Closed-Loop Transfer Functions Explained: Sensitivity, Disturbances & Noise

In this video, we break down the structure of a standard feedback control loop and derive all key closed-loop transfer functions from scratch. We start with the classic block diagram — reference input r, controller G_R(s), plant G_S(s), process disturbance d, and measurement noise n — and systematically derive how each signal affects the output y, control error e, control input u, and intermediate signal v. Key topics covered: The complementary sensitivity function T(s) = G_R·G_S / (1 + G_R·G_S) The sensitivity function S(s) = 1 / (1 + G_R·G_S) Process disturbance sensitivity and noise sensitivity functions Why S(s) + T(s) = 1 and what it means for control design Whether you're studying automatic control, signal processing, or just want to deeply understand how feedback loops behave, this video gives you the mathematical foundation you need. 📌 Topics: Control Theory | Feedback Control | Transfer Functions | Sensitivity Function | Laplace Domain