In this system, the setpoint $x_{sp}$ is tracked using a feedback loop, but the system compensates for disturbances $x_1$ using a feedforward loop. To accomplish this, the output $P_{FB}$ from the feedback controller $G_c$ is added to the output $P_{FF}$ from the feedforward controller $G_f.$

Please derive an equation relating the system output, $x,$ to the setpoint and the disturbance. Your result should have the form

$x=G_{FB}{x}_{sp}+G_{FF}{x}_1$

Answer:

$x=[\frac{K_m{G}_C{G}_v{G}_P}{1+G_C{G}_v{G}_P{G}_m}]x_{SP}+[\frac{G_d+G_t{G}_f{G}_v{G}_P}{1+G_C{G}_v{G}_P{G}_m}]x_1$