This paper examines the synchronization performance of two widely used&nbsp;chaos synchronization techniques: active control and backstepping control. It is shown&nbsp;that the two methods have excellent performance, with the active control marginally&nbsp;outperforming the backstepping control in terms of transient analysis. However, the&nbsp;complexity of active controllers suggests that the backstepping control would be more&nbsp;attainable in engineering applications

Vincent, U. E.

English

Vincent, U. E. E.

Nonlinear Analysis: Modelling and Control

Nonlinear Analysis: Modelling and Control, 2008, Vol. 13, No. 2, 253–261Chaos Synchronization Using Active Control andBackstepping Control: A Comparative AnalysisU. E. VincentDepartment of PhysicsOlabisi Onabanjo UniversityP.M.B. 2002, Ago-Iwoye, Ogun State, Nigeriaue vincent@yahoo.comReceived: 29.09.2007 Revised: 22.03.2008 Published online: 02.06.2008Abstract. This paper examines the synchronization performance of two widely usedchaos synchronization techniques: active control and backstepping control. It is shownthat the two methods have excellent performance, with the active control marginallyoutperforming the backstepping control in terms of transient analysis. However, thecomplexity of active controllers suggests that the backstepping control would be moreattainable in engineering applications.Keywords: chaos, synchronization, backstepping, active control.1 IntroductionThe control and synchronization of chaotic systems are extensively studied fields in non-linear dynamics that were introduced in 1990 by Ott et al. [1] using a scheme now knownas OGY closed-loop method; and Pecora and Carroll [2] using a scheme called APDmethod respectively. Various linear and nonlinear methods have emerged thereafter insearch of more efficient algorithms for controlling and synchronizing identical and non-identical chaotic systems. Two of the most recently proposed methods are the activecontrol [3, 4] and backstepping control [5, 6].Chaos synchronization using active control was proposed by Bai and Lonngren [3,4]and has recently been widely accepted as an efficient technique for the synchronizationof chaotic systems, because it can be used to synchronize non-identical systems as well;a feature that gives it an advantage over other synchronization methods. This method hasbeen applied to many practical systems such as spatiotemporal dynamical systems [7],the Rikitake two-disc dynamo – a geophysical system [8], nonlinear Bloch equationsmodeling nuclear magnetic resonance [9], Van der Pol Duffing oscillators [10], electriccircuits modeling “jerk” equation [11], Chua’s circuits [12], complex dynamos [13], non-linear equations of acoustic gravity waves [14], Qi system [14,15], parametrically excitedsystems [16] and RCL-shunted Josephson junctions [17]. In addition, we have veryrecently presented a novel technique using active control based synchronization scheme253U. E. Vincentfor controlling directed transport arising from co-existing attractors in non-equilibriumphysics – the so-called ratchets [18].Backstepping design on the other hand was originally used as a building block foradaptive control of chaotic systems [19] and was recently extended to the synchronizationof chaotic systems [5, 6]. Backstepping design has been employed recently to controla third-order phase-locked loops [20], permanent magnet reluctance machine [21], ahydraulic servo system [22]; and to synchronize the Genesio chaotic systems [23] aswell as demonstrate the control of directed transport in inertial ratchets [24]; among otherapplications. The method is a systematic design approach and consists in a recursiveprocedure that skillfully interlaces the choice of a Lyapunov function with the control.Indeed, backstepping control can guarantee global stability, tracking and transient perfor-mance for a broad class of strict-feedback nonlinear systems [19, 25].In this paper, we design active controllers and a recursive backstepping controlthat will guarantee global synchronization between two identical chaotic systems andcompare simulation results of the two methods. The rest of the paper is organized asfollows: In Section 2, synchronization via active control is presented; while in Section 3,backstepping control design for chaos synchronization is presented. Section 4 deals withcomparison of numerical simulation results. Finally, the paper is concluded in Section 5.2 Design of active control for synchronizationTo present the synchronization problem, we consider the same system used by Harb andZohdy to study bifurcation and chaos control based on backstepping [26]. The drive-response system is given by:x˙1 = −z1,y˙1 = x1 − y1,z˙1 = ax1 + y21+ bz1(1)for the drive system andx˙2 = −z2 + u1,y˙2 = x2 − y2 + u2,z˙2 = ax2 + y22+ bz2 + u3(2)for the response system, where ui (i = 1, 2, 3) are active control functions to be deter-mined; a and b are the parameters of the system. Defining the error states for the statevariables asex = x2 − x1; ey = y2 − y1; ez = z2 − z1, (3)and following the procedures of active control design, we subtract equation (1) fromequation (2) and using the definitions in equation (3), we obtain the error dynamics254Chaos Synchronization Using Active Control and Backstepping Control: A Comparative Analysisequation given by:e˙x = −ez + u1,e˙y = ex − ey + u2,e˙z = aex +(y22− x22)+ bez + u3.(4)Re-defining the control functions as follows:u1 = v1,u2 = v2u3 = v3 −(y22− x22) (5)the error dynamics equation (4) becomese˙x = −ez + v1,e˙y = ex − ey + v2,e˙z = aex + bez + v3.(6)In the active control method, we choose a constant matrix A which will control the errordynamics (6) such that[v1, v2, v3]T = A[e1, e2, e3]T . (7)Several choice of A can satisfy system (7). Here, we choose the following matrix thatsatisfy the Routh-Hurwtiz criteria for the stability of the synchronized state:A =λ1 0 1−1 λ2 + 1 0−a 0 λ3 − b , (8)which immediately yields the control functionsu1 = λ1ex + ez,u2 = −ex + (λ2 + 1)ey,u3 = −aex + (λ3 − b)ez − ey(ey + 2y1)(9)provided the eigenvalues (λ1, λ2, λ3) are negative. Here, we have chosen (λ1, λ2, λ3) =(−1,−1,−1) for simplicity.3 Design of backstepping control for synchronizationIn [26], the author presented a recursive backstepping control for chaos control. Here,we present backstepping control for chaos synchronization. This will enable us to give areliable performance comparison of the two control methods for chaos synchronization.255U. E. VincentTo treat this problem, we follow the method used by Tan et al. [5]. Consider the drive-response systems given below:x˙1 = −z1,y˙1 = x1 − y1,z˙1 = ax1 + y21+ bz1(10)for the drive system andx˙2 = −z2,y˙2 = x2 − y2,z˙2 = ax2 + y22+ bz2 + u,(11)where u is a control function to be determined. In backstepping, only one controller isrequired. Subtracting equation (5) from (6) and using the error states definition (3), weobtaine˙x = −ez,e˙y = ex − ey,e˙z = aex + bez + ey(ey + 2y1) + u.(12)In the absence of the control u, equation (12) would have an equilibrium (0, 0, 0). Ifu were chosen such that the equilibrium (0, 0, 0) is unchanged, then the problem ofsynchronization of the drive-response system would be reduced to that of asymptoticstability of system (12). Thus, the goal is to find a control law u such that system (12) isstabilized at the origin. Considering the stability of system (13) given below:e˙x = −ez (13)and regarding ez as a virtual control, an estimate stabilizing function α1(ex) can bedesigned for the virtual control ez. Choosing a Lyapunov functionV1(ex) =12e2x. (14)The derivative isV˙1(ex) = exe˙x. (15)For V1(ex) to be negative definite, then, e˙x = −ex, so thatV˙1(ex) = −e2x < 0. (16)Thus α1(ex) = −ex. Note that the function α1(ex) is an estimate control function whenez is considered as a controller. Letw2 = ey − α1(ex) (17)256Chaos Synchronization Using Active Control and Backstepping Control: A Comparative Analysisand consider the subspace (ex, w2) given bye˙x = −ez,e˙y = 2ex − w2.(18)Let ez be a virtual controller in system (18) and assume that when ez = α1(ex, w2),system (18) is made asymptotically stable. Choose the Lyapunov functionV2(ex, w2) = V1(ex) +12w2x (19)for subspace above. The derivative of (19) is given byV˙2(ex, w2) = V˙1(ex) + w2w˙2 = −e2x − w22+ w2(2ex + ez). (20)If α1(ex, w2) = −2ex, then ez = −2ex andV˙2(ex, w2) = −e2x − w22< 0 (21)is negative definite.Define the error dynamics w3 asw3 = ez − α2(exw2). (22)We now study the full dimension or the complete space (ex, w2, w3)e˙x = 2ex,w˙2 = 2ex − w2,w˙3 = (a + 2)ex + b(w2 − 2ex) + ey(ey + 2y1) + u.(23)Choose a Lyapunov functionV3(ex, w2) = V2(ex, w2) +12w23. (24)Ifu = −(a + 2)ex − b(w2 − 2ex)− ey(ey + 2y1)− w3, (25)then,V˙3(ex, w2) = −e2x − w22− w23< 0 (26)is negative definite and according to LaSalle-Yoshizawa theorem [19], the error dynamics(ex, ey, ez) will converge to zero as t → ∞, while the equilibrium (0, 0, 0) remainsasymptotically stable. Thus, the synchronization of the drive-response system is achieved.4 Numerical resultsIn the following numerical results, we employed the 4th order Runge-Kutta algorithmwith time grid of 0.05, a = 3.1, b = 0.5 and the initial conditions x1(0) = y1(0) =z1(0) = 0, x2(0) = 0.05, y2(0) = 0.01, z2(0) = 0.06. In Fig. 1 we display the same257U. E. Vincentchaotic attractor as obtained in [26] as well as the synchronization errors when control isde-activated.-5-4-3-2-1012-8 -6 -4 -2 0 2 4y1x1(a)-10-8-6-4-202468100 20 40 60 80 100 120 140 160extime(b)02468101214160 20 40 60 80 100 120 140 160etime(c)Fig. 1. Dynamics of the chaotic system when control is de-activated: (a) chaoticattractor in the (x1-y1) plane; (b) error dynamics (ex) for the state variable x and(c) average error dynamics (e).To examine and compare the synchronization performance, we compute the synchro-nization quantity, defined as the average error propagation on the system state variables[8, 27, 28] given bye =√e2x + e2y + e2z. (27)258Chaos Synchronization Using Active Control and Backstepping Control: A Comparative AnalysisFig. 2 shows the synchronization error (e) for the two methods when controls areactivated at t = 0 for both methods. We find that at t = 5 s, the synchronization wasalready attained for active control while synchronization was attained at a later time (t =13 s) for backstepping control, the time delay being 8 s. Though it is clear that the activecontrol performs better and is much easier to design, there are three controllers required inthe design, while only one controller is needed for the backstepping. Thus, the controllersin active control are more complex for practical implementation. Comparing equation(1) defining the chaotic system and equation (4) defining the active controllers, one canreadily observe that the controller is more complex than the system itself.-0.03-0.02-0.0100.010.020.030.040.050 5 10 15 20 25 30 35 40extime(a)00.010.020.030.040.050.060.070.080 5 10 15 20 25 30 35 40etime(b)00.010.020.030.040.050.060.070.080 2 4 6 8 10 12 14etime(c)Fig. 2. Error dynamics in the synchronized state when controls have been activated att = 0. Solid (backstepping method), dashed line (active control method): (a) ex; (b)average error e and (c) zoom of the initial transient in (b).259U. E. VincentThe problem of controller complexity is a very crucial issue in the practical im-plementation [29], for example in electronics and engineering applications. Two funda-mental issues in this direction are (a) the cost implication and density requirement fordesigning controllers and (b) the need to make the complexity of the controller to becomparable with, or less than, the device being controlled, if the controlling technique isdesired to achieve a useful end and not merely a scientific curiosity.5 ConclusionsThis paper has examined the performance of two control schemes for chaos synchro-nization: active control and recursive backstepping control. It has been shown that thetwo schemes have excellent transient performance; with the active control marginallyoutperforming the backstepping. However, the complexity of the active controllers withregard to the system being synchronized suggests that the backstepping controller, withone control function would be more attainable in applications; owing to cost effectivenessand density requirement.References1. E. Ott, C. Grebogi, J. A. Yorke, Controlling Chaos, Phys. Rev. Lett., 64, pp. 1196–1199, 1990.2. L. M. Pecora, T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64,pp. 821–824, 1990.3. E. W. Bai, K. E. Lonngren, Synchronization of two Lorenz systems using active control, Chaos,Solitons & Fractals, 8, pp. 51–58, 1997.4. E. W. Bai, K. E. Lonngren, Sequential synchronization of two Lorenz systems using activecontrol, Chaos, Solitons & Fractals, 11, pp. 1041–1044, 2000.5. X. Tan, J. Zhang, Y. Yang, Synchronizing chaotic systems using backstepping design, Chaos,Solitons & Fractals, 16, pp. 37–45, 2003.6. Y. Yu, S. Zhang, Adaptive backstepping synchronization of uncertain chaotic systems, Chaos,Solitons & Fractals, 21, pp. 643–649, 2004.7. S. Codreanu, Synchronization of spatiotemporal nonlinear dynamical systems by an activecontrol, Chaos, Solitons & Fractals, 15, pp. 507–510, 2003.8. U. E. Vincent, Synchronization of Rikitake chaotic attractor via active control, Phys. Lett. A,343, pp. 133–138, 2005.9. A. Ucar, E. W. Bai, K. E. Lonngren, Synchronization of chaotic behavior in nonlinear Blochequations, Phys. Lett. A, 314, pp. 96–101, 2003.10. A. N. Njah, U. E. Vincent, Chaos synchronization between single and double wells DuffingVan-der Pol oscillators using active control, Chaos, Solitons & Fractals, 37, pp. 1356–1361,2008.260Chaos Synchronization Using Active Control and Backstepping Control: A Comparative Analysis11. E. W. Bai, K. E. Lonngren, J. C. Sprott, On the synchronization of a class of electronic circuitsthat exhibits chaos, Chaos, Solitons & Fractals, 13, pp. 1515–1521, 2002.12. F. Tang, L. Wang, An adaptive active control for the modified Chua’s circuit, Phys. Lett. A, 346,pp. 342–346, 2006.13. G. M. Mahmoud, S. A. Aly, A. A. Farghaly, On chaos synchronization of a complex twocoupled dynamos system, Chaos, Solitons & Fractals, 33, pp. 178–187, 2007.14. U. E. Vincent, Synchronization of identical and non-identical 4-D chaotic systems via activecontrol, Chaos, Solitons & Fractals, 37, pp. 1065–1075, 2008.15. Y. Lei, W. Xu, W. Xie, Synchronization of two chaotic four-dimensional systems using activecontrol, Chaos, Solitons & Fractals, 32, pp. 1823–1829, 2007.16. Y. Lei, W. Xu, J. Shen, T. Fang, Global synchronization of two parametrically excited systemsusing active control, Chaos, Solitons & Fractals, 28, pp. 428–436, 2006.17. A. Ucar, K. E. Lonngren, E. W. Bai, Chaos synchronization in RCL-shunted Josephson junctionvia active control, Chaos, Solitons & Fractals, 31, pp. 105–111, 2007.18. U. E. Vincent, J. A. Laoye, Synchronization and control of directed transport in inertial ratchetsvia active control, Phys. Lett. A, 363, pp. 91–95, 2007.19. M. Kristic, I. Kanellakopoulo, Kotkotovic, Nonlinear and Adaptive Control Design, JohnWiley, New York, 1995.20. A. M. Harb, B. A. Harb, Chaos control of third-order phase-locked loops using backsteppingnonlinear controller, Chaos, Solitons & Fractals, 20, pp. 719–723, 2004.21. A. M. Harb, Nonlinear chaos control in permanent magnet reluctance machine, Chaos, Solitons& Fractals, 19, pp. 1217–1224, 2004.22. H. Yu, Z. Feng, X. Wang, Nonlinear control for a class of hydraulic servo system, J. ZhejiangUniv. SCI., 11(14), pp. 13–1417, 2004.23. J. H. Park, Synchronization of Genesio chaotic system via backstepping approach, Chaos,Solitons & Fractals, 27, pp. 1369–1375, 2006.24. U. E. Vincent, Controlling Directed transport in inertia ratchets using adaptive backsteppingcontrol, Acta Physica Polonica B, 38, pp. 1001–1011, 2007.25. S. S. Ge, C. Wang, T. H. Lee, Adaptive backstepping control of a class of chaotic systems, Int.Journal of Bifurcation & Chaos, 10, pp. 1149–1156, 2000.26. A. M. Harb, M. M. Zohdy, Chaos and bifurcation control using nonlinear recursive controller,Nonlinear Analysis: Modelling and Control, 7, pp. 37–43, 2002.27. G. L. Baker, J. A. Blackburn, H. J. T. Smith, intermittent Synchronization in a pair of chaoticpendula, Phys. Rev. Lett., 81, pp. 554–557, 1998.28. G. L. Baker, J. A. Blackburn, H. J. T. Smith, A stochastic model of synchronization for chaoticpendulums, Phys. Lett. A, 252, pp. 191–197, 1999.29. N. J. Corron, S. D. Pethel, B. A. Hopper, Controlling chaos using simple limiters, Phys. Rev.Lett., 84, pp. 3835–3838, 2000.261

Chaos Synchronization Using Active Control and Backstepping Control: A Comparative Analysis

https://www.journals.vu.lt/nonlinear-analysis/article/download/14583/13503

Chaos Synchronization Using Active Control and Backstepping Control: A Comparative Analysis

Abstract

Similar works

Full text

Available Versions

Nonlinear Analysis: Modelling and Control