Supervisor Reconfiguration for Deadlock Prevention by Resources Reallocation

Algorithm 2

Controlled system design for (𝑁,𝑀₀).

Input: an ordinary M-net (𝑁,𝑀₀) with 𝑁=(𝑃⁰∪𝑃_𝐴∪𝑃_𝑅,𝑇,𝐹,𝑊)

Output: controlled system (𝑁^𝑐,𝑀^𝑐)

begin {

design a controlled system (𝑁^1𝑐,𝑀^1𝑐) for (𝑁¹,𝑀¹) by Algorithm 1, then

𝑁^𝑐∶=𝑁^1𝑐

∀𝑣∈𝑃_𝑉, 𝑀^𝑐(𝑣)∶=𝑀^1𝑐(𝑣)

∀𝑝∈𝑃⁰∪𝑃_𝐴∪𝑃_𝑅, 𝑀^𝑐(𝑝)∶=𝑀₀(𝑝)

𝑃_𝑖∶=(^••𝐻(𝑣_𝑖))∩𝑃_𝑅, 𝑖∈{1,2,…,𝑛}

∀𝑝∈𝑃_𝑖, 𝑀^𝑐(𝑣_𝑖)∶=∑_{𝑝∈𝑃𝑖}𝑀(𝑝)−1, 𝑖∈{1,2,…,𝑛}

^→𝜇∶=[𝑀^𝑐(𝑣₁),𝑀^𝑐(𝑣₂),…,𝑀^𝑐(𝑣_𝑗),…,𝑀^𝑐(𝑣_𝑛)]^𝑇, 𝑗∈{1,2,…,𝑛}

While   {^→𝜇 makes (𝑁^𝑐,𝑀^𝑐) not live, which can be decided by the MIP-based deadlock detection

method in [32]}  do

 for  𝑗=1 to 𝑛  do

  𝑀^𝑐(𝑣_𝑗)−−;

 end for

end while

^→𝑙∶=^→𝜇      /∗ ^→𝑙 denotes the marking vector

[𝑀^𝑐_𝑙(𝑣₁),𝑀^𝑐_𝑙(𝑣₂),…,𝑀^𝑐_𝑙(𝑣_𝑗),…,𝑀^𝑐_𝑙(𝑣_𝑛)]^𝑇,𝑗∈{1,2,…,𝑛}∗/

for   𝑗=1 to 𝑛   do

 𝑀^𝑐(𝑣_𝑗)++;

end for

^→ℎ∶=^→𝜇      /∗ ^→ℎ denotes the marking vector

[𝑀^𝑐_ℎ(𝑣₁),𝑀^𝑐_ℎ(𝑣₂),…,𝑀^𝑐_ℎ(𝑣_𝑗),…,𝑀^𝑐_ℎ(𝑣_𝑛)]^𝑇,𝑗∈{1,2,…,𝑛}∗/

while (1) do

 for   𝑗=1 to 𝑛   do

  if   (𝑀^𝑐_ℎ(𝑣_𝑗)==𝑀^𝑐_𝑙(𝑣_𝑗))

   𝑀^𝑐_ℎ(𝑣_𝑗)∶=𝑀^𝑐(𝑣_𝑗);

  else

   𝑀^𝑐_ℎ(𝑣_𝑗)−−;

   break;

  end if

 end for

 if   {^→ℎ makes (𝑁^𝑐,𝑀^𝑐) live, which can be decided by the MIP-based deadlock detection

 method in [32]}do

   break;

 end if

end while

^→𝜇∶=^→ℎ

∀𝑝∈𝑃⁰∪𝑃_𝐴∪𝑃_𝑅, 𝑀^𝑐(𝑝)∶=𝑀₀(𝑝)

𝑁^𝑐∶=𝑁^1𝑐

output (𝑁^𝑐,𝑀^𝑐)

} end of the algorithm