Supervisory Control - The Unknown Component Problem: Theory and Applications - page 246

a

b , c

b

a

c 0 c 1 c 1

c

a

c

b

b

c 0 c 1

b a

b , c

b

a

c 0 c 1 c 1

a c

Fig. 15.6 Illustration of Example 15.21 .( a )Plant P ;( b ) Specification S and controller C under

partial controllability; ( c ) A weak controller C w under partial controllability

When characterizing all controllers C , Pref .S / is a lower bo un d (if there is a

Pref .When

controller), whereas a tight upper bound o n C can be less than .P [ S/

C w , 2
whereas Fig. 15.21 shows a weak controller whose language is incomparable with
Pref

Pref

characterizing all weak controllers C w , .P [ S/ is still an upper bound on

.
Weak controllers arise in the equation solving approach when modeling composi-
tion with priorities. However, when interpreted within the framework of supervisory
control, this notion models actions which the plant can execute with the environment
and which are neither controllable nor observable by the controller. This should
be further investigated in the context of modeling disturbances with both partial
controllability and observability.

.S /

15.4 Supervising a Cat and a Mouse in a Maze

In the next example we derive in some detail the controller for the cat and mouse
from [119].

Example 15.22. There are two mobile agents, say a cat and mouse moving in a
maze. The maze is shown in Fig. 15.7 ; it has five rooms communicating through
doorways. each of which is traversed only in the indicated direction exclusively

2 If

C w is a weak controller, then C w C * ˙ u c

w

.P [ S/

Pref .