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Lexicographic Perturbation for Multiparametric Linear Programming


C.N. Jones

Optimization and Applications Seminar, ETH Zurich

Optimal control problems for constrained linear systems with a linear cost can be posed as multiparametric linear programs and solved explicitly offline. Degeneracy occurs when the control input, or optimiser, is non-unique and can cause chattering of the control input and overlap of the polyhedral regions of the explicit solution. Degenerate situations can occur naturally in control problems, such as outputs that are constrained to lie within a zone, but for which no particular value is preferred. This presentation introduces a new and efficient approach to the computation of the solution to a multiparametric linear program in the presence of degeneracy. Rather than solve the degenerate problem directly, we solve a lexicographically (symbolically) perturbed version of it that is guaranteed to be non-degenerate and whose solution is optimal for the non-perturbed problem. Furthermore, it is continuous, unique and defined over a set of non-overlapping polyhedral regions. The proposed algorithm is particularly suited for the calculation of the explicit solution to a class of constrained optimal control problems, since it ensures that the control input is everywhere continuous and unique, thereby removing the danger of chattering. The algorithm is compared through example to existing proposals and a significant complexity improvement is demonstrated.


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