We define the transfer function of a black box model as
,
where is the Laplace transform of the input and
is the Laplace transform of the output. This yields:
.
is a rational function and therefore has zeroes and roots (removable poles and roots are removed) and therefore so does
:
.
These are the zeroes of
and the
are the poles of
. This means that we have (assuming
) a partial fraction expansion:
.
Applying the Inverse Laplace transform we have an input:
.
These are complex numbers and depending on their nature we get different behaviours.
A positive real pole is a pole such that
. This corresponds to an output
which tends to infinity as
. This is divergent or unstable behaviour.
A negative real pole with
corresponds to an output
as
. This is convergent or stable behaviour.
A zero pole yields an output
which is a constant output (which is considered stable).
A purely imaginary pole is, via Euler Formula, corresponds to oscillatory behaviour:
.
A genuinely complex pole (
) with a positive real part (in the right-half plane) corresponds to the following output
.
While the component is oscillations, the
goes to zero so we get behaviour that looks like:
Note that this behaviour is unstable.
When we have complex pole (
) with a strictly negative real part (a pole in the left-half plane), then we have
. Note that we have oscillations but modulated by a decreasing
. This is the motion of an underdamped harmonic oscillator:
This is inherently stable behaviour.
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