Gain Scheduling with a Neural Operator for a Transport PDE with Nonlinear Recirculation

TL;DR

The paper addresses stabilizing a nonlinear hyperbolic PDE with state-dependent recirculation using gain-scheduled backstepping, where the scheduling kernel is learned by a Neural Operator (DeepONet) to enable real-time implementation. It develops both an exact GS controller and a neural-operator-approximated GS controller, proving local exponential (in $H^1$) stability for each and achieving about $10^3\times$ speedups in kernel computation. A rigorous Lyapunov analysis demonstrates robustness of the perturbed target system to NO approximations, and simulations with nonlinear Chebyshev recirculation validate stabilization and speedups. The work advances real-time learning-based nonlinear PDE control by tightly coupling GS backstepping with neural-operator approximations, and provides open-source code to facilitate replication and extension.

Abstract

To stabilize PDE models, control laws require space-dependent functional gains mapped by nonlinear operators from the PDE functional coefficients. When a PDE is nonlinear and its "pseudo-coefficient" functions are state-dependent, a gain-scheduling (GS) nonlinear design is the simplest approach to the design of nonlinear feedback. The GS version of PDE backstepping employs gains obtained by solving a PDE at each value of the state. Performing such PDE computations in real time may be prohibitive. The recently introduced neural operators (NO) can be trained to produce the gain functions, rapidly in real time, for each state value, without requiring a PDE solution. In this paper we introduce NOs for GS-PDE backstepping. GS controllers act on the premise that the state change is slow and, as a result, guarantee only local stability, even for ODEs. We establish local stabilization of hyperbolic PDEs with nonlinear recirculation using both a "full-kernel" approach and the "gain-only" approach to gain operator approximation. Numerical simulations illustrate stabilization and demonstrate speedup by three orders of magnitude over traditional PDE gain-scheduling. Code (Github) for the numerical implementation is published to enable exploration.

Figures (5)

• Figure 1: Open-loop (U=0) simulation of the PDE with the modified Chebyshev polynomial functions $\beta(x, u(0, t)) = 5 \cos((\gamma+u(0, t))\cos^{-1}(x))$ with initial conditions $u(x, 0) = 0.38, 0.04$ and parameters $\gamma=3, 5$ for the let and right images respectively.
• Figure 2: Analytical solution with gain scheduling for the modified Chebyshev polynomial functions $\beta(x, \mu) = 5 \cos ((\gamma+\mu) \cos^{-1}(x))$ with parameters $\gamma=3, 5$ for the left and right images respectively. The top row shows initial conditions 0.37 (left) and 0.03 (right) respectively and the bottom row shows increased initial conditions of 0.39 (left) and 0.05(right). Naturally, the PDE becomes harder to stabilize and for larger initial conditions, gain scheduling fails to control the PDE.
• Figure 3: PDE stabilization using a controller based on a linearization at the origin, with no gain scheduling, $U(t) = \int_0^x k(1-y, 0)u(y, t) dy$, for the PDE's corresponding to Figure \ref{['fig:gs-openloop']}
• Figure 4: Neural operator kernels when controlling the PDE in Figure \ref{['fig:gs-openloop']}(top row), and the difference in kernel error between the resulting gain scheduling kernels (bottom row).
• Figure 5: Stabilization of the plants in Figure \ref{['fig:gs-openloop']} with the neural operator approximated kernels in the gain scheduling feedback law.

Theorems & Definitions (28)

• Lemma 1
• proof
• Theorem 1: DeepOnet universal approximation theorem lu2021advectionDeepONet
• Definition 1
• Lemma 2
• proof
• Theorem 2
• Theorem 3
• Lemma 3
• proof