Energy-optimal state reconstruction for deterministic nonlinear dynamical systems

This thesis is concerned with the Mortensen observer – also called minimum energy estimator – designed for the reconstruction of the state of disturbed nonlinear dynamical systems based on model information and measured data. The concept that can be interpreted as a generalization of the well-known Kalman filter to nonlinear systems relies on energy minimization by means of optimal control theory. In particular the well-posedness of the concept is studied. The fact that both the definition and the dynamical characterization of the observer depend on the value function associated with the optimal control problem calls for an in-depth study of said problem. Under the assumption of sufficiently small disturbances an application of classical tools from optimal control theory and sensitivity analysis shows that for finite-dimensional systems driven by quadratic dynamics the value function locally is a classical solution to the associated Hamilton Jacobi Bellman equation. A further analysis evolving around differential Riccati equations ensures that the second spatial derivative exists and is invertible at all times leading to well-posedness of the observer. Subsequently the observer is generalized to an infinite-dimensional system. Using the example of a nonlinear wave equation it is demonstrated that the concept can be applied to partial differential equations as well. A similar analysis of the associated optimal control problem based on mild solutions and evolution operators ensures that an energy minimizing state exists at any point in time. In contrast to the finite-dimensional case, however, the lack of regularity in time prohibits generalizing the proof that the energy minimizer satisfies the observer equation. Instead local well-posedness of said equation is shown via an application of a fixed point argument. Since the approximation of value functions associated with systems of medium or high state-space-dimension is a daunting task, only very few numerical schemes for the realization of the Mortensen observer are available. None of them seem to exceed dimension three. To accompany the theoretical results of well-posedness a scheme for the approximation of the value function associated with the Mortensen observer and a subsequent realization of the estimator is proposed. The thesis is concluded by a numerical study of said scheme using various systems of dimension up to 40.