Step Response
First-Order System
\(\displaystyle G(s) = \frac{K}{\tau s + 1}\)
- \(K\) — steady-state gain
- \(\tau\) — time constant
- Time constant and settling time annotated
Step Response
Second-Order System
\(\displaystyle G(s) = \frac{K\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\)
- \(\omega_n\) — natural frequency, \(\zeta\) — damping ratio
- Underdamped, critically damped, overdamped regimes
- Rise time, peak time, overshoot, and settling time annotated
Pole-Zero Map
Pole-Zero Map Simulator
\(\displaystyle G(s) = K\frac{\prod(s-z_i)}{\prod(s-p_i)}\)
- \(K\) gain, pole pair location, and optional zero
- Stable, oscillatory, slow, and unstable configurations
- s-plane map linked to step and impulse responses
Controller Design
PID Controller
\(\displaystyle C(s) = K_p + \frac{K_i}{s} + K_d s\)
- \(K_p\), \(K_i\), \(K_d\) — proportional, integral, derivative gains
- Configurable plant and reference signal
- Real-time animated closed-loop response