Discussion on Supervisory Control by Solving Automata Equation

1. Discussion on Supervisory Control by Solving Automata Equation Victor Bushkov, Nina Yevtushenko, Tiziano Villa, Tomsk State University (Russia), University of Verona (Italy).

2. Problem S S P P C ? P XS C

3. Supervisor Example Standard realizationInit(S) – prefix-closure of S Plant P Specification S Standard realization of supervisor C Init(S)

4. Largest Solution Largest solution (P S)pref of P XS is a solution which includes every possible solution of P XS. Largest solution Plant P Specification S C  (P S)pref

5. Progressive Solution Solution C of P XS is a progressivesolution if in P Ca final state could be reached from every state Non-progressive solution Cnon-prog Progressive solution Cprog Plant P P Cnon-prog P Cprog 2 Specification S

6. Partial Controllability • =c  uc • uc-extensionCucofCis obtained by adding self-loops under everya  ucat every statecofCif there are no transitions fromcunder a Plant P , uc = {c} Largest solution C  (P S)pref Largest solution under partial controllability C  C Cuc Specification S

7. Partial Controllability: Solvable Equations Supervisory Control Automata Equations 1. Init(L(S))ucInit(L(P))Init(L(S)) 2. L(S) = Init(L(S))  L(P) Init(L(S))(uc)* L(P)L(S) L(S), L(P) – languages ofS andP Init(L(S)) – prefix-closure ofL(S) Plant P , uc = {a} Specification S

8. Partial Controllability: Progressive Solutions Solution C of P XS is a progressivesolution under partial controllability if Cuc is a progressive solutionofP XS Plant P , uc={a} Solution under partial controllability C Cuc  P Specification S

9. Partial Observability • =o  uo • uo-foldingCuoofCis obtained by replacing each transition at every statecofCunder a  uoby a self-loop C1 C2 Solution under partial observability C1 Plant P , uo = {a} Solution under partial observability C2 (C1 C2)uo Specification S

10. Conclusions and Future Works • The complexity of solving supervisory control problem by automata equation approach is polynomial • By solving automata equation we can find a largest supervisor, from which an optimal supervisor can be extracted • The automata equation approach can deal with more general topologies

11. Thank you for yourattention Questions?