Stochastic Model Validation and Estimation for Linear Discrete-Time Systems with Partial Prior Information

The problem of recursive estimation and model validation for linear discrete-time systems with partial prior information is examined. More specifically, an underlying linear discrete-time system is considered where the statistics of the driving noise is assumed to be known only partially; i.e. a class of noise inputs is given from which the underlying actual noise is assumed to be chosen. A set-valued estimator is then derived and the conditional expectation is shown to belong to an ellipsoidal set consistent with the measurements and the underlying noise description. When the underlying noise is consistent with the underlying partial model and a sequence of realized measurements is given then the ellipsoidal, set-valued, estimate is computable using a Kalman filter-type algorithm. The estimator inherently solves a stochastic model validation problem whereby it is possible to estimate the consistency between the assumed model, knowledge on the partial prior noise statistics and the measured data.