National Instruments Xmath Multiplicative Robustness Result

Chapter 3 Multiplicative Error Reduction

Xmath Model Reduction Module 3-2 ni.com

Multiplicative Robustness Result

Suppose C stabilizes , that has no jω-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 Equation3 -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(jω). 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

ˆΔGG

–()G

ˆ1–

Δjω()G

ˆjω()Cjω()IG

ˆjω()Cjω()+[]

1–1<

max Δjω()G

ˆjω()Cjω()IG

ˆjω()Cjω()+[]

1–

{}

ˆCI G

ˆC+()

1–

max Δjω() Δjω()

∞

ˆCI G

ˆC+()

1–

maxωΔjω()

Δjω()