link to my thesis

Chapter 11
H 2 Model-order reduction from order
N to N-3
When the order of the model is reduced from order N to N − 3, the co-order k =3
case, the degree of the polynomial q(s) in Equation (8.20) is smaller than or equal to
k − 1=2: q(s) =q 0 + q 1 s + q 2 s 2 0. For q 0 0, the polynomial ρ(s) in Equation
(8.20) becomes: ρ(s) = q(s)
q 0
Then the polynomial ρ(δ i ) becomes 1 + ρ 1 δ i + ρ 2 δi
2
=1+ρ 1 s + ρ 2 s 2 0, where ρ 1 = q1
q 0
and ρ 2 = q2
q 0
.
for i =1,...,N in the system
of equations (8.44). This yields a system of quadratic equations containing the two
unknown parameters ρ 1 and ρ 2 with the following structure:
⎛
⎜
⎝
(1+ρ 1δ 1+ρ 2δ 2 1 )
e(δ 1)
x 2 1
(1+ρ 1δ 2+ρ 2δ 2 2 )
e(δ 2)
x 2 2
(1+ρ 1δ N +ρ 2δ 2 N )
e(δ N )
.
x 2 N
⎞
⎛
− M(δ 1 ,...,δ N )
⎟
⎜
⎠
⎝
x 1
x 2
.
x N
⎞ ⎛ ⎞
0
0
=
⎟ ⎜
⎠ ⎝ .
⎟
⎠
0
(11.1)
where M(δ 1 ,...,δ N )=V (−δ 1 ,...,−δ N )V (δ 1 ,...,δ N ) −1 .
As discussed in Section 8.5, there are two constraints on the coefficients of the
polynomial ã(s) in the co-order k = 3 case which have to be satisfied in order to find
feasible approximations G(s) of order N − 3. Using Equation (8.45) for k = 3, these
constraints are denoted 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
(11.2)
⎩
ã N−2 (x 1 ,x 2 ,...,x N )=γ 2,1 x 1 + γ 2,2 x 2 + ...+ γ 2,N x N =0
Note that the quantities γ i,1 ,...,γ i,N
δ 1 ,...,δ N .
(for i =1, 2) in these constraints depend on
195