Proportional (P) Controller

 




In proportional (P) mode, linear relationship between the controller
output and the error exist in proportional band (PB).
y =Kpe +yo Kp =100
PB
Kp = proportional gain between error and controller output
yo = manipulated variable at operating point
If error is zero, output is a constant equal to yo .
In case of error, for every 1% of error, a correction of Kp% is
added or subtracted from yo , depending on error sign.
There is a band of error about zero of magnitude PB within which
output is not saturated at 0% or 100%.
When a load changes, a permanent residual error occurs.



Integral (I) Controller 

Integral control action is used to fully correct system deviations at
any operating point. As long as the error is nonzero, the integral
action will cause the value of the manipulated variable to change.
In I control mode, the value of the manipulated variable is
changed proportional to the integral of the error e
y =KI
Z
e dt +y(0) with :KI =1
Tn
The higher the integral action coefficient KI , the greater the
integral action of an I controller.
No steady state error
Sluggish response at low KI
At high KI , the control loop tends to oscillate/may become instable




 Derivative (D) Controller 

-D controllers generate the manipulated variable from the rate of
change of the error and not - as P controllers - from their
amplitudes.
-These react much faster than P controllers: even if the error is
small, derivative controllers generate - by anticipation - large
control amplitudes as soon as a change in amplitude occurs.
-A steady-state error signal, however, is not recognized by D
controllers, because regardless of how big the error, its rate of
change is zero.
-Therefore, derivative-only controllers are rarely used in practice.
They are usually found in combination with other control
elements, mostly in combination with proportional control.

 PI controllers are often used in practice. If properly designed, they
combine the advantages of both controller types (stability and
rapidity; no steady-state error), so that their disadvantages are
compensated for at the same time.
The manipulated variable of PI controllers is calculated as follows

 



 PD Controller

In PD controllers with proportional-derivative control action, the
manipulated variable results from the addition of the individual P
and D control elements:





The control response for steady-state error in PD controllers is
just as it occurs in P controllers. Due to the immediate control
action whenever there is a change in the error signal, the control
dynamics is faster than with P controllers.


PI Controller
PI controllers are often used in practice. If properly designed, they
combine the advantages of both controller types (stability and
rapidity; no steady-state error), so that their disadvantages are
compensated for at the same time.
The manipulated variable of PI controllers is calculated as follows:
y =Kp e +KI
Z
edt with :KI =Kp
Tn



PID Controller


If a D component is added to PI controllers, the result is an
extremely versatile PID controller. If properly tuned, this
controller causes the controlled variable to reach its set point more
quickly, thus reaching steady-state more rapidly.
The manipulated variable of PID controllers is calculated as follow :