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25. 24) is called an algebraic Constrained Least Square method, abbreviated, the algebraic CLS method. We show examples for the algebraic CLS method. We can show that the analytic approximate realization algorithm given in the reference [Hasegawa, 2008] produces the same systems as the ones obtained by the algebraic CLS method in the sense of the numerical calculation. In addition, we compare the method with the common method for noise processing which is called AIC. 26. 3 ⎦ , h = [10, 5, −5]. 41 Let an added noise be given in Fig.

3 Let F ∈ Rn×n be given as below. Let g0 be g0 = e1 , where e1 = [1, 0, · · · , 0]T ∈ Rn . 4 Let hs be hs = [Ia (1 )− I¯a (1 ), Ia (1)− I¯a (1),· · ·, Ia (0n1 −2 |1)− I¯a (0n1 −2 |1)]. ⎡ ⎤ 0 · · · 0 α1 ⎢ .. ⎥ ⎢ 1 . α2 ⎥ ⎢ ⎥ F0 = ⎢ . ⎥. ⎣ .. . 0 .. ⎦ 0 1 αn [proof] By 1), the approximate part of the data can be excluded in the sense of the norm of Hankel matrix Ha (p,p) ¯ . 14) corresponds to the matrix composed of eigenvectors of Ha (p,p) ¯ . 17). 17) in the sense of a linear combination. ˆ a (n +1,p) Therefore, we obtain the approximate Hankel matrices H ¯ (n1 + 1 ˆ 1, 0).

Ia (i + ¯i) ⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠ where i ≤ p and ¯i ≤ p¯. i ˆ a (p,p) Note that the column vectors of H ¯ is represented by S l Ia . When we actually treat the approximate and noisy realization problem, we will use a notation Ha (n1 ,N−n1 ) expressed as follows: Ha (n1 ,N−n1 ) = [Ia , · · · , Sln1 −1 Ia ]. 14. Let the rank of a finite-sized Hankel matrix H ¯ be n. 9) for S nl Ia = 2) Let e1 be e1 = [1, 0, · · · , 0]T . 3) Let hs be hs = [Ia (1), Ia (2), · · · , Ia (n)]. n i=1 αi S i−1 Ia . l [proof] It is obvious from the definition of behavior of the system.

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