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Paper details
Number 3 - September 2021
Volume 31 - 2021
A computationally inexpensive algorithm for determining outer and inner enclosures of nonlinear mappings of ellipsoidal domains
Andreas Rauh, Luc Jaulin
Abstract
A wide variety of approaches for set-valued simulation, parameter identification, state estimation as well as reachability,
observability and stability analysis for nonlinear discrete-time systems involve the propagation of ellipsoids via nonlinear
functions. It is well known that the corresponding image sets usually possess a complex shape and may even be nonconvex
despite the convexity of the input data. For that reason, domain splitting procedures are often employed which help to
reduce the phenomenon of overestimation that can be traced back to the well-known dependency and wrapping effects
of interval analysis. In this paper, we propose a simple, yet efficient scheme for simultaneously determining outer and
inner ellipsoidal range enclosures of the solution for the evaluation of multi-dimensional functions if the input domains
are themselves described by ellipsoids. The Hausdorff distance between the computed enclosure and the exact solution set
reduces at least linearly when decreasing the size of the input domains. In addition to algebraic function evaluations, the
proposed technique is—for the first time, to our knowledge—employed for quantifying worst-case errors when extended
Kalman filter-like, linearization-based techniques are used for forecasting confidence ellipsoids in a stochastic setting.
Keywords
bounded uncertainty, guaranteed enclosures, ellipsoidal enclosures, inner and outer approximations, nonlinear systems, confidence intervals