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t
2
with
t>
0 both map to the same qualitative value (0
..
inf
,
+) denoting some
unknown function increasing in the interval (0
,
3
·
). Hence, the base areas of the
two tanks in Equation 1 are not considered by the qualitative model.
Starting from a given initial state, a qualitative inference engine like QSIM [13]
derives all possible behaviors which may evolve over time. This qualitative in-
ference process results in a transition system. It represents the set of all possible
behaviors, which is the set of all state sequences starting from the initial state,
referred to as traces.
Behavior inference relies on solving constraints between the quantities ex-
pressed as QDEs and on transition rules between the current state and succes-
sor states imposed by the qualitative theory. For example, the earlier mentioned
continuity law
preventing jumps in the qualitative evolution is such a transi-
tion rule. A behavior is extended with a new state if at least one of the model
quantities changes its value and the new state satisfies the QDEs and transition
constraints.
∞
Full
Empty
Zero
Full
Reserve
Empty
Zero
Fig. 5.
Qualitative evolution of the two-tank system
Figure 5 shows an example evolution of our system. The upper diagram shows
a concrete evolution while the lower diagram shows the corresponding qualitative
evolution. In hybrid systems theory, a continuous region is denoted as mode. The
scenario comprises four different modes. Discrete controller actions may cause
non-continuous changes in the system as can be observed for the quantity
x
1
at
the change from mode
P1 ON
to
P1 OFF
in the qualitative trace. Here, the
water level
x
1
changes directly from decreasing to increasing. Of course, in a
real system the changes would be continuous as mass inertia requires changes to
take some time.
