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**Delay-dependent robust \(H_\infty \) controller synthesis for discrete singular delay systems.**
*(English)*
Zbl 1237.93054

Summary: The problems of delay-dependent robust stability analysis, robust stabilization and robust \(H_{\infty }\) control are investigated for uncertain discrete-time singular systems with state delay. First, by making use of the delay partitioning technique, a new delay-dependent criterion is given to ensure the nominal system to be regular, causal and stable. This new criterion is further extended to singular systems with both delay and parameter uncertainties. Then, without the assumption that the considered systems being regular and causal, robust controllers are designed for discrete-time singular time-delay systems such that the closed-loop systems have the characteristics of regularity, causality and asymptotic stability. Moreover, the problem of robust \(H_{\infty }\) control is solved following a similar line. The obtained results are dependent not only on the delay, but also on the partitioning size and the conservatism is non-increasing with reducing partitioning size. These results are shown, via extensive numerical examples, to be much less conservative than the existing results in the literature.

### MSC:

93B50 | Synthesis problems |

93D09 | Robust stability |

93C55 | Discrete-time control/observation systems |

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\textit{Z. Feng} et al., Int. J. Robust Nonlinear Control 21, No. 16, 1880--1902 (2011; Zbl 1237.93054)

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### References:

[1] | Hale, Introduction to Functional Differential Equations (1993) · Zbl 0787.34002 |

[2] | Verghese, A generalized state-space for singular systems, IEEE Transactions on Automatic Control 24 pp 811– (1981) · Zbl 0541.34040 |

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

[4] | Shi, Optimal guaranteed cost control of uncertain discrete time-delay systems, Journal of Computational and Applied Mathematics 157 pp 435– (2003) · Zbl 1029.93044 |

[5] | Wang, H filtering for uncertain stochastic time-delay systems with sector-bounded nonlinearities, Automatica 44 pp 1268– (2008) · Zbl 1283.93284 |

[6] | Gu, Stability of Time-Delay Systems (2003) · Zbl 1039.34067 |

[7] | Niculescu, Delay Effects on Stability: A Robust Control Approach (2001) |

[8] | Yue, Delay-dependent robust exponential stability of uncertain descriptor systems with time-varying delays, Dynamics of Continuous, Discrete and Impulsive Systems B: Applications and Algorithms 12 pp 129– (2005) |

[9] | Shi, On H control design for singular continuous-time delay systems with parametric uncertainties, Nonlinear Dynamics and Systems Theory 4 pp 59– (2004) · Zbl 1073.93020 |

[10] | Sun, Delay-dependent robust stabilization for a class of uncertain singular delay systems, International Journal of Innovative Computing, Information and Control 5 pp 1231– (2009) |

[11] | Lam, Robust H control of descriptor discrete-time Markovian jump systems, International Journal of Control 80 pp 374– (2007) · Zbl 1120.93057 |

[12] | Zhang, New bounded real lemma for discrete-time singular systems, Automatica 44 pp 886– (2008) · Zbl 1283.93179 |

[13] | Xu, Robust stability and stabilization for singular systems with state delay and parameter uncertainty, IEEE Transactions on Automatic Control 47 pp 1122– (2002) · Zbl 1364.93723 |

[14] | Zhu S Cheng Z Feng J Delay-dependent robust stability criterion and robust stabilization for uncertain singular time-delay systems |

[15] | Gao HL Xu BG Zhang YG Hou XL A new delay-dependent stability criterion for singular time-delay systems 2252 2256 |

[16] | Fridman, Stability of linear descriptor systems with delay: a Lyapunov-based approach, Journal of Mathematical Analysis and Applications 273 pp 24– (2002) · Zbl 1032.34069 |

[17] | Zhu, Delay-dependent robust stability criterion for two classes of uncertain singular time-delay systems, IEEE Transactions on Automatic Control 52 pp 880– (2007) · Zbl 1366.93478 |

[18] | Fridman, H control of linear state-delay descriptor systems: an LMI approach, Linear Algebra and its Applications 315-352 pp 271– (2002) · Zbl 1006.93021 |

[19] | Xu, An improved characterization of bound realness for singular delay systems and its applications, International Journal of Robust and Nonlinear Control 18 (2008) · Zbl 1284.93117 |

[20] | Qiu, New robust passive stability criteria for uncertain singularly Markov jump systems with time delays, ICIC Express Letters 3 pp 651– (2009) |

[21] | Du, Delay-dependent stabilization of uncertain singular systems with multiple state delays, International Journal of Innovative Computing, Information and Control 5 pp 1655– (2009) |

[22] | Chen, Stability robustness of linear discrete singular time-delay systems with structured parameter uncertainties, IEE Proceedings of Control Theory and Applications 150 pp 296– (2003) |

[23] | Chen, Robust stability of discrete time-delay uncertain singular systems, IEE Proceedings of Control Theory and Applications 151 pp 45– (2003) |

[24] | Ji X Su H Chu J Delay-dependent robust stability of uncertain discrete singular time-delay systems 3843 3848 |

[25] | Wu, Robust stabilization for uncertain discrete singular systems with state delay, International Journal of Robust and Nonlinear Control 18 pp 1532– (2008) · Zbl 1151.93426 |

[26] | Wang, Robust H control for a class of nonlinear discrete time-delay stochastic systems with missing measurements, Automatica 45 pp 684– (2009) · Zbl 1166.93319 |

[27] | Rehm A Allgower F An LMI approach towards H control of discrete-time descriptor systems 614 619 |

[28] | Wang HS Yung CF Chang FR Bound real lemma and H control for descriptor systems 3843 3848 |

[29] | Kim JH Lee JH Park HB Robust H control of singular systems with time delays and uncertainties via LMI approach 620 621 |

[30] | Zhu S Li Z Cheng Z Delay-dependent robust resilient H control for uncertain singular time-delay systems 1373 1378 · Zbl 1150.93361 |

[31] | Xu, H state-feedback control for discrete singular systems, IEEE Transactions on Automatic Control 45 pp 1405– (2000) · Zbl 0988.93026 |

[32] | Xu, Robust H control for discrete singular systems with state delay and parameter uncertainty, Dynamics of Continuous Discrete and Impulsive Systems. Series B: Applications and Algorithms 9 pp 539– (2002) |

[33] | Ma, Delay-dependent robust H control for uncertain discrete-time singular systems with time-delays, Journal of Computational and Applied Mathematics 217 pp 194– (2008) · Zbl 1142.93011 |

[34] | Xu, Control and Filtering of Singular Systems (2006) · Zbl 1114.93005 |

[35] | Wang R Liu Y Asymptotic stability and robustness for discrete-time singular systems with multiple time-delays |

[36] | Gouaisbaut F Peaucelle D Delay-dependent stability analysis of linear time delay systems · Zbl 1293.93589 |

[37] | Du B Lam J Shu Z A delay-partitioning projection approach to stability analysis of neutral systems |

[38] | Du, A delay-partitioning projection approach to stability analysis of continuous systems with multiple delay components, IET Control Theory and Applications 3 pp 383– (2009) |

[39] | Wang, A delay fractioning approach to global synchronization of delayed complex networks with stochastic disturbances, Physics Letters A 372 pp 6066– (2008) · Zbl 1223.90013 |

[40] | Zhou W Lu HQ Delay-dependent robust H stability for uncertain discrete-time singular system 655 659 |

[41] | Wang, Delay-dependent robust control for uncertain discrete singular systems with time-varying delay, Journal of Zhejiang University SCIENCE A 5 pp 1655– (2009) |

[42] | Wu, Delay-dependent robust H control for uncertain singular time-delay systems, IEE Proceedings of Control Theory and Applications 1 pp 1234– (2007) |

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