Chaotic Synchronizing Systems with Zero Time

This research not only dedicated a less restrictive method of iteration-varying function for 8 a learning control law to design a controller but also synchronize two nonlinear systems with free 9 time-delay. In addition, the mathematical theory of system synchronization has proved rigorously 10 and the theory verified through an example to demonstrate the behavior of each parameter in the 11 theory. The design of a controller using the iterative learning control law is significant for robotic 12 tracking. The controller in this research generates a feed-forward control input using the error 13 dynamics among the drive-response systems. The error dynamics satisfies the Lyapunov function 14 and the combination of output errors, which respectively represented relative estimated differences 15 of the drive-response systems. The iterative learning control rule serves the function of a filter 16 adding previous control error after the end of each iteration. The numerical example of a 17 synchronous system is given a Lorenz system for driving and another with the iterative learning 18 control law for response under different initial condition. The results verify and demonstrate the 19 proposed mathematical theory. The simulation exhibits consistency in the behavior of each 20 parameter to match mathematical theory. 21