Convergence Properties of the Kalman Inverse Filter

The Kalman filter has a long history of use in input deconvolution where it is desired to estimate structured inputs or disturbances to a plant from noisy output measurements. However, little attention has been given to the convergence properties of the deconvolved signal, in particular the conditions needed to estimate inputs and disturbances with zero bias. The paper draws on ideas from linear systems theory to understand the convergence properties of the Kalman filter when used for input deconvolution. The main result of the paper is to show that, in general, unbiased estimation of inputs using a Kalman filter requires both an exact model of the plant and an internal model of the input signal. We show that for unbiased estimation, an identified subblock of the Kalman filter that we term the plant model input generator (PMIG) must span all possible inputs to the plant and that the robustness of the estimator with respect to errors in model parameters depends on the eigenstructure of this subblock. We give estimates of the bias on the estimated inputs/disturbances when the model is in error. The results of this paper provide insightful guidance in the design of Kalman filters for input deconvolution.