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${H}_{\infty }$ guaranteed cost control for uncertain Markovian jump systems with mode-dependent distributed delays and input delays. (English) Zbl 1185.93036
Summary: This paper investigates the ${H}_{\infty }$ guaranteed cost control problem for mode-dependent time-delay jump systems with norm-bounded uncertain parameters. Both distributed delays and input delays appear in the system model. Based on a matrix inequality, a sufficient condition for the existence of robust ${H}_{\infty }$ guaranteed cost controller is derived, which stabilizes the considered system and guarantees that both the ${H}_{\infty }$ performance level and a cost function have upper bounds for all admissible uncertainties. By the cone complementary linearization approach, the desired state-feedback controller can be constructed. A numerical example is provided to show the effectiveness of the proposed theoretical results.
##### MSC:
 93B36 ${H}^{\infty }$-control 60J75 Jump processes 60H10 Stochastic ordinary differential equations 34H05 ODE in connection with control problems
##### References:
 [1] Chang, S. S. L.; Peng, T. K. C.: Adaptive guaranteed cost control of systems with uncertain parameter, IEEE trans. Autom. control 17, 474-483 (1972) · Zbl 0259.93018 · doi:10.1109/TAC.1972.1100037 [2] Chen, W.; Xu, J.; Guan, Z.: Guaranteed cost control for uncertain Markovian jump systems with mode-dependent time-delays, IEEE trans. Autom. control 48, 2270-2276 (2003) [3] Boukas, E. K.; Liu, Z. K.; Sunni, F. A.: Guaranteed cost control of a Markov jump linear uncertain system using time-multiplied cost function, J. optim. Theory appl. 116, 183-204 (2003) · Zbl 1046.93049 · doi:10.1023/A:1022170404978 [4] Chen, W.; Guan, Z.; Lu, X.: Delay-dependent guaranteed cost control for uncertain discrete-time systems with both state and input delays, J. franklin inst. 341, 419-430 (2004) · Zbl 1055.93054 · doi:10.1016/j.jfranklin.2004.04.003 [5] Dhawan, A.; Kar, H.: Optimal guaranteed cost control of 2-D discrete uncertain systems: an LMI approach, Signal process. 87, 3075-3085 (2007) · Zbl 1186.94105 · doi:10.1016/j.sigpro.2007.06.001 [6] Chen, Q.; Yu, L.; Zhang, W.: Delay-dependent output feedback guaranteed cost control for uncertain discrete-time systems with multiple time-varying delays, IET control theory appl. 1, 97-103 (2007) [7] Y. Ma, Q. Zhang, Y. Ren, X. Zhang, Hnbsp; guaranteed cost control for time-delay uncertain discrete systems, in: International Conference, ICARCV 9, 2006, pp. 1 – 6. [8] Wu, H.; Cai, K.: H2 guaranteed cost fuzzy control design for discrete-time nonlinear systems with parameter uncertainty, Automatica 42, 1183-1188 (2006) · Zbl 1117.93347 · doi:10.1016/j.automatica.2006.02.025 [9] De Oliveiraa, P. J.; Oliveirab, R. C. L.F.; Leitec, V. J. S.; Montagnerb, V. F.; Peres, P. L. D.: H$\infty$ guaranteed cost computation by means of parameter-dependent Lyapunov functions, Automatica 40, 1053-1061 (2004) · Zbl 1110.93021 · doi:10.1016/j.automatica.2004.01.025 [10] Kosmidou, O. I.; Boutalis, Y. S.: A linear matrix inequality approach for guaranteed cost control of systems with state and input delays, IEEE trans. Syst. man cybern. A 36, 936-942 (2006) [11] Zhou, S.; Li, T.: Robust stabilization for delayed discrete-time fuzzy systems via basis-dependent Lyapunov – krasovakii function, Fuzzy sets syst. 151, 139-153 (2005) · Zbl 1142.93379 · doi:10.1016/j.fss.2004.08.014 [12] Niu, Y.; Ho, D. W. C.; Lam, J.: Robust integral sliding mode control for uncertain stochastic systems with time-varying delay, Automatica 41, 873-880 (2005) · Zbl 1093.93027 · doi:10.1016/j.automatica.2004.11.035 [13] Xie, L.; Fridman, E.; Shaked, U.: Robust H$\infty$ control of distributed delay systems with application to combustion control, IEEE trans. Autom. control 46, 1930-1935 (2001) · Zbl 1017.93038 · doi:10.1109/9.975483 [14] Xu, S.; Chen, T.: Robust H$\infty$ output feedback control for uncertain distributed delay systems, Eur. J. Control 9, 562-570 (2003) [15] Xu, S.; Chen, T.: An LMI approach to the H$\infty$ filter design for uncertain systems with distributed delays, IEEE trans. Circuits syst. 51, 195-201 (2004) [16] Xu, S.; Lam, J.; Zou, Y.: Delay-dependent guaranteed cost control for uncertain systems with state and input delays, IEE proc. Control theory appl. 153, 307-313 (2006) [17] Chen, J. D.: Delay-dependent robust H$\infty$ control of uncertain neutral systems with state and input delays: LMI optimization approach, Chaos solitons fractals 33, 595-606 (2007) · Zbl 1136.93018 · doi:10.1016/j.chaos.2006.01.024 [18] Czornik, A.; ${}^{\text{'}}$swierniak, A.: On direct controllability of discrete time jump linear system, J. franklin inst. 341, 491-503 (2004) [19] Cao, Y.; Lam, J.: Robust control of uncertain Markovian jump systems with time-delay, IEEE trans. Autom. control 45, 77-83 (2000) · Zbl 0983.93075 · doi:10.1109/9.827358 [20] K. Gu, An integral inequality in the stability problem of time-delay systems, in: Proceedings of the 44th IEEE Conference on Decision Control, Sydney, Australia, December 2000, pp. 2805 – 2810. [21] Mao, X.: Exponential stability of stochastic delay interval systems with Markovian switching, IEEE trans. Autom. control 47, 1604-1612 (2002) [22] El Ghaoui, L.; Oustry, F.; Rami, M. Ait: A cone complementarity linearization algorithm for static outputfeedback and related problems, IEEE trans. Autom. control 42, 1171-1176 (1997) · Zbl 0887.93017 · doi:10.1109/9.618250