User's Manual

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 C
w
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 ν
i
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 ν
i
≥ν
i +1
≥ ⋅⋅⋅ being assumed). If ν
k
= ν
k +1
=...=ν
k + r
for some
k, (that is, ν
k
has multiplicity greater than unity), then ν
k
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 ν
i
less than
ν
nsr
. There are restrictions on nsr. nsr cannot exceed the dimension
of a minimal realization of G(s); although ν
i
≥
i +1
⋅⋅⋅, nsr must obey
n
nsr
> n
nsr+1
; and while 1 ≥ ν
i
for all i, it is necessary that 1> ν
nsr +1
. (The
number of ν
i
equal to 1 is the number of right half plane zeros of G(s).
They must be retained in G
r
(s), so the order of G
r
(s), nsr, must at least
be equal to the number of ν
i
equal to 1.) The software checks all these
conditions. The minimum order permitted is the number of Hankel
QA A′QCB′
w
Q–()′DD′()
1–
CB
w
′
Q–()++ 0=
C
w
D
1–
CB′
w
Q–()=
QA A′QC′
w
C
w
++ 0=
Fs() C
w
sI A–()
1–
B=
Fs() C
w
sI A–()
1–
B=
v
nsr 1+
G
1–
GG
r
–()
∞
1 v
j
+()1–
jnsr1+=
ns
∏
≤≤