alt="equation"/>
which has a pair of closed-loop poles determined by the solutions of the polynomial equation:
These poles are at
Now, assuming that we will additionally take the derivative of the feedback error signal into the control signal calculation, this leads to
where
The Laplace transfer function of (1.8) is calculated as
Figure 1.3 Proportional plus derivative feedback control system (
This is what we called a proportional plus derivative (PD) controller.
The closed-loop feedback control configuration for a PD controller is shown in Figure 1.3. For the double integrator system (1.7), with the derivative term included in the controller, then the closed-loop transfer function becomes
(1.9)
(1.10)
The closed-loop poles are determined by the solutions of the characteristic polynomial equation as
which are
Clearly, we can choose the values of
It is worthwhile emphasizing that almost without exception, the derivative term is different from the original form
A commonly used derivative filter is a first order filter and has its time constant linked as a percentage to the actual derivative gain
where
With the derivative filter
(1.12)
Figure 1.4 PD controller structure in implementation.
where
(1.13)
Figure 1.4 shows the block diagram used for implementation of a PD controller with a filter.
If the derivative filter was not considered in the design, there is a certain degree of performance uncertainty due to the introduction of the filter. This may not be ideal for many applications. Designing a PD controller with the filter included will be discussed in Section 3.4.1.
1.2.3 Proportional Plus Integral Controller
A proportional plus integral (PI) controller is the most widely used controller among PID controllers. With the integral action, the steady-state error that had existed with the proportional controller alone (see Example 1.1) is completely eliminated. The output of the controller
(1.14)
where
