arXiv:2611.15307 · v1

Floating-point, exact and certified computation for control: paradigms, pitfalls, and how to combine them

首次提交 2026-11-11 18:42

版本更新 2026-11-11 18:42

雷达观测 2026-11-13 11:11

摘要

Many questions in linear systems theory have a discrete answer: is a polynomial Hurwitz, is a 2-D system structurally stable, is the peak gain finite, which solution of a Riccati equation is stabilizing? Algorithms implemented in floating point usually answer such questions, but often with a limited, or even no, guarantee on the result, which depends on a tolerance fixed a priori. Computer algebra and interval arithmetic provide other ways of computing, which give guarantees on the result, but usually at a higher cost. In this paper, we propose a short guide to these approaches: floating-point, exact, multiprecision and interval arithmetic, certified root isolation and certified Newton methods. Using small examples from control theory (each of them runs in a few seconds), we show what each approach guarantees and in which situations it fails, and how several approaches can be combined, for example by certifying a numerical solution, or by reducing a problem exactly before solving it with certified real solving. We then revisit the structural stability of 2-D systems (with and without parameters) and the computation of the $H_\infty$ norm, and we list the available implementations. The main examples are given as reproducible worksheets based on the PACE ecosystem.