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**Passivity analysis and passive control of fuzzy systems with time-varying delays.**
*(English)*
Zbl 1221.93063

Summary: This paper is concerned with the passive controller design problem for a class of continuous-time Takagi-Sugeno (T-S) fuzzy systems with both state and input delays. The delays are assumed to be time-varying and differentiable. A notion of very-strict passivity is adopted. The purpose is to design a state-feedback fuzzy controller such that the resulting closed-loop system is Very-Strictly Passive (VSP). Delay-dependent conditions for the solvability of the addressed problem are obtained by applying recently developed techniques for time-delay systems and fuzzy systems. These conditions are expressed by means of strict Linear Matrix Inequalities (LMIs) that can be easily solved. A numerical example and simulation results are provided to demonstrate the effectiveness of the proposed method.

### Keywords:

delay-dependent conditions; passive control; Takagi-Sugeno (T-S) fuzzy systems; time-varying delays; very-strict passivity
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\textit{B. Zhang} et al., Fuzzy Sets Syst. 174, No. 1, 83--98 (2011; Zbl 1221.93063)

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### References:

[1] | Bemporad, A.; Bianchini, G.; Brogi, F., Passivity analysis and passification of discrete-time hybrid systems, IEEE transactions on automatic control, 53, 4, 1004-1009, (2008) · Zbl 1367.93337 |

[2] | Bernal, M.; Guerra, T.M.; Kruszewski, A., A membership-function-dependent approach for stability analysis and controller synthesis of takagi – sugeno models, Fuzzy sets and systems, 160, 2776-2795, (2009) · Zbl 1176.93042 |

[3] | Cao, Y.Y.; Frank, P.M., Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach, IEEE transactions on fuzzy systems, 8, 2, 200-211, (2000) |

[4] | Cao, Y.Y.; Frank, P.M., Stability analysis and synthesis of nonlinear time-delay systems via linear takagi – sugeno fuzzy models, Fuzzy sets and systems, 124, 213-229, (2001) · Zbl 1002.93051 |

[5] | Delmotte, F.; Guerra, T.M.; Kruszewski, A., Discrete takagi – sugeno’s fuzzy models: reduction of the number of LMI in fuzzy control techniques, IEEE transactions on systems, man, and cybernetics—B: cybernetics, 38, 5, 1423-1427, (2008) |

[6] | Chen, B.; Liu, X.; Lin, C.; Liu, K., Robust \(H_\infty\) control of takagi – sugeno fuzzy systems with state and input time delays, Fuzzy sets and systems, 160, 403-422, (2009) · Zbl 1175.93119 |

[7] | Chen, B.; Liu, X.; Tong, S.; Lin, C., Guaranteed cost control of T-S fuzzy systems with state and input delays, Fuzzy sets and systems, 158, 2251-2267, (2007) · Zbl 1122.93049 |

[8] | Dong, J.; Yang, G.H., Static output feedback \(H_\infty\) control of a class of nonlinear discrete-time systems, Fuzzy sets and systems, 160, 2844-2859, (2009) · Zbl 1176.93044 |

[9] | Feng, G., A survey on analysis and design of model-based fuzzy control systems, IEEE transactions on fuzzy systems, 14, 5, 676-697, (2006) |

[10] | Fridman, E.; Shaked, U., On delay-dependent passivity, IEEE transactions on automatic control, 47, 4, 664-669, (2002) · Zbl 1364.93370 |

[11] | Gao, H.; Chen, T.; Chai, T., Passivity and passification for networked control systems, SIAM journal on control and optimization, 46, 4, 1299-1322, (2007) · Zbl 1140.93425 |

[12] | Gao, H.; Liu, X.; Lam, J., Stability analysis and stabilization for discrete-time fuzzy systems with time-varying delay, IEEE transactions on systems, man, and cybernetics—B: cybernetics, 39, 2, 306-317, (2009) |

[13] | Gao, H.; Zhao, Y.; Lam, J.; Chen, K., \(H_\infty\) fuzzy filtering of nonlinear systems with intermittent measurements, IEEE transactions on fuzzy systems, 17, 2, 291-300, (2009) |

[14] | Guerra, T.M.; Vermeiren, L., LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the takagi – sugeno’s form, Automatica, 40, 823-829, (2004) · Zbl 1050.93048 |

[15] | Hill, D.; Moylan, P., The stability of nonlinear dissipative systems, IEEE transactions on automatic control, 21, 5, 708-711, (1976) · Zbl 0339.93014 |

[16] | Lam, J.; Zhou, S., Dynamic output feedback \(H_\infty\) control of discrete-time fuzzy systems: a fuzzy-basis-dependent Lyapunov function approach, International journal of systems science, 38, 1, 25-37, (2007) · Zbl 1111.93017 |

[17] | Lee, K.R.; Kim, J.H.; Jeung, E.T.; Park, H.B., Output feedback robust \(H_\infty\) control of uncertain fuzzy dynamic systems with time-varying delay, IEEE transactions on fuzzy systems, 8, 6, 657-664, (2000) |

[18] | Li, C.G.; Zhang, H.B.; Liao, X.F., Passivity and passification of fuzzy systems with time delays, Computers and mathematics with applications, 52, 1067-1078, (2006) · Zbl 1122.93368 |

[19] | Li, C.G.; Zhang, H.B.; Liao, X.F., Passivity and passification of uncertain fuzzy systems, IEE Proceedings circuits, device and systems, 152, 6, 649-653, (2005) |

[20] | Li, H.; Chen, B.; Zhou, Q.; Qian, W., Robust stability for uncertain delayed fuzzy Hopfield neural networks with Markovian jumping parameters, IEEE transactions on systems, man, and cybernetics—B: cybernetics, 39, 1, 94-102, (2009) |

[21] | Li, Y.; Xu, S.; Zhang, B.; Chu, Y., Robust stabilization and \(H_\infty\) control for uncertain fuzzy neutral systems with mixed time-delays, Fuzzy sets and systems, 159, 2730-2748, (2008) · Zbl 1170.93343 |

[22] | Lin, C.; Wang, Q.G.; Lee, T.H., Delay-dependent LMI conditions for stability and stabilization of T-S fuzzy systems with bounded time delay, Fuzzy sets and systems, 157, 1229-1247, (2006) · Zbl 1090.93024 |

[23] | Lin, C.; Wang, Q.G.; Lee, T.H.; He, Y.; Chen, B., Observer-based \(H_\infty\) fuzzy control design for T-S fuzzy systems with state delays, Automatica, 44, 868-874, (2008) · Zbl 1283.93164 |

[24] | Lozano, R.; Brogliato, B.; Egeland, O.; Maschke, B., Dissipative systems analysis and control: theory and applications, (2007), Springer-Verlag London, UK · Zbl 1121.93002 |

[25] | Nguang, S.K.; Shi, P., \(H_\infty\) fuzzy output feedback control design for nonlinear systems: an LMI approach, IEEE transactions on fuzzy systems, 11, 3, 331-340, (2003) |

[26] | Nguang, S.K.; Shi, P., Fuzzy \(H_\infty\) output feedback control of nonlinear systems under sampled measurements, Automatica, 39, 2169-2174, (2003) · Zbl 1041.93033 |

[27] | Niculescu, S.I.; Lozano, R., On the passivity of linear delay systems, IEEE transactions on automatic control, 46, 3, 460-464, (2001) · Zbl 1056.93610 |

[28] | Tanaka, K.; Wang, H.O., Fuzzy control system design and analysis: A linear matrix inequality approach, (2001), Wiley New York |

[29] | Tanaka, K.; Ohtake, H.; Wang, H.O., Guaranteed cost control of polynomial fuzzy systems via a sum of squares approach, IEEE transactions on systems, man, and cybernetics—B: cybernetics, 39, 2, 561-567, (2009) |

[30] | Tian, E.; Yue, D.; Zhang, Y., Delay-dependent robust \(H_\infty\) control for T-S fuzzy system with interval time-varying delay, Fuzzy sets and systems, 160, 1708-1719, (2009) · Zbl 1175.93134 |

[31] | Tong, S.; Wang, W.; Qu, L., Decentralized robust control for uncertain T-S fuzzy large-scale systems with time-delay, International journal of innovative computing, information and control, 3, 657-672, (2007) |

[32] | Tong, S.; Zhang, Q., Decentralized output feedback fuzzy \(H_\infty\) tracking control for nonlinear interconnected systems with time-delay, International journal of innovative computing, information and control, 4, 3385-3398, (2008) |

[33] | Xie, L.; Fu, M.; Li, H., Passivity analysis and passification for uncertain signal processing systems, IEEE transactions on signal processing, 46, 9, 2394-2403, (1998) |

[34] | Xu, S.; Lam, J., Robust \(H_\infty\) control for uncertain discrete-time-delay fuzzy systems via output feedback controllers, IEEE transactions on fuzzy systems, 13, 1, 82-93, (2005) |

[35] | Xu, S.; Lam, J., A survey of linear matrix inequality techniques in stability analysis of delay systems, International journal of systems science, 39, 12, 1095-1113, (2008) · Zbl 1156.93382 |

[36] | Xu, S.; Song, B.; Lu, J.; Lam, J., Robust stability of uncertain discrete-time singular fuzzy systems, Fuzzy sets and systems, 158, 2306-2316, (2007) · Zbl 1122.93065 |

[37] | Zhang, B.; Xu, S., Delay-dependent robust \(H_\infty\) control for uncertain discrete-time fuzzy systems with time-varying delays, IEEE transactions on fuzzy systems, 17, 4, 809-823, (2009) |

[38] | Zhang, B.; Lam, J.; Xu, S.; Shu, Z., Robust stabilization of uncertain T-S fuzzy time-delay systems with exponential estimates, Fuzzy sets and systems, 160, 1720-1737, (2009) · Zbl 1175.93200 |

[39] | Zhang, B.; Xu, S.; Zong, G.; Zou, Y., Delay-dependent stabilization for stochastic fuzzy systems with time delays, Fuzzy sets and systems, 158, 2238-2250, (2007) · Zbl 1122.93051 |

[40] | Zhang, H.B.; Shen, Y.; Feng, G., Delay-dependent stability and \(H_\infty\) control for a class of fuzzy descriptor systems with time-delay, Fuzzy sets and systems, 160, 1689-1707, (2009) · Zbl 1175.93138 |

[41] | Zhao, Y.; Gao, H.; Lam, J.; Du, B., Stability and stabilization of delayed T-S fuzzy systems: a delay partitioning approach, IEEE transactions on fuzzy systems, 17, 4, 750-762, (2009) |

[42] | Zhao, Y.; Lam, J.; Gao, H., Fault detection for fuzzy systems with intermittent measurements, IEEE transactions on fuzzy systems, 17, 2, 398-410, (2009) |

[43] | Zhou, S.; Lam, J.; Zheng, W.X., Control design for fuzzy systems based on relaxed nonquadratic stability and \(H_\infty\) performance conditions, IEEE transactions on fuzzy systems, 15, 2, 188-199, (2007) |

[44] | Zhou, S.; Li, T., Robust stabilization for delayed discrete-time fuzzy systems via basis-dependent lyapunov – krasovskii function, Fuzzy sets and systems, 151, 139-153, (2005) · Zbl 1142.93379 |

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.