Probabilistic Performance Analysis of Fault Diagnosis Schemes

Case 2. Suppose that ∆ belongs to the set ∆ 2,ltv (γ) and assume that G 11,ϑ = 0 (i.e., ∆ does
not experience feedback). Then, applying Theorem 5.9 yields the following optimization:
ˆµ ⋆ = maximize
β
subject to
‖W 1/2 ˆR‖ ∞
ˆR i = r unc
k f +i−1 , i = 1,..., N − k f + 1,
r unc = F 2 G 21,ϑ β + F 2 G 23,ϑ f (ϑ) + (F 2 G 24,ϑ + F 1 )u
α = G 13,ϑ f (ϑ) +G 14,ϑ u
‖τ l β‖ 2 ≤ γ‖τ l α‖ 2 ,
l = 0,1,...,k.
As in Case 1, the objective is a weighted pointwise maximum of affine functions of β, which
implies that it is convex. Since the signal α is fixed, each of the k + 1 inequality constraints
is quadratic in β and the optimization problem is a socp.
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