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On the existence of finite state supervisors for arbitrary supervisory control problems

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1 Author(s)
Sreenivas, R.S. ; Dept. of Gen. Eng., Illinois Univ., Urbana, IL, USA

Given two prefix closed languages K, L ⊆ Σ*, where K ⊆ L represents the desired closed-loop behavior and L is the open-loop behavior, there exists a finite-state supervisor that enforces K in the closed loop if and only if there is a regular, prefix-closed language M ⊆ Σ*, such that: 1) MΣu∪L⊆M, and 2) M∪L=K. In this paper, we show that this is equivalent to: 1) the controllability of sup{P⊆K∪L¯|pr(P)=P} with respect to Σ*; and 2) the regularity of sup{P⊆K∪L¯|pr(P)=P}, where L¯=Σ*-L:and pr(·) is the set of prefixes of strings in the language argument. We use this property to investigate the issue of deciding the existence of a finite-state supervisor for different representations. We also present some properties of the language sup{P⊆K∪L¯|pr(P)=P}, along with implications to the synthesis of solutions to the supervisory control problem with the fewest states

Published in:

Automatic Control, IEEE Transactions on  (Volume:39 ,  Issue: 4 )