Transfer Functions, Poles, and Zeros

A transfer function describes how a linear system turns inputs into outputs. Its poles, the roots of the bottom polynomial, decide whether it is stable, how fast it responds, and whether it rings. Its zeros, the roots of the top, shape the response further.

Include a zero Zero position

Drag the red pole in the s-plane. Its mirror image follows.

The system

$$G(s) = k\,\frac{s - z}{(s - p)(s - \bar p)}, \qquad p = \sigma + j\omega_d$$

The gain $k$ is chosen so that the output settles at 1 for a step input whenever the system is stable.

What to notice

Move the pole left and the response settles faster, since $\sigma$ sets the decay rate $e^{\sigma t}$. Move it up and the response rings faster, since $\omega_d$ is the oscillation frequency. Drag it across into the right half and the response grows without bound: unstable. A zero in the left half speeds up the start of the response and can add overshoot. A zero in the right half, a non-minimum-phase zero, makes the response first go the wrong way, which happens in real systems like a bicycle's steering.