Journal of Applied Mathematics Volume 01, Article ID 607491, 16 pages doi:10.1155/01/607491 Research Article Robust Adaptive Finite-Time Synchronization of Two Different Chaotic Systems with Parameter Uncertainties Yun-An Hu, Hai-Yan Li, Chun-Ping Zhang, and Liang Liu Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai, Shandong 64001, China Correspondence should be addressed to Yun-An Hu, hya507@sina.com Received 10 April 01; Revised 0 July 01; Accepted July 01 Academic Editor: Laurent Gosse Copyright q 01 Yun-An Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper is concerned with the finite-time synchronization problem for two different chaotic systems with parameter uncertainties. Using finite-time control approach and robust control method, an adaptive synchronization scheme is proposed to make the synchronization errors of the systems with parameter uncertainties zero in a finite time. On the basis of Lyapunov stability theory, appropriate adaptive laws are derived to deal with the unknown parameters of the systems. And the convergence of the parameter errors is guaranteed in a finite time. The proposed method can be applied to a variety of chaos systems. Numerical simulations are given to demonstrate the efficiency of the proposed control scheme. 1. Introduction In the past few decades, chaos synchronization has gained much attention from various fields 1 3, since Pecora and carroll 4 introduced a method to synchronize two identical chaotic systems with different initial conditions in 1990. Most of the works on chaos synchronization have focused on two identical chaotic systems 5 11. However, in many real world applications, there are no exactly two identical chaotic systems. Therefore, the problem of chaos synchronization between two different chaotic systems with uncertainties is an important research issue 1. Different synchronization control methods for two different chaotic systems, such as adaptive control 13 1, nonlinear feedback control, backstepping 3, 4, fuzzy technique 5 7, and sliding mode control 8 30, have been proposed to solve the synchronization problem. Since some systems parameters cannot be exactly known in advance, many efforts have been devoted to adaptive synchronization. In 18, 31, Huang discussed the
Journal of Applied Mathematics synchronizations between Lorenz-Stenflo LS system and CYQY system, and between LS system and hyperchaotic Chen system with fully uncertain parameters. Wang et al. 15 designed a general adaptive robust controller and parameter update laws which made the drive-response systems with different structures asymptotically synchronized. In 16, the sufficient conditions for achieving synchronization between generalized Henon-Heiles system and hyperchaotic Chen system with unknown parameters were derived based on Lyapunov stability theory. A new adaptive synchronization scheme by pragmatical asymptotically stability theorem was proposed for two different uncertain chaotic systems 17, but the unknown signals were used in the controller. Chaos synchronization between two different chaotic systems with uncertainties in both master and slave chaotic systems remains a challenging problem 30. Most methods only guarantee the asymptotic stability of the synchronization error dynamics, namely, the trajectories of the slave system approach the trajectories of the master system as t. From a practical point of view, however, it is more valuable that the synchronization objective is realized in a finite time 8. In recent years, some researchers have applied finite-time control techniques, such as nonsingular terminal sliding mode control method 3, CLF-based method 33, 34, sliding mode control method 8 30, and the finite-time stability theory-based method 8, 35, 36, to realize synchronization. Compared with the existing results in the literature, there are three advantages which make our approach attractive. First, based on the finite-time control technique, adaptive control, and robust control, a new synchronization method is presented for a wide class of nonlinear systems. Second, it guarantees that all the errors are driven to zero in a finite time even for the systems with parameter uncertainties. Third, it guarantees that all the parameter errors converge to zero in a finite time. In this paper, an adaptive finite-time synchronization scheme is proposed for a class of chaotic systems. The rest of the paper is organized as follows. In Section, we introduce the chaotic systems considered in this paper and preliminary lemmas. In Section 3, the proposed finite-time controller is designed to synchronize two different chaotic systems. We give the simulation results and the conclusions in Sections 4 and 5, respectively.. System Description Consider the following master chaotic system: ẋ A 1 ΔA 1 x B 1 ΔB 1 f 1 x,.1 where x x 1,x,...,x n T R n denotes a state vector, f 1 is a nonlinear continuous vector function, A 1 and B 1 are n n nominal coefficient matrices, ΔA 1 and ΔB 1 are unknown parts of n n coefficient matrices. The slave system is given with ẏ A ΔA y B ΔB f ( y ) u,. where y y 1,y,...,y n T R n denotes a state vector, f is a nonlinear continuous vector function, A and B are n n nominal coefficient matrices, ΔA and ΔB are unknown parts
Journal of Applied Mathematics 3 of n n coefficient matrices, and u u 1 t,u t,...,u n t T R n is a control input vector to be designed. Subtracting.1 from. yields the error dynamical system as follows: ė A ΔA y B ΔB f ( y ) A1 ΔA 1 x B 1 ΔB 1 f 1 x u,.3 where e y x. Note that only a part of elements of the coefficient matrices unknown, without loss of generality, we assume that the number of the unknown elements of the ith row of ΔA 1 is N A1i,thatofΔA is N Ai,thatofΔB 1 is N B1i, and that of ΔB is N Bi. Then.3 can be rewritten as ė A y B f ( y ) A1 x B 1 f 1 x u N A1 N B1 N A11 N B11 δa 1i y 1i δb 1i f 1i δa 11i x 1i δb 11i f 11i...., N An N Bn N A1n N B1n δa ni y ni δb ni f ni δa 1ni x ni δb 1ni f 1ni.4 where δa ji are nonzero elements of the jth row of ΔA, y ji are corresponding elements of y, δb ji are nonzero elements of the jth row of ΔB and f ji are corresponding elements of f, j 1,...,n. Assumption.1. The unknown parameters are norm-bounded, that is, δa ji da ji, δb ji db ji,.5 where d a ji and d b ji are known positive constants. Definition. see 8. Consider the master and slave chaotic systems described by.1 and., respectively. If there exists a constant T Te0 > 0, such that lim et 0.6 t T and et 0, if t T, then the chaos synchronization between the systems.1 and. is achieved in a finite time. Lemma.3 see 8. Consider the system ẋ fx, f0 0, x R n,.7 where f : D R n is continuous on an open neighborhood D R n.
4 Journal of Applied Mathematics Suppose there exists a continuous differential positive-definite function V x : D R, real numbers p>0, 0 <η<1, such that V x pv η x 0, x D..8 Then, the origin of system.7 is a locally finite-time stable equilibrium, and the settling time, depending on the initial state x0 x 0,satisfies Tx 0 V 1 η x 0 p ( 1 η )..9 In addition, if D R n and V x is also radially unbounded i.e., V x as x, then the origin is a globally finite-time stable equilibrium of system.7. Lemma.4 see 8. Suppose a 1, a,...,a n, and 0 < q < are all real numbers, then the following inequality holds: a 1 q a q a n q ( a 1 a a n) q/..10 3. Synchronization of Two Different Chaotic Systems with Parameter Uncertainties Consider two different chaotic systems.1 and. from different initial states. The aim of controller design is to determine appropriate u such that lime 0. 3.1 t T Now we are ready to give the design steps. Define Lyapunov function V 1 et e 1 N n Aj ( ) 1γ 1 1 kdj aji δã ji 1 N n A1j 1 N n Bj ( ) 1γ 1 1 kdj bji δ b ji 1 N n B1j ( 1 kdj a1ji δã 1ji ( 1 kdj b1ji δ b 1ji, 3. where δã ji δâ ji δa ji, δã 1ji δâ 1ji δa 1ji, δ b ji δ b ji δb ji, δ b 1ji δ b 1ji δb 1ji,and δâ ji, δâ 1ji, δ b ji, δ b 1ji are estimation values of δa ji, δa 1ji, δb ji, δb 1ji, respectively, and γ 1 aji, γ 1 a1ji, γ 1 bji, γ 1 b1ji, k dj are constants greater than zero.
Journal of Applied Mathematics 5 Taking the time derivative of 3. gives V e T ė N n Aj N n Bj ( 1 kdj aji δã jiδ â ji ( 1 kdj bji δ b ji δ bji N n A1j N n B1j ( 1 kdj a1ji δã 1jiδ â 1ji ( 1 kdj b1ji δ b 1ji δ b1ji. 3.3 Design the control law as u Ke K D ė α s [ c 1 sgne 1 e 1 α,...,c n sgne n e n α] T μ n N Aj N ( ) δâ ji 1α n A1j ( ) d aji δâ 1ji 1α d a1ji N n Bj ( ) N 1α n B1j ( ) 1α δ bji dbji δ b1ji db1ji α e 3.4 [ NA11 ] N T A1n δâ 11i y 1i,..., δâ 1ni y ni [ NB1 ] N T A1n δ b 1i f 1i,..., δ b ni f ni [ NA1 N An [ NB11 N B1n ] T δâ 1i y 1i,..., δâ ni y ni ] T δ b 11i f 11i,..., δ b 1ni f 1ni, where K diag{k 1,k,...,k n }, k i > 0, K D diag{k d1,k d,...,k dn }, k di > 0, and c i > 0are constants, 0 <α<1 is a constant, α s α s1... α sn T and α si is given in 3.5 V ( VA ) A/ e i f Ai K Ai / e i f Ai VA / e i 4, if V A / 0 α si V A / e i e i 0, if V A 0, i 1,...,n, e i 3.5 where V A 1/e T e, f A A y B f ya 1 x B 1 f 1 x and K Ai > 0. And α e α e1... α en T and α ei is given in 1, if e i e σ α ei e i sgne i, if e i <e σ, e σ 3.6 where e σ > 0 is a small constant.
6 Journal of Applied Mathematics Substituting 3.4 into.4 gives N A1 1 k d1 1 N B1 δã 1i y 1i 1 k d1 1 δ b 1i f 1i ė KI K D 1 e.. N An 1 k dn 1 N Bn δã ni y ni 1 k dn 1 δ b ni f ni N A11 1 k d1 1 N B11 δã 11i x 1i 1 k d1 1 δ b 11i f 11i f A α s.. N A1n 1 k dn 1 N B1n δã 1ni x ni 1 k dn 1 δ b 1ni f 1ni μi K D 1 n N Aj N ( ) δâji 1α n A1j ( ) daji δâ1ji 1α da1ji N n Bj ( ) N 1α n B1j ( ) 1α δ bji dbji δ b1ji db1ji α e 3.7 I K D 1[ c 1 sgne 1 e 1 α,...,c n sgne n e n α] T. Case 1 e i e σ. Substituting 3.7 into 3.3 yields n V e T KI K D 1 e c i 1 k di 1 e i 1α μk d n N Aj N ( ) δâji 1α n A1j ( ) daji δâ1ji 1α da1ji n j1 n N n Bj ( ) N 1α n B1j ( ) 1α δ bji dbji δ b1ji db1ji ( ) 1δãji n 1 kdj y ji N Aj e j N A1j e j N Bj e j ( ) n 1 kdj 1δ bji f ji N B1j e j ( ) 1δã1ji 1 kdj x ji ( ) 1 kdj 1δ b1ji f 1ji
Journal of Applied Mathematics 7 N n Aj ( 1 kdj aji δã jiδ â ji N n A1j ( 1 kdj a1ji δã 1jiδ â 1ji N n Bj ( 1 kdj bji δ b ji δ bji N n B1j ( 1 kdj b1ji δ b 1ji δ b1ji, 3.8 where k d min{1 k d1 1, 1 k d 1,...,1 k dn 1 }. Choosing the updating law as δ â ji { γaji e j y ji, if âji <daji 0, otherwise, { γa1ji e j x ji, if â1ji <da1ji δ â 1ji 0, otherwise, δ bji γ bji e j f ji, if b ji <dbji 0, otherwise, δ b1ji γ b1ji e j f 1ji, if b 1ji <db1ji 0, otherwise. 3.9 Substituting 3.9 into 3.8 yields n V e T KI K D 1 e c i 1 k di 1 e i 1α μk d n N Aj N ( ) δâji 1α n A1j ( ) daji δâ1ji 1α da1ji N n Bj ( ) N 1α n B1j ( ) 1α δ bji dbji δ b1ji db1ji. 3.10 Since δâ ji δa ji δâ ji δa ji δâ ji d aji, δâ1ji δa 1ji δâ1ji δa1ji δâ1ji da1ji, δ b ji δb ji δ bji δbji δ bji dbji, δ b 1ji δb 1ji δ b1ji δb 1ji δ b 1ji db1ji 3.11
8 Journal of Applied Mathematics hold, one can conclude that δâ ji d aji 1α δâ ji δa ji 1α, ( ) δâ1ji 1α da1ji δâ1ji δa 1ji 1α, ( ) 1α 3.1 1α δ bji dbji δ bji δb ji, and δ b 1ji d b1ji 1α δ b 1ji δb 1ji 1α. Therefore, the inequality 3.10 can be rewritten as n V e T KI K D 1 e c i 1 k di 1 e i 1α μk d n N Aj N n δâji δa ji 1α A1j δâ1ji δa 1ji 1α N n Bj N 1α n B1j 1α δ b ji δb ji δ b 1ji δb 1ji c μ n N n Aj N e i n A1j δâji δa ji δâ1ji δa 1ji N n Bj N n B1j δ b ji δb ji δ b 1ji δb 1ji 1α/ 3.13 c μ V 1α/, where c μ min{c i 1 k di 1,k i 1 k di 1,μk d,i 1,...,n}. According to Lemma.3, e B eσ in a finite time, where B eσ {e e i e σ,i 1,...,n}. Case e i <e σ.using3.3 3.7 and 3.9, it is easy to show that n V e T KI K D 1 e c i 1 k di 1 e i 1α 3.14 holds. According to Barbalat s lemma 37, we can conclude that e 0ast. From the discussion above, we have the following result. Theorem 3.1. For the systems.1 and., under Assumption.1, if the control law is designed as 3.4, updating laws are chosen as 3.9, then e will converge to B eσ in finite time, e 0 as t, and δã ji, δã 1ji, δ b ji, and δ b 1ji remain bounded. Remark 3.. Since the control signal 3.4 contains the discontinuous sign functions, as a hard switcher, it may cause undesirable chattering. In order to avoid the chattering, the sgn function can be replaced by a continuous function tanh to remove discontinuity.
Journal of Applied Mathematics 9 4. Numerical Simulation In this section, we present numerical results to verify the proposed synchronization approach. Consider the following master chaotic system: ẋ 1 ax x 1 a a 0 x 1 0 0 0 0 ẋ bx 1 cx 1 x 3 b 0 0 x 0 c 0 x 1 x 3 ẋ 3 gx 3 hx 1 0 0 g x 3 0 0 h x 1 a 0 a 0 0 x 1 δa 0 δa 0 0 x 1 b 0 0 0 x δb 0 0 0 x 0 0 g 0 x 0 0 δg 0 x 3 }{{}}{{}}{{}}{{} A 1 x ΔA 1 0 0 0 0 0 0 0 0 0 c 0 0 x 1 x 3 0 δc 0 0 x 1 x 3, 0 0 h 0 x 1 0 0 δh 0 x 1 }{{}}{{}}{{}}{{} B 1 f 1x ΔB 1 x f 1x 4.1 where a a 0 δa 0, b b 0 δb 0, c c 0 δc 0, g g 0 δg 0, h h 0 hg 0, a 0 8, δa 0, b 0 35, δb 0 5, c 0 0.7, δc 0 0.3, g 0.0, δg 0 0.5, h 0 0.8, and δh 0 0.. The slave system is given with ( ) ẏ 1 a 1 y y 1 y y 3 a 1 a 1 0 y 1 a 1 0 0 y y 3 ẏ b 1 y c 1 y 1 y 3 0 b 1 0 y 0 c 1 0 y 1 y 3 ẏ 3 g 1 y h 1 y 3 0 g 1 h 1 y 3 0 0 0 0 a 10 a 10 0 y 1 δa 10 δa 10 0 y 1 0 b 10 0 y 0 δb 10 0 y 0 g 10 h 10 y 3 0 δg 10 δh 10 y 3 }{{}}{{}}{{}}{{} A y ΔA a 10 0 0 y y 3 δa 10 0 0 y y 3 u 1 0 c 10 0 y 1 y 3 0 δc 10 0 y 1 y 3 u, } 0 0 {{ 0 0 }}{{} } 0 0 {{ 0 0 }}{{} u 3 B f y ΔB f y y 4. where a 1 a 10 δa 10, b 1 b 10 δb 10, c 1 c 10 δc 10, g 1 g 10 δg 10, h 1 h 10 hg 10, a 10 0.8, δa 10 0., b 10.0, δb 10 0.5, c 10 0.7, δc 10 0.3, g 10 0.7, δg 10 0.3, h 10 3.0, δh 10 1.0. x0 1.8, y0 1., and z0 1.5. The initial states in master system 4.1 are x 1 0 1.8, x 0 1., x 3 0 1.5. The initial states in slave system 4. are y 1 0 1.5, y 0 1., y 3 0 1.1. The initial parameter estimation values of the systems.1 and. are δâ 0 0, δ b 0 0, δĉ 0 0, δĝ 0 0, δĥ0 0, δâ 10 0, δ b 10 0, δĉ 10 0, δĝ 10 0, and δĥ10 0.
10 Journal of Applied Mathematics 150 100 x3 50 0 0 15 10 5 0 5 10 15 0 5 x 1 Figure 1: Chaotic behavior of the master chaotic system under the proposed parameters. According to Remark 3., the control law 3.4 is modified as follows: u Ke K D ė α s C A C B C A1 C B1 [ ( ) μ δâ 0 d a0 1α 1α δ b0 db0 δĉ0 d c0 1α ( ) ( ) δĝ0 1α δ 1α ( ) dg0 ĥ 0 dh0 3 δâ10 d a10 1α 1α δ b10 db10 δĉ 10 d c10 1α ( δĝ10 dg10 ) 1α ( δ ĥ 10 dh10 ) 1α ]α e [ c 1 tanhεe 1 e 1 α,...,c n tanhεe n e n α] T, 4.3 where C A C B [ δâ 10 y y 3, δĉ 10 y 1 y 3, 0 ] T, C B1 [ δâ 10 ( y y 1 ),δ b10 y,δĝ 10 y δĥ10y 3 ] T, CA1 [ δâ 0 x x 1,δ b 0 x 1, δĝ 0 x 3 ] T, [ 0, δĉ 0 x 1 x 3,δĥ0x 1] T, K diag{70, 54, 30}, K D diag{0.93, 0.75, 0.1}, μ diag{1, 0., 0.01}, K A diag{.3,.1,.3}d a0, d b0 10, d c0 1, d g0 1, d h0, d a10 1, d b10, d c10, d g10, d h10, ε 40, c 1 1, c 3, c 3 1. 4.4
Journal of Applied Mathematics 11 y3 3 1 0 1 3 4 5 0 5 10 15 0 5 30 Figure : Chaotic behavior of the slave chaotic system under the proposed parameters. y 1 Choosing the updating law as δ â 0 { 0.30e1 x 1 x, if â 0 <d a0 0, otherwise, δ b0 0.90e x 1, if b 0 <db0 0, otherwise, δ ĉ 0 { { 0.001e x 1 x 3, if ĉ 0 <d c0 0, otherwise, δ ĝ 0.00e3 x 3, if ĝ0 <dg0 0 0, otherwise, δ ĥ 0.018e 3 x 1 0, if { ( ) ĥ 0 <dh0 0.0006e1 y1 y y y 3, if â10 <d a10, δ â 10 0, otherwise 0, otherwise, { δ b10 0.08e y, if b 10 <db10 0.0015e y 1 y 3, if ĉ 10 <d c10, δ ĉ 10 0, otherwise 0, otherwise, { 0.1e3 y, if ĝ 10 <d g10 0.017e 3 y 3, if ĥ10 <d h10 δ ĝ 0, δ ĥ 10 0, otherwise 0, otherwise. 4.5 Chaotic behavior of the master chaotic system under the proposed parameters is shown in Figure 1. Chaotic behavior of the slave chaotic system under the proposed parameters is shown in Figure. From Figures 1 and, we know that the two systems are still chaotic under adopted uncertain parameters. The synchronization errors between two different chaotic systems are illustrated in Figures 3, 4, and 5, where the control inputs are activated at t 1s. One can see that the synchronization errors converge to the zero in a finite time, which implies that the chaos synchronization between the two different chaotic systems
1 Journal of Applied Mathematics 15 10 5 0 e1 5 10 15 0 0 10 0 30 40 50 60 t Figure 3: Synchronization error e 1. 40 30 0 10 e 0 10 0 30 0 10 0 30 40 50 60 t Figure 4: Synchronization error e. is realized. The time responses of parameter estimations â 0, b 0, ĉ 0, ĝ 0,andĥ0 are depicted in Figure 6. The time responses of parameter estimations â 10, b 10, ĉ 10, ĝ 10,andĥ10 are depicted in Figure 7. According to the simulations, it has been shown that the proposed control algorithm provides stable behavior when using online adaptive laws. The control performance is satisfactory and the chattering phenomenon has been successfully improved by using tanh functions. In addition, it is easy to see that the parameter estimation values approach their real values in a finite time.
Journal of Applied Mathematics 13 0 0 40 e3 60 80 100 10 0 10 0 30 40 50 60 t Figure 5: Synchronization error e 3. 1 10 Parameter estimations 8 6 4 0 0 10 0 30 40 50 60 t Estimation for δa 0 Estimation for δb 0 Estimation for δc 0 Estimation for δg 0 Estimation for δh 0 Figure 6: Curves of parameter estimations for the master system. 5. Conclusions In this paper, we have studied chaos synchronization of two different chaotic systems with parameter uncertainties. The two different chaotic systems with parameter uncertainties are synchronized via robust adaptive control based on the Lyapunov stability theory and finitetime theory. The proposed method can be applied to a variety of chaos systems. It guarantees that all the error states are driven to zero in a finite time. Numerical simulations are given to show the proposed synchronization approach works well for synchronizing two different
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