Supervisory Control - The Unknown Component Problem: Theory and Applications - page 244

Indeed, if the controllability condition holds for C then each uncontrollable action

at each state of P reached from the initial state by means of a string in the language

S .

As a direct consequence, an algorithm follows for deriving a largest controller
under partial controllability,

is enabled by C

Pref

1. Step 1: D erive the largest solution .P [ S/ of the equation P \ X D S .

Pref

If P \ .P [ S/ S th en S is not controllable with respect to P .If P \ .P [

Pref .
2. Step 2: Delete iteratively from

Pref

S/ D S then L D .P [ S/

L each state l such that the product P \ L has

a state .p; l/ for which some uncontrollable event is defined at p 2 P ,butis

not defined at l 2 L .If P \ L D S then L is the largest solution under partial

controllability.

Proposition 15.16. If , the above algorithm returns the largest controller

under partial controllability, if such a controller exists. Instead, if

L S

L S then there

is no controller under partial controllability.

Proof. We delete only states which require that the controller should disable an
uncontrollable event, thus we get the largest controller. t

In Sect. 15.4 , we discuss the well-known example of the maze with a cat and a
mouse, and we discuss the solution obtained by applying the previous algorithm.
In the sequel we will show that we can get a larger class of controllable
languages if we introduce the notion of composition with priorities (originally
proposed in [79]), where for the plant each uncontrollable action has a higher
priority compared with controllable actions.

Definition 15.3.3. Given two automata P and C their composition with priorities

\ P yields an automaton P \ p C obtained as follows.

a
! p

a
! c

0 in 0 in

If there are transitions p P and c C , then there is a transition

a
! .p

0 0

.p; c/ ;c / in P \ p C .

a
6! c

a
! p

0 in 0

If a 2 ˙ u c , there is a transition p P and there is no transition c

a
! .p

0

in C , then there is a transition .p; c/ ;c/ in P \ p C .

This motivates the definition of a weak controller as a prefix-closed solution of the
equation P \ p C D S .

Definition 15.3.4. A prefix-closed language C over ˙ is a weak controller for S ,

with respect to a plant P and a set of uncontrollable events ˙ u c ,if P \ p C D S .

Corollary 15.17. A controller that satisfies Definition 15.3.2 is a weak controller
that satisfies Definition 15.3.4 .

Definition 15.3.5. Given an automaton C over the alphabet ˙ and ˙ u c ˙ ,we

C * ˙ u c of obtain the ˙ u c - extension C by adding at each state of C a self-loop

labeled with each action a 2 ˙ u c such that there is no transition from this state

under action a .