Probabilistic Performance Analysis of Fault Diagnosis Schemes

Table 4.1. Time-complexity of computing the performance metrics using Algorithms 4.1–4.4. The
column labeled “Simulations” indicates the number of times the recurrence for the conditional mean
of the residual (equation (4.17)) must be simulated.
Problem Type Simulations Total Complexity Algorithm
General O ( (m + 1) N+1) O ( N (m + 1) N+1) 4.1
Structured O(N m ) O(N m+1 ) 4.1
ltv Special Case O(LN 2 ) O(LN L ) 4.2 & 4.3
lti Special Case O(LN ) O(LN L ) 4.4 & 4.3 (shifted)
to Proposition 4.35.
Corollary 4.36. The time to compute the performance metrics for the lti special case using
Algorithm 4.4 and a time-shifted version of Algorithm 4.3 is O(LN L ).
Proof. By Proposition 4.33, the running time of the time-shifted version of Algorithm 4.3 is
O(LN L ), which dominates the running time of Algorithm 4.4.
The time-complexity results established in Propositions 4.32–4.35 and Corollary 4.36 are
summarized in Table 4.1.
4.6 Comments on Continuous-Time Models
In Chapter 3, as well as the present chapter, the model G θ and the residual generator F
are assumed to be discrete-time dynamic systems. Generally speaking, there is no reason
to assume that the model G θ is discrete. Indeed, continuous-time jump-Markov linear
systems are treated in detail in [65] and [66], and more general hybrid stochastic differential
equations are considered in [105]. The biggest difficulty in using continuous-time models is
extending the Markov chain {θ k } to the more general class of jump processes [105]. In practice,
however, the residual generator F only has access to discrete observations { y(t k ) } k≥0
of the output signal, where {t k } k≥0 is a sequence of discrete observation times. Hence, the
problem is greatly simplified by assuming that G θ is a discrete-time system, as well.
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