For complex mechanical systems modelled as distributed parameter systems, the description of the system’s behaviour can be reduced to determining some of the system’s unknown parameters. This is the case for instance in mechanical cardiac modelisation, but any other dynamical system with unknown parameters could be considered. Given the computational complexity of distributed parameter systems, reduced parameter estimation is applied, parallelly with a state estimator, which is also known as adaptive filtering. Mortensen examined the optimal state estimation problem from a deterministic viewpoint, whose solution requires the solution of an optimal control problem (in particular a Hamilton-Jacobi-Bellman equation), which results in an optimal state estimator. However this is computationally demanding even for small dimensional systems. Our goal was to obtain the equations for the reduced version of this filter, and implement it. Such an optimal filter, is suited for state estimation problems of nonlinear systems, the MACS team project develops such filters for cardiac diagnostics.
In this thesis we state and solve the problem for reduced state estimation. We examine implementation strategies, and present the chosen method. We analyse the problems arising from discretisation. Finally we present the results of numerical simulations, and compare them to reduced versions of other well-known filtering methods, such as the Extended Kalman Filter (ROEKF) and the Unscented Kalman Filter (ROUKF). The work has been implemented in the Verdandi data assimilation library, which is a generic library for identification problems, mostly aimed at high dimensional systems.