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Variable abstraction and Approximations in Supervisory Control Synthesis

Variable abstraction and Approximations in Supervisory Control Synthesis. Marcelo Teixeira (mt@das.ufsc.br) José E. R. Cury (cury@das.ufsc.br) Max H. de Queiroz (max@das.ufsc.br). Robi Malik (robi@waikato.ac.nz). Outline. Supervisory Control Theory - SCT

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Variable abstraction and Approximations in Supervisory Control Synthesis

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  1. Variable abstraction and Approximations in Supervisory Control Synthesis Marcelo Teixeira (mt@das.ufsc.br) José E. R. Cury(cury@das.ufsc.br) Max H. de Queiroz (max@das.ufsc.br) Robi Malik (robi@waikato.ac.nz)

  2. Outline • Supervisory Control Theory - SCT • Describes a formal way to obtain controllers for DES • Grounded on Finite-state Automata – FA Limitations of FA Modeling complexity Computational complexity • Difficulty faced to express control specifications • Data dependency • Computational cost to process large automata • State-space explosion Proposed alternative • Extending FA with variables (EFA) • State-space remains large • Abstracting variable from automata • Irrelevant variables are removed from synthesis

  3. Motivating Example Not modular Solution depends on n Expensive Overflow and Underflow control Plant Models Extended Finite-state Automaton – EFA = Extends FA with VARIABLES ? It’s inevitable to take the whole state-space into account (n+1 states) How to model these specifications? ...

  4. Motivating Example Defining variables b1 b2 0 0 1 1 5 10

  5. Modeling the plant with variables Plant Models

  6. Modeling the plant with variables Specification Models ? It takes the whole state-space into account (n+1 states) ... Modeling complexity has been outlined!

  7. Synthesis of control with variables State-spaces of the EFA used in synthesis Variables are unfolded • Gv = R||M1 ||M2 : 8 • Ev = O1 ||U1 ||O2 ||U2 : 1 • Gv || Ev : 528 • supCv(Ev, Gv) : 484 • Variable Abstraction State-space Oh No!!!

  8. Variable abstraction • Let: • Gv be the EFA for the plant model • V be the set of variables and • W ⊆ V • Define: • W Gvas the abstraction for Gv. • W Gv: is constructed by removing updates implemented with variables in W • No updates, no unfolded states • But, what does it mean for the control solution?

  9. Synthesis with variable abstraction This may be feasible!!! But we don’t know about optimality • Theorem 1: = = Upper bound approximation This is complex!!! We cannot just compare • Theorem 1 implies: • 1) the abstracted solution is a solution • 2) yet, it may be suboptimal, i.e., it may be more restrictive. • So, how to state optimality?

  10. Upper bound construction • It bases on the idea of -controllability • Definition 1: Ev is -controllable wrt Gv if: • for every uncontrollable event eligible in Gv, Ev allows to reach a next state with at lest one variable value • This differs from the standard V-controllability for EFA • Definition 2: Ev is V-controllable wrt Gv if: • for every uncontrollable event eligible in Gv, Ev allows to reach a next state with all variable values

  11. Stating least restrictiveness • Main Theorem: =

  12. Synthesis with variable abstraction ? ? State-spaces of the EFA used in synthesis • Gv = R || M1 || M2 : 8 • Ev = O1 || U1 || O2 || U2 : 1 • Gv || Ev : 528 • supCv(Ev, Gv) : 484 • b1 b2Gv : 8 • Ev = O1 || U1 || O2 || U2 : 1 • b1b2Gv|| Ev : 8 • supCv(Ev, b1b2Gv) : 6 • supC(Ev, b1b2Gv) : 8 ≠ Oh No!!! M1 is not starting...

  13. Synthesis with partial abstraction State-spaces of the EFA used in synthesis • Gv = R || M1 || M2 : 8 • Ev = O1 || U1 || O2 || U2 : 1 • Gv || Ev : 528 • supCv(Ev, Gv) : 484 • b1Gv : 8 • Ev = O1 || U1 || O2 || U2 : 1 • b1Gv|| Ev : 48 • supCv(Ev, b1Gv) : 44 • supC(Ev, b1Gv) : 44 • Solution: • supCv(Ev, b1Gv) || Gv = supCv(Ev,Gv) = Oh Yes!!! It’s working now!

  14. Comments • EFA have been used in SCT: • as a way to simplify modeling tasks • Variable abstractions have been used in model checking: • as a way to reduce computational cost • In this paper: • We show that they both can be combined in synthesis • We present an algorithm to compute supervisors from abstractions • Our next steps include: • to provide a way to inform which variable is required in synthesis • This could replace the upper approximation, for example. Thank you!

  15. Variable abstraction and Approximations in Supervisory Control Synthesis Marcelo Teixeira (mt@das.ufsc.br) José E. R. Cury(cury@das.ufsc.br) Max H. de Queiroz (max@das.ufsc.br) Robi Malik (robi@waikato.ac.nz)

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