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Do Val, J. B. R.; Costa, E. F.

Numerical solution for linear-quadratic control problems of Markov jump linear systems and weak detectability concept. (English) Zbl 1026.93056

J. Optimization Theory Appl. 114, No. 1, 69-96 (2002).
A quadratic control problem for weakly detectable continuous time Markov jump linear systems is considered. A method of solving the associated coupled algebraic Riccati equations, based on recursive solution of a set of uncoupled algebraic Riccati equations, is proposed. It is proved that the method converges if and only if the system is mean-square stabilizable. Numerical examples and comparisons illustrating the results are presented.
Reviewer: H.Pragarauskas (Vilnius)

Cited in 3 Documents

MSC:

93E20 Optimal stochastic control
60J75 Jump processes (MSC2010)
93E15 Stochastic stability in control theory
93D15 Stabilization of systems by feedback

Keywords:

numerical methods for stochastic systems; detectability and observability of stochastic systems; optimal control; Markov jump linear systems; coupled algebraic Riccati equations
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References:

[1] ABOU-KANDIL, H., FREILING, G., and JANK, G., Solution and Asymptotic Behavior of Coupled Riccati Equations in Jump Linear Systems, IEEE Transactions on Automatic Control, Vol. 39, pp. 1631–1636, 1994. · Zbl 0925.93387 · doi:10.1109/9.310038
[2] DO VAL, J. B. R., GEROMEL, J. C., and COSTA, O. L. V., Solutions for the Linear-Quadratic Control Problem of Markov Jump Linear Systems, Journal of Optimization Theory and Applications, Vol. 103, pp. 283–311, 1999. · Zbl 0948.49018 · doi:10.1023/A:1021748618305
[3] GAJIC, Z., and BORNO, I., Lyapunov Iterations for Optimal Control of Jump Linear Systems at Steady State, IEEE Transactions on Automatic Control, Vol. 40, pp. 1971–1975, 1995. · Zbl 0837.93073 · doi:10.1109/9.471227
[4] GAJIC, Z., and LOSADA, R., Monotonicity of Algebraic Lyapunov Iterations for Optimal Control of Jump Parameter Linear Systems, Systems and Control Letters, Vol. 41, pp. 175–181, 2000. · Zbl 0985.93017 · doi:10.1016/S0167-6911(00)00051-7
[5] RAMI, M. A., and GHAOUI, L. E., LMI Optimization for Nonstandard Riccati Equations Arising in Stochastic Control, IEEE Transactions on Automatic Control, Vol. 41, pp. 1666–1671, 1996. · Zbl 0863.93087 · doi:10.1109/9.544005
[6] JI, Y., and CHIZECK, H. J., Controllability, Stabilizability, and Continuous Time Markovian Jump Linear-Quadratic Control, IEEE Transactions on Automatic Control, Vol. 35, pp. 777–788, 1990. · Zbl 0714.93060 · doi:10.1109/9.57016
[7] COSTA, E. F., and DO VAL, J. B. R., On the Detectability and Observability of Continuous-Time Markov Jump Linear Systems, Proceedings of the 2001 European Control Conference, 2001. · Zbl 0986.93008
[8] COSTA, O. L. V., and FRAGOSO, M. D., Discrete-Time LQ-Optimal Control Problems for Infinite Markov Jump Parameter Systems, IEEE Transactions on Automatic Control, Vol. 40, pp. 2076–2088, 1995. · Zbl 0843.93091 · doi:10.1109/9.478328
[9] JI, Y., and CHIZECK, H. J., Jump Linear-Quadratic Gaussian Control: Steady-State Solution and Testable Conditions, Control Theory and Advanced Technology, Vol. 6, pp. 289–319, 1990.
[10] COSTA, E. F., and DO VAL, J. B. R., On the Detectability and Observability of Continuous-Time Markov Jump Linear Systems, SIAM Journal on Control and Optimization (submitted). · Zbl 1029.93007
[11] COSTA, O. L. V., DO VAL, J. B. R., and GEROMEL, J. C., Continuous-Time State-Feedback H 2-Control of Markovian Jump Linear Systems via Convex Analysis, Automatica, Vol. 35, pp. 259–268, 1999. · Zbl 0939.93041 · doi:10.1016/S0005-1098(98)00145-9
[12] DE JAGER, B., Linear and Nonlinear Control: An Example, Proceedings of the 1997 Automatic Control Conference, Albuquerque, New Mexico, pp. 396–397, 1997.
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