Stochastic chaos synchronization using Unscented Kalman–Bucy Filter and sliding mode control

Available online at www.sciencedirect.com

Mathematics and Computers in Simulation 81 (2011) 1770–1784

Original article

Stochastic chaos synchronization using Unscented Kalman–Bucy Filter and sliding mode control Mahdi Heydari, Hassan Salarieh ∗ , Mehdi Behzad Center of Excellence in Design, Robotics, and Automation (CEDRA), Department of Mechanical Engineering, Sharif University of Technology, P.B. 11555-9567, Azadi Ave., Tehran, Iran Received 10 November 2009; received in revised form 4 September 2010; accepted 14 January 2011 Available online 22 February 2011

Abstract This paper presents an algorithm for synchronizing two different chaotic systems by using a combination of Unscented Kalman–Bucy Filter (UKBF) and sliding mode controller. It is assumed that the drive chaotic system is perturbed by white noise and shows stochastic chaotic behavior. In addition the output of the system does not contain the whole state variables of the system, and it is also affected by some independent white noise. By combining the UKBF and the sliding mode control, a synchronizing control law is proposed. Simulation results show the ability of the proposed method in synchronizing chaotic systems in presence of noise. © 2011 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Chaos synchronization; Unscented Kalman–Bucy Filter; Sliding mode; White noise

1. Introduction Chaos synchronization between two identical or different chaotic systems is an attracting field of study in nonlinear dynamics and chaos [5,8,9]. Pecora and Carrol introduced the idea of synchronization between two identical chaotic systems [9,10,31]. Various methods for chaos synchronization have been presented and applied theoretically and experimentally to many chaotic systems [12,21,22]. A basic configuration for chaos synchronization is the driveresponse pattern, where the response chaotic system must track the drive chaotic trajectory. In [4,15,23,24,28,32] synchronization of hyper-chaotic systems were investigated and a generalized method for chaos synchronization was proposed [4,28]. Active linear and nonlinear control methods have been used for chaos synchronization between two identical or non-identical systems [5,11,14,15,27,29,38,42,45,52]. Parametric adaptive control has been greatly used in chaos synchronization [38,45]. Besides, many techniques are investigated based on combination of observer and control systems [6,34,40] and applied for both chaos control and chaos synchronization. In [6] the Extended Kalman–Bucy Filter (EKBF) is used as an observer to estimate the states of the drive system. In [19] the problem of chaos synchronization is investigated using adaptive observer design and Lyapunov stability theory. Also some robust nonlinear methods such as variable structure [25,53], H∞ [1,42] and impulsive [41] robust controllers are used for chaos synchronization. Recently synchronization between two stochastic chaotic systems, i.e. two chaotic systems ∗

Corresponding author. Tel.: +98 21 6616 5586; fax: +98 21 6600 0021. E-mail address: [email protected] (H. Salarieh).

0378-4754/$36.00 © 2011 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2011.01.013

M. Heydari et al. / Mathematics and Computers in Simulation 81 (2011) 1770–1784

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perturbed by white noise, has been investigated and some control algorithms based on adaptive control [2,3,35,36,48] and modified sliding mode control [7,13,26,37] have been proposed. The Unscented Kalman Filter (UKF) developed by Julier et al. [16,18] is a good nonlinear state estimator like Extended Kalman Filter (EKF). Discrete-time form of the UKF is considered for nonlinear state estimation [17,18,20,46,47]. Continuous-time form of the UKF is developed in [39,44] which is called the Unscented Kalman–Bucy Filter (UKBF). An adaptive UKF formulation for orientation estimation of aircraft and UAV utilizing low-cost attitude and heading reference systems is developed in [33]. In [54] the UKF is used to estimate the parameters of a rotary inverted pendulum (RIP). In practice, it hardly occurs that both the drive and response systems have exactly the same configurations whose output signals are not influenced by measurement noises. In this paper, the sliding mode controller is used to synchronize the behavior of two different chaotic systems. The sliding mode control has been frequently used to control uncertain dynamical systems [51]. Here it is assumed that both drive and response systems have some mathematical uncertainties. In addition, the drive system used in this study and the output signal are affected by some independent white noises which are generated by differentiating Wiener processes or Brownian motions. In [49] it is shown that how the sliding mode control method can be utilized for controlling stochastic systems. In this paper the output signal or the measurement of the drive system is assumed to be incomplete. There are some works about combing the sliding mode control and a state estimator or observer to controlling uncertain systems with incomplete measurements [6,50]. Therefore a nonlinear estimator or observer along with a UKBF is designed to estimate all of the drive state variables. A modified sliding mode control algorithm is designed to synchronize the whole states of the response system to the stochastic states of the drive system. The method presented in this paper is applied to a Lur’e like and the Genesio chaotic systems as the drive and response systems. The main objective of the presented paper is extending the previous work of the authors about chaos synchronization in which a combination of the EKBF and sliding mode control were utilized for chaos synchronization between two different chaotic systems with bounded stochastic uncertainties [6]. All other mentioned previous works about chaos synchronization either consider complete measurement of the state variables instead of incomplete noisy state measurement or ignore the uncertainties and noisy excitations which affect on behavior of the system in real world applications. In present work the Extended Kalman–Bucy Filter (EKBF) of [6] is replaced by Unscented Kalman–Bucy Filter which has better performance than EKBF in state estimation of nonlinear dynamical systems. Besides, in present work, the stochastic uncertainty of system is modeled by a standard white noise produced by Wiener process or Brownian motion which is more accurate than bounded noise model utilized in [6]. Simulation results show effectiveness of the proposed method in chaos synchronization of two different uncertain chaotic systems with incomplete measurement in presence of white Gaussian noise. 2. Problem statement A system with following equation is considered as the drive system: x(n) = f (x, t) + L(t)α˙

(1-a) +

where x = = (x1 , x2 , . . . , xn ) ∈ R is the state vector, f : R × R → R is a nonlinear, Lipschitz and sufficiently smooth function, ␣ is a Brownian motion with bounded diffusion Qc (t) , |Qc (t)| ≤ Qcm . α˙ is a white Gaussian noise process with spectral density Qc (t). L(t) is a real bounded and continuous function, i.e. |L(t)| ≤ Lm and Lm > 0 and Qc (t) are assumed to be known. Indeed Eq. (1-a) is an Ito differential equation. In Ito calculus if α is a Brownian motion or Wiener process, sometimes the Ito differential form of a stochastic differential equation is written like an ordinary differential equation by dividing the differential form to dt. In this case α˙  dα/dt is considered as a white Gaussian noise which is only a notation in stochastic differential equation and does not have any sense in ordinary calculus. Eq. (1-a) is rewritten in the standard Ito stochastic form of: (x, x˙ , . . . , x(n−1) )

dx(n−1) = f (x, t)dt + L(t)dα

n

n

(1-b)

The measurement equation is: z = h (x, t) + V (t) β(t)

(2)

where z = (z1 , z2 , ..., zm ) is the measurement vector, ␤(t) is a vector of independent white noise process with bounded ˙ ...eig is the matrix eigen-norm. h : Rn × R+ → spectral density Rc (t) , Rc (t)eig ≤ Rcm which is independent of α,

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Rm is a sufficiently smooth function, and V (t) is a continuous bounded function, i.e. ||V(t)||eig ≤Vm and Vm >0 is assumed to be known. The response system that must be controlled for synchronization is given by: y(n) = g(y, t) + b(y, t)u

(3)

where y = (y, y˙ , . . . , y(n−1) ) = (y1 , y2 , . . . , yn ) ∈ Rn is the state vector, u ∈ R is the control variable of the system, and g : Rn × R+ → R and b : Rn × R+ → R are sufficiently smooth functions. The functions f, g and b have some ˆ It is also assumed that function b is positive definite uncertainties and their nominal values are denoted by fˆ , gˆ and b. which has a strictly positive lower bound bm : |g(y, t) − gˆ (y, t)| < G(y, t)   f (x, t) − fˆ (x, t) < F (x, t)

(4)

ˆ t) > bm (y, t) > 0 b(y, t), b(y,

(6)

(5)

Note that functions f and g may not be the same. The synchronization problem is to design a controller u which synchronizes the states of both the drive and the response systems, in such a way that the response trajectories follow the drive trajectories. 3. Unscented Kalman–Bucy Filter 3.1. Unscented transform (UT) The unscented transform is used when a random variable y as a nonlinear function of Gaussian random variable x is to be approximated by a Gaussian distribution. According to Julier and Uhlmann [16,17] and Julier et al. [18], the UT is based on the intuition that it is simpler to approximate a Gaussian distribution than to approximate an arbitrary nonlinear function. To define the UT at first the concept of “sigma points” should be defined. 3.1.1. Sigma points and theirs associated weights Consider an n-dimensional random variable x whose mean and covariance are x¯ and Pxx .2n + 1 sigma points associated to x are given by: x(0) = x¯ x(i) = x¯ +



x(i+n) = x¯ − where

√

(n + λ) Pxx , i = 1, ..., n



i

(8)



(n + λ) Pxx , i = 1, ..., n i

(9)

 √ (n + λ) Pxx i is the i th row or column of the matrix (n + λ) Pxx and the associated weights are defined as:

(m)

W0

λ (n + λ)

(10)

λ  (n + λ) + 1 − θ 2 + ϕ

(11)

=

1 , i = 1, ..., 2n {2 (n + λ)}

(12)

=

1 , i = 1, ..., 2n {2 (n + λ)}

(13)

=

(c)

W0 = (m)

Wi

(c)

Wi

(7)



where λ ∈ R is a scaling parameter and is defined as: λ = θ 2 (n + κ) − n

(14)

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The positive constants θ, ϕ and κ are used as parameters of the UKF method and in this paper they are chosen 1, 0 2n

(m or c) and 0 respectively. The weights can be positive or negative but they must satisfy Wi = 1. 0

Unscented transform: Suppose that the random variable y is calculated by a nonlinear function f as: y = f (x), f : Rn → Rn

(15)

UT is used to approximate the mean y¯ and covariance Pyy of y through following steps. (I) The transformed set of sigma points are calculated as:  y(i) = f x(i) , i = 0, ..., 2n

(16)

(II) The mean and covariance of y is approximated by: y¯ =

2n

(m) (i)

Wi

y

2n

(c)

(17)

0

Pyy =



Wi

 T y(i) − y¯ y(i) − y¯

(18)

0

(III) The cross covariance of x and y is computed by the following equation: Pxy =

2n

(c)

Wi



 T x(i) − x¯ y(i) − y¯

(19)

0

The UT can also be written in matrix form as follows:   √   √ √ Pxx − Pxx X = x¯ ... x¯ + c 0 Y = f (X)

(21)

y¯ = Ywm Pyy = YWY

(20)

(22) T

(23)

Pxy = XWY T

(24)

where X is the sigma points matrix, c = and vector wm and matrix W are defined as follows:   (0) T wm = Wm ... Wm(2n)      T   W = I − wm wm wm × diag Wc(0) ... Wc(2n) × I − wm wm wm θ 2 (n + κ) ,

(25) (26)

where I is the identity matrix and  diag(. . .) is a diagonal matrix. The unscented transform can be seen as a function (or functional) from (f, x¯ , Pxx ) to y¯ , Pyy , Pxy :   y¯ , Pyy , Pxy = UT (f, x¯ , Pxx ) (27) 3.2. Unscented Kalman Filter [38] The Unscented Kalman Filter (UKF) [17,18,47] is a discrete-time filtering algorithm, which utilizes the unscented transform for computing approximate solutions to the filtering problems of the form xk = fd (xk−1 , k − 1) + qk−1 yk = hd (xk , k) + rk ,

(28)

where xk ∈ Rn is the state vector, yk ∈ Rm is the measurement vector, qk−1 ∈ Rn is a Gaussian noise process whose mean is zero and its covariance matrix is denoted by Qk−1 , i.e. qk−1 ∼N (0, Qk−1 ) , and rk ∈ Rm is a Gaussian measurement

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noise, i.e. rk ∼N (0, Rk ) . The mean and covariance of the initial state x0 are x¯ 0 and Pxx0 , respectively. The Unscented Kalman Filter has two main steps: prediction and update. In the prediction step the predicted state mean, denoted by − xˆ − k and its covariance matrix, denoted by Pxxk are calculated by using the UT transform as:  −   xˆ k , P˜ xxk = UT fd , xˆ k−1 , Pxxk−1 (29) P − = P˜ xxk + Qk−1 In the update step, at first the predicted mean of the measurement and its covariance, which are denoted by yˆ k and Pyyk , are computed.    − yˆ k , P˜ yyk , Pxyk = UT hd , xˆ − k , Pxxk (30) Pyy = P˜ yyk + Rk Then using the following equations called the equations of UKF, the estimation of xk , which is shown by xˆ k , and its covariance Pxxk is computed. −1 Kk = Pxyk Pyy

  ˆk xˆ k = x− k + Kk yk − y

Pxxk =

− Pxx k

(31)

− Kk Pyy KkT

The initial conditions of the above recursive equations are xˆ k=0 = x¯ 0 and Pxx0 . Comparing to the EKF algorithm as a linear sub-optimal filter, the UKF has more accuracy and it can estimate the states of the system like a second order nonlinear estimator, hence it can deal with nonlinearity better than the EKF algorithm [16,17]. 3.3. Continuous-time Unscented Kalman–Bucy Filter (UKBF) For continuous time system: dx(t) = F (x(t), t)dt + L(t)dα(t)

(32)

with measurement equation: z(t) = h(x(t), t)) + V (t)β(t)

(33)

the Unscented Kalman–Bucy Filter equations, that is the continuous-time version of the Unscented Kalman Filter, are given by [39]:  −1 (34) K (t) = X (t) WhT (x (t) , t) V (t) Rc (t) V T (t) d xˆ (t) (35) = F (X (t) , t) wm + K (t) [z (t) − h (X (t) , t) wm ] dt dP (t) = X (t) WF T (X (t) , t) + F (X (t) , t) WXT (t) + L (t) Qc (t) LT (t) − K (t) V (t) Rc (t) V T (t) KT (t) (36) dt where X(t) is the sigma point matrix and is defined as:    √  √ √ P(t) − P(t) (37) X (t) = xˆ (t) ... xˆ (t) n×(m+1) + c 0 ˙ d␣(t) is the Wiener differential and ␤(t) is a white Gaussian noise. Qc (t) and Rc (t) are the spectral densities of α(t) and ␤(t). xˆ (t) is the estimation of x(t) and P(t) is the covariance of the estimation error, i.e. x(t) − xˆ (t). As mentioned in the previous section the initial condition of the UKBF Filter is the mean value of x(0) and its covariance matrix. 4. Observer design for chaotic drive system based on UKBF Consider again equations of the drive system and its measurement: x(n) = f (x, t) + L(t)α˙ or dx(n−1) = f (x, t)dt + L(t)dα

(38)

M. Heydari et al. / Mathematics and Computers in Simulation 81 (2011) 1770–1784

z = h (x, t) + V (t) β(t)

1775

(39)

In this section using the concept  of UKBF a nonlinear observer is designed to estimate the states of drive system, i.e. x(t) = x(t), x˙ (t), ..., x(n−1) (t) , by observing the z = (z1 (τ), z2 (τ), ..., zm (τ)) , τ ≤ t. Define: T  F (x, t) = x˙ , x¨ , ..., x(n−1) , f (x, t)

 0(n−1)×(n−1) 0(n−1)×1 ˜L(t) = 01×(n−1) L(t)

(40)

˙ T : n × 1 vector α˙ = [0· · ·0 α(t)]

 0(n−1)×(n−1) 0(n−1)×1 ˜ Qc = 01×(n−1) Qc (t) Eq. (38) is rewritten as: ˜ α˙ x˙ = F (x, t) + L(t)

or

˜ dx = F (x, t)dt + L(t)dα

(41)

According to (34)–(36) an Unscented Kalman–Bucy Filter is designed as a nonlinear observer to estimate the states of the drive system based on the measurement dynamics (39):  −1 K (t) = X (t) WhT (x (t) , t) V (t) Rc (t) V T (t)

(42)

d xˆ (t) = F (X (t) , t) wm + K (t) [z (t) − h (X (t) , t) wm ] dt

(43)

dP (t) ˜ c (t) L ˜ (t) Q ˜ T (t) − K (t) V (t) Rc (t) V T (t) KT (t) = X (t) WF T (X (t) , t) + F (X (t) , t) WXT (t) + L dt (44) where xˆ (t) is the estimate of x(t), P(t) is the estimation covariance, and X(t) is the sigma point matrix defined in (37). The initial condition for Eqs. (43) and (44) are the mean value of x(t) and the covariance matrix of x(t) at the first instance, i.e. t=0. If the matrix V (t) Rc (t) V T (t) is not invertible, one must use its pseudo-inverse instead of  −1 V (t) Rc (t) V T (t) . 5. Chaos synchronization To design a synchronizing control action, the error dynamics between the master and the slave systems is written as: x(n) − y(n) = f (x, t) + L(t)α˙ − g(y, t) − b(y, t)u

(45)

Regarding the function uncertainties denoted by Eqs. (4) and (5) the sliding mode control method with some modifications is used to obtain the control law u. To this end the sliding surface is defined as: S(t) 



d dt

+λ

n−1

e(t) =

n−1

n−1 n−1−m (m) Cm λ e (t) = 0,

e = y − x,

m=0

The differential of S(t) function satisfies the following equation:  n−1

n−1 d dS(t) = de(t) = Cn λn−1−m de(m) (t) +λ m=0 m dt

n−1 Cm =

(n − 1)! m!(n − 1 − m)!

(46)

(47)

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Substituting Eq. (45) into Eq. (47) results in: dS(t) =

n−1

n−1 n−1−m (m) Cm λ e (t) + de(n−1)

m=1

=

n−1

 n−1 n−1−m (m) Cm λ e (t) + (f (x, t) − g(y, t)) − b(y, t)

dt + L(t)dα

(48)

m=1

Since the above equation is a stochastic differential equation, S(t) is a stochastic process. The estimation error is denoted by ␦ as: δ = xˆ − x

(49)

The error between the estimated states, xˆ , and the drive states which is called the estimated synchronization error is defined by: eˆ = y − xˆ

(50)

Using the above mentioned definitions of error, an estimated sliding surface is defined based on the estimated synchronization error:  ˆ  S(t)

n−1

d +λ dt

eˆ (t) =

n−1

n−1 n−1−m (m) Cm λ eˆ (t) = 0

(51)

m=0

ˆ functions and considering the estimation error δ(t), one can write: Comparing the S(t) and S(t) ˆ + S(t) = S(t)

n−1

n−1 n−1−m (m) Cm λ δ (t)

(52)

m=0

ˆ is denoted by: The difference between the S(t) and S(t) ˆ = ε(t) = S(t) − S(t)

n−1

n−1 n−1−m (m) Cm λ δ (t)

(53)

m=0

Eq. (52) is written briefly as: ˆ + ε(t) S(t) = S(t)

(54)

ˆ = 0 construct two manifolds in the state space and estimated state space, It should be noted that S(t) = 0 and S(t) ˆ however S(t) and S(t) are only two functions of the states and estimated states respectively and they are not manifolds. The expressions in (53) and (54) define the difference between two functions not two manifolds which is used to design the controller. Using triangular inequality of norms the following relation will be established between ε(t) and δ(t) :  n−1  n−1 



 n−1 n−1−m (m)  n−1 n−1−m ε(t)s =  δ(t)s = Mδ(t)s Cm λ δ (t) ≤ Cm λ (55)   m=0

s

m=0

  1/2 where ε(t)s = E ε(t)T ε(t) is the mean square or L2 norm which is used for stochastic variables and as it is obvious M > 0 is a known positive constant. Since S(t) is a stochastic process, the conventional Lyapunov function 1 2 2 S (t) which is usually used to obtain the control law, should be substituted by: J(t) =

1  2  1 E S (t) = S(t)2s 2 2

(56)

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Similar to the conventional sliding mode control, to find a stabilizing control, one should try to satisfy a reaching condition as: ˙ < −θ |S(t)| J(t)

(57)

where θ > 0 is chosen arbitrarily. As it will be shown, due to the effects of random noise, the above conventional reaching condition could not be satisfied. Using the Ito derivative formula, the time derivative of the Lyapunov function J(t) is obtained as: (S(t)dS(t) + dS(t)dS(t)) ˙ = E J(t) dt      n−1

1 n−1 n−1−m (m) 2 = E S(t) Cm λ e (t) + (f (x, t) − g(y, t)) − b(y, t)u dt + L(t)dα + (L(t)dα) dt m=1   

n−1  

1 n−1 n−1−m (m) 2 ˆ Cm λ e (t) + (f (x, t) − g(y, t)) − b(y, t)u + L(t) = E S(t) + ε(t) dt m=1  

n−1  

1 n−1 n−1−m (m) ˆ + ε(t) Cm λ e (t) + (f (x, t) − g(y, t)) − b(y, t)u + L(t)2 = E S(t) dt m=1  

n−1

1 n−1 n−1−m (m) ˆ = E S(t) Cm λ e (t) + (f (x, t) − g(y, t)) − b(y, t)u dt m=1 

n−1 

1 n−1 n−1−m (m) (58) Cm λ e (t) + (f (x, t) − g(y, t)) − b(y, t)u + L(t)2 + E ε(t) dt m=1

Since L(t)2 is a positive function and the control effort cannot affect on it, the reaching condition of Eq. (57) could not be satisfied. Here a modified reaching condition is replaced which results in some steady state errors and certifies the convergence of synchronization error toward zero in L2 norm. As it is observed, j has three parts: 

n−1 

n−1 n−1−m (m) ˆ I1 = E S(t) Cm λ e (t) + (f (x, t) − g(y, t)) − b(y, t)u  I2 = E ε(t)

m=1

n−1



n−1 n−1−m (m) Cm λ e (t) + (f (x, t) − g(y, t)) − b(y, t)u

(59)

m=1

I3 = L(t)2 As mentioned, the third part I3 could not be affected by u. The control law can be designed easily to make the first part, I1 , negative. To this end it is enough to let

 n−1

1 n−1 n−1−m (m) ˆ u= fˆ (ˆx, t) − gˆ (y, t) + Cm λ eˆ (t) + Ksgn(S(t)) (60) bm (y, t) m=1

where K is a positive gain that will be defined later. Substituting u into I1 results in: 

n−1

1 n−1 n−1−m (m) ˆ I1 = E S(t) Cm λ e (t) + (f (x, t) − g(y, t)) − b(y, t) bm (y, t) m=1 

n−1

n−1 n−1−m (m) ˆ ˆ Cm λ eˆ (t) + Ksgn(S(t)) × f (ˆx, t) − gˆ (y, t) + m=1

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 ˆ = E S(t)

n−1

n−1 n−1−m (m) Cm λ δ (t) + f (x, t) − f (ˆx, t) + f (ˆx, t) − fˆ (ˆx, t) − g(y, t) + gˆ (y, t)

m=1

   

n−1 b(y, t) b(y, t) n−1 n−1−m (m) ˆ + 1− Cm λ eˆ (t) + fˆ (ˆx, t) − gˆ (y, t) − Ksgn(S(t)) bm (y, t) bm (y, t)

(61)

m=1

Since f is a Lipschitz function we have |f (x, t) − f (ˆx, t)| ≤ L1 x − xˆ 

(62)

where L1 is the Lipschitz coefficient. Manipulating some calculations on Eq. (61) results in

 n−1

   n−1 n−1−m (m) ˆ ˆ I1 = E S(t) C λ δ (t) + E S(t) f (x, t) − f (ˆx, t) + f (ˆx, t) − fˆ (ˆx, t) − g(y, t) + gˆ (y, t) m

m=1

  

n−1 b(y, t) n−1 n−1−m (m) ˆ ˆ + E S(t) 1 − Cm λ eˆ (t) + f (ˆx, t) − gˆ (y, t) bm (y, t) m=1         b(y, t) ˆ  (L1 δs + F (ˆx, t) + G(ˆy, t)) + S(t) ˆ  |1 ˆ ˆ  Mδs + S(t) ˆ Ksgn(S(t)) ≤ S(t) − E S(t) bm (y, t)    n−1    b(y, t)   n−1 n−1−m (m)  ˆ  − Cm λ eˆ (t) + fˆ (ˆx, t) − gˆ (y, t) − K S(t)    bm (y, t) 

(63)

m=1

Now if we choose K such that

    n−1   b(y, t)   n−1 n−1−m (m)   ˆ K > (M + L1 ) δe + F (ˆx, t) + G(ˆy, t) + 1 − C λ e ˆ (t) + f (ˆ x , t) − g ˆ (y, t)  + θ (64)  m  bm (y, t)   m=1

we have

  ˆ  I1 ≤ −θ S(t)

(65)

Note that the whole terms in the right hand side of Eq. (63) are available except δs .δs is the square root of the estimation error variance. Since the UKBF algorithm is locally stable, the variance of estimation error is bounded so we have  1/2 δs = var(δ)1/2 = E δT δ ≤Δ (66) where Δ > 0 is the error variance bound. Therefore by choosing K as     n−1     n−1 n−1−m (m) b(y, t)  ˆ (ˆx, t) − gˆ (y, t) + θ K = (M + L1 ) Δ + F (ˆx, t) + G(ˆy, t) + 1 − C λ e ˆ (t) + f m  bm (y, t)   m=1

The condition of Eq. (64) will be satisfied and hence Eq. (65) will be established. The second term of Eq. (59) satisfies the following inequality:

 n−1

   n−1 n−1−m (m) I2 = E ε(t) Cm λ δ (t) + E ε(t) f (x, t) − f (ˆx, t) + f (ˆx, t) − fˆ (ˆx, t) − g(y, t) + gˆ (y, t) m=1

 

 n−1 b(y, t) n−1 n−1−m (m) Cm λ eˆ (t) + fˆ (ˆx, t) − gˆ (y, t) + E ε(t) 1 − bm (y, t)

m=1

(67)

M. Heydari et al. / Mathematics and Computers in Simulation 81 (2011) 1770–1784

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  b(y, t) 2 2 ˆ − E ε(t) Ksgn(S(t)) ≤ Mδs + ML1 δs + F (ˆx, t) + G(y, t) bm (y, t)    

    n−1 n−1 n−1−m (m)  b(y, t)   ˆ δs ≤ M(1 + L1 )δ2s Cm λ eˆ (t) + fˆ (ˆx, t) − gˆ (y, t) − Ksgn(S(t)) + 1 −  bm (y, t)   m=1   ˆ δs + K 1 − sgn(S(t)) (68) So an upper bound for time derivative of the Lyapunov function (58) is obtained:     ˆ  + M(1 + L1 )Δ2 + K 1 − sgn(S(t)) ˆ J˙ ≤ I1 + I2 + I3 ≤ −θ S(t) Δ + L(t)2

(69)

The above inequality implies that the following region in the phase space is an attractive region.   ˆ   M(1 + L1 )Δ2 + K 1 − sgn(S(t)) Δ + L(t)2 S(t) ˆ ≤ (70) θ The trajectories of the system will converge toward the above mentioned region as time converges to infinity. ˆ = 0 has an exponentially stable dynamics for eˆ , by choosing sufficiently large values Since the sliding surface S(t) of θ, one can decrease the steady state error of eˆ = y − xˆ steadily. It must be noticed that the synchronization error e = y − x depends on the estimated synchronization error, i.e. eˆ , and the estimation error ␦. Eq. (70) implies that eˆ may become arbitrarily small, however the value of estimation error depends essentially on the filtering algorithm used for estimation. So by using more efficient filtering algorithms the synchronization error can be decreased more than presented method. 6. Simulation and results Here the synchronization algorithm is applied to an example to investigate its performance in stochastic chaos synchronization. The Lur’e dynamic system has been selected as the drive chaotic system. Dynamic equations of this system can be written as, x˙ 1 = x2 + d sin t.α˙ 1 (t) x˙ 2 = x3 + d sin t.α˙ 2 (t) x˙ 3 = a1 x1 + a2 x2 + a3 x3 + 12ϕ(x1 ) + d sin t.α˙ 3 (t) where



ϕ(x1 ) =

kx1

|x1 | < 1/k

sign(x1 )

(71)

(72)

otherwise

For d = 0, a1 = −7.4, a2 = −4.1, a3 = −1, and k = 3.6 the behavior of the system is chaotic [42,43]. In this example L(t) = dsint, and α˙ 1−3 (t) are white noise processes with spectral density Qc (t) . The Gensio chaotic system is chosen as the response system. The dynamic equation of the Genesio system is y˙ 1 = y2 y˙ 2 = y3 y˙ 3 =

b1 y1 + b2 y2 + b3 y3 + y12

(73) + (0.5 + |y1 |)u

For u = 0, b1 = −5.6, b2 = −2.74, b3 = −1.1 the dynamic behavior of the Genesio system is chaotic [30]. It is assumed that the measuring signal is x1 (t) which is perturbed by some noise, so the measurement equation is, z (t) = x1 (t) + V (t) β (t)

(74)

Besides, the nominal values of bi s are shown by bˆ i s and their values are set to bˆ 1 = −4.6, bˆ 2 = −3.24 and bˆ 3 = −2.1. ˆ t) and bm used in sections Based on Eqs. (71) and (73), functions f(x,t), F1 (x,t), L(t), V(t), h(x,t), g(y,t), b(x,t), gˆ (y, t), b(x, 4 and 5 are:

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x ,y

-5 0 5 0 -5 -10 0 10

u

x ,y

3 3

x ,y

0

2 2

1 1

5

Lur'e Genesio 5

10

15

20

5

10

15

20

5

10

15

20

10

15

20

0 -10 0 20 0 -20 0

5

Time(sec)

Fig. 1. Synchronization results between the Lur’e and the Genesio dynamic system. xi ’s are the states of the Lur’e system and yi ’s are the states of the Genesio system.

 f (x, t) = a1 x1 + a2 x2 + a3 x3 + 12φ(x1 ), F1 (x, t) = x2

x3

f (x, t)

T

, h(x, t) = x2 , L(t) = 0.2 sin t

V (t)

= 0.2 g(y, t) = b1 y1 + b2 y2 + b3 y3 + y12 , gˆ (y, t) = bˆ 1 y1 + bˆ 2 y2 + bˆ 3 y3 + y12 b(y, t) ˆ t) = 1 + 2 |y1 | , bm = 1, θ = 0.1, Rc = 0.04, Qc = 0.04 = 0.5 + |y1 | , b(y, Using the method presented in sections 4 and 5, the synchronization process between the Lur’e and the Genesio systems as drive and response systems is done. The results of synchronization are illustrated in Figs. 1–8. It can be seen that the tracking process and hence synchronization of the drive-response systems is properly achieved. Fig. 2 shows that the filtering process acts properly and the states of the system are estimated after about 1 s. Time series of the drive and response systems and the synchronization error are shown in Figs. 1 and 3. It is observed that despite the noise and other random uncertainties the estimated signal approach the actual signal. Figs. 2 and 4 show time series of the estimated states of drive system and the estimation error. The mean value of the synchronization error and its variance are illustrated in Figs. 5 and 6. Figs. 7 and 8 show the mean value and variance of the estimation error. The presented results prove that the synchronization error converges to zero in its mean value, and its variance converges to a small bounded region around zero. Therefore the time series of the synchronization error should converge to zero and fluctuate around it with small bounded amplitude. Finally, Fig. 9 shows the variance of synchronization errors by using EKBF/Sliding mode method [6]. Comparing the results of UKBF/Sliding mode method with the results of EKBF/Sliding mode method shows that the variances converge faster and to smaller values by using the novel presented method of this paper.

Fig. 2. State estimation of the Lur’e dynamic system using the UKBF method.

M. Heydari et al. / Mathematics and Computers in Simulation 81 (2011) 1770–1784

x1-y1

2 0

x2-y2

-2 0 4

5

10

15

20

5

10

15

20

10

15

20

0

x3-y3

-4 0 10 0

-10 0

5

Time(sec)

Fig. 3. Error of synchronization between the Lur’e and the Genesio dynamic systems using UKBF/Sliding mode method.

Fig. 4. Errors in state estimation of the Lur’e dynamic system using the UKBF method.

1

1

E(x -y )

0 -1

3

3

E(x -y )

2

2

E(x -y )

-2 0 1

5

10

15

20

5

10

15

20

10

15

20

0 -1 0 6 4 2 0 -2 0

5

Time(sec)

Fig. 5. Mean values of synchronization errors between the Lur’e and the Genesio dynamic systems using UKBF/Sliding mode method.

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1 0.5 0 2

5

10

15

20

5

10

15

20

10

15

20

1.5

2

2

E((x -y )2)

1

1

E((x -y )2)

2 1.5

1 0.5 0 5

3

3

E((x -y )2)

1782

0 0

5

Time(sec)

Fig. 6. Variances of synchronization errors between the Lur’e and the Genesio dynamic systems.

Fig. 7. Mean values of the state estimation errors of the Lur’e dynamic system using the UKBF method.

Fig. 8. Variances of the state estimation errors of the Lur’e dynamic system using the UKBF mode method.

2 2

E((x -y )2)

1 1

E((x -y )2)

M. Heydari et al. / Mathematics and Computers in Simulation 81 (2011) 1770–1784 3 2 1 0

3 3

5

10

15

20

25

30

35

40

5

10

15

20

25

30

35

40

5

10

15

20

25

30

35

40

2 1 0

E((x -y )2)

1783

4 2 0

Time(sec)

Fig. 9. Variances of synchronization errors between the Lur’e and the Genesio dynamic systems using EKBF/Sliding mode method [6].

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