TY - JOUR

T1 - Characterization of the dual problem of linear matrix inequality for H-infinity output feedback control problem via facial reduction

AU - Waki, Hayato

AU - Sebe, Noboru

N1 - Funding Information:
The first author would like to thank Dr. Yoshiyuki Sekiguchi at Tokyo University of Marine Science and Technology for a fruitful comment for improving the presentation of this paper. Also, the first author would like to thank a reviewer of a paper for pointing out 1 of Remark 3.
Funding Information:
This work is a full version of [], 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.
Publisher Copyright:
© 2020, The Author(s).

PY - 2020/9/1

Y1 - 2020/9/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s00498-020-00261-z

DO - 10.1007/s00498-020-00261-z

M3 - Article

AN - SCOPUS:85087762239

VL - 32

SP - 361

EP - 384

JO - Mathematics of Control, Signals, and Systems

JF - Mathematics of Control, Signals, and Systems

SN - 0932-4194

IS - 3

ER -