##
**Robust finite-time \(H_\infty\) control for Markovian jump systems with partially known transition probabilities.**
*(English)*
Zbl 1293.93773

Summary: This paper is concerned with the problem of robust finite-time \(H_\infty\) control for Markovian jump systems with partially known transition probabilities. Sufficient conditions ensuring the finite-time boundedness, \(H_\infty\) finite-time boundedness and finite-time \(H_\infty\) state feedback stabilization are, respectively, developed for the given system for the first time. A robust finite-time \(H_\infty\) state feedback controller is also designed, which guarantees the \(H_\infty\) finite-time boundedness of the closed-loop system. Seeking computational convenience, all the conditions are cast in the format of linear matrix inequalities. Finally a numerical example is provided to demonstrate the effectiveness of the main results.

### MSC:

93E15 | Stochastic stability in control theory |

93D15 | Stabilization of systems by feedback |

60J75 | Jump processes (MSC2010) |

93B35 | Sensitivity (robustness) |

### Keywords:

robust finite-time \(H_\infty\) control; Markovian jump systems; partially known transition probabilities; finite-time boundedness; state feedback stabilization
PDFBibTeX
XMLCite

\textit{G. Zong} et al., J. Franklin Inst. 350, No. 6, 1562--1578 (2013; Zbl 1293.93773)

Full Text:
DOI

### References:

[1] | Chen, W. H.; Xu, J. X.; Guan, Z. H., Guaranteed cost control for uncertain Markovian jump systems with mode-dependent time-delays, IEEE Transactions on Automatic Control, 48, 12, 2270-2277 (2003) · Zbl 1364.93369 |

[2] | Shen, L. J.; Buscher, U., Solving the serial batching problem in job shop manufacturing systems, European Journal of Operational Research, 221, 1, 14-26 (2012) · Zbl 1253.90119 |

[3] | Assawinchaichote, W.; Nguang, S. K.; Shi, P., Robust \(H_\infty\) fuzzy filter design for uncertain nonlinear singularly perturbed systems with Markovian jumps: an LMI approach, Information Sciences, 177, 7, 1699-1714 (2007) · Zbl 1113.93078 |

[4] | Athans, M., Command and control theorya challenge to control science, IEEE Transactions on Automatic Control, 32, 4, 286-293 (1987) |

[5] | Wu, H. N.; Cai, K. Y., Mode-independent robust stabilization for uncertain Markovian jump nonlinear systems via fuzzy control, IEEE Transactions on Systems, Man and Cybernetics, Part BCybernetics, 36, 3, 509-519 (2006) |

[6] | Wu, H. N.; Cai, K. Y., Robust fuzzy control for uncertain discrete-time nonlinear Markovian jump systems without mode observations, Information Sciences, 177, 6, 1509-1522 (2007) · Zbl 1120.93337 |

[7] | Liu, F.; Cai, Y., Passive analysis and synthesis of Markovian jump systems with norm bounded uncertainty and unknown delay, Dynamics of Continuous, Discrete & Impulsive Systems. Series AMathematical Analysis, 13A, 1, 157-166 (2006) |

[8] | Liu, M.; Ho, D. W.C.; Niu, Y. G., Stabilization of Markovian jump linear system over networks with random communication delay, Automatica, 45, 2, 416-421 (2009) · Zbl 1158.93412 |

[9] | Zhang, L. X.; Boukas, E. K., Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities, Automatica, 45, 2, 463-468 (2009) · Zbl 1158.93414 |

[10] | Luan, X. L.; Liu, F.; Shi, P., Finite-time filtering for non-linear stochastic systems with partially known transition jump rates, IET Control Theory & Applications, 4, 5, 735-745 (2010) |

[11] | Yin, Y. Y.; Liu, F.; Shi, P., Finite-time gain-scheduled control on stochastic bioreactor systems with partially known transition jump rates, Circuits, Systems and Signal Processing, 30, 3, 609-627 (2011) · Zbl 1213.93162 |

[12] | Zhang, Y.; He, Y.; Wu, M.; Zhang, J., Stabilization for Markovian jump systems with partial information on transition probability based on free-weighting matrices, Automatica, 47, 1, 79-84 (2011) · Zbl 1209.93162 |

[14] | Weiss, L.; Infante, E. F., Finite-time stability under perturbing forces and on product spaces, IEEE Transactions on Automatic Control, 12, 1, 54-59 (1967) · Zbl 0168.33903 |

[16] | Amato, F.; Ariola, M., Finite-time control of discrete-time linear systems, IEEE Transactions on Automatic Control, 50, 5, 724-729 (2005) · Zbl 1365.93182 |

[17] | Zhang, Y. S.; Xu, S. Y.; Zhang, J. H., Delay-dependent robust \(H_\infty\) control for uncertain fuzzy Markovian jump systems, International Journal of Control, Automation and Systems, 7, 4, 520-529 (2009) |

[18] | Gao, J.; Huang, B.; Wang, Z., LMI-based robust \(H_\infty\) control for uncertain linear Markovian jump systems with time-delays, Automatica, 37, 7, 1141-1146 (2001) · Zbl 0989.93029 |

[19] | He, S. P.; Liu, F., Unbiased \(H_\infty\) filtering for neutral Markov jump systems, Applied Mathematics and Computation, 206, 1, 175-185 (2008) · Zbl 1152.93052 |

[20] | He, S. P.; Liu, F., Stochastic finite-time stabilization for uncertain jump systems via state feedback, Journal of Dynamic Systems, Measurement and Control, 132, 3, 034504 (2010) |

[21] | He, S. P.; Liu, F., Robust finite-time \(H_\infty\) control of stochastic jump systems, International Journal of Control, Automation and Systems, 8, 9, 1336-1341 (2010) |

[22] | Luan, X. L.; Liu, F.; Shi, P., Neural-network-based finite-time \(H_\infty\) control for extended Markov jump nonlinear systems, International Journal of Adaptive Control and Signal Processing, 24, 7, 554-567 (2010) · Zbl 1200.93125 |

[23] | Li, H. Y.; Zhou, Q.; Chen, B.; Liu, H. H., Parameter-dependent robust stability for uncertain Markovian jump systems with time delay, Journal of the Franklin Institute, 348, 4, 738-748 (2011) · Zbl 1227.93126 |

[24] | Shi, Y.; Yu, B., Output feedback stabilization of networked control systems with random delays modeled by Markov chains, IEEE Transactions on Automatic Control, 54, 7, 1668-1674 (2009) · Zbl 1367.93538 |

[25] | Shi, Y.; Yu, B., Robust mixed \(H_2 / H_\infty\) control of networked control systems with random time delays in both forward and backward communication links, Automatica, 47, 4, 754-760 (2011) · Zbl 1215.93045 |

[26] | Li, H.; Shi, Y., Robust \(H_\infty\) filtering for nonlinear stochastic systems with uncertainties and Markov delays, Automatica, 48, 1, 159-166 (2012) · Zbl 1244.93158 |

[27] | Mao, X. R., Stability of stochastic differential equations with Markovian switching, Stochastic processes and their Applications, 79, 1, 45-67 (1999) · Zbl 0962.60043 |

[28] | Wang, Y. Y.; Xie, L. H.; Souza, C. E., Robust control of a class of uncertain nonlinear systems, Systems and Control Letters, 19, 2, 139-149 (1992) · Zbl 0765.93015 |

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.