International Journal of Bifurcation and Chaos, Vol. 10, No. 11 (2000) 2611–2617 c  World Scientific Publishing Company FEEDBACK SYNCHRONIZATION USING POLE-PLACEMENT CONTROL ROBERTO TONELLI ∗ Institute for Plasma Research, University of Maryland, College Park, MD 20742, USA INFM-Physics Department, University of Cagliari, 09100 Cagliari, Italy YING-CHENG LAI Department of Mathematics, Department of Electrical Engineering, Center for Systems Science and Engineering Research, Arizona State University, Tempe, AZ 85287, USA CELSO GREBOGI ∗ Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA Received August 23, 1999; Revised November 15, 1999 Synchronization in chaotic systems has become an active area of research since the pioneering work of Pecora and Carroll. Most existing works, however, rely on a passive approach: A cou- pling between chaotic systems is necessary for their mutual synchronization. We describe here a feedback approach for synchronizing chaotic systems that is applicable in high dimensions. We show how two chaotic systems can be synchronized by applying small feedback perturbations to one of them. We detail our strategy to design the control based on the pole-placement method, and give numerical examples. 1. Introduction Chaotic systems are characterized by a sensitive dependence on initial conditions: two trajectories starting from slightly different initial conditions diverge exponentially in time. As such, synchro- nization between even identical chaotic systems be- comes a highly intriguing problem. It has been rec- ognized, however, since 1983 that synchronization can indeed occur in chaotic systems [Fujisaka & Yamada, 1983; Afraimovich et al., 1986; Pecora & Carroll, 1990; Chua et al., 1993; Heagy et al., 1994; Kocarev & Parlitz, 1996; Pecora et al., 1997]. It was Pecora and Carroll [1990] who first pointed out that synchronous chaos could be utilized for non- linear digital communication [Parlitz et al., 1992; Cuomo & Oppenheim, 1993; Cuomo et al., 1993; Short, 1994, 1996]. Since then, synchronization in chaotic systems has received a tremendous amount of attention and it remains to be one of the most active research areas in chaotic dynamics [Ditto & Showalter, 1997]. Most existing approaches to synchronizing chaotic systems are passive in the sense that some predesigned coupling scheme is utilized to war- rant synchronization. After the coupled system is switched on so that synchronization is achieved, no external control or perturbation is necessary to keep the coupled subsystems synchronized. This ap- proach can guarantee robust synchronization when a proper coupling scheme is employed, and it has been quite successful indeed [Pecora & Carroll, 1990; Heagy et al., 1994; Ditto & Showalter, 1997]. However, in some situations, it is difficult to de- vise a coupling scheme to achieve synchronization. For concreteness, say we have a chaotic system 2611