Optimisation of the synchronisation of a class of chaotic systems: combination of sliding mode and feedback control

This paper addresses the problem of optimisation of the synchronisation for a class of uncertain chaotic systems from a control theoretic point of view. A robust adaptive feedback which accomplishes the synchronisation of chaotic systems using an optimal tuning scheme based on Riccatti equations is successfully adapted. The underlying idea is to optimise the synchronisation of chaotic systems by accounting the control effort despite the uncertainties. The approach developed considers incomplete state measurements and no detailed model of the systems to guarantee robust stability. This approach includes a high-order sliding mode estimator and leads to a robust adaptive feedback control scheme. A finite horizon can be arbitrarily established by ensuring that the chaos synchronisation is achieved at established time. An advantage is that the studied scheme accounts the energy wasted by the controller and the closed-loop performance on synchronisation. Both mathematic proof and numerical simulations are presented to show the feasibility of the optimisation strategy for establishing the synchronisation of chaotic systems even if there are some modelling mismatches and parametric variations.