Identification of Lorenz chaotic system based on synchronization using fractional order calculus

In this paper the goal is to identify the parameters of the Lorenz chaotic system، based on synchronization of identical systems using fractional calculus. The method which is used for synchronization is come from Lyapunov stability theorem and then by using fractional dynamics، control laws are improved. To this end، a Lyapunov function is presented and based on the Lyapunov stability theory and asymptotic stability criteria، some adaptation laws to estimate unknown parameters of the system are proposed. The introduced method is applied to the Lorenz chaotic system and since the goal is identification، all the parameters of the system are taken unknown. Using invariant set theory، it is proved that the parameter estimation errors converge to zero. Then the results of numerical simulations are shown and discussed and it is observed that fractional calculus has an essential effect on reducing fluctuations and settling time of the parameters convergence. At the end، the stability of the system by using fractional adaptation law is discussed. It is shown that the asymptotic stability of the synchronization error dynamics is proved using the fractional adaptation law، and this is confirmed through simulation.