Interval and ellipsoidal uncertainty models

In this Chapter, we present results derived in the context of state estimation of a class of real-life systems that are driven by some poorly known factors. For these systems, the representation of uncertainty as confidence intervals or the ellipsoids offers significant advantages over the more traditional approaches with probabilistic representation of noise. While the filtered-white-Gaussian noise model can be defined on grounds of mathematical convenience, its use is necessarily coupled with a hope that an estimator with good properties in idealised noise will still perform well in real noise. With good knowledge of the plant and its environment, a sufficiently accurate approximation to the probability density function can be obtained, but shortage of prior information or excessive computing demands normally rule out this option. A more realistic approach is to match the noise representation to the extent of prior knowledge. The relative merits of interval and ellipsoidal representations of noise are discussed in a set theoretic setting and are illustrated using both a simple synthetic example and a real-life scenario of state estimation of a water distribution system.