# A Chaotic Secure Communication System Design Based on Iterative Learning Control Theory
## 1 Introduction  
Iterative Learning Control theory (ILC; the staple of ILC can be found in a U.S. patent filled in 1967 and available in 1971 [1]) is basically a control strategy to improve transient response (or similarly, the controller performance) of dynamical systems that operate repetitively (also called _multi-pass_ or _repetitive_ dynamic processes). This is done by adjusting the system control input(s) during the system cycle operations [1; 2; 3; 4]. Because the system executes the same task multiple times, the control law may learn from the previous system action (or iteration). Hence, the central challenger of ILC design is to use this information from the previous operation to learn and improve the controller execution over the next iteration by following a control objective. Basically, a repetitive dynamic system is a process that has the following two properties:  
* Repetitive action, and
* Iteration.  
The first one is the process action that it is periodically repeated, and the second one is the natural iteration between the plant vector state and the repetitive process action.  
On the other hand, ILC control strategy has been widely employed in many industrial applications, including manufacturing, robotics, chemical process, among many others [1]. Furthermore, the main benefits of iterative learning control framework are its low transient tracking error, despite large model uncertainty and disturbances [1; 5; 6], its short a priori knowledge about the system to be controlled, and its low computational effort for control realization [7].  
To note, ILC theory has also been used in system modeling, two-dimensional systems analysis, linear matrix inequality system design, and in adaptive and robust control innovation (see, for instance, [5; 6] and references therein). In the meantime, some other ILC strategies are mainly based on [PERSON]'s theory [8; 9; 10].  
Finally, many of the available ILC approaches require identical _resetting initial conditions_ (at the beginning of each iteration: the well known _resetting condition_); however, in real applications, the perfect resetting condition may be not realizable [6; 8]. Therefore, under different tests on resetting initial conditions, the boundedness along the time evolution and asymptotic stability on each iterationof an ILC system were well proven in [8]. Actually, this is an important robust property of the ILC tool to design new engineering developments, such as secure communication systems subject to noisy environments.  
In this paper, motivated by the _canonical_ ILC structure stated in [8], a chaotic secure communication scheme is developed. This design is based on the chaotic logistic-map. Therefore, the proposed design uses the ILC method as a base tool to encrypt and decrypt a message, recalling that the development of new chaotic secure communication systems is still an important open topic [11; 12; 13; 14; 15].  
The remainder of this paper is organized as follows. In Section 2, the canonical ILC frame is retrieved from [8]. Section 3 presents our ILC chaotic secure communication system design. Performance numerical evaluation of our scheme is evidenced in Section 4. Finally, the conclusions are given in Section 5, including future issues on the subject.