Probabilistic Performance Analysis of Fault Diagnosis Schemes

malfunction — an intermittent irregularity in the fulfilment of a system’s desired
function.
disturbance — an unknown (and uncontrolled) input acting on a system.
residual — a fault indicator, based on a deviation between measurements and
model-equation-based computations.
fault detection — determination of the faults present in a system and the time
of detection.
fault isolation — determination of the kind, location and time of detection of a
fault. Follows fault detection.
fault identification — determination of the size and time-variant behaviour of a
fault. Follows fault isolation.
fault diagnosis — determination of the kind, size, location and time of detection
of a fault. Follows fault detection. Includes fault identification.
reliability — ability of a system to perform a required function under stated
conditions, within a given scope, during a given period of time.
safety — ability of a system not to cause danger to persons or equipment or the
environment.
availability — probability that a system or equipment will operate satisfactorily
and effectively at any point of time.
2.4.2 Brief Survey of Fault Diagnosis
In this section, we present a brief survey of the vast field of fault diagnosis. For a thorough
treatment, see Chen and Patton [9], Ding [24], or Isermann [48]. Consider the general fault
diagnosis problem in Figure 2.2. The system G is affected by known inputs u, stochastic
noises v, unknown deterministic disturbances w, and an exogenous signal f representing
a fault. The fault diagnosis scheme is comprised of two parts: a residual generator F and
a decision function δ. The residual generator F uses the known input u and the measured
output y to produce a residual r , which carries information about the occurrence of faults.
The decision function δ evaluates the residual and determines what type of fault, if any, has
occurred. The output of the residual generator, d, is called the decision issued by the fdi
scheme. Typically, d takes values in some finite set of decisions D. This separation of a fault
diagnosis scheme into two stages was first proposed in [13].
There are a number of approaches to constructing meaningful residual signals. In a
structured residual set, the residual r is a vector such that each component r i is sensitive to
a subset of faults. If each residual component r i is sensitive to a single component f i of the
fault vector, then r is said to be a dedicated residual set. Another approach is to make each
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