Poles and Zeros
Poles set a system's natural modes and stability; zeros shape how those modes appear in the output, sometimes causing undershoot.
Roots That Define Behavior
For a transfer function G(s) = N(s)/D(s), the poles are the roots of D(s) and the zeros are the roots of N(s). Poles are where G(s) becomes infinite and zeros are where it becomes zero. Together they determine, up to a gain, everything about the system's dynamics.
What poles do
- A real pole at s = -a gives a decaying exponential e^(-a*t); larger a means faster decay.
- A complex-conjugate pole pair gives a damped sinusoid; its imaginary part sets frequency, its real part sets decay.
- A pole in the right half-plane grows without bound: the system is unstable.
- A pole at the origin acts as a pure integrator.
What zeros do
Zeros do not create new modes, but they weight how strongly each pole's mode appears in a particular output. A zero near a pole can nearly cancel that mode's contribution. Zeros also add phase: a left-half-plane zero adds phase lead, useful for stability.
Non-minimum-phase zeros
A zero in the right half-plane makes the system non-minimum-phase. Such systems often show initial undershoot: the output first moves the wrong way before recovering. They are fundamentally harder to control because achievable bandwidth is limited; you cannot make the loop arbitrarily fast without instability.
Dominant poles
When one pole (or pair) sits much closer to the imaginary axis than the rest, it dominates the response and the system behaves approximately as first or second order. This dominant-pole approximation underlies many quick design estimates for rise time and overshoot.
Reading a pole-zero map is a fast diagnostic: a glance tells you whether a system is stable, how oscillatory it is, and roughly how fast it settles, before any simulation is run.