Question

Find the closed-loop transfer function, T(s)-C(s)/R(s) for the following systems using block diagram reduction R(s)+ G1 G2 G8 C(s) G2 G4 G7 G3 G1 G2 G3 G4. C(s) R(s)+ G5 G6 G7

Answer 1

Find T(9) = C(s)/R(s) using block diagram Reduction shifting the takeoff point placed before block G_1 to a position after block G_1 eliminate the summing point C(s)/R(s) = (G_1/1 + H_1 G_1) middot (G_2 G_1 + 1/G_1) middot G_3 = (1 + G_1 G_2) G_3/1 + H_1 G_1 C(s)/R(s) = T(s) = G_3 (1 + G_1 G_2)/1 + H_1 G_1 \r\n eliminate summing point placed before block G_4, G_7 \r\n split takeoff point after block G_5 shifting takeoff point (1) after block G_5 eliminate summing point (2) and cascade. \r\n split summing point eliminate summing point (2) over all Transfer function T(s) = C(s)/R(s) T(s) = G_1 G_5/1 + G_1 G_2 + G_1 G_3 G_5 (G_4 + G_6 G_7)/1 + G_1 G_5 G_8/1 + G_1 G_2 + G_3 G_1 G_5 (G_4 + G_6 G_7) \r\n T(s) = G_1 G_5/(1 + G_1 G_2 + G_1 G_3 G_5 (G_4 + G_6 G_7) + G_1 G_5 G_8)