The synchronization of chaotic systems with different dimensions by a robust generalized active control

Active control strategy is a powerful control technique in synchronizing chaotic/hyperchaotic systems. Until now, active control techniques have been employed to synchronize chaotic systems with the same orders. The present study overcomes the limitations of synchronization of chaotic systems of similar dimensions using active control. In this article, the authors investigate the synchronization problem for a drive-response chaotic system with different orders under the effect of both unknown model uncertainties and external disturbance. Based on the Lyapunov stability theory and Routh–Hurwitz criterion, a robust generalized active control approach is proposed and sufficient algebraic conditions are derived to compute a suitable linear controller gain matrix that guarantees the globally exponentially stable synchronization. Two examples are presented to illustrate the main results, namely reduced-order synchronization between the hyperchaotic Lu and the unified chaotic systems and the increased-order synchronization between the unified chaotic and the hyperchaotic Lu systems. There are three main contributions of the present study: (a) generalization of the active control for synchronization of a class of chaotic systems with different orders; (b) a recursive approach is proposed to compute a suitable linear controller gain matrix and (c) reduced (increased) order synchronization under the effect of both unknown model uncertainties and external disturbances. A comparative study has been done with our results and previously published work in terms of synchronization speed and quality. Future applications of the proposed reduced (increased) order synchronization approach are discussed. Finally, numerical simulations are given to verify the effectiveness of the proposed reduced (increased) order active synchronization approach.