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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