Probabilistic Performance Analysis of Fault Diagnosis Schemes

where the matrix Q is chosen to appropriately weight the estimation errors. The idea behind
observer-based methods is to construct the observer F and the weighting matrix Q such that
the residual is sensitive to faults. Early presentations of the observer-based method (e.g., [3])
assumed that there were no disturbances or noises affecting the system. For such systems, F
consists of a Luenberger observer [64] with weighted estimation error. For systems affected
by noises, a Kalman filter [50–53] may be used to obtain an estimate of Ly that minimizes
the mean-squared estimation error [67]. For systems affected by a disturbance, an unknown
input observer is used to decouple the residual from the effect of the disturbance [10, 58].
Typically, unknown input observers are not full-order and the remaining degrees of freedom
may be used to address some other design objective. For example, in systems affected
by disturbances and noise, the remaining degrees of freedom may be selected such that
mean-squared estimation error is as small as possible [8, 9].
Parity Equation-Based Methods
The parity equation approach is similar to the notion of physical redundancy, in the sense
that the residual is formed by comparing the system outputs y. For simplicity, assume that
the output y ∈ R m is given by
y = C x + v + f ,
where v is a noise process and f is a fault signal. Note that parity equation methods typically
assume that there are no disturbances affecting the system. The residual is defined as
r := Q y,
where Q ≠ 0 is chosen such that QC = 0. Hence, the residual can be written as
r = Q(v + f ) = q 1 (v 1 + f 1 ) + ··· + q m (v m + f m ),
where q i is the i th column of Q. Since each fault f i enters the residual in the direction of
the vector q i , faults are isolated by choosing the largest component (in magnitude) of the
vector Q T r . See [78] for an early survey of parity equation-based methods.
Of course, the requirement that QC = 0 can only be met with a nonzero Q when C has a
nontrivial null space. For systems where this requirement is not met, a form of temporal
redundancy may be used [14,68]. This approach is usually restricted to discrete-time systems
with no disturbances or noises. Suppose that the system is of the form
x k+1 = Ax k + B k u k + R 1 f k
y k = C x k + D k u k + R 2 f k .
16