Recursive EM-filter algorithms for Wiener process-based ...



Supplementary material to: A degradation path-dependent approach for remaining useful life estimation with an exact and closed-form solution

Xiao-Sheng Sia,b, Wenbin Wangc*, Mao-Yin Chenb, Chang-Hua Hua, Dong-Hua Zhoub*[1]

aDepartment of Automation, Xi'an Institute of High-Tech, Xi'an, Shaanxi 710025, P. R. China,

bDepartment of Automation, TNLIST, Tsinghua University, Beijing 100084, P. R. China,

cDongling School of Economics and Management, University of Science and Technology Beijing, China.

APPENDICES

A. Proof of Theorem 1

We only need to check that the following properties hold for [pic] (Karlin and Taylor, 1975, pp. 376): (i) [pic] is a Gaussian process with continuous path; (ii) [pic]; and (iii) [pic]The first two are easy to check via the properties of the standard BM [pic]. Here, we only need to show that the last property is true for [pic]. We have

[pic] (A-1)

This completes the proof.

B. Proof of Theorem 2

By the Markov property of the Wiener process,

[pic]. (B-1)

The second term of theorem 2 follows immediately. This completes the proof.

C. Proof of Theorem 3

From Eq. (25), we can learn that [pic] obtained by Eqs. (26) and (27) is the only solution satisfying [pic]. Consequently, taking [pic], the following is obtained,

[pic], (C-1)

with

[pic]. (C-2)

We show that the matrix in (C-2) is negative definite at [pic], by calculating the order principal minor determinant as follows,

[pic]. (C-3)

Then, at [pic], the followings are obtained,

[pic],[pic],[pic]. (C-4)

This proves that the matrix in (C-1) is negative definite at [pic]. This conclusion plus the result that [pic] is the only solution satisfying [pic] verifies Theorem 3.

D. Proof of Theorem 4

(1) Through some algebraic manipulations, using Lemma 1, we have

[pic] (D-1)

This completes the proof of the first equation in Theorem 4.

(2) Due to the limited space, we only summarize the main results below:

[pic], (D-2)

where [pic] and [pic] can be formulated separately as follows,

[pic]. (D-3)

with [pic]. In a similar way, [pic] can be written as

[pic] (D-4)

Then, the final result can be written directly

[pic]. (D-5)

This completes the proof of the second equation in Theorem 4.

E. Proof of Theorem 5

Using Eqs. (15), (16), (22), (23) and the total law of probability, we have

[pic], (E-1)

The last equation is implied by the second result of Theorem 4.

[pic]. (E-2)

Following the first result in Theorem 4, it is straightforward to obtain Eq. (32). This completes the proof.

F. Proof of Theorem 6

As mentioned in Remark 2, in Gebraeel et al. (2005), they directly used [pic] to estimate the RUL distribution. As a result, the CDF of the estimated RUL can be written as

[pic]. (F-1)

Compared with Eq. (32), it is obviously observed that the following holds,

[pic]. (F-2)

Namely, we have

[pic]. (F-3)

Then according the given definition by Eq. (33), the proof is completed.

G. Proof of Theorem 7

From Eq. (35), we directly have

[pic]. (G-1)

For the third term of the denominator in the above equation, we get

[pic]. (G-2)

Replacing the third term in the denominator of Eq. (G-1) with above equation will complete the proof.

H. Proof of Theorem 8

Based on Theorem 7, we have the following formulas of [pic] and [pic],

[pic] (H-1)

and

[pic]. (H-2)

From Theorem 6, we have [pic]. Then we have

[pic], (H-3)

and

[pic]. (H-4)

As a result, the proof is completed.

I. Proof of Theorem 10

1) For [pic], we have

[pic]. (I-1)

Therefore, we first have

[pic] (I-2)

with [pic].

From Eq. (I-1), we have

[pic], (I-3)

with

[pic]. (I-4)

Then using the second result in Theorem 4 and simplifying the expression can complete the proof of the first equation in Theorem 10.

2) For [pic], we have,

[pic]. (I-5)

Therefore, we first calculate [pic] as

[pic], (I-6)

where [pic] are defined as follows

[pic]. (I-7)

Then, using Theorem 4, it is straightforward to show,

[pic]. (I-8)

Simplifying above expression completes the proof of the second equation in Theorem 10.

J. Proof of Theorem 11

As Gebraeel et al. (2005), following the procedure [pic], we have

[pic]. (J-1)

As a result, the CDF of the estimated RUL can be written as

[pic]. (J-2)

Compared with Eq. (54), we have,

[pic]. (J-3)

Namely,

[pic]. (J-4)

Then according to the given definition by Eq. (33), the proof is completed.

References

1. Karlin, S., H.M. Taylor. 1975, A first course in stochastic processes, the 2nd edition, Academic press, New York.

2. Gebraeel, N., M.A. Lawley, R. Li, J.K. Ryan. 2005. Residual-life distributions from component degradation signals: A Bayesian approach. IIE Transaction 37(6) 543-557.

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[1]Corresponding authors. Tel.: +86 010 62794461; fax: +86 010 62786911

Email: wangwb@ustb.(W. Wang);zdh@mail.tsinghua. (D.-H. Zhou)

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