Comparison of recent advances in set-membership techniques: Application to state estimation, fault detection and collision avoidance

Abstract

There has been a vast body of research on set-membership techniques in recent years. These algorithms compute convex sets that contain the state of a dynamical system given bounds on disturbance and noise signals. Recently, a thorough comparison of zonotopes-based methods against interval arithmetic and ellipsoids has been presented in the literature. However, two main issues were left unexplored: (i) added conservatism in the presence of bounds of different kinds such as ${\ell }_2$-norm for the disturbance and an ${\ell }_{\infty }$-norm bound on the noise: (ii) set-membership methods can be used in different settings apart from guaranteed state estimation, such as fault detection and isolation and collision avoidance of autonomous vehicles. In this paper, we extend this comparison by considering state estimation, fault detection and isolation, and collision avoidance for interval arithmetic, ellipsoids, zonotopes, constrained zonotopes, polytopes and constrained convex generators in the presence of various combination of bounds for the exogenous signals. The main objective is to compare accuracy, computation time and the scalability of the growth of the data structures required by each set representation. The results indicate that intervals, ellipsoids and zonotopes have a much worse accuracy. The recently introduced Constrained Convex Generators have a negligible increase in computation time in comparison with constrained zonotopes but have a better accuracy when bounds for disturbances, noise and initial conditions are heterogeneous or at least not polytopic.