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1、An Introduction To Fuzzy Control Systems Translated by Yang Zhenshou This document is an introduction to fuzzy control systems. It provides an overview of their theory of operation, followed by elementary examples of their use. [1] INTRODUCTION TO FUZZY LOGIC &

2、FUZZY CONTROL "Fuzzy logic" has become a common buzzword in machine control. However, the term itself inspires a certain skepticism, sounding equivalent to "half-baked logic" or "bogus logic". Some other nomenclature might have been preferable, but it's too late now, and fuzzy logic is actually ve

3、ry straightforward. Fuzzy logic is a way of interfacing inherently analog processes, that move through a continuous range of values, to a digital computer, that likes to see things as well-defined discrete numeric values. For example, consider an antilock braking system, directed by a microcontrol

4、ler chip. The microcontroller has to make decisions based on brake temperature, speed, and other variables in the system. The variable "temperature" in this system can be divided into a range of "states", such as: "cold", "cool", "moderate", "warm", "hot", "very hot". Defining the bounds of these

5、states is a bit tricky. An arbitrary threshold might be set to divide "warm" from "hot", but this would result in a discontinuous change when the input value passed over that threshold. The way around this is to make the states "fuzzy", that is, allow them to change gradually from one state to the

6、 next. You could define the input temperature states using "membership functions" such as the following: 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

7、value in the next. At any one time, the "truth value" of the brake temperature will almost always be in some degree part of two membership functions: 0.6 nominal and 0.4 warm, or 0.7 nominal and 0.3 cool, and so on. The input variables in a fuzzy control system are in general mapped into by sets o

8、f membership functions similar to this, known as "fuzzy sets". The process of converting a crisp input value to a fuzzy value is called "fuzzification". A control system may also have various types of switch, or "ON-OFF", inputs along with its analog inputs, and such switch inputs of course will a

9、lways have a truth value equal to either 1 or 0, but the scheme can deal with them as simplified fuzzy functions that are either one value or another. Given "mappings" of input variables into membership functions and truth values, the microcontroller then makes decisions for what action to take ba

10、sed on a set of "rules", each of the form: IF brake temperature IS warm AND speed IS not very fast THEN brake pressure IS slightly decreased. In this example, the two input variables are "brake temperature" and "speed" that have values defined as fuzzy sets. The output variable, "brake

11、pressure", is also defined by a fuzzy set that can have values like "static", "slightly increased", "slightly decreased", and so on. His rule by itself is very puzzling since it looks like it could be used without bothering with fuzzy logic, but remember the decision is based on a set of rules:

12、All the rules that apply are invoked, using the membership functions and truth values obtained from the inputs, to determine the result of the rule. This result in turn will be mapped into a membership function and truth value controlling the output variable. These results are combined to give a

13、 specific ("crisp") answer, the actual brake pressure, a procedure known as "defuzzification". This combination of fuzzy operations and rule-based "inference" describes a "fuzzy expert system". Traditional control systems are based on mathematical models in which the the control system is descri

14、bed using one or more differential equations that define the system response to its inputs. Such systems are often implemented as "proportional-integral-derivative (PID)" controllers. They are the products of decades of development and theoretical analysis, and are highly effective. If PID and oth

15、er traditional control systems are so well-developed, why bother with fuzzy control? It has some advantages. In many cases, the mathematical model of the control process may not exist, or may be too "expensive" in terms of computer processing power and memory, and a system based on empirical rules m

16、ay be more effective. Furthermore, fuzzy logic is well suited to low-cost implementations based on cheap sensors, low-resolution analog-to-digital converters, and 4-bit or 8-bit one-chip microcontroller chips. Such systems can be easily upgraded by adding new rules to improve performance or add ne

17、w features. In many cases, fuzzy control can be used to improve existing traditional controller systems by adding an extra layer of intelligence to the current control method. [2] FUZZY CONTROL IN DETAIL Fuzzy controllers are very simple conceptually. They consist of an input stage, a processing

18、 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 appropriate rule and generates a result for each, then combines the results of the rules. Fi

19、nally, the output stage converts the combined result back into a specific control output value. The most common shape of membership functions is triangular, although trapezoids and bell curves are also used, but the shape is generally less important than the number of curves and their placement. F

20、rom three to seven curves are generally appropriate to cover the required range of an input value, or the "universe of discourse" in fuzzy jargon. 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 "

21、antecedent" and the THEN part is called the "consequent". Typical fuzzy control systems have dozens of rules. Consider a rule for a thermostat: IF (temperature is "cold") THEN (heater is "high") This rule uses the truth value of the "temperature" input, which is some truth value of "cold", t

22、o 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, the greater the truth value of "cold", the higher the truth value of "high", though this does no

23、t 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 adjectives. Common hedges include "about", "near", "close to", "approximately", "very", "slightly", "t

24、oo", "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 func

25、tion. "Extremely" cubes the values to give greater narrowing, while "somewhat" broadens the function by taking the square root. 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

26、vary: AND, in one popular definition, simply uses the minimum weight of all the antecedents, while OR uses the maximum value. There is also a NOT operator that subtracts a membership function from 1 to give the "complementary" function. There are several different ways to define the result of a ru

27、le, but one of the most common and simplest is the "max-min" inference method, in which the output membership function is given the truth value generated by the premise. Rules can be solved in parallel in hardware, or sequentially in software. The results of all the rules that have fired are "defu

28、zzified" to a crisp value by one of of several methods. There are dozens in theory, each with various advantages and drawbacks. 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 valu

29、e of the biggest contributor. The centroid method favors the rule with the output of greatest area, while the height method obviously favors the rule with the greatest output value. The example below demonstrates max-min inferencing and centroid defuzzification for a system with input variables "x

30、", "y", and "z" and an output variable "n". Note that "mu" is standard fuzzy-logic nomenclature for "truth value": 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 ma

31、ximum value is used and values are not added, and the results are then combined using a centroid calculation. Fuzzy control system design is based on empirical methods, basically a methodical approach to trial-and-error. The general process is as follows: Document the system's operational speci

32、fications and inputs and outputs. Document the fuzzy sets for the inputs. Document the rule set. Determine the defuzzification method. Run through test suite to validate system, adjust details as required. Complete document and release to production. As a general example, consider the de

33、sign of a fuzzy controller for a steam turbine. The block diagram of this control system appears as follows: There are two input variables, temperature and pressure, and a single output variable, the turbine throttle setting. The turbine's operation can be reversed, so the throttle setting can b

34、e positive or negative. The fuzzy set mappings are shown below: The throttle settings are defined as follows: N3: Large negative. N2: Medium negative. N1: Small negative. Z: Zero. P1: Small positive. P2: Medium positive. P3: Large positive. The rule

35、set includes such rules as: rule 1: IF temperature IS cool AND pressure IS weak, THEN throttle is P3. rule 2: IF temperature IS cool AND pressure IS low, THEN throttle is P2. rule 3: IF temperature IS cool AND pressure IS ok, THEN throttle is Z.

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

37、e mappings of the two input variables, as the examples above do, the minimum of the two is 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 th

38、en defuzzified. For an 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. Rule 2 is evaluated as follows: Rule 3 is evaluated as follows: The two outputs are then combined: T

39、he output value will adjust the throttle and then the control cycle will begin again to generate the next value. [3] BUILDING A FUZZY CONTROLLER Consider implementing with a microcontroller chip a simple feedback controller: A fuzzy set is defined for the input error variable "e", and the der

40、ived change in error from the previous error input, "delta", as well as the "output", as follows: LP: large positive SP: small positive ZE: zero SN: small negative LN: large negative If the error ranges from -1 to +1, with the analog-to-digital converter used having a reso

41、lution 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 the truth values for each membership function arranged in rows beneath: mu(LP) 0

42、 0 0 0 0 0 0.3 0.7 mu(SP) 0 0 0 0 0.3 0.7 1 0.7 0 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.

43、3 0 0 0 0 0 0 _____________________________________________________________________ This table is a little tricky to interpret. It simply defines each of the elements of the fuzzy set ("LP" through "LN") in terms of the values ("mu") that they have relative to the giv

44、en values in the top row. It is important to remember that for the error / delta variables, the top row values are used to generate the mu values, while for the output variable, the mu values are used to generate the top row values. Suppose this fuzzy system has the following rule base: rul

45、e 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 These rules are typical for control applications in that the antecedents consist of th

46、e logical combination of the error and delta signals, while the consequent is a control command output. These rules can also be a little tricky to interpret. For example, what rule 1 says is that if the error value is the ZE fuzzy set element and the delta, or change from last error, value is the

47、ZE fuzzy set element, then the output is also the ZE fuzzy set element. What is very tricky is that plugging actual numeric values into this rule does not give, at least directly, an actual value for the output setting. The value for the output setting is defined by rule 1 as "ZE". Looking at the

48、fuzzy value table shows that "ZE" has its maximum value corresponding to the value of "0" on the top row of the table, and so this rule always gives an output setting of 0. Similarly, rule 2 always gives an output setting of "SN" or -0.5, rule 3 always gives an output setting of "LP" or 1, and rule

49、4 always gives an output setting of "LN" or -1. The thing to remember is that the actual output is a combination of the outputs of these four rules, weighted by the values by the appropriate mu values provided by the rule calculation, using the centroid calculation: SUM( I = 1 TO 4 OF ( mu(I) *

50、output(I) ) ) / SUM( I = 1 TO 4 OF mu(I) ) The result of this calculation gives the actual output setting. Now, suppose that at a given time we have: Then the corresponding mu values can be obtained by simply taking them from the appropriate columns of the fuzzy set table: mu(SP

51、) 0.7 1 mu(SN) 0 0 mu(LN) 0 0 These values can then be plugged into the rules to give output values. Taking the first rule: rule 1: IF e = ZE AND delta = ZE THEN output = ZE The weighting for the output, "mu(1)", is produced with the

52、 simple calculation: As mentioned above, the output value of this rule is always 1: output(1) = 0 The other rules give: rule 2: IF e = ZE AND delta = SP THEN output = SN mu(2) = MIN( 0.7, 1 ) = 0.7 rule 3: IF e = SN AND delta = SN THEN output = LP mu(3) = MIN(

53、0.0, 0.0 ) = 0 output(3) = 1 rule 4: IF e = LP OR delta = LP THEN output = LN mu(4) output(4) = -1 The centroid computation yields: = = = -0.65 / 1.3 -- for the final control output. Of course

54、 the hard part is figuring out what rules actually work correctly in practice. 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 X0 is the ce

55、nter of gravity, Xi is the location of each mass, and Mi is each mass, this gives: 0 = ( X1 - X0 ) * M1 + ( X2 - X0 ) * M2 + ... + ( Xn - X0 ) * Mn 0 = ( X1 * M1 + X2 * M2 + ... + Xn * Mn ) - X0 * ( M1 + M2 + ... + Mn ) X0 * ( M1 + M2 + ... + Mn ) = ( X1 * M1 + X2 * M2 + ... + Xn * Mn )

56、 ( X1 * M1 + X2 * M2 + ... + Xn * Mn ) X0 = In our example, the values of mu correspond to the masses, and the values of X to location of the masses. 模糊控制系統(tǒng)介紹 楊鎮(zhèn)首翻譯 本文介紹了模糊控制系統(tǒng),其中概述了模糊控制理論和并介紹了一些簡單的應用例子。 （1） 介紹模糊邏輯與模糊控制 模糊邏輯在機器控制中已經(jīng)成為一個普通的詞語。但是

57、，它容易讓人有一定的疑心，其聽起來像〞半生不熟的邏輯〞或“假邏輯〞，其他一些術語可能也是可取的，但為時已晚，現(xiàn)在實際上的模糊邏輯很簡單。模糊邏輯是一種固有的模擬接口程序 ，它在一定的范圍內(nèi)連續(xù)變化，再接到數(shù)字計算機，即得到離散的定義數(shù)值。 例如，一個防抱死制動系統(tǒng)，由微機控制芯片執(zhí)行。微控制器有以下幾方面決定制動：溫度，速度，和系統(tǒng)中的其他變化量。溫度在這個系統(tǒng)中可以分為一系列的狀態(tài)，如：冷，涼，溫和，溫暖。熱，極熱。但是定義這些狀態(tài)的范圍有點難，任意狀態(tài)可能會被設置為溫和或溫暖，但是，當輸入值超過這一范圍時，將會出現(xiàn)一個不連續(xù)的變化，用這種方法就要使狀態(tài)模糊化，也就是說允許他們從一個狀

58、態(tài)變化到另外一個狀態(tài)。你可以用隸屬函數(shù)來定義輸入溫度狀態(tài)。如以下內(nèi)容： 有了這個定義，輸入變量的狀態(tài)不再從一個狀態(tài)突然變化到另外一個狀態(tài)。取而代之的，隨著溫度的變化，在一個隸屬函數(shù)中，當增值到下一個狀態(tài)時，它將在這個狀態(tài)中失去。在任何一個時候，制動溫度的真值，幾乎一般是在一定程度上的兩個隸屬函數(shù)：“正常〞、“溫暖〞或“常溫〞和“涼〞等等。 模糊控制系統(tǒng)的輸入一般都映射到了一系列隸屬函數(shù)，被稱為“模糊集〞。將輸入真值轉換為模糊值的過程就是所謂的模糊化。 控制系統(tǒng)也可能不同類型的或開關量，以模擬量的輸入或以開關量的輸入，它們的真值要相當于“1”或“0”，但是方案只是一種簡單的

59、模糊化功能將輸入值轉換為“1”或“0”。 給定輸入變量到隸屬函數(shù)的映射和輸入真值，微控制器將決定采取何種行動，基于一系列規(guī)那么，例如以下形式： 如果制動溫度“溫暖〞和速度“不是很快〞,那么制動壓力略下降。 在這個例子中，兩個輸入變量是制動溫度，和速度的真值定義為模糊集，輸出變量，制動壓力，也由模糊集定義為有值量，如“靜止〞，“略有上升〞、““略有下降〞等等。 這一規(guī)那么本身是非常另人費解，因為它看起來好似與模糊邏輯無關，但請記住這決定是基于一套規(guī)那么： 所以被引用的規(guī)那么都適用，利用隸屬函數(shù)和輸入真值，以確定規(guī)那么的結果。 這一結果將被映射成一個隸屬函數(shù)和真值作為控

60、制輸出。 這些結果相結合，給出具體的答案，實際的制導壓力，這一程序被稱為“去模糊化〞。 這種模糊器的組合和一規(guī)那么為根底的推理描述了一個“模糊專家系統(tǒng)〞。 傳統(tǒng)的控制系統(tǒng)是基于數(shù)學模型，其中控制系統(tǒng)的描述是使用一個或多個微分方程，它們定義了系統(tǒng)對其輸入的響應，這種系統(tǒng)往往采用“比例積分微分控制器〞。它們是幾十年的開展和理論分析產(chǎn)生的，而且非常有效的。 如果PID和其他傳統(tǒng)的控制系統(tǒng)都這么興旺，為什么要加進模糊控制呢？它有一定優(yōu)勢，在多數(shù)情況下，控制過程的數(shù)學模型可能不存在，或可能在計算機處理能力和內(nèi)存大小多達不到要求，但基于經(jīng)驗規(guī)那么的系統(tǒng)可能更有效。 此外，模糊邏

61、輯很適合用廉價傳感器來實現(xiàn)，低分辨率的模數(shù)轉換器，以及4位或8位的單片機。這種系統(tǒng)可以很容易通過增加新的規(guī)那么來升級，一改善性能或增加新的功能。在許多情況下，模糊控制可被用來改善現(xiàn)有的傳統(tǒng)的控制的控制系統(tǒng)，增加了一層額外的智能方法。 （2） 模糊控制的詳細內(nèi)容 模糊控制是非常簡單的概念。它們包括一個輸入、處理、輸出。輸入即為傳感器或其他它輸入，如交換機等等映射到適當?shù)碾`屬函數(shù)和真值。處理階段調用每個適當?shù)囊?guī)那么后產(chǎn)生的結果，然后結合所有的結果，最后，輸出級把所以結構結合轉換成特定的控制輸出值。 最常見的隸屬函數(shù)類型是三角形的，雖然貝爾曲線也被使用，但這些類型都是比擬不重要的，從3到7

62、曲線大約適用于所有的輸入值的要求范圍，或在模糊術語中的“宇宙論〞。 如上述，處理階段是基于一“IF-THEN〞這種形式的邏輯規(guī)那么，其中IF局部被稱為先行，THEN局部被稱為結果。典型的模糊控制系統(tǒng)有幾十個規(guī)那么。 我們來考慮自動調溫器的一個規(guī)那么： 如果溫度是“冷〞，然后加熱器是“高〞 本規(guī)那么使用真值溫度為輸入，也是冷的在某意義上真值，在模糊控制集上產(chǎn)生一結果為加熱器的輸出，“高〞的真值。這一結果與其它規(guī)那么的結果結合一起使用，最后生成清晰的復合輸出。顯然，“冷〞的真值越大，“高〞的真值就越高。雖然這并不一定意味著輸出本身將被設置為“高〞，因為這只是許多規(guī)那么中的一個。

63、 在某些情況下，隸屬函數(shù)可以被修改的“籬笆〞。通常包括“關于〞、“近〞、“接近〞、“大約〞、“非常〞、“輕微〞、“太〞、“極端〞和“有點〞。這些行動可能有精確的定義，但當考慮到不同的實現(xiàn)時就有不同的意義。“非常〞，作為一個例子，廣泛隸屬函數(shù)；因為隸屬函數(shù)值始終小于1，這縮小了隸屬函數(shù)值?！皹O其〞的立方根就給以了更大的縮小，而“有點〞擴大了其值，采取了平方根。 在實踐中，模糊規(guī)那么集通常有幾個先例，這是結合了利用模糊運算，如與，或，非，雖然他們再次定義往往各不相同。與，流行定義里的一個是它只使用所有之中的最小值，或使用時采用了最高值，非運算就是用1去減相應的隸屬函數(shù)值。 有幾種不

64、同的方式來定義規(guī)那么的結果，但最常見，最簡單的是“最大最小〞推理方法，其中輸出隸屬函數(shù)給出的真值規(guī)那么可以在硬件上并行處理，或軟件上的順序處理。所以地規(guī)那么的結果，都用許多方式中的某一種反模糊化成為一清晰的真值。在理論上有幾十種，每種都有優(yōu)缺點。 重心法的方法很受歡送，其中的結果重心提高了清晰的真值。另一種方法是“高〞的發(fā)放，考慮其中最大的，重心法主要應用大于的領域時，而高度法顯然有利于該細那么的最大值。 下面的例子顯示出最大最小推理和重心法去模糊的系統(tǒng)，其中輸入變量“X〞，“Y〞、“Z〞 、一個輸出變量“n〞，請注意，“mu〞是標準模糊邏輯命名為真值： 請注意每個規(guī)那么是

65、怎樣為特別的隸屬函數(shù)提供一真值作為結果輸出。在使用重心法去模糊化中，用或運算，也就是使用了最大值，然后其結果是由重心法計算。 模糊控制系統(tǒng)的設計是基于經(jīng)驗的方法，根本上是一個有條理的方法，反復實現(xiàn)。其一般過程如下： 系統(tǒng)的操作規(guī)那么，輸入和輸出 輸入的模糊集 規(guī)那么集 確定去模糊化的方法 連接測試，以驗證系統(tǒng)，根據(jù)需要調整細節(jié)。 完整的文件并投入生產(chǎn)中 作為一般的例子，考慮設計一個模糊控制其的汽輪機，控制系統(tǒng)的框圖顯示如下： 有兩個輸入變量，溫度和壓力，和一個單一的輸出變量，渦輪油門，汽輪機的運行可以扭轉的，所以油門可以被加大或關小。模糊集的映射如下： 油門設定如下

66、： N3：負大 N2：負中 N1：負小 Z:零 P1：正小 P2：正中 P3：正大 該規(guī)那么設定包括以下規(guī)那么： 規(guī)那么1: 如果溫度“冷〞與壓力“弱〞 那么油門“正大〞； 規(guī)那么2：如果溫度“涼〞與壓力“低〞 那么油門“正中〞； 規(guī)那么3：如果溫度“涼〞與壓力“適中〞 那么油門“零〞； 規(guī)那么4如果溫度“涼〞與壓力“強〞 那么油門“負大〞； 在實踐中，控制器接受輸入,然后把它們映射到隸屬函數(shù)和真值。這些映射結果然后輸入規(guī)那么，如果該規(guī)那么指定一個和關系在映射的兩個輸入變量間，如上述的例子，兩個中較低的作為結合真值；如果一個或規(guī)那么，那么使用較高值。相應的輸出狀態(tài)是挑選和指定一個值作為真值，然后將真值反模糊化。 舉個例子，假設溫度是在“涼〞狀態(tài)，壓力在“低〞和“適中〞狀態(tài)。壓力值確定只有規(guī)那么2和規(guī)那么3適合。規(guī)那么2如下： 規(guī)那么3如下： 然后兩個輸出結合在一起： 輸出值將調整油門，然后控制系統(tǒng)來循環(huán)，又產(chǎn)生下一個真值。 （3） 建立一個模糊控制器 考慮帶微控制器芯片來實現(xiàn)的一個簡單的反應控制器： 模糊集的定