# Chaotic Synchronizing Systems with Zero Time Delay and Free Couple via Iterative Learning Control
## 1 Introduction  
The method of Iterative Learning Control (ILC) was applied to mechanical robots in 1984 for performance improvement. The improvement is made possible by designing an on-line iterative learning scheme based on a classical linear system that approximates well the original nonlinear system of the robots. The approach of this on-line-learning ILC is of course very different from other known approaches, such as repetitive control, adaptive control, and the neural network [1, 2]. The essence of on-line learning lies in the attribute of the controller that adapts its controller effort continuously based on the error feedback from the output of the system in an on-line fashion [2, 3, 4]. This ILC control method is applicable to many industrial systems for manufacturing, robotics, and assembly line in mass production [2, 3]. Recently, the ILC theory was employed for chaos synchronization to stabilize communication protocols of encrypting and decrypting a message using dynamical filters, which are bi-directional over unidirectional coupling schemes of Bernoulli units, towards the tap-proof transmission of information via chaotic laser systems [5, 6]. Due to the aforementioned value, the ILC for chaos synchronization was also intensively studied by academics [5, 6]. Different types of controllers were developed, such as a proportional-integral-derivate (PID)-type ILC applied for improved robotic operations [3, 7]. Also, ILC was employed for dynamical system synchronization based on classical linear system theory [8, 9, 10, 11]. The Lyapunov criterion that served to guarantee the designed slave system in asymptotical synchronization with the master system in [12] is a common strategy that is used to study the stabilityof a control system. The principle of the learning operator is described in [PERSON] [13], and the Lyapunov function of the synchronization system pursues the method by [PERSON] [14] for stability. The Lyapunov function exhibited in [15; 16; 17; 18; 19; 20; 21] forms the criteria of the restrictive system with couple and time-delay to synchronize, the existence and stability conditions of two different continuous chaotic systems, and the synchronization manifold of two unidirectional systems, respectively. The synchronized system analyzing was similar to the generalized Lorenz systems by coupling linear state error feedback control with coupling matrix to eliminate the given chaotic dynamics synchronization [22]. The controller is considered the linear combination of error dynamics feedback without the coupling matrix and previous learning rule to approach chaos synchronization [23]. The stable synchronization of a multi-chaotic network system was a single direction of the response system with error feedback non-coupling and was dependent on the eigenvalues of different time-delay coupling matrices between systems in [24]. The Iterative Feedback Tuning (IFT) method [25; 26; 27] was addressed to approach the optimization of the parameters of the controller and avoid replacing the bias information of the signal of the system. Therefore, the isochronal phenomenon of the non-linear secure communication system synchronization has been investigated in, along with several activities in nature [25]. This article employed an ILC law to design the controller, which is similar to adding a filter before the input of the system, and to synchronize two nonlinear systems with a distinct initial condition. The feed-forward control signal is an appropriate tracking reference that could achieve a low tracking error between systems to approach the synchronization of systems. The ILC method has only been applied to trace the trajectory of the robotic system and academic field over the past few decades. The goal of this paper was not only to study the characteristics of the parameters in the synchronization process of the system, but also to apply the results into a secure communication system, as well as a traced robotic movement.