Maximal and stabilizing hermitian solutions for discrete-time coupled algebraic Riccati equations. (English) Zbl 0928.93047

Costa, O. L. V.; Marques, R. P.

Math. Control Signals Syst. 12, No. 2, 167-195 (1999).

Discrete-time coupled algebraic Riccati equations (CARE) that arise in quadratic optimal control and \(H_\infty\)-control of Markovian jump linear systems are considered in this paper. In Section 3, the CARE that arise from the quadratic optimal control problem are considered. The matrix cost is only assumed to be Hermitian. A sufficient condition for the existence of a maximal solution is presented (Theorem 1). Theorem 2 establishes a link between the LMI optimization problem and a maximal solution. Section 4 deals with necessary and sufficient conditions for the existence of a stabilizing solution (Theorem 3 and 4). Section 5 presents a recursive procedure for obtaining a stabilizing solution of the CARE that arise in the \(H_\infty\)-problem, whenever it exists (Theorem 6).

Reviewer: Kunihiko Ichikawa (Tokyo)

MSC:

93D15	Stabilization of systems by feedback
93E15	Stochastic stability in control theory
15A24	Matrix equations and identities
93C55	Discrete-time control/observation systems
93B36	\(H^\infty\)-control
49N10	Linear-quadratic optimal control problems

Keywords:

discrete-time coupled algebraic Riccati equations; quadratic optimal control; \(H_\infty\)-control; Markovian jump linear systems; maximal solution; LMI optimization