Robust delay-range-dependent stabilization for Markovian jump systems with mode-dependent time delays and nonlinearities.

*(English)*Zbl 1204.93104Summary: This paper discusses the robust stabilization problem for a class of Markovian jump systems with nonlinear disturbances and time delays, which are time-varying in intervals and depend on system mode. By exploiting a new Lyapunov-Krasovskii functional, which takes into account the range of delay and by making use of novel techniques, a mean-square exponential stability result is proposed. Based on the obtained stability condition, a sufficient condition for state feedback controller which stabilizes the system and maximizes the bound on nonlinear perturbations is derived in terms of linear matrix inequalities involving a convex optimization. Finally, illustrative examples are presented to show the benefits and effectiveness of the proposed approaches.

##### MSC:

93D21 | Adaptive or robust stabilization |

93C10 | Nonlinear systems in control theory |

93D09 | Robust stability |

60J75 | Jump processes (MSC2010) |

93E03 | Stochastic systems in control theory (general) |

##### Keywords:

Markovian jump systems; delay-range-dependent; nonlinear disturbances; mode-dependent time delays
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\textit{G. Wang} et al., Optim. Control Appl. Methods 31, No. 3, 249--264 (2010; Zbl 1204.93104)

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##### References:

[1] | 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) |

[2] | Xu, Delay-dependent H control and filtering for uncertain Markovian jump systems with time-varying delays, IEEE Transactions on Circuits and Systems 54 (9) pp 2070– (2007) · Zbl 1374.93134 |

[3] | Mahmoud, Simultaneous H2/H control of uncertain jump systems with functional time-delays, International Journal of Robust and Nonlinear Control 18 (3) pp 296– (2008) |

[4] | 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 |

[5] | Chen, Delay-dependent stability and H control of uncertain discrete-time Markovian jump systems with mode-dependent time delays, Systems and Control Letters 52 (5) pp 361– (2004) · Zbl 1157.93438 |

[6] | Shao, Delay-range-dependent robust H filtering for uncertain stochastic systems with mode-dependent time delays and Markovian jump parameters, Journal of Mathematical Analysis and Applications 342 (2) pp 1084– (2008) · Zbl 1141.93025 |

[7] | Chen, Guaranteed cost control for uncertain Markovian jump systems with mode-dependent time-delays, IEEE Transactions on Automatic Control 48 (12) pp 2270– (2003) · Zbl 1364.93369 |

[8] | Boukas EK, Liu ZK. Output feedback robust stabilization of jump linear systems with mode-dependent time-delays. Proceedings of the American Control Conference, Arlington, 25-27 June 2001; 4683-4688. |

[9] | Sun, Robust exponential stabilization for Markovian jump systems with mode-dependent input delay, Automatica 43 (10) pp 1799– (2007) · Zbl 1119.93068 |

[10] | Wang, Design of reduced-order H filtering for Markovian jump systems with mode-dependent time delays, Signal Processing 86 (2) pp 187– (2009) · Zbl 1155.94337 |

[11] | Siljak, Robust stabilization of non-linear systems: the LMI appraoch, Mathematical Problems in Engineering 6 (5) pp 461– (2000) |

[12] | Stipanovic, Robust stability and stabilization of discrete-time nonlinear systems: the LMI approach, International Journal of Control 74 (9) pp 873– (2001) |

[13] | Zuo, Robust stabilization for non-linear discrete-time systems, International Journal of Control 77 (4) pp 384– (2004) |

[14] | Ho, Robust stabilization for a class of discrete-time non-linear systems via output feedback: the unified LMI approach, International Journal of Control 76 (2) pp 105– (2003) · Zbl 1026.93048 |

[15] | Lam, Stability analysis for continuous systems with two additive time-varying delay components, Systems and Control Letters 56 (1) pp 16– (2007) · Zbl 1120.93362 |

[16] | Gao, A new delay system approach to network-based control, Automatica 44 (1) pp 39– (2008) · Zbl 1138.93375 |

[17] | Jiang, A new H stabilization criterion for networked control systems, IEEE Transactions on Automatic Control 53 (4) pp 1025– (2008) |

[18] | Wang, Exponential filtering for uncertain Markovian jump time-delay systems with nonlinear disturbances, IEEE Transactions on Circuits and Systems 51 (5) pp 262– (2004) |

[19] | Wei, Robust H control of stochastic time-delay jumping systems with nonlinear disturbances, Optimal Control Applications and Methods 27 (5) pp 255– (2006) |

[20] | Yue, Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian swithcing, IEEE Transactions on Automatic Control 50 (2) pp 217– (2005) · Zbl 1365.93377 |

[21] | Mahmoud, Robust control for Markovian jump linear discrete-time systems with unkown nonlinearities, IEEE Transactions on Automatic Control 49 (4) pp 538– (2002) |

[22] | Li, Delay-range-dependent robust stability and stabilization for uncertain systems with time-varying delay, International Journal of Robust and Nonlinear Control 18 (13) pp 1372– (2008) · Zbl 1298.93263 |

[23] | Jiang, On H control for linear systems with interval time-varying delay, Automatica 41 (12) pp 2099– (2005) |

[24] | He, Delay-range-dependent stability for systems with time-varying delay, Automatica 43 (2) pp 371– (2007) · Zbl 1111.93073 |

[25] | Shao, Improved delay-dependent stability criteria for systems with a delay varying, Automatica 44 (12) pp 3215– (2008) · Zbl 1153.93476 |

[26] | Davis, Markov Models and Optimization (1993) · Zbl 0780.60002 · doi:10.1007/978-1-4899-4483-2 |

[27] | Mao, Exponential stability of stochastic delay interval systems with Markovian switching, IEEE Transactions on Automatic Control 47 (10) pp 1604– (2002) · Zbl 1364.93685 |

[28] | Wu, Delay-dependent robust stability and H control for jump linear systems with delays, Systems and Control Letters 55 (11) pp 939– (2006) |

[29] | Gao, New results on stability of discrete-time systems with time-varying state delay, IEEE Transactions on Automatic Control 52 (2) pp 328– (2007) · Zbl 1366.39011 |

[30] | Yakukbovich, The S-procedure in nonlinear control theory, Vestnik Leningrad University Mathematics 4 pp 73– (1977) |

[31] | Boyd, Linear Matrix Inequalities in System and Control Theory (1994) · Zbl 0816.93004 · doi:10.1137/1.9781611970777 |

[32] | Wei, A delay-dependent approach to H filtering for stochastic delayed jumping systems with sensor non-linearities, International Journal of Control 80 (6) pp 885– (2007) · Zbl 1124.93056 |

[33] | Wang, Robust filtering for discrete-time Markovian jump delay systems, IEEE Signal Processing Letters 11 (8) pp 659– (2004) |

[34] | Wang, On stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters, IEEE Transactions on Automatic Control 47 (4) pp 640– (2002) · Zbl 1364.93672 |

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