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

Yahia Salaheldin Shaaban, Abdelrahman Sayed Sayed, M. Umar B. Niazi, Karl Henrik Johansson

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.

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

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 () is proposed with two input-conditioning strategies. First, an augmented observer approach () adds input-dependent corrections to the latent observer dynamics while retaining static transformation maps. Second, a dynamic observer approach () 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.

Paper Structure

This paper contains 15 sections, 2 theorems, 28 equations, 2 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Literature Gap
  3. Our Contributions
  4. Problem Formulation
  5. HyperKKL Framework
  6. Data Generation and the Operating Domain
  7. $\text{HyperKKL}_\text{obs}$: Augmented Observer
  8. $\text{HyperKKL}_\text{dyn}$: Dynamic Transformation Maps
  9. Comparison of $\text{HyperKKL}_\text{obs}$ and $\text{HyperKKL}_\text{dyn}$
  10. Theoretical Analysis and Design Trade-offs
  11. Numerical Validation
  12. Nonlinear Systems
  13. Baselines and Implementation
  14. Results and Discussion
  15. Conclusion

Key Result

Lemma 1

Let the state space $\mathsf X$ and the set of admissible inputs $\mathfrak U$ be compact. Assume the neural network architectures $\mathcal{N}_\text{enc}, \mathcal{N}_\text{dec}$, and $\mathcal{H}_\psi$ utilize continuously differentiable activation functions. Then, the following statements hold:

Figures (2)

  • Figure 2: $\text{HyperKKL}_\text{dyn}$ training architecture. The shared GRU processes the input window $u_{[t_k-\omega, t_k]}$ to produce a context embedding $h_t$, from which separate MLP heads predict weight perturbations $\widetilde{\theta}(t_k)$ and $\widetilde{\eta}(t_k)$. These are added to the frozen base weights $\overline\theta$ and $\overline\eta$, yielding an input-conditioned encoder and decoder. The losses $\mathcal{L}_\text{pde}$ and $\mathcal{L}_\text{rec}$ are backpropagated only through the hypernetwork parameters $\psi$.
  • Figure 3: Estimation results across all benchmark systems over 100 trials with random initial conditions. (a) $\text{HyperKKL}_\text{obs}$ tracks the ground truth across input transitions, while the Autonomous baseline accumulates persistent phase drift. (b)--(e) SMAPE distributions confirm statistically robust improvements.

Theorems & Definitions (5)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Remark 1