Control and Synchronization of Chaos in ... - Semantic Scholar

ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.5(2008) No.1,pp.11-19

Control and Synchronization of Chaos in Nonlinear Gyros via Backstepping Design B. A. Idowu1 ∗ , U. E. Vincent2,† , A. N. Njah3 1

2

Department of Physics, Lagos State University, Ojo, Nigeria. Department of Physics, Olabisi Onabanjo University, P.M.B 2002, Ago-Iwoye, Nigeria. 3 Department of Physics, University of Agriculture, Abeokuta, Nigeria. (Received 14 July 2007, accepted 6 November 2007)

Abstract:In this paper, we investigate the control and synchronization of chaos in nonlinear gyros. The control analysis is based on backstepping approach that interlace the appropriate choice of a Lyapunov function, while the synchronization is based on adaptive backstepping design. The designed control functions ensures stable controlled and synchronized states for two identical nonlinear gyros. Numerical simulations are implemented to verify the results. Key words: Synchronization, Nonlinear Gyros, Backstepping, Control PACS: 05.45.-a,05.45.Pq,05.45.Ac

1

Introduction

Since the pioneering work of chaos control by Ott, Grebogi and York (OGY) [1] in 1990 and presentation of synchronization of chaotic systems with each other by Pecora and Carroll [2] in the same year, chaos control and synchronization has received increasing attention [3–7] and has become a very active topic in nonlinear science. Over the last decade various effective methods have been proposed and utilized [8–20] to achieve the control and stabilization of chaotic systems, even to experimental systems like laser, power electronics etc. The idea of synchronizing two identical chaotic systems that start from different initial conditions consists of linking the trajectory of one system to the same values in the other so that they remain in step with each other, through the transmission of a signal. The control of physical systems is an important subject in engineering, thus, in some applications, chaos can be useful while in others it might be detrimental, for example, chaos in power systems [21–23] and in mechanical systems is objectionable. On the other hand, the idea of chaos synchronization was utilized to build communication systems to ensure the security of information transmitted [24–32]. Several attempts have been made to control and synchronize chaotic systems [2, 16, 18, 32]. Some of these method need several controllers to realize synchronization. The OGY method, for instance, have been successfully applied to many chaotic systems like the periodically driven pendulum [33] and parametric pendulum [34]. Also, the Pyragas time-delayed auto-synchronization (TDAS) method [35, 36] has been shown to be an efficient method that has been realized experimentally in electronic chaos oscillators [37], lasers [38] and chemical systems [39]. In addition, the delayed feedback control, addition of periodic force and adaptive control algorithm (ACA) have been utilized to control chaos in a symmetric gyro with linear-pluscubic damping [40]. Notwithstanding the success recorded by these methods, some drawbacks associated with their applications have been identified. For instance, the restriction to stabilization of unstable periodic orbits (UPO) to stable periodic orbits (PO) in the OGY method rules out the fact that steady state solutions are the most practical operation modes in many chaotic systems, especially electronic oscillators [41] and lasers [42]. Additionally, the Pyragas TDAS method relies on the torsion of neighbouring trajectories in ∗

Corresponding author.

E-mail address: [email protected]

†

E-mail address:ue− [email protected]

c Copyright°World Academic Press, World Academic Union IJNS.2008.02.15/119

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International Journal of Nonlinear Science,Vol.5(2008),No.1,pp. 11-19

phase space [43]. In order to address these drawbacks, many linear and nonlinear control methods have emerged over the years. In particular, backstepping design and active control have been recognised as two powerful design methods to control and synchronize chaos. It has been reported [44–46] that backstepping design can guarantee global stability, tracking and transient performance for a broad class of strict-feedback nonlinear systems. In recent time, it has been employed for controlling, tracking and synchronizing many chaotic systems [47– 51] as well as hyperchaotic systems [44]. According to ref [48], some of the advantages in the method include applicability to a variety of chaotic systems whether they contain external excitation or not; needs only one controller to realize synchronization between chaotic systems and finally there are no derivatives in the controller. Zhang [44] states that the controller is singularity free from the nonlinear term of quadratic type, gives flexibilty to construct a control law which can be extended to higher dimensional hyperchaotic systems and the closed-loop system is globally stable, while ref [52] adds that it requires less control effort in comparison with the differential geometric method. The goal of this paper are two fold; in the first instance we use a recursive backstepping approach to control chaos in a nonlinear gyro system. The method is a systematic design approach and consists in a recursive procedure that skillfully interlaces the choice of a Lyapunov function with the design of the control. Secondly, we employed adaptive backstepping approach to synchronize two identical nonlinear gyros evolving from different initial conditions. The paper is organized as follows: In section 2, a brief description of the gyro system is introduced. Sections 3 and 4 discuss the design of the backstepping controller for supressing chaotic motion as well as the synchronization of chaos in the gyro. Numerical simulation is given for illustration and verification of the effectiveness and feasibility of the control technique. In section 5 conclusion is presented.

2

System description

The model system which we study is the gyroscope which has attributes of great utility to navigational, aeroneutical and space engineering [40], and have been widely studied. Generally, gyros are understood to be devices which rely on inertial measurement to determine changes in the orientation of an object. Gyros are recently finding application in automotive systems for Smart Braking System, in which different brake forces are applied to the rear tyres to correct for skids. Recently, Chen [40], presented the dynamic behaviour of a symmetric gyro with linear-plus-cubic damping, which is subjected to a harmonic excitation and the Lyapunov direct method was used to obtain the sufficient conditions of the stability of the equilibrium points of the system. The equation governing the motion of the gyro after necesssary transformation is given by [40] x˙1 = y1 y˙1 = g(x1 ) − ay1 − by13 + β sin x1 + f sin ωt sin x1

(1)

)2

x1 where g(x1 ) = −α2 (1−cos . sin3 x1 The nonlinear gyro given by equation (1) exhibits varieties of dynamical behaviour including chaotic motion - displayed in Fig. 1a - for the following parameters α2 = 100, β = 1, a = 0.5, b = 0.05, ω = 2, and f = 35.5. In Fig. 1b, the chaotic attractor, which configuration looks like slanting ’S’ is presented and is in agreement with previously obtained results in [15, 40]. The time history for the parameters with initial condition (x, y) = (1, −1) and (1, −1.2) are displayed in Fig. 2.

3 Controlling chaos in gyroscope system In the following we will use backstepping approach to design a controller for supressing chaos in gyroscope system. To achieve this, we introduce a control signal u(t) to (1) and rewrite the system in strict-feedback form as follows: x˙1 = y1 y˙1 = g(x1 ) − ay1 − by13 + β sin x1 + f sin ωt sin x1 + u(t) IJNS email for contribution: [email protected]

(2)

B. A. Idowu, U. E. Vincent, A. N. Njah: Control and Synchronization of Chaos in · · · 4

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1.5 ’’

’gyro.ttd’

3 1 2 0.5

x2

1

0

-1

0

-0.5

-2 -1 -3

-4 -1.5

-1.5 -0.4 -1

-0.5

0

0.5

1

-0.3

-0.2

-0.1

1.5

(a)

0 x1

0.1

0.2

0.3

0.4

(b)

Figure 1: (a) The phase portrait of the chaotic gyro, (b) The Poincare Map of the chaotic gyro attractor. In the recursive backstepping scheme, its required that the state space x1 and y1 are to take desired values, say x1,d and y1,d respectively, where x1,d = 0 is an arbitrarily chosen scalar input corresponding to a stable fixed point or an equilibrium point of the system (1) and y1,d = c1 e1 is recursively introduced; c1 being arbitrary control parameter to be chosen. Let the error signals be given as e1 = x1 − x1,d ; e2 = y1 − y1,d ,

(3)

and x1,d and y1,d are as defined previously. Using the definition in Eq.(3), we have e˙ 1 = e2 + c1 e1 e˙ 2 = g(e1 ) − a(e2 + c1 e1 ) − b(e2 + c1 e1 )3 + β sin e1 + f sin ωt sin e1 + u(t), 2

(4)

2

e1 ) where g(e1 ) = −α (1−cos . sin3 e1 We choose the following Lyapunov function 2

V =

1X 2 ki ei , 2

(5)

i=1

where ki (i = 1, 2) are arbitrary control gains. The time derivative of Eq. (5) is V˙ = k1 e1 e˙ 1 + k2 e2 e˙ 2 .

(6)

Substituting Eq. (4) in (6) we obtain the control signal u(t) as u(t) = −e2 − g(e1 ) + ae2 + be32 − β sin e1 − f sin ωt sin e1 ,

(7)

where ki , i = 1, 2 and u(t) have been chosen such that V˙ = 0; in which case k1 = c1 = 0, k2 6= 0. With Eq. (7), V˙ = −e22 < 0 is negative definite and according to LaSalle-Yoshizawa theorem [46], it follows that, all the solutions of system (2) converge to the manifold xi (i = 1, 2) = 0 as t → ∞ and the equilibrium (0,0) is global asymptotically stable. Thus, the control goal is achieved. In Fig. 3, we present numerical simulations to verify the effectiveness of the proposed controller (7). When the control is switched on, it is very clear that the chaotic behavior have been controlled.

4

Synchronization via adaptive backstepping design

Here we synchronize two identical nonlinear gyro systems evolving from different initial states. Let the drive system be given by (1) and the slave system be given by (8), where u(t) is the control that is required to drive system (8), the slave, to a synchronized state with system (1), the driver. IJNS homepage:http://www.nonlinearscience.org.uk/

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1.5

(a)

1

x1

0.5 0 -0.5 -1 -1.5 0

10

20

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50

60

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80

(b)

1

x2

0.5 0 -0.5 -1 -1.5 0

10

20

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50

60

70

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Figure 2: The time history of the chaotic gyro with initial conditions of (a) (x1 , y1 ) = (1, −1) and (b) (x2 , y2 ) = (1, −1.2). This shows the exponential divergence due to small difference in initial conditions. The slave system is given as x˙2 = y2 y˙2 = g(x2 ) − ay2 − by23 + β sin x2 + f sin ωt sin x2 + u(t)

(8)

Let the error state between (1) and (8) be ex = x2 − x1 ; ey = y2 − y1 . Using the above definition, we have the following error dynamics equation for the drive-response system: e˙ x = ey e˙ y = g(x2 ) − g(x1 ) − a(y2 − y1 ) − b(y23 − y13 ) + β(sin x2 − sin x1 ) +f sin ωt(sin x2 − sin x1 ) + u(t).

(9)

The objective is to find a control law so that system (9) is stabilized at the origin. Starting from the first equation of system (9), an estimative stabilizing function α1 (ex ) has to be designed for the virtual control IJNS email for contribution: [email protected]

x

B. A. Idowu, U. E. Vincent, A. N. Njah: Control and Synchronization of Chaos in · · ·

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

15

(a)

0

20

40

60

80

100

Time, t

6 (b)

4 2 y

0 -2 -4 -6 -8 0

20

40

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Time, t Figure 3: Dynamics of nonlinear gyro in the controlled state.(a) x variable, (b) y variable. ey in order to make the derivative of V1 (ex ) = (1/2)ex 2 , negative definite when α1 (ex ) = −ex . Define the error variable w2 as w2 = ey − α1 (ex ). Considering the (ex , w2 ) subspace given by e˙ x = w2 − ex w˙ 2 = g(x2 ) − g(x1 ) − a(ey ) − b(y23 − y13 ) + β(sin x2 − sin x1 ) +f sin ωt(sin x2 − sin x1 ) + u(t). which form the complete system. Choosing the Lyapunov function 1 V2 (ex , w2 ) = V1 (ex ) + w22 , 2 the derivative of Eq. (10) is £ V˙ 2 = −e2x + w2 g(x2 ) − g(x1 ) − a(ey ) − b(y23 − y13 ) + β(sin x2 − sin x1 ) +f sin ωt(sin x2 − sin x1 ) + w2 − ex ] + u(t) IJNS homepage:http://www.nonlinearscience.org.uk/

(10)

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International Journal of Nonlinear Science,Vol.5(2008),No.1,pp. 11-19

0.04

(a)

0.02 e1

0 -0.02 -0.04 -0.06 -0.08 0

20

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80

100

Time, t

0.05 (b) 0

e2

-0.05 -0.1 -0.15 -0.2 0

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Time, t Figure 4: Synchronization dynamics when the controller is applied, (a) e1 variable, (b) for e2 variable If u(t) = −g(x2 ) + g(x1 ) + a(ey ) + b(y23 − y13 ) − (β + f sin ωt)(sin x2 − sin x1 ) −2w2 + ex , then, V˙ 2 = −e2x − w22 < 0 is negative definite and according to Lasalle-Yoshizawa theorem [46], the error dynamics will converge to zero as t → ∞, while the equilibrium (0, 0) remains global asymptotically stable. Thus, the synchronization between two gyro system is achieved via the adaptive backstepping design. We performed numerical simulations for the system parameters displayed in Fig. 1 with the initial conditions (x1 , y1 ) = (1, −1) , (x2 , y2 ) = (1, 1.2). In Fig. 4, the control has been activated at t = 0. It is very clear that the synchronization has been achieved since the error dynamics (e1 , e2 ) between the drive-response system approaches zero as t → ∞. IJNS email for contribution: [email protected]

B. A. Idowu, U. E. Vincent, A. N. Njah: Control and Synchronization of Chaos in · · ·

5

17

Conclusion

Research on control and synchronization of chaotic systems has been a rapidly advancing topical issue in the last decade. Emphasis has been on control design techniques which result in prescribed nonlinear performance dynamics for practical controlled processes. In this work we have examined the problem of chaos control and synchronization in the nonlinear gyroscope. In the design of the controller for chaos control, we employed the recursive nonlinear backstepping approach. To synchronize two identical gyroscopes, we employed adaptive backstepping design to formulate a control function which is capable of driving the output of the responding gyro to trace that of the driver gyro. The methods we have applied allow for flexibilty in the controller design and global stability based on the appropriate choice of Lyapunov functions. Our results, complimented with numerical simulations, show that the methods are effective and feasible.

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