Abstract
In this paper we consider some new algorithms for computing the Kalman-Bucy gain in stationary systems requiring a number of equations of ordern (rather thann 2) whenever the ordern of the system is much larger than the dimension of the output. These equations were independently obtained by Kailath and Lindquist in continuous and discrete time respectively. We briefly discuss the relations with some recent related results due to Casti, Kalaba & Murthy and Rissanen. Some of the reasons for these reductions are inherent in the properties of general stationary processes, and therefore a considerable portion of the paper is devoted to exploring the connections with some previous work by Levinson, Whittle and Wiggins & Robinson, and also with the Szegö theory of polynomials orthogonal on the unit circle and some continuous analogs due to Krein. We demonstrate that the Bellman-Krein formula is the fundamental relation in continuous time, the trick being to introduce a “reversed time” counterpart of the weighting function (Fredholm resolvent). This is suggested by the “forward and backward innovation” approach in a previous paper by the author, the essential relations of which we reformulate in terms of Fredholm integral equations (in continuous time) and Toeplitz equations (in discrete time). Therefore we also derive the discrete-time Bellman-Krein formulas of which there are actually two—one corresponding to the one-step predictor and one to the pure filter. In this way we shall be able to pin down the reasons for the striking discrepancies between the continuous-time and the discrete-time cases. Finally we clarify the relations between Levinson's equations and Chandrasekhar'sX- andY-functions.
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Communicated by W. Fleming
This research was carried out while the author held a visiting position at the Division of Applied Mathematics, Brown University, Providence, Rhode Island, and was supported by the National Science Foundation under grant NSF-46614.
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Lindquist, A. On Fredholm integral equations, Toeplitz equations and Kalman-Bucy filtering. Appl Math Optim 1, 355–373 (1975). https://doi.org/10.1007/BF01447958
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DOI: https://doi.org/10.1007/BF01447958