Backstepping Backstepping adaptatif pour le contrôle la poursuite et la synchronisation des systèmes dynamiques non linéaires chaotiques

Abstract

Since the early 1990s, the problems of control of chaos attract attention of the researchers and engineers. Numerous publications have appeared over the recent decade. The chaotic systems represent a class of indeterminacy models differing from the stochastic models, which is used widely since then. Various mathematical definitions of chaos are known, but all of them express close characteristics of the dynamic systems that are concerned with super sensitivity to the initial conditions. the robust control scheme based on the Backstepping the adaptive Backstepping design is used to control, the track and the synchronization of the chaotic systems arising from the autonomous second order strict feedback form such as the Duffing and Van der pool oscillators, to the autonomous third order strict feedback form such as the Lorenz, Chua and Rosslor Chaotic systems. The design procedure is recursive at the ith step, the ith subsystem is stabilized with respect to a Lyapunov function Vi by the design of a stabilizing function αi for the Backstepping and the tuning function τi, and the update law θi for the unknowns' adaptive parameters estimates, the feedback control u is designed at the final step. This procedure possesses strong properties of global stability and tracking which are built into the nonlinear system in a number of steps, which is never higher than the system order, without any growth restrictions on nonlinearities. An important feature of backstepping is that nonlinearities can be dealt with in several ways: useful nonlinearities, which act stabilizing, can be retained, and sector bounded nonlinearities may be dealt with using linear control, retaining nonlinearities instead of canceling them requires less precise models and may also require less control effort. Further, the resulting control laws can sometimes be shown to be optimal with respect to a meaningful performance index, which guarantees certain robustness properties. Some simulation results for the chaotic system cited above are shown to illustrate the parameter convergence with backstepping and the adaptive backstepping design for the control, the track and the synchronization of chaotic systems.