# Rate Estimation of Identical Synchronization by Designing Controllers

Authors: Mitul Islam; Bipul Islam; Nurul Islam
Journal: Journal of Mathematics
 (2013)

---

## Abstract

This paper investigates the synchronization rate for identical synchronization of chaotic dynamical systems, achieved by using controllers. The paper
stresses on the hybrid feedback control technique and the tracking control
technique and determines their corresponding maximum, minimum, and average
synchronization rates. The results obtained are applied on the Shimizu-Morioka
chaotic system, and some necessary and sufficient conditions for synchronization
are obtained. Comparison of the two controllers is undertaken on the basis of
their synchronization rates, in the context of the Shimizu-Morioka system. The
results are analyzed both theoretically and numerically. Moreover, a method
of graphical analysis is proposed to completely characterize the set of hybrid
controllers for a given system.


---

## Body

## 1. Introduction

Chaos control and synchronization has got much attention of the scientists and engineers due to its applicability to various disciplines after the pioneering works by Pecora and Carroll [1] and Ott et al. [2]. It is applied in various fields like information processing, secure communication, neural networks, chemical reactions, biological systems, and so on. Sun and Cao [3] proposed the synchronization between two different noise perturbed chaotic systems. Researchers have proposed different synchronization methods and applied them successfully to control chaos and obtain synchronized chaotic system. Notable methods among them are non-linear feedback control [4], active control [5], adaptive control [6], backstepping design [7], hybrid feedback control [8], and so forth. When the two identical chaotic oscillators are mutually coupled or when one of them drives the other, the synchronization that occurs in this case is known as identical synchronization. Let (x1(t),x2(t),…,xn(t)) and (y1(t),y2(t),…,yn(t)) denote the variables describing the states of the first and second identical oscillators, respectively, depending on time t. For a set of initial conditions (x1(0),x2(0),…,xn(0)) and (y1(0),y2(0),…,yn(0)) of the two systems, if limt→∞[yi(t)-xi(t)]=0, for i=1,2,3,…,n, then the identical synchronization occurs.In this paper, two synchronization schemes for chaotic dynamical systems, using two different controllers [9], are primarily discussed in Section 2. Section 3 deals with measurement of the rate of synchronization [10], when synchronization is achieved using controllers. A quantitative estimate of the rate of convergence of the two systems towards each other in the phase space is obtained. In the following section, the methods stated earlier are applied on coupled Shimizu-Morioka chaotic dynamical system. Synchronization of the chaotic system is achieved both via hybrid feedback control and tracking control, and the results are discussed both analytically and numerically. Finally, the corresponding rates of synchronization are determined and compared. Notably, Section 4.3 explores a method of graphical analysis for determining the possible controller parameters for effective synchronization. It proves to be an efficient and elegant tool for determination of possible controllers where traditional mathematical treatment is rendered inadequate. In fact, it enables us to completely characterize the set of all hybrid controllers for any given problem.


## 2. Description of the Controllers

### 2.1. Description of the Hybrid Controller

System of ordinary differential equations can be expressed as(1)X˙=AX+BΦ(X),
where X∈ℜn, A∈ℜn×n, B∈ℜn×n, and Φ:ℜn→ℜn is the nonlinear part of the system.In the following way, a new system which is identical with the system (1) has been constructed as
(2)Y˙=AY+B[Φ(Y)+U].U is known as the controller which controls the motion of system (2). Feedback controller U should be chosen appropriately to get the identical synchronization of the systems (1) and (2). The controller U is said to be hybrid if it is a combination of linear controller and non-linear controller.Here the controller is chosen asU=u1+u2, where
(3)u1=Φ(X)-Φ(Y)
is the non-linear controller and
(4)u2=K(X-Y)
is the linear controller, with K=diag(k1,k2,…,kn) as the feedback matrix.The synchronization error between the systems (1) and (2) is defined as e=(e1,e2,…,en)T=X-Y, where X=(x1,x2,…,xn)T and Y=(y1,y2,…,yn)T. It is easy to observe that the time evolution of the synchronization error obeys the dynamical equation given by
(5)e˙=(A-BK)e,
where K is the feedback matrix.Yang et al. [8] derived the sufficient condition that if the eigenvalues of the matrix A-BK have negative real parts, then the error dynamical system (5) will be asymptotically stable at origin and the synchronization between the systems (1) and (2) will occur.It is noted that the error dynamical system is of the form(6)e˙=He,
where H∈ℜn×n.


### 2.2. Description of the Tracking Controller

As already noted in Section2.1, any system of first-order non-linear differential equation can be represented as (1). In the context of synchronization of chaotic systems through introduction of tracking controller, a chaotic system of the form (1) is termed as the drive system. Introducing the control vector U∈ℜn, the controlled response system is taken as
(7)Y˙=AY+BΦ(Y)+U,
where Y∈ℜn is the state vector of the response system. The aim of this technique is to design the controller U which will synchronize the states of both the drive and the response systems.Thus, the error dynamical system has a representation(8)e˙=Ae+B(Φ(Y)-ϕ(X))+U,
where e=Y-X is the error vector. With a proper choice of controller U, the previous system can be put in the form
(9)e˙=He,
where H∈ℜn×n. Section 4.5 illustrates a method of choosing the controller suitably.Let us now construct a Lyapunov functionV(e) of the form V(e)=(1/2)eTe. Evidently, V(e) is positive definite. The controller U has to be chosen suitably so that V˙<0 which implies that limt→∞∥e(t)∥=0, and, hence, global synchronization between the states of the drive and response system is achieved asymptotically via the tracking controller U.


## 2.1. Description of the Hybrid Controller

System of ordinary differential equations can be expressed as(1)X˙=AX+BΦ(X),
where X∈ℜn, A∈ℜn×n, B∈ℜn×n, and Φ:ℜn→ℜn is the nonlinear part of the system.In the following way, a new system which is identical with the system (1) has been constructed as
(2)Y˙=AY+B[Φ(Y)+U].U is known as the controller which controls the motion of system (2). Feedback controller U should be chosen appropriately to get the identical synchronization of the systems (1) and (2). The controller U is said to be hybrid if it is a combination of linear controller and non-linear controller.Here the controller is chosen asU=u1+u2, where
(3)u1=Φ(X)-Φ(Y)
is the non-linear controller and
(4)u2=K(X-Y)
is the linear controller, with K=diag(k1,k2,…,kn) as the feedback matrix.The synchronization error between the systems (1) and (2) is defined as e=(e1,e2,…,en)T=X-Y, where X=(x1,x2,…,xn)T and Y=(y1,y2,…,yn)T. It is easy to observe that the time evolution of the synchronization error obeys the dynamical equation given by
(5)e˙=(A-BK)e,
where K is the feedback matrix.Yang et al. [8] derived the sufficient condition that if the eigenvalues of the matrix A-BK have negative real parts, then the error dynamical system (5) will be asymptotically stable at origin and the synchronization between the systems (1) and (2) will occur.It is noted that the error dynamical system is of the form(6)e˙=He,
where H∈ℜn×n.


## 2.2. Description of the Tracking Controller

As already noted in Section2.1, any system of first-order non-linear differential equation can be represented as (1). In the context of synchronization of chaotic systems through introduction of tracking controller, a chaotic system of the form (1) is termed as the drive system. Introducing the control vector U∈ℜn, the controlled response system is taken as
(7)Y˙=AY+BΦ(Y)+U,
where Y∈ℜn is the state vector of the response system. The aim of this technique is to design the controller U which will synchronize the states of both the drive and the response systems.Thus, the error dynamical system has a representation(8)e˙=Ae+B(Φ(Y)-ϕ(X))+U,
where e=Y-X is the error vector. With a proper choice of controller U, the previous system can be put in the form
(9)e˙=He,
where H∈ℜn×n. Section 4.5 illustrates a method of choosing the controller suitably.Let us now construct a Lyapunov functionV(e) of the form V(e)=(1/2)eTe. Evidently, V(e) is positive definite. The controller U has to be chosen suitably so that V˙<0 which implies that limt→∞∥e(t)∥=0, and, hence, global synchronization between the states of the drive and response system is achieved asymptotically via the tracking controller U.


## 3. Theoretical Analysis: Rate Measure for Identical Synchronization via Controllers

As observed already in Section2, the proper choice of controllers for the coupled chaotic dynamical systems produces an error dynamical systems of the form
(10)e˙=He,
where e∈ℜn and H∈ℜn×n. Let us define the Euclidean norm for all n×1 real matrices G as GTG, to be represented, henceforth, as ∥G∥.Definition 1.
If there exist positive constantsC1, C2, r1, and r2 such that C1exp(-r1t)≤∥e(t)∥≤C2exp(-r2t), then r1 is defined as the minimum rate of synchronization, r2 as the maximum rate of synchronization, and their mean r=(r1+r2)/2 as the average rate of synchronization.
If the Euclidean norm of error, that is,∥e∥, can be written in the form C1exp(-r1t)≤∥e(t)∥≤C2exp(-r2t), where C1, C2 are positive constants and r1, r2 are positive, then ∥e(t)∥→0 as t→∞. Clearly, ∥e(t)∥ tends to zero at least as fast as exp(-r2t), but not faster than exp(-r1t). Clearly, for large r2, the rate of convergence of ∥e∥ to zero is very rapid while it is much slower for smaller values of r2. The trend for r1 is also the same. It is noted that if r1 and r2 are close to each other, then ∥e∥ lies within a narrow window between two exponentially decaying curves. Moreover, its value is approximately the same as that of exp(-rt), where r=(r1+r2)/2. When r1 and r2 are far apart, then exp(-rt) determines a mean curve and, hence, is a reasonable approximation for ∥e(t)∥. In this sense, we define r1 as the minimum rate of synchronization, r2 as the maximum rate of synchronization, and r=(r1+r2)/2 as the mean rate of synchronization.Theorem 2.
The average rate of synchronizationr is given by r=(r1+r2)/2, where r1≥sup{-σ(t)∣t≥0} and r2≤inf{-Σ(t)∣t≥0}, σ(t)  and  Σ(t) being the minimum and the maximum eigenvalues of the symmetric matrix G(t)=(H(t)+H(t)T)/2; H(t) is defined in (10).Proof.
Simple mathematical calculation yields(11)12ddt∥e(t)∥2=eTHe=eTHTe.
It yields an estimate of∥e(t)∥ as
(12)∥e(t0)∥exp(∫t0t‍σ(u)du)≤∥e(t)∥≤∥e(t0)∥exp(∫t0t‍Σ(u)du),
where σ(t) and Σ(t) are the minimum and the maximum eigenvalues of the symmetric matrix G=(H+HT)/2, respectively.
Choosingr1 and r2 such that
(13)-r1≤σ(t)≤Σ(t)≤-r2
for all t≥0, we obtain a fairly accurate estimate of the average rate of synchronization r=(r1+r2)/2.Note.In case G becomes a constant matrix, it is easy to obtain the least upper bound and the greatest lower bound of the eigenvalues of G as follows:
(14)r1=sup{-σ(t)∣t≥0}≤inf{-Σ(t)∣t≥0}=r2,
whereby we obtain the average rate of synchronization r=(r1+r2)/2.


## 4. Application of the Results on the Shimizu-Morioka Chaotic System

### 4.1.  The Shimizu-Morioka Chaotic System

Shimizu-Morioka dynamical system is as follows:(15)x˙1=x2,x˙2=x1-λx2-x1x3,x˙3=-αx3+x12,
where x1, x2, x3 are the state variables and λ, α are the parameters. Studies by Shil’nikov [11] revealed that the system exhibits Lorentz-like attractors for α=0.608, λ=1.0499 and for α=0.549, λ=0.605.The previous system of (15) can be written as X˙=AX+BΦ(X) where
(16)X=(x1,x2,x3)T,A=(0101-λ000-α),B=(100010001),Φ(X)=(0,-x1x3,x12)T.


### 4.2. Synchronization of Shimizu-Morioka Chaotic System Using Hybrid Controller

For the feedback matrixK=(k1000k2000k3), we have
(17)A-BK=(-k1000-λ-k2000-α-k3)=H.Characteristic equation of the matrixA-BK is
(18)μ3+(k1+k2+k3+λ+α)μ2+{k1(k2+λ)-(λ+k1+k2)(-α-k3)-1}μ+{-α-k3-k1(λ+k2)(-α-k3)}=0.By the previous Routh-Hurwitz criterion, the roots of the above equation inμ will have negative real parts if and only if
(19)k1+k2+k3+λ+α>0,k1(k2+λ)-(λ+k1+k2)(-α-k3)-1>0,-α-k3+k1(λ+k2)(α+k3)>0,(k1+k2+k3+λ+α)×{k1(k2+λ)-(λ+k1+k2)(-α-k3)-1}+α+k3-k1(λ+k2)(α+k3)>0.Using (3) and (4), the hybrid controller for the Shimizu-Morioka system is given as U=(k1(x1-y1)-x1x3+y1y3+k2(x2-y2)x12-y12+k3(x3-y3)). Therefore, the response system is
(20)y˙1=y2+k1(x1-y1),y˙2=y1-λy2-x1x3+k2(x2-y2),y˙3=-αy3+x12+k3(x3-y3).


### 4.3. Complete Characterization of Control Parameters for Hybrid Controller

The necessary and sufficient conditions for synchronization are given in (19), involving the three control parameters k1, k2, and k3. Due to the complicated nature of the expressions involved, we adopt an alternate path of graphical analysis for explicit determination of “feasible” control parameters, that is, the set of values of (k1,k2,k3) for which the inequalities in (19) are simultaneously satisfied. In this sense, we can claim to have completely characterized the set of hybrid controllers for the synchronization of chaotic coupled Shimizu-Morioka system. The same procedure can be applied to any coupled dynamical system, thus completely characterizing the set of hybrid controllers for it. The method presented also has potential practical applications in controller design.The set of figures presented in Figures1 and 2 illustrate the effectiveness of the method of graphical analysis. Since the problem involves three parameters, a three-dimensional graphical analysis would have been most general. But for the sake of clarity and understandability, we keep one parameter fixed and generate the two-dimensional graphs. The blue regions in all the graphs are the regions of “controllability.” For any set (k1,k2,k3), values within the blue region produce a hybrid controller that drives the response into synchrony with the drive system. Figure 1 fixes k1 at different values and shows the possible values of  k2 and k3. Clearly, k3>-α always for a successful controller design. However, the lower limit on values of k2 decreases with increase of k1. In Figure 2, when k2 is fixed, again k3>-α always and k1 assumes much lower values as k2 is increased. Thus, high values of k1 and k2 always increases the region of “controllability,” while k3 is constrained to lie above -α always.Region of “controllability” for different fixed values ofk1.
(a)
k
1
=
0.1
(b)
k
1
=
0.2
(c)
k
1
=
0.6
(d)
k
1
=
2.0Region of “controllability” for different fixed values ofk3 (a), (b) and k2 (c), (d).
(a)
k
3
=
0.55
(b)
k
3
=
6.0
(c)
k
2
=
0.2
(d)
k
2
=
2.0


### 4.4. Rate of Synchronization Using Hybrid Controller

The symmetric matrixG is given by (-k1101-(λ+k2)000-α-k3). It is clear that the problem has become largely simplified because the matrix G has become constant because of a suitable controller choice. The characteristic equation for this matrix is
(21)(α+k3+μ){(k1+μ)(λ+k2+μ)-1}=0
which has roots -μ1, -μ2, and -μ3, where
(22)μ1=k3+α,μ2=12{λ+k1+k2+(λ+k2-k1)2+4},μ3=12{λ+k1+k2-(λ+k2-k1)2+4}.With the conditionsk3+α>0 and k1(λ+k2)>1, it is observed that μ2>μ3. Thus, Σ(t)=-min{μ1,μ3} and σ(t)=-max{μ1,μ2} for all t≥0. Using Theorem 2,(1)
minimum rate of synchronization =r1=max{μ1,μ2};(2)
maximum rate of synchronization =r2=min{μ1,μ3};(3)
mean rate =r=(r1+r2)/2.Simple calculations yield the maximum possible value ofr given by rmaxH=(μ1+μ2)/2. It is possible to attain r=rmaxH if the following inequality holds: μ2>μ3>μ1.


### 4.5. Synchronization of Shimizu-Morioka Chaotic System via Tracking Controller

The dynamical system (15) is the drive system. Its identical controlled response system is taken as
(23)y˙1=y2+u1,y˙2=y1-λy2-y1y3+u2,y˙3=-αy3+y12+u3,
where  U=(u1,u2,u3)T is the controller to be determined and Y=(y1,y2,y3)T is the response system. Let us define ei=yi-xi(i=1,2,3). Motivated by the observation stated in the outline of this controller design, let us choose the controller as
(24)u1=-e1-e2,u2=-e1+y1y3-x1x3,u3=-(-α+1)e3-y12+x12.Using systems (15), (23), and (24), the error dynamical system takes the form e˙=He, where
(25)H=(-1000-λ000-1).Some trite calculations yieldV˙=-(e12+λe22+e32)<0. Hence, the tracking controller given by (24) leads to global synchronization between the drive and the response systems.


### 4.6. Rate of Synchronization Using Tracking Controller

The symmetric matrixG is given by (-1000-λ000-1). It is clear that the problem has again become largely simplified because the matrix G is a constant diagonal matrix due to proper controller choice. Here, Σ(t)=-min{1,λ} and σ(t)=-max{1,λ} for all t≥0. By Theorem 2,(i)
Case I:0<λ<1, r1=λ, and r2=1;(ii)
Case II:λ=1, r1=1, and r2=1;(iii)
Case III:λ>1, r1=1, and r2=λ.In all the cases, average rate of synchronization is always a constant, wherer=(1+λ)/2; that is,  r=rmaxT=(1+λ)/2.


## 4.1.  The Shimizu-Morioka Chaotic System

Shimizu-Morioka dynamical system is as follows:(15)x˙1=x2,x˙2=x1-λx2-x1x3,x˙3=-αx3+x12,
where x1, x2, x3 are the state variables and λ, α are the parameters. Studies by Shil’nikov [11] revealed that the system exhibits Lorentz-like attractors for α=0.608, λ=1.0499 and for α=0.549, λ=0.605.The previous system of (15) can be written as X˙=AX+BΦ(X) where
(16)X=(x1,x2,x3)T,A=(0101-λ000-α),B=(100010001),Φ(X)=(0,-x1x3,x12)T.


## 4.2. Synchronization of Shimizu-Morioka Chaotic System Using Hybrid Controller

For the feedback matrixK=(k1000k2000k3), we have
(17)A-BK=(-k1000-λ-k2000-α-k3)=H.Characteristic equation of the matrixA-BK is
(18)μ3+(k1+k2+k3+λ+α)μ2+{k1(k2+λ)-(λ+k1+k2)(-α-k3)-1}μ+{-α-k3-k1(λ+k2)(-α-k3)}=0.By the previous Routh-Hurwitz criterion, the roots of the above equation inμ will have negative real parts if and only if
(19)k1+k2+k3+λ+α>0,k1(k2+λ)-(λ+k1+k2)(-α-k3)-1>0,-α-k3+k1(λ+k2)(α+k3)>0,(k1+k2+k3+λ+α)×{k1(k2+λ)-(λ+k1+k2)(-α-k3)-1}+α+k3-k1(λ+k2)(α+k3)>0.Using (3) and (4), the hybrid controller for the Shimizu-Morioka system is given as U=(k1(x1-y1)-x1x3+y1y3+k2(x2-y2)x12-y12+k3(x3-y3)). Therefore, the response system is
(20)y˙1=y2+k1(x1-y1),y˙2=y1-λy2-x1x3+k2(x2-y2),y˙3=-αy3+x12+k3(x3-y3).


## 4.3. Complete Characterization of Control Parameters for Hybrid Controller

The necessary and sufficient conditions for synchronization are given in (19), involving the three control parameters k1, k2, and k3. Due to the complicated nature of the expressions involved, we adopt an alternate path of graphical analysis for explicit determination of “feasible” control parameters, that is, the set of values of (k1,k2,k3) for which the inequalities in (19) are simultaneously satisfied. In this sense, we can claim to have completely characterized the set of hybrid controllers for the synchronization of chaotic coupled Shimizu-Morioka system. The same procedure can be applied to any coupled dynamical system, thus completely characterizing the set of hybrid controllers for it. The method presented also has potential practical applications in controller design.The set of figures presented in Figures1 and 2 illustrate the effectiveness of the method of graphical analysis. Since the problem involves three parameters, a three-dimensional graphical analysis would have been most general. But for the sake of clarity and understandability, we keep one parameter fixed and generate the two-dimensional graphs. The blue regions in all the graphs are the regions of “controllability.” For any set (k1,k2,k3), values within the blue region produce a hybrid controller that drives the response into synchrony with the drive system. Figure 1 fixes k1 at different values and shows the possible values of  k2 and k3. Clearly, k3>-α always for a successful controller design. However, the lower limit on values of k2 decreases with increase of k1. In Figure 2, when k2 is fixed, again k3>-α always and k1 assumes much lower values as k2 is increased. Thus, high values of k1 and k2 always increases the region of “controllability,” while k3 is constrained to lie above -α always.Region of “controllability” for different fixed values ofk1.
(a)
k
1
=
0.1
(b)
k
1
=
0.2
(c)
k
1
=
0.6
(d)
k
1
=
2.0Region of “controllability” for different fixed values ofk3 (a), (b) and k2 (c), (d).
(a)
k
3
=
0.55
(b)
k
3
=
6.0
(c)
k
2
=
0.2
(d)
k
2
=
2.0


## 4.4. Rate of Synchronization Using Hybrid Controller

The symmetric matrixG is given by (-k1101-(λ+k2)000-α-k3). It is clear that the problem has become largely simplified because the matrix G has become constant because of a suitable controller choice. The characteristic equation for this matrix is
(21)(α+k3+μ){(k1+μ)(λ+k2+μ)-1}=0
which has roots -μ1, -μ2, and -μ3, where
(22)μ1=k3+α,μ2=12{λ+k1+k2+(λ+k2-k1)2+4},μ3=12{λ+k1+k2-(λ+k2-k1)2+4}.With the conditionsk3+α>0 and k1(λ+k2)>1, it is observed that μ2>μ3. Thus, Σ(t)=-min{μ1,μ3} and σ(t)=-max{μ1,μ2} for all t≥0. Using Theorem 2,(1)
minimum rate of synchronization =r1=max{μ1,μ2};(2)
maximum rate of synchronization =r2=min{μ1,μ3};(3)
mean rate =r=(r1+r2)/2.Simple calculations yield the maximum possible value ofr given by rmaxH=(μ1+μ2)/2. It is possible to attain r=rmaxH if the following inequality holds: μ2>μ3>μ1.


## 4.5. Synchronization of Shimizu-Morioka Chaotic System via Tracking Controller

The dynamical system (15) is the drive system. Its identical controlled response system is taken as
(23)y˙1=y2+u1,y˙2=y1-λy2-y1y3+u2,y˙3=-αy3+y12+u3,
where  U=(u1,u2,u3)T is the controller to be determined and Y=(y1,y2,y3)T is the response system. Let us define ei=yi-xi(i=1,2,3). Motivated by the observation stated in the outline of this controller design, let us choose the controller as
(24)u1=-e1-e2,u2=-e1+y1y3-x1x3,u3=-(-α+1)e3-y12+x12.Using systems (15), (23), and (24), the error dynamical system takes the form e˙=He, where
(25)H=(-1000-λ000-1).Some trite calculations yieldV˙=-(e12+λe22+e32)<0. Hence, the tracking controller given by (24) leads to global synchronization between the drive and the response systems.


## 4.6. Rate of Synchronization Using Tracking Controller

The symmetric matrixG is given by (-1000-λ000-1). It is clear that the problem has again become largely simplified because the matrix G is a constant diagonal matrix due to proper controller choice. Here, Σ(t)=-min{1,λ} and σ(t)=-max{1,λ} for all t≥0. By Theorem 2,(i)
Case I:0<λ<1, r1=λ, and r2=1;(ii)
Case II:λ=1, r1=1, and r2=1;(iii)
Case III:λ>1, r1=1, and r2=λ.In all the cases, average rate of synchronization is always a constant, wherer=(1+λ)/2; that is,  r=rmaxT=(1+λ)/2.


## 5.  Discussion and Conclusion

Numerical simulations are done with the parameter valuesα=0.549 and λ=0.605, and the initial conditions are taken as X(0)=(1,3,4)T and Y(0)=(-2,1,2)T. In hybrid control technique, the control parameters k1, k2, k3 play an important role for the synchronization of the drive and the response systems. Time evolution of the synchronization errors is plotted in Figure 3. All the errors vanish with time but the rapidity of vanishing of errors depend on the values of ki(i=1,2,3). The synchronization rates are depicted in Figures 4 and 5 for hybrid and tracking controllers, respectively. In both figures, the minimum, maximum, and the average rates of synchronization are shown in addition to ∥e(t)∥. In hybrid control technique, the rate of synchronization is highly dependent on the control parameters ki(i=1,2,3) as depicted in Figures 3 and 4.Convergence of∥e(t)∥ for different parameter choices of the hybrid controller.
(a)
(b)Comparison of the proposed “rate” curves with∥e(t)∥ for synchronization using the hybrid control technique (c=∥e(0)∥).
(a)
(b)Figure 5
Comparison of the proposed “rate” curves with∥e(t)∥ for synchronization using the tracking control technique (c=∥e(0)∥).It is seen from extensive numerical experiments that the average synchronisation rate function, that is,∥e(0)∥exp(-rt), lies extremely close to ∥e(t)∥ (refer to Figures 4 and 5). Clealry, the rate estimation is highly accurate for both the tracking controller and the hybrid controller, illustrating the power of our method of rate estimation. Thus, by and large, the average synchronisation rate function ∥e(0)∥exp(-rt), where r= average synchronisation rate, can be used as a good approximation to ∥e(t)∥ for practical purposes.Figure6 compares the rate of synchronization for hybrid control and the tracking control techniques. It is evident that the hybrid controller may be made better than the tracking controller by adjusting the parameters of the hybrid controller. It is seen from numerical experiments that if k1, k2 and k3 have significantly large values, then hybrid controller is by and large better than the tracking controller in the context of synchronizing the chaotic Shimizu-Morioka system. But choice of arbitrarily low values of k1 and k2 from the region of “controllability” (refer to Section 4.3) causes the tracking controller to work better than the hybrid controller. Thus, we can safely conclude that the choice of comparatively larger parameter values is a sufficient condition for the hybrid controller to be better than the tracking controller. The average rate of synchronization for hybrid controller can be modified at will through parameter adjustments. It is this flexibility that makes hybrid control technique important in chaos synchronization. This is in contrast to tracking controller whose average rate of synchronisation is fairly constant.Comparison of the hybrid and the tracking control techniques in the context of chaotic Shimizu-Morioka system.
(a)
(b)
(c)
(d)As already noted, maximum possible average rate of hybrid control synchronization isrmaxH=(μ1+μ2)/2 and that of tracking control synchronization is rmaxT=(1+λ)/2, which is a constant. Thus, a more rigorous sufficient condition for the hybrid controller to be made superior over tracking controller is
(26)μ1+μ2>1+λ.



---

*Source: 590462-2013-05-26.xml*