Feedback Control
Interactive PID Tuner & Step-Response Sandbox
Adjust proportional, integral, and derivative gains in real-time. Visualize transient response curves, overshoot, and rise times with classic Ziegler-Nichols presets.
🎛️ PID Controller Gains
Auto-Tune Presets:
Proportional Gain (K_p)
2.50
Generates control effort directly proportional to current error.
Integral Gain (K_i)
1.20
Accumulates historical error to eliminate steady-state offset.
Derivative Gain (K_d)
0.35
Dampens system oscillations based on error rate of change.
⚙️ Physical Plant Parameters
Plant Inertia / Mass (J)
1.0
Plant Damping Friction (b)
0.5
External Disturbance Step
0.0
Inject constant torque disturbance (e.g. wind gust or slope gravity).
📈 Live Step-Response Simulation
― Setpoint (r = 1.0)
― System Output (y)
― Control Effort (u)
Peak Overshoot (M_p)
8.4 %
Peak value: 1.08
Rise Time (10% to 90%)
0.42 s
Settling (2%): 1.85s
System Stability & Convergence
✓ STABLE (Closed-loop poles in LHP)
Steady-state error: 0.000 (Exact tracking)
PID Control Law (Continuous Time):
u(t) = K_p · e(t) + K_i · ∫ e(τ) dτ + K_d · (de(t) / dt)
Plant Equation of Motion:
J · ÿ(t) + b · ẏ(t) = u(t) + d(t)
u(t) = K_p · e(t) + K_i · ∫ e(τ) dτ + K_d · (de(t) / dt)
Plant Equation of Motion:
J · ÿ(t) + b · ẏ(t) = u(t) + d(t)