Chaos Synchronization of Nonlinear Bloch Equations Based on Input-to-State Stable Control

Commun. Theor. Phys. (Beijing, China) 53 (2010) pp. 308 312 c Chinese Physical Society and IOP Publishing Ltd Vol. 53, No. 2, February 15, 2010 Chaos Synchronization of Nonlinear Bloch Equations Based on Input-to-State Stable Control Choon Ki Ahn Faculty of the Division of Electronics and Control Engineering, Wonkwang University, 344-2 Shinyong-dong, Iksan 570-749, Korea (Received February 27, 2009) Abstract In this paper, we propose a new input-to-state stable (ISS) synchronization method for chaotic behavior in nonlinear Bloch equations with external disturbance. Based on Lyapunov theory and linear matrix inequality (LMI) approach, for the first time, the ISS synchronization controller is presented to not only guarantee the asymptotic synchronization but also achieve the bounded synchronization error for any bounded disturbance. The proposed controller can be obtained by solving a convex optimization problem represented by the LMI. Simulation study is presented to demonstrate the effectiveness of the proposed synchronization scheme. PACS numbers: 05.45.-a, 05.45.Xt Key words: chaos synchronization, input-to-state stable (ISS) control, nonlinear Bloch equations, linear matrix inequality (LMI), Lyapunov theory 1 Introduction During the last two decades, synchronization in chaotic dynamic systems has received a great deal of interest among scientists from various research fields since Pecora and Carroll [1] introduced a method to synchronize two identical chaotic systems with different initial conditions. It has been widely explored in a variety of fields including physical, chemical, and ecological systems. [2] In the literature, various synchronization schemes, such as variable structure control, [3] OGY method, [4] parameters adaptive control, [5 6] observerbased control, [7] active control, [8 9] time-delay feedback approach, [10 11] fuzzy logic approach, [11 12] backstepping design technique, [13 14] exponential control, [15] H method, [16 17] and so on, have been successfully applied to the chaos synchronization. The concept of synchronization has been extended to generalized synchronization, [18 19] projective synchronization, [20] phase synchronization, [21] and lag synchronization, [22] even anti-synchronization. [16,23] The dynamics of an ensemble of spins usually described by the nonlinear Bloch equation is very important for the understanding of the underlying physical process of nuclear magnetic resonance. The basic process can be viewed as the combination of a precession about a magnetic field and of a relaxation process. Abergel demonstrated that the set of nonlinear Bloch equations would admit chaotic solutions for a certain set of numerical values assigned to the system constants and initial conditions. [24] Ucar et al. [25] extended the calculation of Abergel [24] and demonstrated that an active control method can synchronize two of these nonlinear Bloch equations. Recently, some control schemes, such as adaptive control [26] and stability criterion method, [27] were proposed for synchronizing chaotic behavior in nonlinear Bloch equations. It is well known that real physical systems are often affected by noise, such as perturbations in control or errors on observation. Thus, control systems are required not only to be stable, but also to have the property of input-to-state stability (ISS). A control system is called input-to-state stable (ISS), it means that no matter what the initial state is, if the inputs are uniformly small then the state of the control system must eventually be small. ISS is an interesting concept first introduced in Ref. [28] to nonlinear control systems. It has been widely accepted as an important concept in control engineering and many research results have been reported in recent years. [29 36] To the best of our knowledge, however, for the ISS synchronization for chaos synchronization of nonlinear Bloch equations, there is no result in the literature so far, which still remains open and challenging. In this paper, a new controller for the ISS synchronization of chaotic behavior in nonlinear Bloch equations is proposed. This controller is a new contribution to the topic of chaos synchronization. Theoretical proof reveals that the use of the proposed controller could make the closed-loop error system asymptotically synchronized and always bounded for any bounded disturbance. Based on Lyapunov method and linear matrix inequality (LMI) approach, an existence criterion for the proposed controller is represented in terms of the LMI. The LMI problem can be solved efficiently by using recently developed convex optimization algorithms. [37] This paper is organized as follows. In Sec. 2, the basic concept of ISS is introduced. In Sec. 3, we introduce nonlinear Bloch equations and formulate the problem. In Sec. 4, an LMI problem for the ISS synchronization of chaotic behavior in nonlinear Bloch equations is proposed. In Sec. 5, a numerical example is given, and finally, conclusions are presented in Sec. 6. E-mail: hironaka@wonkwang.ac.kr

No. 2 Chaos Synchronization of Nonlinear Bloch Equations Based on Input-to-State Stable Control 309 2 Basic Concept of Input-to-State Stability Consider the following differential equation: Ẋ(t) = F(X(t), U(t)), (1) where X(t) R n is the state variable, U(t) R m is the external input. F : R n R m R n is continuously differentiable and satisfies F(0, 0) = 0. Throughout this paper, we will use the following definitions: Definition 1 A function γ : R 0 R 0 is a K function if it is continuous, strictly increasing and γ(0) = 0. Definition 2 A function γ : R 0 R 0 is a K function if it is a K function and also γ(s) as s. Definition 3 A function β : R 0 R 0 R 0 is a KL function if, for each fixed t 0, the function β(, t) is a K function, and for each fixed s 0, the function β(s, ) is decreasing and β(s, t) 0 as t. The notion of ISS can be described as follows: Definition 4 The system (1) is said to be input-tostate stable if there exist a K function γ(s) and a KL function β(s, t), such that, for each input U(t) L m (sup t 0 U(t) < ) and each initial state X(0) R n, it holds that X(t) β( X(0), t) + γ( U(t) ), (2) for each t 0. It is noted that, if a system is input-to-state stable, the behavior of the system should remain bounded when its inputs are bounded. Now we introduce a useful result that is employed for obtaining the ISS synchronization controller. Lemma 1 [31] A continuous function V ( ) : R n R 0 is called an ISS-Lyapunov function for the system (1), if there exist K functions α 1, α 2, α 3, and α 4 such that α 1 ( X(t) ) V (X(t)) α 2 ( X(t) ), (3) for any X(t) R n and V (X(t)) α 3 ( X(t) ) + α 4 ( U(t) ), (4) for any X(t) R n and any U(t) L m. Then the system (1) is input-to-state stable if and only if it admits an ISS-Lyapunov function. 3 Nonlinear Bloch Equations and Problem Formulation In dimensionless units, the dynamic model of nonlinear modified Bloch equations with feedback field [24] is given by ẋ d (t) = δy d (t) + λz d (t)(x d (t)sin ψ y d (t)cosψ) x d(t), ẏ d (t) = δx d (t) z d (t) + λz d (t) (x d (t)cosψ + y d (t)sin ψ) y d(t), ż d (t) = y d (t) λsin ψ(x 2 d(t) + y 2 d(t)) z d(t) 1 τ 1, (5) where δ, λ, and ψ are the system parameters and τ 1 and are the longitudinal time and transverse relaxation times, respectively. The subscript d indicates that the system will be considered as the drive (or master) system. Abergel has extensively investigated the dynamics of the system (5) for a fixed subset of the system parameters (δ, λ, τ 1, ) and for a space area range of the radiation damping feedback ψ. [24] In particular, the regions of the radiation damping feedback ψ that would admit chaotic behavior were obtained. For details of other dynamic properties of the system (5), refer to Refs. [24 25]. The synchronization problem of system (5) is considered by using the drive-response configuration. The system (5) is considered as the drive system. According to the drive-response concept, the controlled response (or slave) system is given by ẋ r (t) = δy r (t) + λz r (t)(x r (t)sin ψ y r (t)cos ψ) x r(t) + u 1 (t) + d 1 (t), ẏ r (t) = δx r (t) z r (t) + λz r (t)(x r (t)cosψ + y r (t)sin ψ) y r(t) ż r (t) = y r (t) λsin ψ(x 2 r (t) + y2 r (t)) + u 2 (t) + d 2 (t), z r(t) 1 τ 1 + u 3 (t) + d 3 (t), (6) where u 1 (t), u 2 (t), and u 3 (t) are the nonlinear controllers, and d 1 (t), d 2 (t), and d 3 (t) are the external disturbances. Define the synchronization error as e 1 (t) = x r (t) x d (t), e 2 (t) = y r (t) y d (t), e 3 (t) = x r (t) z d (t). (7) Then we obtain the following synchronization error system: ė 1 (t) = δe 2 (t) 1 e 1 (t) + λz r (t)(x r (t)sin ψ y r (t)cosψ) λz d (t)(x d (t)sin ψ y d (t)cosψ) + u 1 (t) + d 1 (t), ė 2 (t) = δe 1 (t) e 3 (t) 1 e 2 (t) + λz r (t) (x r (t)cosψ + y r (t)sin ψ) λz d (t) (x d (t)cosψ + y d (t)sin ψ) + u 2 (t) + d 2 (t), ė 3 (t) = e 2 (t) 1 τ 1 e 3 (t) λsin ψ(x 2 r (t) + y2 r (t)) + λsin ψ(x 2 d (t) + y2 d (t)) + u 3(t) + d 3 (t), (8) which is rewritten as ė(t) = Ae(t) + f(t) + u(t) + d(t), (9) where e(t), u(t), d(t), A, and f(t) are defined by

310 Choon Ki Ahn Vol. 53 1 δ 0 e 1 (t) u 1 (t) d 1 (t) e(t) e 2 (t), u(t) u 2 (t), d(t) d 2 (t), A δ 1 1 e 3 (t) u 3 (t) d 3 (t), 0 1 1 τ 1 λz r (t)(x r (t)sin ψ y r (t)cosψ) λz d (t)(x d (t)sin ψ y d (t)cosψ) f(t) λz r (t)(x r (t)cos ψ + y r (t)sin ψ) λz d (t)(x d (t)cos ψ + y d (t)sin ψ). λsin ψ(x 2 r (t) + y2 r (t)) + λsin ψ(x2 d (t) + y2 d (t)) Definition 5 (Asymptotic synchronization) The error system (9) is asymptotically synchronized if the synchronization error e(t) satisfies lim e(t) = 0. (10) t Definition 6 (ISS synchronization) The error system (9) is ISS from the disturbance input to the synchronization error if there exist a K function γ(s) and a KL function β(s, t), such that, for each disturbance input d(t) L k and each initial synchronization error e(0) R n, it holds that e(t) β( e(0), t) + γ( d(t) ), (11) for each t 0. The purpose of this paper is to design the controller u(t) guaranteeing the ISS synchronization if there exists the disturbance input d(t). In addition, this controller u(t) will be shown to guarantee the asymptotic synchronization when the disturbance input d(t) disappears. 4 Main Results The LMI problem for achieving the ISS synchronization is presented in the following theorem. Theorem 1 For a given Q = Q T > 0, if there exist X = X T > 0 and Y such that AX + XA T + Y + Y T I X I I 0 < 0, (12) X 0 Q 1 then the ISS synchronization is achieved and the controller is given by u(t) = Y X 1 e(t) f(t) x r (t) x d (t) λz r (t)(x r (t)sin ψ y r (t)cos ψ) λz d (t)(x d (t)sin ψ y d (t)cos ψ) = Y X 1 y r (t) y d (t) λz r (t)(x r (t)cos ψ + y r (t)sin ψ) λz d (t)(x d (t)cos ψ + y d (t)sin ψ). (13) z r (t) z d (t) λsin ψ(x 2 r (t) + y2 r (t)) + λsin ψ(x2 d (t) + y2 d (t)) Proof The closed-loop error system with the control input u(t) = Ke(t) f(t), where K R 3 3 is the gain matrix of the control input u(t), can be written as Consider a Lyapunov function ė(t) = (A + K)e(t) + d(t). (14) V (e(t)) = e T (t)pe(t), (15) where P = P T > 0. Note that V (e(t)) satisfies the following Rayleigh inequality: [38] λ min (P) e(t) 2 V (e(t)) λ max (P) e(t) 2, (16) where λ min ( ) and λ max ( ) are the maximum and minimum eigenvalues of the matrix. The time derivative of V (e(t)) along the trajectory of Eq. (14) is V (e(t)) = ė(t) T Pe(t) + e T (t)pė(t) = e T (t)[a T P + PA + PK + K T P]e(t) + e(t) T Pd(t) + d T (t)pe(t). If we use the inequality X T Y +Y T X X T ΛX+Y T Λ 1 Y, which is valid for any matrices X R n m, Y R n m, Λ = Λ T > 0, Λ R n n, we have e(t) T Pd(t)+d T (t)pe(t) d T (t)d(t)+e(t) T PPe(t). (17) Using Eq. (17), we obtain V (e(t)) e T (t)[a T P + PA + PK + K T P + PP] e(t) + d T (t)d(t). If the following matrix inequality is satisfied we have A T P + PA + PK + K T P + Q + PP < 0, (18) V (e(t)) < e T (t)qe(t) + d T (t)d(t) (19) λ min (Q) e(t) 2 + d(t) 2. (20) Define functions α 1 (r), α 2 (r), α 3 (r), and α 4 (r) as α 1 (r) λ min (P)r 2, (21) α 2 (r) λ max (P)r 2, (22) α 3 (r) λ min (Q)r 2, (23) α 4 (r) r 2. (24) Note that α 1 (r), α 2 (r), α 3 (r), and α 4 (r) are K functions. From Eqs. (16) and (20), we can obtain α 1 ( e(t) ) V (e(t)) α 2 ( e(t) ), (25) V (e(t)) α 3 ( e(t) ) + α 4 ( d(t) ). (26) According to Lemma 1, we can conclude that V (e(t)) is an ISS-Lyapunov function and the ISS synchronization is

No. 2 Chaos Synchronization of Nonlinear Bloch Equations Based on Input-to-State Stable Control 311 achieved. From Schur complement, the matrix inequality (18) is equivalent to A T P + PA + PK + K T P P I P I 0 < 0. (27) I 0 Q 1 Pre- and post-multiplying Eq. (27) by diag(p 1, I, I) and introducing change of variables such as X = P 1 and Y = KP 1, Eq. (27) is equivalently changed into the LMI (12). Then the gain matrix of the control input u(t) is given by K = Y X 1. Corollary Without the disturbance input, if we use the control input u(t) proposed in Theorem 1, the asymptotic synchronization is obtained. Proof When d(t) = 0, we obtain V (e(t)) < e T (t)qe(t) 0, (28) from Eq. (19). This guarantees lim e(t) = 0, (29) t from Lyapunov theory. Remark The LMI problem given in Theorem 1 is to determine whether the solution exists or not. It is called the feasibility problem. The LMI problem can be solved efficiently by using recently developed convex optimization algorithms. [37] In this paper, in order to solve the LMI problem, we utilize MATLAB LMI Control Toolbox, [39] which implements state-of-the-art interiorpoint algorithms. 5 Numerical Example In this section, to verify and demonstrate the effectiveness of the proposed method, we discuss the simulation result for synchronizing nonlinear Bloch equations under different initial conditions. For the numerical simulation, the system parameters ψ, τ 1,, δ, and λ are fixed as 0.173, 5, 2.5, 0.4π, and 35, respectively, so that the Bloch equations exhibit a chaotic behavior. Let Q = I where I R 3 3 is an identity matrix. Applying Theorem 1 to the Bloch equations yields 0.6926 0.0000 0.0000 X = 0.0000 0.6926 0.0000, 0.0000 0.0000 0.6926 0.7619 3.5458 16.9214 Y = 3.5458 0.7619 0.5317. (30) 16.9214 0.5317 0.9004 Figure 1 shows state trajectories for drive and response systems when the initial conditions are given by x d (0) y d (0) = z d (0) 0.5 0.5 0, x r (0) y r (0) = z r (0) 0.5 0.5 0.3, (31) and the disturbance input d i (t) (i = 1, 2, 3) is given by { w(t), 0 t 10, d i (t) = 0, otherwise, where w(t) means a Gaussian noise with mean 0 and variance 1. Fig. 1 State trajectories. Fig. 2 Synchronization errors. From Fig. 1, it can be seen that drive and response systems are indeed achieving chaos synchronization. Figure 2 shows, by the proposed ISS synchronization method, that the synchronization error e(t) is bounded on the interval

312 Choon Ki Ahn Vol. 53 where the disturbance input d(t) exists. In addition, it is shown that the synchronization error e(t) goes to zero after the disturbance input d(t) disappears. 6 Conclusion In this paper, we propose a new ISS synchronization scheme for chaotic behavior in nonlinear Bloch equations. Based on Lyapunov theory and LMI approach, the proposed method guarantees the asymptotic synchronization between the drive and response systems and the bounded synchronization error for any bounded disturbance input. Furthermore, a numerical simulation is given to illustrate the effectiveness of the proposed scheme. References [1] L.M. Pecora and T.L. Carroll, Phys. Rev. Lett. 64 (1996) 821. [2] G. Chen and X. Dong, From chaos to Order, World Scientific, Singapore (1998). [3] C.C Wang and J.P. Su, Chaos, Solitons and Fractals 20 (2004) 967. [4] E. Ott, C. Grebogi, and J.A. Yorke, Phys. Rev. Lett. 64 (1990) 1196. [5] J.H. Park, Int. J. Nonlinear Sci. Numer. Simul. 6 (2005) 201. [6] J.H. Park, Chaos, Solitons and Fractals 25 (2005) 333. [7] X.S. Yang and G. Chen. Chaos, Solitons and Fractals 13 (2002) 1303. [8] E. Bai and K. Lonngen, Phys. Rev. E 8 (1997) 51. [9] E.W. Bai and K.E. Lonngren, Chaos, Solitons and Fractals 11 (2000) 1041. [10] O.M. Kwon and J.H. Park, Chaos, Solitons and Fractals 23 (2005) 445. [11] C.K. Ahn, Nonlinear Analysis: Hybrid Systems 4 (2010) 16. [12] C.K. Ahn, Nonlinear Dyn. in press (2010), doi:10. 1007/s11071-009-9560-6. [13] X. Wu and J. Lu, Chaos, Solitons and Fractals 18 (2003) 721. [14] J. Hu, S. Chen, and L. Chen, Phys. Lett. A 339 (2005) 455. [15] J.H. Park, Physica Script 76 (2007) 617. [16] C.K. Ahn, Phys. Lett. A 373 (2009) 1729. [17] C.K. Ahn, Nonlinear Dyn. 59 (2010) 319. [18] L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76 (1996) 1816. [19] S.S. Yang and C.K. Duan, Chaos, Solitons and Fractals 9 (1998) 1703. [20] J.H. Park, Chaos, Solitons and Fractals 34 (2007) 1552. [21] M.G. Rosenblum, A.S. Pikovsky, and J. Kurths. Phys. Rev. Lett. 76 (1996) 1804. [22] W. Yu and J. Cao, Physica A 375 (2007) 467. [23] Q. Song and J. Cao, Chaos, Solitons and Fractals 33 (2007) 929. [24] D. Abergel, Phys. Lett. A 302 (2002) 17. [25] A. Ucar and K.E. Lonngren, and E.W. Bai, Phys. Lett. A 314 (2003) 96. [26] J.H. Park, Chaos, Solitons and Fractals 27 (2006) 357. [27] D. Ghosh, A.R. Chowdhury, and P. Saha, Commun. Nonlinear Sci. Numer. Simulat. 13 (2008) 1461. [28] E.D. Sontag, IEEE Trans. Autom. Contr. 34 (1989) 435. [29] E.D. Sontag, IEEE Trans. Autom. Contr. 35 (1990) 473. [30] Z.P. Jiang, A. Teel, and L. Praly, Math. Contr. Signal Syst. 7 (1994) 95. [31] E.D. Sontag and Y. Wang, Syst. Contr. Lett. 24 (1995) 351. [32] P.D. Christofides and A.R. Teel, IEEE Trans. Autom. Contr. 41 (1996) 1645. [33] E.D. Sontag, Syst. Contr. Lett. 34 (1998) 93. [34] A.R. Teel, IEEE Trans. Autom. Contr. 43 (1998) 960. [35] W. Xie, C. Wen, and Z. Li, IEEE Trans. Autom. Contr. 46 (2001) 1111. [36] D. Angeli and D. Nesic, IEEE Trans. Autom. Contr. 46 (2001) 1298. [37] S. Boyd, L.E. Ghaoui, E. Feron, and V. Balakrishinan, Linear Matrix Inequalities in Systems and Control Theory, SIAM, Philadelphia (1994). [38] G. Strang, Introduction to Applied Mathematics, Wellesley Cambridge Press, Cambridge (1986). [39] P. Gahinet, A. Nemirovski, A.J. Laub, and M. Chilali, LMI Control Toolbox, The Mathworks Inc., Natick (1995).