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...

Abstract-In this paper we study robustness of consensus in networks of coupled single integrators driven by white noise. Robustness is quantified as the H2 norm of the closed-loop system. In particular we investigate how robustness depends on the properties of the underlying (directed) communication graph. To this end several classes of directed and undirected communication topologies are analyzed and compared. The trade-off between speed of convergence and robustness to noise is also investigated.

In this paper, we study the behavior of a network of N agents, each evolving on the circle. We propose a novel algorithm that achieves synchronization or balancing in phase models under mild connectedness assumptions on the (possibly time-varying and unidirectional) communication graphs. The global convergence analysis on the N -torus is a distinctive feature of the present work with respect to previous results that have focused on convergence in the Euclidean space.

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.

This paper provides synchronization conditions for networks of nonlinear systems. The components of the network (referred to as "compartments" in this paper) are made up of an identical interconnection of subsystems, each represented as an operator in an extended L 2 space and referred to as a "species". The compartments are, in turn, coupled through a diffusion-like term among the respective species. The synchronization conditions are provided by combining the input-output properties of the subsystems with information about the structure of network. The paper also explores results for state-space models, as well as biochemical applications. The work is motivated by cellular networks where signaling occurs both internally, through interactions of species, and externally, through intercellular signaling. The theory is illustrated providing synchronization conditions for networks of Goodwin oscillators.

Flocks of starlings exhibit a remarkable ability to maintain cohesion as a group in highly uncertain environments and with limited, noisy information. Recent work demonstrated that individual starlings within large flocks respond to a fixed number of nearest neighbors, but until now it was not understood why this number is seven. We analyze robustness to uncertainty of consensus in empirical data from multiple starling flocks and show that the flock interaction networks with six or seven neighbors optimize the trade-off between group cohesion and individual effort. We can distinguish these numbers of neighbors from fewer or greater numbers using our systems-theoretic approach to measuring robustness of interaction networks as a function of the network structure, i.e., who is sensing whom. The metric quantifies the disagreement within the network due to disturbances and noise during consensus behavior and can be evaluated over a parameterized family of hypothesized sensing strategies (here the parameter is number of neighbors). We use this approach to further show that for the range of flocks studied the optimal number of neighbors does not depend on the number of birds within a flock; rather, it depends on the shape, notably the thickness, of the flock. The results suggest that robustness to uncertainty may have been a factor in the evolution of flocking for starlings. More generally, our results elucidate the role of the interaction network on uncertainty management in collective behavior, and motivate the application of our approach to other biological networks.

Previously, we showed using a computational agent-based model that a group of animals moving together can make a collective decision on direction of motion, even if there is a conflict between the directional preferences of two small subgroups of "informed" individuals and the remaining "uninformed" individuals have no directional preference. The model requires no explicit signaling or identification of informed individuals; individuals merely adjust their steering in response to socially acquired information on relative motion of neighbors. In this paper, we show how the dynamics of this system can be modeled analytically, and we derive a testable result that adding uninformed individuals improves stability of collective decision making. We first present a continuous-time dynamic model and prove a necessary and sufficient condition for stable convergence to a collective decision in this model. The stability of the decision, which corresponds to most of the group moving in one of two alternative preferred directions, depends explicitly on the magnitude of the difference in preferred directions; for a difference above a threshold the decision is stable and below that same threshold the decision is unstable. Given qualitative agreement with the results of the previous simulation study, we proceed to explore analytically the subtle but important role of the uninformed individuals in the continuous-time model. Significantly, we show that the likelihood of a collective decision increases with increasing numbers of uninformed individuals.collective behavior | Kuramoto | coordinated movement E xplaining the ability of animals that move together in a group to make collective decisions requires an understanding of the mechanisms of information transfer in spatially evolving distributions of individuals with limited sensing capability (1-6). In groups such as fish schools and large insect swarms, it is likely that individuals can sense only the relative motion of near neighbors and may not have the capacity to distinguish a wellinformed neighbor from the less well informed (2, 3). Further, it is increasingly becoming recognized that the emergent intelligence of a collective may be more reliable than the intelligence provided by a few leaders or well-informed individuals (7-11). This result suggests a subtle but important role in collective decision making for those individuals that have no particular information or preference.In this paper we define and analyze a continuous-time dynamical system model to examine collective decision making in moving groups of informed and uninformed individuals that are limited to sensing the relative motion of neighbors and adjusting their steering in response. Informed individuals have a preference for one of two alternative directions of motion, whereas uninformed individuals have no preference. The preferences are representative of knowledge of the direction to a food source or of a migration route, etc. In the discrete-time model of ref. 1 there is no signaling, no identification of t...

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