HyperKKL: Learning KKL Observers for Non-Autonomous Nonlinear Systems via Hypernetwork-Based Input Conditioning

TL;DR

This paper introduces HyperKKL, a novel neural network framework for non-autonomous nonlinear systems that improves state estimation accuracy by conditioning the observer on exogenous inputs using hypernetworks.

Contribution

It proposes two input-conditioning strategies for KKL observers using hypernetworks, extending their applicability to controlled and non-autonomous systems.

Findings

Input conditioning improves estimation accuracy by 29% SMAPE on average.

HyperKKL outperforms static autonomous maps in nonlinear benchmark systems.

Theoretical worst-case error bounds are derived for the proposed methods.

Abstract

Kazantzis-Kravaris/Luenberger (KKL) observers are a class of state observers for nonlinear systems that rely on an injective map to transform the nonlinear dynamics into a stable quasi-linear latent space, from where the state estimate is obtained in the original coordinates via a left inverse of the transformation map. Current learning-based methods for these maps are designed exclusively for autonomous systems and do not generalize well to controlled or non-autonomous systems. In this paper, we propose two learning-based designs of neural KKL observers for non-autonomous systems whose dynamics are influenced by exogenous inputs. To this end, a hypernetwork-based framework ($HyperKKL$) is proposed with two input-conditioning strategies. First, an augmented observer approach ($HyperKKL_{obs}$) adds input-dependent corrections to the latent observer dynamics while retaining static transformation maps. Second, a dynamic observer approach ($HyperKKL_{dyn}$) employs a hypernetwork to generate encoder and decoder weights that are input-dependent, yielding time-varying transformation maps. We derive a theoretical worst-case bound on the state estimation error. Numerical evaluations on four nonlinear benchmark systems show that input conditioning yields consistent improvements in estimation accuracy over static autonomous maps, with an average symmetric mean absolute percentage error (SMAPE) reduction of 29% across all non-zero input regimes.