Closed-Loop Feedback PID Controller — 3D Lab Simulation
Proportional-integral-derivative automatic feedback control with dynamic plant tracking
Theoretical Foundation & Historical Context
PID control dominates over 95% of industrial closed-loop regulation. It continuously computes error e(t) between process variable and target setpoint, synthesizing control effort via proportional, integral, and derivative terms.
Law: PID Regulation Algorithm & Minorsky Closed-Loop Stability | Scientist: Nicolas Minorsky | Year: 1922
Key Learning Objectives & Competencies
- ✓ Analyze the individual contributions of Proportional (Kp), Integral (Ki), and Derivative (Kd) gains.
- ✓ Demonstrate how the Integral term eliminates steady-state offset error (e_ss → 0).
- ✓ Tune PID parameters to minimize settling time, prevent oscillatory overshoot, and maintain stability.
Mathematical Formulations & Differential Equations
u(t) = K_p e(t) + K_i \int_0^t e(\tau) d\tau + K_d \frac{de(t)}{dt}
Proportional, integral, and derivative control effort
m \ddot{y} + c \dot{y} = u(t)
Second-order physical plant response equation
Physical Parameters & Units of Measurement
| Symbol | Physical Quantity | SI Unit | Default |
|---|---|---|---|
| setpoint | Target Setpoint | m | 3 |
| Kp | Proportional Gain Kp | — | 8 |
| Ki | Integral Gain Ki | 1/s | 1.5 |
| Kd | Derivative Gain Kd | s | 4 |
| m | Plant Mass m | kg | 1 |
| c | Viscous Damping c | N·s/m | 0.5 |
Real-World & Industrial Engineering Applications
Drone flight quadcopter attitude auto-leveling and gimbal stabilization.
Automotive cruise control and electric vehicle battery temperature regulation.
Industrial manufacturing CNC milling motor positioning and petrochemical thermal reactors.
3-Step Controlled Experimental Protocol
- Step 1: Baseline Setup — Initialize the simulation in standard conditions and observe undisturbed system dynamics.
- Step 2: Parameter Variation — Gradually modulate physical parameters using the control panel sliders.
- Step 3: Verification & Phase Space Analysis — Compare phase space trajectories and conservation metrics against analytical ODE solutions.
Frequently Asked Questions (FAQ)
What specific problem does each term in a PID controller solve?
P (Proportional) reacts to present error to drive the system forward; I (Integral) accumulates past errors to eliminate steady-state offset; D (Derivative) predicts future error slope to dampen overshoot.
What is 'Integral Windup' and how do engineers prevent it?
When actuators saturate (e.g. valve fully open), the integral term continues accumulating error, causing massive overshoots once the setpoint is reached. Anti-windup clamping circuits halt integration during saturation.
Why is pure derivative control rarely used without low-pass filtering in practical engineering?
Because differentiation amplifies high-frequency sensor noise (d/dt of sin(ωt) = ω·cos(ωt)), producing violent actuator chattering unless the signal is smoothed with a low-pass filter.
Interactive Simulation: Interactive real-time 60 FPS WebGL simulation is active in your browser. Rotate, zoom, and inspect physical apparatus components with full camera freedom.