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

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To begin, examine the given complex conjugate poles of each second-order system: -3 ± j4 for the first, and -6 ± j8 for the second. For a second-order system, the damping ratio ζ is defined as the ratio of the real part to the natural frequency, where ω_n = sqrt(σ^2 + ω^2) and the pole is at -σ ± jω.

First pair: σ = 3 and ω = 4. Compute ω_n = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5. Then ζ = σ / ω_n = 3 / 5 = 0.6.

Second pair: σ = 6 and ω = 8. Compute ω_n = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10. Then ζ = σ / ω_n = 6 / 10 = 0.6.

Observing both results, the damping ratios are equal (0.6 for each system). Hence, despite the different absolute pole locations, the ratio defining damping is identical between the two systems.

A useful intuition: the second pair appears to be a scaled version of the first (both σ and ω are doubled), and since the damping ratio is a scale-invariant ratio of the real part to the magnitude of the pole, it stays the same.

If there were other statements to evaluate, we would check them similarly by computing σ, ω, and ω_n for each option and comparing the resulting ζ values. However, based on the provided information, the key takeaway is that the two systems share the same damping ratio because the ratio 3/5 equals 6/10, yielding ζ = 0.6 in both cases.

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 −3±j4[math]-3\pm j4 and −6±j8[math]-6\pm j8. The following can be said about these two systems.

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