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Chapter 10
H 2 Model-order reduction from order
N to N-2
When reducing the model-order of a given system H(s) from order N to N −2, hence
k = 2, the degree of the polynomial ρ(s) in Equation (8.20) is smaller than or equal
to one:
ρ(s) =1+ρ 1 s 0. (10.1)
Using this polynomial, (8.44) yields the following system of equations for the co-order
k = 2 case:
⎛
⎜
⎝
1+ρ 1δ 1
e(δ 1) x2 1
1+ρ 1δ 2
e(δ 2) x2 2
.
1+ρ 1δ N
e(δ N )
x2 N
⎞
⎛
− M(δ
⎟ 1 ,...,δ N )
⎜
⎠
⎝
x 1
x 2
.
x N
⎞ ⎛ ⎞
0
0
=
⎟ ⎜
, (10.2)
⎠ ⎝ .
⎟
⎠
0
where M(δ 1 ,...,δ N )=V (−δ 1 ,...,−δ N ) V (δ 1 ,...,δ N ) −1 . This system of equations
is parameterized rationally by the parameter ρ 1 .
As stated by Equation (8.45), there is one additional constraint on the coefficients
of ã(s) involved in the co-order k = 2 case. This constraint is defined by:
ã N−1 (x 1 ,x 2 ,...,x N )=γ 1,1 x 1 + γ 1,2 x 2 + ...+ γ 1,N x N =0. (10.3)
Note that the quantities γ 1,1 ,...,γ 1,N in this constraint depend on the poles δ 1 ,...,δ N
and that it does not contain the parameter ρ 1 . Note furthermore that the system of
equations (10.2) yields N equations in the quantities x 1 ,...,x N , and the parameter
ρ 1 . Together with the additional constraint ã N−1 (x 1 ,x 2 , ...,x N ) = 0, there are in
total N + 1 equations in the N + 1 unknowns.
To solve the H 2 model-order reduction problem for the co-order k = 2 case, we
need to compute the solutions (x 1 ,x 2 ,...,x N ,ρ 1 ) of the system of equations (10.2)
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