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Inner approximations for polynomial matrix inequalities and robust stability regions

Didier Henrion(henrion***at***laas.fr)
Jean-Bernard Lasserre(lasserre***at***laas.fr)

Abstract: Following a polynomial approach, many robust fixed-order controller design problems can be formulated as optimization problems whose set of feasible solutions is modelled by parametrized polynomial matrix inequalities (PMI). These feasibility sets are typically nonconvex. Given a parametrized PMI set, we provide a hierarchy of linear matrix inequality (LMI) problems whose optimal solutions generate inner approximations modelled by a single polynomial sublevel set. Those inner approximations converge in a strong analytic sense to the nonconvex original feasible set, with asymptotically vanishing conservatism. One may also impose the hierarchy of inner approximations to be nested or convex. In the latter case they do not converge any more to the feasible set, but they can be used in a convex optimization framework at the price of some conservatism. Finally, we show that the specific geometry of nonconvex polynomial stability regions can be exploited to improve convergence of the hierarchy of inner approximations.

Keywords: Polynomial matrix inequality, linear matrix inequality, robust optimization, linear controller design, moments, positive polynomials

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Robust Optimization


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Entry Submitted: 04/26/2011
Entry Accepted: 04/26/2011
Entry Last Modified: 04/26/2011

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