Alban P.M. Tsui and Antonia J. Jones

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 favorably 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.

Satellite The chaotic satellite system is defined by the following six differential equations: where , , are angles which are successive clockwise rotations about inertial axes I, J and K respectively and , and are the angular velocities, , and are the principal moments of inertia with respect to body axes; , and are the three control torques produced by the thrusters; and are the perturbing torques. We take = 3, = 2 and = 1 and = = = 0, i.e. no control, for the system to be chaotic. The attractor is demonstrated in the two interactive figures shown below in Fig. 1.

(158K) (157K) Fig 1. Phase portraits of the angular velocites and of the attitude angles. Click the above figures to load a 3D view (LiveGraphics3D) of the attractor which allows rotation and zoom etc. These figures were generated by Mathematica. (required java 1.1 enabled browser)

Control

Fig. 2 and Fig. 3 show the resulting controlled state of the chaotic satellite using delayed feedback of one variable of the system. Amazingly, we can ignore 5 out of the 6 senses with a minimal effort to achieve the control of this chaotic sytem.

The control strategy is simply performed by adding a simple perturbation feedback term to one of the dynamic equation . Our strategy is equivalent to by setting the control torque

and keep the other control torques to zero in our satellite system. Note that the control torque is calculated with only the information of . Here the parameters = 0.5 and the delay = 2.12 which is about the period of the stabilised periodic orbit.

See the full version of the paper for further information.