Reachability of Consensus and Synchronizing Automata

We consider the problem of determining the existence of a sequence of matrices driving a discrete-time consensus system to consensus. We transform this problem into one of the existence of a product of the transition (stochastic) matrices that has a positive column. We then generalize some results from automata theory to sets of stochastic matrices. We obtain as a main result a polynomial-time algorithm to decide the existence of a sequence of matrices achieving consensus.Comment: Update after revie

Sensing and Control in Symmetric Networks

In engineering applications, one of the major challenges today is to develop reliable and robust control algorithms for complex networked systems. Controllability and observability of such systems play a crucial role in the design process. The underlying network structure may contain symmetries -- caused for example by the coupling of identical building blocks -- and these symmetries lead to repeated eigenvalues in a generic way. This complicates the design of controllers since repeated eigenvalues might decrease the controllability of the system. In this paper, we will analyze the relationship between the controllability and observability of complex networked systems and graph symmetries using results from representation theory. Furthermore, we will propose an algorithm to compute sparse input and output matrices based on projections onto the isotypic components. We will illustrate our results with the aid of two guiding examples, a network with $D_4$ symmetry and the Petersen graph

The Observability Radius of Networks

This paper studies the observability radius of network systems, which measures the robustness of a network to perturbations of the edges. We consider linear networks, where the dynamics are described by a weighted adjacency matrix, and dedicated sensors are positioned at a subset of nodes. We allow for perturbations of certain edge weights, with the objective of preventing observability of some modes of the network dynamics. To comply with the network setting, our work considers perturbations with a desired sparsity structure, thus extending the classic literature on the observability radius of linear systems. The paper proposes two sets of results. First, we propose an optimization framework to determine a perturbation with smallest Frobenius norm that renders a desired mode unobservable from the existing sensor nodes. Second, we study the expected observability radius of networks with given structure and random edge weights. We provide fundamental robustness bounds dependent on the connectivity properties of the network and we analytically characterize optimal perturbations of line and star networks, showing that line networks are inherently more robust than star networks.Comment: 8 pages, 3 figure

Strong Structural Controllability of Systems on Colored Graphs

This paper deals with structural controllability of leader-follower networks. The system matrix defining the network dynamics is a pattern matrix in which a priori given entries are equal to zero, while the remaining entries take nonzero values. The network is called strongly structurally controllable if for all choices of real values for the nonzero entries in the pattern matrix, the system is controllable in the classical sense. In this paper we introduce a more general notion of strong structural controllability which deals with the situation that given nonzero entries in the system's pattern matrix are constrained to take identical nonzero values. The constraint of identical nonzero entries can be caused by symmetry considerations or physical constraints on the network. The aim of this paper is to establish graph theoretic conditions for this more general property of strong structural controllability.Comment: 13 page

Symmetry-Induced Clustering in Multi-Agent Systems using Network Optimization and Passivity

This work studies the effects of a weak notion of symmetry on diffusively-coupled multi-agent systems. We focus on networks comprised of agents and controllers which are maximally equilibrium independent passive, and show that these converge to a clustered steady-state, with clusters corresponding to certain symmetries of the system. Namely, clusters are computed using the notion of the exchangeability graph. We then discuss homogeneous networks and the cluster synthesis problem, namely finding a graph and homogeneous controllers forcing the agents to cluster at prescribed values.Comment: 7 pages, 4 figure

The Observability Radius of Networks

This paper studies the observability radius of network systems, which measures the robustness of a network to perturbations of the edges. We consider linear networks, where the dynamics are described by a weighted adjacency matrix and dedicated sensors are positioned at a subset of nodes. We allow for perturbations of certain edge weights with the objective of preventing observability of some modes of the network dynamics. To comply with the network setting, our work considers perturbations with a desired sparsity structure, thus extending the classic literature on the observability radius of linear systems. The paper proposes two sets of results. First, we propose an optimization framework to determine a perturbation with smallest Frobenius norm that renders a desired mode unobservable from the existing sensor nodes. Second, we study the expected observability radius of networks with given structure and random edge weights. We provide fundamental robustness bounds dependent on the connectivity properties of the network and we analytically characterize optimal perturbations of line and star networks, showing that line networks are inherently more robust than star networks