Polezero

Polezero refers to the poles and zeros of a system’s transfer function, a concept used in control theory and signal processing to characterize input-output behavior. A transfer function H(s) for continuous-time systems or H(z) for discrete-time systems is a rational function of the complex frequency variable. Poles are the roots of the denominator; zeros are the roots of the numerator. The location of poles and zeros in the complex plane determines stability, transient response, and frequency response.

In continuous-time systems, poles with negative real parts produce decaying responses, while zeros shape the response by attenuating or canceling certain frequencies. In discrete-time systems, poles inside the unit circle imply stability, and zeros similarly influence magnitude and phase. A pole-zero plot visually represents these locations and is a common tool for analysis and design of filters and controllers. The arrangement of poles and zeros affects resonance, damping, overshoot, and bandwidth. Poles near the imaginary axis or zeros near the origin have pronounced effects. Cancellations between poles and zeros can simplify models but may obscure dynamics and impact real-world realizability.

Poles and zeros are obtained by factoring the transfer function, solving polynomial equations, or deriving them from differential equations (continuous time) or difference equations (discrete time). They underpin tasks such as IIR and FIR filter design, controller synthesis, and system identification. Limitations include non-minimum phase zeros, pole-zero cancellation due to modeling errors, and sensitivity to coefficient perturbations.

Example: H(s) = (s−2)/(s^2+3s+2) has a zero at s=2 and poles at s=−1 and s=−2, illustrating how poles determine stability and zeros shape the response.