A differential flatness theory approach to adaptive fuzzy control of chaotic dynamical systems

A solution to the problem of control of nonlinear chaotic dynamical systems, is proposed with the use of differential flatness theory and of adaptive fuzzy control theory. Considering that the dynamical model of chaotic systems is unknown, an adaptive fuzzy controller is designed. By applying differential flatness theory the chaotic system´s model is written in a linear form, and the resulting control inputs are shown to contain nonlinear elements which depend on the system´s parameters. The nonlinear terms which appear in the control inputs of the transformed dynamical model are approximated with the use of neuro-fuzzy networks. It is proven that a suitable learning law can be defined for the aforementioned neuro-fuzzy approximators so as to preserve the closed-loop system stability. Moreover, with the use of Lyapunov stability analysis it is proven that the proposed adaptive fuzzy control scheme results in H_∞ tracking performance, which means that the influence of the modeling errors and the external disturbances on the tracking error is attenuated to an arbitrary desirable level. Simulation experiments confirm the efficiency of the proposed adaptive fuzzy control method, using as a case study the model of the Lorenz chaotic oscillator.