Probabilistic Performance Analysis of Fault Diagnosis Schemes

For p ∈ [1,∞], the l p -norm ball centered at u ◦ ∈ l n p with radius γ > 0 is defined as
B n p (u◦ ,γ) := { u + u ◦ ∈ S n : ‖u‖ p < γ } .
We may write B p (u ◦ ,γ) when the dimension of the sequence is clear from context or of
little significance. Given an input-output operator G : l n p → lm p , with p ∈ [1,∞], define the
induced norm
‖Gu‖ p
‖G‖ i p := sup .
u≠0 ‖u‖ p
For p ∈ [1,∞] and γ > 0, define the set of norm-bounded operators
∆ m×n
p (γ) := { ∆: S n → S m : ‖∆‖ i p < γ } .
Similarly, for p ∈ [1,∞], γ > 0, and q ∈ N, define the set of block-structured norm-bounded
operators
{
ˆ∆ m×n
p (γ) := ∆ = diag{∆ 1 ,∆ 2 ,...,∆ q } : ∆ i ∈ ∆ m i ×n i
p (γ),
q∑
m i = m,
i=1
}
q∑
n i = n .
We may write ∆ p (γ) or ˆ∆ p (γ) when the dimension of the operator is clear from context or of
little significance.
For each s ∈ N, define the s-step truncation operator
The one-step shift operator z is defined as
τ s : S n → S n : u → { u 0 ,u 1 ,...,u s−1 ,u s ,0,0,... } .
z : S n → S n : u → { 0,u 0 ,u 1 ,... } .
An operator G : S n → S m is said to be time-invariant if
Otherwise, G is said to be time-varying.
Gz = zG.
i=1
5.1.2 Types of Uncertainty Considered
Although there are many distinct ways to include uncertainty in the fault detection problem,
we consider the following four types of uncertainty:
1. Families of Inputs: In Chapters 3 and 4, the performance metrics are computed
for a single fixed input sequence u. Since this input sequence affects the values of
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