Zhang, Lixian; Boukas, El-Kébir \(H_\infty\) control for discrete-time Markovian jump linear systems with partly unknown transition probabilities. (English) Zbl 1166.93320 Int. J. Robust Nonlinear Control 19, No. 8, 868-883 (2009). Summary: The problem of \(H_\infty\) control for a class of discrete-time Markovian jump linear system with partly unknown transition probabilities is investigated. The class of systems under consideration is more general, which covers the systems with completely known and completely unknown transition probabilities as two special cases. Moreover, in contrast to the uncertain transition probabilities studied recently, the concept of partly unknown transition probabilities proposed in this paper does not require any knowledge of the unknown elements. The \(H_\infty\) controllers to be designed include state feedback and dynamic output feedback, since the latter covers the static one. The sufficient conditions for the existence of the desired controllers are derived within the matrix inequalities framework, and a cone complementary linearization algorithm is exploited to solve the latent equation constraints in the output-feedback control case. Two numerical examples are provided to show the validness and potential of the developed theoretical results. Cited in 90 Documents MSC: 93B36 \(H^\infty\)-control 60J75 Jump processes (MSC2010) 93C05 Linear systems in control theory 93E03 Stochastic systems in control theory (general) Keywords:Markovian jump linear systems; \(H_\infty\) control; partly unknown transition probabilities; linear matrix inequality (LMI) × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Boukas, Stochastic Switching Systems: Analysis and Design (2005) [2] Costa, Discrete-time Markovian Jump Linear Systems (2005) · doi:10.1007/b138575 [3] Tanikawa, On new smoothing algorithms for discrete-time linear stochastic systems with unknown disturbances, International Journal of Innovative Computing, Information and Control 4 (1) pp 15– (2008) [4] Xu, Robust H control for uncertain discrete-time stochastic bilinear systems with Markovian switching, International Journal of Robust and Nonlinear Control 15 (5) pp 201– (2005) · Zbl 1078.93025 [5] Xu, Robust H filtering for uncertain Markovian jump systems with mode-dependent time-delays, IEEE Transactions on Automatic Control 48 (5) pp 900– (2003) · Zbl 1364.93816 [6] Xu, Robust H filtering for a class of nonlinear discrete-time Markovian jump systems, Journal of Optimization Theory and Applications 122 (3) pp 651– (2004) [7] Karan, Transition probability bounds for the stochastic stability robustness of continuous- and discrete-time Markovian jump linear systems, Automatica 42 (12) pp 2159– (2006) · Zbl 1104.93056 [8] Xiong, On robust stabilization of Markovian jump systems with uncertain switching probabilities, Automatica 41 (5) pp 897– (2005) · Zbl 1093.93026 [9] Krtolica, Stability of linear feedback systems with random communication delays, International Journal of Control 59 (4) pp 925– (1994) · Zbl 0812.93073 [10] Seiler, An H approach to networked control, IEEE Transactions on Automatic Control 50 (3) pp 356– (2005) · Zbl 1365.93147 [11] Zhang, A new method for stabilization of networked control systems with random delays, IEEE Transactions on Automatic Control 50 (8) pp 1177– (2005) [12] Wang, Exponential stability analysis for discrete-time switched linear systems with time-delay, International Journal of Innovative Computing, Information and Control 3 (6) pp 1557– (2007) [13] Zhang, Robust H filtering for switched linear discrete-time systems with polytopic uncertainties, International Journal of Adaptive Control and Signal Processing 20 (6) pp 291– (2006) · Zbl 1127.93324 [14] El Ghaoui, A cone complementarity linearization algorithm for static output-feedback and related problems, IEEE Transactions on Automatic Control 42 (8) pp 1171– (1997) · Zbl 0887.93017 [15] Boukas, Robust H control of discrete-time Markovian jump linear systems with mode-dependent time-delays, IEEE Transactions on Automatic Control 46 (12) pp 1918– (2001) [16] Cao, Robust H control of uncertain Markovian jump systems with time-delay, IEEE Transactions on Automatic Control 45 (1) pp 77– (2000) 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.