National Instruments NI MATRIXx Xmath - Manual (Page 48)

Chapter 3 Multiplicative Error Reduction
Xmath Model Reduction Module 3-2 ni.com
Multiplicative Robustness Result
Suppose C stabilizes , that has no j
ω-axis poles, and
that G has the same number of poles in Re[s]
≥ 0 as . If for all ω,
(3-1)
then C stabilizes G.
This result indicates that if a controller C is designed to stabilize a nominal
or reduced order model
, satisfaction of Equation 3-1 ensures that the
controller also will stabilize the true plant G.
In reducing a model of the plant, there will be concern not just to have this
type of stability property, but also concern to have as little error as possible
between the designed system (based on
) and the true system (based
on G). Extrapolation of the stability result then suggests that the goal
should be not just to have Equation 3-1, but to minimize the quantity on the
left side of Equation 3-1, or its greatest value:
However, there are difficulties. The principal one is that if we are reducing
the plant without knowledge of the controller, we cannot calculate the
measure because we do not know C(j
ω). Nevertheless, one could presume
that, for a well designed system, will be close to I over the
operating bandwidth of the system, and have smaller norm than 1 (tending
to zero as
ω→∞ in fact) outside the operating bandwidth of the system.
This suggests that in the absence of knowledge of C, one should carry out
multiplicative approximation by seeking to minimize:
This is the prime rationale for (unweighted) multiplicative reduction of a
plant.
Two other points should be noted. First, because frequencies well beyond
the closed-loop bandwidth,
will be small, it is in a sense,
wasteful to seek to have
Δ(jω) small at very high frequencies. The choice
of as the index is convenient, because it removes a
requirement to make assumptions about the controller, but at the same time
it does not allow
to be made even smaller in the closed-loop
Gˆ Δ G Gˆ –
( )Gˆ
1 –
=
Gˆ
Δ jω ( ) Gˆ jω ( )C jω ( ) I Gˆ jω ( )C jω ( )
+
[ ]
1 –
1 <
Gˆ
Gˆ
max Δ jω ( ) Gˆ jω ( )C jω ( ) I Gˆ jω ( )C jω ( )
+
[ ]
1 – { }
ω
GˆC I GˆC +
( ) 1 –
max Δ jω ( ) Δ jω ( )
∞
=
ω
GˆC I GˆC +
( ) 1 –
max
ω
Δ jω ( )
Δ jω ( )