State estimation analysed as an inverse problem

Abstract: 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 estima-tion. Since in general only noisy data are given, state estimation yields an ill-posed inverse problem. Observability guarantees a unique least squares solution. While well-posedness as well as observability is a qualitative behaviour, the quantitative behaviour can be described using the concept of condition numbers. They depend, like stability, crucially on the chosen norms. In this context we shortly review on ill-posed problems, observability and conditioning, and introduce, as a quantification, an observ-ability measure based on condition numbers. For the linear case we show the connection to the well known observability Gramian. For state estimation regularization techniques concerning the initial data are commonly applied in addition to the least squares ansatz. However, we show that the least squares formulation is well-posed and avoids otherwise possibly occuring bias. The introduced observ-ability measure gives a lower bound on the conditioning of this problem formulation. Consequently, inspite of well-posedness a low observability measure causes bad conditioning. Introducing possible model error functions we leave the finite dimensional setting. To analyse in detail the influence of the regularization parameters and of the coefficients of the model, we study, a