Probabilistic Performance Analysis of Fault Diagnosis Schemes

4.2.2 Special Case: Fault Model Based on Component Failures
Consider a system with L components (e.g., sensors and actuators), and suppose that each
component may fail independently of the others. The term fail is used to indicate that the
component stops working altogether and never resumes normal function. The status of
each component (failed or not) at each time k is encoded by a binary variable b, where
b = 0 indicates that the component has not failed at or before time k, while b = 1 indicates
otherwise. Thus, the status of all L components at each time k is encoded by a L-bit binary
string b k ∈ {0,1} L . One possible parameter space for this model is the set of 2 L nonnegative
integers whose binary representations require no more than L bits. That is,
Θ = {0,1,...,2 L − 1}.
Converting each element of Θ into its binary representation reveals which component
failures are encoded by that state.
Proposition 4.24. Let θ be the stochastic process taking values in Θ, such that θ k represents
which components have failed at or before time k. Then, θ is a Markov chain.
Proof. Let k > 0 and ϑ 0:k ∈ Θ k+1 . Consider the conditional probability
P(θ k = ϑ k | θ 0:k−1 = ϑ 0:k−1 ). (4.11)
Let i 1 ,i 2 ,...,i l be the indices of the components whose failure is encoded by the state
ϑ k−1 . Also, let i l+1 ,i l+2 ,...,i l+j be the components whose failure is encoded by ϑ k but
not ϑ k−1 . Since a failed component must remain in a failed state, the probability (4.11) is
determined by the probability that components i l+1 ,...,i l+j fail at time k, given {θ 0:k−1 =
ϑ 0:k−1 }. Although the event {θ 0:k−1 = ϑ 0:k−1 } indicates at what times components i 1 ,...,i l
failed, this information is irrelevant, since the failure times are independent. The only
meaningful information contained in the event {θ 0:k−1 = ϑ 0:k−1 } is the fact that components
i l+1 ,...,i l+j fail at time k, which is also indicated by the event {θ k−1 = ϑ k−1 }. Therefore,
P(θ k = ϑ k | θ 0:k−1 = ϑ 0:k−1 ) = P(θ k = ϑ k | θ k−1 = ϑ k−1 ),
which implies that θ is a Markov chain.
Proposition 4.25. The transition probability matrices {Π k } for the Markov chain θ are uppertriangular.
Proof. Suppose that θ transitions from i ∈ Θ to j ∈ Θ at time k. Let b i and b j be the binary
representations of i and j , respectively. The transition from i to j has zero probability unless
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