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Application of Facial Reduction to $H_\infty$ State Feedback Control Problem

Hayato Waki(waki***at***imi.kyushu-u.ac.jp)
Noboru Sebe(sebe***at***ai.kyutech.ac.jp)

Abstract: One often encounters numerical difficulties in solving linear matrix inequality (LMI) problems obtained from $H_\infty$ control problems. We discuss the reason from the viewpoint of optimization, and provide necessary and sufficient conditions for LMI problem and its dual not to be strongly feasible. Moreover, we interpret them in terms of control system. In this analysis, facial reduction, which was proposed by Borwein and Wolkowicz, plays an important role. We show that a necessary and sufficient condition closely related to the existence of invariant zeros in the closed left-half plane in the system, and present a way to remove the numerical difficulty with the null vectors associated with invariant zeros in the closed left-half plane. Numerical results show that the numerical stability is improved by applying it.

Keywords: $H_\infty$ control, linear matrix inequality, state feedback control, facial reduction, invariant zeros

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

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )

Citation:

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Entry Submitted: 06/10/2016
Entry Accepted: 06/11/2016
Entry Last Modified: 06/10/2016

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