By Joe DeFranco
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We shall first show that Y + C (S f l U-) + (S fl U ' ) . 3 above we have (Hwu)(tJEXCS, so that for t 2 O necessarily (W'u)(t) = y ( t ) - (Hwzr)(t) E S as well. In fact (Hwzr)(t) E S f l U- and ( W + u ) ( t )E n U+, given the explicit dependence on the past and future of 14. Taking finite linear combinations of the form E a i y ( t k ) , O ~ E E ' ~ , tk 2 0, and then closing in the Hilbert space norm of second-order random variables gives immediately the inclusion we want. Second, by projecting obliquely x'EX+ onto the direct sum (40), we obtain s x+ = E,,u"x+ I u-] + EI,U-[x+ 1 U+].
We shall now discuss the geometric construction of the state space of a deterministic realization of y . To this purpose we shall introducea technical assumption of “sufficient richness” of the input process. e. ,(e’”) > c l , c > 0, or to all canonical angles between the past and future subspaces of U being strictly positive (or, in turn, to all canonical correlation coefficients between past and future of the input process being strictly less than one). A slightly stronger version of this condition is found in , Chapter 11, Sect.
16. P Faurre, P Chataigner. Identification en temp reel et en temp differee par factorisation de matrices de Hankel French-Swedish colloquium on process control, IRIA Roquencourt, 1971. 17. P Faurre, JP Marmorat. CR Acad Sc Paris, A 268:978-981, 1969. 18. P Faurre, M Clerget, F Germain. OpQateurs Rationnels Positifs. Dunod, 1979. 19. PA Fuhrmann. Linear Operators and Systems in Hilbert Space. McGraw Hill, 1981. 20. MR Gevers, BD0 Anderson. Intern J Control 33:777-809, 1981. 34 Picci 21. MR Gevers, BD0 Anderson.
Built like a Badass by Joe DeFranco