KKL Observer Synthesis for Nonlinear Systems via Physics-Informed Learning

TL;DR

The paper addresses nonlinear state estimation by learning a uniformly injective transform $\mathcal{T}: \mathcal{X} \to \mathcal{Z}$ that satisfies the KKL PDE $\frac{\partial \mathcal{T}}{\partial x}(x) f(x)=A\mathcal{T}(x)+B h(x)$ to realize a KKL observer, and by learning the left inverse $\mathcal{T}^*$. It introduces a sequential training pipeline where a physics-informed neural network (PINN) enforces the PDE constraint to learn the forward map $\hat{\mathcal{T}}_\theta$, followed by training a feedforward network to learn the inverse map $\hat{\mathcal{T}}_\eta^*$ using reconstruction data, avoiding conflicting gradients. The authors derive non-asymptotic generalization bounds that relate the inverse-map error $R_{\mathcal{T}^*}$ to the forward and reconstruction errors, and prove input-to-state stability of the estimation error under bounded disturbances. Numerical experiments on benchmark nonlinear and chaotic systems demonstrate strong out-of-domain generalization and robustness compared to state-of-the-art learning-based observers, validating the effectiveness of the physics-informed, sequential approach and its practical impact for nonlinear state estimation.

Abstract

This paper proposes a novel learning approach for designing Kazantzis-Kravaris/Luenberger (KKL) observers for autonomous nonlinear systems. The design of a KKL observer involves finding an injective map that transforms the system state into a higher-dimensional observer state, whose dynamics is linear and stable. The observer's state is then mapped back to the original system coordinates via the inverse map to obtain the state estimate. However, finding this transformation and its inverse is quite challenging. We propose learning the forward mapping using a physics-informed neural network, and then learning its inverse mapping with a conventional feedforward neural network. Theoretical guarantees for the robustness of state estimation against approximation error and system uncertainties are provided, including non-asymptotic learning guarantees that link approximation quality to finite sample sizes. The effectiveness of the proposed approach is demonstrated through numerical simulations on benchmark examples, showing superior generalization capability outside the training domain compared to state-of-the-art methods.

Figures (4)

• Figure 1: Overview of the proposed KKL observer learning algorithm.
• Figure 2: State estimation with learned KKL observer under noisy measurements. The solid black line shows the true state trajectory, while the dashed magenta line is the state estimate obtained from the learned KKL observer.
• Figure 3: Estimation performance of KKL observers learned from different methods when the system is initialized outside the training domain.
• Figure 4: Comparison of quantitative error metrics averaged over the 100 trials.