On Kalman filtering for conditionally Gaussian systems with random matrices
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We consider linear stochastic systems with additive white Gaussian noise, with the added generality that the system matrices are random and adapted to the observation process. The main result of this paper is that in order for the standard Kalman filter to generate the conditional mean and conditional covariance of the conditionally Gaussian distributed state, it is sufficient for the random matrices to be finite with probability one at each time instant. This generalizes the best previous results available to date, to our knowledge, which require the more stringent hypothesis that the entries of the random matrices should possess finite second moments at each time instant. A significant application of the results of this paper is to the problem of recursive identification of the unknown parameters of a controlled linear stochastic system. In such problems, the observation matrix is typically generated by complicated nonlinear feedback, as for example in adaptive control, and the finiteness of the second moments is difficult, if not impossible, to establish, while the finiteness with probability one has been established in many applications. 1989.