Symbolic and Numerical Tools for $L_{\infty}$-Norm Calculation

TL;DR

This paper reviews symbolic computation methods for calculating the $L_ abla$-norm of linear systems, emphasizing their advantages in accuracy and exact solutions over numerical methods, especially in parametric cases.

Contribution

It surveys key symbolic techniques like Sturm-Habicht sequences, RUR, and CAD, analyzing their theoretical basis, practical use, and comparative performance in $L_ abla$-norm computation.

Findings

Symbolic methods achieve exact $L_ abla$-norm solutions.

Symbolic approaches outperform numerical methods in parametric scenarios.

Benchmark results highlight trade-offs between symbolic and numerical techniques.

Abstract

The computation of the $L_\infty $-norm is an important issue in $H_{\infty}$ control, particularly for analyzing system stability and robustness. This paper focuses on symbolic computation methods for determining the $L_{\infty} $-norm of finite-dimensional linear systems, highlighting their advantages in achieving exact solutions where numerical methods often encounter limitations. Key techniques such as Sturm-Habicht sequences, Rational Univariate Representations (RUR), and Cylindrical Algebraic Decomposition (CAD) are surveyed, with an emphasis on their theoretical foundations, practical implementations, and specific applicability to $ L_{\infty} $-norm computation. A comparative analysis is conducted between symbolic and conventional numerical approaches, underscoring scenarios in which symbolic computation provides superior accuracy, particularly in parametric cases. Benchmark evaluations reveal the strengths and limitations of both approaches, offering insights into the trade-offs involved. Finally, the discussion addresses the challenges of symbolic computation and explores future opportunities for its integration into control theory, particularly for robust and stable system analysis.