Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Adaptive Neural-Operator Backstepping Control of a Benchmark Hyperbolic PDE

Maxence Lamarque, Luke Bhan, Yuanyuan Shi, Miroslav Krstic

TL;DR

The paper addresses real-time adaptive control of a 1-D hyperbolic PDE with an unknown coefficient by replacing the kernel PDE solution with a neural-operator (NO) approximation of the backstepping kernel. It presents two NO-based designs—one Lyapunov-based with a full-kernel NO and another modular passive-identifier approach—establishing global stability under accurate NO approximations and providing rigorous Lyapunov analyses. Numerical results show Newton-like speedups up to $10^3\times$ in computing the adaptive gain and successful stabilization with online coefficient updates, albeit with potential non-convergence of the true coefficient due to limited excitation. The work demonstrates the practicality of NOs for real-time adaptive PDE control of hyperbolic systems and offers publicly available code and DeepONet training data for reproducibility and further research.

Abstract

To stabilize PDEs, feedback controllers require gain kernel functions, which are themselves governed by PDEs. Furthermore, these gain-kernel PDEs depend on the PDE plants' functional coefficients. The functional coefficients in PDE plants are often unknown. This requires an adaptive approach to PDE control, i.e., an estimation of the plant coefficients conducted concurrently with control, where a separate PDE for the gain kernel must be solved at each timestep upon the update in the plant coefficient function estimate. Solving a PDE at each timestep is computationally expensive and a barrier to the implementation of real-time adaptive control of PDEs. Recently, results in neural operator (NO) approximations of functional mappings have been introduced into PDE control, for replacing the computation of the gain kernel with a neural network that is trained, once offline, and reused in real-time for rapid solution of the PDEs. In this paper, we present the first result on applying NOs in adaptive PDE control, presented for a benchmark 1-D hyperbolic PDE with recirculation. We establish global stabilization via Lyapunov analysis, in the plant and parameter error states, and also present an alternative approach, via passive identifiers, which avoids the strong assumptions on kernel differentiability. We then present numerical simulations demonstrating stability and observe speedups up to three orders of magnitude, highlighting the real-time efficacy of neural operators in adaptive control. Our code (Github) is made publicly available for future researchers.

Adaptive Neural-Operator Backstepping Control of a Benchmark Hyperbolic PDE

TL;DR

The paper addresses real-time adaptive control of a 1-D hyperbolic PDE with an unknown coefficient by replacing the kernel PDE solution with a neural-operator (NO) approximation of the backstepping kernel. It presents two NO-based designs—one Lyapunov-based with a full-kernel NO and another modular passive-identifier approach—establishing global stability under accurate NO approximations and providing rigorous Lyapunov analyses. Numerical results show Newton-like speedups up to in computing the adaptive gain and successful stabilization with online coefficient updates, albeit with potential non-convergence of the true coefficient due to limited excitation. The work demonstrates the practicality of NOs for real-time adaptive PDE control of hyperbolic systems and offers publicly available code and DeepONet training data for reproducibility and further research.

Abstract

To stabilize PDEs, feedback controllers require gain kernel functions, which are themselves governed by PDEs. Furthermore, these gain-kernel PDEs depend on the PDE plants' functional coefficients. The functional coefficients in PDE plants are often unknown. This requires an adaptive approach to PDE control, i.e., an estimation of the plant coefficients conducted concurrently with control, where a separate PDE for the gain kernel must be solved at each timestep upon the update in the plant coefficient function estimate. Solving a PDE at each timestep is computationally expensive and a barrier to the implementation of real-time adaptive control of PDEs. Recently, results in neural operator (NO) approximations of functional mappings have been introduced into PDE control, for replacing the computation of the gain kernel with a neural network that is trained, once offline, and reused in real-time for rapid solution of the PDEs. In this paper, we present the first result on applying NOs in adaptive PDE control, presented for a benchmark 1-D hyperbolic PDE with recirculation. We establish global stabilization via Lyapunov analysis, in the plant and parameter error states, and also present an alternative approach, via passive identifiers, which avoids the strong assumptions on kernel differentiability. We then present numerical simulations demonstrating stability and observe speedups up to three orders of magnitude, highlighting the real-time efficacy of neural operators in adaptive control. Our code (Github) is made publicly available for future researchers.
Paper Structure (12 sections, 8 theorems, 139 equations, 3 figures, 2 tables)

This paper contains 12 sections, 8 theorems, 139 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Exact Adaptative PDE Backstepping for a Hyperbolic PDE with Recirculation
  3. Backstepping Kernel Properties
  4. Neural Operator Approximation of Backstepping Kernel
  5. DeepONet-Approximated Lyapunov Adaptive PDE Backstepping Design
  6. Lyapunov Analysis
  7. A Modular Design with a Passive Identifier
  8. Simulations
  9. Conclusion
  10. Backstepping Transformation and Involution Operator for the Kernel
  11. Perturbed Target System with Approximate Estimated Kernel
  12. Perturbed Observer Target System with Exact Estimated Kernel

Key Result

Lemma 1

[Existence and upper bound for kernel and its derivative] Let $B > 0, \hat{\beta} \in \mathcal{C}^0([0, 1] \times \mathbb{R}^+, \mathbb{R})$ such that $\|\hat{\beta}\|_{\infty} \leq B$ and consider the Volterra equation eq-exact-adapt-Volterra-eqn, reiterated here for convenience, There exist a unique $\mathcal{C}^0 ([0, 1] \times \mathbb{R}^+, \mathbb{R})$ solution $\breve{k}$ that satisfies If

Figures (3)

  • Figure 1: Adaptive neural operator controller applied to the PDE governed by \ref{['eq:ut_def']}, \ref{['eq:u1_def']} where $\beta(x)=5\cos(\sigma \cos^{-1}(x))$ with $\sigma=2.9$ and initial condition $u(x, 0)=1$. The initial guess for $\hat{\beta}$ was $\hat{\beta}(x, 0)=1 \quad \forall x \in [0, 1]$ and the control update law \ref{['eq-update-betahat-approx']}, \ref{['eq-tau-approx']}, \ref{['eq-bkst-w-khat']}, \ref{['eq-norm-w']} has parameter $c=1$.
  • Figure 2: Left: $\hat{\beta}$ estimates when controlling the PDE in Figure \ref{['fig:adaptiveControlStabilization']} using neural operator approximated kernels: true $\beta$ (blue) and final estimated $\hat{\beta}$ (red); Right: comparison between the true $\beta$ value, the initial guess $\hat{\beta}(\cdot, 0)=1$, and the final estimated $\hat{\beta}$ at $t=13$.
  • Figure 3: Neural operator approximated kernels when controlling the PDE in Figure \ref{['fig:adaptiveControlStabilization']}(left), and the difference in kernel error between the approximated kernel and the analytical kernel (right).

Theorems & Definitions (8)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8