unscented kalman filter applied to noisy synchronization of rossler chaotic system
TRANSCRIPT
2011 3rd International Conference on Advanced Computer Control (ICACC 2011)
Unscented Kalman Filter Applied to Noisy Synchronization of Rossler Chaotic System
Komeil Nosrati Electrical Engineering Ferdowsi University Mashhad, Iran
Ali Shokouhi Rostami Electrical Engineering Ferdowsi University
Mashhad, Iran [email protected]
Abstract- Extended Kalman Filter (EKF) has been widely used as an important tool in practical applications to estimate
states of nonlinear systems. There are a number of deficiencies in EKF such as biased estimation, complexity in calculation and inefficacity in not being able to compute analytical derivatives affect its application in many fields. In this paper, Unscented Kalman Filter (UKF) is employed for estimation of the state variables of the chaotic dynamical system. The chaotic synchronization is implemented by the UKF in the presence of processing noise and measurement noise. The results of the simulation on the Rossler chaotic system by UKF and its
comparison with EKF show that the UKF has more accuracy and efficiency than EKF.
Keywords-Extended Kalman Filter; Unscented Kalman Filter; Chaos Synchronization; Rossler
I. INTRODUCTION
Chaotic behavior is an attractive phenomenon appearing in nonlinear systems and in the last three decades it has been received more attentions to study. A chaotic system is a nonlinear deterministic system and its behavior is complex and unpredictable. Two of the most important characteristics of a chaotic system are the sensitivity to the initial conditions and the variations of the system's parameters that make the chaotic synchronization problems much more complex and important. Pecora and Carroll [I] in their pioneering work addressed the synchronization of chaotic system using a drive-response conception. The idea is to use the output of the driving system to control the response system so that the trajectories of the response's outputs can synchronize those of drive system and they oscillate in a synchronized manner. Heretofore, many synchronization schemes have been developed such as inverse system approach [2], system approach [3], linear and nonlinear feedback control [4-6], and system decomposition approach [1,7]. Recently, many efforts have been made to show that the synchronization problem of chaotic systems could be solved through observer design approach [8-11], in which only the input and output information of drive system are used to construct part or all of the state information of drive system, and many beneficial
978-1-4244-8810-0111/$26.00 mOl I IEEE 378
Asad Azemi Electrical Engineering Penn State University
Pennsylvania, USA [email protected]
Naser Pariz Electrical Engineering Ferdowsi
University Mashhad, Iran [email protected]
methods have been developed. For example, several kinds of nonlinear observer design methods are summarized and their adaptations to chaotic synchronizations are discussed in [8] and in [12] a sliding-mode adaptive observer synchronization method for chaotic system is developed.
EKF as an optimal observer is a stochastic estimation scheme for estimating of nonlinear state and tracking applications [13]. In this method, Kalman filtering [14] is used to the linearized nonlinear function. The first order Taylor series expansions is applied to linearization.
Application of EKF to synchronization of chaotic systems is studied in [IS] and synchronization is obtained of transmitter and receiver dynamics in case the receiver is given via an extended Kalman filter driven by a noisy drive signal from the transmitter. However, a chief drawback of EKF is the error in function approximation because the EKF uses first order Taylor series for approximating the nonlinearities. So, large errors may be happened when it is used to systems with higher order nonlinearities. For overcoming the drawbacks associated with the approximation errors, many alternatives to EKF have been offered. UKF, as recently proposed by Julier and Uhlman [16], could in theory improve upon EKF for state estimation since linearization is avoided by an unscented transformation and at least second order accuracy is provided.
This last point is achieved by carefully choosing a set of sigma points, which captures the true mean and covariance of a given distribution and then passing the means and covariances of estimated states through a nonlinear transformation. As a result UKF is capable of estimating the posterior mean and covariances accurately to a high order [17].
In this paper, the UKF are applied for the synchronization of the chaotic system. The chaotic synchronization is implemented by the UKF in the presence of processing noise and measurement noise, and performance according to estimation error is evaluated in comparison with the EKF. Here, Rossler system is used for applying UKF and EKF in simulation. The results of the simulation on the Rossler chaotic system by UKF and its comparison with EKF show that the UKF has more accuracy and efficiency than EKF.
2011 3rd International Conference on Advanced Computer Control (ICACC 2011)
This paper is organized as follows: In Section 2, a brief review of Kalman filtering, principles and algorithms of EKF and UKF is presented. The results of simulation on Rossler system with using of UKF and EKF is provided in Section 3. Section 4 deals with concluding remarks.
II. KALMAN FILTER
The Kalman Filter (KF) is a recursive filtering tool which has been developed for estimating the trajectory of a system from a series of noisy and/or incomplete observations of the system's state which has the following specifications: the estimation process is formulated in the system's state space; the solution is obtained by recursive computation and it uses an adaptive algorithm which can be directly applied to stationary and non-stationary environment. In the Kalman filtering algorithm, every new estimate of the state is retrieved from the previous one and the new input in a way that only the previous estimated result need to be stored. Thus, the Kalman filter is more effective in computation than those which use all or considerable amount of the previous data directly in each estimation [18].
If the system is nonlinear, the Kalman filter cannot be applied directly, but two nonlinear Kalman filtering methods, namely, EKF and UKF are applied for stochastic nonlinear system estimation.
A. Extended Kalman Filter The Extended Kalman Filter (EKF) is a set of
mathematical equations which makes an estimate of the current state of a system using an underlying process model and then corrects the estimate using any available sensor measurements. Using this predictor-corrector mechanism, it approximates an optimal estimate due to the linearization of the process and measurement models [19]. Because of limited scope of this paper, the EKF is represented briefly. Therefore, we omit some theoretical considerations and present a more algorithmic description.
To illustrate the principle behind the EKF, Let a nonlinear system be represented by the following standard discrete time equations:
X k +1 = f (x k ) +W k Y k =Hkx k +V k
(1)
where kEN is discrete time and N denotes the set of
natural numbers. x k E jRLXl is the state, and Y k E jRMXl is
the measurement. The nonlinear mapping f 0 is assumed to
be continuously differentiable with respect to x k and H k is
a measurement matrix. Moreover, W k E jRJ>Xl and
v k E jRMXl are uncorrelated zero-mean Gaussian white
sequences and have the following characteristics:
379
To estimate signal, the error covariance matrix can be expressed as
(3)
where x k is the true value of the states.
Using the method of optimal linearization, the propagation of the error covariance matrix and the Kalman gain K can be expressed as
/tlk-l = Fk-l/t-llk-1Fk�1 + Qk
K = Pklk_1CT [ CPklk_1CT + Rk r1
Xk1k = Xk1k-1 + K [Yk - HkXklk_l ] /t1k = (1 - KC)/tlk_l
(4)
where F = of (x )1 is the Jacobian matrix and k -l ox . x =Xk_l
Xk1k-1 = f(xk-llk-I) . Fig.1 illustrates that how the Extended Kalman Filter
linearizes a nonlinear function around the mean of a Gaussian distribution, and then propagates the mean and covariance through this linearized model.
B. Unscented Kalman Filter The problem of propagating Gaussian random variables
through a nonlinear function can also be approached using another technique, namely the unscented transform (UT). Instead of linearization required by the EKF, a new approximate method UT is used in the Unscented Kalman Filter (UKF) [16].
A set of weighted sigma points is deterministically chosen for matching the sample mean and sample covariance of these points with those of a priori distribution. The nonlinear function is applied to each of these points sequently to yield transformed samples, and the predicted mean and covariance are calculated from the transformed samples as shown in Fig.2. This strategy typically both decrease the computational complexity, and at the same time increase estimate accuracy, yielding faster, more accurate results.
The fundamental difference between EKF and UKF lies in the way in which Gaussian random variables (GRV) are represented in the process of propagating through the system dynamics. Basically, the UKF captures the posterior mean and covariance of GRV accurately to the third order (in terms of Taylor series expansion) for any form of nonlinearity, whereas the EKF only achieves first-order accuracy. Moreover, since no explicit Jacobian or Hession calculations are necessary in the UKF algorithm, the computational complexity of UKF is comparable to EKF.
The algorithm for implementing the UKF can be summarized as follows [20].
2011 3rd International Conference on Advanced Computer Control (ICACC 2011)
Consider the nonlinear discrete-time system represented in (1) with same assumptions.
Step 1: The L-dimensional random variable x k _I with
mean X k-I and covariance Pk-I is approximated by sigma
points which are computed with the following equations:
y y=f(x)
x
Figure 1. The principle of the linearization in EKF and propagating the mean and covariance through this linearized model.
y y=f(x)
x
Figure 2. The principle of the unscented transform and the propagating of sigma points through a nonlinear function.
Xi,k-I =xk_1' i =0
Xi,k-I =iH + (a�LPk-l )i' i =1,2, . . . , L
X"k-I =ik_I- (a�LPk_I ) . ' i =L +1, . . . , 2L I-I.
(5)
where a E R is a tuning parameter denoting the spread
of the sigma points around ik_1 and (a�LPk_l)i is the ith
column of the matrix square root of LPk-l. The parameter
is often set to a small positive value. Step 2: Prediction. Each point is instantiated through the
process model to yield a set of transformed samples as (6).
Xi,k lk-I =!(X,k-I)' i =O,1, ... , 2L (6)
The predicted mean and covariance are computed as
2L
xk1k-1 = LW iXi,k lk-1 (7) i=O
380
where
2L
L [ Wi (Xi,klk-I -Xk1k-1 )(Xi,klk-I -Xk1k-1 fJ + Qk (8)
i=O
(9)
Step 3: Update. As the measurement equation is linear, measurement update can be performed with the same equations as the classical Kalman filter as (10).
Yk = HkXklk-1 � � T Pyy = Hk�lk-lHk + Rk � � T �y =�lk-IHk
� �-I K = PXyPyy
(10)
Step 4: Repeat steps 1 to 3 for the next sample. Clearly, the implementation of the UKF is extremely convenient, because Jacobian matrix is not needed to be evaluated which is necessary in the EKF.
III. SIMULA nON RESULS
In this section an example is presented to investigate the performance of the UKF in comparison with EKF. Here, Rossler chaotic system is used for applying UKF and EKF in simulation. This system is described in (11).
dx _I =ax +x dt 1 2
dx2 dt =-X1- X3
dx3 - =b-CX3+X2X3 dt
(11)
When a=O.l , b =0.1 and c =14 system behaves chaotically. As the materials given in section II for the design EKF and UKF are for discrete-time systems, while the applications are given for the continuous-time Rossler system, we have employed sampling method for numerical simulation in MATLAB with sampling time 0.001. For the simulation problem, process noise has been added to chaotic system states and the first chaotic state is employed for synchronization. Therefore, the output measurement matrix can be represented by (12).
Hk =[1 0 0] (12)
2011 3rd International Conference on Advanced Computer Control (ICACC 2011)
The initial conditions for the chaotic system, the EKF and UKF are
f x (0) = ( -1.0032 2.3545 1;(0)=(20 15 15/
-0.087t (13)
and the variances of the process and measurement noise used in the EKF and UKF are
R=0.2, Q=[0�8 0.�8 � l o 0 0.18 J
(14)
In Fig.3, the attractor of the chaotic Rossler system, using the aforementioned parameters, is illustrated.
The following results are obtained for chaotic synchronization of the Rossler system, using EKF and UKF methods. FigA - Fig.6 represent and its estimate, and its estimate and and its estimate respectively using UKF. These figures show that UKF is capable of achieving synchronization for the system and the synchronization is done with very low error and high speed.
In order to further investigate the efficacy of UKF vs. EKF, we have included plots of absolute estimation errors for the three states. Fig.7 - Fig.9 illustrate these results. The results show absolute estimation errors for the three states are significantly lower in the UKF than the EKF, after the first couple of seconds. In order to further illustrate the superior performance of UKF over EKF, for this simulation, we have calculated the mean squared error (MSE) for the three states using UKF and EKF (Table I). The MSE in state estimation is
(14) where x k (i) and; k (i) are the kth state variable and its
estimate at instant of i respectively. As it can be observed in Table I, the UKF method has more accuracy than EKF.
TABLE!. MSE (0-50 SEC) FOR THREE STATE VARIABLES
Synchronization method EKF UKF ············································1 ········· .. ............... ........... . . ....... .
0.0094 0.0097
4.2596
2 0289
3.5309
0.1554
381
� :::. � �
<:I;
Xl
y(1) .2() >«1)
Figure 3. Attractor of Rossler dynamical system.
30
25
20
15
10
5
0
·5
-10
-15
-20 0 5 W " � H m H • a �
Time (Sec)
Figure 4. First state of Rossler system and its estimate.
30�----�------�====�======�====� -- X2 25 -, _. - Estimation of X 2 (Xhat 2)
20
15
-15
_20 L-____ � ______ � ______ -L ______ -L ______ � o W � m �
Time (Sec)
Figure 5. Second state of Rossler system and its estimate.
2011 3rd International Conference on Advanced Computer Control (ICACC 2011)
20 �----�----�====�==========�
18 1 -_ -._- .. �S�imation of X 3 (Xhat 3) 1 16
14
12
4
: p..---_--.-J U . 20�----�1�0------�2�0------�3� 0-------4�0-------"50
Time (Sec)
Figure 6. Third state of Rossler system and its estimate.
0.1 ,-------�------__ ------__ ----�------_,
0.09
0.08
0.07
15 0.06
� 0.05
;:; 0.04
0.03
0.02
0.01
10 20 30 Time (Sec)
40 50
Figure 7. Absolute error in estimation of the first state.
6
2
i� i! il i � . � i I i I
I I
! � � I \ ,. . i i
\. i °O�--��1�O�----�2�O�-----3�O�--��40����50
Time (Sec)
Figure 8. Absolute error in estimation of the second state.
382
0.9
0.8
0.7 "" 0.6 1ii � 0.5
"" 0.4 ><:
0.3
0.2
0.1 0
I UKF I -- ·EKF
t.
.�, i 5 10 15 20 25 30 35 40 45 Time (Sec)
Figure 9. Absolute error in estimation of the third state.
IV. CONCLUSION
I I i � J i r
50
This paper showed the synchronization of Rossler chaotic system using the UKF. The synchronization of the state variables has been done with high accuracy and high speed. The UKF method has been compared with the EKF method to show the improvement of synchronization act and its growth in the performance in regard to accuracy in decreasing state variable estimation error. The calculation of absolute estimation error value for each state variable showed that its value in the UKF method is less than the EKF method. To more illustrate, the mean square error (MSE) for two methods has been compared. The results of simulation indicate that the UKF method is more accurate than EKF because of the lower MSE in the UKF method.
REFERENCES
(I] LM. Pecora and TL. Carroll, "Synchronization in chaotic systems," Phys Rev Lett, pp. 821-824, 1990.
[2] M. Hasle, "Synchronization of chaotic systems and transmission of information," Jnt J Bifur Chaos, pp. 647-659, 1998.
[3] Grassi and S. Mascolo, "Nonlinear observer design to synchronize hyperchaotic systems via a scalar signal," IEEE Trans Circ Syst I, pp. 1011-1014,1997 .
[4] Cc. Hwang and JY. Hsieh and RS. Lin, "A linear continuous feedback control of Chua's circuit," Chaos, Solitons & Fractals, pp. 1507-1515,1997.
[5] Lu and JA Lv, "Controlling uncertain Lu system using linear feedback," Chaos, Solitons & Fractals, pp. 127-33,2003.
[6] M. Chen and Z Han, "Controlling and synchronization chaotic Genesio system via nonlinear feedback control," Chaos, Solitons & Fractals, pp. 709-716,2003.
[7] L. Kocarev and U. Parlitz, "General approach for chaotic synchronization with application to communication," Phys Rev Lett, pp. 5028-5031, 1995.
[8] H. Nijmeijer and I. Mareels, "An observer looks at synchronization," IEEE Trans Circ Syst - I, pp. 882-90, 1997.
[9] L. T. Lu and T. S Hwa, "Adaptive synchronization of chaotic systems and its application to secure communications," Chaos, Solitons & Fractals, pp. 1387-1396,2000.
[10] M. Feki and B. Robert, "Observer-based chaotic synchronization in the presence of unknown inputs," Chaos, Solitons & Fractals, pp. 831-840,2003.
2011 3rd International Conference on Advanced Computer Control (ICACC 2011)
[II] S. Bowong and M. FM and H. Fotsin, "A new adaptive observerbased synchronization scheme for private communication," Phys Lett A, pp. 193-201,2006.
[12] A. Azemi and E. E. Yaz, "Sliding-Mode Adaptive Observer Approach to Chaotic Synchronization," Trans. of the ASME J. of Dynamic Systems, Measurement, and Control, pp. 758-765,2000.
[13] M. S. Grewal and A. P. Andrews, Kalman filtering: Theory and practice using MATLAB, 2nd Ed., John Wiley & Sons, 2001.
[14] R. E. Kalman, "A new approach to linear filtering and prediction problems," Trans. of the ASME J. of Basic Eng., pp. 35-45, 1960.
[15] C Cruz and H. Nijmeijer, "Synchronization through filtering," Int. J. of Bifurc. and Chaos, vol. 10, No. 4, pp. 763-775, 2000.
[16] S. J. Julier and 1. K Uhlman, "Unscented Kalman filtering and nonlinear estimation," Proc. of the IEEE. vol. 92, pp. 401-421,2004.
383
[17] C C Qu and 1. Hahn, "Process monitoring and parameter estimation via unscented Kalman filtering," Journal of Loss Prevention in the Process Industries pp. 703-709,2009.
[18] 1. C Feng and C K. Tse, "Reconstruction of Chaotic Signals with Applications to Chaos-Based Communication," Tsinghua University Press, 2007.
[19] 1. Joseph and Jr. La Viola, " A comparision of Unscented and Extended Kalman Filtering for Estimating Quatemion Motion ".
[20] E. A. Wan and R. V. D. Merwe, " The unscented Kalman filter for nonlinear estimation [A]. Adaptive systems for signal processing ", communications, and control symposium, pp. 153-158,2000.