Probabilistic Performance Analysis of Fault Diagnosis Schemes

where
[
T (β) T T (G 11,0 ) T T (z) + T (z) T T (G 11,0 )T (β) + T (z) T T (z) T (β) T
J (β) :=
( ) −1
T (β)
1
].
I − T (G
γ 2 11,0 ) T T (G 11,0 )
Note that the subscript N has been omitted from the operators T and T for clarity.
Since u and f (0) are known, r unc is an affine function of β. Also, the signal z is fixed, so
the function J (β) is linear in β, and the constraint J (β) ≽ 0 is a linear matrix inequality
(lmi). Therefore, this optimization can be cast as a semidefinite program (sdp), which is a
type of convex program that is readily solved with numerical optimization packages, such as
SeDuMi [90].
Case 2. Suppose that ∆ belongs to the set ∆ 2,ltv (γ) and assume that G 11,0 = 0 (i.e., ∆ does
not experience feedback). Then, for k = 0,1,..., N , applying Theorem 5.9 yields the following
optimization:
ˆr ⋆ k = maximize
β
∣ r
unc∣
k
subject to r unc = F 2 G 21,0 β + F 2 G 23,0 f (0) + (F 2 G 24,0 + F 1 )u
α = G 13,0 f (0) +G 14,0 u
‖τ l β‖ 2 ≤ γ‖τ l α‖ 2 ,
l = 0,1,...,k.
As in Case 1, r unc is affine in β. Since the k + 1 inequality constraints are quadratic in
k
β 0:N , this optimization problem is a socp. As previously mentioned, socps are readily solved
with numerical optimization packages.
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). Recall that the worst-case probability of detection is
Pd ⋆ = 1 − max min P( )
|r k | < ε k | θ 0:k = ϑ 0:k .
∆∈P ∆ k f ≤k≤N
By Proposition 5.2, the optimum values of ∆ and k are obtained by solving
ˆµ ⋆ = min
∆∈P ∆
max
k f ≤k≤N
| ˆr k |
Σk
.
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