\(H_\infty\) control for discrete-time Markovian jump linear systems with partly unknown transition probabilities.

*(English)*Zbl 1166.93320Summary: 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.

##### 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)
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\textit{L. Zhang} and \textit{E.-K. Boukas}, Int. J. Robust Nonlinear Control 19, No. 8, 868--883 (2009; Zbl 1166.93320)

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