56 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
i) State-space upper bound:
inf
Ds∈Ds
σmax[DsGssD−1
s]<1;
ii) Frequency domain, constant D, upper bound:
inf
Dp∈Dp
max
ω∈[0,2π]σmax[DpFu(Gss,eωInx)D−1
p]<1;
iii) Frequency domain upper bound:
max
ω∈[0,2π]inf
Dp∈Dp
σmax[DpFu(Gss,eωInx)D−1
p]<1.
In both the state-space and frequency domain, constant D, upper bound, a single D
scale is selected to guarantee robust performance over all frequencies. These two tests
(items i)andii) above) are equivalent. In the frequency domain upper bound test
(item iii)) a different D-scale is selected at each frequency.
The relationship between all of these tests is summarized by the following:
inf
Ds∈Ds
σmax[DsGssD−1
s]<1 State-space upper bound
m
inf
Dp∈Dp
max
ω∈[0,2π]σmax[DpFu(Gss,eωInx)D−1
p]<1 Frequency domain, constant D,
upper bound
⇓
max
ω∈[0,2π]inf
Dp∈Dp
σmax[DpFu(Gss,eωInx)D−1
p]<1 Frequencydo main upper bound
⇓
max
ω∈[0,2π]µ∆p[Fu(Gss,eωInx)] <1 Frequencydomain µtest
m
µ∆s[Gss]<1 State-space µtest
In the two cases where there are one way implications, there are real gaps. We have
already seen that there is a gap between the frequency domain µtest and its upper
bound for four or more full blocks. This is a computational issue.