Ulrich Münz

Consensus in Multi-Agent Systems With Coupling Delays and Switching Topology

The robustness of consensus in single integrator multi-agent systems (MAS) to coupling delays and switching topologies is investigated. It is shown that consensus is reached for arbitrarily large constant, time-varying, or distributed delays if consensus is reached without delays. This delay robustness holds under the weakest possible connectivity assumptions on the underlying graph, i.e., as long as the graph is uniformly quasi-strongly connected and switches with a dwell-time. The proof is based on a contraction argument and allows to remove technical assumptions that were used in previous publications. Moreover, the result also applies to non-scalar single integrators, an important extension toward consensus and rendezvous in higher dimensions.

Delay-dependent rendezvous and flocking of large scale multi-agent systems with communication delays

We study the stability of multi-agent system (MAS) formations with delayed exchange of information between the agents. The agents are described by second order systems. They communicate via a symmetric connected communication topology with constant, heterogeneous, symmetric delays between any two neighboring agents. We consider two different tasks for the MAS: rendezvous, where all agents meet at an arbitrary point, and flocking, where all agents reach a given formation and move in a predefined direction. Therefore, we propose a decentralized control algorithm with position coupling gains kji. We prove that the MAS achieves rendezvous for any constant delay if the communication topology is connected and the coupling gain is sufficiently small. For larger gains, rendezvous and flocking are delay-dependent, i.e., they are reached for any delay smaller than a bound which depends on kji. Thereby, the controllers can be tuned in a totally decentralized fashion, i.e., only based on the communication delays to their neighbors and not considering the delays in the rest of the network. For the analysis, we use both frequency and time domain methods to prove delay-independent and delay-dependent rendezvous and flocking, respectively.

Delay Robustness in Non-Identical Multi-Agent Systems

We investigate the robustness of consensus protocols on sensor networks to different types of feedback delays. We provide set-valued conditions for consensus of linear Multi-Agent Systems (MAS) with nonidentical agent dynamics and heterogeneous delays. These conditions consider different feedback delay configurations and are robust and scalable to unknown, arbitrary large topologies and unknown but bounded self-delays and communication delays. Moreover, we provide consensus conditions for relative degree two MAS with different feedback delays in order to illustrate the applicability of these set-valued conditions.

Consensus reaching in multi-agent packet-switched networks with non-linear coupling

An important task for multi-agent systems (MAS) is to reach a consensus, e.g. to align their velocity vectors. Recent results propose appropriate consensus protocols to achieve such tasks, but most of them do not consider the effect of communication constraints such as the presence of time-delays in the exchange of information between the agents. In this article, we provide conditions for a non-linear, locally passive MAS of arbitrary size to reach a consensus, when the agents communicate over a packet-switched network that is characterised by a given topology. Both the cases of constant and switching topologies are considered. The nature of the communication channel imposes constraints that are modelled using stochastic delays of arbitrary distribution. We first embed this model in another, distributed but deterministic delay model and provide conditions for the error introduced by this simplification. In our main result, we provide conditions for the locally passive MAS with distributed delays to reach a consensus. In the case of a fixed topology, the underlying directed graph has to contain a spanning tree. In the case of a switching topology, only the union graph of all graphs that persist over time is required to contain a spanning tree. These conditions are independent of the distribution and the size of the packet delays. To show attractivity of the consensus set, we use an invariance principle for systems described by functional differential equations based on an appropriate Lyapunov–Razumikhin function. This methodological approach is the main contribution of this work and can also be applied to other consensus problems with delays. We illustrate our results by numerical simulations showing synchronisation of non-linear Kuramoto oscillators over a digital network.

Motivation and Learning Progress Through Educational Games

In engineering, we are often faced with the problem of how to teach complex theoretical material to students who are mainly interested in solving practical problems. This gap between the application-oriented expectation of the learner and the theory-focused material chosen by the lecturer may end up causing high barriers for the learning performance of the students. This becomes obvious if we recall that motivation is a cornerstone for good learning. One way to close this gap between theory and practice is educational games. In this paper, we show how educational games can exemplarily help to motivate and teach undergraduate university students in a basic automatic control course. The success of this approach can, for example, be seen from the course evaluations and feedback from our students.

Stability Analysis of Time-Delay Systems With Incommensurate Delays Using Positive Polynomials

We present a new stability condition for linear time-delay systems (TDS) with multiple incommensurate delays based on the Rekasius substitution and positive polynomials. The condition is checked by testing the positivity of multivariate polynomials in certain domains. For this purpose, we propose two alternative algorithms based on linear programming and sum of squares methods, respectively. The efficiency and accuracy of the algorithms is compared in an example to alternative stability conditions taken from the literature.

Model predictive control of constrained nonlinear time-delay systems

This paper proposes a model predictive control scheme for a class of constrained nonlinear time-delay systems with guaranteed closed-loop asymptotic stability. Asymptotic stability of the closed-loop is guaranteed by utilizing an appropriate terminal cost functional and an appropriate terminal region. A novel structured procedure is derived to determine the terminal cost and the terminal region offline. For this purpose, a combination of Lyapunov-Krasovskii and Lyapunov-Razumikhin arguments is used to compute a locally stabilizing controller. The resulting conditions are formulated in terms of linear matrix inequalities.

Generalized Nyquist consensus condition for high-order linear multi-agent systems with communication delays

The generalized Nyquist criterion is used to derive necessary and sufficient consensus conditions for large high-order linear multi-agent systems (MAS) with heterogeneous communication delays. The new consensus conditions are set-valued graphical conditions, from which inequality conditions are derived for single integrator MAS. The obtained conditions provide a unification and generalization of several results from the literature. Moreover, convergence rate conditions for single integrator MAS with delays are derived.

Frequency synchronization and phase agreement in Kuramoto oscillator networks with delays

We investigate frequency synchronization and phase agreement in networks of non-identical Kuramoto oscillators with delayed coupling. We provide delay-dependent lower bounds for the coupling gain so that a network with all-to-all coupling and constant heterogeneous delays achieves synchronization. The given bounds are determined explicitly by the system parameters. Furthermore, we show that frequency synchronization and phase agreement are possible in general directed networks of non-identical 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, because phase agreement is not possible in undelayed non-identical Kuramoto oscillator networks. In addition to the sufficient conditions for synchronization, we investigate the connection between phase agreement and the existence of solutions of a necessary self-consistency condition.

Nonlinear Multi-Agent System Consensus with Time-Varying Delays

Abstract Most consensus protocols for Multi-Agent Systems (MAS) presented in the past do not consider communication constraints such as delays in the exchange of information between the agents. In this paper, we provide conditions for a nonlinear, locally passive MAS with time-varying communication delays to reach a consensus, using a recently presented method based on an invariance principle for Lyapunov-Razumikhin functions. We consider both the cases of fixed and switching topologies. In the case of a fixed topology, the underlying directed graph has to contain a spanning tree. In the case of a switching topology, only the union graph of all graphs that persist over time is required to contain a spanning tree.