Optimal feedback control of the forced van der Pol system

A simple feedback control strategy for chaotic systems is investigated using the forced van der Pol system as an example. The strategy regards chaos control as an optimization problem, where the maximum magnitude Floquet multiplier of a target unstable periodic orbit (UPO) is used as a cost function that needs to be minimized. Thus, the method obtains the optimal control gain in terms of the stability of the target UPO. This strategy was recently proposed for the proportional feedback control (PFC) method. Here, it is extended to the highly popular delayed feedback control (DFC) method. Since the DFC method treats the system as a delay-differential equation whose phase space is infinite-dimensional, the characteristic multipliers are found through a truncation in the number of delayed states. Control of a target UPO is achieved for several values of the forcing amplitude. We compare the DFC and PFC methods in terms of stability of the controlled orbit, steady state error and control effort.