Synchronization in networks of identical linear systems

Abstract

The paper investigates the synchronization of a network of identical linear state-space models under a possibly time-varying and directed interconnection structure. The main result is the construction of a dynamic output feedback coupling that achieves synchronization if the decoupled systems have no exponentially unstable mode and if the communication graph is uniformly connected. The result can be interpreted as a generalization of classical consensus algorithms. Stronger conditions are shown to be sufficient -but to some extent, also necessary -to ensure synchronization with the diffusive static output coupling often considered in the literature. synchronization to a common value. The convergence of such consensus algorithms has attracted much attention in the recent years. It only requires a weak form of connectivity for the communication graph [1,2,3,4,5].In the synchronization literature, the emphasis is on the individual dynamics rather than on the communication limitations: the communication graph is often assumed to be complete (or all-to-all), but in the absence of communication, the time-evolution of the systems' variables can be oscillatory or even chaotic. The system dynamics can be modified through the information exchange, and, as in the consensus problem, the goal of the interconnection is to reach synchronization to a common solution of the individual dynamics [6,7,8,9].Coordination problems encountered in the engineering world can often be rephrased as consensus or synchronization problems in which both the individual dynamics and the limited communication aspects play an important role. Designing interconnection control laws that can ensure synchronization of relevant variables is therefore a control problem that has attracted quite some attention in the recent years [10,11,12,13,14].The present paper deals with a fairly general solution of the synchronization problem in the linear case. Assuming N identical individual agents dynamics each described by the linear state-space model (A, B, C), the main result is the construction of a dynamic output feedback controller that ensures exponential synchronization to a solution of the linear systemẋ = Ax under the following assumptions: (i) A has no exponentially unstable mode, (ii) (A, B) is stabilizable and (A, C) is detectable, and (iii) the communication graph is uniformly connected. The result can be interpreted as a generalization of classical consensus algorithms, studied recently, corresponding to the particular case A = 0 [1,2]. The generalization includes the non-trivial examples of synchronizing harmonic oscillators or chains of integrators.The proposed dynamic controller structure proposed in this paper differs from the static diffusive coupling often considered in the synchronization literature, which requires more stringent assumptions on the communication graph. For instance, the results in the recent paper [15] assume a time-invariant topology. The paper also provides sufficient conditions for synchronization by static diffusive couplin...

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