|
Epimenides of Knossos from Crete – a semi-mythical poet and philosopher , who lived in the 6th century BC, once declared: “All Cretans are liars!” Since he himself was a Cretan, he is remembered as the inventor of the so-called Cretan paradox.
In terms of Aristotelian logic, where a statement cannot be both true and false, such self-denials make no sense. If they are true, they are false, but if they are false, they are true.
This is where fuzzy logic comes into play, where variables can be partial members of sets. Truth or falsity cease to be absolute – statements can be partially true and partially false. Using this approach, it is possible to strictly mathematically prove that the Epimenides paradox is exactly 50% true and 50% false.
Thus, fuzzy logic is fundamentally incompatible with Aristotelian logic, especially with regard to the law of Tertium non datur (there is no third – Latin), which is also called the law of excluded middle1. In short, it sounds like this: if a statement is not true, then it is false. These postulates are so basic that they are often simply taken on faith.
A more trivial example of the usefulness of fuzzy logic can be found in the context of the concept of cold. Most people can answer the question, «Are you cold right now?» In most cases (unless you are talking to a physics PhD student), people understand that we are not talking about absolute temperature on the Kelvin scale. While a temperature of 0 oK can certainly be called cold, many would not consider a temperature of +15 oC to be cold.
But machines are not capable of making such a fine gradation. If the standard for defining cold is «temperature below +15 oC», then +14.99 oC would be considered cold, but +15 oC would not be.
Fuzzy Set Theory
Although fuzzy logic is usually only taught in upper-division colleges, its basic concepts are surprisingly simple.
Let's look at Figure 1. It shows a graph that helps us understand how humans perceive temperature. A temperature of +60 oF (+12 oC) is perceived by humans as cold, and a temperature of +80 oF (+27 oC) is perceived as hot. A temperature of +65 oF (+15 oC) is considered cold by some, and quite comfortable by others. We call this group of definitions a membership function for sets that describe a person's subjective perception of temperature.
It is just as easy to create additional sets that describe a person's perception of temperature. For example, you can add sets such as «very cold» and «very hot.» Similar functions can be described for other concepts, such as «open» and «closed» states, the temperature in a cooler, or the temperature in a cooling tower.
That is, fuzzy systems can be used as a universal approximator (averager) of a very wide class of linear and nonlinear systems. This not only makes control strategies more reliable in nonlinear cases, but also allows using expert assessments to build computer logic circuits.
Fuzzy Operators
To apply algebra to fuzzy values, we need to define the operators used. Boolean logic typically uses only a limited set of operators, which are used to perform other operations: NOT, AND, and OR.
There are many definitions for these three basic operators, three of which are given in the table. By the way, all definitions are equally valid for Boolean logic (to check, just substitute 0 and 1 into them).
In Boolean logic, FALSE is equivalent to 0, and TRUE is equivalent to 1. Similarly, in fuzzy logic, truth values range from 0 to 1, so Cold is true to the power of 0.1, and NOT(Cold) is 0.9.
You can go back to the Epimenides paradox and try to solve it (mathematically, it is expressed as A = NOT(A), where A is the truth value of the corresponding statement). If you want a more challenging problem, try solving the question of the sound of a clap made by one hand…
Solving Problems with Fuzzy Logic
Only a few valves are capable of opening «a little bit». Equipment operation usually uses crisp values (for example, in the case of a bimodal signal of 0-10 V), which can be obtained using so-called «fuzzy logic problem solving». This approach allows the semantic knowledge contained in the fuzzy system to be transformed into an implementable control strategy.
This can be done using various techniques, but to illustrate the process as a whole, we will consider just one example.
In the height defuzzification method, the output is the sum of the peaks of the fuzzy sets, calculated using weights. This method has several drawbacks, including poor performance with asymmetric membership functions, but it has the advantage of being the easiest to understand.
Suppose a set of rules governing the opening of a valve gives us the following output:
«Valve closed»: 0.1
«Valve partially closed»: 0.2
«Valve partially open»: 0.7
«Valve open»: 0.3
If we use the height defuzzification method to determine the degree of valve openness, we get the result:
(0.1*0% + 0.2*25% + 0.7*75% + 0.3*100%)//(0.1 + 0.2 + 0.7 + 0.3) = (0% + 5% + 52.5% + 30%)/(1.3) = 87.5/1.3 = 67.3%,
i.e. the valve must be opened by 67.3%.
Practical application of fuzzy logic
When the theory of fuzzy logic first appeared, scientific journals could be found with articles devoted to its possible areas of application. As developments in this field progressed, the number of practical applications for fuzzy logic began to grow rapidly. At present, this list would be too long, but here are some examples that will help to understand how widely fuzzy logic is used in control systems and expert systems.
– Devices for automatic maintenance of vehicle speed and increasing the efficiency/stability of vehicle engines (Nissan, Subaru).
– Handwriting recognition systems in PDAs (Sony).
– Improving safety systems for nuclear reactors (Hitachi, Bernard, Nuclear Fuel Div.).
– Robot control (Toshiba, Fuji Electric, Omron).
– Industrial control systems (Aptronix, Omron, Meiden, Sha, Micom, Nisshin-Denki, Mitsubishi, Oku-Electronics, etc.).
To show where fuzzy logic is used in automation systems, I will give several rules that allow increasing the efficiency of utility systems in buildings or can help identify faults. Parameters that are fuzzy values are highlighted.
Fuzzy logic in building automation systems
While fuzzy logic may indeed be the next step in building automation systems (BAS), it is important to remember that meaningful use of fuzzy logic requires that the software that controls the BAS be designed from the start to take advantage of these features. It is not enough to have a language for defining rules, to be integrated into the software, and to be consistent with the principles of operation. It is also necessary to clearly distinguish between standard control resources and methods (e.g. PID control, scheduling, warning messages, etc.) and fuzzy logic control methods. Incomplete integration or incorrectly constructed rules for conversion between standard and fuzzy logic can lead to errors in programs whose cause will be very difficult to determine.
In addition, much of the information needed for effective operation of fuzzy logic systems should be automatically collected during the initial startup of the software that controls the system. Leaving the programming of the system to building engineers would be imprudent on the part of BAS designers and would result in BAS not being used properly in most buildings. For example, membership functions (see Figures 1 and 2) must be defined for quantities commonly used in BAS. If air temperature sensors or VAV controllers are added, the computer program must be able to automatically define the appropriate membership functions for the various control points.
As energy prices continue to rise and the need for advanced fault detection systems becomes apparent, building designers and property managers should be on the lookout for innovative solutions. Fuzzy logic is a term that is on everyone’s lips, and when used correctly, it can live up to the hype.
Fig. 1. Fuzzy Temperature Detection
Fig. 2. Description of the Fuzzy Valve Operation
The Epimenides Paradox
In its most simplified form, the Epimenides paradox can be expressed as follows: «This sentence is false.» Expressed in algebraic terms, it would look like this:
A = NOT(A).
Thus, if we limit ourselves to the concepts of Boolean logic, this equation takes a paradoxical form:
0 = 1.
However, using fuzzy logic, this equation can be solved as follows:
A = NOT(A)
A = 1— A
2A = 1
A = 0.5.
Thus, the sentence from the Epimenides paradox turns out to be exactly 50% true and 50% false.
Common Misconceptions About Fuzzy Logic
Fuzzy logic is imprecise: fuzzy logic is no more imprecise than standard arithmetic at its core. In fact, it is much more precise when dealing with imprecise information.
Fuzzy logic is based on probabilistic reasoning.Probability deals with the chances of certain events occurring, while fuzzy logic deals with the events themselves. Fuzzy logic typically deals with ambiguity,
not uncertainty.
Fuzzy logic is built on a number of heuristic assumptions.
Although the intuitive nature of fuzzy logic may make it seem at first glance that the rules underlying it are arbitrary or based only on common sense, in fact these rules have been rigorously proven to be correct.
Lotfi Zadeh, the Father of Fuzzy Logic
Born in 1921, Dr. Lotfi Zadeh is considered the founding father of the use of fuzzy logic. After graduating from the University of Tehran with a degree in electrical engineering in 1942, he went to the United States, where he studied at the Massachusetts Institute of Technology (1946) and Columbia University (1949), where he later taught systems theory.
His seminal paper on fuzzy logic was published in 1965 and was not well received by some quarters of the academic community. Even now, forty years later, there is still some controversy in the field, but fuzzy logic methods have become one of the tools used by engineers in designing instrumentation and control systems.
Although Dr. Lotfi Zadeh is best known for developing the principles of fuzzy logic, he was also a pioneer in the development of the Z-transform for discrete signal processing and in systems analysis, which should be familiar to anyone interested in using microcontrollers to process digital signals and implement digital control functions.
[1] Two other laws of Aristotelian logic are also refuted by fuzzy logic: the law of identity, which states that «if a statement is true, then it is true»; and the law of contradiction: «a statement cannot be both true and false.»
[2] An algorithm based on fuzzy logic that typically outperforms PID control is the Mamdani controller:
http://esru.strath.ac.uk/Reference/concepts/fuzzy/fuzzy.htm
[3] A more complete list can be found at the following address:
http://esru.strath.ac.uk/Reference/concepts/fuzzy/fuzzy_appl.10.htm
Enver Bashi
Senior Research Engineer, Computrols, Inc.
Based on materials from the AutomatedBuildings website |
