Application of facial reduction to H_∞ state feedback control problem

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.

KEYWORDS:

• H _∞ control
• linear matrix inequality
• state feedback control
• facial reduction
• invariant zeros

Acknowledgments

The first author was supported by JSPS KAKENHI grant numbers JP22740056 and JP26400203. We would like to thank Prof. Shinji Hara in Tokyo University and Dr Yoshio Ebihara in Kyoto University for a fruitful discussion and significant comments for improving the presentation of the manuscript. Also, we would like to thank the anonymous reviewers for providing us with significant comments and suggestions for improving the presentation of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. An algorithm implemented in SeDuMi, which is one of the well-used LMI softwares, can deal with non-strongly feasible LMI problems, theoretically. Nevertheless, we will observe from our numerical experiments in this manuscript that the accuracy of the solutions computed is often not enough.

Additional information

Funding

JSPS KAKENHI [grant number JP22740056, JP26400203].