HyperKKL: Learning KKL Observers for Non-Autonomous Nonlinear Systems via Hypernetwork-Based Input Conditioning
Authors: Yahia Salaheldin Shaaban, Abdelrahman Sayed Sayed, M. Umar B. Niazi, Karl Henrik Johansson
Organizations: 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.
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 (HyperKKLobs) adds input-dependent corrections to the latent observer dynamics while retaining static transformation maps. Second, a dynamic observer approach (HyperKKLdyn) 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.
Figures & tables
Fig. 1 : HyperKKLdyn training architecture. The shared GRU processes the input window u[tk−ω,tk] to produce a context embedding ht , from which separate MLP heads predict weight perturbations θ(tk) and η(tk) . These are added to the frozen base weights θ and η , yielding an input-conditioned encoder and decoder. The losses Lpde and Lrec are backpropagated only through the hypernetwork parameters ψ .
Fig. 2 : Estimation results across all benchmark systems over 100 trials with random initial conditions. (a) HyperKKLobs tracks the ground truth across input transitions, while the Autonomous baseline accumulates persistent phase drift. (b)–(e) SMAPE distributions confirm statistically robust improvements.
Duffing
Van der Pol
Method
Zero
Const
Sin
Sqr
Zero
Const
Sin
Sqr
Autonomous
19.3
74.6
37.1
40.7
17.7
32.6
20.2
22.2
Curriculum
33.0
80.5
67.3
71.9
51.4
76.3
81.2
81.4
HyperKKL obs
5.6 ↓
21.1 ↓
22.7 ↓
21.3 ↓
5.3 ↓
26.0 ↓
15.5 ↓
17.9 ↓
HyperKKL dyn
8.2 ↓
57.3 ↓
29.7 ↓
30.7 ↓
5.0 ↓
32.7
18.8 ↓
20.8 ↓
Rössler
FitzHugh–Nagumo
TABLE I : State estimation SMAPE averaged over 100 trials. Lower is better. Bold : best per column. ↓ : improvement over Autonomous.
Dynamical systems studies how states evolve over time, including stability, feedback, chaos, and transitions in physical, biological, and engineered systems.
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.
Yahia Salaheldin Shaaban, Salem Lahlou, Abdelrahman Sayed Sayed
MBZUAI, Abu Dhabi, UAE · Univ Gustave Eiffel, COSYS-ESTAS, F-59657 Villeneuve d’Ascq, France
This paper proposes a computable state-estimation error bound for learning-based Kazantzis--Kravaris/Luenberger (KKL) observers. Recent work learns the KKL transformation map with a physics-informed neural network (PINN) and a corresponding left-inverse map with a conventional neural network. However, no computable state-estimation error bounds are currently available for this approach. We derive a state-estimation error bound that depends only on quantities that can be certified over a prescribed region using neural network verification. We further extend the result to bounded additive measurement noise and demonstrate the guarantees on nonlinear benchmark systems.
Hannah Berin-Costain, Harry Wang, Kirsten Morris +1
Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada · University of Toronto, Toronto, ON M5S 1A1, Canada
In many safety-critical applications, control of uncertain dynamical systems relies on observers that estimate states and external disturbances. Neural network observers can improve estimation accuracy, but certifying their Lyapunov stability via Linear Matrix Inequality (LMI) constraints leads to large-scale semidefinite programs (SDPs) that are difficult to solve for large networks. To overcome this scalability bottleneck, we propose a novel two-stage training framework for provably stable neural network observers. Our approach decouples the optimization into a point-guided Lyapunov pre-training phase, which rapidly achieves high estimation accuracy and local stability over sampled states, followed by an LMI fine-tuning phase that efficiently satisfies a strict global Lyapunov stability certificate. We provide formal theoretical guarantees for local stability radii and probabilistic coverage over a prescribed compact error-state domain under specified regularity and sampling assumptions. Experiments on nonlinear control benchmarks and X-29 aircraft ablations show that our LMI-certified neural network observers train significantly faster than direct LMI-based methods and generalize robustly across diverse systems, achieving improved tracking accuracy over a range of observer baselines. The code is available at https://github.com/Berry-Myon/LearningNeuralNetworkObserver.
Zhangyi Wang, Jiaxu Liu, Song Chen +2
College of Control Science and Engineering, Zhejiang University, China · School of Science, Huzhou Normal University, China · Department of Mathematics, National University of Singapore, Singapore