An Ordinary Differential Equation Technique for Continuous Time Parameter Estimation

An ordinary differential equation technique is developed via averaging theory and weak convergence theory to analyze the asymptotic behavior of continuous time recursive stochastic parameter estimators. This technique is an extension of L. Ljung´s work in discrete time. Using this technique, the following results are obtained for various continuous time parameter estimators. The recursive prediction error method, with probability one, converges to a minimum of the likelihood function. The same is true of the gradient method. The extended Kalman filter fails, with probability one, to converge to the true values of the parameters in a system whose state noise covariance is unknown. An example of the extended least squares algorithm is analyzed im detail. Analytic bounds are obtained for the asymptotic rate of convergence of all these estimators applied to this example.