Fuzzy control system

1855:+---------+ +-------+ analog --4-->| analog | | mux / +--4--> analog in | mux | | SH | out +----+----+ +-------+ | ^ V | +-------------+ +--+--+ | ADC / latch | | DAC | +------+------+ +-----+ | ^ | | 8 +-----------------------------+ | | | | V | | +-----------+ +-------------+ | +-->| fuzzifier | | defuzzifier +--+ +-----+-----+ +-------------+ | ^ | +-------------+ | | | rule | | +->| processor +--+ | (50 rules) | +------+------+ | +------+------+ | parameter | | memory | | 256 x 8 | +-------------+ ADC: analog-to-digital converter DAC: digital-to-analog converter SH: sample/hold 610:| XXXXXXX XXXXXXX : : : | -0.5 | XXX XXXXXXXXXX XXX : : | -0.25 | : XXXXXXX XXXXXXX : : | 0.0 | : XXX XXXXXXXXXX XXX : | 0.25 | : : XXXXXXX XXXXXXX : | 0.5 | : : XXX XXXXXXXXXX XXX | 0.75 | : : : XXXXXXX XXXXXXX | 1.0 | : : : XXX XXXXXXXXXX | | | +------------------------------------------------------------------+ 455:

the greater the truth value of "cold", the higher the truth value of "high", though this does not necessarily mean that the output itself will be set to "high" since this is only one rule among many. In some cases, the membership functions can be modified by "hedges" that are equivalent to adverbs. Common hedges include "about", "near", "close to", "approximately", "very", "slightly", "too", "extremely", and "somewhat". These operations may have precise definitions, though the definitions can vary considerably between different implementations. "Very", for one example, squares membership functions; since the membership values are always less than 1, this narrows the membership function. "Extremely" cubes the values to give greater narrowing, while "somewhat" broadens the function by taking the square root.

349:, for temperature measurement; a conductivity sensor, to measure detergent level from the ions present in the wash; a turbidity sensor that measures scattered and transmitted light to measure the soiling of the wash; and a magnetostrictive sensor to read spin rate. The system determines the optimum wash cycle for any load to obtain the best results with the least amount of energy, detergent, and water. It even adjusts for dried-on foods by tracking the last time the door was opened, and estimates the number of dishes by the number of times the door was opened. 43: 264:(CCD) to measure the clarity of the image in six regions of its field of view and use the information provided to determine if the image is in focus. It also tracks the rate of change of lens movement during focusing, and controls its speed to prevent overshoot. The camera's fuzzy control system uses 12 inputs: 6 to obtain the current clarity data provided by the CCD and 6 to measure the rate of change of lens movement. The output is the position of the lens. The fuzzy control system uses 13 rules and requires 1.1 kilobytes of memory. 574:* * | * * * * | * 222222222 | * 22222222222 | 333333332222222222222 +---33333333222222222222222--> ^ +150 __________________________________________________________________ 242:" experiment. This is a classic control problem, in which a vehicle tries to keep a pole mounted on its top by a hinge upright by moving back and forth. Yamakawa subsequently made the demonstration more sophisticated by mounting a wine glass containing water and even a live mouse to the top of the pendulum: the system maintained stability in both cases. Yamakawa eventually went on to organize his own fuzzy-systems research lab to help exploit his patents in the field. 606:

0 0 0 0 0.3 0.7 1 0.7 0.3 mu(ZE) 0 0 0.3 0.7 1 0.7 0.3 0 0 mu(SN) 0.3 0.7 1 0.7 0.3 0 0 0 0 mu(LN) 1 0.7 0.3 0 0 0 0 0 0 _______________________________________________________________________ —or, in graphical form (where each "X" has a value of 0.1):

105: 553:

used as the combined truth value; if an OR is specified, the maximum is used. The appropriate output state is selected and assigned a membership value at the truth level of the premise. The truth values are then defuzzified. For example, assume the temperature is in the "cool" state, and the pressure is in the "low" and "ok" states. The pressure values ensure that only rules 2 and 3 fire:

1851:

This system can be implemented on a standard microprocessor, but dedicated fuzzy chips are now available. For example, Adaptive Logic INC of San Jose, California, sells a "fuzzy chip", the AL220, that can accept four analog inputs and generate four analog outputs. A block diagram of the chip is shown

605:

_______________________________________________________________________ -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 _______________________________________________________________________ mu(LP) 0 0 0 0 0 0 0.3 0.7 1 mu(SP)

573:

__________________________________________________________________ | Z P2 1 -+ * * | * * * * | * *

458:

In practice, the fuzzy rule sets usually have several antecedents that are combined using fuzzy operators, such as AND, OR, and NOT, though again the definitions tend to vary: AND, in one popular definition, simply uses the minimum weight of all the antecedents, while OR uses the maximum value. There

436:

The most common shape of membership functions is triangular, although trapezoidal and bell curves are also used, but the shape is generally less important than the number of curves and their placement. From three to seven curves are generally appropriate to cover the required range of an input value,

609:

LN SN ZE SP LP +------------------------------------------------------------------+ | | -1.0 | XXXXXXXXXX XXX : : : | -0.75

454:

This rule uses the truth value of the "temperature" input, which is some truth value of "cold", to generate a result in the fuzzy set for the "heater" output, which is some value of "high". This result is used with the results of other rules to finally generate the crisp composite output. Obviously,

1847:

In our example, the values of mu correspond to the masses, and the values of X to location of the masses (mu, however, only 'corresponds to the masses' if the initial 'mass' of the output functions are all the same/equivalent. If they are not the same, i.e. some are narrow triangles, while others

601:

If the error ranges from -1 to +1, with the analog-to-digital converter used having a resolution of 0.25, then the input variable's fuzzy set (which, in this case, also applies to the output variable) can be described very simply as a table, with the error / delta / output values in the top row and

432:

Fuzzy controllers are very simple conceptually. They consist of an input stage, a processing stage, and an output stage. The input stage maps sensor or other inputs, such as switches, thumbwheels, and so on, to the appropriate membership functions and truth values. The processing stage invokes each

245:

Japanese engineers subsequently developed a wide range of fuzzy systems for both industrial and consumer applications. In 1988 Japan established the Laboratory for International Fuzzy Engineering (LIFE), a cooperative arrangement between 48 companies to pursue fuzzy research. The automotive company

2041:

Fuzzy control systems are suitable when the process complexity is high including uncertainty and nonlinear behavior, and there are no precise mathematical models available. Successful applications of fuzzy control systems have been reported worldwide mainly in Japan with pioneering solutions since

1903:

With this scheme, the input variable's state no longer jumps abruptly from one state to the next. Instead, as the temperature changes, it loses value in one membership function while gaining value in the next. In other words, its ranking in the category of cold decreases as it becomes more highly

1882:

An arbitrary static threshold might be set to divide "warm" from "hot". For example, at exactly 90 degrees, warm ends and hot begins. But this would result in a discontinuous change when the input value passed over that threshold. The transition wouldn't be smooth, as would be required in braking

552:

In practice, the controller accepts the inputs and maps them into their membership functions and truth values. These mappings are then fed into the rules. If the rule specifies an AND relationship between the mappings of the two input variables, as the examples above do, the minimum of the two is

390:

The input variables in a fuzzy control system are in general mapped by sets of membership functions similar to this, known as "fuzzy sets". The process of converting a crisp input value to a fuzzy value is called "fuzzification". The fuzzy logic based approach had been considered by designing two

190:

can perform just as well as fuzzy logic in many cases, fuzzy logic has the advantage that the solution to the problem can be cast in terms that human operators can understand, so that their experience can be used in the design of the controller. This makes it easier to mechanize tasks that are

653:

output(1): Value (for rule 1) where the result membership function (ZE) is maximum over the output variable fuzzy set range. That is, in terms of a centroid calculation, the location of the "center of mass" for this individual result. This value is independent of the value of "mu". It simply

473:

The "centroid" method is very popular, in which the "center of mass" of the result provides the crisp value. Another approach is the "height" method, which takes the value of the biggest contributor. The centroid method favors the rule with the output of greatest area, while the height method

488:

Notice how each rule provides a result as a truth value of a particular membership function for the output variable. In centroid defuzzification the values are OR'd, that is, the maximum value is used and values are not added, and the results are then combined using a centroid calculation.

286:

robot helicopters (hovering is a "balancing act" rather similar to the inverted pendulum problem); rehabilitation robotics to provide patient-specific solutions (e.g. to control heart rate and blood pressure ); control of flow of powders in film manufacture; elevator systems; and so

953: 423:

In this example, the two input variables are "brake temperature" and "speed" that have values defined as fuzzy sets. The output variable, "brake pressure" is also defined by a fuzzy set that can have values like "static" or "slightly increased" or "slightly decreased" etc.

620:

These rules are typical for control applications in that the antecedents consist of the logical combination of the error and error-delta signals, while the consequent is a control command output. The rule outputs can be defuzzified using a discrete centroid computation:

444:

As discussed earlier, the processing stage is based on a collection of logic rules in the form of IF-THEN statements, where the IF part is called the "antecedent" and the THEN part is called the "consequent". Typical fuzzy control systems have dozens of rules.

1843: 1522: 1683: 207:

in a 1965 paper. He elaborated on his ideas in a 1973 paper that introduced the concept of "linguistic variables", which in this article equates to a variable defined as a fuzzy set. Other research followed, with the first industrial application, a cement

181:

Fuzzy logic is widely used in machine control. The term "fuzzy" refers to the fact that the logic involved can deal with concepts that cannot be expressed as the "true" or "false" but rather as "partially true". Although alternative approaches such as

469:

Rules can be solved in parallel in hardware, or sequentially in software. The results of all the rules that have fired are "defuzzified" to a crisp value by one of several methods. There are dozens, in theory, each with various advantages or drawbacks.

398:, or "ON-OFF", inputs along with its analog inputs, and such switch inputs of course will always have a truth value equal to either 1 or 0, but the scheme can deal with them as simplified fuzzy functions that happen to be either one value or another. 1898: 1349: 1076: 275:, a compressor valve, and a fan motor. Compared to the previous design, the fuzzy controller heats and cools five times faster, reduces power consumption by 24%, increases temperature stability by a factor of two, and uses fewer sensors. 528: 483: 477:

The diagram below demonstrates max-min inferencing and centroid defuzzification for a system with input variables "x", "y", and "z" and an output variable "n". Note that "mu" is standard fuzzy-logic nomenclature for "truth value":

1973:

the fuzzy function arising of an IF-THEN systems of rules. Then this system can be translated into a fuzzy program P containing a series of rules whose head is "Good(x,y)". The interpretation of this predicate in the least fuzzy

636:

________________________ e delta ________________________ mu(LP) 0 0.3 mu(SP) 0.7 1 mu(ZE) 0.7 0.3 mu(SN) 0 0 mu(LN) 0 0 ________________________

2021:

isn't able to determine the correct follow up state, but the system will only guess what will happen if the action was taken. The Fuzzy qualitative simulation can't predict the exact numerical values, but it's using imprecise

1907:

At any sampled timeframe, the "truth value" of the brake temperature will almost always be in some degree part of two membership functions: i.e.: '0.6 nominal and 0.4 warm', or '0.7 nominal and 0.3 cool', and so on.

355:

has developed the first auto-tuner for the fuzzy logic controller's knowledge base known as edeX. This technology was tested by Mohawk College and was able to solve non-linear 2x2 and 3x3 multi-input multi-output

616:

rule 1: IF e = ZE AND delta = ZE THEN output = ZE rule 2: IF e = ZE AND delta = SP THEN output = SN rule 3: IF e = SN AND delta = SN THEN output = LP rule 4: IF e = LP OR delta = LP THEN output = LN

1106:

If you have problems figuring out the centroid equation, remember that a centroid is defined by summing all the moments (location times mass) around the center of gravity and equating the sum to zero. So if

675: 590: 1878:

The variable "temperature" in this system can be subdivided into a range of "states": "cold", "cool", "moderate", "warm", "hot", "very hot". The transition from one state to the next is hard to define.

1848:

maybe wide trapezoids or shouldered triangles, then the mass or area of the output function must be known or calculated. It is this mass that is then scaled by mu and multiplied by its location X_i).

565: 558: 376:" software systems, with the ultimate goal of building "self-learning" fuzzy-control systems. These systems can be employed to control complex, nonlinear dynamic plants, for example, human body. 2033:

notation, for example . After converting the set notation back into numerical values the accuracy get worse. This makes Fuzzy qualitative simulation a bad choice for practical applications.

253:

accordingly. Hitachi washing machines use fuzzy controllers to load-weight, fabric-mix, and dirt sensors and automatically set the wash cycle for the best use of power, water, and detergent.

433:

appropriate rule and generates a result for each, then combines the results of the rules. Finally, the output stage converts the combined result back into a specific control output value.

1687: 1353: 1526: 2725:"Intelligent Modeling and Control of Washing Machine Using Locally Linear Neuro-Fuzzy (LLNF) Modeling and Modified Brain Emotional Learning Based Intelligent Controller (Belbic)" 1994:

which accepts an action from a player and calculates, if the action is valid. After the action was executed, the game is in follow up state. If the aim isn't only to play

1193: 249:

Japanese consumer goods often incorporate fuzzy systems. Matsushita vacuum cleaners use microcontrollers running fuzzy algorithms to interrogate dust sensors and adjust

1890:. That is, allow them to change gradually from one state to the next. In order to do this, there must be a dynamic relationship established between different factors. 2636: 957: 352: 1966:. A rigorous logical justification of fuzzy control is given in Hájek's book (see Chapter 7) where fuzzy control is represented as a theory of Hájek's basic logic. 1186: 1159: 1132: 1101: 2291:

Sarabadani Tafreshi, Amirehsan; Klamroth-Marganska, V.; Nussbaumer, S.; Riener, R. (2015). "Real-time closed-loop control of human heart rate and blood pressure".

2018: 2334: 1998:

but determine the actions for real world applications, the most obvious bottleneck is, that no game rules are available. The first step is to model the domain.

650:

mu(1): Truth value of the result membership function for rule 1. In terms of a centroid calculation, this is the "mass" of this result for this discrete case.

534:

N3: Large negative. N2: Medium negative. N1: Small negative. Z: Zero. P1: Small positive. P2: Medium positive. P3: Large positive.

2958:"Multi objective optimization of performance parameters of a single cylinder diesel engine running with hydrogen using a Taguchi-fuzzy based approach" 2026:

to speculate about the future. It takes the current situation plus the actions from the past and generates the expected follow up state of the game.

519:

As a general example, consider the design of a fuzzy controller for a steam turbine. The block diagram of this control system appears as follows:

3432: 2402: 1990:

is needed. For video games, the model is equal to the game rules. From the programming perspective, the game rules are implemented as a

595:

A fuzzy set is defined for the input error variable "e", and the derived change in error, "delta", as well as the "output", as follows:

307:

has studied fuzzy control for automated space docking: simulations show that a fuzzy control system can greatly reduce fuel consumption.

334:

have worked on fuzzy logic for use in low-power refrigerators, improved automotive transmissions, and energy-efficient electric motors.

271:

designed by Mitsubishi uses 25 heating rules and 25 cooling rules. A temperature sensor provides input, with control outputs fed to an

948:{\displaystyle {\frac {mu(1)\cdot output(1)+mu(2)\cdot output(2)+mu(3)\cdot output(3)+mu(4)\cdot output(4)}{mu(1)+mu(2)+mu(3)+mu(4)}}} 1897: 2435: 2989:"Application of fuzzy logic and data mining techniques as tools for qualitative interpretation of acid mine drainage processes" 1103:—for the final control output. Simple. Of course the hard part is figuring out what rules actually work correctly in practice. 2014: 3349: 2010: 238:

In 1987, Takeshi Yamakawa demonstrated the use of fuzzy control, through a set of simple dedicated fuzzy logic chips, in an "

2603: 3449: 204: 3330: 2583:. European Conference on Symbolic and Quantitative Approaches to Reasoning and Uncertainty. Springer. pp. 600–612. 291:

Work on fuzzy systems is also proceeding in North America and Europe, although on a less extensive scale than in Japan.

296: 3372:

Jan Jantzen, "Tuning Of Fuzzy PID Controllers", Technical University of Denmark, report 98-H 871, September 30, 1998.

3395: 3373: 1838:{\displaystyle X_{0}={\frac {X_{1}\cdot M_{1}+X_{2}\cdot M_{2}+\ldots +X_{n}\cdot M_{n}}{M_{1}+M_{2}+\ldots +M_{n}}}} 1517:{\displaystyle 0=(X_{1}\cdot M_{1}+X_{2}\cdot M_{2}+\ldots +X_{n}\cdot M_{n})-X_{0}\cdot (M_{1}+M_{2}+\ldots +M_{n})} 527: 482: 86: 64: 57: 577:

The output value will adjust the throttle and then the control cycle will begin again to generate the next value.

1969:

In Gerla 2005 another logical approach to fuzzy control is proposed based on fuzzy logic programming: Denote by

1678:{\displaystyle X_{0}\cdot (M_{1}+M_{2}+\ldots +M_{n})=X_{1}\cdot M_{1}+X_{2}\cdot M_{2}+\ldots +X_{n}\cdot M_{n}} 661:

rule 2: IF e = ZE AND delta = SP THEN output = SN mu(2) = MIN( 0.7, 1 ) = 0.7 output(2) = -0.5

246:

Volkswagen was the only foreign corporate member of LIFE, dispatching a researcher for a duration of three years.

462:

There are several ways to define the result of a rule, but one of the most common and simplest is the "max-min"

231:. Their ideas were adopted, and fuzzy systems were used to control accelerating, braking, and stopping when the 2129: 667:

rule 4: IF e = LP OR delta = LP THEN output = LN mu(4) = MAX( 0.0, 0.3 ) = 0.3 output(4) = -1

643:

rule 1: IF e = ZE AND delta = ZE THEN output = ZE mu(1) = MIN( 0.7, 0.3 ) = 0.3 output(1) = 0

17: 3484: 1934:

In spite of the appearance there are several difficulties to give a rigorous logical interpretation of the

1915:

of values from multiple values. This only represents one kind of data, however, in this case, temperature.

664:

rule 3: IF e = SN AND delta = SN THEN output = LP mu(3) = MIN( 0.0, 0.0 ) = 0 output(3) = 1

2197: 2089: 459:

is also a NOT operator that subtracts a membership function from 1 to give the "complementary" function.

3241:

Godoy, Jorge; Pérez, Joshué; Onieva, Enrique; Villagrá, Jorge; Milanés, Vicente; Haber, Rodolfo (2015).

232: 2251: 420:

IF brake temperature IS warm AND speed IS not very fast THEN brake pressure IS slightly decreased.

227:, who in 1985 provided simulations that demonstrated the feasibility of fuzzy control systems for the 1918:

Adding additional sophistication to this braking system, could be done by additional factors such as

369: 187: 3326:

Kevin M. Passino and Stephen Yurkovich, Fuzzy Control, Addison Wesley Longman, Menlo Park, CA, 1998

2523: 1344:{\displaystyle 0=(X_{1}-X_{0})\cdot M_{1}+(X_{2}-X_{0})\cdot M_{2}+\ldots +(X_{n}-X_{0})\cdot M_{n}} 3459: 3429: 1864: 466:

method, in which the output membership function is given the truth value generated by the premise.

272: 51: 2351:

Mamdani, Ebrahim H (1974). "Application of fuzzy algorithms for control of simple dynamic plant".

598:

LP: large positive SP: small positive ZE: zero SN: small negative LN: large negative

3293: 3045: 2957: 2918: 2887: 2856: 2794: 2654: 2379: 279: 3242: 3154: 3084: 2763: 2421:"Development of Vision Based Path Tracking Algorithm with Kinematic Motion and Fuzzy Controller" 2518: 2124: 2099: 1919: 1867:, directed by a microcontroller chip. The microcontroller has to make decisions based on brake 1071:{\displaystyle ={\frac {(0.3\cdot 0)+(0.7\cdot -0.5)+(0\cdot 1)+(0.3\cdot -1)}{0.3+0.7+0+0.3}}} 68: 492:

Fuzzy control system design is based on empirical methods, basically a methodical approach to

122: 2109: 1999: 438: 360:

Research and development is also continuing on fuzzy applications in software, as opposed to

331: 261: 115: 3382: 3479: 3464: 2930: 2473: 1986:

Before an Artificial Intelligence system is able to plan the action sequence, some kind of

1164: 1137: 1110: 1080: 589: 300: 31: 3115: 2724: 2228: 8: 178:

logic, which operates on discrete values of either 1 or 0 (true or false, respectively).

2934: 2477: 282:; optical fuzzy systems; robots, including one for making Japanese flower arrangements; 3223: 3065: 3026: 2744: 2740: 2705: 2628: 2536: 2491: 2316: 2094: 1987: 549:

rule 4: IF temperature IS cool AND pressure IS strong, THEN throttle is N2.

543:

rule 2: IF temperature IS cool AND pressure IS low, THEN throttle is P2.

540:

rule 1: IF temperature IS cool AND pressure IS weak, THEN throttle is P3.

171: 2987:

Aroba, J.; Grande, J. A.; Andújar, J. M.; de la Torre, M. L.; Riquelme, J. C. (2007).

2826:"Toward intelligent machining: hierarchical fuzzy control for the end milling process" 2810: 2670: 2398: 2267: 2211: 2192: 546:

rule 3: IF temperature IS cool AND pressure IS ok, THEN throttle is Z.

174:

variables that take on continuous values between 0 and 1, in contrast to classical or

3391: 3345: 3274: 3215: 3170: 3135: 3100: 3018: 2988: 2779: 2748: 2495: 2420: 2308: 2290: 2216: 1995: 402: 373: 327: 239: 183: 3409: 3227: 3030: 2709: 2540: 2320: 564: 557: 3305: 3264: 3254: 3205: 3197: 3166: 3131: 3127: 3096: 3069: 3057: 3008: 3000: 2969: 2938: 2899: 2868: 2837: 2806: 2793:

Haber, R.E.; Alique, J.R.; Alique, A.; Hernández, J.; Uribe-Etxebarria, R. (2003).

2775: 2736: 2697: 2666: 2618: 2584: 2581:

Generating fuzzy models from deep knowledge: robustness and interpretability issues

2559: 2528: 2481: 2394: 2360: 2300: 2263: 2232: 2224: 2206: 2023: 3441: 3259: 2919:"Arabic handwriting recognition using structural and syntactic pattern attributes" 2903: 2872: 2824:

Haber, R.E.; Peres, C.R.; Alique, A.; Ros, S.; Gonzalez, C.; Alique, J.R. (1998).

3453: 3436: 3334: 3114:

Kelly, Rafael; Haber, Rodolfo; Haber-Guerra, Rodolfo E.; Reyes, Fernando (1999).

2973: 2942: 2632: 493: 414: 410: 268: 130: 3046:"Expert system methodologies and applications—a decade review from 1995 to 2004" 585:

Consider implementing with a microcontroller chip a simple feedback controller:

3446: 3309: 3185: 3061: 2623: 2188: 2029:

The output of the ANFIS system isn't providing correct information, but only a

1991: 1975: 391:

fuzzy systems, one for error heading angle and the other for velocity control.

315: 200: 155: 126: 3342:

Fuzzy control systems design and analysis: a linear matrix inequality approach

3116:"Lyapunov Stable Control of Robot Manipulators: A Fuzzy Self-Tuning Procedure" 3004: 2956:

Bose, Probir Kumar; Deb, Madhujit; Banerjee, Rahul; Majumder, Arindam (2013).

2825: 2685: 2532: 2486: 2461: 2304: 3473: 3327: 3278: 3219: 3201: 3139: 3022: 2220: 2104: 1893:

Start by defining the input temperature states using "membership functions":

365: 319: 283: 250: 228: 223:

Interest in fuzzy systems was sparked by Seiji Yasunobu and Soji Miyamoto of

167: 2855:

Ramı́rez, Mercedes; Haber, Rodolfo; Peña, Vı́ctor; Rodrı́guez, Iván (2004).

2170: 345:

based on a fuzzy controller and a "one-stop sensing module" that combines a

3414: 3243:"A Driverless Vehicle Demonstration on Motorways and in Urban Environments" 2601: 2364: 2312: 175: 3184:

Naranjo, J.E.; Gonzalez, C.; Garcia, R.; dePedro, T.; Haber, R.E. (2005).

2236: 30:"Fuzzy control" and "Fuzzy Control" redirect here. For the rock band, see 2119: 2013:

systems (Adaptive network based fuzzy inference system) for creating the

1978:

of P coincides with f. This gives further useful tools to fuzzy control.

1912: 1868: 406: 163: 159: 3390:

by Kruse, Borgelt, Klawonn, Moewes, Steinbrecher, Held, 2013, Springer,

3269: 3013: 2588: 2556:

Fuzzy qualitative simulation and diagnosis of continuous dynamic systems

2509:

Gerla, Giangiacomo (2005). "Fuzzy logic programming and fuzzy control".

624:

SUM( I = 1 TO 4 OF ( mu(I) * output(I) ) ) / SUM( I = 1 TO 4 OF mu(I) )

602:

the truth values for each membership function arranged in rows beneath:

570:

The two outputs are then defuzzified through centroid defuzzification:

500:

Document the system's operational specifications and inputs and outputs.

2114: 2003: 1926:, set up in dynamic functions, according to the designed fuzzy system. 346: 342: 3210: 2841: 2701: 512:

Run through test suite to validate system, adjust details as required.

417:

for what action to take, based on a set of "rules", each of the form:

278:

Other applications investigated or implemented include: character and

2564: 2030: 463: 385: 257: 2578: 361: 323: 3407: 2462:"Realization of a fuzzy controller with fuzzy dynamic correction" 1923: 523:

The input and output variables map into the following fuzzy set:

224: 2655:"Fuzzy predictive control applied to an air-conditioning system" 3460:

Online interactive demonstration of a system with 3 fuzzy rules

1911:

The above example demonstrates a simple application, using the

395: 338: 311: 2335:"Artificial Intelligence Controllers for Industrial Processes" 2986: 1872: 216: 3183: 2002:

can be realized with precise mathematical equations or with

3186:"Power-Steering Control Architecture for Automatic Driving" 3152: 3113: 2854: 2058:

Control and optimization of industrial processes and system

304: 209: 3442:

Fuzzy Logic in Embedded Microcomputers and Control Systems

3294:"Intelligent rudder control of an unmanned surface vessel" 2792: 2428:

United International Journal for Research & Technology

1929: 474:

obviously favors the rule with the greatest output value.

3388:

Computational Intelligence: A Methodological Introduction

2723:

Lucas, Caro; Milasi, Rasoul M.; Araabi, Babak N. (2008).

3240: 2602:

Liu, Honghai; Coghill, George M; Barnes, Dave P (2009).

2380:"Identifying fuzzy models utilizing genetic programming" 2055:

Domestic appliances (refrigerators, washing machines...)

451:

IF (temperature is "cold") THEN turn (heater is "high")

3339: 3190:

IEEE Transactions on Intelligent Transportation Systems

2955: 2795:"Embedded fuzzy-control system for machining processes" 613:

Suppose this fuzzy system has the following rule base:

3153:

Ollero, A.; García-Cerezo, A.; Martínez, J.L. (1994).

2888:"A survey on industrial applications of fuzzy control" 2652: 2353:

Proceedings of the Institution of Electrical Engineers

1690: 1529: 1356: 1196: 1167: 1140: 1113: 1083: 960: 678: 654:

identifies the location of ZE along the output range.

2916: 2177:(4 ed.). Springer Science & Business Media. 3155:"Fuzzy supervisory path tracking of mobile reports" 2917:Tanvir Parvez, Mohammad; Mahmoud, Sabri A. (2013). 2823: 2653:Sousa, J.M.; Babuška, R.; Verbruggen, H.B. (1997). 405:" of input variables into membership functions and 2045:Some applications reported in the literature are: 1940:IF (temperature is "cold") THEN (heater is "high") 1837: 1677: 1516: 1343: 1180: 1153: 1126: 1095: 1070: 947: 2885: 2722: 3471: 3408:Robert Babuska and Ebrahim Mamdani, ed. (2008). 3292:Larrazabal, J. Menoyo; Peñas, M. Santos (2016). 2252:"Industrial applications of fuzzy logic control" 394:A control system may also have various types of 3291: 2830:IEEE Transactions on Control Systems Technology 2579:Guglielmann, Raffaella; Ironi, Liliana (2005). 1981: 3043: 2611:International Journal of Approximate Reasoning 580: 2886:Precup, Radu-Emil; Hellendoorn, Hans (2011). 2683: 2857:"Fuzzy control of a multiple hearth furnace" 2286: 2284: 2256:International Journal of Man-Machine Studies 515:Complete document and release to production. 215:Fuzzy systems were initially implemented in 3120:Intelligent Automation & Soft Computing 2553: 2459: 2293:IEEE Transactions on Biomedical Engineering 1938:rules. As an example, interpret a rule as 194: 1886:The way around this is to make the states 427: 212:built in Denmark, coming on line in 1976. 191:already successfully performed by humans. 3268: 3258: 3209: 3012: 2764:"An introductory survey of fuzzy control" 2690:IEEE Transactions on Consumer Electronics 2622: 2563: 2522: 2485: 2346: 2344: 2281: 2210: 2156:(2 ed.). Research Studies Press Ltd. 2147: 2145: 87:Learn how and when to remove this message 3362:. IEEE Spectrum, 29:10. pp. 58–61. 3082: 1946:and assume that r is an input such that 50:This article includes a list of general 2686:"Video camera system using fuzzy logic" 2558:(PhD thesis). University of Edinburgh. 2466:Central European Journal of Engineering 2460:Vichuzhanin, Vladimir (12 April 2012). 2418: 2377: 2350: 2165: 2163: 2151: 2017:for a domain has many disadvantages. A 1930:Logical interpretation of fuzzy control 503:Document the fuzzy sets for the inputs. 14: 3472: 3083:Yuan, Yufei; Shaw, Michael J. (1995). 2761: 2341: 2249: 2142: 3369:. IEEE Spectrum, 30:2. pp. 7–31. 2508: 2187: 2169: 1875:, and other variables in the system. 537:The rule set includes such rules as: 509:Determine the defuzzification method. 496:. The general process is as follows: 2160: 627:Now, suppose that at a given time: 368:and integration of fuzzy logic with 205:University of California at Berkeley 98: 36: 3085:"Induction of fuzzy decision trees" 299:has investigated fuzzy control for 27:Method to analyze non-binary inputs 24: 3340:Kazuo Tanaka; Hua O. Wang (2001). 3320: 2741:10.1111/j.1934-6093.2006.tb00290.x 2052:Automatic focus systems in cameras 1858: 1161:is the location of each mass, and 448:Consider a rule for a thermostat: 297:US Environmental Protection Agency 56:it lacks sufficient corresponding 25: 3496: 3401: 2250:Jensen, P. MARTIN (23 May 1979). 670:The centroid computation yields: 640:Plugging this into rule 1 gives: 3298:Expert Systems with Applications 3050:Expert Systems with Applications 2684:Haruki, T.; Kikuchi, K. (1992). 2642:from the original on 2020-05-06. 2604:"Fuzzy qualitative trigonometry" 2408:from the original on 2007-06-12. 1896: 588: 563: 556: 526: 481: 103: 41: 3285: 3234: 3177: 3146: 3107: 3076: 3037: 2980: 2949: 2910: 2879: 2848: 2817: 2786: 2755: 2716: 2677: 2646: 2595: 2572: 2547: 2502: 2453: 2441:from the original on 2021-09-18 2154:Fuzzy control and fuzzy systems 2036: 1904:ranked in the warmer category. 3132:10.1080/10798587.1999.10750611 2554:Shen, Qiang (September 1991). 2412: 2371: 2327: 2243: 2181: 2175:Metamathematics of fuzzy logic 1962:gives a correct control given 1588: 1543: 1511: 1466: 1447: 1363: 1325: 1299: 1274: 1248: 1229: 1203: 1039: 1024: 1018: 1006: 1000: 985: 979: 967: 939: 933: 921: 915: 903: 897: 885: 879: 868: 862: 838: 832: 820: 814: 790: 784: 772: 766: 742: 736: 724: 718: 694: 688: 13: 1: 3430:Introduction to Fuzzy Control 3260:10.3846/16484142.2014.1003406 2904:10.1016/j.compind.2010.10.001 2873:10.1016/j.compind.2003.05.001 2811:10.1016/S0166-3615(03)00022-8 2671:10.1016/S0967-0661(97)00136-6 2399:10.1016/S0165-0114(98)00086-4 2268:10.1016/S0020-7373(80)80050-2 2212:10.1016/S0019-9958(65)90241-X 2135: 2130:Type-2 fuzzy sets and systems 379: 129:in tone and meet Knowledge's 3379:Foundations of Fuzzy Control 3171:10.1016/0967-0661(94)90213-5 3159:Control Engineering Practice 3101:10.1016/0165-0114(94)00229-Z 2974:10.1016/j.energy.2013.10.045 2943:10.1016/j.patcog.2012.07.012 2780:10.1016/0020-0255(85)90026-X 2659:Control Engineering Practice 2419:Nwe Mee, Kyaw (March 2021). 1982:Fuzzy qualitative simulation 341:introduced an "intelligent" 199:Fuzzy logic was proposed by 7: 2083: 1950:is false. Then the formula 1942:by the first order formula 1863:As an example, consider an 630:e = 0.25 delta = 0.5 581:Building a fuzzy controller 10: 3501: 3381:. Wiley, 2007 (209 pages) 3310:10.1016/j.eswa.2016.01.057 3062:10.1016/j.eswa.2004.08.003 2624:10.1016/j.ijar.2009.07.003 2064:Fuel efficiency in engines 1188:is each mass, this gives: 1134:is the center of gravity, 383: 364:, design, including fuzzy 170:input values in terms of 29: 3465:Data driven fuzzy systems 3005:10.1007/s00254-006-0627-0 2533:10.1007/s11225-005-2977-0 2487:10.2478/s13531-012-0003-7 2378:Bastian, Andreas (2000). 2305:10.1109/TBME.2015.2391234 2205:(3). San Diego: 338–353. 3202:10.1109/TITS.2005.858622 2729:Asian Journal of Control 2152:Pedrycz, Witold (1993). 1865:anti-lock braking system 372:and so-called adaptive " 195:History and applications 3344:. John Wiley and Sons. 3044:Shu-Hsien Liao (2005). 2762:Sugeno, Michio (1985). 2198:Information and Control 428:Fuzzy control in detail 353:Xiera Technologies Inc. 280:handwriting recognition 71:more precise citations. 3367:Adaptive fuzzy systems 3089:Fuzzy Sets and Systems 2617:(1). Elsevier: 71–88. 2387:Fuzzy Sets and Systems 2365:10.1049/piee.1974.0328 2125:Fuzzy control language 2100:Function approximation 2019:qualitative simulation 2009:Using Fuzzy logic and 1839: 1679: 1518: 1345: 1182: 1155: 1128: 1097: 1072: 949: 658:The other rules give: 506:Document the rule set. 3358:Cox, E. (Oct. 1992). 2993:Environmental Geology 2892:Computers in Industry 2861:Computers in Industry 2799:Computers in Industry 2110:Fuzzy markup language 2000:System identification 1840: 1680: 1519: 1346: 1183: 1181:{\displaystyle M_{i}} 1156: 1154:{\displaystyle X_{i}} 1129: 1127:{\displaystyle X_{0}} 1098: 1096:{\displaystyle =-0.5} 1073: 950: 439:universe of discourse 262:charge-coupled device 166:system that analyzes 3365:Cox, E. (Feb. 1993) 2768:Information Sciences 1688: 1527: 1354: 1194: 1165: 1138: 1111: 1081: 958: 676: 152:fuzzy control system 123:improve this article 32:Fuzzy Control (band) 3485:Control engineering 3383:(Table of contents) 2935:2013PatRe..46..141T 2923:Pattern Recognition 2589:10.1007/11518655_51 2478:2012CEJE....2..392V 2079:Autonomous vehicles 441:" in fuzzy jargon. 260:camera that uses a 256:Canon developed an 3456:IEC 1131-7 CD1 PDF 3452:2021-03-04 at the 3435:2010-08-05 at the 3360:Fuzzy fundamentals 3333:2008-12-15 at the 2095:Bayesian inference 1996:mathematical games 1958:and therefore any 1835: 1675: 1514: 1341: 1178: 1151: 1124: 1093: 1068: 945: 184:genetic algorithms 3351:978-0-471-32324-2 2842:10.1109/87.664186 2702:10.1109/30.156746 2665:(10): 1395–1406. 2359:(12): 1585–1588. 1833: 1066: 943: 633:Then this gives: 240:inverted pendulum 148: 147: 131:quality standards 97: 96: 89: 16:(Redirected from 3492: 3426: 3424: 3422: 3355: 3314: 3313: 3289: 3283: 3282: 3272: 3262: 3238: 3232: 3231: 3213: 3181: 3175: 3174: 3150: 3144: 3143: 3111: 3105: 3104: 3080: 3074: 3073: 3041: 3035: 3034: 3016: 2984: 2978: 2977: 2953: 2947: 2946: 2914: 2908: 2907: 2883: 2877: 2876: 2852: 2846: 2845: 2821: 2815: 2814: 2790: 2784: 2783: 2759: 2753: 2752: 2720: 2714: 2713: 2681: 2675: 2674: 2650: 2644: 2643: 2641: 2626: 2608: 2599: 2593: 2592: 2576: 2570: 2569: 2567: 2551: 2545: 2544: 2526: 2506: 2500: 2499: 2489: 2457: 2451: 2450: 2448: 2446: 2440: 2425: 2416: 2410: 2409: 2407: 2384: 2375: 2369: 2368: 2348: 2339: 2338: 2331: 2325: 2324: 2299:(5): 1434–1442. 2288: 2279: 2278: 2276: 2274: 2247: 2241: 2240: 2214: 2185: 2179: 2178: 2167: 2158: 2157: 2149: 2049:Air conditioners 2024:natural language 1954:is true for any 1900: 1844: 1842: 1841: 1836: 1834: 1832: 1831: 1830: 1812: 1811: 1799: 1798: 1788: 1787: 1786: 1774: 1773: 1755: 1754: 1742: 1741: 1729: 1728: 1716: 1715: 1705: 1700: 1699: 1684: 1682: 1681: 1676: 1674: 1673: 1661: 1660: 1642: 1641: 1629: 1628: 1616: 1615: 1603: 1602: 1587: 1586: 1568: 1567: 1555: 1554: 1539: 1538: 1523: 1521: 1520: 1515: 1510: 1509: 1491: 1490: 1478: 1477: 1462: 1461: 1446: 1445: 1433: 1432: 1414: 1413: 1401: 1400: 1388: 1387: 1375: 1374: 1350: 1348: 1347: 1342: 1340: 1339: 1324: 1323: 1311: 1310: 1289: 1288: 1273: 1272: 1260: 1259: 1244: 1243: 1228: 1227: 1215: 1214: 1187: 1185: 1184: 1179: 1177: 1176: 1160: 1158: 1157: 1152: 1150: 1149: 1133: 1131: 1130: 1125: 1123: 1122: 1102: 1100: 1099: 1094: 1077: 1075: 1074: 1069: 1067: 1065: 1042: 965: 954: 952: 951: 946: 944: 942: 871: 680: 592: 567: 560: 530: 485: 301:energy-efficient 284:voice-controlled 143: 140: 134: 107: 106: 99: 92: 85: 81: 78: 72: 67:this article by 58:inline citations 45: 44: 37: 21: 3500: 3499: 3495: 3494: 3493: 3491: 3490: 3489: 3470: 3469: 3454:Wayback Machine 3437:Wayback Machine 3420: 3418: 3410:"Fuzzy control" 3404: 3352: 3335:Wayback Machine 3323: 3321:Further reading 3318: 3317: 3290: 3286: 3239: 3235: 3182: 3178: 3151: 3147: 3112: 3108: 3081: 3077: 3042: 3038: 2985: 2981: 2954: 2950: 2915: 2911: 2884: 2880: 2853: 2849: 2822: 2818: 2791: 2787: 2760: 2756: 2721: 2717: 2682: 2678: 2651: 2647: 2639: 2606: 2600: 2596: 2577: 2573: 2552: 2548: 2524:10.1.1.103.1143 2507: 2503: 2458: 2454: 2444: 2442: 2438: 2423: 2417: 2413: 2405: 2382: 2376: 2372: 2349: 2342: 2333: 2332: 2328: 2289: 2282: 2272: 2270: 2248: 2244: 2186: 2182: 2168: 2161: 2150: 2143: 2138: 2086: 2061:Writing systems 2039: 1984: 1952:Cold(r)→High(t) 1944:Cold(x)→High(y) 1932: 1861: 1859:Antilock brakes 1856: 1845: 1826: 1822: 1807: 1803: 1794: 1790: 1789: 1782: 1778: 1769: 1765: 1750: 1746: 1737: 1733: 1724: 1720: 1711: 1707: 1706: 1704: 1695: 1691: 1689: 1686: 1685: 1669: 1665: 1656: 1652: 1637: 1633: 1624: 1620: 1611: 1607: 1598: 1594: 1582: 1578: 1563: 1559: 1550: 1546: 1534: 1530: 1528: 1525: 1524: 1505: 1501: 1486: 1482: 1473: 1469: 1457: 1453: 1441: 1437: 1428: 1424: 1409: 1405: 1396: 1392: 1383: 1379: 1370: 1366: 1355: 1352: 1351: 1335: 1331: 1319: 1315: 1306: 1302: 1284: 1280: 1268: 1264: 1255: 1251: 1239: 1235: 1223: 1219: 1210: 1206: 1195: 1192: 1191: 1172: 1168: 1166: 1163: 1162: 1145: 1141: 1139: 1136: 1135: 1118: 1114: 1112: 1109: 1108: 1104: 1082: 1079: 1078: 1043: 966: 964: 959: 956: 955: 872: 681: 679: 677: 674: 673: 668: 665: 662: 644: 638: 631: 625: 618: 611: 607: 599: 583: 575: 550: 547: 544: 541: 535: 522: 494:trial-and-error 452: 430: 421: 415:makes decisions 411:microcontroller 388: 382: 269:air conditioner 235:opened in 1987. 197: 188:neural networks 144: 138: 135: 120: 108: 104: 93: 82: 76: 73: 63:Please help to 62: 46: 42: 35: 28: 23: 22: 15: 12: 11: 5: 3498: 3488: 3487: 3482: 3468: 3467: 3462: 3457: 3447:IEC 1131-7 CD1 3444: 3439: 3427: 3403: 3402:External links 3400: 3399: 3398: 3385: 3375: 3370: 3363: 3356: 3350: 3337: 3322: 3319: 3316: 3315: 3284: 3253:(3): 253–263. 3233: 3196:(4): 406–415. 3176: 3165:(2): 313–319. 3145: 3126:(4): 313–326. 3106: 3095:(2): 125–139. 3075: 3036: 2999:(1): 135–145. 2979: 2948: 2929:(1): 141–154. 2909: 2898:(3): 213–226. 2878: 2867:(1): 105–113. 2847: 2836:(2): 188–199. 2816: 2805:(3): 353–366. 2785: 2774:(1–2): 59–83. 2754: 2735:(4): 393–400. 2715: 2696:(3): 624–634. 2676: 2645: 2594: 2571: 2546: 2517:(2): 231–254. 2501: 2472:(3): 392–398. 2452: 2411: 2393:(3): 333–350. 2370: 2340: 2326: 2280: 2242: 2180: 2159: 2140: 2139: 2137: 2134: 2133: 2132: 2127: 2122: 2117: 2112: 2107: 2102: 2097: 2092: 2085: 2082: 2081: 2080: 2077: 2074: 2073:Decision trees 2071: 2070:Expert systems 2068: 2065: 2062: 2059: 2056: 2053: 2050: 2038: 2035: 1992:Physics engine 1983: 1980: 1976:Herbrand model 1931: 1928: 1860: 1857: 1854: 1829: 1825: 1821: 1818: 1815: 1810: 1806: 1802: 1797: 1793: 1785: 1781: 1777: 1772: 1768: 1764: 1761: 1758: 1753: 1749: 1745: 1740: 1736: 1732: 1727: 1723: 1719: 1714: 1710: 1703: 1698: 1694: 1672: 1668: 1664: 1659: 1655: 1651: 1648: 1645: 1640: 1636: 1632: 1627: 1623: 1619: 1614: 1610: 1606: 1601: 1597: 1593: 1590: 1585: 1581: 1577: 1574: 1571: 1566: 1562: 1558: 1553: 1549: 1545: 1542: 1537: 1533: 1513: 1508: 1504: 1500: 1497: 1494: 1489: 1485: 1481: 1476: 1472: 1468: 1465: 1460: 1456: 1452: 1449: 1444: 1440: 1436: 1431: 1427: 1423: 1420: 1417: 1412: 1408: 1404: 1399: 1395: 1391: 1386: 1382: 1378: 1373: 1369: 1365: 1362: 1359: 1338: 1334: 1330: 1327: 1322: 1318: 1314: 1309: 1305: 1301: 1298: 1295: 1292: 1287: 1283: 1279: 1276: 1271: 1267: 1263: 1258: 1254: 1250: 1247: 1242: 1238: 1234: 1231: 1226: 1222: 1218: 1213: 1209: 1205: 1202: 1199: 1190: 1175: 1171: 1148: 1144: 1121: 1117: 1092: 1089: 1086: 1064: 1061: 1058: 1055: 1052: 1049: 1046: 1041: 1038: 1035: 1032: 1029: 1026: 1023: 1020: 1017: 1014: 1011: 1008: 1005: 1002: 999: 996: 993: 990: 987: 984: 981: 978: 975: 972: 969: 963: 941: 938: 935: 932: 929: 926: 923: 920: 917: 914: 911: 908: 905: 902: 899: 896: 893: 890: 887: 884: 881: 878: 875: 870: 867: 864: 861: 858: 855: 852: 849: 846: 843: 840: 837: 834: 831: 828: 825: 822: 819: 816: 813: 810: 807: 804: 801: 798: 795: 792: 789: 786: 783: 780: 777: 774: 771: 768: 765: 762: 759: 756: 753: 750: 747: 744: 741: 738: 735: 732: 729: 726: 723: 720: 717: 714: 711: 708: 705: 702: 699: 696: 693: 690: 687: 684: 672: 666: 663: 660: 656: 655: 651: 642: 635: 629: 623: 615: 608: 604: 597: 582: 579: 572: 548: 545: 542: 539: 533: 517: 516: 513: 510: 507: 504: 501: 450: 429: 426: 419: 381: 378: 370:neural-network 366:expert systems 358: 357: 350: 335: 316:General Motors 310:Firms such as 308: 289: 288: 276: 267:An industrial 265: 254: 247: 243: 236: 201:Lotfi A. Zadeh 196: 193: 156:control system 146: 145: 111: 109: 102: 95: 94: 49: 47: 40: 26: 9: 6: 4: 3: 2: 3497: 3486: 3483: 3481: 3478: 3477: 3475: 3466: 3463: 3461: 3458: 3455: 3451: 3448: 3445: 3443: 3440: 3438: 3434: 3431: 3428: 3417: 3416: 3411: 3406: 3405: 3397: 3396:9781447150121 3393: 3389: 3386: 3384: 3380: 3377:Jan Jantzen, 3376: 3374: 3371: 3368: 3364: 3361: 3357: 3353: 3347: 3343: 3338: 3336: 3332: 3329: 3325: 3324: 3311: 3307: 3303: 3299: 3295: 3288: 3280: 3276: 3271: 3266: 3261: 3256: 3252: 3248: 3244: 3237: 3229: 3225: 3221: 3217: 3212: 3207: 3203: 3199: 3195: 3191: 3187: 3180: 3172: 3168: 3164: 3160: 3156: 3149: 3141: 3137: 3133: 3129: 3125: 3121: 3117: 3110: 3102: 3098: 3094: 3090: 3086: 3079: 3071: 3067: 3063: 3059: 3056:(1): 93–103. 3055: 3051: 3047: 3040: 3032: 3028: 3024: 3020: 3015: 3010: 3006: 3002: 2998: 2994: 2990: 2983: 2975: 2971: 2967: 2963: 2959: 2952: 2944: 2940: 2936: 2932: 2928: 2924: 2920: 2913: 2905: 2901: 2897: 2893: 2889: 2882: 2874: 2870: 2866: 2862: 2858: 2851: 2843: 2839: 2835: 2831: 2827: 2820: 2812: 2808: 2804: 2800: 2796: 2789: 2781: 2777: 2773: 2769: 2765: 2758: 2750: 2746: 2742: 2738: 2734: 2730: 2726: 2719: 2711: 2707: 2703: 2699: 2695: 2691: 2687: 2680: 2672: 2668: 2664: 2660: 2656: 2649: 2638: 2634: 2630: 2625: 2620: 2616: 2612: 2605: 2598: 2590: 2586: 2582: 2575: 2566: 2561: 2557: 2550: 2542: 2538: 2534: 2530: 2525: 2520: 2516: 2512: 2511:Studia Logica 2505: 2497: 2493: 2488: 2483: 2479: 2475: 2471: 2467: 2463: 2456: 2437: 2433: 2429: 2422: 2415: 2404: 2400: 2396: 2392: 2388: 2381: 2374: 2366: 2362: 2358: 2354: 2347: 2345: 2336: 2330: 2322: 2318: 2314: 2310: 2306: 2302: 2298: 2294: 2287: 2285: 2269: 2265: 2261: 2257: 2253: 2246: 2238: 2234: 2230: 2226: 2222: 2218: 2213: 2208: 2204: 2200: 2199: 2194: 2191:(June 1965). 2190: 2184: 2176: 2172: 2166: 2164: 2155: 2148: 2146: 2141: 2131: 2128: 2126: 2123: 2121: 2118: 2116: 2113: 2111: 2108: 2106: 2105:Fuzzy concept 2103: 2101: 2098: 2096: 2093: 2091: 2090:Dynamic logic 2088: 2087: 2078: 2075: 2072: 2069: 2066: 2063: 2060: 2057: 2054: 2051: 2048: 2047: 2046: 2043: 2034: 2032: 2027: 2025: 2020: 2016: 2015:forward model 2012: 2007: 2005: 2001: 1997: 1993: 1989: 1979: 1977: 1972: 1967: 1965: 1961: 1957: 1953: 1949: 1945: 1941: 1937: 1927: 1925: 1921: 1916: 1914: 1909: 1905: 1901: 1899: 1894: 1891: 1889: 1884: 1880: 1876: 1874: 1870: 1866: 1853: 1849: 1827: 1823: 1819: 1816: 1813: 1808: 1804: 1800: 1795: 1791: 1783: 1779: 1775: 1770: 1766: 1762: 1759: 1756: 1751: 1747: 1743: 1738: 1734: 1730: 1725: 1721: 1717: 1712: 1708: 1701: 1696: 1692: 1670: 1666: 1662: 1657: 1653: 1649: 1646: 1643: 1638: 1634: 1630: 1625: 1621: 1617: 1612: 1608: 1604: 1599: 1595: 1591: 1583: 1579: 1575: 1572: 1569: 1564: 1560: 1556: 1551: 1547: 1540: 1535: 1531: 1506: 1502: 1498: 1495: 1492: 1487: 1483: 1479: 1474: 1470: 1463: 1458: 1454: 1450: 1442: 1438: 1434: 1429: 1425: 1421: 1418: 1415: 1410: 1406: 1402: 1397: 1393: 1389: 1384: 1380: 1376: 1371: 1367: 1360: 1357: 1336: 1332: 1328: 1320: 1316: 1312: 1307: 1303: 1296: 1293: 1290: 1285: 1281: 1277: 1269: 1265: 1261: 1256: 1252: 1245: 1240: 1236: 1232: 1224: 1220: 1216: 1211: 1207: 1200: 1197: 1189: 1173: 1169: 1146: 1142: 1119: 1115: 1090: 1087: 1084: 1062: 1059: 1056: 1053: 1050: 1047: 1044: 1036: 1033: 1030: 1027: 1021: 1015: 1012: 1009: 1003: 997: 994: 991: 988: 982: 976: 973: 970: 961: 936: 930: 927: 924: 918: 912: 909: 906: 900: 894: 891: 888: 882: 876: 873: 865: 859: 856: 853: 850: 847: 844: 841: 835: 829: 826: 823: 817: 811: 808: 805: 802: 799: 796: 793: 787: 781: 778: 775: 769: 763: 760: 757: 754: 751: 748: 745: 739: 733: 730: 727: 721: 715: 712: 709: 706: 703: 700: 697: 691: 685: 682: 671: 659: 652: 649: 648: 647: 641: 634: 628: 622: 614: 603: 596: 593: 591: 586: 578: 571: 568: 566: 561: 559: 554: 538: 532: 529: 524: 520: 514: 511: 508: 505: 502: 499: 498: 497: 495: 490: 486: 484: 479: 475: 471: 467: 465: 460: 456: 449: 446: 442: 440: 434: 425: 418: 416: 412: 408: 404: 399: 397: 392: 387: 377: 375: 371: 367: 363: 354: 351: 348: 344: 340: 336: 333: 329: 325: 321: 320:Allen-Bradley 317: 313: 309: 306: 302: 298: 294: 293: 292: 285: 281: 277: 274: 270: 266: 263: 259: 255: 252: 251:suction power 248: 244: 241: 237: 234: 230: 229:Sendai Subway 226: 222: 221: 220: 218: 213: 211: 206: 202: 192: 189: 185: 179: 177: 173: 169: 165: 161: 157: 153: 142: 139:February 2010 132: 128: 124: 118: 117: 114:reads like a 112:This article 110: 101: 100: 91: 88: 80: 70: 66: 60: 59: 53: 48: 39: 38: 33: 19: 18:Fuzzy systems 3419:. Retrieved 3415:Scholarpedia 3413: 3387: 3378: 3366: 3359: 3341: 3301: 3297: 3287: 3270:10261/129426 3250: 3246: 3236: 3193: 3189: 3179: 3162: 3158: 3148: 3123: 3119: 3109: 3092: 3088: 3078: 3053: 3049: 3039: 3014:11441/144927 2996: 2992: 2982: 2965: 2961: 2951: 2926: 2922: 2912: 2895: 2891: 2881: 2864: 2860: 2850: 2833: 2829: 2819: 2802: 2798: 2788: 2771: 2767: 2757: 2732: 2728: 2718: 2693: 2689: 2679: 2662: 2658: 2648: 2614: 2610: 2597: 2580: 2574: 2555: 2549: 2514: 2510: 2504: 2469: 2465: 2455: 2443:. Retrieved 2431: 2427: 2414: 2390: 2386: 2373: 2356: 2352: 2329: 2296: 2292: 2271:. Retrieved 2259: 2255: 2245: 2202: 2196: 2193:"Fuzzy sets" 2183: 2174: 2153: 2044: 2040: 2037:Applications 2028: 2008: 1985: 1970: 1968: 1963: 1959: 1955: 1951: 1947: 1943: 1939: 1935: 1933: 1917: 1910: 1906: 1902: 1895: 1892: 1887: 1885: 1883:situations. 1881: 1877: 1862: 1850: 1846: 1105: 669: 657: 645: 639: 632: 626: 619: 612: 600: 594: 587: 584: 576: 569: 562: 555: 551: 536: 525: 521: 518: 491: 487: 480: 476: 472: 468: 461: 457: 453: 447: 443: 435: 431: 422: 407:truth values 400: 393: 389: 359: 303:motors, and 290: 258:autofocusing 233:Namboku Line 214: 198: 180: 164:mathematical 151: 149: 136: 113: 83: 74: 55: 3480:Fuzzy logic 3421:31 December 3328:(522 pages) 3304:: 106–117. 2968:: 375–386. 2273:21 December 2262:(1): 3–10. 2189:Zadeh, L.A. 2171:Hájek, Petr 2120:Neuro-fuzzy 2067:Environment 2004:Fuzzy rules 1913:abstraction 1869:temperature 160:fuzzy logic 125:to make it 69:introducing 3474:Categories 3211:10261/3106 2229:0139.24606 2136:References 2115:Hysteresis 646:-- where: 384:See also: 380:Fuzzy sets 347:thermistor 343:dishwasher 52:references 3279:1648-4142 3247:Transport 3220:1524-9050 3140:1079-8587 3023:0943-0105 2749:109602861 2565:1842/7307 2519:CiteSeerX 2496:123008987 2237:Q25938993 2221:0019-9958 2031:Fuzzy set 1922:, speed, 1817:… 1776:⋅ 1760:… 1744:⋅ 1718:⋅ 1663:⋅ 1647:… 1631:⋅ 1605:⋅ 1573:… 1541:⋅ 1496:… 1464:⋅ 1451:− 1435:⋅ 1419:… 1403:⋅ 1377:⋅ 1329:⋅ 1313:− 1294:… 1278:⋅ 1262:− 1233:⋅ 1217:− 1088:− 1034:− 1031:⋅ 1013:⋅ 995:− 992:⋅ 974:⋅ 842:⋅ 794:⋅ 746:⋅ 698:⋅ 464:inference 386:fuzzy set 356:problems. 332:Whirlpool 158:based on 3450:Archived 3433:Archived 3331:Archived 3228:12554460 3031:15744271 2710:58355555 2637:Archived 2541:14958568 2445:13 March 2436:Archived 2434:(5): 1. 2403:Archived 2321:32000981 2313:25594957 2233:Wikidata 2173:(1998). 2084:See also 2076:Robotics 1920:traction 531:—where: 437:or the " 403:mappings 362:firmware 337:In 1995 324:Chrysler 273:inverter 116:textbook 77:May 2011 3070:6120747 2931:Bibcode 2474:Bibcode 1948:Cold(r) 1936:IF-THEN 1924:inertia 1852:below: 401:Given " 374:genetic 225:Hitachi 203:of the 176:digital 172:logical 127:neutral 121:Please 65:improve 3394: 3348: 3277: 3226: 3218: 3138: 3068: 3029: 3021: 2962:Energy 2747: 2708: 2631: 2539: 2521: 2494: 2319: 2311: 2235: 2227: 2219: 409:, the 396:switch 339:Maytag 330:, and 312:Boeing 168:analog 54:, but 3224:S2CID 3066:S2CID 3027:S2CID 2745:S2CID 2706:S2CID 2640:(PDF) 2633:47212 2629:S2CID 2607:(PDF) 2537:S2CID 2492:S2CID 2439:(PDF) 2424:(PDF) 2406:(PDF) 2383:(PDF) 2317:S2CID 2042:80s. 2011:ANFIS 1988:model 1888:fuzzy 1873:speed 413:then 328:Eaton 217:Japan 154:is a 3423:2022 3392:ISBN 3346:ISBN 3275:ISSN 3216:ISSN 3136:ISSN 3019:ISSN 2447:2021 2309:PMID 2275:2023 2217:ISSN 305:NASA 295:The 210:kiln 186:and 3306:doi 3265:hdl 3255:doi 3206:hdl 3198:doi 3167:doi 3128:doi 3097:doi 3058:doi 3009:hdl 3001:doi 2970:doi 2939:doi 2900:doi 2869:doi 2838:doi 2807:doi 2776:doi 2737:doi 2698:doi 2667:doi 2619:doi 2585:doi 2560:hdl 2529:doi 2482:doi 2395:doi 2391:113 2361:doi 2357:121 2301:doi 2264:doi 2225:Zbl 2207:doi 1091:0.5 1063:0.3 1051:0.7 1045:0.3 1028:0.3 998:0.5 989:0.7 971:0.3 287:on. 162:—a 3476:: 3412:. 3302:55 3300:. 3296:. 3273:. 3263:. 3251:30 3249:. 3245:. 3222:. 3214:. 3204:. 3192:. 3188:. 3161:. 3157:. 3134:. 3122:. 3118:. 3093:69 3091:. 3087:. 3064:. 3054:28 3052:. 3048:. 3025:. 3017:. 3007:. 2997:53 2995:. 2991:. 2966:63 2964:. 2960:. 2937:. 2927:46 2925:. 2921:. 2896:62 2894:. 2890:. 2865:54 2863:. 2859:. 2832:. 2828:. 2803:50 2801:. 2797:. 2772:36 2770:. 2766:. 2743:. 2731:. 2727:. 2704:. 2694:38 2692:. 2688:. 2661:. 2657:. 2635:. 2627:. 2615:51 2613:. 2609:. 2535:. 2527:. 2515:79 2513:. 2490:. 2480:. 2468:. 2464:. 2430:. 2426:. 2401:. 2389:. 2385:. 2355:. 2343:^ 2315:. 2307:. 2297:62 2295:. 2283:^ 2260:12 2258:. 2254:. 2231:. 2223:. 2215:. 2201:. 2195:. 2162:^ 2144:^ 2006:. 1871:, 326:, 322:, 318:, 314:, 219:. 150:A 3425:. 3354:. 3312:. 3308:: 3281:. 3267:: 3257:: 3230:. 3208:: 3200:: 3194:6 3173:. 3169:: 3163:2 3142:. 3130:: 3124:5 3103:. 3099:: 3072:. 3060:: 3033:. 3011:: 3003:: 2976:. 2972:: 2945:. 2941:: 2933:: 2906:. 2902:: 2875:. 2871:: 2844:. 2840:: 2834:6 2813:. 2809:: 2782:. 2778:: 2751:. 2739:: 2733:8 2712:. 2700:: 2673:. 2669:: 2663:5 2621:: 2591:. 2587:: 2568:. 2562:: 2543:. 2531:: 2498:. 2484:: 2476:: 2470:2 2449:. 2432:2 2397:: 2367:. 2363:: 2337:. 2323:. 2303:: 2277:. 2266:: 2239:. 2209:: 2203:8 1971:f 1964:r 1960:t 1956:t 1828:n 1824:M 1820:+ 1814:+ 1809:2 1805:M 1801:+ 1796:1 1792:M 1784:n 1780:M 1771:n 1767:X 1763:+ 1757:+ 1752:2 1748:M 1739:2 1735:X 1731:+ 1726:1 1722:M 1713:1 1709:X 1702:= 1697:0 1693:X 1671:n 1667:M 1658:n 1654:X 1650:+ 1644:+ 1639:2 1635:M 1626:2 1622:X 1618:+ 1613:1 1609:M 1600:1 1596:X 1592:= 1589:) 1584:n 1580:M 1576:+ 1570:+ 1565:2 1561:M 1557:+ 1552:1 1548:M 1544:( 1536:0 1532:X 1512:) 1507:n 1503:M 1499:+ 1493:+ 1488:2 1484:M 1480:+ 1475:1 1471:M 1467:( 1459:0 1455:X 1448:) 1443:n 1439:M 1430:n 1426:X 1422:+ 1416:+ 1411:2 1407:M 1398:2 1394:X 1390:+ 1385:1 1381:M 1372:1 1368:X 1364:( 1361:= 1358:0 1337:n 1333:M 1326:) 1321:0 1317:X 1308:n 1304:X 1300:( 1297:+ 1291:+ 1286:2 1282:M 1275:) 1270:0 1266:X 1257:2 1253:X 1249:( 1246:+ 1241:1 1237:M 1230:) 1225:0 1221:X 1212:1 1208:X 1204:( 1201:= 1198:0 1174:i 1170:M 1147:i 1143:X 1120:0 1116:X 1085:= 1060:+ 1057:0 1054:+ 1048:+ 1040:) 1037:1 1025:( 1022:+ 1019:) 1016:1 1010:0 1007:( 1004:+ 1001:) 986:( 983:+ 980:) 977:0 968:( 962:= 940:) 937:4 934:( 931:u 928:m 925:+ 922:) 919:3 916:( 913:u 910:m 907:+ 904:) 901:2 898:( 895:u 892:m 889:+ 886:) 883:1 880:( 877:u 874:m 869:) 866:4 863:( 860:t 857:u 854:p 851:t 848:u 845:o 839:) 836:4 833:( 830:u 827:m 824:+ 821:) 818:3 815:( 812:t 809:u 806:p 803:t 800:u 797:o 791:) 788:3 785:( 782:u 779:m 776:+ 773:) 770:2 767:( 764:t 761:u 758:p 755:t 752:u 749:o 743:) 740:2 737:( 734:u 731:m 728:+ 725:) 722:1 719:( 716:t 713:u 710:p 707:t 704:u 701:o 695:) 692:1 689:( 686:u 683:m 141:) 137:( 133:. 119:. 90:) 84:( 79:) 75:( 61:. 34:. 20:)

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index