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Moving-Horizon Estimators for Hyperbolic and Parabolic PDEs in 1-D

Luke Bhan, Yuanyuan Shi, Iasson Karafyllis, Miroslav Krstic, James B. Rawlings

TL;DR

This paper addresses the computational burden of real-time PDE state estimation by formulating moving-horizon estimators (MHE) for two PDE classes and deriving explicit, backstepping-based estimators. By transforming hard-to-solve observer PDEs into explicit target systems, the authors obtain MHEs that depend only on the recent input/output over a fixed horizon $T$, avoiding discretization-induced instabilities. They provide explicit MHE formulations and convergence guarantees for both a 1D hyperbolic PIDE and a parabolic PDE, with a simulation demonstrating the parabolic case under noise and excitation. The work offers a practical route to real-time PDE state estimation with potential extensions to data-driven or neural-augmented implementations.

Abstract

Observers for PDEs are themselves PDEs. Therefore, producing real time estimates with such observers is computationally burdensome. For both finite-dimensional and ODE systems, moving-horizon estimators (MHE) are operators whose output is the state estimate, while their inputs are the initial state estimate at the beginning of the horizon as well as the measured output and input signals over the moving time horizon. In this paper we introduce MHEs for PDEs which remove the need for a numerical solution of an observer PDE in real time. We accomplish this using the PDE backstepping method which, for certain classes of both hyperbolic and parabolic PDEs, produces moving-horizon state estimates explicitly. Precisely, to explicitly produce the state estimates, we employ a backstepping transformation of a hard-to-solve observer PDE into a target observer PDE, which is explicitly solvable. The MHEs we propose are not new observer designs but simply the explicit MHE realizations, over a moving horizon of arbitrary length, of the existing backstepping observers. Our PDE MHEs lack the optimality of the MHEs that arose as duals of MPC, but they are given explicitly, even for PDEs. In the paper we provide explicit formulae for MHEs for both hyperbolic and parabolic PDEs, as well as simulation results that illustrate theoretically guaranteed convergence of the MHEs.

Moving-Horizon Estimators for Hyperbolic and Parabolic PDEs in 1-D

TL;DR

This paper addresses the computational burden of real-time PDE state estimation by formulating moving-horizon estimators (MHE) for two PDE classes and deriving explicit, backstepping-based estimators. By transforming hard-to-solve observer PDEs into explicit target systems, the authors obtain MHEs that depend only on the recent input/output over a fixed horizon , avoiding discretization-induced instabilities. They provide explicit MHE formulations and convergence guarantees for both a 1D hyperbolic PIDE and a parabolic PDE, with a simulation demonstrating the parabolic case under noise and excitation. The work offers a practical route to real-time PDE state estimation with potential extensions to data-driven or neural-augmented implementations.

Abstract

Observers for PDEs are themselves PDEs. Therefore, producing real time estimates with such observers is computationally burdensome. For both finite-dimensional and ODE systems, moving-horizon estimators (MHE) are operators whose output is the state estimate, while their inputs are the initial state estimate at the beginning of the horizon as well as the measured output and input signals over the moving time horizon. In this paper we introduce MHEs for PDEs which remove the need for a numerical solution of an observer PDE in real time. We accomplish this using the PDE backstepping method which, for certain classes of both hyperbolic and parabolic PDEs, produces moving-horizon state estimates explicitly. Precisely, to explicitly produce the state estimates, we employ a backstepping transformation of a hard-to-solve observer PDE into a target observer PDE, which is explicitly solvable. The MHEs we propose are not new observer designs but simply the explicit MHE realizations, over a moving horizon of arbitrary length, of the existing backstepping observers. Our PDE MHEs lack the optimality of the MHEs that arose as duals of MPC, but they are given explicitly, even for PDEs. In the paper we provide explicit formulae for MHEs for both hyperbolic and parabolic PDEs, as well as simulation results that illustrate theoretically guaranteed convergence of the MHEs.
Paper Structure (10 sections, 6 theorems, 52 equations, 1 figure)

This paper contains 10 sections, 6 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation: moving-horizon estimators for PDEs
  3. Advances in PDE observers
  4. Moving horizon estimators
  5. Contributions
  6. Notation
  7. 1D Hyperbolic PIDE
  8. 1D Parabolic PDE
  9. Simulation of Parabolic MHE
  10. Conclusion

Key Result

Theorem 1

(proven in BERNARD20142692) For every $f \in C^1(\mathcal{T})$ and $g \in C^1([0, 1])$ the PDE problem eq:hyperbolicKernelCond1, eq:hyperbolicKernelCond2, has a unique $C^1(\mathcal{T})$ solution with the bound

Figures (1)

  • Figure 1: The top row shows system \ref{['eq:parabolicSys1']}, \ref{['eq:parabolicSys2']}, \ref{['eq:parabolicSys3']} with $u_0(x) \equiv 10$ and input $U(t) = 10\cos{(2 \pi t)} + 7 \sin{(16t)}$, which creates an unstable and input-excited system. The second row represents the observer in the explicit MHE form \ref{['eq:parabolicUhatBckstep']}, \ref{['eq:whatsolution']}, \ref{['eq:whatinit']}, with the sliding window as $T=0.1$ and the infinite series in \ref{['eq:whatsolution']} truncated to $N=4$ terms. We add Gaussian measurement noise on the boundary measurement in the form of $\text{Normal}(0, 500)$. The large noise is handled due the frequency of measurements available when using a time step of $1e-6$. In practice, one may not be able to handle such a large deviation with a less rapid measurement frequency. The initial condition for the observer for $t \leq T$ is $\hat{u}(x, \theta) = 20,$$\theta \in [0, T]$. The bottom row showcases the $L^2$ error between the observer system and the true system, $\|\tilde{u}[t]\|_{L^2}$, which decreases precipitously at $t=0.1$ (shown by the grey dotted line), when the observer estimate begins to engage.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • proof