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7.7. Mathematical Model of the Dynamic Semantics
The corresponding connection matrix C for Figure 7.15 must be defined as
⎡
⎤
0 1 0 0
C = ⎢ 0 0 1 0
⎥
⎣ 0 0 0 1 ⎦ (7.27)
1 0 0 0
Finally, the complete connection equation is determined by
⎡ ⎤ ⎡
⎤ ⎡ ⎤
x 1,1,k 0 1 0 0 y 1,1,k
⎢ x 1,2,k
⎥
⎣ x 1,3,k
⎦ = ⎢ 0 0 1 0
⎥ ⎢ y 2,1,k
⎥
⎣ 0 0 0 1 ⎦ ⎣ y 3,1,k
⎦ (7.28)
x 2,1,k 1 0 0 0 y 4,1,k
⎡
⎤ ⎡
⎤
0 1 0 0 x 1,1,k + x 1,2,k + x 1,3,k
= ⎢ 0 0 1 0
⎥ ⎢ s 2,1,k−1
⎥
⎣ 0 0 0 1 ⎦ ⎣ s 3,1,k−1
⎦
1 0 0 0
s 4,1,k−1
It must be remarked that the equations for each element of x k must be calculated in the
order of the elements from top to bottom. This was reasoned in Section 7.6. Thus, the element
order for x k cannot be chosen arbitrarily.
No element x i,j,k may occur at the same time on the left and the right side of the equation to
have meaningful equations for each element of x k . Corresponding to the openETCS model, this
would be the case if an output of an object is directly connected with an own input. This could
be, for example, Figure 7.15 without the “Value” object. From the view of the mathematical
model, such connections are not possible and lead to undefined connection equations because
for the calculation of an output is an input used that is not yet available. In the domain frame
work, for each existing input is a certain default / initial value defined to allow such data
flows anyway. However, the mathematical model suggests that it is in any case better to use
intermediate oVariableStorage objects as “Value” and to model the initial value implicitly. A
virtual storage object has always to be inserted to define connections between input and output
in the mathematical model.
7.7.2. State Machines
The state machine types used in the openETCS meta model do not provide any direct outputs
and accordingly can be called transition systems. Generally, such a transition system T is a
tuple of a set of states S, an initial state s 0 , inputs alphabet Σ, a transition function δ, and a
set of final states F :
T = (S, s 0 , Σ, δ, F ) (7.29)
with
s 0 ∈ S ≡ {s h ; h = 1, 2, 3, . . . , n s } (7.30)
F ⊂ {∅} ∪ S (7.31)
δ : S × Σ → S (7.32)
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