BIBO stabilisation of continuous-time Takagi–Sugeno systems under persistent perturbations and input saturation

Salcedo, J. V. ; Martínez, M. ; García-Nieto, S. ; Hilario, A.

International Journal of Applied Mathematics and Computer Science, Tome 28 (2018) no. 3, pp. 457-472.

Résumé

This paper presents a novel approach to the design of fuzzy state feedback controllers for continuous-time non-linear systems with input saturation under persistent perturbations. It is assumed that all the states of the Takagi–Sugeno (TS) fuzzy model representing a non-linear system are measurable. Such controllers achieve bounded input bounded output (BIBO) stabilisation in closed loop based on the computation of inescapable ellipsoids. These ellipsoids are computed with linear matrix inequalities (LMIs) that guarantee stabilisation with input saturation and persistent perturbations. In particular, two kinds of inescapable ellipsoids are computed when solving a multiobjective optimization problem: the maximum volume inescapable ellipsoids contained inside the validity domain of the TS fuzzy model and the smallest inescapable ellipsoids which guarantee a minimum ^*-norm (upper bound of the 1-norm) of the perturbed system. For every initial point contained in the maximum volume ellipsoid, the closed loop will enter the minimum ^*-norm ellipsoid after a finite time, and it will remain inside afterwards. Consequently, the designed controllers have a large domain of validity and ensure a small value for the 1-norm of closed loop.

Détail

Keywords: linear matrix inequalities, fuzzy system, nonlinear system, input saturation, disturbance
Mots-clés : liniowe nierówności macierzowe, układ rozmyty, układ liniowy, nasycenie wejściowe

Bibliographie

[1] Abdelmalek, I., Goléa, N. and Hadjili, M.L. (2007). A new fuzzy Lyapunov approach to non-quadratic stabilization of Takagi-Sugeno fuzzy models, International Journal of Applied Mathematics and Computer Science 17(1): 39–51, DOI: 10.2478/v10006-007-0005-4.

[2] Abedor, J., Nagpal, K. and Poolla, K. (1996). Linear matrix inequality approach to peak-to-peak gain minimization, International Journal of Robust and Nonlinear Control 6(9–10): 899–927.

[3] Bai, J., Lu, R., Liu, X., Xue, A. and Shi, Z. (2015). Fuzzy regional pole placement based on fuzzy Lyapunov functions, Neurocomputing 167: 467–473.

[4] Benzaouia, A., El Hajjaji, A., Hmamed, A. and Oubah, R. (2015). Fault tolerant saturated control for T–S fuzzy discrete-time systems with delays, Nonlinear Analysis: Hybrid Systems 18: 60–71.

[5] Bezzaoucha, S., Marx, B., Maquin, D. and Ragot, J. (2013). Stabilization of nonlinear systems subject to actuator saturation, Proceedings of the IEEE International Conference Fuzzy Systems (FUZZ-IEEE), Hyderabad, India, pp. 1–6.

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

[7] Chang, W.-J. and Shih, Y.-J. (2015). Fuzzy control of multiplicative noised nonlinear systems subject to actuator saturation and h∞ performance constraints, Neurocomputing 148: 512–520.

[8] Chen, C.W. (2006). Stability conditions of fuzzy systems and its application to structural and mechanical systems, Advances in Engineering Software 37(9): 624–629.

[9] da Silva, J.G., Castelan, E., Corso, J. and Eckhard, D. (2013). Dynamic output feedback stabilization for systems with sector-bounded nonlinearities and saturating actuators, Journal of The Franklin Institute 350(3): 464–484.

[10] Du, H. and Zhang, N. (2009). Fuzzy control for nonlinear uncertain electrohydraulic active suspensions with input constraint, IEEE Transactions on Fuzzy Systems 17(2): 343–356.

[11] Duan, R., Li, J., Zhang, Y., Yang, Y. and Chen, G. (2016). Stability analysis and H∞ control of discrete T–S fuzzy hyperbolic systems, International Journal of Applied Mathematics and Computer Science 26(1): 133–145, DOI: 10.1515/amcs-2016-0009.

[12] El Ghaoui, L. and Niculescu, S. (Eds.) (2000). Advances in Linear Matrix Inequality Methods in Control, SIAM, Philadelphia, PA.

[13] Gahinet, P., Nemirovski, A., Laub, A.J. and Chilali, M. (1995). LMI control toolbox, Technical report, The Mathworks Inc., Natick, MA.

[14] Goh, K.G., Safonov, M.G. and Ly, J.H. (1996). Robust synthesis via bilinear matrix inequalities, International Journal of Robust and Nonlinear Control 6: 1079–1095.

[15] Guerra, T.M., Bernal, M., Guelton, K. and Labiod, S. (2012). Non-quadratic local stabilization for continuous-time Takagi–Sugeno models, Fuzzy Sets and Systems 201: 40–54.

[16] Guerra, T.M., Kruszewski, A., Vermeiren, L. and Tirmant, H. (2006). Conditions of output stabilization for nonlinear models in the Takagi–Sugeno’s form, Fuzzy Sets and Systems 157: 1248–1259.

[17] Jaadari, A., Guerra, T.M., Sala, A., Bernal, M. and Guelton, K. (2012). New controllers and new designs for continuous-time Takagi–Sugeno models, Proceedings of the IEEE Mathematical Conference on Fuzzy Systems (FUZZ-IEEE), Brisbane, Australia, pp. 1–7.

[18] Klug, M., Castelan, E.B., Leite, V.J. and Silva, L.F. (2015). Fuzzy dynamic output feedback control through nonlinear Takagi–Sugeno models, Fuzzy Sets and Systems 263: 92–111.

[19] Liu, X. and Zhang, Q. (2003). Approaches to quadratic stability conditions and H∞ control designs for T–S fuzzy systems, IEEE Transactions on Fuzzy Systems 11(6): 830–838.

[20] Liu, Y., Wu, F. and Ban, X. (2017). Dynamic output feedback control for continuous-time T–S fuzzy systems using fuzzy Lyapunov functions, IEEE Transactions on Fuzzy Systems 25(5): 1155–1167.

[21] Löfberg, J. (2004). YALMIP: A toolbox for modeling and optimization in MATLAB, Proceedings of the CACSD Conference, Taipei, Taiwan, pp. 284–289.

[22] Nguyen, A., Dambrine, M. and Lauber, J. (2014). Lyapunov-based robust control design for a class of switching non-linear systems subject to input saturation: Application to engine control, IET Control Theory and Applications 8(17): 1789–1802.

[23] Nguyen, A., Dequidt, A. and Dambrine,M. (2015). Anti-windup based dynamic output feedback controller design with performance consideration for constrained Takagi–Sugeno systems, Engineering Applications of Artificial Intelligence 40: 76–83.

[24] Nguyen, A.-T., Laurain, T., Palhares, R., Lauber, J., Sentouh, C. and Popieul, J.-C. (2016). LMI-based control synthesis of constrained Takagi–Sugeno fuzzy systems subject to L2 or L∞ disturbances, Neurocomputing 207: 793–804.

[25] Nguyen, A.-T., Márquez, R. and Dequidt, A. (2017). An augmented system approach for LMI-based control design of constrained Takagi–Sugeno fuzzy systems, Engineering Applications of Artificial Intelligence 61: 96–102.

[26] Pan, J., Fei, S., Guerra, T. and Jaadari, A. (2012). Non-quadratic local stabilisation for continuous-time Takagi–Sugeno fuzzy models: A descriptor system method, IET Control Theory and Applications 6(12): 1909–1917.

[27] Qiu, J., Tian, H., Lu, Q. and Gao, H. (2013). Nonsynchronized robust filtering design for continuous-time T–S fuzzy affine dynamic systems based on piecewise Lyapunov functions, IEEE Transactions on Cybernetics 43(6): 1755–1766.

[28] Qiu, J., Wei, Y. and Wu, L. (2017). A novel approach to reliable control of piecewise affine systems with actuator faults, IEEE Transactions on Circuits and Systems II: Express Briefs 64(8): 957–961.

[29] Saifia, D., Chadli, M., Labiod, S. and Guerra, T.M. (2012). Robust h∞ static output feedback stabilization of TS fuzzy systems subject to actuator saturation, International Journal of Control, Automation and Systems 10(3): 613–622.

[30] Salcedo, J. and Martinez, M. (2008). BIBO stabilisation of Takagi–Sugeno fuzzy systems under persistent perturbations using fuzzy output-feedback controllers, IET Control Theory and Applications 2(6): 513–523.

[31] Salcedo, J., Martínez, M., Blasco, X. and Sanchis, J. (2007). BIBO fuzzy stabilization of nonlinear systems under persistent perturbations, Proceedings of the European Control Conference, Kos, Greece, pp. 763–769.

[32] Salcedo, J. V., Martínez, M. and García-Nieto, S. (2008). Stabilization conditions of fuzzy systems under persistent perturbations and their application in nonlinear systems, Engineering Applications of Artificial Intelligence 21(8): 1264–1276.

[33] Sanchez Peña, R.S. and Sznaier, M. (1998). Robust Systems: Theory and Applications, Wiley, New York, NY.

[34] Sturm, J.F. (1999). Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optimization Methods and Software 11–12(1–4): 625–653.

[35] Takagi, T. and Sugeno, M. (1985). Fuzzy identification of systems and its applications to modeling and control, IEEE Transactions on Systems, Man, and Cybernetics 15(1): 116–132.

[36] Tanaka, K., Ikeda, T. and Wang, H. (1998). Fuzzy regulators and fuzzy observers: Relaxed stability conditions and LMI-based designs, IEEE Transactions on Fuzzy Systems 6(2): 250–265.

[37] Tanaka, K. and Wang, H.O. (2001). Fuzzy Control Systems. Design and Analysis. A Linear Matrix Inequality Approach, Wiley, New York, NY.

[38] Teixeira, M.C.M., Assunçao, E. and Avellar, R.G. (2003). On relaxed LMI-based designs for fuzzy regulators and fuzzy observers, IEEE Transactions on Fuzzy Systems 11(5): 613–623.

[39] Tognetti, E.S. and Oliveira, V.A. (2010). Fuzzy pole placement based on piecewise Lyapunov functions, International Journal of Robust and Nonlinear Control 20(5): 571–578.

[40] Tuan, H., Apkarian, P., Narikiyo, T. and Yamamoto, Y. (2001). Parameterized linear matrix inequality techniques in fuzzy control system design, IEEE Transactions on Fuzzy Systems 9(2): 324–332.

[41] Vafamand, N., Asemani,M.H. and Khayatian, A. (2017). Robust l1 observer-based non-PDC controller design for persistent bounded disturbed TS fuzzy systems, IEEE Transactions on Fuzzy Systems 26(3): 1401–1413.

[42] Vafamand, N., Asemani, M.H. and Khayatian, A. (2017b). TS fuzzy robust L1 control for nonlinear systems with persistent bounded disturbances, Journal of The Franklin Institute 354(14): 5854–5876.

[43] Vafamand, N., Asemani, M.H. and Khayatiyan, A. (2016). A robust L1 controller design for continuous-time TS systems with persistent bounded disturbance and actuator saturation, Engineering Applications of Artificial Intelligence 56: 212–221.

[44] Yang, W. and Tong, S. (2015). Output feedback robust stabilization of switched fuzzy systems with time-delay and actuator saturation, Neurocomputing 164: 173–181.

[45] Zhao, Y. and Gao, H. (2012). Fuzzy-model-based control of an overhead crane with input delay and actuator saturation, IEEE Transactions on Fuzzy Systems 20(1): 181–186.