State Estimation analysed as Inverse Problem

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. 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 observability 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 observability 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, as a start, linear state equations as constraints. Linear problems appear nearly always as a subproblem of the nonlinear case, and, hence, their features we have to face solving for the nonlinear model. We show that state estimation formulated as optimization problem omitting regularization of the initial data leads to a well-posed problem with respect to L2and L∞disturbances. However, if the introduced measure of observability is low, the condition numbers of the evolving operators can be arbitrarily large. Small disturbances in the L2-measurement may propagate to large errors in the states. Nevertheless, for the probably in praxis more relevant L∞-norm perturbations yield errors in the initial data bounded independently of the system matrix. Finally, we draw conclusions and emphasize the issue of the appropriate norms for state estimation.