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useful to work from the middle out, i.e., to postulate a design in terms of timed
automata and check upwards and compile downwards.
2Requ emen s
In order to understand requirements for an embedded computer system one must
know a little about the physical plant that it monitors or controls. Such plants
are usually modelled by dynamical systems, i.e. systems with a state that evolves
over time. Within the engineering disciplines, the background for the study of
such systems is control theory. We outline central elements of control theory
below. However, we shall use duration calculus to express top level requirements,
so the next step is to make this notation precise. The nal part of the section then
introduces and discusses the kinds of requirements usually encountered in our
case studies and gives analogies to corresponding dynamical systems properties.
At the end of this section we have thus illustrated an informal link between
control engineering and embedded systems requirements.
2.1
Control Theory
Typically, a model of a plant is given in terms of a vector
x
of state variables
that evolve smoothly over time modelled by the reals. The characteristics of
the particular plant are specied by associating a dierential equation with the
state:
x
(
t
)=
F
(
x
(
t
)) for all time points
t
Here,
x
denotes the component-wise derivative of
x
with respect to time, and
F
is a vector function that characterizes the relationship between the dierent
state components of the plant.
When a plant is to be controlled, the model is extended by superimposing a
control input component
u
such that the model becomes
x
(
t
)=
F
(
x
(
t
)
;
u
(
t
))
There are variations of this framework, e.g. with discrete time, adding dis-
turbance to the model, or with the state as a stochastic variable. However, the
main eort in control engineering is not to modify this framework, but to nd
a mathematically tractable model, investigate its mathematical properties, and
most importantly to check that it is a good enough model of the physical plant.
In formulating requirements to an embedded computer system that controls the
plant, one uses explicitly or implicitly some mathematical model of the plant.
The computer program is in eect observing
x
through sensors and controlling
u
through actuators, and program requirements must unavoidably be formulated
in terms of these or precisely related states.
The key property that one would require of a dynamical system is
stability
,
i.e. if it started in a reasonable state, then it will eventually settle in a bounded
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