Transfer functions, poles and zeros
Poles describe the system's natural modes; zeros reshape how inputs excite them.
First, picture it
Think of a bell: its ring-down is a natural mode. A pole tells you how quickly that mode dies and whether it oscillates. A zero can cancel or redirect part of the input-to-output response, but it does not necessarily remove an internal physical mode.
What the model says
For a rational transfer function G(s)=N(s)/D(s), zeros are roots of N and poles are roots of D after considering cancellations. Stable continuous-time poles lie in the open left half-plane for an internally stable minimal model. In feedback, closed-loop poles come from 1+L(s)=0, where L=C P for a simple unity-feedback loop.
T(s)=L(s)/(1+L(s)); 1+L(s)=0The denominator sets closed-loop modes, not the open-loop denominator by itself.
A pole's real part sets growth or decay; its imaginary part sets oscillation frequency. The boundary itself needs separate analysis.
Make it concrete
For L=K/(s+1), the closed-loop pole is −(1+K). Increasing positive K moves this one pole left, but more complex plants need not behave so simply.
A pole-zero cancellation in an input-output formula can hide an unstable internal mode; do not treat cancellation as proof of physical safety.
Check your understanding+
Which equation determines the closed-loop poles of unity negative feedback?
The characteristic equation 1+L(s)=0.