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

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

IEEE Colloquium, April 4 th BACKGROUND

IEEE Colloquium, April 4 th 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

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

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

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

IEEE Colloquium, April 4 th 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.

IEEE Colloquium, April 4 th INTERPOLATION

IEEE Colloquium, April 4 th 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.

IEEE Colloquium, April 4 th MAS with K1 and K2

IEEE Colloquium, April 4 th 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.

IEEE Colloquium, April 4 th 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.

IEEE Colloquium, April 4 th Weakness of ellipsoidal sets

IEEE Colloquium, April 4 th Feasible regions (figures)

IEEE Colloquium, April 4 th 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).

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

IEEE Colloquium, April 4 th 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.

IEEE Colloquium, April 4 th POLYTOPIC INVARIANT SETS

IEEE Colloquium, April 4 th 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.]

IEEE Colloquium, April 4 th 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.

IEEE Colloquium, April 4 th 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.

IEEE Colloquium, April 4 th Robust and nominal invariant sets

IEEE Colloquium, April 4 th 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.)

IEEE Colloquium, April 4 th 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.

IEEE Colloquium, April 4 th EXPLICIT OR IMPLICIT CONSTRAINT HANDLING

IEEE Colloquium, April 4 th 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 ?

IEEE Colloquium, April 4 th 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

IEEE Colloquium, April 4 th 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

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

IEEE Colloquium, April 4 th 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

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

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

IEEE Colloquium, April 4 th ILLUSTRATIONS

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

IEEE Colloquium, April 4 th 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.

IEEE Colloquium, April 4 th 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!

IEEE Colloquium, April 4 th 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.

IEEE Colloquium, April 4 th Illustration of inequalities N1N2otherTotald.o.f. GIMPC GIMPC RMPC (nc=5) 4485

IEEE Colloquium, April 4 th 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).