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minors of R ; A [ [ R ; p t ] is the Mcl\Iillan degree of the pole pn of R and IVote Added in Proof: Recent work shows that condition (9) rea [ R ] = ~~= l A I R ; p k ] [24]. quires that the return difference I + F&(s) not have a zero of Corollary 1: Let (?be defined by (3) and let L Y~ be defined by Fact 3. transmission at, P, = 1 7 % ' ' . J 1271. Under these conditions, for any k E { 1, 2,. . .,l] for which Re P k > 0, REFEREXCES det X k ( p s ) # 0 if and only if where in the triangular Hankel matrix, t.he Rka's are the coefficient matrices of R k defined by (4). Proof: From (17) and (19)-(20),m ) l ; hence and for any integer Bk > -,t lim (spk)'*det [I + F&(s)] = 0. Hence, by Lemma 2, (9) is true if and only if Q[det [ I f Fk] : p k ] = 7s. Let [ A , B, C , 01 be any minimal realization of the strictly proper element R of CnXn(s) defined by (15); then, because of the coprime factorization (16), det D ( s ) = c det (SI --4) [lti], [20]. S o a d e t (SI - 0). 1 : is still valid for poles on the boundary (i.e., Re pk = 0) whenever Gp(s) is meromorphic in a neighborhood of such p s . This will always be the case for differential delay systems.

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