Block diagrams worksheet with answers

Free printable Block diagrams worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
R(s)+16G₁4G₂C(s)1H−

In the block diagram the forward path has the gains G₁ = 16 and G₂ = 4 in series, and the negative feedback path has the gain H = 1. Find the overall gain T = C/R.

G₁ = 16G₂ = 4H = 1
T =
2.Easy
R(s)+20s + 7G(s)C(s)−

A plant G(s) = 20/(s + 7) is placed in a unity negative feedback loop. Find the time constant τ of the closed loop.

G(s) = 20/(s + 7)unity negative feedback
τ = s
3.Medium
R(s)+1(s + 4)(s + 3)(s + 9)G(s)C(s)−

A unity negative feedback loop has G(s) = 1(s + 4)/((s + 3)(s + 9)). A step of size 10 is applied at the reference. Find the steady-state error ess.

G(s) = 1(s + 4)/((s + 3)(s + 9))R = 10
ess =
4.Medium
R(s)+12s + 1G(s)C(s)0.1H−

A plant G(s) = 12/(s + 1) is controlled in a negative feedback loop whose sensor has the gain H = 0.1. A step of size 10 is applied at R. Find the steady-state output css.

G(s) = 12/(s + 1)H = 0.1R = 10
css =
5.Hard
R(s)+64s(s + 10)G(s)C(s)2H−

The plant G(s) = 64/(s(s + 10)) is in a negative feedback loop with sensor gain H = 2. The closed-loop poles are complex. Find their real part σ.

G(s) = 64/(s(s + 10))H = 2
σ =
6.Hard
R(s)+Kcontroller+D(s)+5s + 9plantC(s)−

A proportional controller K drives the plant G(s) = 5/(s + 9) 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 0.02778.

G(s) = 5/(s + 9)D = 0.5css ≤ 0.02778
K =

Answer key

  1. 1. T = 0.9846
  2. 2. τ = 0.03704 s
  3. 3. ess = 8.71
  4. 4. css = 54.55
  5. 5. σ = −5
  6. 6. K = 16.2

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