Reduced Order Synchronization of Two Non-identical Special Class of Strict Feedback Systems via Back-stepping Control Technique

Abstract In this paper, we have addressed a ROS (Reduced Order Synchronization) problem of two general classes of chaotic or hyperchaotic systems in master (drive) and slave (response) configuration. The master and the slave systems are considered to follow a very specific form of structure; so-called the strict-feedback form. Being a ROS problem, the order of the master system is naturally greater than that of the slave system. But we have taken a particular case where the order difference between both the systems is one. A systematic nonlinear back-stepping control design methodology is developed so to attain the reduced-order synchronization goal. In real practice, only [[EQUATION]] number of states of the response system get synchronized with [[EQUATION]] states of the drive system, incorporating [[EQUATION]] amount of controllers, having considered the order of the drive and response system to be [[EQUATION]] and [[EQUATION]] (and [[EQUATION]] ), respectively. It is shown that, unlike other existing conventional nonlinear control techniques, the back-stepping method requires only a single scalar controller [[EQUATION]] [[EQUATION]][[EQUATION]] to achieve the same task mentioned above. The question arises, what will happen to the [[EQUATION]] state of the master system. Being a chaotic system, it can be said that [[EQUATION]] state will be bounded in nature. But, its stability can not be guaranteed. By appending an extra [[EQUATION]] dynamics to the slave system and adding a suitable nonlinear active controller [[EQUATION]] [[EQUATION]] can tackle the stability challenge of the [[EQUATION]] state, as stated earlier. A sound generalization of the control function [[EQUATION]] and [[EQUATION]] have also been included in the discussion. To prove the asymptotic stability of the error dynamics of the system, the Lyapunov stability direct method has been utilized. Extensive MATLAB simulation results verify the theoretical findings.