Loop shaping

Loop shaping is the dominant design methodology in linear control theory, in which the engineer shapes the open-loop transfer function \(L(s)\) to achieve desired properties in the closed-loop system — specifically, the sensitivity function \(S(s)\) and the complementary sensitivity function \(T(s)\). The practice is part art and part constraint satisfaction: the designer pushes the loop gain high at frequencies where disturbance rejection and reference tracking matter, rolls it off where sensor noise dominates, and maintains adequate phase margin to guarantee stability. The PID controller is the simplest loop-shaping device; H-infinity control and quantitative feedback theory are its sophisticated descendants. Loop shaping is the craft of drawing the right frequency-domain curve and trusting the mathematics to enforce the consequences.