How one can use a real math function on transaction in Hybrid Petri Net fundamental equation?

Question

Say we have a simple HPN with 2 continuous places $A$ and $B$ and one transition. We want a transition not only add and substract $N$ marks from $A$ and add $M$ to $B$ but use mathematical function $F_{in}(\text{NumberOfMarksInA}, \text{NumberOfMarksInB})$ and $F_{out}(\text{NumberOfMarksInA}, \text{NumberOfMarksInB})$. How would (if is at all possible) Hybrid Petri Net fundamental equation look like if a transaction would be not a linear function for in and out?

Answer 1

If the transition (T_0) in the given (Hybrid) Petri Net fires by subtracting from the mark of Place A (P_A) and by adding to the mark of Place B (P_B) using the respective functions F_in and F_out then the logic annotation for updating the mark of Place A is〖 m〗_A=m_A-F_in and the logic annotation for updating the mark of Place B is〖 m〗_B=m_B+F_out.

For comparison, consider the transition of the given (Hybrid) Petri Net that fires by subtracting from the mark of Place A and by adding to the mark of Place B using the respective constants〖 w〗_in and〖 w〗_out; its logic annotation for updating the mark of Place A is m_A=m_A-w_in and its logic annotation for updating the mark of Place B is〖 m〗_B=m_B+w_out.

The logic annotations with constants may be organized as matrices as follows:

The logic annotations with functions may be organized as follows:

For the logic annotations that use functions, the corresponding values of function elements in the matrix must be updated every time a transition fires. In other words, 〖 F〗_in and F_out must be recomputed and W^T updated. For the logic annotations that use only constants, no update is needed when a transition fires; thus the matrix does not have to be recomputed.

References

Chionglo, J. F. (2016). "A Reply to "How on ecan use a real math function on transaction in Hybrid Petri Net fundamental equation" at Math Overflow". Available at http://www.aespen.ca/AEnswers/1455110984.pdf.

“How one can use a real math function on transaction in Hybrid Petri Net fundamental equation?” (2016). Math Overflow. Retrieved on Feb. 9, 2016 at How one can use a real math function on transaction in Hybrid Petri Net fundamental equation?.

Petri Net. (2016) Wikipedia. Retrieved on Feb. 9, 2016 at https://en.wikipedia.org/wiki/Petri_net.