Characterization of the dual problem of linear matrix inequality for H-infinity output feedback control problem via faci
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Characterization of the dual problem of linear matrix inequality for H-infinity output feedback control problem via facial reduction Hayato Waki1
· Noboru Sebe2
Received: 9 October 2019 / Accepted: 2 July 2020 © The Author(s) 2020
Abstract This paper deals with the minimization of H∞ output feedback control. This minimization can be formulated as a linear matrix inequality (LMI) problem via a result of Iwasaki and Skelton 1994. The strict feasibility of the dual problem of such an LMI problem is a valuable property to guarantee the existence of an optimal solution of the LMI problem. If this property fails, then the LMI problem may not have any optimal solutions. Even if one can compute parameters of controllers from a computed solution of the LMI problem, then the computed H∞ norm may be very sensitive to a small change of parameters in the controller. In other words, the non-strict feasibility of the dual tells us that the considered design problem may be poorly formulated. We reveal that the strict feasibility of the dual is closely related to invariant zeros of the given generalized plant. The facial reduction is useful in analyzing the relationship. The facial reduction is an iterative algorithm to convert a non-strictly feasible problem into a strictly feasible one. We also show that facial reduction spends only one iteration for so-called regular H∞ output feedback control. In particular, we can obtain a strictly feasible problem by using null vectors associated with some invariant zeros. This reduction is more straightforward than the direct application of facial reduction. Keywords Linear matrix inequalities · H∞ output feedback control · Facial reduction · Strict feasibility
This work is a full version of [26], which is presented at Proceedings of the SICE International Symposium on Control Systems 2018. The first author was supported by JSPS KAKENHI Grant Numbers JP26400203 and JP17H01700.
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Hayato Waki [email protected] Noboru Sebe [email protected]
1
Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan
2
Department of Intelligent and Control Systems, Kyushu Institute of Technology, Fukuoka, Japan
123
Mathematics of Control, Signals, and Systems
1 Introduction The H∞ control is one of the most important control theories and has attracted much interest from viewpoints of not only the theory, but also application and computation. In particular, two approaches, the algebraic Riccati equations/inequalities approach (e.g., [6]) and linear matrix inequality (LMI) approach (e.g., [8,9,17]) were proposed and investigated thoroughly. These approaches are to find a so-called suboptimal controller, i.e., an admissible controller, so that the H∞ norm of the closed-loop system with a plant and the controller is less than a given value. Naturally, one seeks an optimal controller, i.e., an admissible controller that minimizes the H∞ norm. This paper deals with an LMI formulation of this minimization. The strict feasibility of the dual problem of such an LMI problem is a valu
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