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Stability analysis and \(H_{\infty}\) control of discrete T-S fuzzy hyperbolic systems. (English) Zbl 1336.93121

Summary: This paper focuses on the problem of constraint control for a class of discrete-time nonlinear systems. Firstly, a new discrete T-S fuzzy hyperbolic model is proposed to represent a class of discrete-time nonlinear systems. By means of the parallel distributed compensation (PDC) method, a novel asymptotic stabilizing control law with the ”soft” constraint property is designed. The main advantage is that the proposed control method may achieve a small control amplitude. Secondly, for an uncertain discrete T-S fuzzy hyperbolic system with external disturbances, by the proposed control method, the robust stability and \(H_{\infty}\) performance are developed by using a Lyapunov function, and some sufficient conditions are established through seeking feasible solutions of some linear matrix inequalities (LMIs) to obtain several positive diagonally dominant (PDD) matrices. Finally, the validity and feasibility of the proposed schemes are demonstrated by a numerical example and a Van de Vusse one, and some comparisons of the discrete T-S fuzzy hyperbolic model with the discrete T-S fuzzy linear one are also given to illustrate the advantage of our approach.

MSC:

93D09 Robust stability
93B36 \(H^\infty\)-control
93C42 Fuzzy control/observation systems
93C55 Discrete-time control/observation systems
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[1] Bemporad, A., Borrelli, F. and Morari, M. (2003). Min-max control of constrained uncertain discrete-time linear system, IEEE Transactions on Automatic Control48(9): 1600-1606.; · Zbl 1364.93181
[2] Cao, S.G., Rees, N.W., Feng, G. and Liu, W. (2000). \(H_∞\) control of nonlinear discrete-time systems based on dynamical fuzzy models, International Journal of System Science31(31): 229-241.; · Zbl 1080.93518
[3] Cao, Y.Y. and Frank, P.M. (2000). Robust \(H_∞\) disturbance attenuation for a class of uncertain discrete-time fuzzy systems, IEEE Transactions on Fuzzy Systems8(4): 406-415.;
[4] Chen, B. and Liu, X.P. (2005). Delay-dependent robust \(H_∞\) control for TCS fuzzy systems with time delay, IEEE Transactions on Fuzzy Systems13(4): 544-556.;
[5] Chen, B.-S., Tseng, C.-S. and Uang, H.-J. (2000). Mixed \(H_2/H_∞\) fuzzy output feedback control design for nonlinear dynamic systems: An LMI approach, IEEE Transactions on Fuzzy Systems8(3): 249-265.;
[6] Chen, M.L. and Li, J.M. (2012). Modeling and control of T-S fuzzy hyperbolic model for a class of nonlinear systems, International Conference on Modelling, Identification and Control, Wuhan, China, pp. 57-62.;
[7] Chen, M.L. and Li, J.M. (2015). Non-fragile guaranteed cost control for Takagi-Sugeno fuzzy hyperbolic systems, International Journal of System Science46(9): 1614-1627.; · Zbl 1332.93192
[8] Datta, R., Bittermann, M.S., Deb, K. and Ciftcioglu, O. (2012). Probabilistic constraint handling in the framework of joint evolutionary-classical optimization with engineering application, IEEE Congress on Evolutionary Computation, Brisbane, Australia, pp. 1-8.;
[9] Du, D.S. (2012). Reliable \(H_∞\) control for Takagi-Sugeno fuzzy systems with intermittent measurements, Nonlinear Analysis: Hybrid Systems6(4): 930-941.; · Zbl 1269.93053
[10] Elliott, D.L. (1999). Bilinear systems, Encyclopedia of Electrical Engineering, Wiley, New York, NY.;
[11] Feng, G. (2006). A survey on analysis and design of model-based fuzzy control systems, IEEE Transactions on Fuzzy Systems14(5): 676-697.;
[12] Guan, X. and Chen, C. (2004). Delay-dependent guaranteed cost control for T-S fuzzy system with time delays, IEEE Transactions on Fuzzy Systems12(2): 236-249.; · Zbl 1142.93363
[13] Hsiao, M.Y., Liu, C.H., Tsai, S.H., Chen, P.S. and Chen, T.T. (2010). A Takagi-Sugeno fuzzy-model-based modeling method, IEEE International Conference on Fuzzy Systems, Barcelona, Spain, pp. 1-6.;
[14] Jadbabaie, A., Jamshidi, M. and Titli, A. (1998). Guaranteed-cost design of continuous-time Takagi-Sugeno fuzzy controllers via linear matrix inequalities, IEEE World Congress on Computational Intelligence, Anchorage, AK, USA, Vol. 1, pp. 268-273.;
[15] Kim, S.H., Lee, C.H. and Park, P.G. (2008). Relaxed delay-dependent stabilization conditions for discrete-time fuzzy systems with time delays, IEEE 10th International Conference on Control, Automation, Robotics and Vision, Hanoi, Vietnam, pp. 999-1004.;
[16] Li, J.M., Li, J., and Du, C.X. (2009). Linear Control System Theory and Methods, Xidian University Press, Xian, pp. 10-13.;
[17] Li, J.R., Li, J.M. and Xia, Z.L. (2011). Delay-dependent generalized \(H_2\) control for discrete T-S fuzzy large-scale stochastic systems with mixed delays, International Journal of Applied Mathematics and Computer Science21(4): 583-603, DOI: 10.2478/v10006-011-0046-6.; · Zbl 1283.93255
[18] Li, J.R., Li, J.M. and Xia, Z.L. (2013a). Observer-based fuzzy control design for discrete-time T-S fuzzy bilinear systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems21(3): 435-454.; · Zbl 1323.93046
[19] Li, J.R., Li, J.M. and Xia, Z.L. (2013b). Delay-dependent generalized \(H_2\) fuzzy static-output-feedback control for discrete T-S fuzzy bilinear stochastic systems with mixed delays, Journal of Intelligent and Fuzzy Systems Applications in Engineering and Technology25(4): 863-880.; · Zbl 1303.93112
[20] Li, J.M. and Zhang, G. (2012). Non-fragile guaranteed cost control of T-S fuzzy time-varying state and control delays systems with local bilinear models, IEEE Transactions on Systems, Man and Cybernetics9(2): 43-62.; · Zbl 1260.93102
[21] Li, T.H.S. and Tsai, S.H. (2007). T-S fuzzy bilinear model and fuzzy controller design for a class of nonlinear systems, IEEE Transactions on Fuzzy Systems15(3): 494-506.;
[22] Li, T.H.S. and Tsai, S.H. (2008). Robust \(H_∞\) fuzzy control for a class of uncertain discrete fuzzy bilinear systems, IEEE Transactions on Systems, Man and Cybernetics B: Cybernetics38(2): 510-527.;
[23] Li, T.H.S., Tsai, S.H., Lee, J.Z., Hsiao, M.Y. and Chao, C.H. (2008). Robust \(H_∞\) fuzzy control for a class of uncertain discrete fuzzy bilinear systems, IEEE Transactions on Systems, Man and Cybernetics B: Cybernetics38(2): 510-527.;
[24] Margaliot, M. and Langholz, G. (2003). A new approach to fuzzy modeling and control of discrete-time systems, IEEE Transactions on Fuzzy Systems11(4): 486-494.;
[25] Mohler, R.R. (1973). Bilinear Control Processes, Academic Press, New York, NY.; · Zbl 0343.93001
[26] Park, Y., Tahk, M.J. and Bang, H. (2004). Design and analysis of optimal controller for fuzzy systems with input constraint IEEE Transactions on Fuzzy System12(6): 766-779.;
[27] Qi, R.Y., Tao, G., Jiang, B. and Tan, C. (2012). Adaptive control schemes for discrete-time T-S fuzzy systems with unknown parameters and actuator failures, IEEE Transactions on Fuzzy Systems20(3): 471-486.;
[28] Qiu, J.B., Feng G. and Yang J. (2009). A new design of delay-dependent robust \(H_∞\) filtering for discrete-time T-S fuzzy systems with time-varying delay, IEEE Transactions on Fuzzy Systems17(5): 1044-1058.;
[29] Qiu, J.B., Feng G. and Gao H.J. (2010). Fuzzy-model-based piecewise \(H_∞\) static-output-feedback controller design for networked nonlinear systems, IEEE Transactions on Fuzzy Systems18(5): 919-934.;
[30] Siavash, F.D. and Alireza, F. (2014). Non-monotonic Lyapunov functions for stability analysis and stabilization of discrete time Takagi-Sugeno fuzzy systems, International Journal of Innovative Computing, Information and Control10(4): 1567-1586.;
[31] Su, X.J., Shi P.G., Wu L.G. and Song Y.-D. (2013). A novel control design on discrete-time Takagi-Sugeno fuzzy systems with time-varying delays, IEEE Transactions on Fuzzy Systems21(4): 655-671.;
[32] Su, X. J., Shi P., Wu L. and Basin M.V. (2014). Reliable filtering with strict dissipativity for T-S fuzzy time-delay systems, IEEE Transactions on Cybernetics44(12): 2470-2483, DOI: 10.1109/TCYB.2014.2308983.;
[33] Takagi, T. and Sugeno, M. (1985). Fuzzy identification of systems and its applications to modeling and control, IEEE Transactions on Systems, Man and Cybernetics15(1): 116-132.; · Zbl 0576.93021
[34] Tanaka, K. and Sugeno, M. (1992). Stability analysis and design of fuzzy control systems, Fuzzy Sets and Systems45(2): 135-156.; · Zbl 0758.93042
[35] Tanaka, K. and Wang, H.O. (2001). Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, Wiley, New York, NY.;
[36] Tong, S.C., He, X.L. and Zhang, H.C. (2009). A combined backstepping and small-gain approach to robust adaptive fuzzy output feedback control, IEEE Transactions on Fuzzy Systems17(5): 1059-1069.;
[37] Tong, S.C., Huo, B.Y. and Li, Y.M. (2014). Observer-based adaptive decentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failures, IEEE Transactions on Fuzzy Systems22(1): 1-15.;
[38] Tong, S.C., Liu, C.L. and Li, Y.M. (2010). Fuzzy-adaptive decentralized output-feedback control for large-scale nonlinear systems with dynamical uncertainties, IEEE Transactions on Fuzzy Systems18(5): 845-861.;
[39] Tong, S.C. and Li, Y.M. (2012). Adaptive fuzzy output feedback tracking backstepping control of strict-feedback nonlinear systems with unknown dead zones, IEEE Transactions on Fuzzy Systems20(1): 168-180.;
[40] Tong, S., Yang, G. and Zhang, W. (2011). Observer-based fault-tolerant control against sensor failures for fuzzy systems with time delays, International Journal of Applied Mathematics and Computer Science21(4): 617-627, DOI: 10.2478/v10006-011-0048-4.; · Zbl 1283.93166
[41] Wang, J. (2014). Adaptive fuzzy control of direct-current motor dead-zone systems, International Journal of Innovative Computing, Information and Control10(4): 1391-1399.;
[42] Yan, H.C., Zhang, H., Shi, H.B. and Meng, M.Q.-H. (2010). \(H_∞\) fuzzy filtering for discrete-time fuzzy stochastic systems with time-varying delay, IEEE 29th Chinese Control Conference, Beijing, China, pp. 59993-59998.;
[43] Zhang, G. and Li, J.M. (2010). Non-fragile guaranteed cost control of discrete-time fuzzy bilinear system, Journal of Systems Engineering and Electronics21(4): 629-634.;
[44] Zhang, H.G. (2009). Fuzzy Hyperbolic Model: Modeling Control and Applications, Science Press, Beijing, pp. 121-131.;
[45] Zhang, H.G. and Quan, Y.B. (2001). Modeling, identification and control of a class of nonlinear system, IEEE Transactions on Fuzzy Systems9(2): 349-354.;
[46] Zhang, H., Shi, Y., and Mehr, A.S. (2012). On filtering for discrete-time Takagi-Sugeno fuzzy systems, IEEE Transactions on Fuzzy Systems20(2): 396-401.;
[47] Zhao, Y., and Gao, H.J. (2012). Fuzzy-model-based control of an overhead crane with input delay and actuator saturation Transactions on Fuzzy Systems20(1): 181-186.;
[48] Zhao, T., Xiao, J., Li, Y. and Li, Y.X. (2013). A fuzzy Lyapunov function approach to stabilization of interval type-2 T-S fuzzy systems, IEEE 25th Chinese Control and Decision Conference, Xian, China, pp. 2234-2238.;
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