×

zbMATH — the first resource for mathematics

The \(H_2\)-control for jump linear systems: Cluster observations of the Markov state. (English) Zbl 0991.93125
Summary: The \(H_2\)-norm control problem of discrete-time Markov jump linear system is addressed in this paper when part of, or the total of, the Markov states is not accessible to the controller. The non-observed part of the Markov states is grouped in a number of clusters of observations; the case with a single cluster recovers the situation of when no Markov state is observed. The control action is provided in linear feedback form, which is invariant on each cluster, and this restricted complexity setting is adopted, aiming at computable solutions. We explore a recent result by M. C. de Oliveira J. Bernussou and J. C. Geromel [Syst. Control Lett. 37, 261-265 (1999; Zbl 0948.93058)] involving an LMI characterization to establish an \(H_2\) solution that is stabilizing in the mean square sense. The novelty of the method is that it can handle in LMI form the situation ranging from no Markov state observation to complete state observation. In addition, when the state observation is complete, the optimal \(H_2\)-norm solution is retrieved.

MSC:
93E20 Optimal stochastic control
60J75 Jump processes (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abou-Kandil, H.; Freiling, G.; Jank, G., Solution and asymptotic behavior of coupled Riccati equations in jump linear systems, IEEE transactions on automatic control, 39, 1631-1636, (1994) · Zbl 0925.93387
[2] Caines, P.E.; Zhang, J., On the adaptive control of jump parameter systems via nonlinear filtering, SIAM journal on control and optimization, 33, 1758-1777, (1995) · Zbl 0843.93076
[3] Costa, 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, International journal of control, 66, 557-579, (1997) · Zbl 0951.93536
[4] Costa, O.L.V.; Fragoso, M.D., Stability results for discrete-time linear systems with Markovian jumping parameters, Journal of mathematical analysis and applications, 179, 154-178, (1993) · Zbl 0790.93108
[5] de Farias, D.P.; Geromel, J.C.; do Val, J.B.R.; Costa, O.L.V., Output feedback control of Markov jump linear systems in continuous-time, IEEE transactions on automatic control, 45, 944-949, (2000) · Zbl 0972.93074
[6] de Oliveira, M.C.; Bernussou, J.; Geromel, J.C., A new discrete-time robust stability condition, Systems and control letters, 37, 261-265, (1999) · Zbl 0948.93058
[7] do Val, J.B.R.; Basar, T., Receding horizon control of jump linear systems and a macroeconomic policy problem, Journal of economic dynamics and control, 23, 1099-1131, (1999) · Zbl 0962.91058
[8] Geromel, J.C.; Peres, P.L.D.; Souza, S.R., H2-guaranteed cost control for uncertain discrete-time linear systems, International journal of control, 57, 853-864, (1993) · Zbl 0772.93054
[9] Ji, Y.; Chizeck, H.J., Jump linear quadratic Gaussian control: steady-state solution and testable conditions, Control theory and advanced technology, 6, 289-319, (1990)
[10] Ji, Y.; Chizeck, H.J., Jump linear quadratic Gaussian control in continuous time, IEEE transactions on automatic control, 37, 1884-1892, (1992) · Zbl 0773.93052
[11] Oliveira, M. C., Faria, D. P., & Geromel, J. C. (1997). LMIsol user’s guide. Available at http://dt.fee.unicamp.br/∼carvalho/#soft.
[12] Pan, G.; Bar-Shalom, Y., Stabilization of jump linear Gaussian systems without mode observations, International journal of control, 64, 631-661, (1996) · Zbl 0857.93095
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.