Jan 13, 2012 - From ancient times, some have regulated or even resisted technologies. A Byzantine city in the 530s had zon- ing laws that separated kilns, ...
»
FROM THE EDITOR
The Efficiency of the Power of One (or Zero)
M
any of the practices of control systems engineering are useful in everyday life. Especially useful are practices that increase efficiency. For example, in areas of control systems engineering with few experts, associate editors can have difficulty in obtaining enough detailed technical reviews to have confidence in the correctness of the results. Like most young assistant professors, when asked to review manuscripts for a journal or conference, I provided highly detailed technical reviews. Rather quickly I was reviewing more than 50 control systems papers per year, which was cutting into the time needed to carry out research, teaching, and service duties, so many efficient methods had to be applied for assessing whether a control result could be technically correct. Below are some comments on one of these approaches, because it is still useful today, and the general principle equally applies to control theory and practice. Probably the most commonly used approach to improve efficiency was the power of one, which is the evaluation of whether a theorem could possibly be correct based on whether the result holds for the number one, assuming that the number one is allowed by the stated assumptions of the result. For example, the number one could be inserted in place of the transfer function of a process and/or a controller to assess whether the proposed control systems analysis or controller design method would apply for those particular cases. Digital Object Identifier 10.1109/MCS.2011.2173257
Date of publication: 13 January 2012
6 IEEE CONTROL SYSTEMS MAGAZINE
»
The power of one.
The number one is an effective counterexample for debunking a surprisingly large number of proposed control systems results and often helps to quickly gain insight into at least one important consideration not being addressed. Sometimes a transfer function slightly different from one would be used to show that a result did not hold for all plants, such as 12s 1 12 / 1 s 1 12 , which has a magnitude of one but a phase that is not exactly zero. Often a technical problem with a control systems result
Closed-Loop System r
C(s) –
u
P(s)
y
Block diagram showing a process with transfer function P(s) and a controller with transfer function C (s) in a negative feedback configuration.
FEBRUARY 2012
is associated with implicit assumptions made in the derivations. Other times a technical problem is associated with incorrect logical arguments in the proofs such as commuting two noncommuting operators, nonallowable swapping the order of limits, or not correctly taking into account the behavior of a function at infinity. An advantage of applying the power of one is that the incorrectness of a technical result can be shown quickly without having to wade through pages of a proof to try to locate exactly where mistakes were made. A variation on this approach for efficiently locating problems in control systems results is to employ the power of zero in which a theoretical result is checked whether it is true for the number zero. Sometimes applying this approach only identifies that a simple assumption needs to be inserted to correct a result, but sometimes this approach identifies a serious problem. For example, many papers and control systems textbooks make the incorrect statement that the interconnected system in the figure is stable provided that none of the roots of the characteristic equation 1 1 P 1 s 2 C 1 s 2 5 0 are in the closed right-half plane. A simple counterexample to this result is to consider any unstable process P 1 s 2 with controller C 1 s 2 5 0, demonstrating that zero is an effective counterexample to this incorrect stability condition. The equation 1 1 P 1 s 2 C 1 s 2 5 0 for this process and controller has no roots in the closed right-half plane but the system in the figure is not stable. In particular, a bounded input to the process P 1 s 2 results in its output being unbounded.
Inserting the assumptions P 1 s 2 2 0 and C 1 s 2 2 0 does not fix the incorrect stability condition, as another counterexample is a variation on the power of one, which is P 1 s 2 5 12s112 / 1 s112 and C 1 s 2 5 1 / 12s112 . Basically, this approach employs the simple principle that a theoretical result for some set, no matter how complicated, must also hold for the simplest elements of the set. It is very good to check whether a statement holds for the simplest elements of the set before trying to publish, or recommending to publish, a claim that the statement holds for all elements in the set. The above approach that applies to analyzing control theoretic results applies to other professions and everyday life. Within the context of accounting, a reasonableness test is a “procedure to examine the logic of accounting
The term the power of one is more precise, and a reasonableness test has different definitions in different fields.
information” (from J. G. Siegel and J.K. Shim’s Dictionary of Accounting Terms, fourth edition, 2010). Based on this definition, the power of one is an example of a reasonableness test. Other accounting references define a reasonableness test in a more precise way, requiring that the procedure rely on data that is partly or whole independent of the accounting system. Either definition applies to the power of one. Instead of the generic term reasonableness test, the term the power of one
is more precise, and a reasonableness test has different definitions in different fields. For example, in legal circles, one definition of a reasonableness test is whether enacting a law or setting a precedent would result in outcomes that are unreasonable based on the opinion of a judge. That definition is too subjective of a definition for use in evaluating the correctness of a control systems result. Richard D. Braatz
Early Flouters
F
rom ancient times, some have regulated or even resisted technologies. A Byzantine city in the 530s had zoning laws that separated kilns, blacksmiths, and polluting activities from shops and houses. Medieval French slaughterhouses and tanners polluted rivers and streams, leading to legislation in 1366 to prohibit such pollution of the Seine in Paris. In the same years, demand for wood stripped much of England and France bare of forests. Firewood was burned for cooking and heating, and wood was also the primary building material. As early as 1140 the French had difficulty finding 35-foot beams for building, and architects worked out ingenious ways to use shorter pieces of wood to construct bridges and churches. Iron smelters consumed 25 cubic meters of wood to produce 50 kilograms of iron. A single furnace operating for just 40 days devoured an encircling forest for a onekilometer radius. By 1230 the English were importing wood from Norway, and English coal was sold not only in London but also on the Continent. Mines became ever more extensive, and by the end of the thirteenth century air pollution was a problem. In 1307 London was the first city to prohibit coal burning, in a royal proclamation that was generally ignored. Many contemporary ecological problems—deforestation, fuel shortages, and both air and water pollution—can be traced back in European history at least 700 years. —Technology Matters, Questions to Live with, by David E. Nye, MIT Press, Cambridge, Massachusetts, pp. 96–97.
FEBRUARY 2012
« IEEE CONTROL SYSTEMS MAGAZINE
7
