Probabilistic Performance Analysis of Fault Diagnosis Schemes

As in Section 5.2.2, if the matrix W is defined by equation (5.6), then this optimization may
be written more formally as
ˆµ ⋆ = minimize
ũ, w, f ˜
subject to
∥ W
1/2 ˆR ∥ ∥
∞
ˆR i = r nom
k f +i−1 + r unc
k f +i−1 , i = 1,2,... N − k f + 1,
r nom = (F 1 G 1,ϑ + F 2 )u ◦ + F 1 G 4,ϑ f ◦ ,
r unc = (F 1 G 1,ϑ + F 2 )ũ + F 1 G 3,ϑ w + F 1 G 4,ϑ ˜ f ,
‖ũ‖ p < γ 1 ,
‖w‖ p < γ 2 ,
∥
∥ f ˜ ∥p < γ 3 ,
for p ∈ [1,∞] and γ 1 ,γ 2 ,γ 3 > 0. Since the signal r nom is fixed, ˆR k is an affine function of the
decision variables ũ, w, and ˜ f , for each k. Since the pointwise maximum of convex functions
is convex [5] and the matrix W is fixed, the objective function is convex. For p ∈ [1,∞] the
norm bounds on ũ, w, and ˜ f are convex constraints. Therefore, this optimization is a convex
program. In particular, if p ∈ {1,∞}, this optimization is a linear program (lp), and if p = 2,
this optimization is a second-order cone program (socp). Both lps and socps are readily
solved with optimization packages, such as SeDuMi [90].
5.4 Problems with Model Uncertainty
In this section, we consider systems of the form shown in Figure 5.2, where the linear
operator ∆ represents model uncertainty and the signals u and f are known. Note that this
system is not affected by a disturbance w. If the system G θ is partitioned as
G θ =
[
]
G 11,θ G 12,θ G 13,θ G 14,θ
,
G 21,θ G 22,θ G 23,θ G 24,θ
then the signals labeled in Figure 5.2 are related as follows:
β = ∆α,
α = G 11,θ β +G 12,θ v +G 13,θ f (θ) +G 14,θ u,
y = G 21,θ β +G 22,θ v +G 23,θ f (θ) +G 24,θ u.
Recall that Proposition 5.2 only applies if Assumptions 1–3 of Section 5.2.1 hold. Since the
residual generator F is a known linear operator with no uncertainty, the validity of these
assumptions depends on the manner in which the noise v affects the system output y.
Let T v→y denote the map from v to y. If the interconnection shown in Figure 5.2 is
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