The control of higher dimensional chaos: comparative results for the chaotic satellite attitude control problem

Abstract

Using simulations this paper reports the success and failure of three experiments each using a different technique to control the same higher dimensional chaotic system. The three techniques are: a simple delayed feedback control method originally suggested by Pyragas, the Otani–Jones technique, and a higher dimensional variation of the OGY method. The three methods are applied to a six-dimensional system which describes the attitude dynamics of a satellite (rigid body) subjected to deterministic external perturbations which induce chaotic motion when no control is effected. The attitude of the satellite is controlled by three orthogonal pairs of thrusters aligned with the principal axes and the system can be described by a modified set of Euler equations. The three control methods are compared in terms of the prior calculations required, the real-time computational requirements, and the effectiveness of the method in stabilizing the system. The results show that the method of Pyragas compares very favourably with the other two techniques, requiring no prior calculation and having a very low real-time computational overhead. Pyragas’ method also provides the most satisfactory control solution. The problems in presenting a theoretical justification of Pyragas’ method are briefly discussed and the connection between the method and a recent technique for the location of unstable fixed points in chaotic systems is highlighted.