General Lq Problem For Infinite Jump Linear Systems And The Minimal Solution Of Algebraic Riccati Equations

The paper addresses the LQ control problem for systems with countable Markov jump parameters, and the associated coupled algebraic Riccati equations. The problem is considered in a general optimization setting in which the solution is not required to be stabilizing in any sense. We show that a necessary and sufficient condition for a solution to the control problem to exist is that the Riccati equations have a nonempty set of solutions, which generalizes previous known results requiring stabilizability as a sufficient condition.We clarify the connection between the minimal solution of the Riccati equation and the control problem, showing that the minimal solution provides the synthesis of the optimal control. The derived results strengthen the relations of the theory of Markov jump systems with the one of linear deterministic systems. An illustrative example is included. Copyright © 2005 IFAC.1616Athans, M., Castanon, D., Dunn, K.P., Greene, C.S., Lee, W.H., Sandell, N.R., Willsky, A.S., The stochastic control of the F- 8C aircraft using a multiple model adaptive control (MMAC) method - Part i: Equilibrium flight (1977) IEEE Transactions on Automatic Control, 22, pp. 768-780Costa, E.F., Do Val, J.B.R., On the detectability and observability of continuous-time Markov jump linear systems (2002) SIAM Journal on Control and Optimization, 41 (4), pp. 1295-1314Costa, E.F., Do Val, J.B.R., Weak detectability and the linear quadratic control problem of discrete-time Markov jump linear systems (2002) International Journal of Control. Special Issue on Switched and Polytopic Linear Systems, 75, pp. 1282-1292Costa, E.F., Do Val, J.B.R., An algorithm for solving a perturbed algebraic Riccati equation European Journal of Control, , To appear in theCosta, E.F., Do Val, J.B.R., Fragoso, M.D., A new approach to detectability of discrete-time infinite Markov jump linear systems SIAM Journal on Control and Optimization, , To appear in theCosta, E.F., Do Val, J.B.R., Fragoso, M.D., On a detectability concept of discrete-time infinite Markov jump linear systems Stochastic Analysis and Applications, , To appear in theCosta, O.L.V., Fragoso, M.D., Discrete-time LQ-optimal control problems for infinite Markov jump parameter systems (1995) IEEE Transactions on Automatic Control, AC-40, pp. 2076-2088Costa, O.L.V., Marques, R.P., Robust H 2-control for discrete-time Markovian jump linear systems (2000) International Journal of Control, 73 (1), pp. 11-21Costa, O.L.V., Val, J.B.R.D., Geromel, J.C., A convex programming approach to H 2 control of discrete-time Markovian jump linear systems (1997) Int. J. Control, 66 (4), pp. 557-579Costa, O.L.V., Do Val, J.B.R., Geromel, J.C., Continuous-time state-feedback H 2-control of Markovian jump linear systems via convex analysis (1999) Automatica, 35, pp. 259-268Do Val, J.B.R., Costa, E.F., Numerical solution for the linear-quadratic control problem of Markov jump linear systems and a weak detectability concept (2002) Journal of Optimization Theory and Applications, 114 (1), pp. 69-96Do Val, J.B.R., Geromel, J.C., Costa, O.L.V., Solutions for the linear quadratic control problem of Markov jump linear systems (1999) Journal of Optimization Theory and Applications, 103, pp. 283-311Fragoso, M., Baczynski, J., Optimal control for continuous time LQ problems with infinite Markov jump parameters (2001) SIAM Journal on Control and Optimization, 40 (1), pp. 270-297Ji, Y., Chizeck, H.J., Controlability, stabilizability and continuous time Markovian jump linear quadratic control (1990) IEEE Transactions on Automatic Control, 35 (7), pp. 777-788Kucera, V., On nonnegative definite solutions to matrix quadratic equations (1972) Automatica, 8, pp. 413-423Martensson, K., On the matrix Riccati equation (1971) Inform. Sci., 3, pp. 17-49Molinari, B.P., The time-invariant linear-quadratic optimal control problem (1977) Automatica, 13, pp. 347-357Rami, M.A., Ghaoui, L.E., LMI optimization for nonstandard Riccati equations arising in stochastic control (1996) IEEE Transactions on Automatic Control, 41 (11), pp. 1666-1671Sworder, D.D., Rogers, R.O., An LQ-solution to a control problem associated with a solar thermal central receiver (1983) IEEE Transactions on Automatic Control, 28 (10), pp. 971-97