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$H_{\infty}$-Optimal Estimator Synthesis for Coupled Linear 2D PDEs using Convex Optimization

Declan S. Jagt, Matthew M. Peet

TL;DR

This work develops a convex-optimization-based framework to synthesize $H_{\infty}$-optimal estimators for linear 2D PDEs by representing the PDEs as Partial Integral Equations (PIEs) and encoding outputs and boundary conditions with PI operators. Central to the method is an LPIs-based condition that, when feasible, yields an estimator gain $\mathcal{L} = \mathcal{P}^{-1}\mathcal{W}$ with a guaranteed $L_2$-gain bound $\gamma$ on the estimator error, and an explicit inversion mechanism for a broad class of 2D PI operators to compute the gain. The approach is implemented in the PIETOOLS software and demonstrated on an unstable 2D heat equation with boundary observations, achieving tight $H_{\infty}$ bounds ($\gamma$ values of 0.0476 for $r=4$ and 0.1403 for $r=8$) and validating convergence of the estimator via Galerkin discretization. This framework offers a scalable alternative to projection or backstepping methods, enabling exact PI-based observer synthesis with reduced conservatism through higher-degree polynomial bases. Overall, the paper advances the practical design of optimal infinite-dimensional estimators for 2D PDEs and provides actionable tools for implementation and benchmarking.

Abstract

Any suitably well-posed PDE in two spatial dimensions can be represented as a Partial Integral Equation (PIE) -- with system dynamics parameterized using Partial Integral (PI) operators. Furthermore, $L_2$-gain analysis of PDEs with a PIE representation can be posed as a linear operator inequality, which can be solved using convex optimization. In this paper, these results are used to derive a convex-optimization-based test for constructing an $H_{\infty}$-optimal estimator for 2D PDEs. In particular, we first use PIEs to represent an arbitrary well-posed 2D PDE where sensor measurements occur along some boundary of the domain. An associated Luenberger-type estimator is then parameterized using a PI operator $\mathcal{L}$ as the observer gain. Examining the error dynamics of this estimator, it is proven that an upper bound on the $H_{\infty}$-norm of these error dynamics can be minimized by solving a linear operator inequality on PI operator variables. Finally, an analytical formula is proposed for inversion of a class of 2D PI operators, which is then used to reconstruct the Luenberger gain $\mathcal{L}$. Results are implemented in the PIETOOLS software suite -- applying the methodology and simulating the resulting observer for an unstable 2D heat equation with boundary observations.

$H_{\infty}$-Optimal Estimator Synthesis for Coupled Linear 2D PDEs using Convex Optimization

TL;DR

This work develops a convex-optimization-based framework to synthesize -optimal estimators for linear 2D PDEs by representing the PDEs as Partial Integral Equations (PIEs) and encoding outputs and boundary conditions with PI operators. Central to the method is an LPIs-based condition that, when feasible, yields an estimator gain with a guaranteed -gain bound on the estimator error, and an explicit inversion mechanism for a broad class of 2D PI operators to compute the gain. The approach is implemented in the PIETOOLS software and demonstrated on an unstable 2D heat equation with boundary observations, achieving tight bounds ( values of 0.0476 for and 0.1403 for ) and validating convergence of the estimator via Galerkin discretization. This framework offers a scalable alternative to projection or backstepping methods, enabling exact PI-based observer synthesis with reduced conservatism through higher-degree polynomial bases. Overall, the paper advances the practical design of optimal infinite-dimensional estimators for 2D PDEs and provides actionable tools for implementation and benchmarking.

Abstract

Any suitably well-posed PDE in two spatial dimensions can be represented as a Partial Integral Equation (PIE) -- with system dynamics parameterized using Partial Integral (PI) operators. Furthermore, -gain analysis of PDEs with a PIE representation can be posed as a linear operator inequality, which can be solved using convex optimization. In this paper, these results are used to derive a convex-optimization-based test for constructing an -optimal estimator for 2D PDEs. In particular, we first use PIEs to represent an arbitrary well-posed 2D PDE where sensor measurements occur along some boundary of the domain. An associated Luenberger-type estimator is then parameterized using a PI operator as the observer gain. Examining the error dynamics of this estimator, it is proven that an upper bound on the -norm of these error dynamics can be minimized by solving a linear operator inequality on PI operator variables. Finally, an analytical formula is proposed for inversion of a class of 2D PI operators, which is then used to reconstruct the Luenberger gain . Results are implemented in the PIETOOLS software suite -- applying the methodology and simulating the resulting observer for an unstable 2D heat equation with boundary observations.
Paper Structure (14 sections, 67 equations, 1 figure)

This paper contains 14 sections, 67 equations, 1 figure.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Notation
  4. Algebras of PI Operators on 2D Functions
  5. A PIE Representation of 2D Input-Output PDEs
  6. A Linear PI Inequality (LPI) for $L_2$-gain Analysis
  7. An $H_{\infty}$-Optimal Estimator for 2D PDEs
  8. Representation of Infinite-Dimensional PDE Outputs
  9. An LPI for Optimal Estimation of PIEs
  10. Estimator Synthesis of 2D PDEs using Convex Optimization
  11. A Numerical Example
  12. Estimator for a 2D Heat Equation
  13. Conclusion
  14. Numerical Simulation of PIEs

Figures (1)

  • Figure 1: Error in the estimate of the PDE state $\mathbf{ u}(t)$ and the output $z(t)$ for the PDE in Subsection \ref{['sec:Numerical_Examples:1']} with $r=4$ and $r=8$, for disturbance $w(t)=5e^{-t/2}\sin(\pi t)$ and initial error $\hat{\mathbf{ u}}(0,x,y)-\mathbf{ u}(0,x,y)=5((x-1)^4-1)\sin(0.5\pi y)$. Errors are plotted for estimators achieved both for fixed $\gamma=1$ in the LMI \ref{['eq:estimator_LMI']}, as well as minimizing over the value of $\gamma$ in this LMI.