Probabilistic Performance Analysis of Fault Diagnosis Schemes

Maximizing the Probability of False Alarm
Assume that no faults have occurred (i.e., ϑ = 0). The worst-case probability of false alarm is
P ⋆ f = 1 −
min
(u,w)∈P s
min P( )
|r k | < ε k | θ 0:k = 0 0:k .
0≤k≤N
As explained in Section 5.2.2, the crux of computing P ⋆ f
ˆr ⋆ k = max
∣ ∣r nom
(u,w)∈P
k
+ r unc
∣
k ,
s
is computing
for k = 0,1,..., N . More formally, this optimization can be written as
ˆr ⋆ k = maximize
ũ,w
∣ r
nom
k
+ r unc
∣
k
subject to r nom = (F 1 G 1,0 + F 2 )u ◦ ,
r unc = (F 1 G 1,0 + F 2 )ũ + F 1 G 3,0 w,
‖ũ‖ p < γ 1 ,
‖w‖ p < γ 2 ,
for p ∈ [1,∞] and γ 1 ,γ 2 > 0. Note that the signal r nom is fixed. Since r unc is a linear function
of ũ and w, the mean of the residual ˆr k = r nom + r unc is an affine function of the decision
k k
variables ũ and w. For p ∈ [1,∞], the norm bounds on the decision variables are convex
constraints. Therefore, this optimization can be written as a convex program, for all k. In
particular, if p ∈ {1,∞}, this optimization can be written as a pair of linear programs (lp),
and if p = 2, this optimization can be written as a pair of second-order cone programs (socp).
Both lps and socps are readily solved with optimization packages, such as SeDuMi [90].
Minimizing the Probability of Detection
Let ϑ be a fault parameter sequence such that ϑ N ≠ 0, and let k f be the fault time, as defined
in equation (5.2). The worst-case probability of detection is
P ⋆ d = 1 −
max
(u,w,f (ϑ))∈P s
min P( )
|r k | < ε k | θ 0:k = ϑ 0:k .
k f ≤k≤N
By Proposition 5.2, optimum values of u, w, f , and k are obtained by solving
ˆµ ⋆ = min
(u,w,f (ϑ))∈P s
max
k f ≤k≤N
| ˆr k |
Σk
.
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