GRAFCET and Petri Nets Outline Introduction GRAFCET - EPFL

Reachability graph
Reachability tree
We create a graph of
reachable markings
Exhaustive search
Risk of combinatorial
explosion
t 1
p 1
t 2 t 3
t 1
p 2 p 3
t 3 t 2 t 1
[2,0,0] [1,0,1] [0,1,1] [1,0,0]
If we want to show that
the PN is not bounded,
it might be useless to continue
p 1
This is the case if M 1
that has been reached from M,
M 1 ≥M (∀i: m 1 (p i ) ≥m(p i ))
if M 1 =M, we are back to the
same marking
t 2
if M 1 >M, it is possible to repeat
[1,0,1]
the same firing sequence and
increment again the number of tokens
t 2 t 3
t 1
p 2 p 3
[1,0,0]
t 1
[0,1,1]
t 3
[1,1,0]
Real-Time Programming GRAFCET and Petri nets 137
© J.-D. Decotignie, 2007
Real-Time Programming GRAFCET and Petri nets 138
© J.-D. Decotignie, 2007
Analyis by reachability tree
Matrix analysis
The tree is always finite
The tree permits to check the following properties
Not bounded if there exsists M1>M
Conservative if the number of tokens is constant
L1-Live is each transitions appears at least once in the tree
Reversibility and home state
Complex to use
Uses I and O matrices
Look for a non trivial solution to
| O-I |.x T =0 avec x= | x 1 , ...., x i , ...., x N | (1)
Let M et M 1 be two successive markings
M 1 = M + O[t j ] – I[t j ] after firing t j (2)
Let ρ = | ρ 1 , ...., ρ i , ...., ρ N | be a solution of (1)
Let multiply each term of (2) by ρ i and sum up on i
N
N
N
∑m
pi
i = ∑ im
pi
+ ∑ i O t j pi
− I t j pi
i=
1 1 ρ ρ ρ , ,
i=
1 i=
1
( ) ( ) [ ( ) ( )]
Real-Time Programming GRAFCET and Petri nets 139
© J.-D. Decotignie, 2007
Real-Time Programming GRAFCET and Petri nets 140
© J.-D. Decotignie, 2007