Probabilistic Performance Analysis of Fault Diagnosis Schemes

so the hazard rate of κ converges to the hazard rate of τ as T s → 0.
Proof. For k ≥ 0, the cdf of κ is
P κ (k) = 1 − (e −λ∆ ) k = 1 − e −λk∆ = P τ (k∆).
Since the second-order Taylor approximation of the exponential function is
the hazard rate of κ is approximated by
e −x = 1 − x + x2
2 +O(x3 ),
h(k) = q T s
= 1 − e−λT s
T s
= λ − λ2 T s
2 +O(T 2 s )
Hence, h(k) → λ as T s → 0.
2.4 Fault Diagnosis
This section provides a brief survey of the fault diagnosis literature. To begin, we establish a
lexicon of common fault diagnosis terminology. Then, we briefly review some of the existing
techniques used to design fault diagnosis schemes. Although this dissertation is focused on
performance analysis, rather than design, this survey provides some context for our analysis.
Similarly, we survey some of the ways in which redundancy can be used, in conjunction with
fault diagnosis schemes, to produce more reliable systems. Finally, we discuss the existing
approaches to analyzing the performance of fault diagnosis schemes.
2.4.1 Basic Terminology
Because fault diagnosis research spans many engineering disciplines, there is some disagreement
about even the most basic terminology. In the late 1980s, the International
Federation of Automatic Control (ifac) formed the Technical Committee on Fault Detection,
Supervision, and Safety of Technical Processes (safeprocess). One key contribution of the
ifac safeprocess committee was to establish a set of commonly accepted definitions. The
following list, taken directly from [49], is comprised of these definitions:
fault — an unpermitted deviation of at least one characteristic property or
parameter of the system from the acceptable/usual/standard condition.
failure — a permanent interruption of a system’s ability to perform a required
function under specified operating conditions.
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