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

Authors: Yeong-Jeu Sun
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The concept of practical synchronization is introduced and the chaos synchronization of master-slave chaotic systems with uncertain input nonlinearities is investigated. Based on the differential and integral inequalities (DII) approach, a simple linear control is proposed to realize practical synchronization for master-slave chaotic systems with uncertain input nonlinearities. Besides, 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 are given to demonstrate the feasibility and effectiveness of the obtained result.


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## Body

## 1. 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, time-domain 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 Section2. 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 AT is used to denote the transpose for a matrix A, and ∥x∥≔xT·x denotes the Euclidean norm of the column vector x.


## 2. Problem Formulation and Main Result

In this paper, we consider the following master-slave chaotic systems with uncertain input nonlinearities.Master system is as follows:(2.1a)x˙1(t)=(10+2529a)·[x2(t)-x1(t)],(2.1b)x˙2(t)=(28-3529a)x1(t)+(a-1)x2(t)-x1(t)x3(t),(2.1c)x˙3(t)=(-83-187a)x3(t)+x1(t)x2(t),(2.1d)[x1(0)x2(0)x3(0)]T=[Δx10Δx20Δx30]T,slave system is as follows:(2.2a)z˙1(t)=(10+2529a)·[z2(t)-z1(t)]+Δϕ1(u1),(2.2b)z˙2(t)=(28-3529a)z1(t)+(k-1)z2(t)-z1(t)z3(t)+Δϕ2(u2),(2.2c)z˙3(t)=(-83-187a)z3(t)+z1(t)z2(t)+Δϕ3(u3),(2.2d)[z1(0)z2(0)z3(0)]T=[Δz10Δz20Δz30]T,wherex(t):=[x1(t)x2(t)x3(t)]T∈ℜ3×1 and z(t):=[z1(t)z2(t)z3(t)]T∈ℜ3×1 are state vectors, a is the system parameter with 0≤a<1, [Δx10Δx20Δx30]T is the unknown initial value satisfying ∥x(0)∥≤R, where R≥0 is given, u=[u1u2u3]T∈ℜ3 is the control input, and Δϕi(·):ℜ→ℜ 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=[u1u2u3]T∈ℜ3 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 numbersβ1,β2, and β3 such that
(2.3)βi·ui2≤uiΔϕ(ui),∀i∈{1,2,3}.Remark 2.1.
Generally speaking, if the uncertain input nonlinearity satisfies(2.4)r1·u2≤uΔϕ(u)≤r2·u2,∀u∈ℜ,
we often refer r2 as the gain margin and r1 as the gain reduction tolerance.For brevity, let us define the synchronous error vector as(2.5)e(t):=[e1(t)e2(t)e3(t)]T:=z(t)-x(t).The precise definition of practical synchronization is given as follows.Definition 2.2.
Given anyα>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 ofu(e,α,R) is linear in the synchronous error e.
In this case, the positive numberα 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
(2.6)u(e,α,R)=-ke(t),
where
(2.7)k=(α+(2/2)p)β,(2.8)p≥max{r1,r2},(2.9)β=min{β1,β2,β3},(2.10)r1:=((8/3)+(a/87))·(19-(5a/29))(1-a)·(5/3+88a/87),(2.11)r2:=R+38-10a29.Proof.
From Theorem 1 of [13], one has
(2.12)min{-r1,-p2}≤x2(t)≤max{r1,p2},∀t≥0,min{-r1+38-10a29,-p2+38-10a29}min≤x3(t)≤max{r1+38-10a29,p2+38-10a29},∀t≥0,
with
(2.13)p2=Δx102+Δx202+(Δx30-38+10a29)2.
This implies that
(2.14)|x2(t)|≤max{r1,r2}≤p,∀t≥0,|38-10k29-x3(t)|≤max{r1,r2}≤p,∀t≥0,
in view of ∥x(0)∥≤R and p2≤p. From (2.1a)–(2.1d)–(2.5), we deduce that, for every t≥0,(2.15a)e˙1=(10+2529a)·(e2-e1)+Δϕ1(u1),(2.15b)e˙2(t)=(28-3529a)e1+(a-1)e2-e1e3-x1e3-x3e1+Δϕ2(u2),(2.15c)e˙3=(-83-187a)e3+e1e2+x2e1+x1e2+Δϕ3(u3).
Let(2.16)W(t):=e12(t)+e22(t)+e32(t).
The time derivative ofW(t) along the trajectories of the closed-loop system (2.15a)–(2.15c) with (2.6)–(2.10) is given by
(2.17)dW(t)dt=2e1(10+25a29)(e2-e1)+2e1Δϕ2(u1)+2e2[(28-35a29)e1+(a-1)e2-e1e3]+2e2(-x1e3-x3e1+Δϕ1(u2))+2e3[(-83-a87)e3+e1e2]+2e3(x2e1+x1e2+Δϕ3(u3))=-2(10+25a29)e12-2(1-a)e22-2(83+a87)e32+2(38-10a29-x3)e1e2+2x2e1e3-∑i=132β(α+(2/2)p)uiΔϕi(ui)≤2(10+25k29)e12-2(1-k)e22-2(83+k87)e32+2(38-10k29-x3)e1e2+2x2e1e3-∑i=132β(α+(2/2)p)βiui2=-2(10+25a29)e12-2(1-a)e22-2(83+a87)e32+2(38-10a29-x3)e1e2+2x2e1e3-∑i=132βiβ(α+22p)ei2≤-2(10+25a29)e12-2(1-a)e22-2(83+a87)e32+2(38-10a29-x3)e1e2+2x2e1e3-∑i=13(2α+2p)ei2≤2|38-10a29-x3|·|e1|·|e2|+2|x2||e1||e3|-(2α+2p)(e12+e22+e32)≤2p·|e1|·|e2|+2p|e1||e3|-(2α+2p)(e12+e22+e32)≤22p·e12+2p·e22+22p·e12+2p·e32-(2α+2p)(e12+e22+e32)=-2αW(t),∀t≥0,
in view of (2.14). It is easy to deduce that
(2.18)e2αt·W˙(t)+e2αt·2αW(t)=ddt[e2αt·W(t)]≤0,∀t≥0.
It follows that
(2.19)∫0tddt[e2αt·W(t)]dt=e2αt·W(t)-W(0)≤∫0t0dt=0,∀t≥0.
Thus, from (2.16) and (2.19), it can be readily obtained that
(2.20)∥e(t)∥2=W(t)≤e-2αtW(0),∀t≥0.
Consequently, we conclude that
(2.21)∥e(t)∥≤e-αtW(0),∀t≥0.
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 1.
Chooseβ1,β2, and β3 such that (A1) is satisfied.Step 2.
Determineβ from (2.9).Step 3.
Determiner1 and r2 from (2.10) and (2.11).Step 4.
Determinep from (2.8).Step 5.
Determinek from (2.7).Step 6.
OUPUTu(e,α,R)=-ke(t).


## 3. 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:(3.1a)Δϕ1(u1)=Δa1·u1+Δb1u13,Δϕ2(u2)=(Δa2+Δb2sint)·u2,Δϕ3(u3)=Δa3·u3.
In addition, the uncertain parameters are bounded by
(3.1b)Δb1≥0,|Δb2|≤1,5≤Δai,∀i∈{1,2,3}.Comparison of (3.1a) and (3.1b) with (A1) and (2.9) yields
(3.2)β1=β3=5,β2=β=4.
From (2.8), (2.10), and (2.11), one has
(3.3)r1=45.35,r2=45.86,p=45.9≥45.86=max{r1,r2}.
Furthermore, from (2.7), it is easy to deduce that
(3.4)k=0.25α+8.11.
Thus, we obtain the design controller
(3.5)u(e,α,R)=-(0.25α+8.11)·e(t),
in view of (2.6). Consequently, by Theorem 2.3, we conclude that the system (2.2a)–(2.2d) with the linear control (3.5) practically synchronizes the generalized Lorenz chaotic system (2.1a)–(2.1d), with the guaranteed exponential convergence rate α.
The typical state trajectories of the sytem (2.1a)–(2.1d) with a=0.4, are depicted in Figure 1. Furthermore, the synchronization errors of systems (2.1a)–(2.1d), and (2.2a)–(2.2d) with the linear control (3.5) are depicted in Figure 2. From the foregoing simulation 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.Figure 1
Typical state trajectories of the system (2.1a)–(2.1d) with a=0.4.Figure 2
Synchronization errors of systems (2.1a)–(2.1d) and (2.2a)–(2.2d) with a=0.4 and α=1.Figure 3
Synchronization errors of systems (2.1a)–(2.1d) and (2.2a)–(2.2d) with a=0.4 and Δϕi(ui)=ui,∀i∈{1,2,3}, under the control of (2.6).Figure 4
Synchronization errors of systems (2.1a)–(2.1d) and (2.2a)–(2.2d), with a=0.4 and Δϕi(ui)=ui,∀i∈{1,2,3}, under the control proposed in [8].Example 3.2.
Consider the following secure communication system and the proposed scheme is illustrated in Figure5.Figure 5
Secure communication system.Transmitter is as follows:(3.6a)x˙1(t)=(10+2529a)·[x2(t)-x1(t)],(3.6b)x˙2(t)=(28-3529a)x1(t)+(a-1)x2(t)-x1(t)x3(t),(3.6c)x˙3(t)=(-83-187a)x3(t)+x1(t)x2(t),(3.6d)ϕx(t)=Cx(t),(3.6e)ϕh(t)=Chx(t)+h(t),(3.6f)[x1(0)x2(0)x3(0)]T=[Δx10Δx20Δx30]T,Receiver is as follows:(3.7a)z˙1(t)=(10+2529a)·[z2(t)-z1(t)]+Δϕ1(u1),(3.7b)z˙2(t)=(28-3529a)z1(t)+(k-1)z2(t)-z1(t)z3(t)+Δϕ2(u2),(3.7c)z˙3(t)=(-83-187a)z3(t)+z1(t)z2(t)+Δϕ3(u3),(3.7d)h1(t)=ϕh(t)-Chz(t),(3.7e)[z1(0)z2(0)z3(0)]T=[Δz10Δz20Δz30]T,whereu is designed as (2.6)–(2.11), x(t)≔[x1(t)x2(t)x3(t)]T∈ℜ3×1, z(t):=[z1(t)z2(t)z3(t)]T∈ℜ3×1, [Δx10Δx20Δx30]T is the unknown initial value satisfying ∥x(0)∥≤8, C∈ℜ3×3 is a nonsingular matrix, h(t)∈ℜq×1 is the information vector, Ch∈ℜq×3,h1(t)∈ℜq×1 is the signal recovered from h(t), and Δϕi(·):ℜ→ℜ 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)∥≤k1·e-αt,∀t≥0. Consequently, by (3.6a)–(3.6f) and (3.7a)–(3.7e), one can see that
(3.8)∥h1(t)-h(t)∥=∥ϕh(t)-Chz(t)-ϕh(t)+Chx(t)∥≤∥Ch∥·∥e(t)∥≤k1∥Ch∥e-αt,∀t≥0.This implies that one can recover the messageh(t) in the receiver system, with the guaranteed exponential convergence rate α. In other words, the synchronization of signals h(t) and h1(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 h1(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.Figure 6
Real message ofh(t)∈ℜ2×1 described in the transmitter of (3.6a)–(3.6f).Figure 7
Recoverd message ofh1(t)∈ℜ2×1 described in in the receiver of (3.7a)–(3.7e).Figure 8
Error signal ofh1(t)-h(t).


## 4. 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.



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