On the Quality of Wireless Network Connectivity

Despite intensive research in the area of network connectivity, there is an important category of problems that remain unsolved: how to measure the quality of connectivity of a wireless multi-hop network which has a realistic number of nodes, not necessarily large enough to warrant the use of asymptotic analysis, and has unreliable connections, reflecting the inherent unreliable characteristics of wireless communications? The quality of connectivity measures how easily and reliably a packet sent by a node can reach another node. It complements the use of \emph{capacity} to measure the quality of a network in saturated traffic scenarios and provides a native measure of the quality of (end-to-end) network connections. In this paper, we explore the use of probabilistic connectivity matrix as a possible tool to measure the quality of network connectivity. Some interesting properties of the probabilistic connectivity matrix and their connections to the quality of connectivity are demonstrated. We argue that the largest eigenvalue of the probabilistic connectivity matrix can serve as a good measure of the quality of network connectivity.Comment: submitted to IEEE INFOCOM 201

Matrix Design for Optimal Sensing

We design optimal $2 \times N$ ($2 <N$) matrices, with unit columns, so that the maximum condition number of all the submatrices comprising 3 columns is minimized. The problem has two applications. When estimating a 2-dimensional signal by using only three of $N$ observations at a given time, this minimizes the worst-case achievable estimation error. It also captures the problem of optimum sensor placement for monitoring a source located in a plane, when only a minimum number of required sensors are active at any given time. For arbitrary $N\geq3$, we derive the optimal matrices which minimize the maximum condition number of all the submatrices of three columns. Surprisingly, a uniform distribution of the columns is \emph{not} the optimal design for odd $N\geq 7$.Comment: conferenc

Certifying non-existence of undesired locally stable equilibria in formation shape control problems

A fundamental control problem for autonomous vehicle formations is formation shape control, in which the agents must maintain a prescribed formation shape using only information measured or communicated from neighboring agents. While a large and growing literature has recently emerged on distance-based formation shape control, global stability properties remain a significant open problem. Even in four-agent formations, the basic question of whether or not there can exist locally stable incorrect equilibrium shapes remains open. This paper shows how this question can be answered for any size formation in principle using semidefinite programming techniques for semialgebraic problems, involving solutions sets of polynomial equations, inequations, and inequalities.Comment: 6 pages; to appear in the 2013 IEEE Multiconference on Systems and Contro