Separate CPNs with Additive Faults Detection and Correction

The strategy in this section is used to detect and correct additive faults. The encoded input matrix, encoded output matrix and encoded initial marking matrix here would have the same forms as those in the previous section, but with stricter restrictions. Therefore, let G be a CPN with no more than k place faults, x amount transition faults and k - x colour transition faults, where 0 ≤ x ≤ k, the separate CPN H with additive faults detection and correction capabilities is constructed by adding 2k additional places to G and the colour sets of both CPNs are the same. The input

matrix, output matrix and initial marking matrix of G are

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

= ₋⁻−

−

E DB B B

g g

h ,

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

= ₊⁺−

+

E DB B B

g g

h and _h Q _g

D Q₀ I ⎥ ₀

⎦

⎢ ⎤

⎣

=⎡ ^β respectively, where all the entries in D are

coprime with a prime number, and all the entries in E are multiples of the same prime number. After H is constructed, the place faults, amount treansition faults and colour transition faults occurring on H can be extracted from the syndromes, and the correct marking can be obtained. These properties are proved in Lemma 4.12.

Lemma 4.12: Let G be a CPN which has α transitions, β places, γ colours, input matrixB_g⁻, output matrixB_g⁺ and initial marking matrixQ_0g, and H be a separate CPN

with d additional places with respect to G, input matrix

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

= ₋ −

−

−

E DB B B

g g

h , output

matrix

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

= ₊ −

+ +

E DB B B

g g

h and initial marking matrix _h Q _g UQ _g D

Q₀ I ⎥ ₀ = ₀

⎦

⎢ ⎤

⎣

=⎡ ^β .

„ If there are additive faults on H, it can be detected by a d × (β + d) check matrix W, such that WU = 0d × β, where 0d × β is a d × β matrix with all entries of zero.

The syndrome S = WFp + WB_h⁻(F_a⁻ +F_c⁻) - WB_h⁺(F_a⁺ +F_c⁺) = WFp + WB_h⁻F_t⁻

- WB_h⁺F_t⁺, iff the place fault, pre-condition and post-condition amount transition fault, pre-condition and post-condition colour transition fault indicator matrices are Fp, F , _a⁻ F , _a⁺ F and _c⁻ F respectively. _c⁺ F and _t⁻ F are named as _t⁺

pre-condition transition fault indicator matrix and post-condition transition fault indicator matrix respectively. F = Fp + F - _t⁻ F is named as fault _t⁺ indicator matrix.

„ k place faults, x amount transition faults and k - x colour transition faults on H can be identified and corrected if W =

[

−D I_d

]

, where 0 ≤ x ≤ k, E = j * E', D

= j * i * 1d × β - D', j is a prime number lager than all the entries in E' and D', i is a positive integer lager than all the entries in D', 1d × β is a d × β matrix with all entries of one, any 2k columns of the d × αγ matrix E' are linearly dependent, and any 2k columns of the d × 2β matrix

[

^{D' I}_d

]

are linearly dependent.

Proof:

Assume Qg and Qh are fault-free marking matrices of G and H respectively, and

they have the same firing transition sequence. Hence, by Lemma 4.7, _h Q_g D

Q I ⎥

⎦

⎢ ⎤

⎣

=⎡ ^β ,

and Qh multiplied by W would be WQh = WUQg = 0d × γ. By Lemma 4.1, 4.5 and 4.9, if there are additive faults on H, the faulty marking matrix Qf satisfies Qf = Qh + Fp +

−

− c hF

B - B_h⁺F_c⁺ + B^-F - B_a^{− +}F = UQ_a⁺ g + Fp + B_h⁻F_c⁻ - B_h⁺F_c⁺ + B^-F - B_a^{− +}F . _a⁺ Hence, The syndrome S = WQf = WUQg + WFp + WB_h⁻F_c⁻ - WB_h⁺F_c⁺ + WB^-F - _a⁻ WB⁺F = WF_a⁺ p + WB_h⁻(F_a⁻ +F_c⁻) - WB_h⁺(F_a⁺ +F_c⁺). By the same way, if the syndrome is WFp + WB_h⁻(F_a⁻ +F_c⁻) - WB_h⁺(F_a⁺ +F_c⁺), it will be S = WFp + WB_h⁻(F_a⁻ +F_c⁻) - WB_h⁺(F_a⁺ +F_c⁺) = WFp + WB_h⁻(F_a⁻ +F_c⁻) - WB_h⁺(F_a⁺ +F_c⁺) + WUQg = WFp + WB_h⁻F_c⁻ - WB_h⁺F_c⁺ + WB^-F - WB_a^{− +}F + WQ_a⁺ h = W(Fp + B_h⁻F_c⁻ - B_h⁺F_c⁺ + B^-F - B_a^{− +}F + Q_a⁺ h), where Qh is a fault-free marking matrix of H, and

Fp, F , _a⁻ F , _a⁺ F and _c⁻ F are place fault, pre-condition and post-condition _c⁺ amount transition fault, pre-condition and post-condition colour transition fault indicator matrices respectively. Therefore, a marking matrix of H could be examined if it is a faulty marking matrix by multiplying the marking matrix with the check matrix W.

Assume Qf is a faulty marking matrix of H, which states k place faults, k amount transition faults and k colour transition faults on H. By Lemma 4.1, 4.5 and 4.9, Qf = Qh + Fp + B_h⁻F_c⁻ - B_h⁺F_c⁺ + B^-F - B_a^{− +}F . If the check matrix W = _a⁺

[

−D I_d

]

, the syndrome will be

S =

[

s1 s2 L sγ

]

= WQf

=

[ ] [ ]

( )

[ ]

₊ ( ^{+ +})

+

−

−

−

−

⎥ +

⎥⎦

⎤

⎢⎢

⎣

⎡

− −

−

⎥ +

⎥⎦

⎤

⎢⎢

⎣

⎡

− − +

− _{a c}

g g d

c a g

g d

p

d F F

E DB I B D F

E F DB I B D F

I D

=

[

⁻D Id

]

Fp ⁺E⁽Fa^{+ +}Fc^{+ −}Fa^{− −}Fc⁻⁾

=

[

D^'−j^*i^*1_d×_β I_d

]

F_p + j^*E^'(F_a⁺ +F_c⁺ −F_a⁻ −F_c⁻⁾ =

[

D^'−j^*i^*1_d×_β I_d

]

F_p + j^*E^'(F_t⁺ −F_t⁻⁾

=

[

^D'−j*i*1_d×β ^I_d

]

^F_p ^{+ j*E'F}_t. From S mod j,

Sp =

[

sp1 sp2 L sp_γ

]

= S mod j

=

[

^D'−^j*ⁱ*1d_×β ^Id

]

^Fpmod ^j+ ^j*^E'(^Fa⁺ +^Fc⁺ −^Fa⁻ −^Fc⁻)mod ^j

=

[

^{D' I}_d

]

^F_p

=

[

D' Id

] [

fp1 fp2 L fp_γ

]

is obtained, which contains only place fault part. Since any 2k columns of the matrix

[

D' I_d

]

are linearly dependent and each column of Fp has at most k nonzero entries, the place fault indicator matrix Fp can be be found by solving equations

[

D' I_d

]

f_pn =s_pn , where 1 ≤ n ≤ γ. After F^p is solved out, it can have St =

[

st1 st2 L stγ

]

= S -

[

−D I_d

]

F_p = EF _t

= E

[

ft1 ft2 L ft_γ

]

which contains only amount and colour transition fault part. Since any 2k columns of

the matrix E' are linearly dependent, any 2k columns of the matrix E = j * E' are linearly dependent. Since each column of Ft has at most k nonzero entries, the transition fault indicator matrix Ft can be be found by solving equations

n

n t

t s

Ef = ,

where 1 ≤ n ≤ γ. By Lemma 4.6 and 4.10, the F^a would not have nonzero entries on j

≠ (i-1) mod γ +1: faij, but Fc would have nonzero entries on j ≠ (i-1) mod γ +1:

cij

f . Hence, if an entry on j ≠ (i-1) mod γ +1: ftij is an nonzero entry, this nonzero value is belong to Fc. By Lemma 4.10, the sum of all the entries in each row is zero, all the entries on j = (i-1) mod γ +1: fc_ij can be solved out from this conditions, and all the remining entries in Fc are zero. After Fc is obtained, Fa can be obtained from Fa

= Ft - Fc.

Let G be a CPN which has α transitions, β places, γ colours, the input matrixB_g⁻, the output matrixB_g⁺ and the initial marking matrix

Q₀g . From Lemma 4.12, constructing a separate CPN H which can detect and correct at most k place faults, k amount transition faults and k colour transition faults is concluded as following steps:

(1) designing a d × αγ matrix E' from the check matrix of Reed-Solomon codes, (2) choosing a d × β matrix D' which satisfies any 2k columns of

[

D' I_d

]

are linearly dependent, (3) choosing a prime number j lager than all the entries in E' and D', (4) choosing a positive integer i lager than all the entries in D', (5) constructing matrices E = j * E' and D = j * i * 1d × β - D', (6) constructing the check matrix W from W =

[

−D I_d

]

, and finally (7) deriving H containing input matrix

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

= ₋ −

−

−

E DB B B

g g

h ,

output matrix

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

= ₊ −

+ +

E DB B B

g g

h and initial marking matrix

g

h Q

D Q₀ I ⎥ ₀

⎦

⎢ ⎤

⎣

=⎡ ^β . By

applying the method of Reed-Solomon codes, it would have d = 2k. Whether a marking Qh in H is correct can be examined with the check matrix W, and the faults can be distinguished from the following steps: (1) obtaining the syndrome S from S = WQh, (2) obtaining part of syndrome Sp containing only place fault part from Sp = S mod j, (3) solving the equation Sp =

[

D' I_d

]

F_p to obtain the place fault indicator matrix Fp, and the place faults can be interpreted from Fp, (4) obtaining part of syndrome St containing only amount and colour transition fault part from St = S -

[

−D I_d

]

F_p, (5) solving the equation St = EF to obtain the transition fault indicator _t matrix Ft, (6) obtaining the entries ∀j ≠ (i-1) mod γ +1: fcij in the colour transition fault indicator matrix Fc from f_cij = f_tij , (7) obtaining the entries ∀j = (i-1) mod γ +1: fcij in the colour transition fault indicator matrix Fc from

∑

≠

=

−

= ^γ

j h

h c

cij fih

f

, 1

, (8) obtaining the pre-condition and post-condition colour transition fault indicator matrices, F and _c⁻ F , from F_c⁺ c as the steps in section 4.5.2, and the pre-condition

and post-condition colour transition faults can be interpreted from F and _c⁻ F _c⁺ respectively, (9) obtaining the amount transition fault indicator matrix Fa from Fa = Ft

- Fc, and (10) obtaining the pre-condition and post-condition colour transition fault indicator matrices, F and _a⁻ F , from F_a⁺ a as the steps in section 4.4.2, and the pre-condition and post-condition amount transition faults can be interpreted from F _a⁻ and F respectively. _a⁺

4.6.2 An Example of Identifying and Correcting Additive Faults

This section adopts the same given CPN G in section 4.5.3 to show how to identify and correct additive faults. Assume there are at most k = 2 place faults, x = 1 amount transition faults and k - x = 1 colour transition faults. As described in section 4.6.1, it need design a 4 × 8 matrix E' from the check matrix of Reed-Solomon codes in order to design the separate CPN H first. The matrix E in section 4.4.3 is a 4 × 8 matrix designed from the check matrix of Reed-Solomon codes, thus the matrix E in section 4.4.3 can adopted as the matrix below

E' =

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

7 6 4 8 1 0 0 0

8 7 6 4 8 1 0 0

10 8 7 6 4 8 1 0

0 10 8 7 6 4 8 1

.

Next, choosing a 4 × 2 matrix D' which satisfies any 4 columns of

[

D' I₄

]

are linearly dependent. D' can be declared as

D' =

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

7 1

8 4

10 8

4 8

which is the same as D in section 4.4.3, and any 4 columns of

[

D' I₄

]

are linearly dependent. Then, a prime number j lager than all the entries in E' and D' is chosen, e.g.,

j = 11.

Then, a positive integer i lager than all the entries in D' is chosen, e.g., i = 11.

E = j * E' =

are obtained. Next, computing out E

Therefore, input matrix

−

of separate CPN H are obtained, and the check matrix is

W =

[

−D I₄

]

=

Assume there are place faults, amount transition faults and colour transition faults inside the firing sequence, which is represented as m_0h ⎯⎯→^Fp1 m_1f [t1,

− a1

F > m_2f ⎯⎯→^Fp2 m_3f [t2, F_c⁺₁ > m_4f ⎯^F⎯→^p3 m' , where _f

p1

Q₄f =

respectively.

Following steps are identifying and correcting these faults from the marking matrix Q' and the check matrix W. First, f

is obtained. Second, since

Sp = S mod 11 =

[

D' I_d

]

F_p mod 11 =

the following sets of equations are figured out:

⎪⎪

and there are two more restrictions: (1) each set of equations has at most two nonzero variables and (2) ∀1 ≤ i ≤ 6, 1 ≤ j ≤ 4:

=

By Lemma 4.2, it can be interpreted from Fp that there are two place faults, where one is on p1, which denotes lack of two red tokens, one green token and three blue tokens, and appearing four extra yellow tokens, and the other place fault is on p5, which denotes appearing three extra red tokens, five extra green tokens, four extra blue tokens and two extra yellow tokens. Fifth, since

EFt / 11 mod 11 =

the following sets of equations are figured out:

⎪⎪

⎪⎪

and there are two more restrictions: (1) each set of equations has at most two nonzero variables and (2) ∀1 ≤ i ≤ 8, 1 ≤ j ≤ 4:

and F = _c⁻

interpreted as: There is a post-condition colour transition fault on t2, which denotes c3

changing into c1. Ninth, Fa =

=

Figure 4.5 A CPN with Additive Faults detection and correction capabilities.

Chapter 5. Conclusion and Future Works

This thesis proposed a methodology to determine whether a marking in a coloured Petri net is a faulty marking which can be mapped to a faulty state in a system modeled by a coloured Petri net. The main idea of the methodology is applying the methods of error control coding on coloured Petri nets. In [3], the authors present the methods detecting the faults on Petri nets only. There are more issues to be studied on coloured Petri nets. Since a Petri net can be deem as a coloured Petri net with only single colour, the method for this special case would be the same as [3] in this thesis. Thus, the methods presented in this thesis are more general than those in [3]. If the applied error control coding is Reed-Solomon code, the methodology in this thesis can simultaneously detect and correct k place faults, x amount transition faults and k - x colour transition faults after adding 2k places, where 0 ≤ x ≤ k. There is a corresponding code correction algorithm in Reed-Solomon code, which is Berlekamp-Massey algorithm [16]. By appling Berlekamp-Massey algorithm on the syndrome of a faulty marking, the equation sets obtained from the syndrome can be solved out in time complexity O(kγ(α+β)), and hence the marking can be corrected in time complexity O(kγ(α+β)), where α, β and γ are the number of transitions, places and colour types in a coloured Petri net, respectively.

There are two further research topics which can be extended from this thesis.

First, from the marking, input and output matrices of the CPN with fault detection and correction capability, it can be seen that the values in these matrices are large. The reason is that the Reed-Solomon code has only minimized the length of a code but hasn’t minimized the value of a code word. Thus, one of the future works is to come

out the encoding matrices which can also minimize the values in these matrices.

Second, there are several kinds of high level CPNs extended from basic CPN discussed in this thesis. Thus, the other future work is to extend the methodology in this thesis to these high level CPNs.

Reference

[1] Kurt Jensen, “Coloured Petri Nets: Basic Concepts,” Springer, 1992.

[2] Kurt Jensen, “Coloured Petri Nets: Analysis Methods,” Springer, 1995.

[3] Y. Wu and C. N. Hadjicostis, “Algebraic approaches for fault identification in discrete-event systems,” IEEE Transactions on Automatic Control, vol. 50, no.

12, Dec. 2005, pp. 2048-2055.

[4] L. Li, C. N. Hadjicostis, and R. S. Sreenivas, “Designs of bisimilar petri net controllers with fault tolerance capabilities,” IEEE Transactions on Systems, Man and Cybernetics (Part A), vol. 38, no. 1, pp. 207-217, January 2008.

[5] P. Jancar, “Decidability questions for bisimilarity of Petri nets and some related problems,” Proceedings 11th Annual Symposium on Theoretical Aspects of Computer Science, pp. 581-592, 1994.

[6] C. A. Petri, “Kommunikation mit Automaten,” PhD thesis, University of Bonn, Bonn, Germany, 1962.

[7] Y. Narahari and N. Viswanadham, “On the Invariants of Coloured Petri Nets,”

Advances in Petri Nets 1985, pp. 330-345. Springer, 1986.

[8] V. K. Belikov and Y. F. Rutner, “Coloured Petri Nets and the Matrix Approach,”

Programming and Computer Software, vol. 14, no. 3, pp. 120-124, March 1989.

[9] Y.-H Lin, “A Technique for Reducing Occurrence Graph Building Time,”

National Chiao-Tung University, M.S. Thesis, 2007.

[10] P. Jancar, “Undecidability of bisimilarity for Petri nets and some related problems,” Theoretical Computer Science, vol. 148, no. 2, pp. 281-301, September 1995.

[11] C. N. Hadjicostis and G. C. Verghese, “Monitoring discrete event systems using

Petri net embeddings,” Application and Theory of Petri Nets 1999, Series Lecture Notes in Computer Science, vol. 1639, pp. 188-207, 1999.

[12] E.-R. Olderog, “Strong Bisimilarity on Nets: A New Concept for Comparing Net Semantics,” vol. 354, pp. 549–573, New York: Springer-Verlag, 1988.

[13] V. K. Belikov and Y. F. Rutner, “Matrix Specification and Analysis of Colored Petri Nets,” Soviet Journal of Computer and System Sciences, vol. 26, no. 3, pp.

77-80, March 1988.

[14] V. K. Belikov, “Two Approaches to the Definition of the Model of Colored Petri Net,” published by the editorial board of Izv. Akad. Nauk SSSR, Tekh. Kibern., Moscow (1985). Manuscript available from VINITI, 17.12.85, no. 8669-v85.

[15] Paul Garrett, “The Mathematics of coding Theory,” Prentice Hall, 2004.

[16] E. R. Berlekamp, “Algebraic Coding Theory,” Revised 1984 Edition, Aegean Park Press, Laguna Hills, CA, 1984.

在文檔中 一在Coloured Petri Nets中的錯誤找尋與校正之方法 (頁 104-124)