Mehran Mesbahi

Agreement over random networks

We consider the agreement problem over random information networks. In a random network, the existence of an information channel between a pair of elements at each time instance is probabilistic and independent of other channels; hence, the topology of the network varies over time. In such a framework, we address the asymptotic agreement for the networked elements via notions from stochastic stability. Furthermore, we delineate on the rate of convergence as it relates to the algebraic connectivity of random graphs.

Controllability of Multi-Agent Systems from a Graph-Theoretic Perspective

In this work, we consider the controlled agreement problem for multi-agent networks, where a collection of agents take on leader roles while the remaining agents execute local, consensus-like protocols. Our aim is to identify reflections of graph-theoretic notions on system-theoretic properties of such systems. In particular, we show how the symmetry structure of the network, characterized in terms of its automorphism group, directly relates to the controllability of the corresponding multi-agent system. Moreover, we introduce network equitable partitions as a means by which such controllability characterizations can be extended to the multileader setting.

On State-dependent dynamic graphs and their controllability properties

We consider distributed dynamic systems operating over a graph or a network. The geometry of the networks is assumed to be a function of the underling systems states, giving it a unique dynamic character. Certain aspects of the resulting abstract structure, having a mixture of combinatorial and system theoretic features, are then studied. The author explores an interplay between notions from extremal graph theory and system theory by considering a controllability framework for such state-dependent dynamic graphs.

On the rank minimization problem over a positive semidefinite linear matrix inequality

We consider the problem of minimizing the rank of a positive semidefinite matrix, subject to the constraint that an affine transformation of it is also positive semidefinite. Our method for solving this problem employs ideas from the ordered linear complementarity theory and the notion of the least element in a vector lattice. This problem is of importance in many contexts, for example in feedback synthesis problems, and such an example is also provided.

Edge Agreement: Graph-Theoretic Performance Bounds and Passivity Analysis

This work explores the properties of the edge variant of the graph Laplacian in the context of the edge agreement problem. We show that the edge Laplacian, and its corresponding agreement protocol, provides a useful perspective on the well-known node agreement, or the consensus algorithm. Specifically, the dynamics induced by the edge Laplacian facilitates a better understanding of the role of certain subgraphs, e.g., cycles and spanning trees, in the original agreement problem. Using the edge Laplacian, we proceed to examine graph-theoretic characterizations of the H2 and H∞ performance for the agreement protocol. These results are subsequently applied in the contexts of optimal sensor placement for consensus-based applications. Finally, the edge Laplacian is employed to provide new insights into the nonlinear extension of linear agreement to agents with passive dynamics.

On the controlled agreement problem

Our work examines the controlled agreement problem over a network of interconnected dynamic units. The agreement protocol has recently been a focus of a large number of research work in systems and control community. Most of the existing work in this area, however, consider the uncontrolled agreement protocol. In this work, we consider the controlled agreement problem and introduce algebraic and graph theoretic conditions for its controllability. We then proceed to provide a graphical interpretation of these controllability conditions. In addition, we explore the role of anchored vertex position in the information structure to improve the convergence properties of the controlled agreement protocol

Agreement in presence of noise: pseudogradients on random geometric networks

We consider the agreement problem over realizations of a (Poisson) random geometric network with noisy interconnections. The vertices of random geometric networks are assumed to be uniformly distributed on the unit square; an edge exists between a pair of vertices if the distance between them is less than or equal to a given threshold. Our treatment of the agreement problem in such a setting relies upon notions from stochastic stability. In this venue, we show that the noisy agreement protocol has a guaranteed convergence with probability one, provided that an embedded step size parameter meets certain constraints. These constraints turn out to closely related to the spectra of the underlying graph Laplacian. Moreover, we point out the ramifications of having noisy networks by establishing connections between rate of convergence of the protocol and the range threshold in random geometric graphs.

On the rank minimization problem and its control applications

We present several new results on the rank minimization problem over sets which are defined by linear matrix inequalities. We proceed to apply some of these results to propose a computational procedure for determining lower and upper bounds for the minimal-order dynamic output feedback which stabilizes a given linear time-invariant plant.

State-dependent graphs

We consider distributed dynamic systems operating over state-dependent dynamic graphs or networks. Various aspects of the resulting abstract structure, having a mixture of combinatorial and control theoretic features, are then studied. Along the way, we explore an interplay between notions from the theory of graphs (elementary, extremal, and random), and system theory (controllability, invariance, etc.).

On the Controllability Properties of Circulant Networks

This paper examines the controllability of a group of first order agents, adopting a weighted consensus-type coordination protocol over a circulant network. Specifically, it is shown that a circulant network with Laplacian eigenvalues of maximum algebraic multiplicity q is controllable from q nodes. Our approach leverages on the Cauchy-Binet formula, which in conjunction with the Popov-Belevitch-Hautus test, leads to new insights on structural aspects of network controllability.