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AbstractWe consider an unknown multivariate function representing a system—such as a complex numerical simulator—taking both deterministic and uncertain inputs. Our objective is to estimate the set of deterministic inputs leading to outputs whose probability (with respect to the distribution of the uncertain inputs) of belonging to a given set is less than a given threshold. This problem, which we call Quantile Set Inversion (QSI), occurs for instance in the context of robust (reliability-based) optimization problems, when looking for the set of solutions that satisfy the constraints with sufficiently large probability. To solve the QSI problem we propose a Bayesian strategy, based on Gaussian process modeling and the Stepwise Uncertainty Reduction (SUR) principle, to sequentially choose the points at which the function should be evaluated to efficiently approximate the set of interest. We illustrate the performance and interest of the proposed SUR strategy through several numerical experiments. [ABSTRACT FROM AUTHOR]

A guaranteed fault-estimation algorithm based on interval set inversion observer filtering is proposed for linear discrete-time systems with unknown but bounded disturbance and noise. The minimal conservative interval observer is designed by minimizing the F-norm of the state error. Vector Boolean operations and dimensional operations are used to develop a new interval set inversion algorithm to further contract the guaranteed interval estimation results of the observer. The computational complexity, memory requirements, and accuracy of the proposed algorithm are also analyzed. Finally, simulation examples are provided to verify the efficiency and practicability of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Set inversion and box contraction consist in finding guaranteed solutions to an equation where variables belong to bounded sets. These problems are classic in Euclidean spaces and can be solved using interval analysis. However, they quickly become complex when dealing with non-Euclidean manifolds. This paper investigates set inversion and box contraction problems when variables belong to generic Lie groups (e.g. the set of rotation matrices SO (3)). The concept of Lie group boxes is introduced and discussed. The core idea is to treat Lie group subsets via interval analysis in the Euclidean space of the Lie algebra. The concept of inclusion function on a Lie group is defined as an extension of classic inclusion functions in Euclidean interval analysis. It is shown that set inversion problems on Lie groups can be solved via the Lie algebra using Euclidean interval analysis and that the solution set has a finite volume in the group. On the same principle, the box contraction problem is extended to Lie groups via a Euclidean contractor in the Lie algebra. Numerical simulations demonstrate the high interest of using Lie group boxes instead of classic Euclidean parametrization for nonlinear problems (such as the Euler angles for rotations) both in terms of computational efficiency and accuracy. The proposed approach is of practical interest for state estimation and navigation in robotics since it makes it possible to solve highly nonlinear problems with a limited complexity and a high accuracy, in particular for attitude and pose estimation. [ABSTRACT FROM AUTHOR]