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The extended Kalman filter is an approximate filter for nonlinear systems, based on first-order linearization. Its use for the joint parameter and state estimation problem for linear systems with unknown parameters is well known and widely spread. Here a convergence analysis of this method is given. It is shown that in general, the estimates may be biased or divergent and the causes for this are displayed. Some common special cases where convergence is guaranteed are also given. The analysis gives insight into the convergence mechanisms and it is shown that with a modification of the algorithm, global convergence results can be obtained for a general case. The scheme can then be interpreted as maximization of the likelihood function for the estimation problem, or as a recursive prediction error algorithm.