Projective Synchronization of Oscillation and Chaos in Fractional-Order
Anti-Controlled Rigid Body System
M. Shahiri* T. , A. Ranjbar N.*,
S.H. Hosseinnia*, R. Ghaderi*, S. Momani**
* Noshirvani University of Technology, Faculty of Electrical and Computer Engineering
, P.O. Box 47135-484, Babol, Iran,() ,()
** Department of Mathematics, Mutah University, P.O. Box: 7, Al-Karak, Jordan
Abstract: The main objective of this study is to show the necessity requirements of the stability of an integer and fractional order Euler equations for motion of a rigid body. Although the stability of chaotic Euler dynamic is of the main concern, a projective synchronization (PS) technique is applied to control this chaotic system. PS with both identical and different scaling factors is also realized. This type of controller is also applied to synchronize chaotic fractional-order systems in master–slave structure. The numerical simulation verifies the effectiveness of the proposed controller even for chaotic synchronization task. Copyright © 2008 IFAC
Keywords: Fractional-order Differential Equations (FDEs), Chaos, Euler equation, Anti-control, Projective Synchronization.
1. INTRODUCTION
In 1944, Nadolschi (Leimanis, 1965) showed that the motion of a rigid body obeys true Euler differential equations. In fact, Euler equations of a rigid body motion are a basic and fundamental three-dimensional autonomous system in classical mechanics. Closing a linear feedback with certain gains (to Euler dynamic), a chaotic behaviour is easily occurred (Chen, 2004). It has also been shown (Ahmad, 2003) that nonlinear chaotic systems may keep their chaotic behaviour when their models become fractional. Fractional calculus has 300-year of history. However, applications of fractional calculus in physics and engineering have just begun (Hilfer, 2001). Many systems are known to display fractional order dynamics, such as viscoelastic systems (Bagley, 1991), dielectric polarization, and electromagnetic waves. In the recent years, emergence of effective methods in differentiation and integration of non-integer order equations makes fractional-order systems more and more attractive for the control systems community.
The current work will be stabilization of a fractional order of Euler dynamics and generalization of control systems. There is still lack of report to control and synchronize two identical fractional-order Euler equations with chaotic behaviour. A pioneering work has been presented the concept of “chaotic synchronization” (Pecora and Carroll, 1990). The work has been continued through presentation of a method to synchronize two identical chaotic systems with different initial conditions (Carroll and Pecora, 1991).
Nowadays chaotic synchronization has been intensively studied (Chen, 1993, 1998; Fuh, 1995; Wang et al., 2003, 2001). Different types of chaotic synchronization methods in terms of complete synchronization, generalized synchronization, phase synchronization, lag synchronization (Ho et al, 2002; Michael et al, 1997; Shahverdiev et al, 2002; Shinbrot et al, 1993; Yang, 2001; Yu et al, 2001) have been presented. In recent years, PS has received many attentions, which characterizes the state vectors of synchronization systems to become proportional with a scaling factor (Jia, 2007). Under certain conditions, not only the phase difference between the drive and response systems is locked, but also their amplitudes of state vectors evolve in a proportional scale (Mainieri and Rehacek, 1999). In (Mainieri and Rehacek, 1999), R. Maineiri and J. Rehacek studied PS in partially coupled linear chaotic systems such as the Lorenz system. Later, D. Xuetal. [Xu et al, 2004, 2002] investigated the conditions of PS in partially linear continuous and discrete-time chaotic systems in an arbitrary dimension.
The current work mainly considers PS and control of a fractional order Euler equations for motion of a rigid body system. PS with both identical and different scaling factors between these systems is also realized. As a novel idea, a fractional order dynamic will be controlled in a Projective Synchronization (PS) approach. Meanwhile, it will be shown this type of controller will cope with complexity of the Euler terms, at least in the numerical point of view. It is also promising the PS decreases the time of synchronisation though, this will be done in near future.
This paper is organized as follows:
System description and chaotic behaviour in Equations of motion of rigid body (Euler Equations) will be studied in section 2. PS method will be proposed to synchronize two identical fractional-order Euler Equations in section 3. PS with different scaling factors between these systems are also realized in section 4. Ultimately, the work will be concluded at section 5.
2. SYSTEM DISCRIPTION
Consider the following Euler equations for motion of a rigid body with principle axes located at the centre of mass (Chen,2004):
(1)
where I1, I2, I3 are the principal moment of inertias, ω1, ω2, ω3 are the angular velocities about principal axes fixed at the centre of mass and M1, M2, M3 are applied moments. In our case, the applied moments are realized in a linear feedback i.e. , where:
(2)
Then the equations of motion are represented as:
(3)
Denoting,, , . Differential equations in (3) are rewritten in the state space form of:
(4)
A chaos may appear when the equilibrium point of system (4) is unstable. According to the results of Liu and Chen (Liu and Chen, 2003), the parameters a, b and c must satisfy the following necessary condition such that the system (4) generates chaos.
a > 0; b < 0; c < 0 and 0 < a < -(b+c) (5)
On the other hand, the parameters I1, I2 and I3 need to satisfy
(6)
(7)
Proper variables transformation, the same results will be obtained for the system (4) with conditions (6) and (7). So we can just study the case with conditions (5) and (6). For simplicity, assumption of I3 =3I0, I1 = 2I0 and I2 = I0, where I3 > I1 > I2, transforms system (4) into the following state equations:
(8)
The system (8) exhibits both strange attractors and limit cycles for certain choices of a, b, c. In Fig. 1, for initial conditions (0.2,0.2,0.2) and a = 5, b =-10 and c = -3.8
,-0.38,- 0.038 ,different strange attractors for this system and their chaotic behaviour are displayed.
Although some works have been reported on chaotic Euler system, there are less reported results on fraction order Euler system. In this paper, a PS is applied on fractional order chaotic Euler system. To define the fractional order Euler with oscillation behaviour (equation 8), the equation will be defined as follows:
(9)
in which q varies in accordance with the fractional order of the system in (9). It can be shown that for some range of q, the fractional order Euler system is unstable (Tavazoei, M.S. 2008). A resonance phenomenon of fractional order Euler system is shown in Figure (2) for different values of q=0.98, 0.95 and 0.9 together with initial conditions (0.2,0.2,0.2) , a = 5, b =-10 and c =-3.8. Numerical simulation has carried out using the SIMULINK based on the frequency domain approximation.
Fig. 2: Phase portrait of chaotic Euler system for different values of the fraction parameter. a) q=0.98 b) q=0.95 c) q=0.93. PROJECTIVE SYNCHRONIZATION BETWEEN TWO CHAOTIC EULER SYSTEMS
Some of nonlinear systems represent a chaotic behaviour due to sensitivity to initial conditions. This means two identical distinct systems but with a minor deviation in their initial condition may exhibit completely different. This means having known bounded initial conditions, there is less chance for the dynamic to predict the behaviour. This unpredictable dynamic is called chaos.
3.1 Projective control of integer-order Euler system
In this section a necessity condition of the stability based on the Lyapunov stability requirements will be derived. Additionally this condition will be used to synchronize two chaotic Euler systems.
The drive system is of the form:
(10)
Similarly the response system can be of the same form. Due to lack of identical initial condition, an external input controls add for synchronization. Therefore the model can be expressed as:
(11)
The objective is to find suitable controllers ui (i = 1, 2, 3). To ensure the drive system (10) and the response system (11) approach PS with a scaling factor α. Define the error vector as , , . The error dynamic can be written as:
(12)
If the following Lyapunov function is candidate as:
(13)
The time derivation of the Lyapunov function along with the trajectory is:
(14)
Designing a suitable controller, two chaotic systems will approach PS for any bounded initial conditions. In order to achieve this objective, the following controllers will be chosen:
(15)
Substitution of the controllers (15) into (12) leads to the following dynamic:
(16)
These obtain the time derivation of V as:
(17)
According to the Lyapunov stability theory, the error dynamic (16) is asymptotically stable. Therefore, PS between the system (10) and (11) will be achieved through the controllers (15) (Jia, 2007). Numerical simulations are given to show the feasibility and the effectiveness of the controllers in (15). At this stage a proper value of makes production of the control signal possible. As an example, for a a high amplitude control signal will be produced which is not suitable to be practically built. Therefore the parameter is a normal value.
A 4th order Runge–Kutta solver is employed to integrate the differential equations with time interval t = 0.001 second. Initial conditions of the drive system and the response system are chosen as (50, −50, 30) and (0.2, 0.2, 0.2), respectively. This is because to force two systems behave differently.
Fig. 3: synchronization of states x, y ,z. and Time history of the error dynamical systemIt can be observed from Fig. 3 that starting from different initial conditions, the drive system (10) and the response system (11) have the same chaotic attractor and move onto two completely different orbits before the control is activated. But, the synchronization will be immediately achieved when the control is activated (the controller activated at t=0 sec).
3.2 projective control of fractional-order Euler system
The same procedure is done to achieve PS to the fractional-order of Euler dynamic as:
(18)
The response system defined as:
(19)
Again, the error dynamic can be written as:
(20)
Similarly, the Lyapunov function will be candidate and used to investigate the stability and for the designation.
(21)
(22)
To satisfy the negative definite condition of derivative of the Lyapunov function, it is necessary to define the control according to:
(23)
The state equation of the controlled system can be expressed as the following:
(24)
Substituting the controllers (23) into the time derivation V, we obtain
(25)
According to the Lyapunov stability analysis can be seen that the synchronized system is asymptotically stable. In this section, numerical simulations have carried out using the SIMULINK based on ode45 method solver. The time step size employed in this simulation is 0.001. Initial conditions of the drive and response are chosen (-15,-10,10), (0.2,0.2,0.2) , α=2,(a=5,b=-10,c=-3.8).
Fig. 4: synchronization of states x, y ,z. and Time history of the error for oscillator fractional Euler system for q=0.9 and a=2.4. PROJECTIVE SYNCHRONIZATION WITH DIFFERENT SCALING FACTORS FOR FRACTIONAL-ORDER EULER EQUATION
In the previous section, a PS with the same scaling factor α was realized. In the following, PS with different scaling factors which implies that the three state variables of the drive system are in proportion to that of the response system with three different scaling factors α1, α2 and α3 respectively. In another word, there exists a constant matrix α = diag(α1, α2, α3) such that . This kind of synchronization form which is called ‘modified PS’ [Li, 2007], has recently been considered in chaotic systems [Li ,2007; Park, 2007].
Similarly the same derive–response system is used together with the error definition as:, , .
(26)
The aim is to design suitable controllers to achieve PS with different scaling factors between the drive system (18) and that of the response in (19). The same Lyapunov function as (21) leads to the following differentiation as:
(27)
To guarantee the negative definite condition for the time derivative of the Lyapunov function the following controllers are defined:
(28)
With these choices, the time derivation of the Lyapunov function becomes:
(29)
Therefore, the error dynamic (26) is found asymptotically stable at the origin, according to the Lyapunov stability theory. Meanwhile the drive in (18) and the response in (19) can asymptotically approach PS with different scaling factors if the controller in (28) is used. As a result of the large compression and stretch associated to the different proportions, the shape of the response system attractor become remarkably different from that of in the drive
system attractor (Fig. 5). Figure 6 exhibits the
numerical result of PS using different proportions.
Fig. 5: The shape of the fractional drive and response system for ai=1,-1,1 and q=0.9.
4. CONCLUSION
In this paper, the stability requirements of a PS of Euler dynamic based on the Lyapunov criteria are given. Additionally a controller synchronizes both integer and fractional orders of the Euler system. The work has also been developed for both identical and different scaling factors especially when the behaviour of dynamic is oscillatory. The simulation results verify the significance of the propose synchronization task even when two systems are chaotic.
Fig. 6: synchronization of states x, y ,z. and Time history of the error for fractional Euler system for q=0.9 and a=(5,-1,1).REFERENCES