Application of facial reduction to H∞ state feedback control problem. Issue 2 (1st February 2019)

• Record Type: Journal Article
• Authors: Waki, Hayato; Sebe, Noboru
• Is Part Of: International journal of control. Volume 92:Issue 2(2019)
• Page Start: 303
• Page End: 316
• Publication Date: 2019-02-01
• Subjects: H∞ control -- linear matrix inequality -- state feedback control -- facial reduction -- invariant zeros; Automatic control -- Periodicals; Electronic journals; 629.8
• DOI: 10.1080/00207179.2017.1351625
• Languages: English
• ISSNs: 0020-7179

Abstract

ABSTRACT: One often encounters numerical difficulties in solving linear matrix inequality (LMI) problems obtained from H ∞ control problems. For semidefinite programming (SDP) relaxations for combinatorial problems, it is known that when either an SDP relaxation problem or its dual is not strongly feasible, one may encounter such numerical difficulties. We discuss necessary and sufficient conditions to be not strongly feasible for an LMI problem obtained from H ∞ state feedback control problems and its dual. Moreover, we interpret the conditions in terms of control theory. In this analysis, facial reduction, which was proposed by Borwein and Wolkowicz, plays an important role. We show that the dual of the LMI problem is not strongly feasible if and only if there exist invariant zeros in the closed left-half plane in the system, and present a remedy 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.