Exponential \(H_{\infty }\) filtering for time-varying delay systems: Markovian approach.

*(English)*Zbl 1217.93170Summary: This paper discusses the \(H_{\infty }\) filtering problem for a class of deterministic systems with time-varying delays, where the stochastic property of time-varying delays described by Markovian approach is taken into consideration in filter design. Firstly, the delay interval is separated into several subintervals, which can be described by Markov process. Then, a new \(H_{\infty }\) filtering method for deterministic system with time-varying delay is given, whose filter can switch with time delay in terms of Markov process. Sufficient conditions for the existence of \(H_{\infty }\) filter are obtained as linear matrix inequalities, where the mode transition rates are known exactly or inexactly. Finally, numerical examples are used to demonstrate the utility of the given methods.

##### Keywords:

\(H_{\infty }\) filtering; time delay; Markovian jump systems; linear matrix inequalities (LMIs)
PDF
BibTeX
XML
Cite

\textit{G. Wang} et al., Signal Process. 91, No. 8, 1852--1862 (2011; Zbl 1217.93170)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Shi, P.; Boukas, E. K.; Agarwal, R. K.: Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters, IEEE transactions on automatic control 44, No. 8, 1592-1597 (1999) · Zbl 0986.93066 |

[2] | Yang, F. W.; Wang, Z. D.; Hung, Y. S.: Robust Kalman filtering for discrete time-varying uncertain systems with multiplicative noises, IEEE transactions on automatic control 47, No. 7, 1179-1183 (2002) · Zbl 1364.93817 |

[3] | Xu, S. Y.; Lam, J.; Chen, T. W.; Zou, Y.: A delay-dependent approach robust H\(\infty \) filtering for uncertain distributed delay systems, IEEE transactions on signal processing 53, No. 10, 3764-3772 (2005) · Zbl 1370.93109 |

[4] | Palhares, R. M.; De Souza, C. E.; Peres, P. L. D.: Robust H\(\infty \) filtering for uncertain discrete-time state-delayed systems, IEEE transactions on signal processing 49, No. 8, 1696-1703 (2001) · Zbl 1369.93641 |

[5] | Yue, D.; Han, Q. L.: Robust H\(\infty \) filter design of uncertain descriptor systems with discrete and distributed delays, IEEE transactions on signal processing 52, No. 11, 3200-3212 (2004) · Zbl 1370.93111 |

[6] | Gao, H. J.; Wang, C. H.: A delay-dependent approach to robust H\(\infty \) filtering for uncertain discrete-time state-delayed systems, IEEE transactions on signal processing 52, No. 6, 1631-1640 (2004) · Zbl 1369.93175 |

[7] | Wang, Z. D.; Yang, F. W.: Robust filtering for uncertain linear systems with delayed states and outputs, IEEE transactions on automatic control 49, No. 1, 125-130 (2002) |

[8] | Shi, S. H.; Yuan, Z. H.; Zhang, Q. L.: Fault-tolerant H\(\infty \) filter design of a class of switched systems with sensor failures, International journal of innovative computing, information and control 5, No. 11(A), 3827-3838 (2009) |

[9] | Song, B.; Xu, S. Y.; Zou, Y.: Non-fragile H\(\infty \) filtering for uncertain stochastic time-delay systems, International journal of innovative computing, information and control 5, No. 8, 2257-2266 (2009) |

[10] | Liu, Y. S.; Wang, W.: Fuzzy H\(\infty \) filtering for nonlinear stochastic systems with missing measurements, ICIC express letters 3, No. 3(B), 739-744 (2009) |

[11] | Alcorta, M. A.; Basin, M.; Maldonado, J. J.; Anguiano, S. G.: Sub-optimal risk-sensitive filtering for third degree polynomial stochastic systems, ICIC express letters 2, No. 4, 371-377 (2008) |

[12] | Alcorta, M. A.; Basin, M.; Sanchez, Y. G. A.: Risk-sensitive approach to optimal filtering and control for linear stochastic systems, International journal of innovative computing, information and control 5, No. 6, 1599-1614 (2009) |

[13] | Shi, P.; Boukas, E. K.; Agarwal, R. K.: Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay, IEEE transactions on automatic control 44, No. 11, 2139-2144 (1999) · Zbl 1078.93575 |

[14] | Lee, Y. S.; Moon, Y. S.; Kwon, W. H.; Park, P. G.: Delay-dependent robust H\(\infty \) control for uncertain systems with a state-delay, Automatica 40, No. 1, 65-72 (2004) · Zbl 1046.93015 |

[15] | Basin, M.; Gonzalez, R. R.: Optimal control for linear systems with multiple time delays in control input, IEEE transactions on automatic control 51, No. 1, 91-97 (2006) · Zbl 1366.93275 |

[16] | Lam, J.; Gao, H. J.; Wang, C. H.: Stability analysis for continuous systems with two additive time-varying delay components, Systems and control letters 56, No. 1, 16-24 (2007) · Zbl 1120.93362 |

[17] | Li, H. Y.; Chen, B.; Zhou, Q.; Fang, S. L.: Robust exponential stability for uncertain stochastic neural networks with discrete and distributed time-varying delays, Physics letters A 372, No. 19, 3385-3394 (2008) · Zbl 1220.82085 |

[18] | 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–B: cybernetics 39, No. 1, 94-102 (2009) |

[19] | Gu, K. Q.; Niculescu, S. I.: Further remarks on additional dynamics in various model transformations of linear delay systems, IEEE transactions on automatic control 46, No. 3, 497-500 (2001) · Zbl 1056.93511 |

[20] | Jiang, X. F.; Han, Q. L.: On H\(\infty \) control for linear systems with interval time-varying delay, Automatica 41, No. 12, 2099-2106 (2005) · Zbl 1100.93017 |

[21] | He, Y.; Wang, Q. G.; Lin, C.; Wu, M.: Delay-range-dependent stability for systems with time-varying delay, Automatica 43, No. 2, 371-376 (2007) · Zbl 1111.93073 |

[22] | Mahmoud, M. S.; Ismail, A.: New results on delay-dependent control of time-delay systems, IEEE transactions on automatic control 50, No. 1, 95-100 (2005) · Zbl 1365.93143 |

[23] | Shao, H. Y.: Improved delay-dependent stability criteria for systems with a delay varying in a range, Automatica 44, No. 12, 3215-3218 (2008) · Zbl 1153.93476 |

[24] | Zhang, L. Q.; Shi, Y.; Chen, T. W.; Huang, B.: A new method for stabilization of networked control systems with random delays, IEEE transactions on automatic control 50, No. 8, 1177-1181 (2005) · Zbl 1365.93421 |

[25] | Yue, D.; Tian, E. G.; Zhang, Y. J.; Peng, C.: Delay-distribution-dependent robust stability of uncertain systems with time-varying delays, International journal of robust and nonlinear control 19, No. 4, 377-393 (2008) · Zbl 1157.93478 |

[26] | Huang, D.; Nguang, S. K.: State feedback control of uncertain networked control systems with random time delays, IEEE transactions on automatic control 53, No. 3, 829-834 (2008) · Zbl 1367.93510 |

[27] | Gao, H. J.; Wang, C. H.: Delay-dependent robust H\(\infty \) and L2-L\(\infty \) filtering for a class of uncertain nonlinear time-delay systems, IEEE transactions on automatic control 48, No. 9, 1661-1666 (2003) · Zbl 1364.93210 |

[28] | Gao, H. J.; Meng, X. Y.; Chen, T. W.: A parameter-dependent approach to robust H\(\infty \) filtering for time-delay systems, IEEE transactions on automatic control 53, No. 10, 2420-2425 (2008) · Zbl 1367.93176 |

[29] | Qiu, J. B.; Feng, G.; Yang, J.: A new design of delay-dependent robust H\(\infty \) filtering for continuous-time polytopic systems with time-varying delay, International journal of robust and nonlinear control 20, No. 3, 346-365 (2010) · Zbl 1185.93137 |

[30] | Xiong, J. L.; Lam, J.: Fixed-order robust H\(\infty \) filter design for Markovian jump systems with uncertain switching probabilities, IEEE transactions on signal processing 54, No. 4, 1421-1430 (2006) · Zbl 1373.94736 |

[31] | Wang, G. L.; Zhang, Q. L.; Sreeram, V.: Partially mode-dependent H\(\infty \) filtering for discrete-time Markovian jump systems with partly unknown transition probabilities, Signal processing 90, No. 2, 548-556 (2010) · Zbl 1177.93086 |

[32] | Wang, Z. D.; Lam, J.; Liu, X. H.: Exponential filtering for uncertain Markovian jump time-delay systems with nonlinear disturbances, IEEE transactions on circuits and systems II: Express briefs 51, No. 5, 262-268 (2004) |

[33] | Liu, H. P.; Sun, F. C.; He, K. Z.; Sun, Z. Q.: Design of reduced-order H\(\infty \) filter for Markovian jumping systems with time delay, IEEE transactions on circuits and systems II: Express briefs 51, No. 11, 607-612 (2004) |

[34] | Boukas, E. K.; Shi, P.; Nguang, S. K.: Robust H\(\infty \) control for linear Markovian jump systems with unknown nonlinearities, Journal of mathematical analysis and applications 282, No. 1, 241-255 (2003) · Zbl 1029.93064 |

[35] | Shi, P.; Mahmoud, M.; Nguang, S. K.; Ismail, A.: Robust filtering for jumping systems with mode-dependent delays, Signal processing 86, No. 1, 140-152 (2006) · Zbl 1163.94387 |

[36] | Xu, S. Y.; Lam, J.; Mao, X. Y.: Delay-dependent H\(\infty \) control and filtering for uncertain Markovian jump systems with time-varying delays, IEEE transactions on circuits and systems I: Regular paper 54, No. 9, 2070-2077 (2007) · Zbl 1374.93134 |

[37] | Xu, S. Y.; Chen, T. W.; Lam, J.: Robust H\(\infty \) filtering for uncertain Markovian jump systems with mode-dependent time delays, IEEE transactions on automatic control 48, No. 5, 900-907 (2003) · Zbl 1364.93816 |

[38] | 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, No. 2, 187-196 (2009) · Zbl 1155.94337 |

[39] | Shao, H. Y.: Delay-range-dependent robust H\(\infty \) filtering for uncertain stochastic systems with mode-dependent time delays and Markovian jump parameters, Journal of mathematical analysis and applications 342, No. 2, 1084-1095 (2008) · Zbl 1141.93025 |

[40] | Davis, M. H. A.: Markov models and optimization, (1993) · Zbl 0780.60002 |

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.