DII-Based Linear Feedback Control Design for Practical Synchronization of Chaotic Systems with Uncertain Input Nonlinearity and Application to Secure Communication

and Applied Analysis 3 Master system is as follows: ẋ1 t ( 10 25 29 a ) · x2 t − x1 t , 2.1a ẋ2 t ( 28 − 35 29 a ) x1 t a − 1 x2 t − x1 t x3 t , 2.1b ẋ3 t ( − 3 − 1 87 a ) x3 t x1 t x2 t , 2.1c x1 0 x2 0 x3 0 T Δx10 Δx20 Δx30 T , 2.1d slave system is as follows: ż1 t ( 10 25 29 a ) · z2 t − z1 t Δφ1 u1 , 2.2a ż2 t ( 28 − 35 29 a ) z1 t k − 1 z2 t − z1 t z3 t Δφ2 u2 , 2.2b ż3 t ( − 3 − 1 87 a ) z3 t z1 t z2 t Δφ3 u3 , 2.2c z1 0 z2 0 z3 0 T Δz10 Δz20 Δz30 T , 2.2d where x t : x1 t x2 t x3 t T ∈ 3×1 and z t : z1 t z2 t z3 t T ∈ 3×1 are state vectors, a is the system parameter with 0 ≤ a < 1, Δx10 Δx20 Δx30 T is the unknown initial value satisfying ‖x 0 ‖ ≤ R, where R ≥ 0 is given, u u1 u2 u3 T ∈ 3 is the control input, and Δφi · : → is the uncertain input nonlinearity for every i ∈ {1, 2, 3}. It is noted that the system 2.1a – 2.1d , displays chaotic behavior for each 0 ≤ a < 1 21 . The original Lorenz system is a special case of system 2.1a – 2.1d , with a 0. The objective of this paper is to search a tracking control law u u1 u2 u3 T ∈ 3 such that the states z1, z2, and z3 of the slave system 2.2a – 2.2d track, respectively, the states x1, x2, and x3 of the master system 2.1a – 2.1d , with any desired exponential convergence rate. Throughout this paper, the following assumption is made: A1 There exist positive numbers β1, β2, and β3 such that βi · ui ≤ uiΔφ ui , ∀ i ∈ {1, 2, 3}. 2.3 Remark 2.1. Generally speaking, if the uncertain input nonlinearity satisfies r1 · u2 ≤ uΔφ u ≤ r2 · u2, ∀u ∈ , 2.4 we often refer r2 as the gain margin and r1 as the gain reduction tolerance. 4 Abstract and Applied Analysis For brevity, let us define the synchronous error vector as e t : e1 t e2 t e3 t T : z t − x t . 2.5 The precise definition of practical synchronization is given as follows. Definition 2.2. Given any α > 0 and R ≥ 0, the slave system 2.2a – 2.2d practically synchronizes the master system 2.1a – 2.1d provided that there exist a suitable control u e, α, R and a positive number k1 such that the following conditions are satisfied: i the synchronous error satisfies ‖e t ‖ ≤ k1 · e−αt, ∀t ≥ 0; ii the control law of u e, α, R is linear in the synchronous error e. In this case, the positive number α is called the exponential convergence rate. In other words, the practical synchronization means that there exists a linear control law such that the state of the slave system can track the state of the master system with any desired exponential convergence rate. Now we present the main result for the practical synchronization between system 2.1a – 2.1d and system 2.2a – 2.2d . Theorem 2.3. The uncertain slave system 2.2a – 2.2d practically synchronizes the master system 2.1a – 2.1d with the exponential convergence rate α, under the linear feedback control u e, α, R −ke t , 2.6


Introduction
Chaotic system is a kind of nonlinear dynamic system with unpredictable and irregular behavior. These characteristics may cause difficulties in controlling the system or may deteriorate the system performance. Besides, chaotic systems are extremely sensitive to their initial conditions, so that they are not readily synchronized. However, if these characteristics can be applied skillfully, there are some merits that may be utilized, for instance, applying the chaotic synchronization scheme to chaotic secure communications. On the other hand, linear feedback controllers have the following advantages over nonlinear controllers: i low implementation complexity; ii easily realized in hardware; iii reduced sensitivity to parameter variations; iv improved reference tracking performance 1-4 . Consequently, how to design a linear feedback controller instead of a complicated nonlinear controller is a main issue in the field of chaos synchronization.
In recent years, a wide variety of methodologies in the synchronization of chaotic systems have been proposed, such as Lyapunov's stability theory, adaptive control approach, variable structure control VSC approach, H ∞ control approach, adaptive sliding mode control approach, backstepping control approach, projective synchronization approach, timedomain approach, and others. For more detailed knowledge, one can refer to 5-12 . Over the past decades, generalized Lorenz systems, which are much more useful than traditional Lorenz system in practical applications, have been received a great deal of interest due to theoretical interests and successful applications in numerous areas; see, for example, 6, 8, 10, 12-15 . In 8 , by means of linearization and Lyapunov's stability theory, a linear state error feedback control has been presented to guarantee the uniform chaos synchronization of master-slave identical generalized Lorenz systems without any uncertainties. In 14, 15 , two kinds of state observers for the generalized Lorenz chaotic system have been developed to guarantee the global exponential stability of the resulting error system. Besides, based on the adaptive sliding mode control approach, a single nonlinear control has been proposed in 6 to ensure the synchronization of master-slave identical generalized Lorenz systems without any uncertainties. In 10 , using Lyapunov's stability theory, a linear feedback controller has been developed to realize the exponential synchronization of master-slave identical generalized Lorenz systems without any uncertain input nonlinearity. Meanwhile, some control strategies have been established in 12 to guarantee the coexistence of antiphase and complete synchronization in master-slave identical generalized Lorenz systems without any uncertainties. Besides, based on the time domain approach, the upper solution bound and lower solution bound of the generalized Lorenz chaotic system have been offered in 13 .
Owing to unavoidable tolerances and uncontrollable and unpredictable environmental conditions, it seems to be difficult and impossible to maintain the parameter values e.g., resistors, inductors, and capacitors of the controllers as fixed values. Therefore, uncertain input nonlinearity always exists in dynamic control systems. Over the past decades, researchers have been concerned with various uncertain input nonlinearities common in nonlinear systems, such as deadzones, saturation, hysteresis, relays, and others; see, for instance, 5, 7, 9, 16-20 and the references therein. In this paper, motivated by the discussion mentioned above, the chaos synchronization of master-slave identical generalized Lorenz systems with uncertain input nonlinearities will be investigated. Using the DII methodology, a linear feedback control is proposed to realize practical synchronization for such master-slave systems with any prespecified exponential convergence rates. Applications of proposed master-slave chaotic synchronization technique to secure communication as well as several numerical simulations are given to demonstrate the feasibility and effectiveness of the obtained result. This paper is organized as follows. The problem formulation and main result are presented in Section 2. Several numerical simulations are given in Section 3 to illustrate the main result. Finally, conclusion is made in Section 4. Note that throughout the remainder of this paper, the notation A T is used to denote the transpose for a matrix A, and x : √ x T · x denotes the Euclidean norm of the column vector x.

Problem Formulation and Main Result
In this paper, we consider the following master-slave chaotic systems with uncertain input nonlinearities.

Abstract and Applied Analysis 3
Master system is as follows:ẋ 1c slave system is as follows: 2c where x t : x 1 t x 2 t x 3 t T ∈ 3×1 and z t : z 1 t z 2 t z 3 t T ∈ 3×1 are state vectors, a is the system parameter with 0 ≤ a < 1, Δx 10 Δx 20 Δx 30 T is the unknown initial value satisfying x 0 ≤ R, where R ≥ 0 is given, u u 1 u 2 u 3 T ∈ 3 is the control input, and Δφ i · : → is the uncertain input nonlinearity for every i ∈ {1, 2, 3}. It is noted that the system 2.1a -2.1d , displays chaotic behavior for each 0 ≤ a < 1 21 . The original Lorenz system is a special case of system 2.1a -2.1d , with a 0. The objective of this paper is to search a tracking control law u u 1 u 2 u 3 T ∈ 3 such that the states z 1 , z 2 , and z 3 of the slave system 2.2a -2.2d track, respectively, the states x 1 , x 2 , and x 3 of the master system 2.1a -2.1d , with any desired exponential convergence rate. Throughout this paper, the following assumption is made: A1 There exist positive numbers β 1 , β 2 , and β 3 such that Remark 2.1. Generally speaking, if the uncertain input nonlinearity satisfies we often refer r 2 as the gain margin and r 1 as the gain reduction tolerance.

Abstract and Applied Analysis
For brevity, let us define the synchronous error vector as e t : e 1 t e 2 t e 3 t T : z t − x t .

2.5
The precise definition of practical synchronization is given as follows.
Definition 2.2. Given any α > 0 and R ≥ 0, the slave system 2.2a -2.2d practically synchronizes the master system 2.1a -2.1d provided that there exist a suitable control u e, α, R and a positive number k 1 such that the following conditions are satisfied: i the synchronous error satisfies e t ≤ k 1 · e −αt , ∀t ≥ 0; ii the control law of u e, α, R is linear in the synchronous error e.
In this case, the positive number α is called the exponential convergence rate. In other words, the practical synchronization means that there exists a linear control law such that the state of the slave system can track the state of the master system with any desired exponential convergence rate.

2.13
This implies that a e 3 e 1 e 2 x 2 e 1 x 1 e 2 Δφ 3 u 3 .

2.16
The time derivative of W t along the trajectories of the closed-loop system 2.15a -2.15c with 2.6 -2.10 is given by  This completes the proof.

Remark 2.4.
Based on the adaptive sliding mode control approach, a single nonlinear control has been proposed in 6 to realize the synchronization of master-slave identical generalized Lorenz systems without any uncertainties. It is seen that our designed control 2.6 is a simple linear form which is much simpler than the nonlinear form proposed in 6 . Obviously, the proposed linear feedback control form is much more simply implemented.
Remark 2.5. In this paper, the merits of DII approach can be stated as follows.
i Based on the DII approach, the proposed control law has certain intrinsic robustness properties, in particular, infinite gain margin.
ii Based on the DII approach, the proposed feedback control can be easily implemented owing to the linearity of 2.6 .
iii Based on the DII approach, not only the exponential synchronization can be realized but also the guaranteed exponential convergence rate can be prespecified.

Remark 2.6.
In what follows, we present an algorithm to find the linear control law of 2.6 stated in Theorem 2.3.

INPUT
The master-slave chaotic systems with uncertain input nonlinearities 2.1a -2.1d -2.2a -2.2d the parameters a, α, and R.
OUPUT linear control of 2.6 .
Step 3. Determine r 1 and r 2 from 2.10 and 2.11 .

Numerical Examples and Simulations
In what follows, we provide two examples to illustrate the main results.
Example 3.1. Consider the uncertain master-slave systems 2.1a -2.1d , and 2.2a -2.2d with a 0.4, R 8, and uncertain input nonlinearities: In addition, the uncertain parameters are bounded by  results, it is seen that the controlled uncertain master-slave systems 2.1a -2.1d and 2.2a -2.2d realize the practical synchronization under the linear control 3.5 . It is noted that 8 has proposed a linear control to achieve the synchronization of the systems 2.1a -2.1d and 2.2a -2.2d without any uncertain input nonlinearity, but the design control only guarantees that the synchronization error system is asymptotically stable. The comparisons of the error systems' trajectories are shown in Figures 3 and 4.
Example 3.2. Consider the following secure communication system and the proposed scheme is illustrated in Figure 5. Channel Channel Transmitter is as follows: 3.6e Receiver is as follows: 3.7d where u is designed as 2.6 -2.11 , x t : 3×1 , Δx 10 Δx 20 Δx 30 T is the unknown initial value satisfying x 0 ≤ 8, C ∈ 3×3 is a nonsingular matrix, h t ∈ q×1 is the information vector, C h ∈ q×3 , h 1 t ∈ q×1 is the signal recovered from h t , and Δφ i · : → is the uncertain input nonlinearity satisfying 3.1a -3.1b , with the system parameter a 0.4 and q ∈ N. Setting the control u as 3.5 , by Example 3.1, we have e t z t − x t ≤ k 1 · e −αt , ∀t ≥ 0. Consequently, by 3.6a -3.6f and 3.7a -3.7e , one can see that This implies that one can recover the message h t in the receiver system, with the guaranteed exponential convergence rate α. In other words, the synchronization of signals h t and h 1 t for the proposed secure communication 3.6a -3.6f and 3.7a -3.7e can always be achieved with any prespecified convergence rate α. With, for example, α 6, the real message h t , the recovered message h 1 t , and the error signal are depicted in Figures 6, 7, and 8, respectively, which clearly indicates that the real message h t is recovered after 0.2 seconds.

Conclusion
In this paper, the notion of practical synchronization has been introduced and the chaos synchronization of master-slave chaotic systems with uncertain input nonlinearities has been investigated. Based on the DII approach, a simple linear control has been proposed to realize the practical synchronization for the master-slave chaotic systems with uncertain input nonlinearities. Furthermore, the guaranteed exponential convergence rate can be prespecified. Applications of proposed master-slave chaotic synchronization technique to secure communication as well as several numerical simulations have also been given to demonstrate the feasibility and effectiveness of the obtained result.