Estimation of signals in an interconnection of LTI systems and unknown static maps

We consider the estimation of unknown signals in structured models that are interconnections of known linear dynamic systems and unknown static maps, and contain unmeasured exogenous disturbances. A main motivation for considering this is a system identification problem in which such an interconnection exists, and the static maps are to be identified when the inputs and/or outputs of the maps themselves are not available. Our approach is to search for estimates of the unmeasured signals based on three main types of criteria, these being that they are consistent with the linear dynamic system, that stochastic assumptions for disturbance processes are met, and that input-output pairs of the static maps are consistent with there being a static relationship between them. We consider various candidate criteria for enforcing the staticness consideration; they are essentially smoothness or regularizing criteria. These are what make our formulation different from other common estimation methods, for instance Kalman smoothing. We compare and contrast different methods using an example.