Nonlinear observer design - a data-driven optimal control perspective

• Applicant: Professor Dr. Tobias Breiten
• Subject Area: Mathematics ; Automation, Mechatronics, Control Systems, Intelligent Technical Systems, Robotics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 504768428

A mathematical description of complex dynamical processes is often built upon simplifying and idealized assumptions. Unknown perturbations of the underlying mathematical models, e.g., inaccurate sensor measurements, thus have to be incorporated accordingly. In this context, two common research fields are that of filter (stochastic viewpoint) and observer (deterministic viewpoint) design. In this project, we will follow the latter framework and focus on an optimal control formulation for nonlinear observer design.The main emphasis will be on a dynamic programming formalism which leads to a nonlinear high-dimensional time-dependent Hamilton-Jacobi-Bellman (HJB) equation. Recent theoretical and numerical results have shown that, despite the well-known "curse of dimensionality", the HJB equation can nowadays be approximately solved for considerably large dimensions. In particular, the combination of local approximation concepts, low rank tensor formats and novel data-driven approaches has lead to remarkable success in the field of optimal feedback control. The goal of this project is to exploit a rather natural feedback perspective on (optimal) observer design and incorporate appropriate approximation techniques for the corresponding HJB equation. Several challenges arise in this context as the HJB equation of interest is generically time-dependent and non homogeneous. For example, the optimal observer is based (implicitly) on the Hessian matrix of the value function thereby requiring different regularity (w.r.t. differentiability as well as invertibility) results. The project consists of two main work packages (each subdivided) which are grouped according to theoretical (WP1) and numerical (WP2) focus areas. For the theoretical considerations, we will utilize and extend some of our recent results on local approximations of minimal value functions for abstract optimal control problems. Our numerical contributions will benefit from low rank and model reduction methods for tensor structured (non-)linear equations.

• DFG Programme: Research Grants