Locating a target from uncertain data: convex supersets based on linear-fractional representations

We address the problem of locating a target by using measurements from a set of sensors called anchors. Each anchor has a known location and measures its distance to the target, the measurement being contaminated with additive noise. The noise vector, which contains the noise realizations across the anchors, is naturally unknown, but we assume it to be drawn from a known bounded uncertainty set.Because the noise vector is unknown, the location of the target is not uniquely defined from the available vector of measurements: uncountably many pairs of target positions and noise vectors can account for the measurements. In fact, the set of all possible target positions (consistent with the measurements) can be nonconvex and even disconnected. We consider the problem of finding a simple convex set-a rectangle-that encloses all possible target locations. We use ideas from linear-fractional representations of uncertainty (LFR) to create convex optimization problems that yield the rectangle. Numerical examples indicate that our LFR approach gives a smaller enclosing rectangle (thus, a tighter delimitation of the target) than a standard convex relaxation, for most of the randomly generated scenarios.