stability

Stability is the single most important question you ask of any control loop: if you give it a bounded push, does it settle back down, or does it run away? A stable system is like a marble in a bowl — nudge it and it rolls back to the bottom. An unstable one is a marble balanced on a dome — the tiniest disturbance sends it off forever. Formally, a system is BIBO-stable (bounded-input, bounded-output) if every bounded input produces a bounded output, with no signal growing without limit.

For a linear system, stability has a beautifully crisp test: every pole of the closed-loop transfer function must lie strictly in the left half of the s-plane (negative real part). One pole creeping into the right half — and the system oscillates with ever-growing amplitude until something saturates, clips, or breaks. This is why the central act of control design is pole placement: you wrap feedback around a plant not just to make it accurate, but first and foremost to drag every closed-loop pole safely into the left half-plane. An aircraft autopilot, a power-grid inverter, a Segway — all are useless, even dangerous, if the loop isn't stable first.