Control Engineering Basics
Which of the following is the range of the damping ratio ζ for which the step response of the second-order system G(s) = ωn²/(s² + 2ζωn s + ωn²) overshoots the final value and approaches it while oscillating?
- Aζ > 1
- Bζ = 1
- C0 < ζ < 1
Show answer and explanation
Answer: C. 0 < ζ < 1
For 0 < ζ < 1 the poles are a complex-conjugate pair and the response approaches the final value while oscillating (underdamped). For ζ = 1 the poles are a repeated negative real root, and the response settles fastest among responses without overshoot (critically damped). For ζ > 1 the poles are two negative real numbers, and the response approaches slowly without overshoot (overdamped). For ζ = 0 the oscillation continues without decaying, and for ζ < 0 the system is unstable.
- A. This is overdamped: the response approaches the final value slowly, with neither overshoot nor oscillation.
- B. This is critically damped, the fastest case among responses without overshoot.
- C. Correct. This range is called underdamped.