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By Bruce A. Francis

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4 (2b) Now define T 1 := G 11 + G (3a) 21 G12M2 (3b) T3:= M2G21. (3c) w The first statement the second statement z Tj (i =1-3) 1. The matrices the transfer matrix from Substitute Y2G T2;= Theorem Proof. 12M 2 = [Gu belong to RHCXJ' With K given by (2) to z equals T c T 2 Q T 3' follows from the realizations to be given below. For we have + GdI-KG G 22 = N 2M 2-1 22t1KG (4) 2dw and (2b) into (1-KG 22t1 and use (1) to get Thus from (2b) again Substitute this into (4) and use the definitions of Tj to get For computations matrices T, (i G (s) = = 1-3).

Such The inverse bilateral Laplace transform of F (s ) is = -Ce At B, f (t) = 0, in (all eigenvalues in Re s >0). t <0 t ~O . 5 Jf y(t)= (t-r)u(r)dr, t

But the latter condition implies left-coprimeness M, (ii) =? of [~]. (i): Choose, by right-coprimeness and Lemma 2, matrices X and Y 1Il RHoo such that Also, choose, by left-coprimeness, matrices Rand T in RHoo such that (1) Now define U:= TX (2a) eh. 4 34 V := Y - [0 I]NRX . (2b) Then we have from (1) and (2) that [I -RX] x] [[0 MI]N 0 Y I (3) v The two matrices on the left in (3) have inverses in RH()(), hence so does the matrix on the right in (3). The next step is to show that [ [0 MI]N [I [~]u1 V = all the matrices noting that ~ ~ G 21 G 22 in (4) at S We have [~]u [MO] V 0 [0 I] G [ Evaluate V is nonsingular.

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