Trajectory Tracking between Josephson Junction and Classical Chaotic System via Iterative Learning Control  
[PERSON]  
1 Institute of Electrical and Control Engineering, National Chiao Tung University, Hsinchu 300, Taiwan; [EMAIL_ADDRESS]  
[PERSON]  
1 Institute of Electrical and Control Engineering, National Chiao Tung University, Hsinchu 300, Taiwan; [EMAIL_ADDRESS]  
Footnote 1: email: {chchu-  
algorithms based on the measured data without any internal partial knowledge to reconstruct chaotic systems and demonstrates the numerical results in Reference [17].  
In many studies, synchronization systems in the state space model for superconductor devices were described in identical RCLs-Josephson junction systems. Classical non-identical system synchronization is not meant to handle a superconducting system. Accordingly, many trajectory tracking studies have rarely been directly based on the RCLs-JJ system and classical chaotic systems together. This article focuses on the trajectory tracking between two non-identical systems in which one of the systems is the classical chaos system of the Rossler and the other is the RCLs-JJ model in state space representing a mesoscopic system. In this research, two stages tracking the trajectory method uses the RCLs-JJ system in the state space model to trace the trajectory of the Rossler system. The first tracking stage utilizes the active control technique [18] to transfer the RCLs-JJ system into the quasi-Rossler system. In the next stage, the proposed iterative learning control law approaches the trajectory of the Rossler system by correcting the repetition tolerance from the preceded output information between the quasi-Rossler system and the Rossler system [12]. The RCLs-JJ system becomes the quasi-Rossler system and employs the iterative learning control (ILC) procedure to control signals from the Rossler system to trace the trajectory of the drive system. However, most ILC research update rules are designed based on the linear ILC law. The varying reference input [19] is very different from the chaotic signals, which are unexpected. Nevertheless, chaotic systems are unpredictable and can be synchronized [20; 21] by designing a controller. The challenges involve extracting the message embedded within the chaotic signal from the normal signal of the regular system such as a digital encoding system. A chaotic signal cannot implement the code by the obvious method. The chaotic synchronization should lead to a secure communication system by using a synchronized chaotic electronic circuit [22].  
In this study, the chaotic tracking trajectory that utilized the ILC method based on the Josephson junction chaos can also employ other methods such as a backstepping controller [7], an active control [8], a common chaos driving by Rossler [9], and LMI [16] and we examine the results. This article provides two main contributions. The first is a real-time feedforward procedure that uses iterative learning to modify the trajectory error between systems for tracking two non-identical systems. A few studies have been published on two different chaotic systems directly tracking or synchronizing. However, reports are especially rare for Josephson Junction systems and classical chaotic systems. Another contribution is the utility of the synchronization or minimum tracking error leads to the secure communication system by applying the Josephson Junction systems into quantum chaotic systems to deliver the message in the communication system.  
The organization of this paper is explained. The next section describes the Rossler Chaotic and RCL-Shunted Josephson Junction system. Section 3 outlines the simulation results and provides an example to demonstrate the proposed learning control law in the RCLs-JJ system to trace the path of the Rossler system. Lastly, this paper highlights the future applications and conclusions in Section 4.