Poles & zeros · controls

A linear system's poles tell you everything about how it responds to a kick. Where each pole sits in the complex plane is the system's personality: how fast it forgets the past (distance from the imaginary axis), and whether it likes to ring (height above the real axis). Drag the two × marks around. The right pane is the corresponding step response, recomputed every frame.

Drag the poles

status —

ζ (damping) —

ω_n —

dominant τ —

poles —

what to try

Drag a pole horizontally toward the imaginary axis. Watch the decay get slower and the envelope swell.
Drag a pole across the imaginary axis (into the red zone). The trace explodes — the system is now unstable.
Bring the two poles toward the real axis. The conjugate pair collides — you've reached critical damping, the fastest non-oscillating response. Continue past it: two real poles, slower than critical.
Slide a pole up the imaginary axis (purely imaginary). The response becomes a pure sinusoid — undamped natural oscillation at ω_n.

show the math

With two conjugate poles at $s = - z e t a o m e g a_{n} p mj o m e g a_{n} s q r t 1 - z e t a^{2}$ , the transfer function is

G(s) = rac{omega_n^2}{s^2 + 2zetaomega_n s + omega_n^2}

Distance from the imaginary axis equals $z e t a o m e g a_{n}$ — the decay rate. Height above the real axis equals $o m e g a_{n} s q r t 1 - z e t a^{2}$ — the damped natural frequency at which the system rings.

Once $z e t a g e q 1$ , the radical goes imaginary; the two poles become real and the response stops ringing entirely. Once any pole crosses into the right half-plane $(s i g ma > 0)$ , the response grows without bound: instability.