Subject Training

Scientific Context In many real-world dynamical systems (industrial processes, energy systems, biomedical systems, dynamic networks), the internal state is not fully measurable and must be reconstructed from the available outputs. When these systems are nonlinear, subject to unknown inputs (disturbances, faults, attacks), and affected by modeling uncertainties, the state estimation problem becomes particularly challenging. Classical observer-based approaches such as Unknown Input Observers (UIO), robust observers (H?), and sliding mode observers provide strong theoretical guarantees but rely heavily on accurate model knowledge. However, in many practical situations, certain dynamics are poorly known or difficult to model. The emergence of machine learning techniques (deep neural networks, dynamic residual networks, Physics-Informed Neural Networks (PINNs)) now makes it possible to compensate for unknown dynamics. Nevertheless, purely data-driven approaches often lack formal stability guarantees. The scientific challenge is therefore to combine analytical rigor with the flexibility of machine learning.

Keywords : Nonlinear observer, State estimation, Unknown inputs, Hybrid observer, Machine learning, Neural networks, PINNs, Robustness.

Research Question : How can we design a nonlinear observer with unknown inputs that: preserves stability guarantees provided by model-based approaches, compensates for unknown dynamics through a learning module, improves robustness against disturbances and uncertainties.

Scientific Objectives : The work will be structured around three main axes : Theoretical Study Local observability analysis of nonlinear systems Study of unknown input observers Analysis of the limitations of classical approaches under strong uncertainties Design of a Hybrid Observer Propose an observer structure of the form: : \frac{d\hat{x}}{dt}=f\left(\hat{x},u\right)+L\left(y-\hat{y}\right)+N\left(\hat{x},u,y\right), where: f\left(\hat{x},u\right) represents the known nominal model, L\left(y-\hat{y}\right)+ is the classical correction term, N\left(\hat{x},u,y\right) is a neural network compensating for: unknown inputs, unmodeled dynamics, parametric uncertainties. Numerical Validation Implementation in MATLAB or Python on a nonlinear test system (e.g., a perturbed Lorenz system). A comparative analysis will be conducted between: a classical observer, a robust observer and the proposed hybrid observer. Performance will be evaluated in terms of estimation error, robustness to disturbances, sensitivity to uncertainties and computational cost.

Required Profile Master's degree (M2) in Control Engineering or Applied Mathematics Strong background in nonlinear dynamical systems Interest in state estimation and machine learning Good command of MATLAB or Python