The classical three-term PID controller

We have seen in section 7.1 that proportional feedback control can reduce error responses but that it still allows a non-zero steady-state error for a proportional system. In addition, proportional feedback increases the speed of response but has a much larger transient overshoot. When the controller includes a term proportional to the integral of the error, then the steady-state error can be eliminated, as shown in section 7.2. But this comes at the expense of further deterioration in the dynamic response. Addition of a term proportional to the derivative of the error can damp the dynamic response. Combined, these three kinds of actions form the classical PID controller, which is widely used in industry.

This principle mode of action of the PID controller can be explained by the parallel connection of the P, I and D elements shown in Figure 8.1. From this diagram the transfer function of the PID controller is

Figure 8.1: Block diagram of the PID controller

The controller variables are

Eq. (8.1) can be rearranged to give

These three variables , and are usually tuned within given ranges. Therefore, they are often called the tuning parameters of the controller. By proper choice of these tuning parameters a controller can be adapted for a specific plant to obtain a good behaviour of the controlled system.

If follows from Eq. (8.2) that the time response of the controller output is

Using this relationship for a step input of , i.e. , the step response of the PID controller can be easily determined. The result is shown in Figure 8.2a. One has to observe that the length of the arrow of the D action is only a measure of the weight of the impulse.

Figure 8.2: Step responses (a) of the ideal and (b) of the real PID controller

In the previous considerations it has been assumed that a D behaviour can be realised by the PID controller. This is an ideal assumption and in reality the ideal D element cannot be realised (see section 3.3). In real PID controllers a lag is included in the D behaviour. Instead of a D element in the block diagram of Figure 8.1 a element with the transfer function

is introduced. From this the transfer function of the real PID controller or more precisely of the controller follows as

Introducing the controller tuning parameters

it follows

The step response of the controller is shown in Figure 8.2b. This response from gives a large rise, which declines fast to a value close to the P action, and then migrates into the slower I action. The P, I and D behaviour can be tuned independently. In commercial controllers the 'D step' at can often be tuned 5 to 25 times larger than the 'P step'. A strongly weighted D action may cause the actuator to reach its maximum value, i.e. it reaches its 'limits'.

As special cases of PID controllers one obtains for:

a)

the PI controller with transfer function

b)

the ideal PD controller with the transfer function

and the controller with the transfer function

c)

and the P controller with the transfer function

The step responses of these types of controllers are compiled in Figure 8.3. A pure I controller may also be applied and this has the transfer function

Figure 8.3: Step responses of the PID controller family