Probabilistic Performance Analysis of Fault Diagnosis Schemes

β
∆
α
β
∆ 1
. ..
∆q
α
(a)
(b)
β
∆
α
β
∆ 1
. ..
∆q
α
M
M 1
. ..
Mq
(c)
z
(d)
z
Figure 5.3. Block diagrams for the interpolation results. Theorems 5.4 and 5.9 apply to diagram (a).
Corollaries 5.5 and 5.10 apply to diagram (b). Theorem 5.6 applies to diagram (c) and Theorem 5.7
applies to diagram to diagram (d).
Theorem 5.4. Given sequences α ∈ l n 2 and β ∈ lm 2
and constants γ > 0 and N ∈ N, there exists
an operator ∆ ∈ ∆ 2,lti (γ), such that
τ N β = τ N ∆α
if and only if
T ∗ N (β)T N (β) ≼ γ 2 T ∗ N (α)T N (α).
For many applications, it is appropriate to impose additional structure on the interpolating
operator ∆. One structure that appears frequently in the robust control literature
[28, 86, 110] is the class of block-diagonal operators, which we denote ˆ∆ p (γ). As shown
in [11], Theorem 5.4 is extended to operators in set ˆ∆ 2,lti (γ) by simply treating each blockpartition
separately. Hence, we state this extension as a corollary of Theorem 5.4.
Corollary 5.5. Given sequences α ∈ l n 2 and β ∈ lm 2
and constants γ > 0 and N ∈ N, there exists
an operator ∆ = diag{∆ 1 ,...,∆ q } ∈ ˆ∆ 2,lti (γ), such that
τ N β = τ N ∆α
if and only if
T ∗ N (β i )T N (β i ) ≼ γ 2 T ∗ N (α i )T N (α i ),
for i = 1,2,..., q, where α and β are partitioned such that β i = ∆ i α i .
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