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. G2(s) is the transfer function of the system being controlled and G1(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 Gc(s) here
The standard Controllers are (P) Proportional Gc(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.