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via mutation testing in Section 5 and show how to select test cases in Section 6.
Finally, we draw our conclusions, discuss related work and give an outlook to
future work in Section 7.
2 Hybrid Systems
Hybrid systems provide a closed-
loop view on control programs oper-
ating in their environment. This
closed-loop view, which incorporates
continuous (environment) and discrete
(controller) behavior, allows to draw
more conclusions than by looking at
the controller in isolation. For exam-
ple, properties about the system's sta-
bility or its long term behavior.
Consider the example hybrid sys-
tem of Figure 3 and the task to auto-
matically derive tests. The first step
towards a hybrid system model is to
know the (informal)
system require-
ments
: in the two-tank system in
Figure 3 tank
T
1 is on a lower level
than the tank
T
2.
T
1 is being filled with water having some inflow rate
in
.Both
tanks (
T
1,
T
2) are connected by the pump
P
1 that is controlled such that: if
the water level in
T
2 decreases below a certain
Reserve
mark and
T
1 is full,
pump
P
1 starts pumping water until
T
2isfullor
T
1 gets empty. In addition,
the controller needs to control the pump
P
2 that is pumping water out of
T
2:
P
2 shall be turned on as long as a button
WaterRequest
is pressed and there
is enough water in
T
2(
T
2not
Empty
). Note that the signal
WaterRequest
and
the inflow rate
in
are not controllable, hence
T1
may overflow.
Given these requirements, one is able to derive a formal model. In this exam-
ple, the continuous dynamics of the system is expressed by two coupled differ-
ential equations:
Fig. 3.
A
Two-Tank
System with
two
pumps
x
1
=(
in
−
inout
)
/A
1
and
x
2
=(
inout
−
out
)
/A
2
.
(1)
Here,
A
1
and
A
2
are the base areas of the two tanks and
x
1
and
x
2
denote the
current level in the tanks. The variables
in
,
inout
,and
out
denote the flow rates
into
T
1, between
T
1and
T
2, and out of
T
2 respectively.
Observe that for testing whether some given implementation of this two-tank
system conforms to the stated requirements we do not need to know the exact
numerical value of the water level at all times, nor do we care about the exact
time information. We would also need to incorporate additional knowledge, such
as the exact base areas of the tanks in order to solve these equations numerically:
in reality we might not have all information that is required for such a detailed
