# Introduction to Controllers

Cuthbert Nyack

In the followng pages only single loop control systems similar
to the one below are considered. *R(s)* is the transfer function
of the input, *C(s)* is the transfer function of the output and
*H(s)* is the transfer function of the feedback element.
*G*_{2}(s) is the transfer function of the system being
controlled and *G*_{1}(s) is the
transfer function of the controller
for P, PD, PI and PID controllers. For Tachometer control the
controller is incorporated into *H(s)*.
The transfer function between input and output is given by :-
The Transfer function between the error and the input is given by:-
If a disturbance *N(s)* occurs in the system then the loop becomes:-
With *R(s)* = 0, the transfer function between *N(s)* and
*C(s)* is given by:-
In the above expressions *c(t)* can be found by substituting the
appropriate Transfer functions and Laplace Transforms into the
right hand side and then finding the inverse Laplace Transform.
The transfer function of the controller is also referred to as G_{c}(s) here
The standard Controllers are (P) Proportional G_{c}(s) = K,
(P + I or PI) Proportional plus Integral. (P + D or PD) Proportional + Derivative and (PID) Proportional + Integral + Derivative. The Transfer functions are shown below.
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COPYRIGHT © 1996 Cuthbert A. Nyack.