Robust fuzzy control for uncertain discrete-time nonlinear Markovian jump systems without mode observations. (English) Zbl 1120.93337
Summary: This paper studies the robust fuzzy control problem of uncertain discrete-time nonlinear Markovian jump systems without mode observations. The Takagi and Sugeno (T — S) fuzzy model is employed to represent a discrete-time nonlinear system with norm-bounded parameter uncertainties and Markovian jump parameters. As a result, an uncertain Markovian jump fuzzy system (MJFS) is obtained. A stochastic fuzzy Lyapunov function (FLF) is employed to analyze the robust stability of the uncertain MJFS, which not only is dependent on the operation modes of the system, but also directly includes the membership functions. Then, based on this stochastic FLF and a non-parallel distributed compensation (non-PDC) scheme, a mode-independent state-feedback control design is developed to guarantee that the closed-loop MJFS is stochastically stable for all admissible parameter uncertainties. The proposed sufficient conditions for the robust stability and mode-independent robust stabilization are formulated as a set of coupled linear matrix inequalities (LMIs), which can be solved efficiently by using existing LMI optimization techniques. Finally, it is also demonstrated, via a simulation example, that the proposed design method is effective.
MSC:
| 93C42 | Fuzzy control/observation systems |
| 93C55 | Discrete-time control/observation systems |
| 93C41 | Control/observation systems with incomplete information |
| 93C10 | Nonlinear systems in control theory |
| 93E15 | Stochastic stability in control theory |
| 93B12 | Variable structure systems |
Keywords:
uncertain nonlinear systems; Markovian jump parameters; robust control; fuzzy control; linear matrix inequality (LMI); stochastic stabilitySoftware:
LMI toolboxReferences:
| [1] | Boukas, E. K.; Shi, P., Stochastic stability and guaranteed cost control of discrete-time uncertain systems with Markovian jumping parameters, Int. J. Robust Nonlinear Control, 8, 13, 1155-1167 (1998) · Zbl 0918.93060 |
| [2] | Boyd, S.; Ghaoui, L. E.; Feron, E.; Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory (1994), SIAM: SIAM Philadelphia, PA · Zbl 0816.93004 |
| [3] | Cao, S. G.; Rees, N. W.; Feng, G.; Liu, W., \(H_∞\) control of nonlinear discrete-time systems based on dynamical fuzzy models, Int. J. Syst. Sci., 31, 229-241 (2000) · Zbl 1080.93518 |
| [4] | Cao, Y. Y.; Frank, P. M., Robust \(H_∞\) disturbance attenuation for a class of uncertain discrete-time fuzzy systems, IEEE Trans. Fuzzy Syst., 8, 4, 406-415 (2000) |
| [5] | Chen, B.-S.; Tseng, C.-S.; Uang, H.-J., Mixed \(H_2/H_∞\) fuzzy output feedback control design for nonlinear dynamic systems: an LMI approach, IEEE Trans. Fuzzy Syst., 8, 3, 249-265 (2000) |
| [6] | Costa, O. L.V.; do Val, J. B.R.; Geromel, J. C., A convex programming approach to \(H_2\)-control of discrete-time Markovian jump linear systems, Int. J. Control, 66, 4, 557-579 (1997) · Zbl 0951.93536 |
| [7] | Costa, O. L.V.; Fragoso, M. D., Stability results for discrete-time linear systems with Markovian jumping parameters, J. Math. Anal. Appl., 179, 2, 154-178 (1993) · Zbl 0790.93108 |
| [8] | Costa, O. L.V.; Marques, R. P., Mixed \(H_2/H_∞\) control of discrete-time Markovian linear systems, IEEE Trans. Automat. Contr., 43, 1, 95-100 (1998) · Zbl 0907.93062 |
| [9] | El Ghaoui, L.; Aitrami, M., Robust state-feedback stabilization of jump linear systems via LMIs, Int. J. Robust Nonlinear Control, 6, 9-10, 1015-1022 (1996) · Zbl 0863.93067 |
| [10] | Fragoso, M. D.; do Val, J. B.R.; Pinto, D. L., Jump linear \(H_∞\) control: the discrete-time case, Control Theory and Adv. Technol., 10, 4, 1459-1474 (1995) |
| [11] | Gahinet, P.; Nemirovski, A.; Laub, A. J.; Chilali, M., LMI Control Toolbox (1995), Math Works Inc.: Math Works Inc. Natick, MA |
| [12] | Gao, H.; Wang, Z.; Wang, C., Improved \(H_∞\) control of discrete-time fuzzy systems: a cone complementarity linearization approach, Inform. Sci., 175, 1-2, 57-77 (2005) · Zbl 1113.93308 |
| [13] | 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 |
| [14] | Ji, Y.; Chezick, H. J., Controllability, observability and discrete-time Markovian jump linear quadratic control, Int. J. Control, 48, 2, 481-498 (1988) · Zbl 0669.93007 |
| [15] | Ji, Y.; Chezick, H. J., Jump linear quadratic Gaussian control: steady-state solution and testable conditions, Control Theory Adv. Technol., 6, 3, 289-319 (1990) |
| [16] | Ji, Y.; Chezick, H. J.; Feng, X.; Loparo, K. A., Stability and control of discrete-time jump linear systems, Control Theory Adv. Technol., 7, 2, 247-270 (1991) |
| [17] | Lee, H. J.; Park, J. B.; Chen, G., Robust fuzzy control of nonlinear systems with parametric uncertainties, IEEE Trans. Fuzzy Syst., 9, 2, 369-379 (2001) |
| [18] | Mariton, M., Jump Linear Systems in Automatic Control (1990), Marcel Dekker: Marcel Dekker New York |
| [19] | Pan, G.; Bar-Shalom, Y., Stabilization of jump linear gaussian systems without mode observations, Int. J. Control, 64, 4, 631-661 (1996) · Zbl 0857.93095 |
| [20] | Seiler, P.; Sengupta, R., A bounded real lemma for jump systems, IEEE Trans. Automat. Contr., 48, 9, 1651-1654 (2003) · Zbl 1364.93223 |
| [21] | Shi, P.; Boukas, E. K., \(H_∞\)-control for Markovian jumping linear systems with parametric uncertainty, J. Optim. Theory Appl., 95, 1, 75-99 (1997) · Zbl 1026.93504 |
| [22] | Shi, P.; Boukas, E. K.; Agarwal, R. K., Robust control for Markovian jump discrete-time systems, Int. J. Syst. Sci., 30, 8, 787-797 (1999) · Zbl 1112.93313 |
| [23] | Takagi, T.; Sugeno, M., Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybern., 15, 1, 116-132 (1985) · Zbl 0576.93021 |
| [24] | Tanaka, K.; Ikeda, T.; Wang, H. O., Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, \(H_∞\) control theory, and linear matrix inequalities, IEEE Trans. Fuzzy Syst., 4, 1, 1-13 (1996) |
| [25] | Tanaka, K.; Hori, T.; Wang, H. O., A multiple Lyapunov function approach to stabilization of fuzzy control systems, IEEE Trans. Fuzzy Syst., 11, 4, 582-589 (2003) |
| [26] | 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 Syst., 2, 2, 119-134 (1994) |
| [27] | Tanaka, K.; Wang, H. O., Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach (2001), Wiley: Wiley New York |
| [28] | Tuan, H. D.; Apkarian, P.; Narikiyo, T.; Yamamoto, Y., Parameterized linear matrix inequality techniques in fuzzy control system design, IEEE Trans. Fuzzy Syst., 9, 2, 324-332 (2001) |
| [29] | Wang, H. O.; Tanaka, K.; Griffin, M. F., An approach to fuzzy control of nonlinear systems: stability and design issues, IEEE Trans. Fuzzy Syst., 4, 1, 14-23 (1996) |
| [30] | Wang, Y.; Xie, L.; de Souza, C. E., Robust control for a class of uncertain nonlinear systems, Syst. Contr. Lett., 19, 139-149 (1992) · Zbl 0765.93015 |
| [31] | Wu, H.-N., Reliable LQ fuzzy control for nonlinear discrete-time systems via LMIs, IEEE Trans. Syst. Man Cybern.: Part B, 34, 2, 1270-1275 (2004) |
| [32] | Wu, H.-N.; Cai, K.-Y., \(H_2\) guaranteed cost fuzzy control for uncertain nonlinear systems via linear matrix inequalities, Fuzzy Sets Syst., 148, 411-429 (2004) · Zbl 1057.93030 |
| [33] | Wu, H.-N.; Zhang, H.-Y., Reliable mixed \(L_2/H_∞\) fuzzy static output feedback control for nonlinear systems with sensor faults, Automatica, 41, 11, 1925-1932 (2005) · Zbl 1086.93034 |
| [34] | Xiong, J.; Lam, J.; Gao, H.; Ho, D. W.C., On robust stabilization of Markovian jump systems with uncertain switching probabilities, Automatica, 41, 11, 1925-1932 (2005) |
| [35] | Zhou, S.; Feng, G.; Lam, J.; Xu, S., Robust \(H_∞\) control for discrete-time fuzzy systems via basis-dependent Lyapunov functions, Inform. Sci., 174, 3-4, 197-217 (2005) · Zbl 1113.93038 |
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