The governor that needed friction to work

Turning is an acceleration found that anything moving in a circle needs a force pointing inward, of size $m v^2/r$, even at constant speed, and that without such a force the body leaves the circle along a straight line. A ball on a string swung round the head is held on its circle by the string’s tension; a ball on a hinged arm attached to a spinning shaft is held by the arm, which lets it swing outward until the arm’s pull, partly inward and partly upward, both supplies the circular force and holds the ball’s weight.

That second arrangement is the centrifugal governor, the device in front of nearly every steam engine of the nineteenth century, spinning on top of the engine with two heavy balls on hinged arms. It was adapted to steam engines by James Watt and Matthew Boulton in 1788 from a mechanism already used in windmills, and it was the first widely used automatic feedback controller. It is a piece of circular motion made into a measuring instrument and then into a regulator — and the way it misbehaved led directly to the mathematics of stability that every control system since has been designed with.

A speedometer made of a conical pendulum #

Two balls of mass $m$ on arms of length $\ell$, hinged at the top of a shaft turning at angular speed $\Omega$, are a pair of conical pendulums. Each ball moves on a horizontal circle of radius $\ell\sin\varphi$, where $\varphi$ is the arm’s angle from the shaft. The arm’s tension has an upward part that must balance the weight and an inward part that must supply the circular force, and the two balance at

$$\cos\varphi = \frac{g}{\ell\,\Omega^2}.$$

Seen from the shaft, turning with it, the same balance reads differently: in that frame the balls are at rest, and they stand out because of an outward centrifugal force, $m\Omega^2 r$, balanced by the arm’s inward pull. The forces that are not there found that such forces exist only in a turning frame and are the price of describing motion from one; they give the right answer here, and they give the device its name, but from the ground nothing pushes the balls outward. They are trying to go straight and the arms will not let them.

Two features make this a good speedometer. Below a critical speed, $\sqrt{g/\ell}$, there is no solution with the arms raised: the balls hang straight down and turn on the shaft without moving outward. Above it, the angle rises steadily with speed. And the height of the balls below the hinge, $\ell\cos\varphi = g/\Omega^2$, depends on the speed alone — not on the mass of the balls, not on the length of the arms. A governor running at sixty revolutions a minute holds its balls 24.8 centimetres below the hinge whatever they are made of, which made it easy to set: choose the height at which the throttle should be half open, and the speed follows.

The mass of the balls does matter for something else. Heavy balls can push hard on the linkage that moves the valve, so a governor with heavy balls can operate a large valve against its friction and the pressure of the steam, and the governors on big mill engines carried balls of tens of kilograms.

Holding a speed, nearly #

Connect the arms to the throttle so that rising arms close it. If the engine speeds up, the balls rise, the throttle closes, the steam supply falls and the engine slows. If it slows, they drop, the throttle opens and it speeds up. That is negative feedback, and at a steady state it holds the engine at the speed where the balls’ height gives just the throttle opening the load needs.

But the speed it holds depends on the load. A heavier load needs a wider throttle, a wider throttle needs lower arms, and lower arms need a lower speed. A governor of this kind cannot hold an engine at exactly the same speed under different loads; it can only hold it at a speed that falls as the load rises, a property called droop. Making the linkage stiffer — more throttle movement per degree of arm movement — reduces the droop, so that a small fall in speed opens the throttle a lot. That is the sensitivity a mill owner wanted, and it is where the trouble began.

The flywheel is the other half of the loop. An engine’s speed changes at a rate set by the difference between the steam’s torque and the load’s, divided by the flywheel’s moment of inertia, which the mass and where it sits found growing with the square of the distance of the rim from the axle. A heavy rim makes the speed sluggish: it changes slowly whatever the governor does, which gives the governor time to respond before the speed has run far away from its target. The flywheel is a filter in the loop as much as an energy store, and its size will turn out to set how much the governor can be trusted.

Hunting #

By the 1860s, with engines larger and governors made more sensitive to hold speed more tightly, many installations began to hunt. Instead of settling at a new speed after a change of load, the engine would speed up and slow down in a slow oscillation, sometimes steady, sometimes growing until the balls hit their stops. Engineers tried adding weights, springs and dashpots, with mixed success, and nobody could say in advance which governor would hunt.

The simulation shows all three behaviours with one governor. The load rises by a small step. With no friction in the governor’s linkage, the engine slows, the balls drop, the throttle opens, the engine speeds up — and the balls, swinging on their arms, overshoot, close the throttle too far, the engine slows again, and each swing is larger than the last until the arms hit their stops. With a little friction it does the same, more slowly. With enough friction, the oscillation dies and the engine settles.

The reason is a delay that the feedback cannot see. Any loop in which a correction arrives late can feed its own oscillation, a fact the answer that cannot come first traced back to causality itself: a response must follow its cause, so every real system adds some phase lag, and the lag grows with frequency until, at some frequency, the correction pushes in step with the error instead of against it. The throttle’s position follows the balls, and the balls are pendulums: when the speed changes, they do not move instantly to their new height but swing towards it, with their own natural frequency, and they lag. The engine’s speed, in turn, does not respond to the throttle instantly, because the flywheel takes time to speed up or slow down. Two lags in a loop make a correction that arrives late, and a late correction overshoots. The balls’ own swinging is the larger of the two. A governor’s arms are a pendulum whose restoring force comes partly from gravity and partly from the spin, and like any pendulum it has a natural frequency at which it will happily oscillate on its own; the swing that is pumped, not pushed found how readily such a pendulum takes up energy from anything that varies at the right rate. A governor without friction is a pendulum coupled to an engine whose speed is fed back into it, and the loop can find a rate at which it pumps itself. With nothing to absorb the swinging of the balls — no friction in the hinges and linkage — each overshoot feeds the next.

Maxwell’s equations of a governor #

Maxwell had a practical reason to care. In the early 1860s he was part of the British Association committee fixing a standard of electrical resistance, and its method spun a coil of wire in the Earth’s magnetic field and measured the current induced in it — the induction of the field that makes the other put to work as a measurement. The result depended on the coil’s speed, which had to be held constant for minutes at a time, and the committee’s apparatus carried a governor that Maxwell, Fleeming Jenkin and their colleagues had to make behave. James Clerk Maxwell’s paper “On Governors”, read to the Royal Society in 1868, is usually taken as the beginning of control theory. Maxwell wrote the equations of motion for the engine and its governor, linearised them about the steady state, and observed that whether the steady state is stable depends on the roots of a polynomial: the motion is a sum of terms $e^{st}$, one for each root $s$, and the engine settles only if every root has a negative real part.

For the governor drawn here, three quantities enter: the flywheel’s moment of inertia $J$, the gain $K$ of the linkage, and the damping $\beta$ of the balls’ swinging. The linearised loop gives the cubic

$$s^3 + \beta s^2 + c\,s + \frac{aK}{J} = 0,$$

where $a$ and $c$ are fixed by the governor’s geometry at its operating speed. A cubic’s roots all have negative real parts if and only if all its coefficients are positive and $\beta c > aK/J$ — a criterion found in general form by Edward Routh in 1877 and Adolf Hurwitz in 1895, and stated for this governor by Ivan Vyshnegradsky in 1876. With no damping, $\beta = 0$, the condition fails whatever the other quantities are: a frictionless governor cannot settle.

The stability boundary explains the 1860s at once. The friction needed is proportional to the gain divided by the flywheel’s inertia. Engineers had been making governors more sensitive — raising the gain — to reduce the droop, and at the same time making them better, with polished hinges and lighter linkages that reduced the friction, and fitting lighter flywheels to faster engines. Every one of those changes moved the operating point towards the hunting side of the line. Vyshnegradsky’s rule of thumb, which he wrote down for designers, was that friction is necessary for stable regulation, and that an unduly sensitive governor is unstable.

A heavier flywheel buys time #

The boundary also says what else could be done, short of adding friction: make the flywheel heavier. Doubling the moment of inertia halves the damping the governor needs, because the engine’s speed then changes half as fast, and a governor whose balls lag by the same amount is late by a smaller fraction of each oscillation.

With the heavier flywheel the governor that hunted with a little friction now settles, and the one with no friction still hunts, more slowly than before. This was the other remedy mill engineers found by trial: a heavier flywheel steadied a hunting engine. It costs something too — a heavier flywheel makes the engine slower to respond to a genuine change of load, so the speed wanders further before the governor catches it — and the choice between friction, a dashpot and inertia was one engineers made by feel until Vyshnegradsky gave them the inequality.

The lesson generalises beyond engines. Wherever a correction has to pass through something that lags — the swing of a pendulum, the heating of a room, the playing of a loudspeaker — the loop is only as stable as the margin between its gain and its lag allows. The silence that fits in a tenth of a wavelength found a fixed delay limiting what a noise canceller can do at high frequency; the governor’s lags are mechanical rather than electronic, but they bound it in the same way, and the cure is the same: slow the loop down where it cannot keep up, or damp it where it would overshoot.

The trade-off in every controller since #

Behind the particular equations is a general conflict that every feedback controller faces. A controller that reacts strongly to a small error holds its target closely in the steady state, and the same strong reaction, arriving late through whatever lags the system contains, overshoots and can feed an oscillation. The cure is a term that resists rapid change — damping, in the governor’s case friction in the linkage or a dashpot added on purpose — or, in later controllers, a term proportional to the rate of change of the error, which anticipates where the error is going. The proportional–integral–derivative controller that runs most of the world’s industrial processes is the governor’s problem solved with three adjustable coefficients, and the rules for tuning it descend from the stability arguments Maxwell and Vyshnegradsky made.

The same balance appears elsewhere in this collection without being called control. The three ways of coming to rest found an oscillator returning fastest with just enough damping, neither too much nor too little; here the oscillator is the whole loop, and too little damping does not merely slow the return but makes it grow. The swing that dies in a straight line found that the friction in a real linkage is often dry, not viscous; dry friction gives a governor a dead band, a range of speeds over which the balls do not move at all, and dead bands in feedback loops produce their own slow limit cycles, which governor designers of the period called “hunting” too.

What the picture leaves out #

The load is not a step. A real mill’s load changes irregularly as machines are engaged and released, and a governor that settles cleanly after a single step can still be driven into large swings by a load that happens to fluctuate near the loop’s natural frequency. Damping protects against that too, since it lowers the loop’s resonance as well as stabilising it.

The steam does not respond instantly. A real engine’s torque lags the throttle, because steam must flow through pipes and cylinders, adding another lag to the loop and lowering the gain the system can tolerate further. Maxwell’s analysis included some such effects; the drawing here has one lag in the balls and one in the flywheel only.

Governors were modified to remove the droop. Adding a term that integrates the speed error over time, so that the throttle keeps moving as long as the speed is wrong, removes the steady error entirely. Maxwell analysed governors of that kind too — he called them “governors” as opposed to “moderators” — and found that the integral action makes stability harder still.

The arms are not the only pendulum. In a real governor the balls can also swing sideways relative to the shaft’s rotation, the linkage has its own flexibility, and the governor’s drive from the engine may slip. Each adds modes that the simple model ignores.

Still open: how much stability a controller needs #

The governor’s question has never gone away; it has moved. Every aircraft autopilot, every power grid’s frequency regulation, every insulin pump and every speed controller in a hard disk faces the same trade between how closely it holds its target and how robustly it stays stable when its model of the system is wrong. Modern control theory measures the margin by how much extra gain or extra delay a loop can tolerate before it starts to oscillate, and designs for a margin rather than for the boundary itself.

How large that margin must be, and how it should be shared between performance and safety, has no general answer. It depends on how uncertain the system is, on what happens when it fails, and on what is being regulated. The power grids of the 2020s pose a new form of it: large rotating generators, whose flywheel inertia steadies the grid’s frequency the way an engine’s flywheel steadied Watt’s, are being replaced by solar and wind sources connected through electronics that have no inertia at all, and how much artificial inertia and damping the electronics must supply to keep the grid from hunting is being worked out on live systems. The answer Maxwell gave for one governor is the template. A loop with delay needs damping to settle, and the more tightly it is asked to hold, the more damping it needs.