ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.15(2013) No.1,pp.45-50
Nonlinear Control of Chaotic Rikitake Two-Disk Dynamo Ahmad Harb ∗ , Nabil Ayoub School of Natural Resources Engineering, German Jordanian University, Amman Jordan (Received 11 April 2012, accepted 5 October 2012)
Abstract:In this paper the modern nonlinear theory, bifurcation and chaos theory, is used to investigate and analyze the dynamics of the Rikitake two-disk dynamo system. The mathematical model of the Rikitake system consists of three nonlinear differential equations, which found to be the same as the mathematical model of the well known Lorenz system. The study showed that the system is experiencing a chaotic behavior at certain value of the control parameter. The experienced chaotic oscillation can be simulates the reversal of the Earth magnetic field. Finally, a nonlinear controller based on the backstepping theory is designed to control the chaotic behaviors. The designed controller was so effective in controlling the unstable chaotic oscillations. Keywords: chaos theory; nonlinear control; magnetic theory
1
Introduction
First of all let us discuss the physics behind Rikitake two-disk dynamo [1]. Figure 1 shows the Rikitake two-disk dynamo. A common torque of magnitude G is applied to both conducting disks D1 and D2 . Both disks rotate in the same sense, one with an angular velocity of ω1 around the axis of rotation A1 and the other with angular velocity ω2 around the axis of rotation A2 . A current I1 circulates the current loop L1 where the current loop is coaxial with D2 with axis A2 and L1 is located below D2 , and causes a magnetic field B2 to pass through the rotating disk D2 . Loop L1 is connected to Disk D1 and its axis of rotation A1 by conducting brushes and the current along A1 is upward and in D1 it is radially outward. Similarly a current I2 circulates the current loop L2 where the current loop is coaxial with D1 with axis A1 and L2 is located above D1 , and causes a magnetic field B1 to pass through the rotating disk D1 . Loop L2 is connected to Disk D2 and its axis of rotation A2 by conducting brushes and the current along A2 is upward and in D2 it is radially inward. Since D1 is rotating B1 caused by L2 will create an induced emf between the center of D1 and its rim causing an induced inward current I’1 to occur in the opposite direction to I1 where the total current becomes less than the original current I1 . Similarly since D2 is rotating B2 caused by L1 will create an induced emf between the center of D2 and its rim causing an induced outward current I’2 to occur in the opposite direction to I2 where the total current becomes less than the original current I2 . This process continues until we achieve current reversal in both loops which causes a reversal in the total magnetic field. Under particular initial conditions this process becomes chaotic. Many researchers have discussed the dynamics of Rikitake system. Historically E. C. Bullard [4-6] has extensively discussed the behavior of earth’s magnetic fieldand its simulation with dynamos. He first discussed the magnetic field within the earth [4], then the similar behavior between a set of homogeneous dynamos and terrestrial magnetism [5] and in 1955 [6] discussed the stability of a homopolar dynamo. Liu Xiao-Jun, et.al. [7] analyzed the dynamics of Rikitake two-disk dynamo to explain the reversals of the Earth’s magnetic field. They concluded that the chaotic behavior of the system can be used to simulate the reversals of the geomagnetic field. The Rikitake chaotic attractor was studied by several authors. T. McMillen [8] has studied the shape and dynamics of the Rikitake attractor. J. Llibre et al [9] used the Poincare compactification to study the dynamics of the Rikitake system at infinity. Chien- Chih Chen et al [10] have studied the stochastic resonance in the periodically forced Rikitake dynamo.
In the past decade, many researchers start working on controlling the chaotic behaviors. Harb and Harb [11] have designed a nonlinear controller to control the chaotic behavior in the phase-locked loop by means of nonlinear control. ∗ Corresponding
author.
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International Journal of Nonlinear Science, Vol.15(2013), No.1, pp. 45-50
Figure 1: Rikitake Two-Disk Dynamo [2,3].
Harb and Abdel -Jabber [12] used both linear and nonlinear control to mitigate the chaotic oscillations in electrical power system. Nayfeh et al. [13], Chiang et al. [14] and Abed et al. [15] have controlled the chaotic oscillations in power systems using nonlinear control. Chang and Chen [16] investigated bifurcation characteristics of nonlinear systems under a PID controller. The main objective of control is to stabilize and delay a chaotic oscillation and reduce the amplitude of bifurcation solution. In [17] they used a control law to transform an unstable subcritical bifurcation point into a stable supercritical bifurcation point. Ott [18] and Ott et. al. [19] used other methods to control the chaotic behavior. In Hubler [20], Hubler and Luscher [21], and Jackson [22] they used methods based on classical control. Harb et. al. [23], and Zaher et. al. [24 and 25] used the backstepping recursive nonlinear controller. Calvo and Cartwright [26], and Mann et. al. [27] introduced the use of fuzzy theory control in chaotic systems. Lately, the control aspect of the Rikitake chaotic attractor was studied through self coupling of single state variable and nonlinear feedback controller with partial systems states [28]. In this paper we used the nonlinear control based on backstepping technique to control the chaotic oscillations of the Rikitake two-disk dynamo.
2
Mathematical model
As shown in Fig. 1, the Rikitake two-disk system consists of two conducting rotating disks. These disks are connected into two coils. The current in each coil feeds the magnetic field of the other. The self inductance (L) and resistance (R) are the same in each circuit. An external constant mechanical torque (G) for each circuit is applied on the axis to rotate with an angular velocity. The RL circuit equations as shown in Fig. 1 are given as: RI1 + L
dI1 = M I2 ω1 dt
(1)
RI2 + L
dI2 = M I1 ω2 dt
(2)
where M I2 ω1 and M I1 ω2 are the rotation voltages V1 and V2 . M is the mutual inductance. If the moment of inertia of each disk, C is considered, the system mathematical model can be written after [2] as follows:
√ where: a = R
LC GM
and p = (ω1 − ω2 )
√
x˙ 1 = x2 x3 − ax1
(3)
x˙ 2 = (x3 − p)x1 − ax2
(4)
x˙ 3 = 1 − x1 x2
(5)
CM GL
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Ahmad Harb, Nabil Ayoub: Nonlinear Control of Chaotic Rikitake Two-Disk Dynamo
3 3.1
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Simulation results and discussion Equilibrium and dynamic solutions
The equilibrium solution or constant solution of the system can be found if we let the set of Eqs. (3-5) to be zero. One will end up with a set of nonlinear algebraic equations. One of the solutions of this set of equations willbe function of one of the control parameters.The stability of this constant solution, can be studied by using eigenvalue analysis which is based on the Jacobian matrix derived based on linearization. The eigenvalues of the Jacobian matrix of the set of Eqs. (3-5) evaluated at the equilibrium point. In this paper, we used our own program for calculating and analyzing the stability of the fixed points and their bifurcations rather than using software packages in the market. On the other hand, the stability of the periodic solution is studied by means of the Floquet theory. At the value of the control parameter p = 3.4641, we found that the system is experiencing chaotic oscillations as shown in Figs.2(a-c).In this paper the main objective is to find a way to eliminate the chaotic oscillations. So, a nonlinear controller based on backstepping nonlinear theory control is designed. In the next section, the nonlinear controller design will be discussed.
Figure 2: Time history (Chaotic behavior, without controller).
Figure 3: State space trajectory projection (chaotic behavior, without controller).
3.2
Figure 4: State space trajectory projection (chaotic behavior, without controller).
Backstepping nonlinear controller
In this section, a nonlinear controller will be designed based on the well known backstepping nonlinear control. For more details, once can see Khalil [29]. To this end, let us rewrite the Eqs. (3) – (5) as follows: x˙ 1 = x2 x3 − ax1
(6)
x˙ 2 = (x3 − p)x1 − ax2 + u
(7)
x˙ 3 = 1 − x1 x2
(8)
where u is the control signal need to be designed.
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International Journal of Nonlinear Science, Vol.15(2013), No.1, pp. 45-50
First, let us define the error signals as follows: e1 = x1 − x1d ,
(9)
e2 = x2 − c1 e1 ,
(10)
e3 = x3 − c2 e1 − c3 e2 ,
(11)
where c1 , c2 and c3 are constants to be defined later, and x1d is the desired constant value. Derive the above error signals, Eqs. (9) - (11), one obtains; e˙ 1 = x˙ 1 ,
(12)
e˙ 2 = x˙ 2 − c1 e˙ 1 ,
(13)
e˙ 3 = x˙ 3 − c2 e˙ 1 − c3 e˙ 2 .
(14)
Now define the Lyapunov function V =
1 2
3 ∑ 1
ki e2i .
(15)
The time derivative for Eq. (15) is: V˙ = k1 e1 e˙ 1 + k2 e2 e˙ 2 + k3 e3 e˙ 3 .
(16)
Substitute Eqs. (6)-(14) into Eq. (16), and choosing the following parameters to be: c1 = 0, c2 = 1, c3 = 0, k1 = k3 = 0 and k2 = 1, we end up with the control signal as follows; u = −k4 e2 + ae2 − e1 e2 − c2 e21 + pe1
(17)
This control law guarantees the negative definitiveness of V in Eq. (15). As mentioned previously, the Rikitake system is experiencing a chaotic behavior as shown in Figs. 2(a-c), and this was when no control signal added to the system. But once we add the designed control signal (u) to the mathematical model of the Rikitake system, one can see that all the chaotic oscillations have been eliminated as shown in Figs. 3(a-d). Figs.3(c-d) show the time history of the system when the designed controller u, is applied after 400 sec. One can see how effective the controller in eliminating the chaotic oscillations.
4
Conclusions
The modern nonlinear theory, bifurcation and chaos theory was used to investigate dynamics of the Rikitake two-disk dynamo. The mathematical model of the Rikitake system consists of three nonlinear differential equations, which found to be the same as the mathematical model of the well known Lorenz system. The study showed that the system is experiencing a chaotic behavior at certain value of the control parameter. The experienced chaotic oscillation can be simulates the reversal of the Earth magnetic field. To control chaotic behavior a nonlinear controller based on the backstepping theory was designed. It is found that the designed controller was so effective to eliminate the unstable chaotic oscillations of the Rikitake two–disk dynamo.
References [1] T. Rikitake. Oscillations of a system of disk dynamos. Proc. Camb. Phil. Soc., 54(1958):89-105. [2] Marius-F. Danca, StelianaCodreanu. Finding the Rikitake’s attractors by parameter switching., arXiv:1102.2164v1 10(2011). [3] Marius-F. Danca, StelianaCodreanu. Modeling numerically the Rikitake’s attractors. Journal of the Franklin Institute, 349(3) (2012):861-878. [4] E. C. Bullard. The magnetic field within the earth. Proc. Roy. Soc., A197(1949): 433. [5] E. C. Bullard, H. Gellman. Homogeneous dynamos and terrestrial magnetism. Phil. Trans., A247(1954): 89. [6] E. C. Bullard. The stability of a homopolar dynamo. Proc. Camb. Phil. Soc., 51(1955):744.
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Ahmad Harb, Nabil Ayoub: Nonlinear Control of Chaotic Rikitake Two-Disk Dynamo
Figure 5: Time history (with controller).
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Figure 6: State space trajectory projection (with controller).
Figure 8: Comparison of state space trajectory projecFigure 7: State space trajectory projection (with con- tion between two cases (with and without controller, troller). and the controller applied after 400 sec).
[7] Liu Xiao-jun, Li Xian-feng, Chang Ying-xiang, Zhang Jian-gang. Chaos and Chaos Synchronism of the Rikitake Two-Disk Dynamo.Fourth International Conference on Natural Computation, IEEE computer Society,DOI 10.1109/ICNC.2008.706:613-617. [8] T. McMillen. The shape and dynamics of the Rikitake attractor. The Nonlinear Jour., 1(1999):1-10. [9] J. Llibre, M. Messias. Global dynamics of the Rikitake system.Physica D, 238(2009):241-252. [10] C.-C. Chen, C.-Y. Tseng. A study of stochastic resonance in the periodically forced Rikitake dynamo. Terr. Atmos. Ocean. Sci., 18(4)(2007):671-680. [11] Ahmad Harb, Bassam Harb. Chaos control of third-order phase-locked loops using backstepping nonlinear controller.Chaos, Solitons & Fractals, 20(4)(2004). [12] Ahmad Harb, Nabil Abedl-Jabbar. Controlling Hopf bifurcation and chaos in a small power system. Chaos, Solitons & Fractals, 18(5)(2003). [13] A. H. Nayfeh, A. M. Harb, C-M. Chin. Bifurcation in a Power System Model.International Journal of Bifurcation and Chaos, 6(3)(1996). [14] Abed EH, Alexander JC, Wang H, Hamdan AH, Lee HC. Dynamic bifurcation in a power system model exhibiting voltage collapse.In: Proceedings of the 1992 IEEE International Symposium on Circuits and Systems, San Diego,CA,(1992). [15] Chiang H-D, Dobson I, Thomas RJ, Thorp JS, Fekih-Ahmed L. On voltage collapse in electric power systems. IEEE Trans Power Syst., 5(1990):601-611. [16] Chang H-C, Chen L-H. Bifurcation characteristics of nonlinear systems under conventional PID control. Chem Eng Science, 39(1984):1127-1142. [17] Abed EH, Fu J-H. Local feedback stabilization and bifurcation control. I. Hopf bifurcation. Syst Control Lett., 7(1986):11-17. [18] E. Ott. Chaos in Dynamical Systems. Cambridge University Press, Cambridge, England, (1993). [19] E. Ott, C. Grebogi, J. A. Yorke. Controling chaotic dynamical systems. in CHAOS: Soviet-American Prespectives on Nonlinear Science, D. K. Campbell, ed., 2a(1990):153-172. [20] A. Hubler. A daptive control of chaotic systems. Helv, Phys. Acta, 62(1989): 343-346, [21] A. Hubler, E. Luscher. Resonant stimulation and control of nonlinear oscillators.Naturwissenschaften, 76(1989):6769 .
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