P, PI Control 1PT0 Step Response

Cuthbert Nyack

The applet below shows the step response of P and PI controllers and the output from the P and I contributions to the controller output.
For P control, the output from the P controller becomes equal to the output for large t.
For PI control at time zero the output from the integral controller is zero while that from the P controller is K. As t increases the output from the P controller reduces to zero while that from the I controller becomes equal to the output. For the parameters (5.0, 1.0, 3.0, 1.0) the outputs from the P and I controllers become equal at t ~ 1.14s.
It takes approximately 5 integral times(Ti) for the output to reach its final value(assuming Ti > T1). The rate of increase of the integral controller output is proportional to the error.
Open and Closed loop Transfer function for P control is
G(s) = K/(T1s + 1), C(s)/R(s) = K/(T1s + (1 + K))
And for PI control is
G(s) = K(Tis + 1)/(Tis(T1s + 1)), C(s)/R(s) = K(Tis + 1)/(T1Tis² + (1 + K)Tis + K)

The P controller has an offset of 1/(K + 1) while with the PI controller, the offset goes to zero. The effect is illustrated below. It also shows that if Ti is made less than T1 then the output will overshoot.

Gif image below show how 1 of the applets should appear.

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