Local Control of Chaotic Systems — A Lyapunov Approach

Abstract

In this paper a method for local control of an unstable equilibrium point in chaotic systems is presented. Linear state feedback to stabilize the equilibrium is employed which is only active in a bounded region around the desired point: the area of control action. Size and shape of the area of control action are determined by a Lyapunov function of the controlled chaotic system such that it belongs to the basin of attraction of the equilibrium point. We give the design procedure for both continuous-time and discrete-time systems.

Adaptive Control of Accumulative Error for Nonlinear Chaotic Systems

“…It is the product of error and its integral in essence, but not the proportional integral (PI) control. It is significantly different from the usual linear feedback in [5,12], where εi is a constant or piecewise constant. Especially, the fixed control strength is used in linear feedback wherever the initial points start, thus the strength must be maximal, which means a kind of waste in practice [13] .…”

“…Since Ott et al [1] firstly proposed the method of chaos control in 1990, chaos control has attracted great attention and lots of successful experiments have been reported [2−17] , such as feedback control [5,6] , impulsive control [7] , backstepping method [8] , adaptive control [9−14] , sliding mode control [15,16] , fuzzy control [17] , finite-time control and so on. However, most methods aim at a kind of chaotic system and they are complex both in design and implementation.…”

Stabilizing Unstable Equilibria of Chaotic Systems From a State Observer Approach

“…By using some idea borrowed from the state observer approach, the controller is constructed in the following simple form: (3) where is the control gain matrix, is the output of system (1) with a constant matrix , is the observation of the targeted equilibrium point , and if else (4) where denotes the attraction region of the targeted (unstable) equilibrium point , with , where is the attraction region of the chaotic system (1).…”

“…It has been noticed that the OGY method [3], which is a local control scheme, and some other similar methods [4], [5], even with precise state feedback control, may fail when the system has parameter variations. This is due to the requirement of precise information about the system's steady-state state and the system parameter values.…”

Adaptive control of discrete-time chaotic systems: a fuzzy control approach

“…Many effective methods have been developed for chaos control during the past decade (e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]) since the wellknown OGY control method [16], which stabilizes unstable periodic orbits embedded in chaotic attractors using small control input signals. These methods include linear state feedback, feedback-linearization, dynamic delayed feedback, invariant-manifold, and back-stepping techniques, to name just those that are familiar to control engineers.…”