Separate CPNs with Place Faults Detection and Correction

, , 22 1

1 ≠

hn in h

i h

i p p

p f f

f L .

Proof:

By Definition 4.10, n place faults denote that there are n places with place faults.

Hence, by Lemma 4.1, there are n matrices,F_pⁱ¹,F_pⁱ²,L,F_pⁱⁿ, indicate these place faults, where i1 ≠ i² ≠ … ≠ iⁿ, and Qf =Q+F_pⁱ¹ +F_pⁱ² +L+F_pⁱⁿ= Q + Fp. Therefore,

the entries in Fp should satisfy ∃1 ≤ i¹, i2, …, in ≤ β, i1 ≠ i² ≠ … ≠ iⁿ, ∀1 ≤ j ≤

γ: _j ∈

in j

i j

i p p

p f f

f , , ,

2

1 L ℤ, ∀1 ≤ l ≤ γ, 1 ≤ k ≤ β, k ≠ i¹≠ i²≠ … ≠ iⁿ: f_pkl =0, and ∃1

≤ h1, h2, …, hn ≤ γ: 0, , ,

2 2 1

1 ≠

hn in h

i h

i p p

p f f

f L .

A matrix Fp which indicates n place faults would have n rows with nonzero entries, and the other rows have only entries of zero. Each row with nonzero entries in Fp represents a place fault. The relation equation, Qf = Q + Fp, could be proved by deriving from Lemma 4.1.

4.3.2 Separate CPNs with Place Faults Detection and Correction Capabilities

In order to detect and correct place faults, the strategy in this thesis encodes a CPN with additional tokens as a separate CPN firstly. Definitely, Let G be a CPN with no more than k place faults, the separate CPN with place faults detection and correction capabilities is constructed by adding 2k additional places to G and the

colour sets of both CPNs are the same. Assume that G has α transitions, β places, γ colours, input matrixB_g⁻, output matrixB_g⁺ and initial marking matrix

Q₀g. The separate CPN H with place faults detection and correction capabilities in respect to G would have α transitions, β + 2k places, γ colours, input matrixB_h⁻, output matrixB_h⁺ and initial marking matrix

Q₀h . Besides, ⁻ ⎥ ⁻

⎦

⎢ ⎤

⎣

=⎡ _g

h B

D

B I^β , ⁺ ⎥ ⁺

⎦

⎢ ⎤

⎣

=⎡ _g

h B

D

B I^β and

g

h Q

D Q₀ I ⎥ ₀

⎦

⎢ ⎤

⎣

=⎡ ^β , where D is a 2k × β matrix, and Iβ denotes a β × β identity matrix.

After H is constructed, the place faults occurring on H can be identified and corrected from the syndromes. These properties are proved in Lemma 4.3 and 4.4.

Lemma 4.3: 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.

If the CPN H, constructed by adding d additional places to G, has the same colour set with G, α transitions, β + d places, γ colours, input matrix

−

− ⎥

⎦

⎢ ⎤

⎣

=⎡ _g

h B

D

B I^β ,

output matrix

+

+ ⎥

⎦

⎢ ⎤

⎣

=⎡ _g

h B

D

B I^β

and initial marking matrix

g

h Q

D Q₀ I ⎥ ₀

⎦

⎢ ⎤

⎣

=⎡ ^β

, where D is a d × β matrix, all the entries in D are nonnegative, d ∈ ℕ, and I^β denotes a β × β identity matrix, H has following two properties.

„ H is a separate CPN with respect to G.

„ If a reachable marking matrix Qg of G has the same firing transition sequence

with a reachable marking matrix Qh of H, _h Q_g D

Q I ⎥

⎦

⎢ ⎤

⎣

=⎡ ^β .

Proof:

First, Let Pg and Ph be the place sets of G and H respectively. Consider conditions 2 and 3 in Definition 4.4, CPN H is composed of CPN G and d additional places, and thus Pg⊇ P^h. Since H and G have the same colour set, Cg = Ch where Cg

and Ch are the colour sets of G and H respectively.

Next, let i and j be two integers, where 1 ≤ i ≤ β and 1 ≤ j ≤ γ.

⎥⎥

⎦

⎤

⎢⎢

⎣

=⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

=⎡

⎥⎦

⎢ ⎤

⎣

=⎡

g g

g g g

h DQ

Q DQ

Q Q I

D Q I

0 0 0

0 0

0

β β , thus

gij

hij q

q₀ = ₀ , for all possible i and j. By

Definition 3.1, q₀ m_0h(p_i)(c_j)

hij = and q₀ m_0g(p_i)(c_j)

gij = , and hence

) )(

( )

)(

( :

, _{j g 0 i j 0 i j}

g

i P c C m p c m p c

p ∈ ∀ ∈ h = g

∀ . By Definition 2.2, since Cg = Ch,

) ( )

(

: _{0 i 0 i}

g

i P m p m p

p ∈ h = g

∀ . Therefore, condition 4 in Definition 4.4 is satisfied.

Next, since H has no additional transition compared to G,

Tg = Th, (1) where Tg and Th are the transition sets of G and H respectively. Since

g g

h Q UQ

D

Q₀ I ⎥ ₀ = ₀

⎦

⎢ ⎤

⎣

=⎡ ^β , there is a matrix V =

[

I_β 0_β×d

]

such that

[ ]

_{g g}

g

h Q Q

D I I

VUQ

VQ_{0 0} 0 _d ⎥ ₀ = ₀

⎦

⎢ ⎤

⎣

= ⎡

= _{β β×} ^β , where 0β×d is a β×d matrix with all

entries of zero. By Definition 3.1,

∑

∑

= =

=

=

= ^{β β}

1 0 1

0 0

0 ( )( ) ( )( )

l

j l il

l il j

i c q u q u m p c

p

m _g

glj hij

h . Hence, there is a linear

transformation set F, such that

) Assume there are two markings

m₁g and

m₁h in G and H respectively. The marking matrices of

m₁g and

m₁h are

Q₁g and

Q₁h respectively. Assume that

g enabled by marking

m₁g if and only if ∀ ≤ ≤ ≤ ≤ ≥ ⁻ The marking after firing transition tr by marking

m₁g is

Therefore, m_2h =F(m_2g) (5) and m_2g =H(m_2h). (6) In the same way, if m_1h =F(m_1g) , m_1g =H(m_1h) and m_1h[t_r >m_2h , then

g

g t m

m₁ [ _r > ₂ such that m_2h =F(m_2g) and m_2g =H(m_2h). (7) By (1), condition 1 in Definition 4.3, the definition of redundant relation, is satisfied. By (2) and (3), condition 2 in Definition 4.3 is satisfied. By (4), (5) and (6), condition 3 in Definition 4.3 is satisfied. By (7), condition 4 in Definition 4.3 is satisfied. Therefore, G ≃ H which satisfies condition 1 in Definition 4.4.

Finally, since Qg and Qh are reachable marking matrices of G and H respectively, and Qg and Qh have the same firing transition sequence, Qg =

g g g

gX B X

B

Q_0g − ⁻ + ⁺ and Qh = Q_0h −B_h⁻X_h +B_h⁺X_h , where Xg = Xh . Since

−

− ⎥

⎦

⎢ ⎤

⎣

=⎡ _g

h B

D

B I^β , ⁺ ⎥ ⁺

⎦

⎢ ⎤

⎣

=⎡ _g

h B

D

B I^β and _h Q _g

D Q₀ I ⎥ ₀

⎦

⎢ ⎤

⎣

=⎡ ^β , Qh =

g g

g g g h

g h

g Q

D X I

B X B D Q

X I D B X I

D B Q I

D I

g

g ⎥

⎦

⎢ ⎤

⎣

=⎡ +

⎥ −

⎦

⎢ ⎤

⎣

=⎡

⎥⎦

⎢ ⎤

⎣ +⎡

⎥⎦

⎢ ⎤

⎣

−⎡

⎥⎦

⎢ ⎤

⎣

⎡ _{β β − β + β − + β}

) ( ₀

0 . Since

g

h Q

D

Q I ⎥

⎦

⎢ ⎤

⎣

=⎡ ^β , it can be proved that ∀p_i∈P_g :m_h(p_i)=m_g(p_i) by the same way

proving p_i P_g :m₀ (p_i) m₀ (p_i)

g

h =

∈

∀ . Hence, condition 5 in Definition 4.4 is

satisfied. Therefore two properties in Lemma 4.3 are proved.

In Lemma 4.3, A CPN H with input matrix ⁻ ⎥ ⁻

⎦

⎢ ⎤

⎣

=⎡ _g

h B

D

B I^β , output matrix

+

+ ⎥

⎦

⎢ ⎤

⎣

=⎡ _g

h B

D

B I^β and initial marking matrix

g

h Q

D Q₀ I ⎥ ₀

⎦

⎢ ⎤

⎣

=⎡ ^β is proved to be a separate

CPN of G. It is also derived that the marking matrix, Qh, in H would be Q_g D I ⎥

⎦

⎢ ⎤

⎣

⎡ _β

all along, where Qg, the marking matrix in G, has the same firing transition sequence as

Qh.

Lemma 4.4: Let G be a CPN which has α transitions, β places, γ colours, and H be a separate CPN with d additional places with respect to G. Besides, if a reachable marking matrix Qg of G has the same firing transition sequence with a reachable marking matrix Qh of H, _{h g} Q_g

D UQ I

Q ⎥

⎦

⎢ ⎤

⎣

=⎡

= ^β .

„ If there are place 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 iff the place fault indicator matrix is Fp.

„ k place faults on H can be identified and corrected if any 2k columns of the check matrix W are linearly dependent.

Proof:

Assume Qh is a fault-free marking matrix of H, then WQh = WUQg = 0d × γ. If there are place faults on H, by Lemma 4.2, the faulty marking matrix Qf satisfies Qf = Qh + Fp = UQg + Fp. Hence, The syndrome S = WQf = WUQg + WFp = WFp. By the same way, if the syndrome is WFp, it will be S = WFp = WFp + 0d × γ = WFp + WUQg = WFp + WQh = W(Fp + Qh), where Qh is a fault-free marking matrix of H, and Fp is a place fault indicator matrix. 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.

If Qf is a faulty marking matrix of H, which states k place faults on H, by Lemma 4.2, Qf = Qh + Fp, where k rows of Fp have nonzero entries. In other words, each column in Fp has at most k nonzero entries, and hence each column in Qf has at

most k incorrect entries. Thus, the syndrome S =

[

^s₁ ^s_{2 L} ^s_γ

]

^{= WQ}f

= W

[

qf1 qf2 L qf_γ

] [

= Wqf1 Wqf2 L Wqf_γ

]

= WFp =

[

fp fp fp_γ

] [

Wfp Wfp Wfp_γ

]

W 1 2 L = 1 2 L , where sn, q_fn and f_pn

represent the nth column in S, Qf and Fp, respectively, and 1 ≤ n ≤ γ. A column in Q^f,

fn

q , can be deemed as a linear code which is of length β. There are two theorems [15]

in error control coding: (1) If a linear code with a check matrix, such that any 2k columns of the check matrix are linearly dependent, the linear code has minimum distance 2k + 1. (2) A code with minimum distance 2k + 1 can identify and correct k errors. Since each column in Qf has at most k incorrect entries, the faults in Qf can be identified and corrected by the check matrix W inside which any 2k columns are linearly dependent. After getting the syndrome by WQf, the place fault indicator matrix Fp can be found by solving equations Wf_p s_n

n = , where 1 ≤ n ≤ γ.

Lemma 4.4 shows that if there are at most k place faults, it needs to find out a check matrix which has any 2k columns are linearly dependent, and the result of multiplying the check matrix with a fault-free marking matrix is a matrix with all entries of zero. By applying the method of Reed-Solomon codes [15], it would find a check matrix of 2k rows and any 2k columns are linearly dependent. Hence, d = 2k. In other word, if there are at most k place faults, 2k additional places is needed in the separate CPN by applying the method of Reed-Solomon codes in order to derive the detection and correction capabilities of place faults. Lemma 4.4 also shows that the place faults can be identified and corrected from the syndrome.

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 matrixQ_0g. From Lemmas 4.3 and 4.4, constructing a separate CPN H which can detect and correct at most k place faults is concluded as following steps: (1) First, constructing a 2k × (β + 2k) check matrix W from the check matrix of Reed-Solomon codes. (2) Second, solving the equation

β β

= ×

⎥⎦

⎢ ⎤

⎣

⎡ D d

W I 0 and getting the entries of D. (3) Finally, deriving the separate CPN H

containing input matrix ⁻ ⎥ ⁻

⎦

⎢ ⎤

⎣

=⎡ _g

h B

D

B I^β , output matrix ⁺ ⎥ ⁺

⎦

⎢ ⎤

⎣

=⎡ _g

h B

D

B I^β and initial

marking matrix _h Q _g D Q₀ I ⎥ ₀

⎦

⎢ ⎤

⎣

=⎡ ^β . A marking Qh in the separate CPN H can be

examined if it is a correct marking by the check matrix W. If the marking Qh is a faulty marking, it can be corrected by solving the place fault indicator matrix Fp from the equation S = WFp, where S is the syndrome from WQh.

4.3.3 An Example of Identifying and Correcting Place

在文檔中 一在Coloured Petri Nets中的錯誤找尋與校正之方法 (頁 48-55)