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

Grace Younes, Alban Quadrat, Fabrice Rouillier

TL;DR

The paper tackles the challenge of computing the L-infinity norm in H-infinity control, arguing that symbolic computation can provide exact, reliable solutions, especially for parameterized systems. It develops a symbolic reformulation that reduces the problem to solving real gamma-critical points via when n(omega, gamma) = 0 and its omega-derivative vanish, and surveys a suite of symbolic tools (Sturm-Habicht sequences, Rational Univariate Representations, CAD) for real root counting, elimination, and parametric analysis. Benchmark results show that symbolic methods offer certified accuracy and robustness in parametric or ill-conditioned settings, while numerical methods excel in speed on fixed-coefficient problems, highlighting a complementary trade-off. The work demonstrates the potential of symbolic-numeric hybrids to advance robust and exact stability analysis in control theory, and outlines concrete directions for integrating these techniques into practical workflows. The findings have practical impact by enabling precise, parametric L-infinity computations that support robust controller design and verification.

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.

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

TL;DR

The paper tackles the challenge of computing the L-infinity norm in H-infinity control, arguing that symbolic computation can provide exact, reliable solutions, especially for parameterized systems. It develops a symbolic reformulation that reduces the problem to solving real gamma-critical points via when n(omega, gamma) = 0 and its omega-derivative vanish, and surveys a suite of symbolic tools (Sturm-Habicht sequences, Rational Univariate Representations, CAD) for real root counting, elimination, and parametric analysis. Benchmark results show that symbolic methods offer certified accuracy and robustness in parametric or ill-conditioned settings, while numerical methods excel in speed on fixed-coefficient problems, highlighting a complementary trade-off. The work demonstrates the potential of symbolic-numeric hybrids to advance robust and exact stability analysis in control theory, and outlines concrete directions for integrating these techniques into practical workflows. The findings have practical impact by enabling precise, parametric L-infinity computations that support robust controller design and verification.

Abstract

The computation of the -norm is an important issue in control, particularly for analyzing system stability and robustness. This paper focuses on symbolic computation methods for determining the -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 -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.

Paper Structure

This paper contains 12 sections, 5 theorems, 13 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The $L_{\infty}$-Norm in Control Theory
  3. Characterization of the $L_{\infty}$-Norm
  4. Overview of Numerical Methods
  5. Challenges and Limitations of Numerical Methods
  6. Symbolic Computation Tools
  7. Symbolic Reformulation of the $L_{\infty}$-Norm Computation
  8. Key Symbolic Computation Tools
  9. Symbolic-Numeric Approach
  10. Symbolic & Numerical Methods
  11. Benchmarking Methods for $H_\infty$-Norm Computation
  12. Conclusion

Key Result

Proposition 1

Let $G\in \mathbb{R}(s)^{u\times v}$ with $G|_{i\mathbb{R}}\in RL_\infty^{u\times v}$ and $\gamma>0$. Define Then, $\gamma > \, \parallel G \parallel_{\infty}$ if and only if $\gamma > \Bar{\sigma}\left(G(i \, \infty)\right)$ and $\det(\Phi_{\gamma}(i \, \omega)) \neq 0$ for all $\omega \in \mathbb{R}$.

Figures (1)

  • Figure 1: Regions and root functions for $\xi>0$.

Theorems & Definitions (18)

  • Definition 1: DoyleZhou
  • Proposition 1: kanno2006validated
  • Definition 2
  • Proposition 2: bouzidi2021computation
  • Theorem 1
  • Example 1
  • Proposition 3
  • Example 2
  • Proposition 4
  • Example 3
  • ...and 8 more