Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices.

*(English)*Zbl 1209.93162Summary: This paper focuses on stability and stabilization for a class of continuous-time Markovian jump systems with partial information on transition probability. The free-connection weighting matrix method is proposed to obtain a less conservative stability criterion of Markovian jump systems with partly unknown transition probability or completely unknown transition probability. As a result, a sufficient condition for a state feedback controller design is derived in terms of linear matrix inequalities. Finally, numerical examples are given to illustrate the effectiveness and the merits of the proposed method.

##### MSC:

93E15 | Stochastic stability in control theory |

60J75 | Jump processes (MSC2010) |

15B48 | Positive matrices and their generalizations; cones of matrices |

##### Keywords:

Markovian jump systems; transition rate; free-connection weighting matrix method; linear matrix inequalities
Full Text:
DOI

##### References:

[1] | Boukas, E.K., Stochastic switching systems: analysis and design, (2005), Birkhauser Basel, Berlin · Zbl 1108.93074 |

[2] | Chen, W.H.; Guan, Z.H.; Lu, X.M., Delay-dependent output feedback stabilisation of Markovian jump systems with time-delay, IEE proceedingsâ€”control theory and applications, 151, 5, 561-566, (2004) |

[3] | Kushner, H.J., Stochastic stability and control, (1967), Academic Press New York · Zbl 0178.20003 |

[4] | Li, H.Y.; Chen, B.; Zhou, Q.; Qian, W.Y., Robust stability for uncertain delayed fuzzy Hopfield neural networks with Markovian jumping parameters, IEEE transactions on systems, man and cybernetics, part B (cybernetics), 39, 1, 94-102, (2009) |

[5] | Lou, X.Y.; Cui, B.T., Stochastic exponential stability for Markovian jumping BAM neural networks with time-varying delays, IEEE transactions on systems, man and cybernetics, part B (cybernetics), 37, 3, 713-719, (2007) |

[6] | Mahmoud, M.S., Delay-dependent \(H_\infty\) filtering of a class of switched discrete-time state delay systems, Signal processing, 88, 11, 2709-2719, (2008) · Zbl 1151.93338 |

[7] | Mao, Z.; Jiang, B.; Shi, P., \(H_\infty\) fault detection filter design for networked control systems modelled by discrete Markovian jump systems, IET control theory & applications, 1, 5, 1336-1343, (2007) |

[8] | Martinelli, F., Optimality of a two-threshold feedback control for a manufacturing system with a production dependent failure rate, IEEE transactions on automatic control, 52, 10, 1937-1942, (2007) · Zbl 1366.90091 |

[9] | Shi, P.; Mahmoud, M.; Nguang, S.K.; Ismail, A., Robust filtering for jumping systems with mode-dependent delays, Signal processing, 86, 1, 140-152, (2006) · Zbl 1163.94387 |

[10] | Shu, Z.; Lam, J.; Xu, S.Y., Robust stabilization of Markovian delay systems with delay-dependent exponential estimates, Automatica, 42, 11, 2001-2008, (2006) · Zbl 1113.60079 |

[11] | Skorohod, A.V., Asymptotic methods in the theory of stochastic differential equation, (1989), American Mathematical Society Providence, RI |

[12] | Tao, F.; Zhao, Q., Synthesis of active fault-tolerant control based on Markovian jump system models, IET control theory & applications, 1, 4, 1160-1168, (2007) |

[13] | Wang, G.L.; Zhang, Q.L.; Sreeram, V., Design of reduced-order \(H_\infty\) filtering for Markovian jump systems with mode-dependent time delays, Signal processing, 89, 2, 187-196, (2009) · Zbl 1155.94337 |

[14] | Wu, Z.; Su, H.; Chu, J., \(H_\infty\) model reduction for discrete singular Markovian jump systems, Proceedings of the institution of mechanical engineers, part I, 223, 7, 1017-1025, (2009) |

[15] | Xu, S.Y.; Mao, X.R., Delay-dependent \(H_\infty\) control and filitering for uncertain Markovian jump system with time-varying delay, IEEE transactions on circuits and systems part I: regular papers, 54, 9, 2070-2077, (2007) · Zbl 1374.93134 |

[16] | Zhang, L.X.; Boukas, E.K., Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities, Automatica, 45, 2, 436-468, (2009) · Zbl 1158.93414 |

[17] | Zhang, L.X.; Boukas, E.K., Mode-dependent \(H_\infty\) filtering for discrete-time Markovian jump linear systems with partly unknown transition probabilities, Automatica, 45, 6, 1462-1467, (2009) · Zbl 1166.93378 |

[18] | Zhang, L.X.; Boukas, E.K., \(H_\infty\) control for discrete-time Markovian jump linear systems with partly unknown transition probabilities, International journal of robust and nonlinear control, 19, 8, 868-883, (2009) · Zbl 1166.93320 |

[19] | Zhang, L.X.; Boukas, E.K.; Lam, J., Analysis and synthesis of Markov jump linear systems with time-varying delays and partially known transition probabilities, IEEE transactions on automatic control, 53, 10, 2458-2464, (2009) · Zbl 1367.93710 |

[20] | Zhang, H.G.; Wang, Y.C., Stability analysis of Markovian jumping stochastic cohen – grossberg neural networks with mixed time delays, IEEE transactions on neural networks, 19, 2, 366-370, (2008) |

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.