Stability is not an issue • Implementation is straight forward • On-line tuning can be provided by the filter time constant

17. Computation of Approximate Inverses - Continued 3. Values of ‘all’ future inputs may be limited to a few in the immediate future. 4. Problem must be solved every so often (at discrete sampling times) when new estimates of disturbance become available. 5. We must limit the size and velocity of control input variations. 6. On-line computations should be kept to a minimum. 7. Smooth transfer between auto/manual should be possible. 8. It should be recognize constraints on inputs. 9. There should be operator adjustable constant(s) to account for plant/model mismatch.

18. Review of least-square problem • Given a set of equations: Ax=b+e • We seed a solution which minimizes iei2 • The solution is given by: x=(ATA)-1ATb • We term (ATA)-1AT to be pseudo inverse of matrix A

19. A Discrete Input Plant Model Note that N/D is actually the impulse response of the system m(z)=1 without delay y(t) h3 h4 h5 h1 h2 h6 h7 t

20. Example: G(s)=1/(s+1)3 global m m=1;TSPAN=[0 1]; Y0=[0 0 0]; Y_real=[];Y_sample=[];T_real=[]; T_sample=[0];Y_model=[0]; for i=1:29 [T,Y] = ODE45('model_3',TSPAN,Y0); TSPAN=[TSPAN(2),TSPAN(2)+1]; Y0(1)=Y(end,1); Y0(2)=Y(end,2); Y0(3)=Y(end,3); TT=T(end); Y_real=[Y_real;Y(:,1)];T_real=[T_real;T]; m=0; T_sample=[T_sample,TT]; Y_model=[Y_model,Y0(1)]; end function dy=model_3(t,y) global m dy(1)=y(2); dy(2)=y(3); dy(3)=-3*y(3)-3*y(2)-y(1)+m; dy=dy';

21. Example: G(s)=1/(s+1)3 - continued TSPAN=[0 1];mm=zeros(1,29); Y0=[0 0 0]; Y_real=[];Y_sample=[];T_real=[]; T_sample=[0];Y_pred=[0]; for i=1:50 m=randn(1,1); for i=1:28 mm(29-i+1)=mm(29-i); end mm(1)=m; [T,Y] = ODE45('model_3',TSPAN,Y0); TSPAN=[TSPAN(2),TSPAN(2)+1]; Y0(1)=Y(end,1); Y0(2)=Y(end,2); Y0(3)=Y(end,3); TT=T(end); Y_real=[Y_real;Y(:,1)];T_real=[T_real;T]; yp=Y_model*mm'; Y_pred=[Y_pred,yp]; T_sample=[T_sample,TT]; end

22. A Discrete Input Plant Model

23. A Discrete Input Plant Model-Continued This expresses y in terms of past inputs m; i.e.:

24. Approximate Inversion Since we cannot make y(t)=yd(t) exactly, we pose the following least square minimization problem: subject to the above process model: No control changes beyond M

25. The Solution • The previous problem can be solved based on a Quadratic Programming solver or using previous pseudo-inverse of matrix approach.

26. MPC-Servo Control (A Feed-forward Approach) Want yk+1=yk+2=…=yd P=4; M=4

27. MPC-Servo Control (A Feed-forward Approach) -Example Y_real Y_sample time time P=4; M=4

28. MPC-Servo Horizon Control (A Feed-forward Approach) Want yk+1=yk+2=…=yd, but mk+1=mk+2=mk+3 P=4; M=2

29. MPC-Servo Horizon Control (A Feed-forward Approach) Want yk+1=yk+2=…=yd, but mk+1=mk+2=mk+3 Response Time P=4; M=2

30. MPC-Regulation Control (A Feedback Approach) P=4; M=4

31. MPC-Regulation Control (A Feedback Approach) P=4; M=2

32. Response Time MPC-Regulation Control (A Feedback Approach) Response Time M=2 M=4

33. Multi-variable Discrete Input Plant Model

34. Hot Cold LT TT Examples of Multivariable Control: Control of a Mixing Tank MV’s: Flow of Hot Stream CV’s: Level in the tank Flow of Cold Stream Temperature in the tank

35. Example- Mixing Tank Problem Height Time

36. Example- Mixing Tank Problem Temperature Time

37. Dynamic Matrix Control (DMC) Response Response a4,…. a3 h2 h3 h4,…. a2 h1 a1 Time Time Step response Pulse response

38. Dynamic Matrix Control (DMC)- Continued

39. Dynamic Matrix Control (DMC)- Continued disturbance Effect of the past

40. Dynamic Matrix Control (DMC)- Continued P=4; M=4

41. Dynamic Matrix Control (DMC)- Continued P=4; M=2

42. Tuning Procedures • Sampling time (T): stability is not affected by T. Larger T leads to less variations in m, but deteriorates system performance in presence of frequent disturbances • Horizon for m (M): Choosing M=P (perfect control) leads to severe oscillation in m(t). Reducing M, leads to a more desired response

43. Tuning Procedures - Continued • Input penalty parameter : Increasing  makes system more sluggish and nonzero  lead to offset, but can be compensated by adding integral control algorithm itself. • Optimization horizon, P: increasing P gets better inverse for system of order n, P>2n is generally sufficient.

44. Summary - Continued • Model Predictive Control (MPC) is the major existed advanced process control in chemical engineering industrial • The modeling in the MPC is crucial • The tuning of MPC using M (horizon of suppression) is the most effective for stability. All other parameters may also frequently implement to improve the control quality