Control using chaos

Abstract

A nonlinear dynamical system with a chaotic attractor will produce motion on the attractor which has random-like properties. This observation leads to a very simple algorithm for bringing a discrete or continuous nonlinear dynamical system to a fixed point. Suppose that the system to be controlled is either naturally chaotic or that chaotic motion can be produced by means of open-loop control. Suppose also that a neighborhood of the desired fixed point can be found, such that, under state variable feedback control, the system is guaranteed to be driven to the fixed point. If this neighborhood has points in common with a chaotic attractor, it may be used as a "controllable target" for the fixed point. The control algorithm consists of first using, if necessary, open-loop control to generate chaotic motion and then waiting for the system to move into the controllable target. At such a time the open-loop control is turned off and the appropriate closed-loop control applied. A basic requirement with this approach is to determine a large enough controllable target so that one does not have to wait too long for the system to reach it. The following method is used here: the system is first linearized about the desired fixed-point solution. If necessary, a feedback controller is then designed so that this reference solution has suitable stability properties. Then a Lyapunov function is obtained based on this stable linear system and a level curve for the Lyapunov function is determined, such that, whenever the state of the nonlinear system is within this level curve, the feedback controller will drive the nonlinear system to the desired equilibrium solution. Such a level curve defines a controllable target provided that it actually does contain points on the chaotic attractor. A multiple step approach for determining the Lyapunov level curve is presented which helps in finding large controllable targets for discrete systems.