The Laplace transform of the controller output is
(1.15)
with
(1.16)
Figure 1.5 shows a block diagram of the PI control system.
The example below is used to illustrate closed-loop control with a PI controller. For comparison purpose, we use the same plant as that used in Example 1.1.
Figure 1.5 PI control system.
Example 1.2
Assume that the plant is a first order system with the transfer function:
(1.17)
the PI controller has the proportional gain , and the integral time constant and 0.5 respectively. Examine the locations of the closed-loop poles. With the reference signal as a unit step signal, find the steady-state value of the closed-loop output .
Solution. We calculate the closed-loop transfer function between the reference and output signals:
(1.18)
With given in (1.16) and in (1.17), we have
(1.19)
The closed-loop poles of this system are determined by the solutions of the closed-loop characteristic equation,
(1.20)
which are
(1.21)
If the quantity
then there are two identical real poles located at
If the quantity
then there are two real poles located at
If the quantity
then there are two complex poles located at
The closed-loop system is stable as long as is positive and .
Applying the final value theorem, we calculate
(1.22)
where the steady-state value is equal to the reference signal, and it is independent of the value of integral time constant . Figure 1.6 shows for the same as in Example 1.1 the closed-loop step response with and , respectively. It is seen that as reduces, the closed-loop response speed becomes faster. Nevertheless, the steady-state responses with both values are equal to one.
It is often the case that the output of a PI control system exhibits overshoot to a step reference signal. The percentage of overshoot increases as higher control performance demanded. This may cause a conflict in the PI control performance specifications: on the one hand a fast control system response is desired, and yet on the other hand, the overshoot is not desirable when step reference changes are performed. The overshooting problem in reference change could be reduced by a small change in the configuration of the PI controller. This small change is to put the proportional control on the output signal
(1.23)
Figure 1.6 Closed-loop step response of a PI control system (Example 1.2).
