## Robust stability and stabilization for uncertain Takagi-Sugeno fuzzy time-delay systems.(English)Zbl 1110.93033

Summary: We consider robust stability and stabilization of uncertain Takagi–Sugeno fuzzy time-delay systems where uncertainties come into the state and input matrices. Some stability conditions and robust stability conditions for fuzzy time-delay systems have already been obtained in the literature. However, those conditions are rather conservative and do not guarantee the stability and robust stability for a wide class of fuzzy systems. This is true in case of designing stabilizing controllers for fuzzy time-delay systems. We first consider robust stability conditions of uncertain fuzzy systems. Conditions we obtain here are delay-dependent conditions that depend on the upper bound of time delay, and are given in linear matrix inequalities (LMIs). An appropriate selection of Lyapunov–Krasovskii function and introduction of free weighting matrices generalize robust stability conditions. Next, we consider the stabilization problem with memoryless and delayed feedback controllers. Based on our generalized robust stability conditions, we obtain delay-dependent sufficient conditions for the closed-loop system to be robustly stable, and give a design method of robustly stabilizing controllers. Finally, we give two examples that illustrate our results. We compare our conditions with other stability conditions and show our conditions are rather generalized.

### MSC:

 93C42 Fuzzy control/observation systems 93D09 Robust stability 93D21 Adaptive or robust stabilization
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### References:

 [1] Cao, Y.-Y.; Frank, P.M., Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach, IEEE trans. fuzzy systems, 8, 200-211, (2000) [2] Chen, B.; Liu, X., Fuzzy guaranteed cost control for nonlinear systems with time-varying delay, IEEE trans. fuzzy systems, 13, 238-249, (2004) [3] Fridman, E.; Shaked, U., An improved stabilization method for linear time-delay systems, IEEE trans. automat. control, 47, 1931-1937, (2002) · Zbl 1364.93564 [4] Gu, K.; Niculescu, S.-I., Additional dynamics in transformed time-delay systems, IEEE trans. automat. control, 45, 572-575, (2000) · Zbl 0986.34066 [5] Gu, K.; Niculescu, S.-I., Further remarks on additional dynamics in various model transformations of linear delay systems, IEEE trans. automat. control, 46, 497-500, (2001) · Zbl 1056.93511 [6] Guan, X.-P.; Chen, C.-L., Delay-dependent guaranteed cost control for T-S fuzzy systems with time delays, IEEE trans. fuzzy systems, 12, 236-249, (2004) · Zbl 1142.93363 [7] He, Y.; Wu, N.; She, J.-H.; Liu, G.P., Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays, Systems control lett., 51, 57-65, (2004) · Zbl 1157.93467 [8] Kolmanovskii, V.B.; Richard, J.-P., Stability of some linear systems with delays, IEEE trans. automat. control, 44, 984-989, (1999) · Zbl 0964.34065 [9] Lee, K.R.; Kin, J.H.; Jeung, E.T.; Park, H.B., Output feedback robust $$H_\infty$$ control of uncertain fuzzy dynamic systems with time-varying delay, IEEE trans. fuzzy systems, 8, 657-664, (2000) [10] Li, X.; de Souza, C.E., Delay-dependent robust stability and stabilization of uncertain linear delay systems: a linear matrix inequality approach, IEEE trans. automat. control, 42, 1144-1148, (1997) · Zbl 0889.93050 [11] Liu, X.; Zhang, H.; Zhang, F., Delay-dependent stability of uncertain fuzzy large-scale with time delays, Chaos solitons fractals, 26, 147-158, (2005) · Zbl 1080.34058 [12] Mahmoud, M.S., Robust control and filtering for time-delay systems, (2000), Marcel Dekker Inc. New York · Zbl 0969.93002 [13] Mahmoud, M.S.; Al-Muthairi, N.F., Quadratic stabilization of continuous time systems with state-delay and norm-bounded time-varying uncertainties, IEEE trans. automat. control, 39, 2135-2139, (1994) · Zbl 0925.93585 [14] Moon, Y.S., Delay-dependent robust stabilization of uncertain state-delayed systems, Internat. J. control, 74, 1447-1455, (2001) · Zbl 1023.93055 [15] Park, P.G., A delay-dependent stability criterion for systems with uncertain time-invariant delays, IEEE trans. automat. control, 44, 876-877, (1999) · Zbl 0957.34069 [16] Phoojaruenchanachai, S.; Furuta, K., Memoryless stabilization of uncertain linear systems including time-varying state delays, IEEE trans. automat. control, 37, 1022-1026, (1992) · Zbl 0767.93073 [17] Shen, J.C.; Chen, B.-S.; Kung, F.-C., Memoryless stabilization of uncertain dynamic delay systems: Riccati equation approach, IEEE trans. automat. control, 36, 638-640, (1991) [18] Takagi, T.; Sugeno, M., Fuzzy identification of systems and its applications to modeling and control, IEEE trans. systems man cybernetics, 15, 116-132, (1985) · Zbl 0576.93021 [19] Tanaka, K.; Ikeda, T.; Wang, H.O., Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, $$H^\infty$$ control theory, and linear matrix inequalities, IEEE trans. fuzzy systems, 4, 1-13, (1996) [20] Tanaka, K.; Sano, M., A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer, IEEE trans. fuzzy systems, 4, 119-134, (1994) [21] Tanaka, K.; Sugeno, M., Stability analysis and design of fuzzy control systems, Fuzzy sets and systems, 45, 135-156, (1992) · Zbl 0758.93042 [22] J. Yoneyama, Output stabilization of fuzzy time-delay systems, Fourth Asian Fuzzy Systems Symposium, 2000, 1057-1061. [23] J. Yoneyama, Robust control analysis for uncertain fuzzy systems with time-delays, Joint First International Conference on Soft Computing and Intelligent Systems and 3rd International Symposium on Advanced Intelligent Systems (SCIS & ISIS 2002), 2002, 22P5-2. [24] J. Yoneyama, Stability and stabilization of fuzzy time-delay systems, Fifth Asian Control Conference, 2004, pp. 1518-1525.
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