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8.6. COMPUTING THE APPROXIMATION G(S) 149
ρ k−1 )[x 1 ,x 2 ,...,x N ]/I(ρ 1 ,...,ρ k−1 ) where I(ρ 1 ,...,ρ k−1 ) denotes the associated
ideal generated by the polynomials in the system of equations (8.44). Applying the
Stetter-Möller matrix method gives rise to matrices A r (ρ 1 ,...,ρ k−1 ) T which are rational
in ρ 1 ,...,ρ k−1 .
Because we require for k>1 any feasible solution to satisfy all the constraints in
(8.45), we consecutively choose r as the polynomials involved in these k −1 additional
constraints ã N−1 , ...,ã N−k+1 . This yields a polynomial eigenvalue problem involving
k − 1 matrices AãN−1 (ρ 1 ,...,ρ k−1 ) T ,..., AãN−k+1 (ρ 1 ,...,ρ k−1 ) T which are rational
in ρ 1 ,...,ρ k−1 but can be made polynomial in these parameters by getting rid of the
(common) denominators.
In the Chapters 9, 10, and 11 we show procedures for the co-order k =1,k =2,
and k = 3 case, respectively, to solve the corresponding eigenvalue problems. These
procedures lead to solutions x 1 ,...,x N , and ρ 1 ,...,ρ k−1 of the systems of equations
(8.44) which satisfy the constraints in (8.45). Subsequently, these solutions yield
approximations G(s) as described in the next section.
8.6 Computing the approximation G(s)
Let x 1 ,...,x N and ρ 1 ,...,ρ k−1 be a solution of the system of equations (8.44), which
also satisfy the additional constraints ã N−1 = ···=ã N−k+1 = 0 in (8.45). Using this
solution an approximation of order N − k can be computed as described in the next
theorem.
Theorem 8.3. (Co-order k) Let H(s) =e(s)/d(s) be a given strictly stable proper
rational transfer function of order N which has N distinct poles δ 1 ,...,δ N . In the coorder
k case, consider the associated linear system of equations (8.44). Let x 1 ,...,x N
and ρ 1 ,...,ρ k−1 constitute a solution to this set of equations, which satisfies the additional
constraints ã N−1 = ···=ã N−k+1 =0in (8.45). Then a feasible approximation
G(s) =b(s)/a(s) of order N − k can be computed as follows, provided that it happens
that q 0 0:
• From Equation (8.27) the polynomial ã(s) is constructed by computing its coefficients:
once x i =ã(δ i ) for i =1,...,N are known, the coefficients ã 0 , ...,
ã N−1 are computed by:
⎛
⎜
⎝
ã 0
.
.
⎞
⎛
⎟
⎠ = V (δ 1 ,...,δ N ) −1 ⎜
⎝
x 1
.
.
⎞
⎟
⎠ . (8.48)
ã N−1
x N
• Feasibility requires that the ‘leading’ coefficients ã N−1 ,...,ã N−k+1 of the polynomial
ã(s) are zero. Subsequently, the parameter q 0 is read off from the coefficient
ã N−k by using: q 0 =(−1) N−k ã N−k .