With the rising complexity of dynamical systems generating ever more data, learning dynamics models appears as a promising alternative to physics-based modeling. However, the data available from physical platforms may be noisy and not cover all state variables. Hence, it is necessary to jointly perform state and dynamics estimation. In this paper, we propose interconnecting a high-gain observer and a dynamics learning framework, specifically a Gaussian process state-space model. The observer provides state estimates, which serve as the data for training the dynamics model. The updated model, in turn, is used to improve the observer. Joint convergence of the observer and the dynamics model is proved for high enough gain, up to the measurement and process perturbations. Simultaneous dynamics learning and state estimation are demonstrated on simulations of a mass-spring-mass system.

This paper presents a first step towards tuning observers for nonlinear systems. Relying on recent results around Kazantzis-Kravaris/Luenberger (KKL) observers, we propose to design a family of observers parametrized by the cutoff frequency of a linear filter. We use neural networks to learn the mapping between the observer and the nonlinear system as a function of this frequency, and present a novel method to sample the state-space efficiently for nonlinear regression. We then propose a criterion related to noise sensitivity, which can be used to tune the observer by choosing the most appropriate frequency. We illustrate the merits of this approach in numerical simulations.

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Identifying dynamical systems from experimental data is a notably difficult task. Prior knowledge generally helps, but the extent of this knowledge varies with the application, and customized models are often needed. We propose a flexible framework to incorporate a broad spectrum of physical insight into neural ODE-based system identification, giving physical interpretability to the resulting latent space. This insight is either enforced through hard constraints in the optimization problem or added in its cost function. In order to link the partial and possibly noisy observations to the latent state, we rely on tools from nonlinear observer theory to build a recognition model. We demonstrate the performance of the proposed approach on numerical simulations and on an experimental dataset from a robotic exoskeleton.