HyperKKL: Enabling Non-Autonomous State Estimation through Dynamic Weight Conditioning

Yahia Salaheldin Shaaban; Salem Lahlou; Abdelrahman Sayed Sayed

eess.SYFeb 26, 2026

HyperKKL: Enabling Non-Autonomous State Estimation through Dynamic Weight Conditioning

Authors: Yahia Salaheldin Shaaban, Salem Lahlou, Abdelrahman Sayed Sayed

Organizations: MBZUAI, Abu Dhabi, UAE · Univ Gustave Eiffel, COSYS-ESTAS, F-59657 Villeneuve d’Ascq, France

Abstract

This paper proposes HyperKKL, a novel learning approach for designing Kazantzis-Kravaris/Luenberger (KKL) observers for non-autonomous nonlinear systems. While KKL observers offer a rigorous theoretical framework by immersing nonlinear dynamics into a stable linear latent space, its practical realization relies on solving Partial Differential Equations (PDE) that are analytically intractable. Current existing learning-based approximations of the KKL observer are mostly designed for autonomous systems, failing to generalize to driven dynamics without expensive retraining or online gradient updates. HyperKKL addresses this by employing a hypernetwork architecture that encodes the exogenous input signal to instantaneously generate the parameters of the KKL observer, effectively learning a family of immersion maps parameterized by the external drive. We rigorously evaluate this approach against a curriculum learning strategy that attempts to generalize from autonomous regimes via training heuristics alone. The novel approach is illustrated on four numerical simulations in benchmark examples including the Duffing, Van der Pol, Lorenz, and Rössler systems.

Figures & tables

Figure 1: Dynamic HyperKKL Architecture (Phase 2). The base encoder $\hat{\mathcal{T}}_{\theta^{\text{base}}}$ and decoder $\hat{\mathcal{T}}^{*}_{\phi^{\text{base}}}$ , pre-trained on autonomous dynamics in Phase 1, are frozen. A shared LSTM encoder processes the input window $u_{[t-w,\,t]}$ and produces a hidden state $h_{t}$ , which is passed to two separate MLP decoder heads that predict chunked weight perturbations $\Delta\theta_{\text{enc}}$ and $\Delta\phi_{\text{dec}}$ . The perturbations are added to the frozen base weights, yielding input-conditioned maps $\hat{\mathcal{T}}(x;\,\theta^{\text{base}}+\Delta\theta)$ and $\hat{\mathcal{T}}^{*}(z;\,\phi^{\text{base}}+\Delta\phi)$ .

Method	Duffing				Van der Pol
Method	Zero	Const	Sin	Sqr	Zero	Const	Sin	Sqr
Autonomous	0.04 (5.6)	0.63 (66)	0.26 (26)	0.33 (31)	0.15 (7.0)	0.35 (23.1)	0.23 (9.8)	0.25 (10.5)
Curriculum	0.27 (33)	0.64 (63)	0.44 (41)	0.57 (46)	1.10 (51.4)	1.00 (54.3)	1.15 (51.5)	1.15 (51.7)
Static HyperKKL	0.04 (5.6)	0.39 $\downarrow$ (38)	0.10 $\downarrow$ (9.3)	0.17 $\downarrow$ (14)	0.12 $\downarrow$ (5.3)	0.26 $\downarrow$ (14.1)	0.24 (10.2)	0.25 (10.8)
Dynamic HyperKKL	0.08 (8.2)	0.56 $\downarrow$ (62)	0.24 $\downarrow$ (25)	0.27 $\downarrow$ (28)	0.12 $\downarrow$ (5.0)	0.38 (25.6)	0.21 $\downarrow$ (8.6)	0.22 $\downarrow$ (9.1)

Table 1: State Estimation Performance (Duffing and Van der Pol): RMSE (SMAPE %) . Lower is better.

Method	Rossler (Chaotic)				Lorenz (Chaotic)
Method	Zero	Const	Sin	Sqr	Zero	Const	Sin	Sqr
Autonomous	1.14 (6.7)	1.75 (8.6)	1.47 (7.6)	1.48 (8.3)	5.56 (18)	5.50 (18)	5.58 (18)	5.55 (18)
Curriculum	5.58 (35)	5.55 (35)	5.94 (37)	5.61 (38)	11.5 (41)	11.4 (41)	11.6 (42)	11.6 (42)
Static HyperKKL	1.14 (6.7)	1.50 $\downarrow$ (9.5)	1.70 (10)	1.75 (12)	5.56 (18)	16.0 (51)	16.3 (52)	16.2 (51)
Dynamic HyperKKL	1.01 $\downarrow$ (5.1)	1.57 $\downarrow$ (7.7)	1.38 $\downarrow$ (6.0)	1.36 $\downarrow$ (6.9)	6.67 (22)	6.64 (22)	6.67 (22)	6.66 (22)

Table 2: State Estimation Performance (Rossler and Lorenz): RMSE (SMAPE %) . Lower is better.

Figure 2: Duffing (Square Input): The Dynamic HyperKKL adapts to discontinuous square waves without the transient spikes seen in static methods.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 3 — Figure 3: Duffing System: State estimation time-series.

Panel of Figure 4 — Figure 4: Van der Pol System: State estimation time-series.

Panel of Figure 5 — Figure 5: Rossler System (Chaotic): State estimation time-series.

Panel of Figure 6 — Figure 6: Lorenz System (Chaotic): State estimation time-series.

State Space Models

State space models represent sequences through evolving latent states, enabling efficient temporal modeling, system identification, and forecasting.

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1Tsinghua University, PR China · 2China University of Petroleum-Beijing at Karamay, PR China · 3Xinchen Qihang Inc., PR China

Mar 31, 2026J/K move · Enter open · S save

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

Mohamed bin Zayed University of Artificial Intelligence (MBZUAI), Abu Dhabi, UAE. · Univ Gustave Eiffel, COSYS-ESTAS, F-59657 Villeneuve d’Ascq, France. · Department of Decision and Control Systems and Digital Futures, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden.

Dynamical Systems

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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.

Abstract

State Space Models

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