where Kp is the proportional gain, Ti
is the integral
time constant and Td is the derivative time constant.
Expression (1) is usually known as the ideal form. Other
forms (usually called series and parallel forms) can also
be employed [3]. In order to be effectively employed in
practical cases, additional functionalities also have to be
implemented. The most important functionalities can be
summarized as follows (details can be found in [52]).
• The derivative action has to be filtered in order to
make the controller proper and to filter the (high
frequency) measurement noise; in addition, the
derivative action is often applied directly to the
process variable instead of to the control error in
order to avoid the so-called derivative kick when
a step signal is applied to the set-point. The
derivative filter has to be taken into consideration in
the overall design of the controller [13, 17].
• The set-point value for the proportional action
can be weighted in order to obtain a two-degreeof-freedom
controller, i.e. in order to reduce the
overshoot in the set-point step response when
the controller is tuned in order to increase the
bandwidth of the system with the aim of increasing
the load disturbance rejection performance. In this
case a suitable choice of the value of the set-point
weight (or the application of a more sophisticated
techniques [47]) can yield a significant increment of
the control performance.
• Suitable techniques (see, e.g. [48]) should be implemented
properly in order to avoid the windup effect