Optimal fuzzy control for a class of nonlinear systems. (English) Zbl 1264.93125

Summary: We present conditions suitable in design giving quadratic performances to stabilizing controllers for given class of continuous-time nonlinear systems, represented by Takagi-Sugeno models. Based on extended Lyapunov function and slack matrices, the design conditions are outlined in the terms of linear matrix inequalities to possess a stable structure closest to LQ performance, if premise variables are measurable. Simulation results illustrate the design procedure and demonstrate the performances of the proposed control design method.


93C42 Fuzzy control/observation systems
93D15 Stabilization of systems by feedback
49N10 Linear-quadratic optimal control problems
Full Text: DOI


[1] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Transactions on Systems, Man and Cybernetics, vol. 15, no. 1, pp. 116-132, 1985. · Zbl 0576.93021
[2] I. Abdelmalek, N. Goléa, and M. L. Hadjili, “A new fuzzy Lyapunov approach to non-quadratic stabilization of Takagi-Sugeno fuzzy models,” International Journal of Applied Mathematics and Computer Science, vol. 17, no. 1, pp. 39-51, 2007. · Zbl 1133.93032
[3] M. Johansson, A. Rantzer, and K. E. Årzén, “Piecewise quadratic stability of fuzzy systems,” IEEE Transactions on Fuzzy Systems, vol. 7, no. 6, pp. 713-722, 1999.
[4] L. A. Mozelli and R. M. Palhares, “Less conservative H\infty fuzzy control for discrete-time Takagi-Sugeno systems,” Mathematical Problems in Engineering, vol. 2011, Article ID 361640, 21 pages, 2011. · Zbl 1213.93111
[5] H. O. Wang, K. Tanaka, and M. F. Griffin, “An approach to fuzzy control of nonlinear systems: stability and design issues,” IEEE Transactions on Fuzzy Systems, vol. 4, no. 1, pp. 14-23, 1996.
[6] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1994. · Zbl 0816.93004
[7] F. Khaber, K. Zehar, and A. Hamzaoui, “State feed-back controller design via Takagi-Sugeno fuzzy model: LMI approach,” International Journal of Computational Intelligence, vol. 2, no. 3, pp. 148-153, 2005.
[8] K. Michels, F. Klawonn, R. Kruse, and A. Nürnberger, Fuzzy Control: Fundamentals, Stability and Design of Fuzzy Controllers, Springer, Berlin, Germany, 2006. · Zbl 1099.93025
[9] M. P. A. Santim, M. C. M. Teixeira, W. A. de Souza, R. Cardim, and E. Assunao, “Design of a Takagi-Sugeno fuzzy regulator for a set of operation points,” Mathematical Problems in Engineering, vol. 2012, Article ID 731298, 17 pages, 2012. · Zbl 1264.93133
[10] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analysis. A Linear Matrix Inequality Approach, John Wiley & Sons, New York, NY, USA, 2001.
[11] B. D. O. Anderson and J. B. Moore, Optimal Control Linear Quadratic Methods, Prentice-Hall, Englewood Cliffs, NJ, USA, 1989. · Zbl 0751.49013
[12] A. E. Bryson Jr. and Y. C. Ho, Applied Optimal Control. Optimization, Estimation, and Control, Taylor & Francis, New York, NY, USA, 1975.
[13] D. E. Kirk, Optimal Control Theory. An Introduction, Prentice Hall, Englewood Cliffs, NJ, USA, 1970.
[14] F. L. Lewis, Optimal Control, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1986.
[15] R. C. Dorf and R. H. Bishop, Modern Control Systems, Pearson Education, Upper Saddle River, NJ, USA, 2011. · Zbl 0163.11104
[16] D. Henry, “A norm-based point of view for fault diagnosis. Application to aerospace missions,” in Proceedings of the 8th European Workshop on Advanced Control and Diagnosis (ACD ’10), pp. 4-16, Ferrara, Italy, 2010.
[17] A. Zolghadri, “A redundancy-based strategy for safety management in a modern civil aircraft,” Control Engineering Practice, vol. 8, no. 5, pp. 545-554, 2000.
[18] D. Krokavec, “Convergence of action dependent dual heuristic dynamic programming algorithms in LQ control tasks,” in Intelligent Technologies-Theory and Application, New Trends in Intelligent Technologies, pp. 72-80, IOS Press, Amsterdam, The Netherlands, 2002.
[19] G. Pipeleers, B. Demeulenaere, J. Swevers, and L. Vandenberghe, “Extended LMI characterizations for stability and performance of linear systems,” Systems & Control Letters, vol. 58, no. 7, pp. 510-518, 2009. · Zbl 1166.93014
[20] W. Xie, “An equivalent LMI representation of bounded real lemma for continuous-time systems,” Journal of Inequalities and Applications, vol. 2008, Article ID 672905, 8 pages, 2008. · Zbl 1395.93239
[21] K. M. Passino and S. Yurkovich, Fuzzy Control, Addison-Wesley Longman, Berkeley, Calif, USA, 1998.
[22] P. Gerland, D. Groß, H. Schulte, and A. Kroll, “Robust adaptive fault detection using global state information and application to mobile working machines,” in Proceedings of the 1st Conference on Control and Fault-Tolerant Systems (SysTol ’10), pp. 813-818, Nice, France, October 2010.
[23] D. Peaucelle, D. Henrion, Y. Labit, and K. Taitz, User’s Guide for SeDuMi Interface 1. 04, LAAS-CNRS, Toulouse, France, 2002.
[24] Q.-G. Wang, Decoupling Control, vol. 285 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 2003. · Zbl 1010.93002
[25] D. Krokavec, A. Filasova, and V. Hladky, “Stabilizing fuzzy control for a class of nonlinear systems,” in Proceedings of the 10th Jubilee International Symposium on Applied Machine Intelligence and Informatics (SAMI ’12), pp. 53-58, Herlany, Slovakia, 2012.
[26] W. H. Ho, J. T. Tsai, and J. H. Chou, “Robust quadratic-optimal control of TS-fuzzy-model-based dynamic systems with both elemental parametric uncertainties and norm-bounded approximation error,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 3, pp. 518-531, 2009.
[27] C. S. Tseng, B. S. Chen, and H. J. Uang, “Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model,” IEEE Transactions on Fuzzy Systems, vol. 9, no. 3, pp. 381-392, 2001.
[28] C. H. Liu, J. D. Hwang, Z. R. Tsai, and S. H. Twu, “An LMI-based stable T-S fuzzy model with parametric uncertainties using multiple Lyapunov function approach,” in Proceedings of the IEEE Conference on Cybernetics and Intelligent Systems, pp. 514-519, Singapore, December 2004.
[29] S. H. Kim and P. G. Park, “Observer-based relaxed H\infty control for fuzzy systems using a multiple Lyapunov function,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 2, pp. 476-484, 2009.
[30] H. K. Lam and M. Narimani, “Quadratic-stability analysis of fuzzy-model-based control systems using staircase membership functions,” IEEE Transactions on Fuzzy Systems, vol. 18, no. 1, pp. 125-137, 2010.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.