Further results on robust fuzzy dynamic systems with LMI D-stability constraints

Assawinchaichote, W.

International Journal of Applied Mathematics and Computer Science, Tome 24 (2014) no. 4, pp. 785-794.

Résumé

This paper examines the problem of designing a robust H∞ fuzzy controller with D-stability constraints for a class of nonlinear dynamic systems which is described by a Takagi–Sugeno (TS) fuzzy model. Fuzzy modelling is a multi-model approach in which simple sub-models are combined to determine the global behavior of the system. Based on a linear matrix inequality (LMI) approach, we develop a robust H∞ fuzzy controller that guarantees (i) the L₂-gain of the mapping from the exogenous input noise to the regulated output to be less than some prescribed value, and (ii) the closed-loop poles of each local system to be within a specified stability region. Sufficient conditions for the controller are given in terms of LMIs. Finally, to show the effectiveness of the designed approach, an example is provided to illustrate the use of the proposed methodology.

Détail

Keywords: fuzzy controller, robust control, LMI approach, D-stability, Takagi-Sugeno fuzzy model
Mots-clés : regulator rozmyty, sterowanie odporne, stabilność, model Takagi-Sugeno

Bibliographie

[1] Assawinchaichote, W. (2012). A non-fragile H∞ output feedback controller for uncertain fuzzy dynamical systems with multiple time-scales, International Journal Computers, Communications Control 7(1): 8–16.

[2] Assawinchaichote, W. and Chayaopas, N. (2013). Robust H∞ fuzzy speed control design for brushless DC motor, International Conference on Computer, Electrical, and Systems Sciences, and Engineering, Tokyo, Japan, pp. 1592–1598.

[3] Assawinchaichote, W. and Nguang, S.K. (2004a). H∞ filtering for fuzzy singularly perturbed systems with pole placement constraints: An LMI approach, IEEE Transactions on Signal Processing 52(6): 1659–1667.

[4] Assawinchaichote, W. and Nguang, S.K. (2004b). H∞ fuzzy control design for nonlinear singularly perturbed systems with pole placement constraints: An LMI approach, IEEE Transactions on Systems, Man, and Cybernetics: Part B 34(1): 579–588.

[5] Assawinchaichote, W. and Nguang, S.K. (2006). Fuzzy H∞ output feedback control design for singularly perturbed systems with pole placement constraints: An LMI approach, IEEE Transactions Fuzzy Systems 14(3): 361–371.

[6] Ball, J.A., Helton, W.J. and Walker, M.L. (1993). H∞ control for nonlinear systems with output feedback, IEEE Transactions on Automatic Control 38(4): 546–559.

[7] Ball, J.A., Helton, W.J. and Walker, M.L. (1994). H∞ control of systems under norm bounded uncertainties in all systems matrices, IEEE Transactions on Automatic Control 39(6): 1320–1322.

[8] Basar, T. and Olsder, G.J. (1982). Dynamic Noncooperative Game Theory, Academic Press, New York, NY.

[9] Bouarar, T., Guelton, K. and Manamanni, N. (2013). Robust non-quadratic static output feedback controller design for Takagi–Sugeno systems using descriptor redundancy, Engineering Applications of Artificial Intelligence 26(42): 739–756.

[10] Boyd, S., Ghaoui, L.E., Feron, E. and Balakrishnan, V. (1994). Linear Matrix Inequalities in Systems and Control Theory, SIAM Books, Philadelphia, PA.

[11] Chayaopas, N. and Assawinchaichote, W. (2013). Speed control of brushless DC mortor with H∞ fuzzy controller based on LMI approach, International Conference Modelling, Indentification and Control, Phuket, Thailand, pp. 21–26.

[12] Chen, B.-S., Tseng, C.-S. and Uang, H.-J. (2000). Mixed H2/H∞ fuzzy output feedback control design for nonlinear dynamic systems: An LMI approach, IEEE Transactions on Fuzzy Systems 8(3): 249–265.

[13] Chilali, M. and Gahinet, P. (1996). H∞ design with pole placement constraints: An LMI approach, IEEE Transactions on Automatic Control 41(3): 358–367.

[14] Chilali, M., Gahinet, P. and Apkarian, P. (1999). Robust pole placement in LMI regions, IEEE Transactions on Automatic Control 44(12): 2257–2270.

[15] Fu, M., de Souza, C.E. and Xie, L. (1992). H∞ estimation for uncertain systems, International Journal of Robust and Nonlinear Control 2(1): 87–105.

[16] Gahinet, P., Nemirovski, A., Laub, A.J. and Chilali, M. (1995). LMI Control Toolbox—For Use with MATLAB, The MathWorks, Inc., Natick, MA.

[17] Han, Z.X. and Feng, G. (1998). State-feedback H∞ controllers design for fuzzy dynamic system using LMI technique, IEEE World Congress on Computational Intelligence, Anchorage, AL, USA, pp. 538–544.

[18] Han, Z.X., Feng, G., Walcott, B.L. and Zhang, Y.M. (2000). H∞ controller design of fuzzy dynamic systems with pole placement constraints, American Control Conference, Chicago, IL, USA, pp. 1939–1943.

[19] Hill, D.J. and Moylan, P.J. (1980). Dissipative dynamical systems: Basic input-output and state properties, Journal of the Franklin Institute 309(5): 327–357.

[20] Isidori, A. and Astolfi, A. (1992). Disturbance attenuation and H∞-control via measurement feedback in nonlinear systems, IEEE Transactions on Automatic Control 37(9): 1283–1293.

[21] Joh, J., Chen, Y.H. and Langari, R. (1998). On the stability issues of linear Takagi–Sugeno fuzzy models, IEEE Transactions on Fuzzy Systems 6(3): 402–410.

[22] Ma, X.J., Qi Sun, Z. and He, Y.Y. (1998). Analysis and design of fuzzy controller and fuzzy observer, IEEE Transactions on Fuzzy Systems 6(1): 41–51.

[23] Mansouri, B., Manamanni, N., Guelton, K., Kruszewski, A. and Guerra, T. (2009). Output feedback LMI tracking control conditions with H∞ criterion for uncertain and disturbed T–S models, Journal of the Franklin Institute 179(4): 446–457.

[24] Park, J., Kim, J. and Park, D. (2001). LMI-based design of stabilizing fuzzy controller for nonlinear system described by Takagi–Sugeno fuzzy model, Fuzzy Sets and Systems 122(1): 73–82.

[25] Rezac, M. and Hurak, Z. (2013). Structured MIMO design for dual-stage inertial stabilization: Case study for HIFOO and Hinfstruct solvers, Physics Procedia 23(8): 1084–1093.

[26] Scherer, C., Gahinet, P. and Chilali, M. (1997). Multiobjective output-feedback control via LMI optimization, IEEE Transactions on Automatic Control 42(7): 896–911.

[27] Tanaka, K., Ikeda, T. and Wang, H.O. (1996). Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: Quadratic stabilizability, H∞ control theory, and linear matrix inequality, IEEE Transactions on Fuzzy Systems 4(1): 1–13.

[28] Tanaka, K. and Sugeno, M. (1992). Stability analysis and design of fuzzy control systems, Fuzzy Sets and Systems 45(2): 135–156.

[29] Tanaka, K. and Sugeno, M. (1995). Stability and stabiliability of fuzzy neural linear control systems, IEEE Transactions on Fuzzy Systems 3(4): 438–447.

[30] Teixeira, M. and Zak, S.H. (1999). Stabilizing controller design for uncertain nonlinear systems using fuzzy models, IEEE Transactions on Fuzzy Systems 7(2): 133–142.

[31] van der Schaft, A.J. (1992). L2-gain analysis of nonlinear systems and nonlinear state feedback H∞ control, IEEE Transactions on Automatic Control 37(6): 770–784.

[32] Vesely, V., Rosinova, D. and Kucera, V. (2011). Robust static output feedback controller LMI based design via elimination, Journal of the Franklin Institute 348(9): 2468–2479.

[33] Wang, H.O., Tanaka, K. and Griffin, M.F. (1996). An approach to fuzzy control of nonlinear systems: Stability and design issues, IEEE Transactions on Fuzzy Systems 4(1): 14–23.

[34] Willems, J.C. (1972). Dissipative dynamical systems, Part I: General theory, Archive for Rational Mechanics and Analysis 45(5): 321–351.

[35] Wonham, W.M. (1970). Random differential equations in control theory, Probabilistic Methods in Applied Mathematics 2(3): 131–212.

[36] Yakubovich, V.A. (1967a). The method of matrix inequalities in the stability theory of nonlinear control system I, Automation and Remote Control 25(4): 905–917.

[37] Yakubovich, V.A. (1967b). The method of matrix inequalities in the stability theory of nonlinear control system II, Automation and Remote Control 26(4): 577–592.

[38] Yeh, K., Chen, C., Chen, C., Lo, D. and Chung, P. (2012). A fuzzy Lyapunov LMI criterion to a chaotic system, Physics Procedia 25(1): 262–269.

[39] Yoneyama, J., Nishikawa, M., Katayama, H. and Ichikawa, A. (2000). Output stabilization of Takagi–Sugeno fuzzy system, Fuzzy Sets and Systems 111(2): 253–266.

[40] Zadeh, L.A. (1965). Fuzzy set, Information and Control 8(3): 338–353.

[41] Zhang, J.M., Li, R.H. and Zhang, P.A. (2001). Stability analysis and systematic design of fuzzy control system, Fuzzy Sets and Systems 120(1): 65–72.