This paper presents a method for synchronizing the unified chaotic systems via a sliding mode controller (SMC). The unified chaotic system and problem formulation are described. Two identical unified chaotic systems can be synchronized using the SMC technique. The switching surface and its controller design are developed in detail. Simulation results show the feasibility of a chaotic secure communication system based on the synchronization of the Lorenz circuits via the proposed SMC

Ben-Yi Liau

Hsin-Chieh Chen

Yi-You Hou

Mathematical Problems in Engineering

This paper presents a method for synchronizing the unified chaotic systems via a sliding mode controller (SMC). The unified chaotic system and problem formulation are described. Two identical unified chaotic systems can be synchronized using the SMC technique. The switching surface and its controller design are developed in detail. Simulation results show the feasibility of a chaotic secure communication system based on the synchronization of the Lorenz circuits via the proposed SMC.</jats:p

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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 632712, 10 pagesdoi:10.1155/2012/632712Research ArticleSynchronization of Unified Chaotic Systems UsingSliding Mode ControllerYi-You Hou,1 Ben-Yi Liau,2 and Hsin-Chieh Chen21 Department of Electrical Engineering, Far East University, Tainan 74448, Taiwan2 Department of Biomedical Engineering, Hungkuang University, Taichung 43302, TaiwanCorrespondence should be addressed to Hsin-Chieh Chen, hcchen@sunrise.hk.edu.twReceived 29 September 2012; Accepted 21 November 2012Academic Editor: Teh-Lu LiaoCopyright q 2012 Yi-You Hou et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.This paper presents a method for synchronizing the unified chaotic systems via a sliding modecontroller SMC. The unified chaotic system and problem formulation are described. Twoidentical unified chaotic systems can be synchronized using the SMC technique. The switchingsurface and its controller design are developed in detail. Simulation results show the feasibility ofa chaotic secure communication system based on the synchronization of the Lorenz circuits via theproposed SMC.1. IntroductionChaos theory is an extensively studied branch of the theory of nonlinear systems. Lorenz pre-sented the first well-known chaotic system, whichwas a third-order autonomous systemwithonly two multiplication-type quadratic terms but displayed very complex dynamical behav-iors 1. A chaotic system is a very complex, dynamical nonlinear system whose responsehas intrinsic characteristics such as broadband noise-like waveforms, diﬃcult predictability,and sensitivity to initial condition variations 2, 3. These properties are advantageous insecure communication systems. Therefore, the synchronization of chaotic circuits for securecommunication has received a lot of research attention 4–7. Studies have shown that it ispossible to set up a chaotic communication system to obtain secure communication 8.The synchronization between master transmitter and slave receiver chaotic sys-tems has potential applications for secure communication 9–13. Several control schemeshave been developed for the synchronization of chaotic systems. Sliding mode control is apopular nonlinear control strategy 14–19. For sliding mode controller SMC design, theLyaponov stability method is applied to keep the nonlinear system under control. The slidingmode approach transforms a higher-order system into a lower-order system, allowing a2 Mathematical Problems in EngineeringSliding modecontrollerSlave chaotic systemsMaster  chaotic systems Informationp(t)xm +−e xsueqFigure 1: Block diagram of SMC-based synchronized chaotic systems.simple control algorithm to be applied, making the system very straightforward and robust.Sliding mode control has been applied to the synchronization of chaotic systems 5, 6, 11.The present study designs an SMC-based chaotic secure communication system. Toachieve this goal, a proportional-integral PI switching surface is first designed for theconsidered error dynamics system in sliding motion, and then, based on it, a sliding modecontroller is derived. This controller is eﬀective and guarantees both the occurrence of slidingmotion and synchronization of the master-slave unified chaotic systems. Finally, an exampleis given to illustrate the usefulness of the proposed SMC.2. Problem Formulation and Main ResultsA sliding mode control system consisting of master and slave chaotic systems and informa-tion pt is shown in Figure 1.Consider the following unified chaotic system:x˙1  25α 	 10x2 − x1x˙2  28 − 35αx1 	 29α − 1x2 − x1x3x˙3  x1x2 − 8 	 α3 x3,2.1where x1, x2, and x3 are states of system 2.1 and α ∈ 0, 1. Obviously, the system 2.1becomes the original Lorenz system for α  0, while the system 2.1 becomes the originalChen system for α  1. When α  4/5, the system 2.1 becomes the critical system. Inparticular, the system bridges the gap between Lorenz system and Chen system. Moreover,the system is always chaotic in the whole interval α ∈ 0, 1. Before the secure communicationsystem can be constructed, the synchronization problem of the system based on the slidingmode control must first be solved. For the unified chaotic system, the master and slavesystems are defined asx˙m1t  25α 	 10xm2t − xm1tx˙m2t  28 − 35αxm1t 	 29α − 1xm2t − xm1txm3t 	 ptx˙m3t  xm1txm2t − 8 	 α3 xm3t,x˙s1t  25α 	 10xs2t − xs1tx˙s2t  28 − 35αxs1t 	 29α − 1xs2t − xm1txs3t 	 utx˙s3t  xm1txs2t − 8 	 α3 xs3t,2.2Mathematical Problems in Engineering 3where ut is the controller output used to synchronize the master and slave systems 2.2,and pt is the embedded message bounded by∣∣pt∣∣ ≤ ψ, ψ > 0. 2.3The control goal is for the two unified chaotic systems 2.2 to be synchronized such that theresulting error vector satisfieslimt→∞‖et‖  limt→∞‖xmit − xsit‖  0, i  1, 2, 3. 2.4The error vectors and error dynamics are defined ase1t  xm1t − xs1te2t  xm2t − xs2te3t  xm3t − xs3t,e˙1t  x˙m1t − x˙s1te˙2t  x˙m2t − x˙s2te˙3t  x˙m3t − x˙s3t.2.5Then, the following error dynamics are obtained:e˙1t  25α 	 10e2t − e1te˙2t  28 − 35αe1t 	 29α − 1e2t − xm1e3t 	 pt − ute˙3t  xm1e2t − 8 	 α3 e3t.2.6Therefore, an SMC must be designed such that the resulting error vector satisfieslimt→∞‖Et‖  limt→∞‖e1te2te3t‖ −→ 0. 2.7To stabilize the error dynamics 2.6 and achieve synchronization, two basic stepsare used: first, an appropriate switching surface is selected such that the sliding motionon the sliding manifold is stable and ensures limt→∞‖Et‖  0 ; second, an SMC lawwhich guarantees the existence of the sliding mode st  0 is established. To guarantee theasymptotic stability of the sliding mode, the PI switching surface st is defined asst  e2t 	∫ t0(25α 	 10e1τ 	 xm1e3τ 	 βe2τ)dτ, 2.84 Mathematical Problems in Engineeringwhere β > 0 is given. It is well known that when the system operates in the sliding mode, thefollowing equation must be satisfied 20, 21:st  e2t 	∫ t0(25α 	 10e1τ 	 xm1e3τ 	 βe2τ)dτ 0.2.9Since st  0, we consequently haves˙t  e˙2t 	(25α 	 10e1t 	 xm1e3τ 	 βe2t) 0.2.10From 2.10, the following is obtained:e˙2t  −(25α 	 10e1t 	 xm1e3t 	 βe2t). 2.11Then, the equivalent sliding mode dynamics is obtained ase˙1t  25α 	 10e2t − e1te˙2t  −25α 	 10e1t 	 xm1e3t 	 βe2te˙3t  xm1e2t − 8 	 α3 e3t.2.12The stability of the sliding mode dynamics 2.11 is analyzed below based on the Lyapunovstability theory.The Lyapunov function is selected as V  0.5e21t 	 e22t 	 e23t, which leads toV˙ t  e1te˙1t 	 e2te˙2t 	 e3te˙3t 25α 	 10e2t − e1te1t− (25α 	 10e1t 	 xm1e3t 	 βe2t)e2t	(xm1e2t − 8 	 α3 e3t)e3t − 25α 	 10e21t − βe22t −8 	 α3e23t≤ 0.2.13According to Lyapunov stability theory, the sliding motion on the sliding manifold isstable and ensures limt→∞‖Et‖  limt→∞‖e1te2te3t‖  0.Mathematical Problems in Engineering 5Having established the appropriate switching surface 2.8, as described above, thenext step is to design an SMC scheme to drive the system trajectories onto the sliding modest  0. This study proposes the following SMC:ut  u1t 	 ηψ[signst], η > 1, 2.14where u1t  38 	 10αe1t 	 29α − 1 	 βe2t.Before the scheme of the controller is given, the reaching condition of the sliding modeis derived below.Main TheoremConsider the error dynamics 2.6. If this system is controlled by ut in 2.14, then thesystem trajectory converges to the sliding surface st  0 and satisfies limt→∞‖Et‖ limt→∞‖e1te2te3t‖  0.Proof. Consider the following Lyapunov function candidate:V t  0.5s2t. 2.15Taking the derivative of 2.15with respect to time and using 2.11, 2.14, and 2.15 yieldsV˙ t  sts˙t st(e˙2t 	 25α 	 10e1t 	 xm1e3t 	 βe2t)≤ |st|ψ − stut.2.16Since ψ > 0 and η > 1, V˙ t  sts˙t < 0 can be derived when s/ 0. Thus, according to Lya-punov stability theory, st always converges to the switching surface s  0. Furthermore,since the error dynamics in the sliding manifold is asymptotically stable according to thediscussion above, the error dynamic response of limt→∞‖Et‖  limt→∞‖e1te2te3t‖ 0 is satisfied. Hence, the proof is achieved completely.After the SMC is designed to ensure limt→∞‖Et‖  limt→∞‖e1te2te3t‖  0,thene˙1t  25α 	 10e2t − e1t  0e˙2t  28 − 35αe1t 	 29α − 1e2t−xm1e3t 	 pt − ut  0e˙3t  xm1e2t − 8 	 α3 e3t  0⇒e˙1t  0e˙2t  0 	 pt − ut  0e˙3t  0.2.17It can be inferred thatlimt→∞(pt − ut)  0 2.186 Mathematical Problems in Engineeringwhich means that the message pt can be approximated by the control ut. From previousstudies 22, 23, the control input ut can be approximated by the following continuousequivalent control ueqt:ueqt  u1t 	 ηψ[st|st| 	 σ], 2.19where σ is an arbitrarily small positive constant. When σ is suﬃciently small, then 2.19willarbitrarily approach 2.14 and input message pt can be recovered using 2.19.3. Numerical Simulation and AnalysisIn this section, the continuous equivalent control ueqt 2.19 is utilized for the synchroniza-tion of unified chaotic circuits 2.1. When α  0, the system becomes the original Lorenzsystem asx˙1  10x2 − x1x˙2  28x1 − x2 − x1x3x˙3  x1x2 − 83x3.3.1As mentioned in 24, when directly implementing nonlinear Lorenz systems withelectronic circuits, the major diﬃculty is that the state variables of the system 3.1 occupy awide dynamic range with values that exceed reasonable power supply limits. However, thisdiﬃculty can be eliminated by a simple transformation of variables. Let the Lorenz equationsbe transformed intox˙m1  10xm2 − xm1x˙m2  28xm1 − xm2 − k1xm1xm3x˙m3  k1xm1xm2 − 83xm3.3.2The system 3.2with k1  10, which is referred to as the transmitter, can bemore easilyimplemented because the state variables never exceed the range of typical power supplylimits.Now consider the following Lorenz circuits:Master Lorenz circuit asx˙m1t  10xm2t − xm1tx˙m2t  28xm1t − xm2t − 10xm1txm3t 	 ptx˙m3t  10xm1txm2t − 83xm3t,3.3Mathematical Problems in Engineering 70 0.5 1 1.5 2 2.5 3012345Master and slave states (V)Time (s)−3−2−1Figure 2: States of master and slave systems.0 2 4 6 8 10Switch surface (V)Time (s)−2−1.8−1.6−1.4−1.2−1−0.8−0.6−0.4−0.200.2Figure 3: Simulation results of switch surface st.Slave Lorenz circuit asx˙s1t  10xs2t − xs1tx˙s2t  28xs1t − xs2t − 10xm1txs3t 	 utx˙s3t  10xm1txs1txs2t − 83xs3t,3.4where x˙m and x˙s denote the derivatives of xm and xs with respect to time t, respectively. Theinput message pt is a sine wave 0.5V, 5Hz embedded into the chaotic transmitter andthe equivalent SMC synchronization scheme 2.9 is given in the receiver. The initial valueconditions xm10,xm20,xm30  0.4 0.1 0.2 and xs10,xs20,xs30  0.1 0.1 0.1are used in this example. The control toolbox of MATLAB was used to simulate the proposed8 Mathematical Problems in Engineering0 1 2 3 4 500.050.10.150.20.250.30.35Errors (V)Time (s)−0.15−0.1−0.05Figure 4: Simulation results of errors between state xm and state xs.0 1 2 3 4 5 6 7 8 9 1003691215Recovered and orignal signal (V)Time (s)−15−12−9−6−3Figure 5: Simulation results of continuous equivalent control ueqt and input message pt.secure commutation system. Figure 2 shows the states of the master and slave systems.Figure 3 shows the simulation results of the switch surface st. Figure 4 shows the simulationresults of errors between state xm and state xs. These figures show that the switching surfacest converges to almost zero within 8 s and that the synchronization errors converge toalmost zero after 0.5 s. The master and slave are then synchronous. Figure 5 shows thesimulation results of the continuous equivalent control ueqt and input message pt. Theinput message pt can be recovered.4. ConclusionThis study presented amethod for synchronizing the unified chaotic systems via an SMC. Thesimulation results show the feasibility of the chaotic secure communication system based onthe synchronization of the Lorenz circuits via an SMC.Mathematical Problems in Engineering 9AcknowledgmentsThe authors thank the National Science Council of Taiwan for supporting this work underGrants NSC 101-2622-E-269-012-CC3 and NSC 101-2221-E-241-002. The authors also wish tothank the anonymous reviewers for providing constructive suggestions.References1 E. N. 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Strogatz, “Synchronization of Lorenz-based chaotic circuitswith applications to communications,” IEEE Transactions on Circuits and Systems II, vol. 40, no. 10, pp.626–633, 1993.Submit your manuscripts athttp://www.hindawi.comHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical Problems in EngineeringHindawi Publishing Corporationhttp://www.hindawi.comDifferential EquationsInternational Journal ofVolume 2014Applied MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical PhysicsAdvances inComplex AnalysisJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014OptimizationJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Operations ResearchAdvances inJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Function SpacesAbstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal of Mathematics and Mathematical SciencesHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AlgebraDiscrete Dynamics in Nature and SocietyHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Decision SciencesAdvances inDiscrete MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Stochastic AnalysisInternational Journal of

Synchronization of Unified Chaotic Systems Using Sliding Mode Controller

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