We consider observability properties of the periodic Toda lattice and show that the system is observable by measuring the impulse of one mass and the exponential of the distance of the same mass to a neighboring mass. Furthermore, we discuss coordinate transformations of the system and apply different local observer design methods to locally observe a specific solution of the periodic Toda lattice. All discussed local observer design methods show poor convergence properties, hence an observability gramian analysis has been performed which reveals the difficulty of the observer design problem for the periodic Toda lattice.

Synchronization of coupled oscillators is an important problem in the analysis and control of coupled dynamical systems. Loosely speaking, oscillators are synchronous if the solutions of all individual oscillators converge towards a common periodic solution. This requirement implies properties on the omega-limit set and the asymptotic phase of the oscillator network. Both properties are barely considered in literature. Here, we consider the class of so called Lyapunov oscillators and derive novel conditions that are sufficient for synchronization of the oscillator networks.

The paper proposes a synchronization mechanism in a set of nonlinear oscillators interconnected through a communication network. In contrast to many existing results, we do not employ strong, diffusive couplings between the individual oscillators. Instead, each individual oscillator is weakly forced by a linear resonator system. The resonator systems are synchronized using results from consensus theory. The synchronized resonator systems force the frequencies of the nonlinear oscillators to a constant frequency and thereby yield synchronization of the oscillators. We prove this result using the theory of small forcings of stable oscillators. This synchronization scheme allows for synchronization of nonlinear oscillators over uniformly connected communication graphs.

Consensus in bistable and multistable multi-agent systems (MAS) is investigated. The considered MAS consists of agents with nonlinear, bistable or multistable dynamics and local coupling. For this MAS, consensus conditions are presented for arbitrary large networks depending on the algebraic connectivity of the underlying graph. In contrast to most publications in the consensus literature, the considered MAS has only a finite number of discrete consensus points instead of a consensus subspace. The differences to classical consensus problems are discussed and the main result is illustrated in a simulation example.

A group coordination problem is studied, where different agents are modelled as double integrator systems. Each agent aims to minimize a convex optimization function on the velocity. The coordination objective, studied in this note, is to find distributed coordination laws, such that the agents achieve a common velocity, which is the minimizer of the sum of all objective functions, while achieving a certain inter-agent spacing. Two different fully distributed coordination algorithms are proposed and analyzed. The algorithms combine ideas from distributed optimization and group coordination.

We present conditions for a nonlinear single integrator multi-agent system to achieve a consensus on the derivatives of the agents` states, e.g. their speed, by comparing the state information, e.g. their position. This consensus property is even more surprising because the states themselves do not necessarily reach a consensus. We prove that consensus is reached even if the communication network between the agents introduces communication delays. As a special case, the considered model includes delay coupled Kuramoto oscillators with nonidentical natural frequencies. We apply our result to these oscillators and show that frequency entrainment is guaranteed for appropriate initial conditions.

We investigate synchronization in networks of nonidentical Kuramoto oscillators with delayed coupling. Similar to the undelayed case, two effects are of primary interest: frequency entrainment and phase locking. We provide delay-dependent lower bounds for the coupling gain to achieve frequency entrainment in the case of a network with all-to-all coupling and heterogeneous delays. The given bounds are determined explicitly by the system parameters. Furthermore, we show that phase locking is possible in general directed networks of nonidentical oscillators which contain a spanning tree if the delays satisfy a self-consistency condition. Networks with delays differ at this point notably from networks without delays, since phase-locking is not possible in undelayed nonidentical Kuramoto oscillator networks. The derivations for our results use tools from nonlinear stability analysis for retarded functional differential equations and give results for the regions of attraction of the considered equilibrium sets. In addition to the sufficient conditions for synchronization, we investigate the connection between synchronization and the existence of solutions of a necessary self-consistency condition.