Switching Fuzzy Model Construction and Controller Design for Dynamical Systems with Input Nonlinearity
Hiroshi Ohtake and Kazuo Tanaka
Department of Mechanical Engineering and Intelligent Systems,
The University of Electro-Communications
1-5-1 Chofugaoka, Chofu, Tokyo 182-8585 Japan
Fuzzy model-based control mainly deals with dynamical systems which affinely depend on control inputs. In this paper, dynamical systems which is permitted to have nonlinearity not only in the states, but also in the inputs is considered. Input nonlinearity makes a nonlinear system complicated and makes the number of fuzzy model rules increase. Thus, switching fuzzy control approach is employed. Firstly, the switching fuzzy model construction with arbitrary linear dividing planes, which is an extension of the ordinary switching fuzzy model construction method with dividing planes corresponding to quadrants, is introduced. Secondly, by applying the switching fuzzy model construction method to the dynamical system with input nonlinearity, the switching fuzzy model with membership functions which depend on control inputs is constructed. Finally, by utilizing the dynamic state feedback control approach, we show that membership functions which depend on control inputs can be calculated. Moreover, by employing augmented system approach, the switching fuzzy dynamic state feedback controller design conditions based on the switching Lyapunov function are derived in terms of linear matrix inequalities. A design example illustrates the utility of this approach.
 H. K. Khalil, “Nonlinear Systems,” Prentice Hall, Upper Saddle River, NJ, Third Edition, 2002.
 K. Tanaka and M. Sugeno, “Stability Analysis and Design of Fuzzy Control System,” Fuzzy Sets And System, Vol.45, No.2, pp. 135-156, 1992.
 H. O. Wang, K. Tanaka, and M. 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.
 K. Tanaka and H. O. Wang, “Fuzzy Control Systems Design and Analysis,” John Wiley & Sons, Inc., 2001.
 D. Filev, “Gain scheduling based control of a class of TSK systems,” in Fuzzy Control, Synthesis and Analysis, edited by S. Farinwata, D. Filev and R. Langari John Wiley & Sons, Ltd, Chichester, England, pp. 321-334, 2000.
 S. K. Hong and R. Langari, “Robust Fuzzy Control of Magnetic Bearing System Subject to Harmonic Disturbances,” IEEE Transactions on Control Systems Technology, Vol.8, No.2, pp. 366-371, March, 2000.
 D. Filev and P. Angelov, “Fuzzy Optimal Control,” Fuzzy Sets and Systems, Vol.47, pp. 151-156, 1992.
 R. Langari, “A Nonlinear Formulation of a Class of Fuzzy Linguistic Control Algorithms,” Proc. of the 1992 American Control Conf., pp. 2273-2278, Chicago, IL, 1992.
 M. Sugeno, “On Stability of Fuzzy Systems Expressed by Fuzzy Rules with Singleton Consequents,” IEEE Transactions on Fuzzy Systems, Vol.7, No.2, pp. 201-224 April, 1999.
 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, pp. 116-132, 1985.
 T. Nebuya, K. Tanaka, and H. Ohtake, “Fuzzy Control of Dynamical Systems with Input Nonlinearity,” Proc. of the Joint 2nd Int. Conf. on Soft Computing and Intelligent Systems and 5th Int. Symposium on Advanced Intelligent Systems, TUP-7-1, Yokohama, 2004.
 K. Tanaka, H. Ohtake, and H. O. Wang, “A practical design approach to stabilization of a 3-DOF RC helicopter,” IEEE Transactions on Control Systems Technology, Vol.12, No.2, pp. 315-325, March, 2004.
 H. Ohtake, K. Tanaka, and H. O. Wang, “Fuzzy Modeling via Sector Nonlinearity Concept,” Integrated Computer-Aided Engineering, Vol.10, No.4, pp. 333-341, 2003.
 H. Ohtake, K. Tanaka, and H. O. Wang, “A Construction Method of Switching Lyapunov Function for Nonlinear Systems,” Journal of Advanced Computational Intelligence and Intelligent Informatics, Vol.10, No.1, pp. 3-10, 2006.
 H. Ohtake, K. Tanaka and H. O. Wang, “Switching Fuzzy Controller Design based on Switching Lyapunov Function for a Class of Nonlinear Systems,” IEEE Transactions on Systems, Man, and Cybernetics Part B, Vol.36, No.1, pp. 13-23, 2006.
 M. Johansson, “Piecewise Linear Control Systems,” Ph.D. Thesis, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden, 1999.
 S. Boyd et al., “Linear Matrix Inequalities in Systems and Control Theory,” SIAM, 1994.