# PID Controller Visualization

Interactive exploration of proportional, integral, and derivative control

> Canonical page: https://elysiatools.com/en/visualizations/pid-controller

- **Category:** Math

## Overview

Interactive visualization of PID control algorithm with real-time parameter tuning and physical system animation. Explore proportional (P), integral (I), and derivative (D) control components through dynamic visualizations including: (1) Real-time response curve plotting with setpoint tracking, error area visualization, and time-axis scrolling display. (2) Physical system animation showing ball/cart tracking target position on track,直观展示振荡、超调、稳态误差 with velocity indicators and position markers. (3) PID components breakdown chart displaying individual P, I, and D contributions over time. (4) Interactive parameter controls with Kp (0-10), Ki (0-5), Kd (0-5) sliders and real-time value display. (5) Multiple test scenarios: Step response (sudden setpoint change), Disturbance response (external interference), and Sine tracking (dynamic reference following). (6) Real-time metrics: current error, integral term, derivative term, PID output, overshoot percentage, and settling time calculation. (7) Educational content covering PID theory with mathematical formula u(t) = Kp·e(t) + Ki·∫e(t)dt + Kd·de(t)/dt, component explanations (P: responds to current error, I: eliminates steady-state error, D: predicts future error), tuning guidelines (start with Kp, add Ki for steady-state error, add Kd to reduce oscillation), and practical applications (temperature control, motor speed, robotics, drones, cruise control, HVAC systems). (8) System settings: adjustable setpoint (0-100), noise level simulation (0-10), and start/pause/reset controls. PID algorithm implementation: error = setpoint - measured_value, integral += error * dt (with anti-windup clamping), derivative = (error - last_error) / dt, output = Kp * error + Ki * integral + Kd * derivative. Physical simulation using second-order mass-spring-damper system with Euler integration. Canvas-based rendering at 60 FPS with smooth animations, gradient-filled error regions, and responsive design for mobile devices. Multi-language support (zh, en, es, fr, de, ru, pt) with complete translations of theory, controls, metrics, and educational content. Perfect for engineering students studying automatic control principles, embedded developers, robotics enthusiasts, and anyone interested in feedback control systems.

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