Mr. Integral
Math notes/Differential Equations/Laplace Transforms

Transfer Functions, Poles and Response — Cheat sheet

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Core rule

G(s)=Y(s)/U(s)with zero initial conditions.G(s)=Y(s)/U(s)\quad\text{with zero initial conditions}.

Watch for

Frequency response uses G(iω)G(i\omega) after stable transients have decayed. A pole-zero cancellation can hide an unstable internal mode, so transfer behavior alone should not be treated as a complete state-stability proof.