Synchronization in complex networks of phase oscillators: A survey <s> Consensus, Contraction, & Convexity <s> We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for all-to-all networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value, the synchronized state is locally asymptotically stable, resulting in convergence of all phase differences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We further explain the behavior of the system as the number of oscillators grows to infinity. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Consensus, Contraction, & Convexity <s> We study the stability properties of linear time-varying systems in continuous time whose system matrix is Metzler with zero row sums. This class of systems arises naturally in the context of distributed decision problems, coordination and rendezvous tasks and synchronization problems. The equilibrium set contains all states with identical state components. We present sufficient conditions guaranteeing uniform exponential stability of this equilibrium set, implying that all state components converge to a common value as time grows unbounded. Furthermore it is shown that this convergence result is robust with respect to an arbitrary delay, provided that the delay affects only the off-diagonal terms in the differential equation. <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Consensus, Contraction, & Convexity <s> We study a simple but compelling model of network of agents interacting via time-dependent communication links. The model finds application in a variety of fields including synchronization, swarming and distributed decision making. In the model, each agent updates his current state based upon the current information received from neighboring agents. Necessary and/or sufficient conditions for the convergence of the individual agents' states to a common value are presented, thereby extending recent results reported in the literature. The stability analysis is based upon a blend of graph-theoretic and system-theoretic tools with the notion of convexity playing a central role. The analysis is integrated within a formal framework of set-valued Lyapunov theory, which may be of independent interest. Among others, it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence, even for the simple models discussed in the present paper. <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Consensus, Contraction, & Convexity <s> Two related problems are treated in continuous time. First, the state agreement problem is studied for coupled nonlinear differential equations. The vector fields can switch within a finite family. Associated to each vector field is a directed graph based in a natural way on the interaction structure of the subsystems. Generalizing the work of Moreau, under the assumption that the vector fields satisfy a certain subtangentiality condition, it is proved that asymptotic state agreement is achieved if and only if the dynamic interaction digraph has the property of being sufficiently connected over time. The proof uses nonsmooth analysis. Second, the rendezvous problem for kinematic point-mass mobile robots is studied when the robots’ fields of view have a fixed radius. The circumcenter control law of Ando e [IEEE Trans. Robotics Automation, 15 (1999), pp. 818-828] is shown to solve the problem. The rendezvous problem is a kind of state agreement problem, but the interaction structure is state dependent. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Consensus, Contraction, & Convexity <s> Abstract Consensus problems have attracted significant attention in the control community over the last decade. They act as a rich source of new mathematical problems pertaining to the growing field of cooperative and distributed control. This paper is an introduction to consensus problems whose underlying state-space is not a linear space, but instead a highly symmetric nonlinear space such as the circle and other relevant generalizations. A geometric approach is shown to highlight the connection between several fundamental models of consensus, synchronization, and coordination, to raise significant global convergence issues not present in linear models, and to be relevant for a number of engineering applications, including the design of planar or spatial coordinated motions. <s> BIB005
The interest of the control community in oscillator networks (1) was initially sparked by BIB001 and BIB003 , who analyzed networks of identical oscillators as nonlinear extensions of the consensus protocol (5). Indeed, for zero natural frequencies ω = 0 n and for angles contained in an open semicircle θ ∈ Arc n (π), the dynamics (29) can be projected onto the real line via the local coordinate map ϕ : ] − π/2, π/2[ n → R n defined by x i = ϕ i (θ i ) = tan(θ i ). With this projection proposed by BIB003 , the dynamics (29) are rewritten as the consensus-type model where b ij (x) = a ij (1 + x 2 i )/(1 + x 2 j ) ≥ 0. In particular, for θ ∈ Arc n (γ) for some γ ∈ [0, π[, we have that b ij (x) ≥ a ij / sec(γ/2) > 0 is strictly positive for all {i, j} ∈ E. A similar viewpoint is taken by BIB001 , where the coupled oscillator model (29) is rewritten aṡ where In both formulations (30) and (31), the dynamics (29) are regarded as a consensus protocol (5) with strictly positive weights whose values are time-varying or statedependent. This interpretation is well defined provided that θ(t) ∈ Arc n (γ) for all t ≥ 0 and for some γ ∈ [0, π[. Different Lyapunov functions can be used to assure this boundedness, for example, the potential function U (θ) or standard quadratic Lyapunov functions used in consensus theory. Generally, the level sets of these Lyapunov functions are hard to characterize and provide poor estimates on the region of attraction. Another natural Lyapunov function is simply the length of the shortest arc containing all oscillators. This approach relies upon the contraction property, it has been developed for general nonlinear consensus systems, and it aims at showing that the convex hull of all states is decreasing, e.g., see BIB002 BIB003 BIB004 BIB005 . Recall the geodesic distance on S 1 and define the continuous function V : If all angles at time t are contained in an arc of length strictly less than π, then the arc length V (θ(t)) = max i,j∈{1,...,n} |θ i (t) − θ j (t)| is a Lyapunov function candidate for phase synchronization, see Fig. 11 . Intuitively, the oscillators θ (t) and θ r (t) at both boundaries are pulled towards their neighbors in the interior Arc n (V (θ(t)), and the Lyapunov function V (θ(t)) is non-increasing. The technical analysis is slightly more subtle since the function V (θ(t)) is continuous but not necessarily differentiable when the maximum geodesic V (θ(t)) θ ℓ (t) θ r (t) Fig. 11 . Illustration of the Lyapunov function candidate V (θ(t)) for angles in an open semicircle θ(t) ∈ Arcn(π). The oscillators at the boundaries of the arc containing all oscillators Arcn(V (θ(t)) are denoted by θ (t) and θr(t). distance is attained by more than one pair of oscillators. In summary, we state the following result, which follows from the analysis of nonlinear consensus protocols, see BIB004 , Theorem 3.6 and 3.7) and BIB003 , Theorems 1 and 2) for continuous and discrete-time results.