Block diagrams worksheet with answers

The signal R splits into two parallel blocks G₂ = 10 and G₃ = 9. 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.

The signal R splits into two parallel blocks G₂ = 15 and G₃ = 4. Their outputs meet at a summing junction with the signs shown (+ and −), and the sum passes through G₁ = 2. Find the overall gain T = C/R.

The plant G(s) = 7/(s + 1) has an inner (minor) negative feedback loop with gain H₁ = 5. A controller gain K = 18 and an outer unity negative feedback loop close the system. Find the closed-loop DC gain T(0).

A unity negative feedback loop has G(s) = 12(s + 8)/((s + 4)(s + 6)). A step of size 2 is applied at the reference. Find the steady-state error e_ss.

Reduce the block diagram with G₁ = 3/(s + 5), G₂ = 5/(s + 2), an inner negative feedback H₁ = 0.5 around G₂ and an outer negative feedback H₂ = 0.25. Find the DC gain T(0) of C(s)/R(s).

A proportional controller K = 16 drives the plant G(s) = 7/(s + 4) in a unity negative feedback loop. A disturbance D enters at the plant input (it is added to the controller output). With R = 0, a disturbance step of size 0.5 is applied. Find the steady-state output c_ss.