Control Theory #41 — Reading Time-Domain Specs from a Step Response (Worked Example 18)

Instructor: Dr. Süleyman Burak ÇELİK

A measurement of a step response — and four time-domain performance specifications to read straight off the graph. Rise time, peak time, percent overshoot, settling time. Then we back out the second-order parameters ζ and ωₙ in two short calculations. Topics covered: - y∞ (final value): horizontal asymptote on the right side of the curve - y_max (peak value): height of the first peak - M_p = (y_max − y∞) / y∞ × 100% — percent overshoot - t_r (rise time): first time the output reaches y∞ (or 95% of it, in the textbook convention) - t_p (peak time): time of the first maximum - t_s (settling time): first time the output enters and stays inside a tolerance band — for both 5% and 2% bands - Closed-form back-out: ζ = −ln(M_p) / √(π² + ln²(M_p)), ω_n = π / (t_p √(1−ζ²)) Problem (HW5 P11): given a measured step response with y∞ = 100 and a peak at y_max = 180 reached at t = 0.5 s. Read-off values: - t_r = 0.3 s (output first reaches y∞ = 100) - t_p = 0.5 s (first peak at y_max = 180) - M_p = 80% (output nearly doubles before settling — system is lightly damped) - t_s(5%) = 1.7 s (output enters [95, 105] and stays) - t_s(2%) = 2.3 s (output enters the tighter [98, 102] window) Reverse-engineering (assume 2nd-order): - ζ ≈ 0.071 (very low damping — matches the 80% overshoot) - ω_n ≈ 6.28 rad/s (the natural frequency of the underlying 2nd-order model) Big picture: time-domain specs are not abstract — they live on the graph, in places you can point to with a pencil. M_p and t_p give you ζ and ω_n in two lines. Reading a step response is the bridge from measurement to model — and that bridge is the heart of every control design problem. Every step shown clearly — no shortcuts. Topics: 00:00 Cover 00:03 Problem — read 4 specs from a measurement 01:10 Definitions — what each spec means on the curve 02:37 Final value, peak, and percent overshoot 03:38 Rise time and peak time 04:56 Settling time — 5% and 2% bands 06:18 Reverse-engineering ζ and ω_n 08:34 Summary Playlist: Control Theory - AcEdumy GitHub: https://github.com/acedumy