🧲 Magnetic Levitation Control: Stabilizing an Unstable System

The attractive force follows F = k·I²/x², which grows as the gap x shrinks. If the ball drifts slightly closer to the coil at a fixed current, F increases, pulling it closer still — and if it drifts slightly farther away, F decreases, letting gravity pull it away further. This is positive feedback: linearizing F around any equilibrium gap gives a restoring-force term with the wrong sign, so the open-loop system has no stable operating point. It will either snap up into the coil or fall away, never settle on its own.

The other Magnetic Levitation simulation is about the underlying physics of levitation itself — the Meissner effect and flux pinning in superconductors, which sidestep Earnshaw's theorem through passive electromagnetic response. This simulation is about the control problem: it uses an ordinary electromagnet and ball, a setup that is definitely unstable on its own, and focuses entirely on the PID feedback loop that has to run continuously to hold it up. The two pages don't duplicate each other; they look at magnetic levitation from physics and control-engineering angles respectively.

Each control tick it measures the position error e = x_set − x (target gap minus measured gap) and computes the coil current as I = I₀ − Kp·e − Ki·∫e·dt − Kd·de/dt, clamped to [0, I_max] with anti-wind-up. I₀ is a feed-forward term — the exact current that would hold the ball motionless at the setpoint — while the Kp/Ki/Kd terms correct any error away from that unstable equilibrium. Because I₀ alone is not enough (the equilibrium it defines is unstable), the feedback terms have to run every tick, not just once.

The simulation holds the coil at exactly the feed-forward current that balances gravity at the current setpoint — the same unstable equilibrium the controller would otherwise correct around. A small simulated perturbation is added each tick, the same way real sensor noise, air currents or power-supply ripple would perturb a real open-loop maglev rig, and the genuine F ∝ I²/x² physics does the rest: the ball drifts toward the coil (and crashes) or away from it (and drops) within a few seconds, exactly as the instability argument predicts.

The derivative term Kd·de/dt reacts to how fast the gap error is changing, adding damping that is essential here because the plant itself has no natural damping — without it, a proportional-only loop tends to oscillate or ring around the setpoint. But Kd amplifies any noise in the velocity estimate, so pushing it too high makes the coil current chatter and can destabilize the loop in a different way; real maglev controllers typically pair a moderate Kd with a filtered velocity estimate rather than raising it without bound.

The feed-forward current is I₀ = √(g·m·x_set²/k): at a fixed target gap, a heavier ball needs a larger magnetic force to support its weight, and since F ∝ I², the required current scales with the square root of mass. This simulation recomputes I₀ live whenever you change the mass or target-gap sliders, which is why resetting the ball after such a change is necessary — the previous equilibrium current no longer matches the new physical situation.

Push down/up applies an instantaneous velocity impulse to the ball, simulating a sudden bump — a hand nudging it, a gust of air, a small mechanical shock. With the controller on, a well-tuned PID loop should measure the resulting gap error, recompute the coil current, and pull the ball back to the setpoint within a fraction of a second. With the controller off, the same push is simply an initial condition for the unstable open-loop dynamics, and the ball will not recover on its own.