Scattering theory and linear state-space estimation (Q1087854)

This paper shows that the estimation theory can be cast in the \(3\times 3\) scattering theory, subject to arbitrary boundary condition. Therefore one can take advantage of many results in scattering theory that have already been established. By using \textit{T. Kailath}'s [part I: IEEE Trans. Autom. Control AC-13, 646-655 (1968); part II (together with \textit{P. Frost}): ibid. AC-13, 655- 660 (1968)] innovations process definition, the state system has been uniquely decomposed into internal state and external state systems. The internal state system governs the smoothed estimate while the external state system controls the filtered estimate. Special cases are the Kalman-Bucy equation and the results of \textit{D. G. Lainiotis} [Inf. Sci. 7, 317-340 (1974; Zbl 0301.62051)] and Kailath when the system is subjected to some special conditions, such as a nonreflective boundary condition and zero inputs. It is also shown that estimation theory is associated with scattering theory with right and left transmission operators being self-adjoint to each other. An approximate solution for such a problem is suggested by the use of the algebraic star-products. A complete numerical analysis will be presented in our next paper in which we will take full advantage of some properties developed here.