Generalised Wiener and Kalman filters for uncertain systems

A linear stationary optimal filtering problem is considered in which the plant dynamics and noise covariances are incompletely known. Unknown plant parameters in the plant model, such as gains and time constants are treated as random variables with specified means and variances. The assumption is made that the parameter variations are small, however, if the model is linear in the unknown parameters, the assumption is unnecessary. Generalised Wiener and Kalman-Bucy filters are derived on the basis of transfer-function matrix or state-space representations of the plant, respectively. These estimators are similar in structure to the case where the plant dynamics are completely determined and significantly extend the uses of such estimators to an important class of uncertain systems. An application of the generalised filter to the LQG optimal control of plants with unknown disturbances is also described and a separation principle is shown to apply.