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Extra info for Advances in Theory and Applications : System Identification and Adaptive Control, Part 2

Sample text

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.

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). Comment 3. An alternative to using the "data-generating model" (l)-(2) as our starting point is to use a "predictor model," that is, the model that generates the one-step-ahead predictions of y(t) given the infinite past.

This is what we SELECTION OF IDENTIFIABLE PARAMETERIZATIONS V. 59 CANONICAL AND PSEUDOCANONICAL FORMS We now describe SS, MFD, and ARMA canonical forms for a p x s transfer function K(z) of order n, assuming that K(z) e V , where μ = (η χ , Recall that these Kronecker indices are linear dependence relations be­ 'V determined by the tween the rows of H, „[K]. For simplicity of notation we as­ sume that y(t) = K(z)m(t) (44) rather than the original model (33) or (1). It is trivial, of course, to split K(z) into [G(z) · H(z) - I ] , to replace m(t) by u(t) e(t).