Next: Optimal tuning of PID
Up: PID control and associated
Previous: PID control and associated
Contents
The classical threeterm PID controller
We have seen in
section 7.1 that
proportional feedback control can reduce error responses but that
it still allows a nonzero steadystate 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 steadystate 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

(8.1) 
Figure 8.1:
Block diagram of the PID controller

The controller variables are

gain 


integral action time 


derivative action time 

Eq. (8.1) can be rearranged to give

(8.2) 
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

(8.3) 
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

(8.4) 
is introduced. From this the transfer function of the real
PID controller or more precisely of the
controller
follows as

(8.5) 
Introducing the controller tuning parameters
and 

it follows

(8.6) 
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

(8.7) 
 b)

the ideal PD controller
with the
transfer
function

(8.8) 
and the
controller with the
transfer
function

(8.9) 
 c)

and
the P controller
with the transfer function

(8.10) 
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

(8.11) 
Figure 8.3:
Step responses of the PID controller family

Next: Optimal tuning of PID
Up: PID control and associated
Previous: PID control and associated
Contents
Christian Schmid 20050509