TY - JOUR

T1 - Reduction of SISO H-infinity output feedback control problem

AU - Waki, Hayato

AU - Ebihara, Yoshio

AU - Sebe, Noboru

N1 - Funding Information:
This study was supported by Toyota Riken Specially Promoted Research Program in 2018 (PI: Yoshio Ebihara ( Toyota Physical and Chemical Research Institute )). The first author was supported by JSPS KAKENHI Grant Numbers JP26400203 , JP17H01700 and JP20K11696 .

PY - 2021/2/1

Y1 - 2021/2/1

N2 - We consider the linear matrix inequality (LMI) problem of H∞ output feedback control problem for a generalized plant whose control input, measured output, disturbance input, and controlled output are scalar. We provide an explicit form of the optimal value. This form is the unification of some results in the literature of H∞ performance limitation analysis. To obtain the form of the optimal value, we focus on the non-uniqueness of perpendicular matrices, which appear in the LMI problem. We use the null vectors of invariant zeros associated with the dynamical system for the expression of the perpendicular matrices. This expression enables us to reduce and simplify the LMI problem. Our approach uses some well-known fundamental tools, e.g., the Schur complement, Lyapunov equation, Sylvester equation, and matrix completion. We use these techniques for the simplification of the LMI problem. Also, we investigate the structure of dual feasible solutions and reduce the size of the dual. This reduction is called a facial reduction in the literature of convex optimization.

AB - We consider the linear matrix inequality (LMI) problem of H∞ output feedback control problem for a generalized plant whose control input, measured output, disturbance input, and controlled output are scalar. We provide an explicit form of the optimal value. This form is the unification of some results in the literature of H∞ performance limitation analysis. To obtain the form of the optimal value, we focus on the non-uniqueness of perpendicular matrices, which appear in the LMI problem. We use the null vectors of invariant zeros associated with the dynamical system for the expression of the perpendicular matrices. This expression enables us to reduce and simplify the LMI problem. Our approach uses some well-known fundamental tools, e.g., the Schur complement, Lyapunov equation, Sylvester equation, and matrix completion. We use these techniques for the simplification of the LMI problem. Also, we investigate the structure of dual feasible solutions and reduce the size of the dual. This reduction is called a facial reduction in the literature of convex optimization.

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U2 - 10.1016/j.laa.2020.09.034

DO - 10.1016/j.laa.2020.09.034

M3 - Article

AN - SCOPUS:85092453733

VL - 610

SP - 321

EP - 378

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -