Synchronizing chaos in an experimental chaotic pendulum using methods from linear control theory

Linear feedback control, specifically model predictive control (MPC), was used successfully to synchronize an experimental chaotic pendulum both on unstable periodic and aperiodic orbits. MPC enables tuning of the controller to give an optimal controller performance. That is, both the fluctuations around the target trajectory and the necessary control actions are minimized using a least-squares solution of the linearized problem. It is thus shown that linear control methods can be applied to experimental chaotic systems, as long as an adequate model is available that can be linearized along the desired trajectory. This model is used as an observer, i.e., it is synchronized with the experimental pendulum to estimate the state of the experimental pendulum. In contrast with other chaos control procedures like the map-based Ott, Grebogi, and York method [Phys. Rev. Lett. 64, 1196 (1990)], the continuous type feedback control proposed by Pyragas [Phys. Lett. A 170, 421 (1992)], or the feedback control method recently proposed by Brown and Rulkov [Chaos 7 (3), 395 (1997)], the procedure outlined in this paper automatically results in a choice for the feedback gains that gives optimum performance, i.e., minimum fluctuations around the desired trajectory using minimum control actions.