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$H_2$-Optimal Estimation of a Class of Linear PDE Systems using Partial Integral Equations

Danio Braghini, Sachin Shivakumar, Matthew M. Peet

Abstract

The $H_2$ norm is a commonly used performance metric in the design of estimators. However, $H_2$-optimal estimation of most PDEs is complicated by the lack of state-space and transfer function representations. To address this problem, we re-characterize the $H_2$-norm in terms of a map from initial condition to output. We then leverage the Partial Integral Equation (PIE) state-space representation of systems of linear PDEs coupled with ODEs to recast this characterization of $H_2$ norm as a convex optimization problem defined in terms of Linear Partial Integral (LPI) inequalities. We then parameterize a class of PIE-based observers and formulate the associated $H_2$-optimal estimation problem. The optimal observer synthesis problem is then recast as an LPI, and the resulting observers are validated using numerical simulation.

$H_2$-Optimal Estimation of a Class of Linear PDE Systems using Partial Integral Equations

Abstract

The norm is a commonly used performance metric in the design of estimators. However, -optimal estimation of most PDEs is complicated by the lack of state-space and transfer function representations. To address this problem, we re-characterize the -norm in terms of a map from initial condition to output. We then leverage the Partial Integral Equation (PIE) state-space representation of systems of linear PDEs coupled with ODEs to recast this characterization of norm as a convex optimization problem defined in terms of Linear Partial Integral (LPI) inequalities. We then parameterize a class of PIE-based observers and formulate the associated -optimal estimation problem. The optimal observer synthesis problem is then recast as an LPI, and the resulting observers are validated using numerical simulation.

Paper Structure

This paper contains 16 sections, 5 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. State Space and Convex Optimization: PIs, PIEs, and LPIs
  4. The Algebra of Partial Integral Operators
  5. Partial Integral Equations
  6. Linear PI Operator Inequalities
  7. Problem Formulation
  8. The $H_2$ norm of a PIE
  9. $H_2$-Optimal Estimators
  10. An LPI for the $H_2$ norm
  11. An LPI for $H_2$-optimal Estimator
  12. Estimator Gain Reconstruction
  13. Numerical Examples
  14. Unstable Reaction-Diffusion Equation
  15. Euler-Bernoulli Beam Equation
  16. ...and 1 more sections

Key Result

Corollary 4

Suppose $A$ is Hurwitz and $\hat{G}(s)=C(sI-A)^{-1}B$ with $B \in \mathbb{R}^{n_x \times n_w}$. Consider solutions of the auxiliary ODE Then

Figures (2)

  • Figure 1: Numerical estimation of an $H_2$-optimal estimator for an unstable reaction-diffusion equation (Eq. \ref{['eq:react-diff']}) using measurement at the boundary along with process and sensor disturbance $w(t)=sin(100t)$ and PDE initial condition $\mathbf{ \xi}(0,s)=-s^2/2$. (a): Evolution of error in estimate of the PDE state $\mathcal{ T} \mathbf{ e}(t)=\mathcal{ T}\tilde{\mathbf{ x}}(t)-\mathbf{ \xi}(t)$. (b): Evolution of the regulated output ($z(t)$) of both estimator and PDE.
  • Figure 2: Numerical estimation of an $H_2$-optimal estimator for a neutrally stable Euler-Bernoulli beam equation (Eq. \ref{['eq:EBSS']}) using velocity measurement at the tip without disturbances and with PDE initial condition $\mathbf{ \partial}_t\eta(0,s)=-s^2/2$. (a): Evolution of error in estimate of the PDE state $\mathcal{ T} \mathbf{ e}(t)=\mathcal{ T}\tilde{\mathbf{ x}}(t)-\mathbf{ \xi}(t)$. (b): Evolution of the regulated output ($z(t)$) of both estimator and PDE.

Theorems & Definitions (12)

  • Definition 1
  • Definition 2: PIE solution
  • Definition 3
  • Corollary 4
  • proof
  • Theorem 5
  • Lemma 6: Schur Complement
  • proof
  • Theorem 7
  • proof
  • ...and 2 more