Maxwell on Governors

Watt's governor had been spinning on engines for eighty years and nobody could say why some of them held a steady speed while others swung back and forth for ever. Engineers knew the cures by feel - add friction, lighten the balls, shorten the linkage - and had no way to tell which cure a given engine needed, or whether a new design would hunt before it was built. James Clerk Maxwell's paper of 1868 stopped describing the mechanism and wrote down its equation. The governor is a mass on a spring with damping, pushed outwards by speed; the engine is an inertia whose acceleration depends on where the governor has moved the throttle. Put the two together and the loop is THIRD order - which is the lowest order that can oscillate while every individual part of it looks perfectly stable on its own. The answer is then a sign. Solve for the roots of the characteristic equation and the machine settles if every root has a negative real part, and swings for ever if any of them does not. For the cubic that condition reduces to something you can read out loud: damping times stiffness times engine inertia must beat gain times governor mass. Maxwell solved the third and fourth order cases and left the general one open; Routh took it in 1876 and Hurwitz in 1895, and the whole of stability theory is downstream. The paper also draws a distinction that is still the first thing anyone learns about controllers. A flyball device only MODERATES - after a load change it settles to a new speed that is permanently a few per cent wrong. Maxwell refused to call that a governor. A real governor keeps moving the throttle while any error remains, which means it acts on the accumulated error: integral action, ninety years before anybody painted PID on a panel.