The performance of two widely used chaos synchronization approaches, active control and backstepping control, is investigated in this study. These two methods are projected to synchronize two chaotic systems (Master/Drive of Rucklidge Systems) that are identical but have different initial conditions. The paper’s significant feature is that based on error dynamics, controllers are designed using the appropriate variable and the time synchronization between master Rucklidge and drive Rucklidge systems using both methods. The control function of the active control method is designed on the proper selection of matrices. The chaotic behavior is controlled using a recursive backstepping design based on the Lyapunov stability theory with a validated Lyapunov function. The effectiveness of the controller in eradicating the chaotic behavior from the state trajectories is also revealed using numerical simulations with Matlab. The backstepping method is superior to the active control method for synchronization of the measured pair of systems, as it takes less time to synchronize while exhausting the first one than the second one with great performance, according to numerical simulation and graphical outcomes.
Natural sciences are founded on the basis of chaos [
Pecora and Carrol demonstrated that two chaotic systems could synchronize in 1990s [
Bai and Lonngren [
The backstepping control algorithm is a type of non-linear controller design. It is very existent for handling mismatched perturbation [
To control chaos in the Rucklidge system, the Active control and Backstepping approaches are applied in this work. To provide global synchronization between two identical chaotic systems, we project active controllers and a recursive backstepping control and compare simulation results of the two strategies. In Section 1, we give the theoretical foundation of our paper. The remainder of the paper will be written as follows: Dynamical Analysis of the Rucklidge System with two Parameters, Section 2; Synchronization via Active Control is presented in Section 3, with numerical results, whereas Backstepping control design for chaos synchronization is presented in Section 4, with numerical results and a comparative research description in Section 5. Finally, Section 6 brings the paper to a close.
In this work, the Rucklidge chaotic system (1) is described by [
x ˙ = − b x + a y − y z y ˙ = x z ˙ = − z + y 2 (1)
where x , y , z and a , b are the state variables and positive constant parameters, respectively. We show in this paper that the system (1) is chaotic when the parameters are a = 6.7 , b = 2 . We use the chaotic system’s (8) initial values for numerical simulations as: x 1 ( 0 ) = 0.001 , y 1 ( 0 ) = 0.001 , z 1 ( 0 ) = 0.001 .
And the chaotic system (9) as: x 2 ( 0 ) = 5 , y 2 ( 0 ) = 8 , z 2 ( 0 ) = 4 . Figures 1-3 demonstrate the Rucklidge System’s (1) 2-D projections on ( x , y ) , ( y , z ) and ( z , x ) space projections, respectively.
The time response of before synchronization states for the Master system ( x 1 , y 1 , z 1 ) and the Drive system ( x 2 , y 2 , z 2 ) is shown in
The error dynamics in the uncontrolled state are shown in
When Master and Drive systems are in the same parameters, achieving synchronization using active control is appropriate and effective. Assume the existence of a Master system as
x ˙ = M x + g ( x ) (2)
where x = ( x 1 , y 1 , ⋯ , z 1 ) T ∈ R n , M ∈ R n × R n , M is a constant system matrix and g ( x ) is a sequence function that is nonlinear.
The Drive system is
y ˙ = M y + g ( y ) + u ( t ) (3)
where y ˙ = ( x 2 , y 2 , ⋯ , z 2 ) T ∈ R n , u ( t ) = ( u i ( t ) ) ∈ R n , where i = ( 1 , 2 , ⋯ , n ) .
If an appropriate controller u ( t ) exists that satisfies ∀ x , y , e ∈ R n , lim t → 0 ‖ y − x ‖ = lim t → 0 ‖ e ‖ = 0 the Master and Drive systems are then designed to be synchronized.
As a result, the error is defined as e = y − x .
Then the error dynamics is
e ˙ = y ˙ − x ˙ = M e + G ( x , y ) + u ( t ) (4)
where G ( x , y ) = g ( y ) − g ( x ) . Without e of system (4), controller u ( t ) may eliminate non-linear section. That is
u ( t ) = v ( t ) − G ( x , y ) (5)
where v ( t ) = K e represents a linear section with error variables. From Equation (5) and Equation (4) we get
e ˙ = M e + v ( t ) (6)
Equation (6) becomes when v ( t ) is a linear section with error variables and v ( t ) = k e is a constant matrix.
e ˙ = ( M + K ) e (7)
Satisfing the requirements needed for diagonal matrix ( M + K ) is λ i ≤ 0 , where λ i is the eigenvalue of matrix ( M + K ) , state vectors of system (7) is asymptotically converge to zero, as a result Master system (2) and Drive system (3) asymptotically synchronize. In this context, the Master and Drive systems are defined as follows:
The Master Rucklidge system is defined by
x ˙ 1 = − b x 1 + a y 1 − y 1 z 1 y ˙ 1 = x 1 z ˙ 1 = − z 1 + y 1 2 (8)
where a and b are the system parameters.
The Drive Rucklidge system is defined by
x ˙ 2 = − b x 2 + a y 2 − y 2 z 2 + u 1 y ˙ 2 = x 2 + u 2 z ˙ 2 = − z 2 + y 2 2 + u 3 (9)
u i = ( 1 , 2 , 3 ) are active control functions for the Master system. In Equation (9), we presented three control functions: u 1 ( t ) , u 2 ( t ) , and u 3 ( t ) which are must be regulated. We subtract Equation (8) from Equation (9) to estimate the control functions. Consider the state errors between the controlling system (9) and the master system (8) that needs to be controlled by using
e x = x 2 − x 1 ; e y = y 2 − y 1 ; e z = z 2 − z 1 ; (10)
Applying the active control design methods, we subtract Equation (8) from Equation (9) and use the definitions in Equation (10) to derive the error dynamics equation:
e ˙ x = − b e x + a e y − y 2 z 2 + y 1 z 1 + u 1 e ˙ y = e x + u 2 e ˙ z = − e z + e y ( y 2 + y 1 ) + u 3 (11)
tracking the re-describing control functions as:
v 1 = u 1 − y 2 z 2 + y 1 z 1 v 2 = u 2 v 3 = u 3 (12)
the error dynamics Equation (11) takes on a new state
e ˙ x = − b e x + a e y + v 1 e ˙ y = e x + v 2 e ˙ z = − e z + e y ( y 2 + y 1 ) + v 3 (13)
The controllable state of error system (11) is a linear system with control effort ( v 1 ( t ) , v 2 ( t ) , v 3 ( t ) ) as a function of error states ( e x , e y , e z ) . The error states ( e x , e y , e z ) converge to zero as time t → ∞ passes if these stabilize the system. This means that active control is used to synchronize the Master and Drive systems. For the control ( v 1 ( t ) , v 2 ( t ) , v 3 ( t ) ) , there are various possible sets. We created a constant matrix A that will govern the error dynamics (11) in accordance with the active control approach, such that
[ v 1 , v 2 , v 3 ] T = A [ e x , e y , e z ] T (14)
where A is a constant matrix. To stabilize the state of the error system, the components of the matrix A must be chosen in such a way that the feedback system has all eigenvalues with negative real portions. In the tracking form, choose the matrix A:
A = ( − b − a 0 − 1 − 1 0 0 − ( y 2 + y 1 ) 0 )
Bring Equation (14) into Equation (13), we may find
e ˙ = K e
( e ˙ x e ˙ y e ˙ z ) = ( − b a 0 1 0 0 0 y 2 + y 1 − 1 ) ( e 1 e 2 e 3 ) + A ( e 1 e 2 e 3 )
K = ( − 2 b 0 0 0 − 1 0 0 0 − 1 )
In this specific selection, the eigenvalues of the system (13) are ( − 2 b , − 1, − 1 ) . Due to the linearity system’s stability theory, this option will result in the convergence of error states ( e x , e y , e z ) to zero as time t approaches infinity, and therefore the synchronization of the two systems is achieved under the control system (15)
u 1 = − b e x − a e y + y 2 z 2 − y 1 z 1 u 2 = − e x − e y u 3 = − ( y 2 + y 1 ) e y (15)
To get numerical results, we apply MatLab Simulink and the Runge-Kutta algorithm of 4th order with a time grid of 0.05. We continue with the initial conditions of the Master system Equation (8) as follows:
x 1 ( 0 ) = 0.001 , y 1 ( 0 ) = 0.001 , z 1 ( 0 ) = 0.001 (16)
and the Drive system’s initial conditions (9) as follows:
x 2 ( 0 ) = 5 , y 2 ( 0 ) = 8 , z 2 ( 0 ) = 4 (17)
The time response of the error states obtained from numerical simulations of synchronization systems (11) is shown in
e = e x 2 + e y 2 + e z 2
and Figures 8-10 demonstrate the response time of the synchronization of the driven system (9) to the driver (8), control signals are activated at the time t = 0, and
The purpose of this section Figures 8-10 is to exhaust active control procedure to synchronize two chaotic chaotic systems (8) and (9) by defining a controller that the Drive system capability to trail the Master System and the states of two chaotic systems (8) and (9) show similar activities for all future states [
This section focuses on Backstepping Control systems because this class offers various examples of chaotic circuits and systems [
x ˙ = f ( x ) + g ( x ) μ 1 (18)
be a responsive affine nonlinear system, with μ 1 ∈ R as the control input, x ∈ R n as the state, and f and g as nonlinear functions, with f ( 0 ) = 0 . The control system 18 can use the usual backstepping design. To stabilize systems that are in strict feedback form, we can use recursive application of backstepping control design as follows.
x ˙ = f ( x ) + g ( x ) μ 1 μ ˙ 1 = f 1 ( x , μ 1 ) + g 1 ( x , μ 1 ) μ 2 ⋮ μ ˙ k − 1 = f k − 1 ( x , μ 1 , ⋯ , μ k − 1 ) + g k − 1 ( x , μ 1 , ⋯ , μ k − 1 ) μ k μ ˙ k = f k ( x , μ 1 , ⋯ , μ k ) + g k ( x , μ 1 , ⋯ , μ k ) u (19)
where μ 1 , μ 2 , ⋯ , μ k and u are scalars [
Backstepping design is a Lyapunov-based control approach that is well-organized. Backstepping control design of systems (18) is a recursive strategy that ensures the system’s global asymptotic stability. The fundamental concept is to extend Lyapunov’s method by breaking down the full system design model (19) into a series of design challenges for lower-order systems. Depending on the state variables, control parameters, and stabilizing functions, incorporate new variables into transformation processes. The ith subsystem may be stabilized with respect to a specific Lyapunov function V i , i = 1 , 2 , ⋯ , n by utilizing the backstepping control design at the ith step. By treating the variable μ 1 as a virtual control input, the approach is used to stabilize the first equation. Similarly, the second equation is stabilized by continuing to use the variable μ 2 as the virtual control, when μ 1 is designed and so on. So the design of the final control input u, which generally depends on x and μ 1 , μ 2 , ⋯ , μ k , is steadily achieved in n steps [
Consider the following Master system of Rucklidge systems:
x ˙ 1 = − b x 1 + a y 1 − y 1 z 1 y ˙ 1 = x 1 z ˙ 1 = − z 1 + y 1 2 (20)
in relation to the Drive system as
x ˙ 2 = − b x 2 + a y 2 − y 2 z 2 + u y ˙ 2 = x 2 z ˙ 2 = − z 2 + y 2 2 (21)
where u is a to-be-defined control function. We required only one controller in backstepping method. By using the error states definition (10) and subtracting Equation (20) from (21), we obtain
e ˙ x = − b e x + a e y − z 2 e y − y 1 e z + u e ˙ y = e x e ˙ z = − e z + e y ( y 2 + y 1 ) (22)
to re-arrange error system (22) into system (23) according to sequence of full system (19)
e ˙ z = − e z + e y ( y 2 + y 1 ) e ˙ y = e x e ˙ x = − b e x + a e y − z 2 e y − y 1 e z + u (23)
The previous system (23) is an equilibrium point ( 0,0,0 ) in the absence of all control u. If u is chosen, then the equilibrium point residues are unchanged, the challenge of synchronization between the Master and Drive systems can be transformed into a problem of asymptotical system stabilization (23). The goal of this project is to find a control law u that will stabilize the system’s error variables (23) at the origin. Since the system comprises three states or three non-linear differential equations, a recursive strategy will have three steps. We split down the system (23) into three subsystems, each with a single input and output. We initiate the formation with the first subsystem of the first non-linear differential equation, and one continues until the last subsystem. In the procedure of formation a change of coordinates w i = ( e x , e y , e z ) is done.
To begin, we use e y as a virtual controller to stabilize the first equation in (23), the object of the control is to drive ( e x , e y , e z ) = ( 0 , 0 , 0 ) i.e.. ( x 1 ; y 1 ; z 1 ) = ( x 2 ; y 2 ; z 2 ) , and define the first new virtual variable of the Backstepping design as w 1 = e z and Considering another new virtual variable as w 2 = e y − α 1 . This last term will not be used in the first step, but it is needs to w 2 to join the first subsystem on w 1 to the next subsystem on w 2 which will be considered in the second step, α 1 is the stabilization function. To find the stabilization function α 1 , consider the Lyapunov function [
V 1 ( w 1 ) = 1 2 ( w 1 ) 2 (24)
substituting the derivative of w 1 for the time derivative of Equation (24) yields:
V ˙ 1 = − ( w 1 ) 2 + w 1 w 2 ( y 2 + y 1 ) + w 1 ( y 2 + y 1 ) α 1 (25)
If the estimated stabilisation function α 1 is chosen as α 1 = 0 , as a result, V ˙ 1 is indefinitely negative. The second term in Equation (25) will be eradicated in the next step.
We stabilize the second equation in (23) for the stability of the second subsystem and its dynamics are calculated by putting the second equation of the system (23) into the second virtual variable with derivatives:
w ˙ 2 = e ˙ y − α ˙ 1
The stabilization function α 2 for the second subsystem is preferred such as the new virtual state variable w 3 = e x − α 2 . Let Lyapunov function for the second subsystem as:
V 2 ( w 1 , w 2 ) = V 1 ( w 1 ) + 1 2 ( w 2 ) 2 (26)
Substituting the derivative of w 2 in time derivative of V ˙ 2 then we get:
V ˙ 2 = − w 1 2 − w 2 2 + w 2 w 3 + w 2 ( w 2 + w 1 ( y 2 + y 1 ) + α 2 ) (27)
The stabilization function α 2 is chosen as:
α 2 = − w 2 − w 1 ( y 2 + y 1 ) (28)
And by substituting α 2 , V ˙ 2 converted:
V ˙ 2 = − w 1 2 − w 2 2 + w 2 w 3 (29)
then V ˙ 2 is negative definite, the third term in Equation (29) will be eradicated in the next step.
To get the third virtual variable state, the time derivative of both the error on the third coordinate state and the stabilization function α 2 . The equation of the third subsystem become:
w ˙ 3 = e ˙ x − α ˙ 2
w ˙ 3 = − b e x + a e y − z 2 e y − y 1 e z + u − α ˙ 2 (30)
We choose the Lyapunov function for the third subsystem is as:
V 3 ( w 2 , w 3 ) = V 2 ( w 2 ) + 1 2 ( w 3 ) 2 (31)
It’s time derivative gives:
V ˙ 3 = − w 1 2 − w 2 2 + w 2 w 3 + w 3 w ˙ 3 V ˙ 3 = − w 1 2 − w 2 2 − b w 3 2 + w 3 ( w 2 − b α 2 + ( a − z 2 ) e y − y 1 e z − α ˙ 2 + u ) (32)
If the control law u is preferred as follows
u = − w 2 + b α 2 − ( a − z 2 ) e y + y 1 e z + α ˙ 2 (33)
Then,
V ˙ 3 = − w 1 2 − w 2 2 − b w 3 2 (34)
is negative definite. The final system of the time derivative of the error model in ( w 1 , w 2 , w 3 ) coordinates in (35), the error dynamics ( e x , e y , e z ) will converge to zero as t → ∞ , whereas the equilibrium ( 0,0,0 ) residues are asymptotically stable, according to the LaSalle-Yoshizawa theorem [
w ˙ 1 = − w 1 + w 2 ( y 2 + y 1 ) w ˙ 2 = e x w ˙ 3 = − b e x + a e y − z 2 e y − y 1 e z + u − α ˙ 2 (35)
The equilibrium ( 0,0,0 ) of system (35) is universally asymptotically stable, as illustrated in
We solved system (22) with the controllers defined in (33) using the fourth-order Runge-Kutta algorithm with initial conditions x 1 ( 0 ) = 0.001 , y 1 ( 0 ) = 0.001 , z 1 ( 0 ) = 0.001 and x 2 ( 0 ) = 5 , y 2 ( 0 ) = 8 , z 2 ( 0 ) = 4 with a time step of 0.01, and fixing the parameter values of a and b as shown in Figures 1-3. To demonstrate the efficiency of the control law, we show numerical findings. The time response of the error state ( e x , e y , e z ) is shown in
obtained from numerical simulations of the synchronization systems (20) and (21) under the controller (33).
From
e = e x 2 + e y 2 + e z 2
After an initial transcience of around t = 6 s , the Master-Drive system is globally synchronized. The exponential convergence of the synchronization quality described by error propagation on average error states also confirms this in
The persistence of this part
We obtained a relationship between synchronization time to investigate and compare the synchronization performance of the two strategies.
The active control and backstepping control techniques are used to create a controller for a three-dimensional autonomous chaotic system, with backstepping control reducing the number of controllers from three to one, and active control works according to dimension. So Backstepping control performs significantly, reducing the controller complexity and cost than the active control method.
The performance of two control techniques for chaos synchronization, active control and recursive backstepping control, was investigated in this work. The two techniques are proven to have outstanding synchronization performance, with the active control slightly outperforming the backstepping. The performance of theoretically designed nonlinear controllers was verified by numerical simulations that confirmed the proposed controller’s effectiveness. The results are presented in graphical style, together with a time history (Figures 1-16). The summaries are as follows:
• Three controllers are found in active control design, and one controller is originated for the Backstepping design;
• In two techniques, the error dynamics converge to zero as t → ∞ , hence the equilibrium point ( 0,0,0 ) remains asymptotically stable;
• Similar activities are shown in both ways for the states of two chaotic systems (Master and Drive);
• The results reveal that the backstepping strategy converges to zero faster than the active control technique for momentary error dynamics.
The authors declare no conflicts of interest regarding the publication of this paper.
Tarammim, A. and Akter, M.T. (2022) A Comparative Study of Synchronization Methods of Rucklidge Chaotic Systems with Design of Active Control and Backstepping Methods. International Journal of Modern Nonlinear Theory and Application, 11, 31-51. https://doi.org/10.4236/ijmnta.2022.112003