Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design |
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Authors: | Didier Henrion Christophe Louembet |
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Affiliation: | 1. CNRS, LAAS , 7 Avenue du Colonel Roche, F-31077 Toulouse , France;2. Université de Toulouse, LAAS , F-31400 Toulouse , France;3. Faculty of Electrical Engineering , Czech Technical University in Prague , Technická 2, CZ – 16626 Prague , Czech Republic henrion@laas.fr;5. Université de Toulouse, UPS , LAAS , F-31400 Toulouse , France |
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Abstract: | We describe an elementary algorithm to build convex inner approximations of nonconvex sets. Both input and output sets are basic semialgebraic sets given as lists of defining multivariate polynomials. Even though no optimality guarantees can be given (e.g. in terms of volume maximisation for bounded sets), the algorithm is designed to preserve convex boundaries as much as possible, while removing regions with concave boundaries. In particular, the algorithm leaves invariant a given convex set. The algorithm is based on Gloptipoly 3, a public-domain Matlab package solving nonconvex polynomial optimisation problems with the help of convex semidefinite programming (optimisation over linear matrix inequalities, or LMIs). We illustrate how the algorithm can be used to design fixed-order controllers for linear systems, following a polynomial approach. |
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Keywords: | polynomials nonconvex optimisation LMI fixed-order controller design |
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