##
**Finite-time analysis and \(H_\infty\) control for switched stochastic systems.**
*(English)*
Zbl 1273.93173

Summary: This paper is concerned with the finite-time stability, boundedness and \(H_\infty\) control problems for a class of switched stochastic systems. Using the average dwell time method and the multiple Lyapunov-like function technique, some sufficient conditions are proposed to guarantee the finite-time properties for the switched stochastic systems in the form of matrix inequalities. Also, a state feedback controller for the finite-time \(H_\infty\) control problem is obtained. An example is employed to verify the effectiveness of the proposed method.

### MSC:

93E15 | Stochastic stability in control theory |

93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |

93B36 | \(H^\infty\)-control |

### Keywords:

finite-time stability; boundedness; \(H_\infty\) control; switched stochastic systems; average dwell time method; multiple Lyapunov-like function technique; state feedback controller
PDFBibTeX
XMLCite

\textit{Z. Xiang} et al., J. Franklin Inst. 349, No. 3, 915--927 (2012; Zbl 1273.93173)

### References:

[1] | A. Balluchi, M. Di Benedetto, C. Pinello, C. Rossi, A. Sangiovanni-Vincentelli, Cut-off in engine control: a hybrid system approach, Proceedings of the 36th IEEE Conference on Decision and Control, 1997, pp. 4720-4725.; A. Balluchi, M. Di Benedetto, C. Pinello, C. Rossi, A. Sangiovanni-Vincentelli, Cut-off in engine control: a hybrid system approach, Proceedings of the 36th IEEE Conference on Decision and Control, 1997, pp. 4720-4725. |

[2] | B.E. Bishop, M.W. Spong, Control of redundant manipulators using logic-based switching, Proceedings of the 37th IEEE Conference on Decision and Control, 1998, pp. 1488-1493.; B.E. Bishop, M.W. Spong, Control of redundant manipulators using logic-based switching, Proceedings of the 37th IEEE Conference on Decision and Control, 1998, pp. 1488-1493. |

[3] | Zhang, W. A.; Yu, L.; Yin, S., A switched system approach to \(H_∞\) control of networked control systems with time-varying delays, Journal of the Franklin Institute, 348, 2, 165-178 (2011) · Zbl 1214.93044 |

[4] | Morse, A. S., Supervisory control of families of linear set-point controllers, Part1:exact matching, IEEE Transactions on Automatic Control, 41, 10, 1413-1431 (1996) · Zbl 0872.93009 |

[5] | Sun, X. M.; Zhao, J.; Hill, D. J., Stability and \(L_2\) gain analysis for switched delay systems: A delay-dependent method, Automatica, 42, 10, 1769-1774 (2006) · Zbl 1114.93086 |

[6] | Phat, V. N., Switched controller design for stabilization of nonlinear hybrid systems with time-varying delays in state and control, Journal of the Franklin Institute, 347, 1, 195-207 (2010) · Zbl 1298.93290 |

[7] | Chiou, J. S.; Wang, C. J.; Cheng, C. M., On delay-dependent stabilization analysis for the switched time-delay systems with the state-driven switching strategy, Journal of the Franklin Institute, 348, 2, 261-276 (2011) · Zbl 1218.34091 |

[8] | Mahmoud, M. S.; Nounou, H. N.; Xia, Y., Robust dissipative control for internet-based switching systems, Journal of the Franklin Institute, 347, 1, 154-172 (2010) · Zbl 1298.93138 |

[9] | Wu, L. G.; Wang, Z. D., Guaranteed cost control of switched systems with neutral delay via dynamic output feedback, International Journal of Systems Science, 40, 7, 717-728 (2009) · Zbl 1291.93138 |

[10] | Zhai, G. S.; Hu, B.; Yasuda, K.; Michel, A. N., Disturbance attenuation properties of time-controlled switched systems, Journal of the Franklin Institute, 338, 7, 765-779 (2001) · Zbl 1022.93017 |

[11] | Sun, Y. G.; Wang, L.; Xie, G. M., Delay-dependent robust stability and \(H_∞\) control for uncertain discrete-time switched systems with mode-dependent time delays, Applied Mathematics and Computation, 187, 2, 1228-1237 (2007) · Zbl 1114.93075 |

[12] | Lin, H.; Antsaklis, P. J., Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results, IEEE Transactions on Automatic Control, 54, 2, 308-322 (2009) · Zbl 1367.93440 |

[13] | Filipovic, V., Exponential stability of stochastic switched systems, Transactions of the Institute of Measurement and Control, 31, 2, 205-212 (2009) |

[14] | Liu, J.; Liu, X. Z.; Xie, W. C., Exponential stability of switched stochastic delay systems with non-linear uncertainties, International Journal of Systems Science, 40, 6, 637-648 (2009) · Zbl 1291.93317 |

[15] | Wu, L.; Ho, D. W.C.; Li, C. W., Stabilisation and performance synthesis for switched stochastic systems, IET Control Theory and Applications, 4, 10, 1877-1888 (2010) |

[16] | Feng, W.; Tian, J.; Zhao, P., Stability analysis of switched stochastic systems, Automatica, 47, 1, 148-157 (2011) · Zbl 1209.93157 |

[17] | Xiang, Z. R.; Wang, R. H.; Chen, Q. W., Robust stabilization of uncertain stochastic switched non-linear systems under asynchronous switching, Proceedings of the Institution of Mechanical Engineers Part I: Journal of Systems and Control Engineering, 225, 1, 8-20 (2011) |

[18] | P. Doroto, Short time stability in linear time-varying systems, Proceeding of the IRE International Convention Record, 1961, pp. 83-87.; P. Doroto, Short time stability in linear time-varying systems, Proceeding of the IRE International Convention Record, 1961, pp. 83-87. |

[19] | Amato, F.; Ariola, M.; Dorato, P., Finite-time control of linear systems subject to parametric uncertainties and disturbances, Automatica, 37, 9, 1459-1463 (2001) · Zbl 0983.93060 |

[20] | Amato, F.; Ariola, M.; Cosentino, C., Finite-time stabilization via dynamic output feedback, Automatica, 42, 2, 337-342 (2006) · Zbl 1099.93042 |

[21] | Zhang, W. H.; An, X. Y., Finite-time control of linear stochastic systems, International Journal of Innovative Computing, Information and Control, 4, 3, 689-696 (2008) |

[22] | Amato, F.; Ariola, M.; Cosentino, C., Finite-time control of discrete-time linear systems: Analysis and design conditions, Automatica, 46, 5, 919-924 (2010) · Zbl 1191.93099 |

[23] | Amato, F.; Ariola, M.; Cosentino, C., Finite-time stability of linear time-varying systems: analysis and controller design, IEEE Transactions on Automatic Control, 55, 4, 1003-1008 (2010) · Zbl 1368.93457 |

[24] | Lin, X.; Du, H.; Li, S., Finite-time boundedness and \(L_2\) gain analysis for switched delay systems with norm-bounded disturbance, Applied Mathematics and Computation, 217, 12, 5982-5993 (2011) · Zbl 1218.34082 |

[25] | Xiang, W.; Xiao, J., \(H_∞\) finite-time control for switched nonlinear discrete-time systems with norm-bounded disturbance, Journal of the Franklin Institute, 348, 2, 331-352 (2011) · Zbl 1214.93043 |

[26] | H. Du, X. Lin, S. Li, Finite-time stability and stabilization of switched linear systems, Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, 2009, pp. 1938-1943.; H. Du, X. Lin, S. Li, Finite-time stability and stabilization of switched linear systems, Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, 2009, pp. 1938-1943. |

[27] | Orlov, Y., Finite time stability and robust control synthesis of uncertain switched systems, SIAM Journal on Control and Optimization, 43, 4, 1253-1271 (2004) · Zbl 1085.93021 |

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.