دانشگاه صنعتي اميركبير دانشكده مهندسي پزشكي Constraints in MPC

2. دانشگاه صنعتي اميركبير دانشكده مهندسي پزشكي Constraints in MPC کنترل پيشبين-دکتر توحيدخواه

3. Feasibility in MPC • Infeasibility implies that, for the current state, the constraints within the MPC algorithm cannot be satisfied. • Without feasibility the MPC optimisation is ill posed and there is no assurance that the answer has any useful meaning. • MPC looks at mechanisms for overcoming or avoiding this.

4. Hard constraints Hard constraints are constraints which must be satisfied.

5. Soft constraints • Soft constraints are those which should be satisfied if possible. Soft constraints can be violated (ignored). • Usually soft constraints are on outputs/states although they could also be applied to inputs. Such violations may have no effect on nominal stability results.

6. Terminal constraints

7. Summary: Constraints are a combination of hard, soft and terminal constraints.

8. Model uncertainty Model uncertainty may cause infeasibility because the actual behaviour differs from the predicted behaviour. Hence, even though the nominal predictions could satisfy constraints over the entire future, a small change in the model will cause the actual behaviour to differ, and the associated predictions at the subsequent sampling instant could then violate constraints.

9. Model uncertainty (cont.) One can form algorithms based on invariant sets (Section 11.11) to handle model uncertainty; however, the results are usually very conservative, as guarantees must allow for the worst case (which will arise with negligible probability). A more pragmatic approach is to accept that guarantees cannot be given where there is significant uncertainty and make other contingencies for the rare occasions where infeasibility arises.

10. The stronger a guarantee you want, the more conservative your control law will be. • In practise there must be a compromise between feasibility assurances and performance. • Feasibility would usually be ensured by a systematic relaxation of soft constraints. This would be determined at a supervisory level.

11. Using artificially tight constraints on future predictions automatically builds in some slack which can be used to retain feasibility in the presence of moderate uncertainty. The slack should be montonically increasing with the horizon. Example: input limits:

12. Example 1. A mathematical model for an undamped oscillator is given by:

13. What happens if the control amplitude is limited to +/- 25 ?

15. closed-loop eigenvalues are at −0.1946, 0, 0.

16. Case A. Closed loop control without saturation

17. Case B- Closed-loop control with saturation

18. If we do not pay attention to the saturation of the control, then in the presence of constraints, the closed-loop control performance could severely deteriorate.

19. Example 2: A common practice in dealing with saturation is to let the model know the difference in Δu(k) when saturation becomes effective.

20. over-shoot in the closed-loop response is significantly reduced.

22. Formulation of Constrained Control Problems

23. There are three major types of constraints frequently encountered in applications: The first two types deal with constraints imposed on the control variables u(k), and the third type of constraint deals with output y(k) or state variable x(k) constraints.

24. Constraints on the Amplitude of the Control Variable

25. Constraints on the Control Variable Incremental Variation در بيهوشی بعنوان مثال:

26. Output Constraints Output constraints are often implemented as ‘soft’ constraints in the way thata slack variable sv > 0 is added to the constraints

27. Constraints in a Multi-input and Multi-output Setting

29. Constraints as Part of the Optimal Solution

34. Numerical Solutions Using Quadratic Programming E is assumed to be symmetric and positive definite.

35. Quadratic Programming for Equality Constraints Example 4. Minimize Solution. The global minimum, without constraint, is at

36. Illustration of constrained optimal solution

38. Lagrange Multipliers

40. Example 5. Minimize subject to:

41. Solution: Without the equality constraints, the optimal solution is:

44. Example 6. what happens to the constrained optimal solution when the linear constraints are dependent. There is no feasible solution of x1 and x2 Matrix MTE−1M is not invertible

45. Solution:

46. Illustration of no feasible solution of the constrained optimization problem. Solid-line x1 + x2 = 1; darker-solid-line 2x1 +2x2 = 6

47. Example 7: How the number of equality constraints is also an issue in the constrained optimal solution? (Ex. 5) We add an extra constraint to the original constraints so that:

48. The only feasible solution: In summary, the number of equality constraints is required to be less than or equal to the number of decision variables (i.e., x). If the number of equality constraints equals the number of decision variables, the only feasible solution is the one that satisfies the constraints and there is no additional variable in x that can be used to optimize the original objective function.

49. Minimization with Inequality Constraints In the minimization with inequality constraints, the number of constraints could be larger than the number of decision variables. An inequality Mix ≤ γi is said to be active if Mix = γi and inactive if Mix < γi.

50. Kuhn-Tucker Conditions