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)+4s + 7G(s)C(s)−

A plant G(s) = 4/(s + 7) is placed in a unity negative feedback loop. A step of size 5 is applied at R. Find the steady-state output css.

G(s) = 4/(s + 7)unity negative feedbackR = 5
css =
2.Easy
R(s)+10G₁10G₂C(s)0.25H−

In the block diagram the forward path has the gains G₁ = 10 and G₂ = 10 in series, and the negative feedback path has the gain H = 0.25. Find the overall gain T = C/R.

G₁ = 10G₂ = 10H = 0.25
T =
3.Medium
R(s)+2(s + 1)(s + 5)(s + 8)G(s)C(s)−

A unity negative feedback loop has G(s) = 2(s + 1)/((s + 5)(s + 8)). A step of size 1 is applied at the reference. Find the steady-state error ess.

G(s) = 2(s + 1)/((s + 5)(s + 8))R = 1
ess =
4.Medium
R(s)+16K+9s + 5G(s)C(s)1H₁−−

The plant G(s) = 9/(s + 5) has an inner (minor) negative feedback loop with gain H₁ = 1. A controller gain K = 16 and an outer unity negative feedback loop close the system. Find the closed-loop DC gain T(0).

G(s) = 9/(s + 5)H₁ = 1K = 16
T(0) =
5.Hard
R(s)+13controller+D(s)+4s + 3plantC(s)−

A proportional controller K = 13 drives the plant G(s) = 4/(s + 3) 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 5 is applied. Find the steady-state output css.

K = 13G(s) = 4/(s + 3)D = 5R = 0
css =
6.Hard
R(s)+14s(s + 4)G(s)C(s)2H−

The plant G(s) = 14/(s(s + 4)) is in a negative feedback loop with sensor gain H = 2. The closed-loop poles are complex. Find their real part σ.

G(s) = 14/(s(s + 4))H = 2
σ =

Answer key

  1. 1. css = 1.818
  2. 2. T = 3.846
  3. 3. ess = 0.9524
  4. 4. T(0) = 0.9114
  5. 5. css = 0.3636
  6. 6. σ = −2

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