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 asymptotically unbiased linear minimum-variance recursive estimator in continuous time. Estimates of the parameters can be obtained simultaneously, but are found to be biased. By augmenting additional linear dynamic equations which represent an asymptotic expansion in the unknown parameters, a linear structure is formed which approximates the original nonlinear system. However, the initial conditions and additive process noise are not Gaussian. The convergence properties of the state variance for this expansion are illustrated analytically by a scalar dynamic system. 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. This approach can be extended to correlated parameter processes.