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.

Introduction

In recent years, consensus, coordination and synchronization problems have been popular subjects in systems and control, motivated by many applications in physics, biology, and engineering. These problems arise in multi-agent systems with the collective objective of reaching agreement about some variables of interest.

In the consensus literature, the emphasis is on the communication constraints rather than on the individual dynamics: the agents exchange information according to a communication graph that is not necessarily complete, nor even symmetric or time-invariant, but, in the absence of communication, the agreement variables usually have no dynamics. It is the exchange of information only that determines the time-evolution of the variables, aiming at asymptotic synchronization to a common value. The convergence of such consensus algorithms has attracted much attention in recent years. It only requires a weak form of connectivity for the communication graph (Jadbabaie et al., 2003, Moreau, 2005, Moreau, 2004, Olfati-Saber and Murray, 2004).

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 (Hale, 1996, Pham and Slotine, 2007, Pogromsky, 1998, Stan and Sepulchre, 2007).

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 recent years (Nair and Leonard, 2008, Sarlette et al., 2007, Scardovi et al., 2008, Scardovi et al., 2007, Sepulchre et al., 2008).

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 system $\dot{x}=Ax$ under the following assumptions: (i) $A$ has no exponentially unstable modes, (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 $A=0$ (Moreau, 2005, Moreau, 2004). The generalization includes the non-trivial examples of synchronizing harmonic oscillators and chains of integrators. In turn, these models, are applicable to distributed coordination of interconnected mobile systems. Models of vehicles often require to take into account second-order dynamics that can be feedback linearized as double integrators. Furthermore clocks synchronization problems have been reduced to synchronization of double integrator models (Carli, Chiuso, Schenato, & Zampieri, 2008).

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 papers (Carli et al., 2008, Ren, 2008, Tuna, 2008) on synchronization of harmonic oscillators and double integrators, assume a time-invariant communication topology. The idea of designing dynamic controllers for synchronization of networked systems was recently proposed in Scardovi et al. (2007) and has been applied to stabilize planar and three-dimensional collective motion (Scardovi et al., 2008, Sepulchre et al., 2008).

The paper also summarizes several sufficient conditions for synchronization by static diffusive coupling and illustrates on simple examples that synchronization may fail under diffusive coupling when the stronger assumptions on the communication graph are not satisfied.

The paper is organized as follows. In Section 2 the notation used throughout the paper is summarized, some preliminary results are reviewed and the synchronization problem is introduced and defined. In Section 3 the main result is presented. In Section 4 we derive the discrete-time counterpart of the main result. Finally, in Section 5, two-dimensional examples are reported to illustrate the role of the proposed dynamic controller in situations where static diffusive coupling fails to achieve synchronization.