Union Of Ellipsoids Research Articles

Fault detection for uncertain LPV systems using probabilistic set-membership parity relation

This paper considers fault detection of uncertain linear parameter varying systems that have polynomial dependence on parametric uncertainties. A conventional set-membership (SM) approach is able to ensure zero false alarm rate (FAR) by using conservative threshold sets, but usually results in a high missed detection rate (MDR) due to equally treating all uncertainty realizations without distinguishing between high and low probability of occurrence. To address this limitation, a probabilistic SM parity relation approach is proposed to exploit probabilistic information on the parametric uncertainties, which results in a reduced MDR by admitting an acceptable FAR. The parity relation is first polynomially parameterized with respect to uncertain parameters. Then, Gaussian mixtures are adopted to efficiently compute uncertainty propagation from stochastic uncertainties to the residual distribution. To achieve an acceptable FAR, a non-convex confidence set of residuals – represented by a union of ellipsoids – is determined for the consistency test. The effectiveness of the proposed approach is illustrated using a continuous stirred tank reactor example including performance comparisons with a deterministic zonotope-based method.

Shape description and representation by ellipsoids

Shape description schemes by decomposition of regions using ellipsoids as primitive elements are discussed from the viewpoints of both amount of data and degree of approximation. Two algorithms of shape decomposition are described. One aims at describing shapes approximately, and uses skeletons of a region to determine the allocation of ellipsoids inside of the region. The other is capable of describing shapes exactly by decomposing a whole region into a union of ellipsoids, and it is formulated with morphological operations. For the respective shape decomposition algorithm, a scheme of structuring a set of ellipsoids obtained as the result of shape description in a hierarchical tree is presented. These tree structures specify the priority of each ellipsoid to represent the spread of the region progressively. Then, for reducing amount of data, a subset of ellipsoids is chosen according to the order given in the tree structure, and regions are reproduced from it. To improve the degree of approximation of the reproduced shape, a scheme of fusing the scattered set of ellipsoids with density distribution functions is presented. Simulation results have shown that our schemes perform effectively both data reduction and progressive approximation of shape.