Block diagrams worksheet with answers

The signal R splits into two parallel blocks G₂ = 5 and G₃ = 12. Their outputs meet at a summing junction with the signs shown (+ and −), and the sum passes through G₁ = 6. Find the overall gain T = C/R.

A plant G(s) = 7/(s + 2) is placed in a unity negative feedback loop. A step of size 1 is applied at R. Find the steady-state output c_ss.

A plant G(s) = 29/(s + 3) is controlled in a negative feedback loop whose sensor has the gain H = 0.5. Find the time constant τ of the closed loop.

A unity negative feedback loop has G(s) = 46/(s(s + 5)). The reference is a ramp r(t) = 2·t. Find the steady-state error e_ss.

A proportional controller K drives the plant G(s) = 4/(s + 5) in a unity negative feedback loop. A disturbance step of size 2 enters at the plant input while R = 0. Find the smallest gain K that keeps the steady-state output at or below 0.016.

The plant G(s) = 78/(s(s + 12)) is in a negative feedback loop with sensor gain H = 2. The closed-loop poles are complex. Find their imaginary part ω_d (the positive value), in rad/s.