A possibility measure can encode a family of probability measures. This fact is the basis for a transformation of a probability distribution into a possibility distribution that generalises the notion of best interval substitute to a probability distribution with prescribed confidence. This paper describes new properties of this transformation, by relating it with the well-known probability inequalities of Bienaymé-Chebychev and Camp-Meidel. The paper also provides a justification of symmetric triangular fuzzy numbers in the spirit of such inequalities. It shows that the cuts of such a triangular fuzzy number contains the “confidence intervals” of any symmetric probability distribution with the same mode and support. This result is also the basis of a fuzzy approach to the representation of uncertainty in measurement. It consists in representing measurements by a family of nested intervals with various confidence levels. From the operational point of view, the proposed representation is compatible with the recommendations of the ISO Guide for the expression of uncertainty in physical measurement.
Consider a plane curve B defined as the projection of the intersection of two surfaces in R^3 or as the apparent contour of a surface. In general, B has node or cusp singular points and thus is a singular curve. Our main contribution is the computation of a data structure answering point location queries with respect to the subdivision of the plane induced by B. This data structure is composed of an approximation of the space curve together with a topological representation of its projection B. Since B is a singular curve, it is challenging to design a method based only on reliable numerical algorithms. In recent work, the authors show how to describe the set of singularities of B as regular solutions of a so-called ball system suitable for a numerical subdivision solver. Here, the space curve is first enclosed in a set of boxes with a certified path-tracker to restrict the domain where the ball system is solved. Boxes around singular points are then computed such that the correct topology of the curve inside these boxes can be deduced from the intersections of the curve with their boundaries. The tracking of the space curve is then used to connect the smooth branches to the singular points. The subdivision of the plane induced by B is encoded as an extended planar combinatorial map allowing point location. We experimented with our method, and we show that our reliable numerical approach can handle classes of examples that symbolic methods cannot.
In a context of predictive control strategy, this paper addresses the computation of admissible control set for a trajectory tracking over a prediction horizon. The proposed method combines numerical methods based on set-membership computation and control methods based on a flatness concept. It makes possible i) to provide a guaranteed computation of admissible controls, ii) to deal with uncertain reference trajectory, iii) to reduce the time complexity of the algorithm compared to the existing approach. Simulations illustrate the efficiency of the developed methods in two different cases. For Single Input-Single Output (SISO) systems, generalized affine forms are computed otherwise a Branch & Prune algorithm with an inner inclusion test is used for Multi Inputs-Multi Outputs (MIMO) systems. The computational time is reduced significantly compared to the one required by the existing approach.