Laplace Transform & System Stability

System Preset

Pole 1 (Manual)

Real part σ -1.0 Imaginary part ω_d 3.0

Complex conjugate pair

Add 2nd real pole

Pole 2 real part σ₂ -3.0

Zero

Add finite zero

Zero real part σ_z -2.0 Zero imag part 0.0

Conjugate zero pair

Input & Stability

STABLE — Poles in LHP

σ_max (dominant)—

Natural freq ω_n—

Damping ζ—

Laplace Transform
H(s) = N(s)/D(s)

BIBO Stability:
ROC includes jω-axis
⇔ all poles in LHP (σ < 0)

2nd Order System:
H(s) = ω_n²/(s²+2ζω_ns+ω_n²)
ζ < 1: underdamped
ζ = 1: critically damped
ζ > 1: overdamped