History and Math of Control theory: Roman aqueducts meet thermostats meet cybernetics

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Be still my ❤︎heart. A culmination of high-level mathematics, history, interdisciplinary application, and references to some of my favorites (like pendulums, Roman aqueducts, toilets, thermostats, and HVAC systems in big buildings…) all to describe a simple concept pervasive: control theory.

The 2003 article Control Theory: History, Mathematical Achievements and Perspectives, found via wikipedia-warp, does just that. (I remember my African religions professor in the early 2000s cautioning against Wikipedia, and here I am believing it with abandon in the 20teens.)

Sidenote: Follow up with this complementary essay, How to Read Mathematics. “The beauty in a mathematics article is in the elegant efficient way it concisely describes precise ideas of great complexity.” If only I took the time (and practice) to heed its advice.

According to the authors, there are 3 key ingredients in Control Theory.

Feedback. “This term was incorporated to Control Engineering in the twenties by the engineers of the Bell Telephone Laboratory but, at that time, it was already recognized and consolidated in other areas, such as Political Economics.”

Fluctuations. Described by H.R. Hall in 1907, “The aim of the mechanical economist, as is that of the political economist, should be not to do away with these fluctuations altogether (for then he does away with the principles of self-regulation), but to diminish them as much as possible, leaving them large enough to have sufficient regulating power.”

Optimization. “This can be regarded as a branch of Mathematics whose goal is to improve a variable in order to maximize a benefit (or minimize a cost).”

“But of course in the development of Control Theory many other concepts have been important. One of them is Cybernetics.”

Examples of “classical” control theory:

  • Pendulums and oscillations. “One can obtain feedback controls, but their effects on the system are not necessarily in agreement with the very first intuition.”
  • Roman aqueducts: “In the ancient Egypt the “harpenodaptai” (string stretchers), were specialized in stretching very long strings leading to long straight segments to help in large constructions.” These were used to build optimal curves, with the understanding that (a) “The shortest distance between two points is the straight line” and (b) “Among all the paths of a given length the one that produces the longest distance between its extremes is the straight line as well.”
  • Windmills: “When the rotational velocity increased, the balls got farther from the axis, acting on the wings of the mill through appropriate mechanisms,” regulating the velocity.
  • Watt’s steam engine: Watt’s regulating system built upon this concept, so that “as the velocity of the balls increases, one or several valves open to let the vapor escape,” reducing pressure and thus velocity. “The goal of introducing and using this mechanism is of course to keep the velocity as close as possible to a constant.”
  • And beyond. “The central ideas of Control Theory gained soon a remarkable impact and, in the twenties, engineers were already preferring the continuous processing and using semi-automatic or automatic control techniques.”

Adding to the classical, contributors to “modern” control theory include:


Control Theory: History, Mathematical Achievements and Perspectives

Authors: E. Fernandez-Cara and E. Zuazua

Link: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.302.5633&rep=rep1&type=pdf

EXCERPTS

The underlying idea that motivated this article is that Control Theory is certainly, at present, one of the most interdisciplinary areas of research. Control Theory arises in most modern applications. The same could be said about the very first technological discoveries of the industrial revolution. On the other hand, Control Theory has been a discipline where many mathematical ideas and methods have melt to produce a new body of important Mathematics. Accordingly, it is nowadays a rich crossing point of Engineering and Mathematics.
The word control has a double meaning. First, controlling a system can be understood simply as testing or checking that its behavior is satisfactory. In a deeper sense, to control is also to act, to put things in order to guarantee that the system behaves as desired.
One of the simplest applications of Control Theory appears in such an apparently simple machine as the tank of our bathroom. There are many variants of tanks and some of the licences go back to 1886 and can be found in [25]*. But all them work under the same basic principles: the tank is supplied of regulating valves, security mechanisms that start the control process, feedback mechanisms that provide more or less water to the tank depending of the level of water in its interior and, finally, mechanisms that avoid the unpleasant flooding in case that some of the other components fail.
The systems of heating, ventilation and air conditioning in big buildings are also very efficient large scale control systems composed of interconnected thermo-fluid and electromechanical subsystems. The main goal of these systems is to keep a comfortable and good quality air under any circumstance, with a low operational cost and a high degree of reliability. The relevance of a proper and efficient functioning of these systems is crucial from the viewpoint of the impact in Economical and Environmental Sciences. The predecessor of these sophisticated systems is the classical thermostat that we all know and regulates temperature at home.
The optimization viewpoint is, at least apparently, humble in comparison with the controllability approach. But it is many times much more realistic. In practice, it provides satisfactory results in many situations and, at the same time, it requires simpler mathematical tools.
  • [25] E.B. Lee and L. Markus, Foundations of Optimal Control Theory, The SIAM Series in Applied Mathematics, John Wiley & Sons, New York 1967.
By iessi (Flickr) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons
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