Probabilistic Performance Analysis of Fault Diagnosis Schemes

if and only if
[ T (βi ) T T (M i ) T T (z i ) + T (z i ) T T (M i )T (β i ) + T (z i ) T T (z i ) T (β i ) T ]
( ) −1
T (β i )
1
I − T (M
γ 2 i ) T ≽ 0,
T (M i )
for i = 1,2,..., q, where β and z are partitioned compatibly with ∆ and M.
Remark 5.8. The statement and proof of Theorems 5.6 and 5.7 involves the relationship
T N (α) = T N (M)T N (β) + T N (z),
which only holds when M is time-invariant. To the best of our knowledge, there is no
extension of these results in which M is time-varying.
The following time-varying extension of Theorem 5.4 is due to Poolla et al. [74] and used
in the model-invalidation context by [27, 87, 92].
Theorem 5.9. Given sequences α ∈ l n 2 and β ∈ lm 2
and constants γ > 0 and N ∈ N, there exists
an operator ∆ ∈ ∆ 2,ltv (γ), such that
τ N β = τ N ∆α
if and only if
‖τ k β‖ 2 ≤ γ‖τ k α‖ 2 ,
for k = 0,1,..., N .
As in Corollary 5.5, this result is easily extended to the case where ∆ is block-diagonal
by considering each block-partition separately. Hence, we have the following corollary of
Theorem 5.9.
Corollary 5.10. 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,ltv (γ), such that
τ N β = τ N ∆α
if and only if
‖τ k β i ‖ 2 ≤ γ‖τ k α i ‖ 2 ,
for k = 0,1,..., N and i = 1,2,..., q, where α and β are partitioned such that β i = ∆ i α i .
Remark 5.11. The condition τ N β = τ N ∆α used in these interpolation theorems implies that
the values α j and β j are irrelevant for j > N. In the model invalidation literature, this
94