Usman Khan

Mohammadreza Doostmohammadian, Hamid R. Rabiee, Houman Zarrabi et al.

Observability of complex systems/networks is the focus of this paper, which is shown to be closely related to the concept of contraction. Indeed, for observable network tracking it is necessary/sufficient to have one node in each contraction measured. Therefore, nodes in a contraction are equivalent to recover for loss of observability, implying that contraction size is a key factor for observability recovery. Here, using a polynomial order contraction detection algorithm, we analyze the distribution of contractions, studying its relation with key network properties. Our results show that contraction size is related to network clustering coefficient and degree heterogeneity. Particularly, in networks with power-law degree distribution, if the clustering coefficient is high there are less contractions with smaller size on average. The implication is that estimation/tracking of such systems requires less number of measurements, while their observational recovery is more restrictive in case of sensor failure. Further, in Small-World networks higher degree heterogeneity implies that there are more contractions with smaller size on average. Therefore, the estimation of representing system requires more measurements, and also the recovery of measurement failure is more limited. These results imply that one can tune the properties of synthetic networks to alleviate their estimation/observability recovery.

Sam Safavi, Usman Khan

In this paper, we develop a \textcolor{black}{\emph{distributed}} algorithm to localize a network of robots moving arbitrarily in a bounded region. In the case of such mobile networks, the main challenge is that the robots may not be able to find nearby robots to implement a distributed algorithm. We address this issue by providing an opportunistic algorithm that only implements a location update when there are nearby robots and does not update otherwise. We assume that each robot measures a noisy version of its motion and the distances to the nearby robots. To localize a network of mobile robots in~$\mathbb{R}^m$, we provide a simple \emph{linear} update, which is based on barycentric coordinates and is linear-convex. We abstract the corresponding localization algorithm as a Linear Time-Varying (LTV) system and show that it asymptotically converges to the true locations~of~the robots. We first focus on the noiseless case, where the distance and motion vectors are known (measured) perfectly, and provide sufficient conditions on the convergence of the algorithm. We then evaluate the performance of the algorithm in the presence of noise and provide modifications to counter the undesirable effects of noise. \textcolor{black}{We further show that our algorithm precisely tracks a mobile network as long as there is at least one known beacon (a node whose location is perfectly known).