An integral controller is used for speed control with a setpoint of12 rpm within a range of 10 to 15 rpm. The controller output is 22% initially. The Integral gain $K_r$ = -0.15% controller output per second per percentage error. If the speed jumps to 13.5 rpm, calculate the controller output after 2s for a constant $e_p$.

Set Point = 12 rpm

Range = 10 to 25 rpm

The controller O/P is 22% initially,

$K_r = -0.15%$

If the speed jump to 13.5 rpm, O/P controller for constant $e_p$,

$$e_p = \frac{setpoint\ - b}{b_{max} - b_{min}} * 100$$

$$e_p = \frac{12 - 13.5}{15 - 10} * 100$$

$$e_p = -30 \%$$

The integral gain $K_1$ = 0.15%

The rate of controller O/P changes is given by,

Therefore, Controller O/P

$$\frac{dP}{dt} = K_I * e_p$$

$$\frac{dP}{dt} = (-30%) * (-0.15)$$

$$\frac{dP}{dt} =4.5\% / Sec$$

$$And,\ P = 31\%. $$