Uncoupled Riccati Iterations For The Linear Quadratic Control Problem Of Discrete-time Markov Jump Linear Systems

This paper deals with recursive methods for solving coupled Riccati equations arising in the linear quadratic control for Markovian jump linear systems. Two algorithms, based on solving uncoupled Riccati equations at each iteration, are presented. The standard method for this problem relies on finite stage approximations with receding horizon, whereas the methods presented here are based on sequences of stopping times to define the terminal time of the approximating control problems. The methods can be ordered in terms of rate of convergence. Comparisons with other methods in the current literature are also presented.431217271733Abou-Kandil, H., Freiling, G., Jank, G., On the solution of discrete-time Markovian jump linear quadratic control problems (1995) Automatica, 31 (5), pp. 765-768Ait-Rami, M., Ghaoui, L.E., LMI optimization for nonstandard Riccati equations arising in stochastic control (1996) IEEE Trans. Automat. Contr., 41, pp. 1666-1671. , NovCosta, O.L.V., Do Val, J.B.R., Geromel, J.C., A convex programming approach to H2-control of discrete-time Markovian jump linear systems (1997) Int. J. Contr., 66, pp. 557-579Costa, O.L.V., Fragoso, M.D., Stability results for discrete-time linear systems with Markovian jumping parameters (1993) J. Math. Analysis and Appl., 179, pp. 154-178Discrete-time LQ-optimal control problems for infinite Markov jump parameter systems (1995) IEEE Trans. Automat. Contr., 40, pp. 2076-2088Davis, M.H.A., Vinter, R.B., (1985) Stochastic Modeling and Control, , London, U.K.: Chapman and HallEl Ghaoui, L., Nikoukhah, R., Delebecque, F., LMITOOL: A Front-End for LMI Optimization - User's Guide, , ftp.ensa.fr/pub/elghaoui/lmitoolFragoso, M.D., Ribeiro Do Val, J.B., Pinto Jr., D.L., Jump linear H∞-control: The discrete-time case (1995) Contr. Th. and Adv. Tech., 10, pp. 1459-1474Gajic, Z., Borno, I., Lyapunov iterations for optimal control of jump linear systems at steady state (1995) IEEE Trans. Automat. Contr., 40 (11), pp. 1971-11075Geromel, J.C., Peres, P.L.D., Souza, S.R., H2-guaranteed cost control for uncertain discrete-time linear systems (1993) Int. J. Contr., 57, pp. 853-864Ji, Y., Chizeck, H.J., Controllability, observability and discrete-time Markovian jump linear quadratic control (1988) Int. J. Contr., 48, pp. 481-498Ji, Y., Chizeck, H.J., Feng, X., Loparo, K.A., Stability and control of discrete-time jump linear systems (1991) Control Th. and Adv. Tech., 7, pp. 247-270Laub, A., Algebraic aspects of generalized eigenvalue problems for solving Riccati equations (1986) Computational and Combinatorial Methods in Systems Theory, pp. 213-227. , C. Byrnes and A. Lindquist, Eds. Amsterdam, The Netherlands: ElsevierMariton, M., (1990) Jump Linear Systems in Automatic Control, , New York: Marcel DekkerVandenberghe, L., Boyd, S., Software for Semidefinite Programming - User's Guide, , isl.stanford.edu/pub/boyd/semidef_progWimmer, H.K., Monotonicity and maximality of solutions of discrete-time algebraic Riccati equations (1992) J. Math. Syst., Estimation and Contr., 2 (2), pp. 219-23