Page 95 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

70 Chapter Three

corresponds to a time domain term of

Poles can be complex but if so then they must occur in conjugate
pairs. Therefore, the factor occurring in a Laplace transform as in
1 1 1
(s- p)(s- p·) (s-(a+ jb}}(s-(a- jb}}

has two complex poles that occur as conjugates. As a consequence,
the factor is purely real which you would want because an imaginary
process transfer function does not make physical sense.
These complex conjugates also correspond to exponential terms
in the time domain except now they occur as

and end up contributing sinusoidal terms in the time domain.
These pairings suggest several things:
1. A pole at s = 0 corresponds to a constant or an offset.
2. When the pole lies on the negative real axis, the corresponding
exponential term will also be real and will die away with time.
3. As the pole's location moves to the left on the negative real axis
the exponential term will die away more quickly. As the pole
moves to the right along the negative real axis in the s-plane it
will soon reach s = 0 at which point it corresponds to a constant
in the time domain. As the pole continues to move into the right-
hand side of the s-plane, still along the real axis, the exponential
component now increases with time without bound.
4. When the poles appear in the s-plane with components dis-
placed from the real axis then the poles are complex and appear
as complex conjugates. The corresponding time domain terms
will contain sinusoidal parts and underdamped bounded
behavior will result if the poles lie in the left half of the s-plane.
5. If the complex poles are purely imaginary they still appear as
conjugates on the imaginary axis and they correspond to
undamped sinusoidal behavior that does not dissipate. As
the imaginary component of the complex conjugate poles
moves away from the real axis (while staying on the imaginary
axis) the frequency of the underdamping will increase.
6. If the transfer function has poles that occur in the right-hand
side of the s-plane, that is, if the poles have positive real parts,
then the process represented by the transfer function will be
unstable.