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CHAPTER 1
ON GOVERNORS
by J. C. Maxwell
Although isolated examples of control concepts occur throughout recorded history, from the irrigation mentioned in the code of Hammurabi to the centrifugal governor discussed by Huygens for the regulation of windmills and water wheels, it was not until the flowering of the English Industrial Revolution that these techniques came into widespread use and became of great significance. The familiar governor of Watt, used for his steam engine, was only one of a number of similar devices which began to play important roles in industry. As a result, serious scientific attention was focused upon the use of governors, regulators, and related mechanisms.
These contrivances are cleverly devised to use the very deviation from desired performance of a system to call upon an actuator to exert a restoring force. This idea must be used with caution, however, since under unfavorable circumstances, the effect which is supposed to be abated can actually be abetted. It appears that Maxwell was the first to realize that these phenomena could be analyzed in mathematical terms, and that there were some subtle mathematical problems contained in the engineering questions. Among the foremost of these is that of the stability of the resulting system.
In the paper that follows, he reviews a number of control devices and transforms the problem of choosing combinations of elements which will yield efficient operation into that of choosing parameters which will produce solutions of a differential equation with desired behavior over time.
Using standard techniques of perturbation theory, Maxwell thus reduces the design problem to that of determining the location of the roots of algebraic polynomials. He resolves it easily enough for quadratic and cubic polynomials, and points out that the problem for nth. order polynomials is nontrivial. The general solution was given independently by Routh and Hurwitz; see the paper by Hurwitz.
This paper sets the stage for much of the subsequent development.
Independently, the importance of control theory was recognized by the Russian engineer, Vyshnegradskii. See:
J. Vyshnegradskii, "Sur la théorie générale des régulateurs," Compt. Rend. Acad. Sei. Paris, Vol. 83, 1876, pp. 318-321.
_______, "Über direkt wirkende Regulatoren," Der Civilingenieur, (2), Vol. 23, 1877, pp. 95-132.
Examples of this work are discussed by Pontryagin in his recent book on differential equations.
I. "Governors." By J. Clerk Maxwell, M.A., F.R.SS.L. & E. Received Feb. 20, 1868
A Governor is a part of a machine by means of which the velocity of the machine is kept nearly uniform, notwithstanding variations in the driving-power or the resistance.
Most governors depend on the centrifugal force of a piece connected with a shaft of the machine. When the velocity increases, this force increases, and either increases the pressure of the piece against a surface or moves the piece, and so acts on a break or a valve.
In one class of regulators of machinery, which we may call moderators, the resistance is increased by a quantity depending on the velocity. Thus in some pieces of clockwork the moderator consists of a conical pendulum revolving within a circular case. When the velocity increases, the ball of the pendulum presses against the inside of the case, and the friction checks the increase of velocity.
In Watt's governor for steam-engines the arms open outwards, and so contract the aperture of the steam-valve.
In a water-break invented by Professor J. Thomson, when the velocity is increased, water is centrifugally pumped up, and overflows with a great velocity, and the work is spent in lifting and communicating this velocity to the water.
In all these contrivances an increase of driving-power produces an increase of velocity, though a much smaller increase than would be produced without the moderator.
But if the part acted on by centrifugal force, instead of acting directly on the machine, sets in motion a contrivance which continually increases the resistance as long as the velocity is above its normal value, and reverses its action when the velocity is below that value, the governor will bring the velocity to the same normal value whatever variation (within the working limits of the machine) be made in the driving-power or the resistance.
I propose at present, without entering into any details of mechanism, to direct the attention of engineers and mathematicians to the dynamical theory of such governors.
It will be seen that the motion of a machine with its governor consists in general of a uniform motion, combined with a disturbance which may be expressed as the sum of several component motions. These components may be of four different kinds : —
1. The disturbance may continually increase.
2. It may continually diminish.
3. It may be an oscillation of continually increasing amplitude.
4. It may be an oscillation of continually decreasing amplitude.
The first and third cases are evidently inconsistent with the stability of the motion; and the second and fourth alone are admissible in a good governor. This condition is mathematically equivalent to the condition that all the possible roots, and all the possible parts of the impossible roots, of a certain equation shall be negative.
I have not been able completely to determine these conditions for equations of a higher degree than the third; but I hope that the subject will obtain the attention of mathematicians.
The actual motions corresponding to these impossible roots are not generally taken notice of by the inventors of such machines, who naturally confine their attention to the way in which it is designed to act; and this is generally expressed by the real root of the equation. If, by altering the adjustments of the machine, its governing power is continually increased, there is generally a limit at which the disturbance, instead of subsiding more rapidly, becomes an oscillating and jerking motion, increasing in violence till it reaches the limit of action of the governor. This takes place when the possible part of one of the impossible roots becomes positive. The mathematical investigation of the motion may be rendered practically useful by pointing out the remedy for these disturbances.
This has been actually done in the case of a governor constructed by Mr. Fleeming Jenkin, with adjustments, by which the regulating power of the governor could be altered. By altering these adjustments the regulation could be made more and more rapid, till at last a dancing motion of the governor, accompanied with a jerking motion of the main shaft, showed that an alteration had taken place among the impossible roots of the equation.
I shall consider three kinds of governors, corresponding to the three kinds of moderators already referred to.
In the first kind, the centrifugal piece has a constant distance from the axis of motion, but its pressure on a surface on which it rubs varies when the velocity varies. In the moderator this friction is itself the retarding force. In the governor this surface is made moveable about the axis, and the friction tends to move it; and this motion is made to act on a break to retard the machine. A constant force acts on the moveable wheel in the opposite direction to that of the friction, which takes off the break when the friction is less than a given quantity.
Mr. Jenkin's governor is on this principle. It has the advantage that the centrifugal piece does not change its position, and that its pressure is always the same function of the velocity. It has the disadvantage that the normal velocity depends in some degree on the coefficient of sliding friction between two surfaces which cannot be kept always in the same condition.
In the second kind of governor, the centrifugal piece is free to move further from the axis, but is restrained by a force the intensity of which varies with the position of the centrifugal piece in such a way that, if the velocity of rotation has the normal value, the centrifugal piece will be in equilibrium in every position. If the velocity is greater or less than the normal velocity, the centrifugal piece will fly out or fall in without any limit except the limits of motion of the piece. But a break is arranged so that it is made more or less powerful according to the distance of the centrifugal piece from the axis, and thus the oscillations of the centrifugal piece are restrained within narrow limits.
Governors have been constructed on this principle by Sir W. Thomson and by M. Foucault. In the first, the force restraining the centrifugal piece is that of a spring acting between a point of the centrifugal piece and a fixed point at a considerable distance, and the break is a friction-break worked by the reaction of the spring on the fixed point.
In M. Foucault's arrangement, the force acting on the centrifugal piece is the weight of the balls acting downward, and an upward force produced by weights acting on a combination of levers and tending to raise the balls. The resultant vertical force on the balls is proportional to their depth below the centre of motion, which ensures a constant normal velocity. The break is: — in the first place, the variable friction between the combination of levers and the ring on the shaft on which the force is made to act; and, in the second place, a centrifugal air-fan through which more or less air is allowed to pass, according to the position of the levers. Both these causes tend to regulate the velocity according to the same law.
The governors designed by the Astronomer Royal on Mr. Siemens's principle for the chronograph and equatorial of Greenwich Observatory depend on nearly similar conditions. The centrifugal piece is here a long conical pendulum, not far removed from the vertical, and it is prevented from deviating much from a fixed angle by the driving-force being rendered nearly constant by means of a differential system. The break of the pendulum consists of a fan which dips into a liquid more or less, according to the angle of the pendulum with the vertical. The break of the principal shaft is worked by the differential apparatus; and the smoothness of motion of the principal shaft is ensured by connecting it with a fly-wheel.
In the third kind of governor a liquid is pumped up and thrown out over the sides of a revolving cup. In the governor on this principle, described by Mr. C. W. Siemens, the cup is connected with its axis by a screw and a spring, in such a way that if the axis gets ahead of the cup the cup is lowered and more liquid is pumped up. If this adjustment can be made perfect, the normal velocity of the cup will remain the same through a considerable range of driving-power.
It appears from the investigations that the oscillations in the motion must be checked by some force resisting the motion of oscillation. This may be done in some cases by connecting the oscillating body with a body hanging in a viscous liquid, so that the oscillations cause the body to rise and fall in the liquid.
To check the variations of motion in a revolving shaft, a vessel filled with viscous liquid may be attached to the shaft. It will have no effect on uniform rotation, but will check periodic alterations of speed.
Similar effects are produced by the viscosity of the lubricating matter in the sliding parts of the machine, and by other unavoidable resistances; so that it is not always necessary to introduce special contrivances to check oscillations.
I shall call all such resistances, if approximately proportional to the velocity, by the name of "viscosity," whatever be their true origin.
In several contrivances a differential system of wheelwork is introduced between the machine and the governor, so that the driving-power acting on the governor is nearly constant.
I have pointed out that, under certain conditions, the sudden disturbances of the machine do not act through the differential system on the governor, or vice versa. When these conditions are fulfilled, the equations of motion are not only simple, but the motion itself is not liable to disturbances depending on the mutual action of the machine and the governor.
Distinction between Moderators and Governors.
In regulators of the first kind, let P be the driving-power and R the resistance, both estimated as if applied to a given axis of the machine. Let V be the normal velocity, estimated for the same axis, and dx/dt the actual velocity, and let M be the moment of inertia of the whole machine reduced to the given axis.
Let the governor be so arranged as to increase the resistance or diminish the driving-power by a quantity F (dx/dt - V), then the equation of motion will be
d/dt (M dx/dt) = P-R-F(dx/dt - V) ... (1)
When the machine has obtained its final rate the first term vanishes, and
dx/dt = V + P - R/F ... (2)
Hence, if P is increased or R diminished, the velocity will be permanently increased. Regulators of this kind, as Mr. Siemens has observed, should be called moderators rather than governors.
In the second kind of regulator, the force F (dx/dt - V), instead of being applied directly to the machine, is applied to an independent moving piece, B, which continually increases the resistance, or diminishes the driving-power, by a quantity depending on the whole motion of B.
If y represents the whole motion of B, the equation of motion of
B is
d/dt(B dy/dt) = F(dx/dt - V), ... (3)
and that of M
d/dt(M dx/dt) = P-R-F(dx/dt - V) + Gy, ... (4)
where G is the resistance applied by B when B moves through one unit of space.
We can integrate the first of these equations at once, and we find
B dy/dt = F (x - Vt); ... (5)
so that if the governor B has come to rest x=Vt, and not only is the velocity of the machine equal to the normal velocity, but the position of the machine is the same as if no disturbance of the driving-power or resistance had taken place.
Jenkin's Governor. — In a governor of this kind, invented by Mr. Fleeming Jenkin, and used in electrical experiments, a centrifugal piece revolves on the principal axis, and is kept always at a constant angle by an appendage which slides on the edge of a loose wheel, B, which works on the same axis. The pressure on the edge of this wheel would be proportional to the square of the velocity; but a constant portion of this pressure is taken off by a spring which acts on the centrifugal piece. The force acting on B to turn it round is therefore
[MATHEMATICAL EXPRESSION OMITTED]
and if we remember that the velocity varies within very narrow limits, we may write the expression
F(dx/dt - V1),
where F is a new constant, and V1 is the lowest limit of velocity within which the governor will act.
Since this force necessarily acts on B in the positive direction, and since it is necessary that the break should be taken off as well as put on, a weight W is applied to B, tending to turn it in the negative direction; and, for a reason to be afterwards explained, this weight is made to hang in a viscous liquid, so as to bring it to rest quickly.
The equation of motion of B may then be written
Bd2y/dt2 = F(dx/dt - V1) - Y dy/dt - W, ... (6)
where Y is a coefficient depending on the viscosity of the liquid and on other resistances varying with the velocity, and W is the constant weight.
(Continues…)
Excerpted from "Classic Papers in Control Theory"
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