Conditions of parameter identification from time series

PHYSICAL REVIEW E 83, 036202 (2011)

Conditions of parameter identification from time series Haipeng Peng, Lixiang Li, Yixian Yang, and Fei Sun Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, P.O. Box 145, Beijing 100876, China; Key Laboratory of Network and Information Attack and Defence Technology of MOE, Beijing University of Posts and Telecommunications, Beijing 100876, China and National Engineering Laboratory for Disaster Backup and Recovery, Beijing University of Posts and Telecommunications, Beijing 100876, China (Received 16 November 2009; revised manuscript received 6 December 2010; published 11 March 2011) We study the problem of synchronization-based parameters identification of dynamical systems from time series. Through theoretical analysis and numerical examples, we show that some recent research reports on this issue are not perfect or even incorrect. Long-time full rank and finite-time full rank conditions of Gram matrix are pointed out, which are sufficient for parameters identification of dynamical systems. The influence of additive noise on the proposed parameter identifier is also investigated. The mean filter is used to suppress the estimation fluctuation caused by the noise. DOI: 10.1103/PhysRevE.83.036202

PACS number(s): 05.45.Ac, 05.45.Gg, 05.45.Vx

I. INTRODUCTION

Since the pioneering works of Ott, Gregogi, and York [1] and Pecora and Carroll [2], there has been increasing interest in developing powerful techniques for chaos control and synchronization. The adaptive method is an available tool to ensure the control and synchronization with uncertain parameters. To identify the uncertain parameters correctly by using the adaptive method, some conditions should be satisfied [3]. However, the theories in Ref. [4] ignored the conditions which guarantee that the estimated parameters converge to the true values. Recently, the adaptive synchronization method has been used to identify the network topological structures, where the authors in Ref. [5] did not pay attention to the conditions which guarantee the effective estimation of network topology. More recently, a similar method is also used to estimate the interaction delays of systems, and the proposed technique in Ref. [6] has the same drawback. In addition, in Ref. [7], the parameter identifier θ˙ˆ = F T (x)e + θ˜ = F T (x)e + (θ − θˆ ) was used to estimate the unknown parameter. θ is the unknown parameter which should not be used in the parameter identifier. Thus, the method in Ref. [7], which also occurs in Ref. [8], is not practically useful. In the traditional adaptive control field [9], the persistent excitation (PE) condition is applied to ensure the estimation of unknown parameters which requires rich enough training information. Recently, in Refs. [10,11], linear independence (LI) was addressed to guarantee that the estimated parameters converge to the true values. The PE condition and the LI condition are especially crucial for parameter identification. However, from the mathematical point of view, there are some drawbacks in Refs. [10,11] which did not realize the restriction on the time domain for defining the LI of the function vector group. In Ref. [12], the authors gave out the correct definition of the LI of the function vector group. Furthermore, based on Gram matrix theory, a special relationship between the LI and PE conditions is proposed for estimating uncertain model parameters [12]. Note that the conditions of PE and LI are sufficient, but not necessary to guarantee that the estimated parameters converge to the true values. In Ref. [10], the authors demonstrated that “parameter identification is almost impossible for systems converging to a fixed 1539-3755/2011/83(3)/036202(8)

point” and they declared that “one cannot estimate the system parameters without the condition of linear independence of the functions on the synchronization manifold.” In the following, we will show that parameter identification is possible for systems converging to a fixed point, which cannot be explained from PE and LI conditions. To explain this phenomenon, some new theoretical analysis is needed. In this paper, based on Lyapunov theory, adaptive controller and parameters identifier are designed to achieve synchronization for different dynamical systems with uncertain parameters. To make sure that the estimated parameters converge to the true values, long-time full rank and finite-time full rank conditions of the Gram matrix are given. We will show that under these sufficient conditions, parameters can be well identified from time series. If the Gram matrix is full rank for a limited period of time only, we find that the parameters remain unchanged after the full rank period is over. Furthermore, we discuss the influence of the system noise. Simulations illustrate the validity and feasibility of the proposed scheme. II. PRINCIPLE OF THE METHOD

Consider the drive system in the form of x˙ = f (x) + F (x)θ,

(1)

where x ∈ R n is the state vector, f : R n → R n is a continuous function, F (x) : R n → R n×m is the function matrix, and θ = (θ1 ,θ2 , . . . ,θm )T ∈ R m is the unknown parameter vector. The response system is shown as follows: y˙ = g(y) + u,

(2)

where y ∈ R n is the state vector, g : R n → R n is a continuous function, and u ∈ R n is the controller. We define the error term e = x − α(t)y,

(3)

where α(t) = diag[α1 (t),α2 (t), . . . ,αn (t)] is a continuously differentiable bounded matrix function and αi (t) = 0 (i = 1,2, . . . ,n) for any t. The goal of this paper is to design a controller and a parameter identifier such that e and θ˜ (θ˜ = θ − θˆ ) converge to

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HAIPENG PENG, LIXIANG LI, YIXIAN YANG, AND FEI SUN

zero as t → ∞, where θˆ is the estimated value of θ . The above synchronization belongs to the projective synchronization [13], and when α(t) = diag(1,1, . . . ,1), it belongs to the identical synchronization [2]. To state our main results, some definitions are given as follows. Long-time full rank. The Gram matrix G is a longtime rank if it is full rank for any t  0, where G =  t+τ full T F [x(s)]F [x(s)]ds is the Gram matrix of F [x(s)] and t τ is a positive constant. Finite-time full rank. The Gram matrix G is finite-time full rank in the time range [t1 ,t2 ], if it is full rank for any t, where 0  t1  t  t2 . tp = t2 − t1 is named as the persistence time of full rank. From Eq. (3), we have e˙ = x˙ − α(t)y˙ − α(t)y. ˙

(4)

The controller and the adaptive laws are constructed as  u(t) = α −1 (t)(f (x) + F (x)θˆ − α(t)g(y) − α(t)y ˙ + e), (5) ˙θˆ = F T (x)e. Substituting Eqs. (1), (2), and (5) into Eq. (4), and noting that θ˙ = 0, we obtain  e˙ = −e + F (x)θ˜ , (6) θ˙˜ = θ˙ − θ˙ˆ = −F T (x)e. Defining a Lyapunov function of the form V = 12 eT e + then we have

1 ˜T ˜ θ θ, 2

T V˙ = eT e˙ + θ˙˜ θ˜ = eT [−e + F (x)θ˜ ] + [−F T (x)e]T θ˜ = − eT e.

It is obvious that V˙ = 0 if and only if e = 0. From the Barbalat’s lemma, we conclude that e → 0 and θ˙˜ → 0 as t → ∞. Thus, when t is large enough, θ˜ is almost constant [14]. Then, we can obtain the largest invariant set M which is described as M = {e ∈ R n ,θ˜ ∈ R m |e = 0, − e + F (x)θ˜ = 0}. In such a circumstance, the following equation can be satisfied F (x)θ˜ = 0.

(7)

Next multiply Eq. (7) by F T (x) for both sides and integrate the equation for a period of time τ , such that  t+τ F T [x(s)]F [x(s)]ds θ˜ = 0, (8) t

and thereby Gθ˜ = 0. If G is full rank for any t  0, which means that G is long-time full rank, Eq. (8) has a unique zero solution. Then the estimated parameters will converge to the true values. To summarize the above analysis, the following conclusions are thus proved. Result 1. For the drive system (1) and response system (2) with the controller and parameter identifier (5), if the Gram matrix G is long-time full rank, then e and θ˜ converge to zero as t → ∞ for any arbitrary initial conditions. If the Gram matrix is a diagonal matrix, then we have (i) when it is long-time full rank, we can identify all the unknown parameters; (ii) when it is not full rank (we assume

PHYSICAL REVIEW E 83, 036202 (2011)

the rank is r, where r < m), we could identify r unknown parameters. Long-time full rank of Gram matrix is a sufficient condition to ensure that the true values of unknown parameters are obtained. In practical applications, the Gram matrix G may not be long-time full rank but finite-time full rank. In what follows, we will show that with some conditions the unknown parameters can be identified even if the Gram matrix G is finite-time full rank. When the estimated parameters run from the initial values to the stable values, they need a transient time tf . When the persistence time tp of finite-time full rank is much larger than the transient time tf , which means there is enough time for driving the estimated parameters from the initial values to the true (stable) values, we can also obtain the true values of unknown parameters. From the energy point of view, if the persistence time of finite-time full rank is large enough, the system may have sufficient energy to drive the estimated parameters to the true values. Thus we have the following result. Result 2. For the drive system (1) and response system (2) with the controller and parameter identifier (5), if the Gram matrix G is finite-time full rank, and if the time difference epf = tp − tf between the persistence time tp and the transient time tf is large enough, then e and θ˜ converge to zero as t → ∞ for any arbitrary initial conditions. For system (6), according to the Lyapunov stable theory and Barbalat’s lemma, we have e → 0 and θ˙˜ → 0 as t → ∞, which means θˆ converges to constant θ  . If G is full rank during some transient (finite-time full rank), and the transient is long enough, then θˆ converges to constant θ  = θ . When the transient is over, since F (x) is bounded, we have θ˙ˆ = F T (x)e → 0 as e → 0, then θˆ remains unchanged and stable at θ  = θ . Similarly, if the transient is not long enough, θˆ converges to constant θ  = θ and keeps stable at θ  . When the synchronization errors run from the initial state to the stable state, they need a transient time te . Due to θ˙ˆ = F T (x)e, we know tf is influenced by te . When tp is much larger than te [in this condition from θ˙ˆ = F T (x)e, we have te ≈ tf < tp ], then tp is much larger than tf , and from Result 2, we have θˆ converge to the true valve of θ . When te is larger than tp [in this condition when full-rank-period is over, if F (x) converges to 0, then based on θ˙ˆ = F T (x)e, we have tp ≈ tf < te , similarly if F (x) does not converge to 0, we have tp < te ≈ tf ], the conditions of Result 2 cannot be satisfied, and we cannot guarantee θˆ converges to the true valve of θ . That is to say, we can distinguish whether θˆ converges to the true valve or not through analyzing te and tp . Now we discuss the relationship between the full rank of the Gram matrix and the linear independence of the function vector group. Linear Independence. The function li (t) is linear independent in the time range [t,t + τ ], if there do not exist nonzero constants ki (i = 1,2, . . . ,N ), such that k1 l1 (t) + k2 l2 (t) + · · · + kN lN (t) = 0 in the time range [t,t + τ ] and τ is a positive constant. Long-time linear independence. The function li (t) is longtime linear independent if there do not exist nonzero constants

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ki (i = 1,2, . . . ,N), such that k1 l1 (t) + k2 l2 (t) + · · · + kN lN (t) = 0 in the time range [t,t + τ ] for any t  0. Finite-time linear independence. The function li (t) is finite-time linear independent, if there do not exist nonzero constants ki (i = 1,2, . . . ,N), such that k1 l1 (t) + k2 l2 (t) + · · · + kN lN (t) = 0 in the time range [t,t + τ ] for any t, where 0  t1  t  t 2 . When the Gram matrix G is full rank, we can easily conclude that the function vector group is linear independent in the range [t,t + τ ] [15]. Long-time full rank and finitetime full rank of the Gram matrix indicate long-time linear independence and finite-time linear independence condition, respectively. The above definitions about linear independence are different from those in Refs. [10,11]. III. SIMULATION EXPERIMENTS

Now we consider the following Lorenz system as the drive system to simulate ⎧ ⎪ ⎨x˙1 = a(y1 − x1 ), y˙1 = (b − z1 )x1 − y1 , (9) ⎪ ⎩z˙ 1 = x1 y1 − cz1 . The Newton-Leipnik system is employed as the response system, which is given by ⎧ ⎪ ⎨x˙r = −0.4xr + yr + 10yr zr + u1 , y˙r = −xr − 0.4yr + 5xr zr + u2 , (10) ⎪ ⎩z˙ r = 0.175zr − 5xr yr + u3 . 1 ˆ 1 − x1 ) − [a(y We design the following controllers u1 = α(t) 1 α(t)(−0.4xr + yr + 10yr zr ) − αx ˙ r + e1 ], u2 = α(t) [−z1 x1 − ˆ 1 − α(t)(−xr − 0.4yr + 5xr zr ) − αy y1 + bx ˙ r + e2 ], u3 = 1 ˆ [x y − cz − α(t)(0.175z − 5x y ) − αz ˙ + e ], and the 1 r r r r 3 α(t) 1 1 ˙ parameter update laws a˙ˆ = (y1 − x1 )e1 , bˆ = x1 e2 , c˙ˆ = −z1 e3 . The condition that Gram matrix G is long-time full rank is important to make sure that the estimated parameters converge to the true values. According to the system (9), we have F (x) = diag(y1 − x1 ,x1 ,z1 ). Then we obtain  t+τ F T [x(s)]F [x(s)]ds t ⎞ ⎛ t+τ (y1 − x1 )2 ds 0 0 t  t+τ 2 ⎟ ⎜ = ⎝ 0 x1 ds 0 ⎠. (11) t  t+τ 0 0 (z1 )2 ds t

In our simulations, we select α(t) = 20 + 3 sin( 2πt ). When 80 3 a = 10,b = 28,c = 8 , system (9) is in the chaotic state and we conclude that G is full rank for any t  0. According to the analysis of this paper, we have both e = (e1 ,e2 ,e3 )T and θ˜ = ˜ c) ˜ b, ˜ converge to zero as t → ∞. Figures 1(a) through 1(c) (a, depict the error curves of synchronization between systems (9) and (10). Figure 1(d) shows the results of parameters estimation. When a = 1, b = 28, and c = 38 , the system (9) is in the stable state. Figure 2(a) displays the state curves of system (9). From Fig. 2(a), we can see that all states of system (9) converge to positive constants and y1 (t) − x1 (t) converges to zero as t →

ˆ cˆ ˆ b, FIG. 1. (Color online) The error curves of e1 ,e2 ,e3 and a, for synchronization between systems (9) and (10), and the curves ˆ cˆ for parameters a,b,c where ˆ b, of parameters estimation results a, a = 10, b = 28, and c = 38 .

∞. Figure 2(b) depicts the curves of synchronization errors and Fig. 2(c) shows the results of parameters  t+τ estimation. In this case, the rank of Gram matrix G = t F T [x(s)]F [x(s)]ds ˆ c(t)] ˆ is 2 for any t  0, so we can use the identifiers [b(t), to achieve the true values of parameters b and c. But the identifier aˆ does not converge to a = 1. Note that, from Fig. 2(a), we can see that the Gram matrix of Eq. (11) is finite-time full rank  t+τ t+τ [in the first several seconds, all of t (y1 − x1 )2 ds, t x12 ds  t+τ and t (z1 )2 ds do not equal] and the persistence time tp is too short to drive aˆ to the true value of a. Now, we consider the following drive system which is constructed based on Lorenz system: ⎧ x˙1 = a(y1 − x1 ), ⎪ ⎪ ⎪ ⎨y˙1 = (b(t) − z1 )x1 − y1 , (12) z˙ 1 = x1 y1 − cz1 , ⎪ ⎪ ⎪ ⎩b(t) ˙ = −kb(t), where a,c are the unknown parameters and the response system is selected as Eq. (10). Similarly, we have the following controllers and the parameters update laws u1 = 1 ˆ 1 − x1 ) − α(t)(−0.4xr + yr + 10yr zr ) − αx [a(y ˙ r + e1 ], α(t) 1 u2 = α(t) [−z1 x1 − y1 + b(t)x1 + α(t)(xr + 0.4yr − 5xr zr ) − 1 ˆ 1 − α(t)(0.175zr − 5xr yr ) − αy ˙ r + e2 ], u3 = α(t) [x1 y1 − cz ˙ αz ˙ r + e3 ], and aˆ = (y1 − x1 )e1 , c˙ˆ = −z1 e3 . When the parameters in the system (12) are set as a = 10, k = 0.005, c = 38 , and b(0) = 28, the system (12) is in the stable state. Figure 3(a) displays the state curves of system (12). Figure 4(a) shows the state curves x1 , y1 , z1 , and y1 − x1 of system (12) when a = 10, k = 0.1,c = 38 , and b(0) = 28.

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FIG. 2. (Color online) The curves of x1 ,y1 ,z1 ,y1 − x1 and ˆ cˆ for ˆ b, e1 ,e2 ,e3 , and the curves of parameters estimation results a, parameters a,b,c where a = 1,b = 28,c = 38 .

PHYSICAL REVIEW E 83, 036202 (2011)

FIG. 3. (Color online) The curves of x1 ,y1 ,z1 ,e1 ,e3 and the curves ˆ cˆ for parameters a,c of system (12) of parameters estimation results a, where a = 10, b(0) = 28, c = 38 , and k = 0.005.

From Figs. 3(a) and 4(a), we can find that all states of system (12) converge to zero after a transient chaotic process. In these cases,  t+τ  2 (y − x ) ds 0 1 1 G= t , (13)  t+τ 0 (z1 )2 ds t where the Gram matrix is not long-time full rank. However, we can see from Fig. 3(a) that in the time range [0,90], the Gram matrix is finite-time full rank, and the persistence time tp = 90 is large enough to drive the estimated parameters to run from the initial values to the true (stable) values of a = 10,c = 38 . Figure 3(b) depicts the curves of synchronization errors e1 and e3 . Figure 3(c) shows the results of parameters estimation. From Figs. 3(b) and 3(c), we can see that the times tf and te are much smaller than tp . From Fig. 4(a), we can see that in the time range [0,9], the Gram matrix is finite-time full rank. Figure 4(b) depicts the curves of synchronization errors e1 and e3 . Figure 4(c) shows the results of parameters estimation and where aˆ converges to a constant (but now at an incorrect value). From Figs. 4(a) through (c), we can see that the persistence time tp = 9 is too small (tp is not much larger than te ) to drive the estimated parameter aˆ from the initial values to the true values. If the Gram matrix is full rank for a limited period of time only, we find that the parameters remain unchanged after the full rank period is over. Figure 5 shows the estimation

FIG. 4. (Color online) The curves of x1 ,y1 ,z1 ,y1 − x1 ,e1 ,e3 and ˆ cˆ for parameters a,c of the curves of parameters estimation results a, system (12) where a = 10, b(0) = 28, c = 38 , and k = 0.1.

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FIG. 5. (Color online) The simulation results of b(t), (x1 ,y1 ,z1 ), ˆ c), ˆ respectively. (e1 ,e2 ,e3 ), and (a,

process in such a condition. When b(t) changes from 28 to 0 as shown in Fig. 5(a), we can see that the full rank period of the Gram matrix is over at time t = 20 [see Fig. 5(b)]. Figures 5(c) and 5(d) show the corresponding curves of ˆ c), ˆ respectively. Figures 6(a) through e = (e1 ,e2 ,e3 ) and (a, (c) show the difference between the true and the estimated parameters on a logarithmic scale when the full rank period of the Gram matrix is over at times t = 20, 40, and 80, respectively. We also do some research on other systems such as the Mackey-Glass (M-G) system and time-delay logistic system [16]. For the M-G system x˙ = −θ x + b(t)x(t − γ )/[1 + x(t − γ )10 ], where γ is the time delay, θ is an unknown parameter, and b(t) is the control parameter which can

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FIG. 7. (Color online) The simulation results of the M-G system. (a) The simulation result of b(t), (b) the curve of the state x, ˆ and (d) the curve of log10 (|θˆ − θ |), where (c) the curves of e and θ, γ = 2,θ = 2.

 t+τ control the full rank period of Gram matrix t x(s)2 ds. We use the following response system and parameter update law y˙ = −θ x + b(t)x(t − γ )/[1 + x(t − γ )10 ] + x − y, θ˙ˆ = −xe. Figure 7 shows the estimation process. When b(t) changes from 10 to 1 as shown in Fig. 7(a), we can see that the full rank period of the Gram matrix is over at time t = 30 [see Fig. 7(b)]. Figure 7(c) depicts the curves of e and θˆ . Figure 7(d) shows the corresponding result of log10 (|θˆ − θ |). Now we consider time-delay logistic system x˙ = −θ x + b(t)x(t − γ )[1 − x(t − γ )], where γ is the time delay, θ is an unknown parameter, and b(t) is the control parameter  t+τwhich can control the full rank period of the Gram matrix t x(s)2 ds. We use the following response system and parameter update law y˙ = −θ x + b(t)x(t − γ )[1 − x(t − γ )] + x − y, θ˙ˆ = −xe. Figure 8 shows the estimation process. When b(t) changes from 104 to 10 as shown in Fig. 8(a), we can see that the full rank period of the Gram matrix is over at time t = 50 [see Fig. 8(b)]. Figure 8(c) depicts the curves of e and θˆ . Figure 8(d) shows the corresponding result of log10 (|θˆ − θ |). From Figs. 7 and 8, we know that the parameters will not drift away after the full rank period is over and where tp > tf ≈ te . The above examples show that parameter identification is possible for systems converging to a fixed point. The conditions in Refs. [10,12] could not be used to explain the phenomenon, while Result 2 of this paper could explain the phenomenon. IV. ANALYSIS IN THE PRESENCE OF NOISE

Noise plays an important role in synchronization and parameters identification of dynamical systems. Noise usually deteriorates the performance of parameter identification and results in the drift of parameter identification around their true values. Here we consider the influence of the noise. Suppose that there is additive noise in drive system (1) in the form of FIG. 6. (Color online) The difference between true and the estimated parameters on a logarithmic scale when the full rank period of the Gram matrix is over at times t = 20, 40, and 80, respectively.

x˙ = f (x) + F (x)θ + η, where η ∈ R n is the zero mean bounded noise.

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FIG. 8. (Color online) The simulation results of time-delay logistic system. (a) The simulation result of b(t), (b) the curve of ˆ (d) the curve of log10 (|θˆ − θ|), the state x, (c) the curves of e and θ, where γ = 1,θ = 26.

Using the same response system (2), the error term (3), and the controller and adaptive laws (5), we easily have  e˙ = −e + F (x)θ˜ + η, (15) θ˙˜ = θ˙ − θ˙ˆ = −F T (x)e. Then we have  dE(e) dt dE(θ˜ ) dt

= −E[e(t)] + E[F (x)θ˜ ], = −E[F T (x)e(t)].

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FIG. 9. (Color online) The simulation results of (x1 ,y1 ,z1 ), ˆ c). ˆ b, ˆ (a) The curves of the states x1 ,y1 ,z1 , (b) the (e1 ,e2 ,e3 ), and (a, error curves of e1 ,e2 ,e3 , (c) the curves of parameters estimation results ˆ cˆ for parameters a,b,c, and (d) the simulation results of a, ˆ cˆ ˆ b, ˆ b, a, when the filter (19) is added.

tems of (9) are disturbed by uniformly distributed random noise with amplitude ranging from −10 to 10. Figures 9(a) and 9(b) show the simulation results of x1 ,y1 ,z1 and e1 ,e2 ,e3 ; ˆ cˆ fluctuate ˆ b, Fig. 9(c) shows that the estimated parameters a, around their true values. When the Gram matrix of E{F [x(s)]} is finite-time full rank, through simulations we find that when the full-rank period is over, the estimated parameters will drift away if the computation is continued. Figure 10 shows the estimation

where E(∇) is the mean value of ∇. Construct Lyapunov ˜ T E(θ˜ ), then we get function V = 12 E[e(t)]T E[e(t)] + 12 E(θ) ˙ + E(θ˙˜ )T E(θ˜ ) V˙ = E[e(t)]T E(e) ˜ − E(e) − {E[F T (x)e]}T E(θ˜ ). = E(e)T [E(F (x)θ] If F (x) is independent to e and θ˜ , then V˙ = −E(e)T E(e), and similarly we have ˜ = 0. E[F (x)]E(θ)

(17)

Then multiply Eq. (17) by E[F (x)]T for both sides and integrate the equation for a period of time τ , such that  t+τ ˜ = 0. E{F [x(s)]}T E{F [x(s)]}dsE(θ) (18) t

 t+τ

If t E{F [x(s)]}T E{F [x(s)]}ds is full rank for any t  0, which means the Gram matrix of E{F [x(s)]} is longtime full rank, Eq. (18) has a unique zero solution. Then the ˜ will converge to 0 mean values of estimated parameters E(θ) as t → ∞, which means that the estimated values for unknown parameters will fluctuate around their true values. To summarize the above analysis, the following conclusion is thus obtained. Result 3. For the drive system (14) with zero mean noise and the response system (2) with the controller and parameter identifier (5), if the Gram matrix of E{F [x(s)]} is long-time full rank, and if F (x) is independent of e and θ˜ , then E(θ˜ ) converges to zero as t → ∞ for any arbitrary initial conditions. As an illustrating example, we revisit the Lorenz system (9) and the slave system (10), and we assume that all the subsys-

FIG. 10. (Color online) The estimation process of aˆ and cˆ in the presence of noise. (a) The curves of the states x1 ,y1 ,z1 , (b) the curves of (y1 − x1 )e1 ,z1 e3 , and (c) the curves of parameters estimation results ˆ cˆ for parameters a,c. a,

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that the estimated parameters θˆ drift away when the full-rank period is over at time t = 100. From Fig. 11(b) we can see that dE(θˆ ) = −E(xe) does not converge to zero when the full-rank dt period is over. To suppress the estimation fluctuation caused by the noise, it is suitable to use mean filters. Here we introduce the following filter t θˆ (s) ds . (19) θˆ = 0 t It is clear to see from Fig. 9(d) that through the introduction of the filter (19), the unknown parameters a, b, and c can be identified with high accuracy even in the environment of large random noise. Note that the condition in Result 1 is equivalent to the conditions of PE and LI in Ref. [12]. The conditions in Results 2 and 3 are novel to analyze the convergence of the estimated parameters. V. CONCLUSION

FIG. 11. (Color online) The simulation result of the M-G system in the presence of noise when the full rank period of the Gram matrix is over at t = 100. (a) The curves of the state x, (b) the curve of xe, and (c) the curve of parameters estimation result θˆ for parameters θ.

process of aˆ and cˆ for system (12) when b(t) changes from 28 to 0 at t = 80, where the uniformly distributed random noise with amplitude ranging from −10 to 10. Figures 10(a) and 10(b) show the simulation results of x1 ,y1 ,z1 and (y1 − x1 )e1 ,z1 e3 ; ˆ cˆ drift away Fig. 10(c) shows that the estimated parameters a, when the full-rank period is over. From Fig. 10(b), we can see that E[(y1 − x1 )e1 ] and E(z1 e3 ) do not converge to zero when ˜ θ) the full-rank period is over. Due to dE( = −E[F T (x)e(t)], we dt

Many interesting problems can be transformed into the adaptive parameters identifications synthesis issue. Therefore, it is important to study the conditions which guarantee that the estimated parameters converge to the true values. In this paper, we systematically investigate the adaptive synchronization and parameter estimation problem. Correspondingly, we give some novel sufficient conditions to ensure the parameters convergence. The results of this paper could be extended to solve other problems about adaptive control, synchronization, and structure identification of dynamical networks. ACKNOWLEDGMENTS

θ) have dE( = E[F T (x)e(t)] does not converge to zero, and this dt is the main reason why the estimated parameters drift away (if the computation is continued) when the full-rank period is over. The simulations on the M-G system and time-delay logistic system show similar results with the above example. Figure 11 shows the estimation process of θˆ for the M-G system in the presence of uniformly distributed random noise with amplitude ranging from −3 to 3. Figures 11(a) and 11(b) show the simulation results of x and xe; Fig. 11(c) shows

The authors would like to thank Gang Hu for useful discussions and all the anonymous reviewers for their helpful advices. This work is supported by the National Natural Science Foundation of China (Grants No. 61070209, 60821001), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (FANEDD) (Grant No. 200951), the National Basic Research Program of China (973 Program) (Grant No. 2007CB310704), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 200800131028) and the Program for New Century Excellent Talents in university of the Ministry of Education of China (NCET-10-0239).

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