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By Cornelius T. Leondes

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In order for (17)-(18) to be a representation of (l)-(2), the following conditions must hold: P"1(D)[Q(D) i R(D)] = [G(D) \ H(D)], _ When th_eve is P(D)y(t) = R(D)e(t) no exogenous input and is called an u(t)3 ARMA (19) Eq. (17) model. becomes SELECTION OF IDENTIFIABLE PARAMETERIZATIONS 45 where 00 1 G(D) Δ GiD" ) = 2, 1 ^D1, and similarly for H(D). (i) (ii) (20) In particular (19) implies that P Q is nonsingular and P Q = R~ , (21) det P(z) and det R(z) have all their roots in |z| > 1. (22) In addition to (21) , we will often want ARMAX models for which P 0 = R0 = I p .

Is a set of models: M {M la G I }, where I is an index set. a' a a = SELECTION OF IDENTIFIABLE PARAMETERIZATIONS 49 Most often the index set is noncountable. The model set is then smoothly parametrized by a parameter vector Θ of dimension d, where Θ G D Definition C R . 4. A model structure M is a differentiable mapd * ping from a subset D__ of R to a model set: M : Θ G D w -► Μ(θ) G M M M where Μ(θ) is defined by any one of the stable parametrized algebraic operators described earlier. Example 1, We illustrate by a scalar example.

Z p } , and let z k D k = D k z k = 1 so that M(z)M(D) = M(D)M(z) = I . Then P M(D) [P(z) · Q(z) ·'R(z)] = [P(D) : Q(D) : R(D)] . (25a) 46 MICHAEL GEVERS AND VINCENT WERTZ Similarly M(z) [ P ( D ) I Q(D) - R(D) ] = [P(z) : Q(z) : R ( z ) ] . (25b) Note that the row degrees of [P(z) j Q(z) · R(z)] are identical to the row degrees of [P(D) j Q(D) · R(D)]; however, the row de­ grees of P(D) are not equal to those of P(z), and deg det P(D) is not equal to the order of the system (see [30] and [31] for details).

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