Block diagrams worksheet with answers

The signal R splits into two parallel blocks G₂ = 5 and G₃ = 2. Their outputs meet at a summing junction with the signs shown (+ and −), and the sum passes through G₁ = 3. Find the overall gain T = C/R.

In the block diagram the forward path has the gains G₁ = 19 and G₂ = 10 in series, and the negative feedback path has the gain H = 0.2. Find the overall gain T = C/R.

A unity negative feedback loop has G(s) = 58/(s(s + 4)). The reference is a ramp r(t) = 2·t. Find the steady-state error e_ss.

The plant G(s) = 9/(s + 1) has an inner (minor) negative feedback loop with gain H₁ = 1. A controller gain K = 19 and an outer unity negative feedback loop close the system. Find the closed-loop DC gain T(0).

A proportional controller K drives the plant G(s) = 5/(s + 5) in a unity negative feedback loop. A disturbance step of size 0.5 enters at the plant input while R = 0. Find the smallest gain K that keeps the steady-state output at or below 0.005.

A proportional controller K = 3 drives the plant G(s) = 6/(s + 8) in a unity negative feedback loop. A disturbance D enters at the plant input (it is added to the controller output). A reference step of size 5 and a disturbance step of size 10 are applied together. Find the steady-state output c_ss.