Learning objective: Connect a closed-loop characteristic equation to performance measures and steady-state tracking error.
Understand the idea
A standard second-order closed-loop denominator is s² + 2ζωns + ωn². Comparing a given characteristic equation with this standard form is a reliable way to identify damping and speed of response. The approximations below apply to the ideal second-order system with an underdamped response.
Worked example: given data
- Unity-feedback system with open-loop G(s) = 25/[s(s + 4)].
- Find its closed-loop poles, damping ratio, natural frequency, approximate percent overshoot and 2% settling time.
- Also determine steady-state error for a unit ramp input.
Detailed solution
Form the closed-loop transfer function
For negative unity feedback, T(s) = G(s)/(1 + G(s)) = 25/(s² + 4s + 25). The characteristic equation is s² + 4s + 25 = 0.
Identify ωn and damping ratio
Comparing with s² + 2ζωns + ωn² gives ωn = 5 rad/s and 2ζωn = 4. Therefore ζ = 0.4.
Find the poles
The roots are s = (−4 ± √(16 − 100))/2 = −2 ± j√21. Their negative real parts correspond to a stable, oscillatory second-order response.
Estimate overshoot
For zero initial conditions and a unit step, Mp = e−πζ/√(1−ζ²) × 100%. With ζ = 0.4 this gives approximately 25.4% peak overshoot.
Estimate settling time and ramp error
The familiar 2% settling-time approximation is Ts ≈ 4/(ζωn) = 2 s. The velocity constant is Kv = lims→0sG(s) = 25/4 = 6.25 s⁻¹, so the unit-ramp steady-state error is 1/Kv = 0.16.
What the answer means
The settling-time formula is an approximation, not an exact crossing time, and it is most useful for predominantly second-order systems. Step and ramp tracking are different questions: a type-1 unity-feedback system has zero steady-state step error but a finite unit-ramp error.
Three mistakes to avoid
- Reading ζ = 4/25 instead of comparing the coefficients of s.
- Confusing percent overshoot with a decimal ratio.
- Using the step-error theorem for a ramp input.
Try a related question
For T(s) = 16/(s² + 8s + 16), identify ζ and ωn and predict whether a step response overshoots.
Show the worked-answer check
ωn = 4 rad/s and 2ζωn = 8, so ζ = 1. It is critically damped and does not overshoot for a standard unit-step response.
How to revise this concept
Practise Routh–Hurwitz stability, root-locus sketches and gain/phase margins. First attempt the example on paper without opening the answer, then compare the intermediate steps. Explain each assumption before memorising a formula.
Content note: This is a newly prepared practice example, not an official GATE previous-year question. It was drafted with AI assistance for this educational website. Students should verify calculations and exam requirements against standard textbooks and official sources; educator review is recommended before publication.