A methodology to design stable nonlinear fuzzy control systems

The stability of nonlinear systems has to be investigated without making linear approaches. In order to do this, there are several techniques based on Lyapunov's second method. For example, Krasovskii's method allows to prove the sufficient condition for the asymptotic stability of nonlinear systems. This method requires the calculation of the Jacobian matrix. In this paper, an equivalent mathematical closed loop model of a multivariable nonlinear control system based on fuzzy logic theory is developed. Later, this model is used to compute the Jacobian matrix of a closed loop fuzzy system. Next, an algorithm to solve the Jacobian matrix is proposed. The algorithm uses a methodology based on the extension of the state vector. The developed algorithm is completely general: it is independent of the type of membership function that is chosen for building the fuzzy plant and controller models, and it allows the compound of different membership functions in a same model. We have developed a MATLAB's function that implements the improved algorithm, together with a series of additional applications for its use. The designed software provides complementary functions to facilitate the reading and writing of fuzzy systems, as well as an interface that makes possible the use of all the developed functions from the MATLAB's environment, which allows to complement and to extend the possibilities of the MATLAB's Fuzzy Logic Toolbox. An example with a fuzzy controller for a nonlinear system to illustrate the design procedure is presented. The work developed in this paper can be useful for the analysis and synthesis of fuzzy control systems.