engineering

Control Systems & Feedback

The mathematics of feedback — PID controllers, Bode plots, root locus diagrams, state-space models, and Nyquist stability analysis for designing robust dynamic systems.

control systemsPID controllerBode plotroot locusstate spaceNyquistfeedbackstability

Control systems engineering is the discipline of designing feedback loops that drive dynamic systems toward desired behavior. From cruise control in your car to the autopilot on an aircraft, feedback controllers measure output, compare it to a reference, and compute corrective inputs — a cycle that repeats thousands of times per second.

These simulations let you tune PID controllers in real time, analyze frequency response via Bode plots, trace root locus paths as gain varies, explore state-space representations, and evaluate stability with Nyquist diagrams — all with interactive, animated visualizations grounded in classical and modern control theory.

5 interactive simulations

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Bode Plot Analyzer

Visualize frequency response of transfer functions — plot magnitude and phase Bode diagrams to analyze gain margin, phase margin, and bandwidth

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Nyquist Stability Diagram

Plot the Nyquist contour of an open-loop transfer function — count encirclements of the critical point to determine closed-loop stability

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PID Controller Tuner

Tune a PID controller in real time — adjust proportional, integral, and derivative gains to minimize overshoot, settling time, and steady-state error

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Root Locus Explorer

Trace how closed-loop poles move in the complex plane as gain varies — visualize stability boundaries, breakaway points, and asymptotic angles

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State-Space Simulator

Explore state-space representations of dynamic systems — visualize state trajectories, eigenvalue placement, and controllability in the phase plane