Two independent closed loop second order systems have the roots of their individual characteristic equations located at [math: −3±j4]-3\pm j4 and [math: −6±j8]-6\pm j8. The following can be said about these two systems.

Question

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The question describes two independent closed-loop second-order systems with poles located at -3 ± j4 for the first and -6 ± j8 for the second.

Option analysis:
- Option B: They both will have the same damping ratio ζ.
  For a second-order system with poles at -σ ± jω_d, the damping ratio is ζ = σ / ω_n, where ω_n = sqrt(σ^2 + ω_d^2).
  - For the first pair: σ = 3 and ω_d = 4, so ω_n = sqrt(3^2 + 4^2) = 5, and ζ = σ/ω_n = 3/5 = 0.6.
  - For the second pair: σ = 6 and ω_d = 8, so ω_n = sqrt(6^2 + 8^2) = 10, and ζ = σ/ω_n = 6/10 = 0.6.
  Since both yield ζ = 0.6, the damping ratios are indeed equal.
  This equality arises because the two pole pairs are scalar multiples of each other in the complex plane (their real and imaginary parts scale by the same factor), which preserves the ratio σ/ω_n = ζ.

Notes on the other options:
- If there were alternative statements suggesting different ζ values or relationships, they would need to be evaluated against the definition ζ = σ/ω_n using the corresponding σ and ω_n for each system. In this case, the computed damping ratios coincide, so any claim asserting a difference would be incorrect.
- Without additional options provided, only the evaluation of the given statement can be performed; the conclusion about equality follows directly from the standard second-order pole form and the provided pole locations.

MTRN3020- Modelling & Control of Mechatronic Systems - 2025 T1

Two independent closed loop second order systems have the roots of their individual characteristic equations located at [math: −3±j4]-3\pm j4 and [math: −6±j8]-6\pm j8. The following can be said about these two systems.

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