Sparse Spectrahedral Shadows for State Estimation and Reachability Analysis: Set Operations, Validations and Order Reductions

TL;DR

This paper introduces spectrahedral shadows as a versatile set representation for state estimation and reachability analysis, providing exact set operations and reduction techniques with polynomial-time algorithms.

Contribution

It generalizes traditional set representations and offers analytical, exact set operations and reduction methods for spectrahedral shadows, enhancing set-based analysis tools.

Findings

Exact set operations implemented without approximation

Spectrahedral shadows generalize multiple set representations

Polynomial-time algorithms for set validation and reduction

Abstract

Set representations are the foundation of various set-based approaches in state estimation, reachability analysis and fault diagnosis. In this paper, we investigate spectrahedral shadows, a class of nonlinear geometric objects previously studied in semidefinite programming and real algebraic geometry. We demonstrate spectrahedral shadows generalize traditional and emerging set representations like ellipsoids, zonotopes, constrained zonotopes and ellipsotopes. Analytical forms of set operations are provided including linear map, linear inverse map, Minkowski sum, intersection, Cartesian product, Minkowski-Firey Lp sum, convex hull, conic hull and polytopic map, all of which are implemented without approximation in polynomial time. In addition, we develop set validation and order reduction techniques for spectrahedral shadows, thereby establishing spectrahedral shadows as a set representation applicable to a range of set-based tasks.