Controllable Targets Near a Chaotic Attractor

A nonlinear system with a chaotic attractor produces random like motion on the attractor. As a consequence, there must exist a neighborhood about any given point on or near the attractor such that the system will visit the neighborhood in finite time. This observation leads to a very simple “Chaotic control” algorithm for bringing nonlinear systems 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, a neighborhood of the desired fixed point can be found, such that, using standard feedback control techniques, the system is guaranteed to be driven to the fixed point. If this neighborhood also has points in common with the chaotic attractor it is then called a “controllable target” for the fixed point. The chaotic control algorithm consists of first using, if necessary, open loop control to generate chaotic motion and then wait 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. The basic requirement with this approach is, of coarse, being able to determine a large controllable target. 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. A Lyapunov function is then obtained based on this stable linear system.