Probabilistic Performance Analysis of Fault Diagnosis Schemes

The following extension of Theorem 5.4, due to Chen and Wang [11], is useful when the
interpolating operator ∆ is in a feedback interconnection with another operator.
Theorem 5.6. Consider the feedback interconnection shown in Figure 5.3(c), in which
β = ∆(I − M∆) −1 z.
Fix γ > 0 and let M : l m 2
→ ln 2 be a linear time-invariant operator, such that ‖M‖ i 2 ≤ 1 γ . Then,
given sequences z ∈ l n 2 and β ∈ lm 2
and N ∈ N, there exists an operator ∆ ∈ ∆ 2,lti(γ), such that
if and only if
τ N β = τ N ∆(I − M∆) −1 z
[ T (β) T T (M) T T (z) + T (z) T T (M)T (β) + T (z) T T (z) T (β) T ]
( ) −1
T (β)
1
I − T (M) T ≽ 0,
T (M)
γ 2
where the subscript N on the operators T and T has been omitted for clarity.
As in Corollary 5.5, Theorem 5.6 can be extended to the case where ∆ is block-diagonal.
However, for general M, the matrix inequality in Theorem 5.6 becomes a nonconvex constraint
on β, and there is no computationally tractable way to check for the existence of a
block-diagonal interpolating operator [11, 92]. However, Chen and Wang [11] show that this
matrix inequality is convex in β if M is sufficiently structured. The necessary structure is
stated in the following theorem.
Theorem 5.7. Consider the feedback interconnection shown in Figure 5.3(d), in which
β = ∆(I − M∆) −1 z.
Fix γ > 0, assume ∆ = diag{∆ 1 ,...,∆ q }, and let M : l m 2
→ ln 2
be a linear time-invariant operator,
such that
M = diag{M 1 , M 2 ,..., M q },
where the dimensions of M i are compatible with ∆ i . Further, assume that ‖M i ‖ i 2 ≤ 1 γ , for
all i . Then, given sequences z ∈ l n 2 and β ∈ lm 2 , there exists an operator ∆ ∈ ˆ∆ 2,lti (γ), such that
τ N β = τ N ∆(I − M∆) −1 z
93