An approximation method for estimation in linear systems with parameter uncertainty

Estimation of the state variables of a linear system with parameter uncertainties is performed using an asymtotically unbiased linear minimum-variance recursive estimator in continuous time. Estimates of the parameters can be obtained simultaneously, but are found to be biased. A linearized structure is formed for this nonlinear problem by augmenting additional linear dynamic equations which represent an asymtotic expansion in the unknown parameters. The convergence properties of the state variance for this expansion are illustrated analytically by a scalar state variable example. The numerical aspects of this example illustrate the behavior of the actual variance of the error in the state estimate and the predicted error variance as the order of the approximation increases. For the vector state problem, only the multidimensional dynamic system in canonical form with a single output is developed. For an n dimensional system with n unknown constant parameters, a first order approximation requires n additional linear equations. It is seen that this approach can be extended to correlated parameter processes.