Probabilistic Performance Analysis of Fault Diagnosis Schemes

the performance metrics, a comprehensive performance analysis would consider all
possible values of u, which is clearly not feasible. One reasonable compromise is to
compute the worst-case performance over a specified family of inputs. To this end,
we consider families of inputs that have the following form:
B n u
p (u ◦ ,γ) = { u + u ◦ ∈ S n u
: ‖u‖ p < γ } ,
where u ◦ ∈ l n u
p is a fixed nominal input, p ∈ [1,∞] specifies the l p -norm, and γ > 0 is
the desired bound.
2. Bounded Disturbances: Thus far, we have assumed that the system G θ is affected by
a noise signal v. It is also useful to consider the case where a deterministic signal w,
called a disturbance, affects the system in such a way that the fault diagnosis scheme
cannot use w to generate a residual. We consider disturbances in the bounded set
B n w
p (0,γ) = { w ∈ S n w
: ‖w‖ p < γ } ,
where p ∈ [1,∞] specifies the l p -norm, and γ > 0 is the desired bound.
3. Uncertain Fault Signals: In Chapters 3 and 4, it is assumed that the fault signal f k
at time k is a known, fixed function of the fault parameter sequence θ 0:k . While this
approach may work for certain types of faults, it often useful to consider the worst-case
performance of a fault diagnosis scheme over a set of possible fault signals. Hence, for
a given parameter sequence ϑ, we assume the fault signal lies in a bounded set of the
form
B n f
p
(
f (ϑ) ◦ ,γ ) = { f + f ◦ (ϑ) ∈ S n f
: ‖f ‖ p < γ } ,
where f ◦ (ϑ) ∈ l n f
p is the nominal value of the fault signal, p ∈ [1,∞] specifies the
l p -norm, and γ > 0 is the desired bound.
4. Model Uncertainty: In model-based fault diagnosis schemes, the residual generator
is usually designed according to the nominal system model G 0 . However, it useful to
consider cases where G 0 does not perfectly model the system or the designer of the
residual generator does not have accurate knowledge of the true model. Both of these
cases are addressed by assuming that the parameterized system G θ is uncertain. In
particular, we assume that the system consists of an interconnection of the system G θ
and an uncertain operator ∆. We consider two classes of uncertain operators. First,
we consider the class norm-bounded linear time-invariant uncertainties
∆ 2,lti (γ) := { ∆ ∈ ∆ 2 (γ) : ∆ is lti, causal, stable } ,
where γ > 0 is the desired bound. Second, we consider the class of norm-bounded
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