Block diagrams worksheet with answers

The signal R splits into two parallel blocks G₂ = 15 and G₃ = 6. Their outputs meet at a summing junction with the signs shown (+ and −), and the sum passes through G₁ = 4. Find the overall gain T = C/R.

An amplifier with gain G = 4 has positive feedback through a path with gain H = 0.1 (the feedback signal is added at the summing junction). Find the closed-loop gain T = C/R.

The plant G(s) = 9/(s + 6) has an inner (minor) negative feedback loop with gain H₁ = 1. A controller gain K = 5 and an outer unity negative feedback loop close the system. Find the closed-loop DC gain T(0).

A plant G(s) = 29/(s + 10) is controlled in a negative feedback loop whose sensor has the gain H = 2. Find the time constant τ of the closed loop.

The plant G(s) = K/(s(s + 2)(s + 7)) is in a unity negative feedback loop. Use the Routh–Hurwitz criterion to find the largest gain K_max for which the closed loop is stable.

A proportional controller K = 8 drives the plant G(s) = 8/(s + 5) 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 2 and a disturbance step of size 1 are applied together. Find the steady-state output c_ss.