# P Control 1st, 2nd, 3rd and 4th order

Cuthbert Nyack
P Controllers Gc(s) = K are the simplest continuous linear controllers. The applets below for step, ramp, step disturbance, sine and triangle response illustrate some of the properties of P controllers. There is an offset 1/(1 + K) at all orders and oscillatory behaviour which increases with K. The oscillatory behaviour and tendency to instability increases with order because of the increasing phase lag.

The term "proportional band" (1/K) is often used to describe P controllers. When expressed as a % it represents the % change in the error which gives an output which is linearly proportional to the error.
Open and Closed loop transfer functions for 1st order is
G(s) = K/(T1s + 1), C(s)/R(s) = K/(T1s + (1 + K))
For 2nd order is
G(s) = K/((T1s + 1)(T2s + 1)), C(s)/R(s) = K/((T1s + 1)(T2s + 1) + K))
For 3rd order is
G(s) = K/((T1s + 1)(T2s + 1)(T3s + 1)), C(s)/R(s) = K/((T1s + 1)(T2s + 1)(T3s + 1) + K))
For 4th order is
G(s) = K/((T1s + 1)(T2s + 1)(T3s + 1)(T4s + 1)), C(s)/R(s) = K/((T1s + 1)(T2s + 1)(T3s + 1)(T4s + 1) + K))
For the parameter values (K, 1.0, 1.0, 1.0, 1.0, 2.0), the 4th order system becomes unstable for K > 4 and the third order becomes unstable for K > 8.0. For the ramp response, since all systems are type 0, then the offset increases linearly with t.

Gif images below show how 2 of the applets should appear.  