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A $μ$-Analysis and Synthesis Framework for Partial Integral Equations using IQCs

Thijs Lenssen, Aleksandr Talitckii, Matthew Peet, Amritam Das

TL;DR

This work extends the integral quadratic constraint (IQC) framework to infinite-dimensional systems described by Partial Integral Equations (PIEs), unifying robust stability and performance analysis with μ-theory via Linear Partial Integral Inequalities (LPIs). By characterizing structured uncertainties through PI operators and multipliers, the authors provide a computational pathway (via PIETOOLS) to compute upper bounds on the structured singular value for PIEs and to synthesize robust observers. Numerical examples on diffusion–reaction PDEs and delay systems demonstrate reduced conservatism compared to unstructured approaches and illustrate systematic trade-offs between stability and performance. The framework thus enables μ-analysis-like guarantees for spatially distributed systems and opens avenues for robust synthesis in the PIE setting.

Abstract

We develop a $μ$-analysis and synthesis framework for infinite-dimensional systems that leverages the Integral Quadratic Constraints (IQCs) to compute the structured singular value's upper bound. The methodology formulates robust stability and performance conditions jointly as Linear Partial Integral Inequalities within the Partial Integral Equation framework, establishing connections between IQC multipliers and $μ$-theory. Computational implementation via PIETOOLS enables computational tools that practically applicable to spatially distributed infinite dimensional systems. Illustrations with the help of Partial and Delay Differential Equations validate the effectiveness of the framework, showing a significant reduction in conservatism compared to unstructured methods and providing systematic tools for stability-performance trade-off analysis.

A $μ$-Analysis and Synthesis Framework for Partial Integral Equations using IQCs

TL;DR

This work extends the integral quadratic constraint (IQC) framework to infinite-dimensional systems described by Partial Integral Equations (PIEs), unifying robust stability and performance analysis with μ-theory via Linear Partial Integral Inequalities (LPIs). By characterizing structured uncertainties through PI operators and multipliers, the authors provide a computational pathway (via PIETOOLS) to compute upper bounds on the structured singular value for PIEs and to synthesize robust observers. Numerical examples on diffusion–reaction PDEs and delay systems demonstrate reduced conservatism compared to unstructured approaches and illustrate systematic trade-offs between stability and performance. The framework thus enables μ-analysis-like guarantees for spatially distributed systems and opens avenues for robust synthesis in the PIE setting.

Abstract

We develop a -analysis and synthesis framework for infinite-dimensional systems that leverages the Integral Quadratic Constraints (IQCs) to compute the structured singular value's upper bound. The methodology formulates robust stability and performance conditions jointly as Linear Partial Integral Inequalities within the Partial Integral Equation framework, establishing connections between IQC multipliers and -theory. Computational implementation via PIETOOLS enables computational tools that practically applicable to spatially distributed infinite dimensional systems. Illustrations with the help of Partial and Delay Differential Equations validate the effectiveness of the framework, showing a significant reduction in conservatism compared to unstructured methods and providing systematic tools for stability-performance trade-off analysis.

Paper Structure

This paper contains 18 sections, 4 theorems, 43 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. PI operators
  5. Linear PI Inequalities (LPIs)
  6. PIETOOLS
  7. IQC-Based framework for robust stabilityand performance analysis
  8. Description of uncertain systems in a Hilbert-space
  9. Robust stability and performance
  10. Theorem for joint robust stability and performance
  11. Connection to $\mu$-theory
  12. Usage of multipliers
  13. Application to PIE systems
  14. Numerical Examples
  15. Stability of the Diffusion-Reaction Equation
  16. ...and 3 more sections

Key Result

Theorem 1

Let $G$, $\Delta\in\mathbf{\Delta}$ be given, and there exists $\mathcal{M}_\mathcal{V}$, such that, Then the interconnection $[G,\Delta]$ is robustly stable and robust performance is achieved with respect to some $\mathcal{M}_{\hat{\mathcal{V}}}$, with $\hat{\mathcal{V}}_{11} \succeq 0$, if, for all $w_\Delta \in \mathbf{L}_{e,[a,b]}^{\mathbf{n}_{w_\Delta}}$, $w\in L_2^{n_w}([0, \infty))$ and $

Figures (4)

  • Figure 1: Block diagram interpretations of robust stability (left) and robust performance (right) in $\mu$-theory. The robust performance case includes the performance operator $\hat{\Delta}$ that enforces specifications on the $z$-$w$ channel.
  • Figure 2: Equivalence between ODE-PDE and PIE Linear Fractional Representations, showing the nominal system interconnected with stability ($\Delta$) and performance ($\hat{\Delta}$) operators.
  • Figure 3: Stability-performance trade-off curve showing the relationship between robustness ($\gamma_V$) and performance ($\gamma_{\hat{V}}$). The curve exhibits asymptotic behavior where performance improvements become minimal despite significant stability compromises.
  • Figure 4: Robust synthesis comparison: (Left) Parameter space with stable (blue) and unstable (green) variables, showing admissible uncertainty sets for structured (solid) and unstructured (dashed) synthesis. (Right) Trajectories simulated using structured-synthesis observer gain, maintaining consistent color coding. At $0.5\leq t \leq 0.75$ a step disturbance was applied to the system.

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Theorem 1: Robust Stability and Performance
  • proof
  • ...and 8 more