Some Computational Tools for Solving a Selection of Problems in Control Theory

TL;DR

PACE.jl provides certified symbolic tools for control-theoretic problems by wrapping discriminant-variety computations, Rational Univariate Representation, and interval arithmetic in a single Julia package. The authors demonstrate three applications—parameter identification from polynomialized ODEs, structural stability analysis of multidimensional systems via CAD on discriminant varieties, and exact computation of the $H_\infty$ norm using Sturm-Habicht—through reproducible experiments, including a NF-kB-like model with eight candidate solutions and a representative stable region in a parametric example. The results illustrate guaranteed correctness and practical feasibility on commodity hardware, with cross-language tool integration (e.g., Gröbner bases and parameter-estimation pipelines). The paper outlines future work toward parametric $H_\infty$ computation and extending to broader system classes, emphasizing performance evaluation.

Abstract

This paper demonstrates how certified computational tools can be used to address various problems in control theory. In particular, we introduce PACE.jl, a Julia package that implements symbolic elimination techniques, including (among others) discriminant varieties and Rational Univariate Representation, while also supporting multi-precision interval computations. We showcase its applications to key control theory problems, including identification, stability analysis, and optimization, for both parameter-dependent and parameter-free systems.

Figures (1)

• Figure 1: Points sampled in each 2d-cell of the decomposition in Example 2. In blue are the points corresponding to unstable regions and in green to stable.

Theorems & Definitions (3)

• Example 1
• Example 2
• Example 3