Chapter 3 Multiplicative Error Reduction
© National Instruments Corporation 3-7 Xmath Model Reduction Module
strictly proper stable part of θ(s), as the square roots of the eigenvalues
of PQ. Call these quantities νi. The Schur decompositions are,
where VA, VD are orthogonal and Sasc, Sdes are upper triangular.
4. Define submatrices as follows, assuming the dimension of the reduced
order system nsr is known:
Determine a singular value decomposition,
and then define transformation matrices:
The reduced order system Gr is:
where step 4 is identical with that used in redschur( ), except
thematrices P, Q which determine VA, VD and so forth, are the
controllability and observability grammians of CW(sI–A)–1B rather
than of C(sI –A)–1B, the controllability grammian of G(s) and the
observability grammian of W(s).
The error formula [WaS90] is:
(3-2)
All νi obey νi ≤ 1. One can only eliminate νi where νi < 1. Hence, if nsr is
chosen so that νnsr+ 1 = 1, the algorithm produces an error message. The
algorithm also checks that nsr does not exceed the dimension of a minimal
VA
′PQVASasc
=VD
′PQVDSdes
=
Vlbig VA
0
Insr
=Vrbig VD
Insr
0
=
UebigSebigVebig Vlbig
′Vrbig
=
Slbig VlbigUebigSebig
12⁄–
=
Srbig VrbigVebigSebig
12⁄–
=
ARSlbig
′ASrbig
=
ARCSrbig
=
BRSlbig
′B=
DRD=ARCSrbig
=
G1–GG
r
–()
∞2vi
1vi
–
-------------
∑
≤