Chapter 3 Multiplicative Error Reduction
Xmath Model Reduction Module 3-16 ni.com
eigenvalues of A– B/D *C with the aid of schur( ). If any real part
of the eigenvalues is less than eps, a warning is displayed.
Next, a stabilizing solution Q is found for the following Riccati
equation:
The function singriccati( ) is used; failure of the nonsingularity
condition of G(jω) will normally result in an error message. To obtain
the best numerical results, singriccati( ) is invoked with the
keyword method="schur".
The matrix Cw is given by .
Notice that Q satisfies , so that P and Q are
the controllability and observability grammians of
This strictly proper, stable transfer function matrix is the strictly
proper, stable part (under additive decomposition) of
θ(s)=W–T(–s)G(s), which obeys the matrix all pass property
θ(s)θ'(–s)=I. It is the phase matrix associated with G(s).
3. The Hankel singular values νi of are
computed, by calling hankelsv( ). The value of nsr is obtained if
not prespecified, either by prompting the user or by the error bound
formula ([GrA89], [Gre88], [Glo86]).
(3-3)
(with νi≥ν
i+1≥ ⋅⋅⋅ being assumed). If νk=νk+1=...=νk+r for some
k, (that is, νk has multiplicity greater than unity), then νk appears once
only in the previous error bound formula. In other words, the number
of terms in the product is equal to the number of distinct νi less than
ν
nsr. There are restrictions on nsr. nsr cannot exceed the dimension
of a minimal realization of G(s); although νi≥i+1
⋅⋅⋅, nsr must obey
nnsr > nnsr+1; and while 1 ≥ νi for all i, it is necessary that 1>
ν
nsr+1
. (The
number of νi equal to 1 is the number of right half plane zeros of G(s).
They must be retained in Gr(s), so the order of Gr(s), nsr, must at least
be equal to the number of νi equal to 1.) The software checks all these
conditions. The minimum order permitted is the number of Hankel
QA A′QCB′wQ–()′DD′()
1–CB
w
′Q–()++ 0=
CwD1–CB′wQ–()=
QA A′QC′wCw
++ 0=
Fs() CwsI A–()
1–B=
Fs() CwsI A–()
1–B=
vnsr 1+G1–GG
r
–()
∞1vj
+()1–
jnsr1+=
ns
∏≤≤