Stein iterations for the coupled discrete-time Riccati equations. (English) Zbl 1190.65066

Markovian jump linear systems (MJLS) are a class of models used to represent discrete jumps in continuous dynamics. This paper deals with iterative algorithms for computing solutions of a set of coupled algebraic Riccati equations that appears in quadratic optimal control problems for discrete-time MJLS. Two algorithms using decoupled Stein matrix equations are presented. A numerical example illustrates the proposed methods.


65F30 Other matrix algorithms (MSC2010)
65F10 Iterative numerical methods for linear systems
93C55 Discrete-time control/observation systems
65K10 Numerical optimization and variational techniques
49N10 Linear-quadratic optimal control problems
15A24 Matrix equations and identities
Full Text: DOI


[1] Costa, O.L.V; Aya, J.C.C., Temporal difference methods for the maximal solution of discrete-time coupled algebraic ricacti equations, J. optim. theory appl., 109, 289-309, (2001) · Zbl 0984.93051
[2] Costa, O.L.V; Marques, R.P., Maximal and stabilizing Hermitian solutions for discrete-time coupled algebraic ricacti equations, Math. control signals systems, 12, 167-195, (1999) · Zbl 0928.93047
[3] Costa, O.L.V; Fragoso, M.D., Stability results for discrete-time linear systems with Markovian jumping parameters, J. math. anal. appl., 179, 154-178, (1993) · Zbl 0790.93108
[4] Rami, M.; Ghaoui, L., LMI optimization for nonstandard Riccati equations arising in stochastic control, IEEE trans. automat. control., 41, 11, 1666-1671, (1996) · Zbl 0863.93087
[5] Gajic, Z.; Qureshi, M., Lyapunov equation in system stability and control, series mathematics in science and engineering, (1995), Academic Press San Diego, republished by Dover Publications, 2008
[6] Borno, I.; Gajic, Z., Parallel algorithm for solving coupled algebraic Lyapunov equations of discrete-time jump linear systems, Comput. math. appl., 30, 1-4, (1995) · Zbl 0837.93075
[7] Ortega, J.; Rheinboldt, W., Iterative solution of nonlinear equations in several variables, (1970), Academic Press New York · Zbl 0241.65046
[8] Judd, K., Numerical methods in economics, (1998), MIT Press Cambridge · Zbl 0924.65001
[9] Freiling, G.; Hochhaus, A., Properties of the solutions of rational matrix difference equations, Comput. math. appl., 45, 1137-1154, (2003) · Zbl 1055.39033
[10] Abou-Kandil, H.; Freiling, G.; Jank, G., Solution and asymptotic behavior of coupled Riccati equations in jump linear systems, IEEE trans. automat. control., 39, 8, 1631-1636, (1994) · Zbl 0925.93387
[11] Gajic, Z.; Borno, I., Lyapunov iterations for optimal control of jump linear systems at steady state, IEEE trans. automat. control, 40, 1971-1975, (1995) · Zbl 0837.93073
[12] Barraud, A.Y., A numerical algorithm to solve \(A^{\operatorname{T}} X A - X = Q\), IEEE trans. automat. control, AC 22, 883-885, (1977) · Zbl 0361.65022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.