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

TL;DR

This work introduces spectrahedral shadows as a unifying, exact set representation for state estimation, reachability analysis, and fault diagnosis. It derives a comprehensive suite of polynomial-time set operations (Affine map, Minkowski sum, intersection, Cartesian product, Minkowski-Firey L_p sum, conic hull, convex hull, and linear set-valued maps) and provides robust set validations and conversion procedures from classical representations. To keep these representations scalable, the paper develops order-reduction and polyhedral-approximation strategies, along with sparsity-based acceleration, enabling practical use in high-dimensional SE/RA/FD tasks. The approach yields more compact, exact, and versatile representations than ellipsotopes or CCGs, and demonstrates improved space efficiency with competitive or improved computation times in mixed-polytopic and quadratic uncertainty settings and in Minkowski-Firey L_p reachability contexts. Overall, spectrahedral shadows offer a principled, scalable, and exact framework for sophisticated set-based analysis in dynamic systems with complex uncertainties.

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.

Figures (8)

• Figure 1: Spectrahedral shadows given in Example \ref{['example of noncompact']}.
• Figure 2: Minkowski-Firey $\textit{L}_\textit{p}$-sums of a zonotope $\mathcal{S}_1$ and an ellipsoid $\mathcal{S}_2$ for $p \in\{ 1, 1.15, 1.4, 2, 4, \infty\}$.
• Figure 3: The conic hull and convex hull of spectrahedral shadows.
• Figure 4: Example of local low rank approximation. $\mathcal{S}_g$ is a spectrahedron to be reduced size ($s_g = 50$). $\mathcal{S}_r = \mathcal{S}_r^1 \cap \mathcal{S}_r^{2} \cap \mathcal{S}_r^{3} \cap\mathcal{S}_r^{4}$ is the reduced-size spectrahedron ($s_r = 40$). $\text{bd}{(\mathcal{S}_r^i)}$ ($1\leq i \leq 4$) is the part of the boundary of $\mathcal{S}_r^i$ that locally approximates $\text{bd}{(\mathcal{S}_g)}$ at $x^i$.
• Figure 5: The box plot for the volume ratio $\Delta V$ of $\mathcal{S}_g$ and $\mathcal{S}_r$. Each row label $(n\,, s_g,\, s_r)$ corresponds to a group of results for $100$ random simulations, where the spectrahedron $\mathcal{S}_g \subset \mathbb{R}^n$ with the size $s_g$ is outer approximated by the spectrahedron $\mathcal{S}_r \subset \mathbb{R}^n$ with the size $s_r$.

Theorems & Definitions (33)

• Definition 1
• Remark 1
• Lemma 1: helton2009
• Example 1
• Lemma 2: Translation
• Lemma 3: Invertible linear map
• Proposition 1: Linear map
• Proposition 2: Linear inverse map
• Proposition 3
• Corollary 1