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National Institute of Technology Hamirpur,. Himachal Pradesh, India [email protected]. Abstract—In this paper an observer based synchronization for a 4-D ...
2013 Annual IEEE India Conference (INDICON)

Observer Based Synchronization of 4-D Modified Lorenz-Stenflo Chaotic System Piyush Pratap Singh,a B K Royb Member IEEE

Himesh Handa

Department of Electrical Engineering, National Institute of Technology Silchar, Assam, India a [email protected] b [email protected]

Department of Electrical Engineering, National Institute of Technology Hamirpur, Himachal Pradesh, India [email protected]

Abstract—In this paper an observer based synchronization for a 4-D Modified Lorenz-Stenflo (MLS) chaotic system is reported. The proposed work uses a scalar coupling signal for synchronization of the 4-D MLS chaotic system in contrary with the conventional diagonal coupling signal. Based on Liapunov’s stability theory, an inequality is derived for the synchronization of proposed system. Simulation is used to demonstrate the effectiveness of the proposed scheme under the Nearestneighborhood coupling and the Global coupling configurations. Simulation results suggest that proposed scheme is working satisfactorily. Keywords—Observer, Chaotic System Synchronization, Linear Matrix Inequality (LMI), Modified Lorenz-Stenflo System, Complex Dynamical Networks.

I.

INTRODUCTION

In the last two and half decades, analysis of behavior in complex networks of interacting systems has been considerably entered. During this time, two parallel branches of research activities have emerged. On the one hand, in 1990, on synchronization of chaotic systems, the paper by Pecora and Carroll [1], opens the subway for the plethora of activity on the synchronization of coupled chaotic systems. In this type of synchronization, the state variables of individual systems converge towards each other, which is known as complete synchronization. On the other hand, different mathematical models have been proposed in order to describe various real world complex dynamical networks, such as the Erdos-Renyi (E-R) random-graph models [2], scale-free models [3],[4], small-world models [5], and so on. In 1998, Watts and Strogatz [5] proposed a model of a small-world network and in 1999. Albert and Barabasi [4] proposed a model of a scale-free network based on preferential attachment. Various networks have been studied, including regular networks [6], [7], small-world, and scale-free networks [4],[8] - [16], in addition to a large amount of work on the classic random-graph networks. The state observer approach has been applied to chaos synchronization between two chaotic systems, and can be used for secure communication [17], [18], Synchronization of

Chaotic Discrete-Time Systems [19], where only one scalar driving signal is used. Practical Implementation of Adaptive Impulsive Observer based Chaotic Synchronization Scheme [20], using the output of the drive system at discrete instant times. A maximum likelihood approach to state estimation of complex dynamical networks [21], with unknown noisy transmission channel has been formulated. Here, the technique of only one driving signal is used to a large-scale complex network, where multiple nodes (each node is a modified Lorenz-Stenflo system) are to be synchronized. Based on the Liapunov’s stability theory and the linear matrix inequality (LMI) technique [22], some criteria are established in the form of LMIs for exponentially synchronization of MLS chaotic systems. These LMIs are solved by existing LMI toolbox. The remaining part of this paper is organized as follows. In Section II, a dynamical network model is discussed. Design of observer for synchronization of such dynamical network and some criteria are then described in Section III. In Section IV, two typical dynamical networks with Global coupling and Nearest-neighbor coupling are simulated. In Section V, results and discussion to illustrate the effectiveness of the theoretical results and validate the criteria derived in the paper. Finally, some concluding remarks are given in Section VI. II.

DYNAMICAL NETWORK MODEL DISCRIPTION

A dynamical network consisting of N linearly coupled and identical nodes are considered. Each node is a ndimensional dynamical system. The proposed dynamical network model [23] is described as 1,2, …

1

,

Where is a state variable corresponding to the node , ,…, . , 1,2, , , , defind as is the scalar output of node . , is observer gain matrix, in , is order to achieve synchronization. , given as coupling configuration matrix which represents topological structure and coupling strength of the network. If there is a connection between node and node , and then,

978-1-4799-2275-8/13/$31.00 ©2013 IEEE

0, otherwise, elements of matrix are defined by

0, and then diagonal 1,2, …

2

,

is The coupling configuration matrix symmetrical and irreducible. This is due to assumption that there is no isolate cluster in the network. By the dynamics of (1) and (2), it can be concluded as 1,2, …

Theorem 1: Considering the dynamics of the network described by (5) and assumes that the Jacobian matrix is exponentially stable. If the given linear, time-varying systems described in (10), are exponentially stable:

3

From (1) or (3), it can be seen that only one scalar signal is needed for coupling between two directly connected nodes in the network, while the other network synchronization methods generally require state variables for coupling between any two directly connected nodes [11]. Let 1,2, …

Based on the stability theory, the given network (3), will lead to asymptotically synchronized if (9) is asymptotically stable about zero.

2,3, … ,

is eigenvalue of the coupling configuration matrix . The network described by (5) is exponentially synchronized. For proof, see [23] please. For a continuous nonlinear function it is assume that (11)

4

where, is defined as observer matrix. Thus, results in the form having linear combination of state variables at node . By the relation of (4) and (3), (1) can be written as 5

where, is a real matrix, linear function having

is a

where, is a constant. From (10) and (11), we obtain error dynamics as 2, … , .

DESIGN OF OBSERVER FOR SYNCHRONIZATION

The given network in (5) will lead to synchronization or asymptotical synchronization [6], if 0 where,

1,2, …

non(12)

,

III.

(10)

6

(13)

Remark 1: Define a Lyapunov function , where is a positive definite and symmetric matrix. As a result, can be obtained as:

satisfies the relation as 7

is known as equilibrium point, or a chaotic orbit. By the dynamics of (7), the stability of network (5) can be obtained about the chaotic orbit. Let the error is defined as, and linearizing the network (5) about , will get the error dynamics as 8 where 1,2, … … and matrix about the equilibrium point

is known as Jacobian . Then,

2 2 Hence, as per Lyapunov stability theory, system described in (13) is stable about equilibrium point. And dynamics (5) will synchronize exponentially. Theorem 2: On the basis of Remark 1, satisfies (11) and (12), and by proper selection of gain matrix so that 2

0

(14)

Then, , pair is observable. Therefore, network (5) is exponentially synchronized. … … … …

9

Lemma 1: Schur Complements [24], results in the form of nonlinear inequalities. Using these results, a nonlinear inequality can be converted into LMI.

, (14) can be written easily

2

0

(15)

Hence, can be calculated by calculating from the Eq. (MATLAB LMI toolbox), which will (15) using ensure that (13) is uniformly stable.

50 40 30 x3

Hence, using Lemma 1, and in the form of LMI as,

20

NUMERICAL EXAMPLE

IV.

10

There are so many systems that can be applied which fall under the non-linear (chaotic) behavior. Here, we explain the proposed scheme for Modified Lorenz-Stenflo (MLS) system.

0 400 100 50

0

Considering the familiarity with the Lorenz system and its behavior, the proposed MLS system, is derived from the LS system which is brought by Stenflo [1996], known as LorenzStenflo system. The MLS system differs from the LS system, provided it has a control parameter 0 along with introducing as a new state variable. Note that, if the control value (or parameter) d is zero, then fourth equation in (16) is decoupled from all the other ones. And then we can get the is replaced by state Lorenz system, provided variable.

0 -50 -200

x2

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(a)

40 20 0 x4

As described in [26], the modified system of LS system is a 4-D dynamical system is known as modified Lorenz-Stenflo (MLS) system. The 4-D chaotic dynamical system is given as:

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0

.

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(16) where, · denoted as signum function, and , , , are constant and real parameters. When, 1, 0.7, 26, 1.5, then the phase plane behavior of (16), is shown in Fig. 1(a) and Fig. 1(b). 0, 0, For equilibrium, (16) has three fixed points, which are

0, and , , and

0, system , given as:

0, 0, 0, 0 , ,

,

,

,

,

,

,

,

Fig. 1.Chaotic attractors (a) on the

,

, 1

,

.

Now, for the stability of these fixed points of the MLS nonlinear system is invariant under the complete change in the axis. Therefore, if appearance, with respect to , , , and , , , , both will be the solution of system (16).

, , plane

plane, (b) on the

,

,

Consider parameters value 1, 0.7, 26, 1.5. 0,0,0,0 , system (16) have thelinear For equilibrium relation and its Jacobian can be calculated as:

1

where,

for

(b)

1 0 0

0 0

0 0

1 1 0 1.5 26 1 0 0 0 0 0.7 0 1 0 0 1 From (3), a chaotic dynamical network (4-D system corresponding to each node), with nodes can be formed as:

V.

, , ∑

(17)

Remark 3: Achieving synchronization is much easier provided the nodes should be stable equilibrium or periodic orbits. A. Global Coupling In Global coupling, any two nodes in any type of networks can be connected directly. The global coupling matrix is given by 1

1

1 1

… …

1

1 1



1

(18) 1

L = [ 0.8193 1.0642 −0.2665 −0.0009]

T

In this case, synchronization in the network (17) can be done easily with a small observer gain matrix according to (15), implying that synchronization is very easy to be achieved. B. Nearest Coupling The nearest-neighbor coupling configuration consists of nodes arranged in a ring and coupled to the nearest neighbors. In this case, the coupling configuration matrix is given as

1 The eigenvalues of

1 2 0

1 0 1

It is noted that, if

REFERENCES

[2]

2

[3] [4] [5]

(19)

∞, implying a sufficient large

network, the second large eigenvalue 4 in (19) will tends to zero. In this case, a sufficiently large is needed, to ensure synchronization in network (17). Therefore, synchronization in network (17) is not easily attained if it consists large number of nodes.

CONCLUSIONS

Observer based synchronization of 4-D MLS system under Global and Nearest-neighbor coupling configuration has been done. The proposed observer based scheme for the 4-D MLS system is better than the conventional coupling because in this scheme of synchronization requires only one scalar signal coupling between any pair of directly connected nodes. Hence, this scheme is more practical and convenient over conventional diagonal coupling in real engineering applications. For the synchronization in the networks some conditions have been established on the basis of Liapunov’s stability theory. By LMI toolbox, suitable observer gains can be easily obtained. Simulation results illustrate the effectiveness of the proposed synchronization scheme. Finally, it is possible that synchronized 4-D MLS chaotic system can be used for the purpose of secure communication. This work is in progress and we will bring it soon in future.

[1]

matrix are

⎧ ⎫ 2 ⎛ pπ ⎞ ⎨ −4 sin ⎜ ⎟ , p = 0,1,…, ( N − 1) ⎬ N ⎝ ⎠ ⎩ ⎭

Two dynamical network configurations, Global and Nearest coupling have been simulated. Each node of the dynamical network is a 4-D MLS system. A four-node network for MLS system is considered. Synchronization of the MLS system under the Global coupling and Nearest-neighbor coupling configuration is shown in Fig. 2 and Fig. 3 respectively. Fig. 2(a) and 3(a) shows the behavior of the first state variable of the first node. Fig. 2(b) and 3(b) shows synchronization of the first state variables between nodes 1 and 2. Similarly, Fig. 2(c) and Fig. 3(c) & Fig. 2(d) and Fig. 3(d) shows synchronization of the first state variables between nodes 1 and 3 & nodes 1 and 4 respectively. Similarly, it has been done for rest of all the three state variables and results for each state variable have been synchronized properly. But here results are given only for first state variable between any nodes for the sake of simplicity. Proposed synchronization scheme reduces the large capacity of connection channel in comparison to conventional diagonal coupling. VI.

0, while rest Coupling matrix has a one eigen value as, 1 for 2, 3, … , . By eigenvalues are, choosing 0.2 and 1 0 0 0 for Modified Lorenz-Stenflo system, , pair is observable. Using MATLAB LMI toolbox, Observer gain matrix is calculated as

2 1

RESULTS AND DISSCUSSION

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[19] G. Grassi, D. Cafagna, P. Vecchio and Damon A. Miller, “A New Scheme to Synchronize Chaotic Discrete-Time Systems via a Scalar Signal,” IEEE 55’th International midwest symposium on Circuits and Systems (MWSCAS), pp. 654-657, 2012. [20] M. Ayati, and H. Khaloozadeh, “Practical Implementation of Adaptive Impulsive Observer based Chaotic Synchronization Scheme,” IEEE International Conference on System Science and Engineering (ICSSE), vol. 44, no. 10, pp. 367-372, June 2011. doi: 10.1109/ICSSE.2011.5961930 [21] H. Zhu, and Henry Leung, “A maximum likelihood approach to state estimation of complex dynamical networks with unknown noisy transmission channel,” IEEE International Symposium on Circuits and Systems (ISCAS), vol. 44, no. 10, pp. 2521-2524, May 2013. doi: 10.1109/ISCAS.2013.6572391 [22] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, “Linear Matrix Inequalities in System and Control Theory,” SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics, vol. 15, Philadelphia, PA, 1994. [23] G. P. Jiang, W. Kit-Sang Tang, and G. Chen, “A State-Observer-Based Approach for Synchronization in Complex Dynamical Networks,” IEEE Transactions On Circuits And Systems, vol. 53, no. 12, pp. 2739-2745, December 2006. [24] R. H. Knudsen, Linear Matrix Inequality for Robust Control, Mechanical and Aerospace Department, University of New York, 2006, pp. 62. [25] W. Tucker, and E. Tommy, A numerical study of the Lorenz and Lorenz-Stenflo systems, Ph.D. Thesis, KTH Engineering Sciences, Stockholm, Sweden, 2005. [26] L. Shan, Z. Liu, and Z. Wang “A New MLS Chaotic System and its Backstepping Sliding Mode Synchronization Control” Journal of Computers, vol. 5, no. 3, pp. 456-463, March 2010.

M. Barahona, and L. M. Pecora, “Synchronization in small-world systems,” Phys. Rev. Lett., vol. 89, no. 5, pp. 054-101,Washington DC, USA, 2002. P. M. Gade, “Synchronization of oscillators with random nonlocal connectivity,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat., Interdiscip. Top., vol. 54, no. 1, pp. 64-70,Washington DC, USA, 1996. P. M. Gade, and C. K. Hu, “Synchronous chaos in coupled map lattices with small-world interactions,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat., Interdiscip. Top., vol. 62, no. 5, pp. 6409-6413, 2000. X. Wang, and G. Chen, “Complex network: Small-world, scale-free, and beyond,” IEEE Circuits Syst. Mag., vol. 3, no. 2, pp. 6-20, February 2003. X. Wang, and G. Chen, “Synchronization in scale-free dynamical networks: Robustness and fragility,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 49, no. 1, pp. 54-62, January 2002. X. Wang, and G. Chen, “Synchronization in small-world dynamical networks,” Int. J. Bifurc. Chaos, vol. 12, no. 1, pp. 187-192, 2002. J. Lü, X. Yu, and G. Chen, “Chaos synchronization of general complex dynamical networks,” Physica A, vol. 334, no. 1-2, pp. 281-302, 2004. J. Lü, X. Yu, G. Chen, and D. Cheng, “Characterizing the synchronizability of small-world dynamical networks,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 4, pp. 787-796, April 2004. X. Li, and G. Chen, “Synchronization and desynchronization of complex dynamical networks: An engineering viewpoint,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 50, no. 11, pp. 1381-1390, November 2003. T. L. Liao, and N. S. Huang, “An observer-based approach for chaotic synchronization with application to secure communications,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 46, no. 9, pp. 11441150, September 1999. H. Nijmeijer, and I. M. Y. Mareels, “An observer looks at synchronization,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 44, no. 10, pp. 882-890, October 1997.

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Fig. 2.Synchronization of the MLS system under the Global coupling configuration. (a) The behavior of the first state variable of the first node. (b) Synchronization of the first state variables between nodes 1 and 2. (c) Synchronization of the first state variables between nodes 1 and 3. (d) Synchronization of the first state variables between nodes 1 and 4.

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Fig. 3.Synchronization of the MLS system under the Nearest-neighbor coupling configuration. (a) The behavior of the first state variable of the first node. (b) Synchronization of the first state variables between nodes 1 and 2. (c) Synchronization of the first state variables between nodes 1 and 3. (d) Synchronization of the first state variables between nodes 1 and 4.