Hankel singular values of phase matrix of gr, Further error bounds, Reduction of minimum phase, unstable g – National Instruments NI MATRIXx Xmath User Manual
Page 55: Hankel singular values of phase matrix of g
Chapter 3
Multiplicative Error Reduction
© National Instruments Corporation
3-9
Hankel Singular Values of Phase Matrix of G
r
The
ν
i
, i = 1,2,...,ns have been termed above the Hankel singular values of
the phase matrix associated with G. The corresponding quantities for G
r
are
ν
i
, i = 1,..., nsr.
Further Error Bounds
The introduction to this chapter emphasized the importance of the error
measures
or
for plant reduction, as opposed to
or
The BST algorithm ensures that in addition to (Equation 3-2), there holds
[WaS90a].
which also means that for a scalar system,
and, if the bound is small:
Reduction of Minimum Phase, Unstable G
For square minimum phase but not necessarily stable G, it also is possible
to use this algorithm (with minor modification) to try to minimize (for G
r
of a certain order) the error bound
G G
r
–
(
)G
r
1
–
∞
G
r
1
–
G G
r
–
(
)
G G
r
–
(
)G
1
–
∞
G
1
–
G G
r
–
(
)
∞
G
r
1
–
G G
r
–
(
)
∞
2
v
i
1 v
i
–
-------------
i
nsr 1
+
=
ns
∑
≤
20log
10
G
r
G
------
8.69
≤
2
v
i
1 v
i
–
-------------
i
nsr 1
+
=
ns
∑
⎝
⎠
⎜
⎟
⎜
⎟
⎛
⎞
dB
phase G
( ) phase G
r
( )
–
v
i
1 v
i
–
-------------
i
nsr 1
+
=
ns
∑
radians
≤
G G
r
–
(
)G
r
1
–
∞