Adaptive nonlinear controller for stabilizing saddle-type steady states of dynamical systems

Abstract

An adaptive feedback method for stabilizing unknown saddle- type steady states of dynamical systems is described. Many other techniques developed so far use unstable linear filters to track and control the saddles. In contrast, the novel controller is an essentially nonlinear unit. It implements the Heaviside step function (piecewise constant function) and minimizes the feedback control force to move the error into a very small tolerance gap. Within the tolerance gap, the reference point has fixed coordinates. In this sense, the method is somewhat similar (except the fact that it automatically locates unknown steady states) to the conventional proportional feedback technique, providing an advantage of a simpler analysis. The method has been applied analytically, numerically and experimentally to two different nonlinear dynamical systems, namely the second-order damped non-driven Duffing system and the third-order chaotic Lorenz system.