Controller Synthesis of Time Petri Nets Using Stopwatch

Algorithm 1

On-the-fly algorithm for the safety control of TPN-Part I.

Function main (TPN 𝒩, Markings bad)

Where 𝒩 is a TPN

bad is a set of bad markings.

Let  𝑇_𝑐  be the set of controllable transitions of  𝒩  and

𝛼₀  the initial state class of  𝒩.

Passed=∅

If  (explore(𝛼₀,𝜖,{𝛼₀})≠∅)  then

   {Controller does not exist}

   return

end if

for all ((𝛼,Ω,LI)∈Passed) do

  Ctrl[𝛼]=⋃_{(𝑡𝑐,𝑡𝑠,BI)∈LI}{(𝑡_𝑐,𝑡_𝑠,INT(𝛼,𝑡_𝑐−𝑡_𝑠)−BI)}

end for

(^∗Remarks:)

𝛼=(𝑀,𝐹);

En_𝑐(𝑀)=En(𝑀)∩𝑇_𝑐;

En⁰_𝑐(𝑀)=En_𝑐(𝑀)∪{𝑡₀};

New_𝑐(𝑀,𝑡)=New(𝑀,𝑡)∩𝑇_𝑐;

New(𝑀₀,𝜖)=En(𝑀₀);

𝑡₀ is a fictitious transition whose time variable is fixed at 0.

Dep(𝛼,𝑡,LI)≡

∃(𝑡_𝑐,𝑡_𝑠,BI)∈LI,𝑡_𝑐∉New(𝑀,𝑡)∧(𝑡_𝑠∉New(𝑀,𝑡)∨

INT(𝛼,𝑡_𝑐−𝑡₀)⊆⋂̸_𝐼∈BI(𝐼⊕INT(𝛼,𝑡_𝑠−𝑡₀)))