Robust H State-Feedback Control for Uncertain Fuzzy Markovian Jump Systems: LMI-Based Design

Authors: Wudhichai Assawinchaichote, Sing Kiong Nguang

Abstract:

This paper investigates the problem of designing a robust state-feedback controller for a class of uncertain Markovian jump nonlinear systems that guarantees the L2-gain from an exogenous input to a regulated output is less than or equal to a prescribed value. First, we approximate this class of uncertain Markovian jump nonlinear systems by a class of uncertain Takagi-Sugeno fuzzy models with Markovian jumps. Then, based on an LMI approach, LMI-based sufficient conditions for the uncertain Markovian jump nonlinear systems to have an H performance are derived. An illustrative example is used to illustrate the effectiveness of the proposed design techniques.

Keywords: Robust H, Fuzzy Control, Markovian Jump Systems, LMI.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1328140

References:

[1] S.P. Sethi and Q. Zhang, Hierarchical Decision Making in Stochastic Manufacturing Systems, Birkhauser, Boston, 1994.
[2] M. Mariton, Jump Linear Systems in Automatic Control, Dekker, New York, 1990.
[3] E.K. Boukas and A. Haurie, "Manufacturing flow control and preventive maintenance: A stochastic control approach", IEEE Trans. Automat. Contr., vol. 35, pp. 1024-1031, 1990.
[4] E.K. Boukas, Q. Zhang, and G. Yin, "Robust production and maintenance planning in stochastic manufacturing systems", IEEE Trans. Automat. Contr., vol. 40, pp. 1098-1102, 1995.
[5] E.K. Boukas and P. Shi, "Stochastic stability and guaranteed cost control of discrete-time uncertain systems with Markovian jumping parameters", Internat. J. Robust Nonlinear Contr., vol. 8, pp. 1155-1167, 1998.
[6] O.L.V. Costa and M.D. Fragoso, "Stability results for discrete-time linear systems with Markovian jumping parameters", J. Math. Anal. Appl., vol. 179, pp. 154-178, 1993.
[7] V. Dragan, P. Shi, and E.K. Boukas, "Control of singularly perturbed systems with Markovian jump parameters: an H approach", Automatica, vol. 35, pp. 1369-1378, 1999.
[8] K. Ito and H.P. McKean, Diffusion Processes and Their Sample Paths, Springer-Verlag, Berlin, 1965.
[9] Y. Ji and H.J. Chizeck, "Controllability, stabilizability, and continuoustime Markovian jump linear quadratic control", IEEE Trans. Automat. Contr., vol. 35, pp. 777-788, 1990.
[10] N.N. Krasovskii and E.A. Lidskii, "Analysis design of controller in systems with random attributes-Part 1", Automat. Remote Contr., vol. 22, pp. 1021-1025, 1961.
[11] P. Shi and E.K. Boukas, "H control for Markovian jumping linear systems with parametric uncertainty", J. Optim. Theory Appl. vol. 95, pp. 75-99, 1997.
[12] E. K. Boukas, Peng Shi, S.K. Nguang, and R.K. Agarwal, "Robust H control of a class of nonlinear systems with markovian jumping parameters", IEEE American Contr. Conf., pp. 970-976, 1999.
[13] C.E. de Souza and M.D. Fragoso, "H control for linear systems with Markovian jumping parameters", Control Theory Adv. Technol. vol. 9, pp. 457-466, 1993.
[14] M.D.S. Aliyu and E.K. Boukas, "H control for markovian jump nonlinear systems", 37th IEEE Conf. on Decision and Contr., pp. 766- 771, 1998.
[15] K. Tanaka and M. Sugeno, "Stability analysis and design of fuzzy control systems", Fuzzy Sets Systs., vol. 45, pp. 135-156, 1992.
[16] K. Tanaka, "Stability and stabiliability of fuzzy neural linear control systems", IEEE Trans. Fuzzy Syst., vol. 3, pp. 438-447, 1995.
[17] K. Tanaka, T. Ikeda, and H. O. Wang, "Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: Quadratic stabilizability, H control theory, and linear matrix inequality", IEEE Trans. Fuzzy Syst., vol. 4, pp. 1-13, 1996.
[18] H. O. Wang, K. Tanaka, and M. F. Griffin, "An approach to fuzzy control of nonlinear systems: Stability and design issues", IEEE Trans. Fuzzy Syst., vol. 4, pp. 14-23, 1996.
[19] S. G. Cao, N. W. Ree, and G. Feng, "Quadratic stability analysis and design of continuous-time fuzzy control systems", Int. J. Syst. Sci., vol. 27, pp. 193-203, 1996.
[20] W. Assawinchaichote and S.K. Nguang, "Fuzzy observer-based controller design for singularly perturbed nonlinear systems: An LMI approach", IEEE Conf. on Decision and Contr., pp. 2165-2170, 2002.
[21] T. Takagi and M. Sugeno, "Fuzzy identification of systems and its applications to modelling and control", IEEE Trans. Syst. Man. Cybern., vol. 15, pp. 116-132, 1985.
[22] C. L. Chen, P. C. Chen, and C. K. Chen, "Analysis and design of fuzzy control system", Fuzzy Sets Systs., vol. 57, pp. 125-140, 1995.
[23] X. J. Ma, Z. Q Sun, and Y. Y He, "Analysis and design of fuzzy controller and fuzzy observer", IEEE Trans. Fuzzy Syst., vol. 6, pp. 41- 51, 1998.
[24] B. S. Chen, C. S. Tseng, and H. J. Uang, "Mixed H2/H fuzzy output feedback control design for nonlinear dynamic systems: An LMI approach", IEEE Trans. Fuzzy Syst., vol. 8, pp. 249-265, 2000.
[25] S.K. Nguang and P. Shi, "Stabilisation of a class of nonlinear time-delay systems using fuzzy models", Proc. 39th IEEE Conf. on Decision and Contr., pp. 4415-4419, 2000.
[26] S. K. Nguang and P. Shi, "H fuzzy output feedback control design for nonlinear systems: An LMI approach", 40th IEEE Conf. on Decision and Contr., pp. 2501-2506, 2001.
[27] S. K. Nguang and P. Shi, "Fuzzy H output feedback control of nonlinear systems under sampled measurements", 40th IEEE IEEE Conf. on Decision and Contr., pp. 120-126, 2001.
[28] M. Teixeira and S. H. Zak, "Stabilizing controller design for uncertain nonlinear systems using fuzzy models", IEEE Trans. Fuzzy Syst., vol. 7, pp. 133-142, 1999.
[29] S. H. Zak, "Stabilizing fuzzy system models using linear controllers", IEEE Trans. Fuzzy Syst., vol. 7, pp. 236-240, 1999.
[30] L. X. Wang, A course in fuzzy systems and control. Englewood Cliffs, NJ: Prentice-Hall, 1997.
[31] S. K. Nguang and P. Shi, "H fuzzy output feedback control design for nonlinear systems: An LMI approach ," IEEE Trans. Fuzzy Syst., vol. 11, pp. 331-340, 2003.
[32] S. K. Nguang and W. Assawinchaichote, "H filtering for fuzzy dynamic systems with pole placement," IEEE Trans. Circuits Systs. I, vol. 50, pp. 1503-1508, 2003.
[33] W.Assawinchaichote and S. K. Nguang, "H filtering for nonlinear singularly perturbed systems with pole placement constraints: An LMI approach", IEEE Trans. Signal Processing, vol. 52, pp. 579-588, 2004.
[34] W. M. Wonham, "Random differential equations in control theory," Probabilistic Methods in App. Math., vol. 2, pp. 131-212, 1970.
[35] E. B. Dynkin, Markov Process, Berlin: Springer-Verlag, 1965.
[36] H. E. Nusse amd C. H. Hommes, "Resolution of chaos with application to a modified Samuelson model," J. of Economic Dyn. Contr., vol. 14, pp. 1-19, 1990.
[37] W. P. Blair, Jr. and D. D. Sworder, "Continuous-time regulation of a class of econometric models," IEEE Trans. Sys. Man and Cybern., vol. 5, pp. 341-346, 1975.