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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

This paper presents the synchronization between the master and slave Lorenz chaotic systems by slide mode controller (SMC)-based technique. A proportional-integral (PI) switching surface is proposed to simplify the task of assigning the performance of the closed-loop error system in sliding mode. Then, extending the concept of equivalent control and using some basic electronic components, a secure communication system is constructed. Experimental results show the feasibility of synchronizing two Lorenz circuits via the proposed SMC.

Chaos theory is a branch of nonlinear system theory and has been intensively studied in the past four decades. In 1963, E. N. Lorenz presented the first well-known chaotic system, which was a third-order autonomous system with only two multiplication-type quadratic terms but which displayed very complex dynamic behaviors [

Recently, the chaos synchronization between master (transmitter) and slave (receiver) chaotic systems has been an attractive topic for its potential applications in secure communication [

To verify the above systems performance, in this paper a SMC-based chaotic secure communication system, which includes two chaotic Lorenz circuits (transmitter and receiver) and a sliding mode controller, is realized by using some electronic components containing operational amplifiers (OPAs), resistors and capacitors.

The aim of this paper was to utilize the unpredictable characteristics of chaos signals, such as broadband noise-like waveform, prediction difficulty and sensitivity to initial condition variations, to construct a secure communication system. Now we consider the following Lorenz circuits, which are typical chaotic systems that have been thoroughly studied [

Master Lorenz chaotic circuit:

Slave Lorenz chaotic circuit:

Obviously, _{m}_{s}_{m}_{s}

The control goal is for the two Lorenz chaotic systems

The error vectors and error dynamics are defined as:

Then, the following error dynamics are obtained:

To stabilize the error dynamics _{t}_{→∞} ‖

Having established the appropriate switching surface _{1}(_{1}(_{2}(

After design the control to ensure lim_{t}_{→∞} ‖_{t}_{→∞} ‖[_{1}(_{2}(_{3} (

Then, we can infer that:
_{eq}

The preceding SMC scheme of synchronization is applied to establish chaotic secure communication systems.

In the following, we use simple electronic components: OPAs, resistors and capacitors to implement the presented secure communication system. In order to speed up the dynamic response of chaotic Lorenz circuit, we rescale the systems _{m}_{s}_{m}_{s}_{1} is a scaling factor. Practical circuits of the chaotic master and slave Lorenz systems with _{1} = 10 and supplied voltages ±15 V are shown in

The circuit of error dynamics _{eq}

In the following, the commercial electronic circuit simulation software Orcad/PSpice 9.0 is used. _{m}_{s}_{m} and state _{s}. _{eq}

This study has been proposed to ensure the synchronization between the master and the controlled slave Lorenz chaotic systems via a sliding mode controller. Furthermore, the proposed scheme has been also successfully applied to a secure communication system. Some basic electronic circuits are used to implement the SMC-based secure communication system. The experimental results verify that the methods are correct and practical.

The authors thank the National Science Council of Taiwan for supporting this work under grants NSC 101-2622-E-269-012-CC3 and NSC 101-2221-E-241-002. The authors also wish to thank the anonymous reviewers for providing constructive suggestions.

Block diagram of SMC-based scheme secure communication system.

Electronic implementation of the master Lorenz circuit.

Electronic implementation of the slave Lorenz circuit.

Electronic implementation of the error dynamics

Electronic implementation of _{eq}

Electronic implementation of

The trajectories of the Lorenz system.

Experimental results of synchronization between state _{m}_{s}

Experimental results of errors between state _{m}_{s}

Experimental result of switch surface

Experimental results of control input _{eq}