Control Theory #40 — Dominant Pole + Over/Undershoot from the s-Plane (Worked Example 17)

Instructor: Dr. Süleyman Burak ÇELİK

Four transfer functions, three questions. For each, we read the dominant pole straight from the s-plane and decide whether the step response will undershoot or overshoot — without running a single time-domain calculation. Topics covered: - Rule 1: smallest |Re| pole dominates (closest to the imaginary axis) - Rule 2: any RHP pole automatically dominates (unstable system) - Rule 3: complex-conjugate pair vs real pole — compare real parts, not magnitudes - Rule 4: pole-zero near cancellation suppresses the slow pole, dominance shifts - Rule 5: RHP zero ⇒ undershoot; slow LHP zero (closer to jω than the dominant pole) ⇒ overshoot Transfer functions (HW5 P9): - G₁(s) = (s + 3.5)(s + 4) / [(s + 3)(s + 10)] - G₂(s) = (s + 8) / [(s − 3)(s + 200)] - G₃(s) = (s − 4) / [(s + 4)(s² + 3s + 3)] - G₄(s) = (s + 2) / [(s + 10)(s + 20)] Key results: - G₁: zero at −3.5 nearly cancels pole at −3 (only 0.5 apart), so −10 takes over → dominant pole = −10 - G₂: factor (s − 3) gives an RHP pole at +3 → unstable; +3 dominates exponentially over the −200 mode - G₃: discriminant of s² + 3s + 3 is −3 → complex pair at −1.5 ± 0.87j; |Re| = 1.5 ‹ 4, so the pair dominates the real pole at −4. Plus RHP zero at +4 ⇒ undershoot - G₄: dominant pole = −10. Zero at −2 is 5× closer to jω than the dominant pole ⇒ overshoot Verification: the 4-panel step-response plot confirms each prediction — G₁ settles fast and monotonic, G₂ blows up, G₃ dips below zero before oscillating around its final value, G₄ shoots above its final value before decaying. (b) Undershoot ⇒ G₃ (RHP zero forces dip below zero) (c) Overshoot ⇒ G₄ (slow LHP zero amplifies the transient peak) Big picture: the s-plane geometry tells you everything. Five rules, infinite systems, no simulator required. Every step shown clearly — no shortcuts. Topics: 00:00 Cover 00:03 Problem — four transfer functions, three questions 01:14 Strategy — five rules in the s-plane 02:43 G₁ — near pole-zero cancellation, dominance shifts to −10 03:48 G₂ — RHP pole at +3, system unstable 04:53 G₃ — complex pair dominates, RHP zero ⇒ undershoot 06:25 G₄ — slow LHP zero ⇒ overshoot 07:40 Verification — four step responses 09:36 Summary Playlist: Control Theory - AcEdumy GitHub: https://github.com/acedumy