Analytical Considerations of State Estimation: Regularization and Error Propagation

Monitoring dynamical processes requires the estimation of the entire state, which is only partly accessible by measurements. Most quantities must be determined via model based state estimation, which in general is an ill-posed inverse problem. Regularization techniques have to be applied. In most of the literature the initial data are also regularized. However, the initial data are typically unknown and only a rough guess is provided by experience or from the previous time window. Hence, to avoid undesired bias we omit the regularization of the initial data. The purpose of this paper is to analyse the effects of this problem formulation. We shortly deduce the problem formulation of state estimation with incorporated model error functions, discuss the ill-posedness and introduce the applied regularization. Then the first order necessary conditions are presented and the optimization problem is reduced by several variables. Hence, e.g. one otherwise necessary regularization parameter is dispensable. To analyse in detail the influence of the regularization parameters and of the coefficients of the model, we study, as a start, linear state equations as constraints. We show that the regularized inverse problem is well-posed with respect to L2-perturbations. However, a large spectral radius of the system matrix can lead to a large condition number and small disturbances in the measurements may propagate to large errors in the initial data. Moreover the issue of observability comes into play, since also eigenvectors of the system matrix which are close to the null space of the output election matrix can cause arbitrary large norms. From the engineering side the perhaps more interesting case of disturbances in the supremumnorm leads to bounds only dependent on the regularization parameters.