Lyapunov exponent-based PID control for noisy chaotic systems

This paper addresses the stabilisation issue of noisy chaotic systems. It proposes a new hybrid control approach that combines the proportional-integral-derivative (PID) method with the delayed feedback technique. The idea is to turn the structural stabilisation problem into an optimisation one. The goal is to determine the PID gains that give the lowest value of the largest-Lyapunov exponent. The difference between the system's current state and its delayed value, by one period, is used as the PID argument. As a result, the systems chaotic orbits are structurally stabilised to the greatest extent possible, the stable zone is extended, and the stabilisation time is reduced. Numerical simulations are carried out to show the effectiveness of the proposed approach on two well-known benchmarks of chaotic systems.