The classic paper, Maxwell's "On governors". (1869) [1] This is the first theoretical analysis of feedback.
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.
That is, in the left half-plane.
(Terminology has changed. Maxwell says "disturbance" where today, the term "error" would be used.
Today, "disturbance" means an input which disturbs stability, while error is an output.)
Maxwell got so much right in that paper, and it was a long time before anybody picked up on that result.
Now, where it looks like the author is going is into economic territory. Basic economics talks about "economic equilibrium". The concept is that restoring forces will bring supply and demand into equilibrium. But basic control theory tells us that may not happen. Any system with delay in it can potentially be unstable. Too much delay, and even simple systems will not stabilize.
In the real world all exponentials are sigmoids eventually, so what we actually get is is a recession with a drastic reallocation of resources (creative destruction).
Not really, there are plenty that are sinusoids which are just complex exponentials.
The real trouble with sigmoids is when the saturation point is beyond physically meaningful quantities of the system. See the tacoma narrows bridge collapse.
Isn't it true to say that a theoretical exponential becomes a practical sigmoid precisely because some property of the system has become saturated and gone nonlinear?
(Just trying to get this clear in my own head by writing it down.)
When a real world system goes non linear (like the Tacoma Narrows case) you don’t get a sigmoid but something catastrophic.
(If you graph the amplitude of the vibrations they increase and increase — and then — if it was a sigmoid they’d level out and stay at the max amplitude… but in reality they go to zero as there is no bridge left to vibrate.)
When a company is “growing exponentially” it may saturate the market and then the growth slows in a nice sigmoid function. That’s common. But if, for example, the investors insist that the company must maintain the growth at all costs… it breaks laws, gets destroyed and there’s no company left to grow. No exponential curve, no sigmoid, no signal at all.
Both the sigmoid and the total collapse are typical real world results of what a simple model would expect to be an unbounded exponential curve.
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.
That is, in the left half-plane.
(Terminology has changed. Maxwell says "disturbance" where today, the term "error" would be used. Today, "disturbance" means an input which disturbs stability, while error is an output.)
Maxwell got so much right in that paper, and it was a long time before anybody picked up on that result.
Now, where it looks like the author is going is into economic territory. Basic economics talks about "economic equilibrium". The concept is that restoring forces will bring supply and demand into equilibrium. But basic control theory tells us that may not happen. Any system with delay in it can potentially be unstable. Too much delay, and even simple systems will not stabilize.
[1] https://en.wikisource.org/wiki/On_Governors