The Small Gain Theorem And Its Application To Chaos Control

As one of the most important achievements of nonlinear science, chaos attracted common attention and had been extensively studied in the last dozens of years. Chaos is a kind of quasi-stochastic behaviors of determinate nonlinear system. It has the property of the extreme sensitivity to initial conditions which was thought to be a"bothering"property. So, people used to want to avert it. Recently, although how to apply chaos has become the researching focus, chaos control is still one of the basic problems of people's concern.To deal with this problem, first of all, three kinds of important methods of chaos control are introduced, which are OGY control, backstepping control and sliding mode control. Secondly, since the small gain theorem is an important theorem in nonlinear system and it has particular effects in judging input-to-state stable, a new method of chaos control is proposed in this paper based on the small gain theorem and its correlative theories. By designing the suitable controller to make the controlled system satisfy the condition of the small gain, its origin will globally asymptotically stable, then the chaotic system will be controlled. One of the most important advantages of the chaos controller by this method is that, it makes the controlled system globally asymptotically stable. And we show how to use this method to design chaos controller by constructing the controllers of Lorenz system and its other two generalized systems. Numerical simulations are presented to show these results.