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

2025-05-14

AI Summary

This work addresses the need for mathematically verifiable solutions to parameter identification, stability analysis, and optimization in control theory. We propose a unified framework integrating symbolic elimination with high-precision interval arithmetic. Implemented as the open-source Julia package PACE.jl, it is the first tool to deeply integrate discriminant variety construction, rational univariate representation (RUR), and arbitrary-precision interval arithmetic—enabling verified numerical solving for both parametric and non-parametric dynamical systems. Compared to conventional numerical methods, our approach significantly improves reliability and accuracy in stability certification and parameter estimation, delivering rigorously error-bounded solutions on benchmark control problems. The core contribution lies in the design and engineering implementation of a symbolic–numeric co-verification mechanism, establishing a new paradigm for trustworthy control algorithm development.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Intelligent Robots: State Estimation

Application Category

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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.

Problem

• Develop computational tools for control theory problems
• Introduce PACE.jl for symbolic elimination techniques
• Apply tools to identification, stability, and optimization

Innovation

• Julia package PACE.jl for control theory
• Symbolic elimination and interval computations
• Handles parameter-dependent and free systems