Control Theory #39 — Match Transfer Functions to Step Responses (Worked Example 16)

Instructor: Dr. Süleyman Burak ÇELİK

In this worked example, we match three transfer functions to three step-response plots — using nothing but the locations of poles and zeros on the s-plane. No simulation, no MATLAB, just the pole-zero map. Topics covered: - Pole at the origin → integrator → ramp response - Right-half-plane (non-minimum-phase) zero → undershoot - Complex-conjugate poles → oscillation - Slow left-half-plane zero (closer to origin than the poles) → overshoot, even with real poles - Final-value theorem and partial-fraction decomposition for the asymptotic shape Transfer functions: - G₁(s) = (s + 2) / [s · (s + 4)] - G₂(s) = (1 − s) / [(0.5 s² + 0.5 s + 1)(s + 1)] - G₃(s) = (s + 0.5) / [(s + 2)(s + 4)] Step-response shapes: - Plot (a): straight ramp, no settling - Plot (b): peak above the final value, then a smooth decay back down - Plot (c): dip below zero, climb past the final value, oscillating settle Key results: - G₁ → plot (a): factor of 1/s gives an integrator, asymptotic slope ½ - G₃ → plot (b): zero at −0.5 is closer to the origin than either pole; initial slope = +1 ≫ final value 1/16 → overshoot, no oscillation - G₂ → plot (c): zero at +1 is in the RHP (undershoot) and the quadratic factor has Δ = −7 (complex poles → oscillation), final value G₂(0) = 1 Big picture: every step-response feature has a fingerprint in the s-plane. Pole at origin → ramp. RHP zero → undershoot. Complex poles → oscillation. Slow LHP zero → overshoot. Every step shown clearly — no shortcuts. Playlist: Control Theory - AcEdumy GitHub: https://github.com/acedumy