1. Feasibility, uncertainty and interpolation J. A. Rossiter (Sheffield, UK)

2. IEEE Colloquium, April 4 th 2005 2 Overview Predictive control (MPC) Interpolation instead of optimisation Invariant sets Combining invariant sets Illustrations Conclusions.

3. IEEE Colloquium, April 4 th 2005 3 BACKGROUND

4. IEEE Colloquium, April 4 th 2005 4 Notation Assume a state space model and constraints Let the control law be Define the maximal admissible set (MAS), that is region within which constraints are met, as

5. IEEE Colloquium, April 4 th 2005 5 Invariant set and closed-loop trajectories

6. IEEE Colloquium, April 4 th 2005 6 Minimise a performance index of the form Can write solutions as Predictive control

7. IEEE Colloquium, April 4 th 2005 7 Impact on invariant sets of adding d.o.f.

8. IEEE Colloquium, April 4 th 2005 8 Observations If terminal control is optimal, then the terminal region may be small. – Need large d.o.f. to get large feasible region. – Good performance If terminal control is detuned, terminal region may be large. – Small d.o.f. to get large feasible region. – Suboptimal performance.

9. IEEE Colloquium, April 4 th 2005 9 INTERPOLATION

10. IEEE Colloquium, April 4 th 2005 10 Alternative strategy Interpolation is known to: 1. Allow efficient (often trivial) optimisations. 2. Combine qualities of different strategies. Interpolate between K1 and K2 where: K1 has optimal performance but possibly a small feasible region K2 has large feasible region.

11. IEEE Colloquium, April 4 th 2005 11 MAS with K1 and K2

12. IEEE Colloquium, April 4 th 2005 12 How to interpolate A simple summary: split the state into 2 components and predict separately through the 2 closed-loop dynamics, then recombine. Decomposition into x 1 and x 2 to ensure constraint satisfaction.

13. IEEE Colloquium, April 4 th 2005 13 Feasible regions with Interpolation Ellipsoidal invariant sets Find max. volume feasible invariant ellipsoid. By necessity conservative in volume. Can be computed easily, even with model uncertainty. Generalised interpolation algorithm takes convex hull of several ellipsoids. SDP solver required. Polytopic invariant sets Can use MAS – maximum possible feasible regions. Easily computed for nominal case only. Various interpolation algorithms for certain case. Still limited to convex hull of underlying sets. Optimisation requires QP or LP.

14. IEEE Colloquium, April 4 th 2005 14 Weakness of ellipsoidal sets

15. IEEE Colloquium, April 4 th 2005 15 Feasible regions (figures)

16. IEEE Colloquium, April 4 th 2005 16 When to use Interpolation? Which is more efficient: – A normal MPC algorithm with d.o.f.? – An interpolation? ONEDOF interpolations have only one d.o.f. but severely restricted feasibility. General interpolation requires nx d.o.f. (nx the state dimension).

17. IEEE Colloquium, April 4 th 2005 17 Feasible regions with general interpolation, ONEDOF and n c d.o.f.

18. IEEE Colloquium, April 4 th 2005 18 Weaknesses of interpolation 1. Algorithms using MAS can only be applied to the nominal case. 2. Easy to show that uncertainty can cause infeasibility and instability. 3. Need modifications to cater for uncertainty. Here we consider changes to cater for LPV systems.

19. IEEE Colloquium, April 4 th 2005 19 POLYTOPIC INVARIANT SETS

20. IEEE Colloquium, April 4 th 2005 20 Polytopic invariant sets (MAS) for nominal systems The computation of these is generally considered tractable. Let constraints be Then the MAS is given as Where for n large enough. [Redundant rows can be removed in general.]

21. IEEE Colloquium, April 4 th 2005 21 Polytopic invariant sets for LPV systems The computation of these is generally considered intractable. Consider a closed-loop LPV system Then computing all possible open-loop predictions. Clearly, there is a combinatorial explosion in the number of terms.

22. IEEE Colloquium, April 4 th 2005 22 Polytopic invariant sets for LPV systems There is a need for an alternative approach. [Pluymers et al, ACC 2005] Specifically, remove redundant constraints from M i before computing M i+1. This will slow the rate of growth and produce a tractable algorithm, if, the actual MAS is of reasonable complexity.

23. IEEE Colloquium, April 4 th 2005 23 Robust and nominal invariant sets

24. IEEE Colloquium, April 4 th 2005 24 Polytopic invariant sets and interpolation MUST USE ROBUST SETS TO ENSURE FEASIBILITY! We can simply use the ‘robust’ invariant sets in the algorithm developed for the nominal case. Proofs of recursive feasibility and convergence carry across easily if the cost is replaced by a suitable upper bound. (A quadratic stabilisability condition is required.)

25. IEEE Colloquium, April 4 th 2005 25 Summary Polytopic invariant sets allow the use of interpolation with LPV systems and hence: 1. Large feasible regions. 2. Robustness. 3. Small computational load. BUT: General interpolation still only applicable to convex hull of underlying regions. This could be too restrictive.

26. IEEE Colloquium, April 4 th 2005 26 EXPLICIT OR IMPLICIT CONSTRAINT HANDLING

27. IEEE Colloquium, April 4 th 2005 27 Extending feasibility of interpolation methods General interpolation does implicit not explicit constraint handling. So: 1. membership of the set implies the trajectories are feasible. 2. non-membership may not imply infeasibility. Therefore, we know that feasibility may be extended beyond the convex hull in general, but how ?

28. IEEE Colloquium, April 4 th 2005 28 Implicit constraint handling With ellipsoidal invariant sets this is obvious. Constraints are converted into an LMI, with some conservatism because of: 1. Asymmetry 2. Conversion of linear inequalities to quadratic inequalities. A trivial example of this might be or

29. IEEE Colloquium, April 4 th 2005 29 Conservatism with linear inequalities Define the invariant sets associated to K 1, K 2,… to be Then, general interpolation first splits x into several components and uses the constraints

30. IEEE Colloquium, April 4 th 2005 30 Conservatism with linear inequalities (b) The constraint enforces feasibility. However, consider the following hypothetical illustration: This implies that

31. IEEE Colloquium, April 4 th 2005 31 Remarks The constraint is necessary with ellipsoidal invariant sets as one can not check predictions explicitly against constraints. This is not the case with polytopic invariant sets. Hence we propose to relax this condition and hence increase feasible regions. Remove the two conditions

32. IEEE Colloquium, April 4 th 2005 32 Relaxed constraints General interpolation can be composed as We propose to replace this as a single inequality: NOTE: No longer any variables!

33. IEEE Colloquium, April 4 th 2005 33 Structure of inequalities (nominal case) Consider the predictions And hence the explicit constraints are

34. IEEE Colloquium, April 4 th 2005 34 ILLUSTRATIONS

35. IEEE Colloquium, April 4 th 2005 35 Illustrations 1. There can be surprisingly large increases in feasibility. 2. Probably because the directionality of trajectories for each controller are different.

36. IEEE Colloquium, April 4 th 2005 36 Extensions to the LPV case Unfortunately, explicit constraint handling requires a direct link between the prediction equations and the inequalities. However, the algorithm for finding polytopic invariant sets in the LPV case, relied, for efficiency, on removing redundant constraints from the predictions.

37. IEEE Colloquium, April 4 th 2005 37 Extensions to the LPV case (b) For the original GIMPC, sets S 1, S 2,.. could be described as efficiently as possible. There was no need for mutual consistency because constraint handling was implicit. Notably, all redundant inequalities could be eliminated. When doing explicit constraint handling, redundant constraints cannot be eliminated from S i, just in case the overall x(k+j) for that row is against a constraint!

38. IEEE Colloquium, April 4 th 2005 38 Constraints for general interpolation with LPV systems Algorithms can be written to formulate the inequalities, but suffer more from the combinatorial growth problems outlined earlier. Assuming the resulting sets are not too large, proofs of convergence and feasibility are straightforward.

39. IEEE Colloquium, April 4 th 2005 39 Illustration of inequalities N1N2otherTotald.o.f. GIMPC30122443 GIMPC2412 0 2 RMPC (nc=5) 4485

40. IEEE Colloquium, April 4 th 2005 40 Conclusions Interpolation is known to facilitate reductions in complexity at times, particular for low dimensional systems. However most work has focussed on the nominal case. Some earlier interpolation algorithms used implicit constraint handling to cater for uncertainty. This could lead to considerable conservatism. We have illustrated: – How interpolation can be modified to overcome this conservatism and the associated issues (recently submitted). – how polytopic robust MAS might be computed and used in MPC (to be published IFAC and ACC, 2005). – how to use polytopic robust MAS with interpolation (recently submitted).