PRACTICAL SYNCHRONIZATION IN COMPLEX NETWORKS OF
NON-IDENTICAL DYNAMICAL NODES ∗
L. CONTEVILLE
AND E. PANTELEY†
Key words. practical stability, synchronization, networked system, convergent systems, semipassivity
AMS subject classifications. 34D06, 37C75, 93D05
EXTENDED ABSTRACT. During the last years synchronization and control
of complex dynamical systems became a very popular topic which attracts attention of
researches from different fields of science and engineering. Behavior and synchronization of power grids and natural networks such as social, biological, neuronal networks
has been focus of particular attention.
In general, asymptotic synchronization of dynamical systems is defined as a the
property of convergence to zero of differences between solutions of the subsystems
or, alternatively synchronization can be viewed as convergence to zero of differences
between solutions corresponding to individual nodes and a solution to some nominal
system which corresponds to some type of averaged dynamics of the system, as it was
done for example in [4, 10].
There exist several approaches in the literature to analyze stability and attractivity of synchronized motion: for example, master stability functions and transverse
Lyapunov exponentials are used for local analysis of the synchronization([5, 6]), while
tools from incremental stability theory [7] or contraction theory [8, 9] are used for
global analysis to ensure stability of the synchronization manifold. It should be noted
here that, majority of results on synchronization concerns networks where all the
nodes have identical dynamics, there are few papers which consider networks with
heterogeneous nodes, see e.g. [10] and references therein.
In this paper we consider problem of synchronization of a network of non-identical
nodes and in doing so we follow an approach proposed in [8] for synchronization of
identical nodes and used e.g. in [11] for synchronization of a network of HindmarshRose neurons.
In particular, we consider a problem of practical synchronization of nonidentical subsystems connected via diffusive coupling. We assume that each subsystem is
described by ordinary differential equations of the form
(0.1)
ẋi = fi (x(xi ) + Bui ,
(0.2)
yi (x) = Cxi
i = 1, ...N
where xi ∈ Rn , ui , yi ∈ Rm are the state, the input and the output of the i-th
subsystem and we assume that the matrix CB is similar to a positive definite matrix
while the N subsystems (0.1) - (0.2) are coupled via diffusive output coupling
(0.3)
ui = −δ
N
X
γij (yi − yj ).
j=1
∗ This work was financially supported by the European Commission through the FP7 NoE HYCON2.
† Laboratoire des Signaux et systems, CNRS - Supelec - Universite Paris-Sud, France
(author@lss.supelec.fr.)
1
2
L. CONTEVILLE AND E. PANTELEY
where δ and γij are nonnegative scalars. Due to the nonsingularity of CB the systems
(0.1)-(0.2) can be rewritten in the form
ẏi = ai (yi , zi ) + CBui ,
żi = qi (yi , zi )
As in [8, 11] we assume that each of the subsystems is semi-passive and that z subsystems are non-critically convergent. In the case of identical nodes these conditions ensure asymptotic sychronization under the assumption that matrix Laplacian
Γ = Γ> = [γij ] has a single zero eigenvalue. In the case of non-indentical nodes we
introduce a nominal system
ẋm = f (xm ),
f (x) =
N
1 X
fi (x)
N i=1
and show that under additional assumptions on the gain δ, smallness of the terms
qi (d, zi ) − qj (d, zj ) and assumption that nominal system has a (semi) globally stable
attractor, the nodes will globally practically synchronize to the motion defined by xm .
REFERENCES
[1] Z. Q. Yang, Z. X. Liu, Z. Q. Chen, and Z. Z. Yuan. Controlled synchronization of complex network
with different kinds of nodes. Journal of Control Theory and Application, 6(1):1115, 2008.
[2] [21] Q. Song, J. Cao, and F. Liu. Synchronization of complex dynamical networks with nonidentical nodes. Physics Letters A, 374:544551, 2010.
[3] A. Pogromsky and T. Yu, Passivity based design of synchronizing systems, Int. J. Bifurcation
and Chaos, vol. 8, no. 2, pp. 295319, 1998.
[4] Z. Liu, Z. Chen, Controlled Synchronization of Complex Network with Nonidentical Nodes and
Time-Varying Delay, Journal of Computational Information Systems 7:4 (2011) 1046-1056
[5] L. M. Pecora and T. L. Carroll, Master Stability Functions for Synchronized Coupled Systems,
Phys. Rev. Lett. 80, 21092112, 1998.
[6] H. Fujisaka and T. Yamada, Stability Theory of Synchronized Motion in Coupled-Oscillator
Systems, Prog. Theor. Phys. Vol. 69, pp. 32-47, 1983.
[7] L. Scardovi, M. Arcak, and E.D. Sontag. Synchronization of interconnected systems with applications to biochemical networks: an input-output approach. IEEE Transactions Autom.
Control, 55:1367-1379, 2010.
[8] A. Pogromsky and H. Nijmeijer, Cooperative Oscillatory Behavior of Mutually Coupled Dynamical Systems, IEEE Trans. on circuit and systems - I, vol. 48, 2, 2001.
[9] E. M. Aylward, P. A. Parrilo, J.-J. E. Slotine, Stability and robustness analysis of nonlinear
systems via contraction metrics and SOS programming, Automatica Vol. 44, Issue 8, 2008,
21632170.
[10] Hamadeh, A.O.; Stan, G.-B.; Goncalves, J., Robust synchronization in networks of cyclic feedback systems, Proc, CDC 2008.
[11] E. Steur, I. Tyukin and H. Nijmeijer, Semi-passivity and synchronization of diffusively coupled
neuronal oscillators, Physica D, 238, 2009 .