mamdani and sugeno

7/28/2019 Mamdani and Sugeno

1/9

MAMDANI-TYPE INFERENCE

Mamdani's fuzzy inference method is the most commonly seen fuzzy methodology.

Mamdani's method was among the first control systems built using fuzzy set theory. It was

proposed in 1975 by Ebrahim Mamdani as an attempt to control a steam engine and boiler

combination by synthesizing a set of linguistic control rules obtained from experienced human

operators. Mamdani's effort was based on Lotfi Zadeh's 1973 paper on fuzzy algorithms for

complex systems and decision processes. Although the inference process described in the next

few sections differs somewhat from the methods described in the original paper, the basic idea is

more or less the same.

Mamdani-type inference, as defined for the toolbox, expects the output membership

functions to be fuzzy sets. After the aggregation process, there is a fuzzy set for each output

variable that needs defuzzification. It is possible, and in many cases much more efficient, to use

a single spike as the output memberships function rather than a distributed fuzzy set. This type

of output is sometimes known as asingleton output membership function, and it can be thought

of as a pre-defuzzified fuzzy set. It enhances the efficiency of the defuzzification process

because it greatly simplifies the computation required by the more general Mamdani method,

which finds the centroid of a two-dimensional function. Rather than integrating across the two-

dimensional function to find the centroid, you use the weighted average of a few data points.

An example of a Mamdani inference system is shown below. To compute the output of this

system for the given inputs following six steps are to be followed:

Determining a set of fuzzy rules Fuzzifying the inputs using the input membership functions, Combining the fuzzified inputs according to the fuzzy rules to establish a rule strength, Finding the consequence of the rule by combining the rule strength and the output

membership function,

Combining the consequences to get an output distribution, and Defuzzifying the output distribution (this step is only if a crisp output (class) is needed).


2/9

The following is a more detailed description of this process.

A two input, two rule Mamdani FIS with crisp inputs

Creating fuzzy rules

Fuzzy rules are a collection of linguistic statements that describe how the FIS should make a

decision regarding classifying an input or controlling an output. Fuzzy rules are always written in

the following form:

i f(input1 is membership function1) and/or(input2 is membership function2) and/or. then

(outputn is output membership functionn).

For example, one could make up a rule that says:

i ftemperature is high andhumidity is high thenroom is hot.

There would have to be membership functions that define what we mean by high temperature

(input1), high humidity (input2) and a hot room (output1). This process of taking an input such

as temperature and processing it through a membership function to determine what we mean by


3/9

"high" temperature is called fuzzification. Also the definition of what we mean by "and" / "or" in

the fuzzy rule is to be given.

Fuzzification

The purpose of fuzzification is to map the inputs from a set of sensors (or features of

those sensors such as amplitude or spectrum) to values from 0 to 1 using a set of input

membership functions. In the example there are two inputs, x0 and y0 shown at the lower left

corner. These inputs are mapped into fuzzy numbers by drawing a line up from the inputs to the

input membership functions above and marking the intersection point.

These input membership functions discussed previously, can represent fuzzy concepts

such as "large" or "small", "old" or "young", "hot" or "cold", etc. For example, x 0 could be the

EMG energy coming from the front of the forearm and y0 could be the EMG energy comingfrom the back of the forearm. The membership functions could then represent "large" amounts of

tension coming from a muscle or "small" amounts of tension. When choosing the input

membership functions, the definition of what we mean by "large" and "small" may be different

for each input.

Fuzzy combinations (T-norms)

In making a fuzzy rule, we use the concept of "and", "or", and sometimes "not". The sections

below describe the most common definitions of these "fuzzy combination" operators. Fuzzy

combinations are also referred to as "T-norms".

The fuzzy rule is computed using two steps:

1) Computing the rule strength by combining the fuzzified inputs using the fuzzycombination. The fuzzy "and" is used to combine the membership functions to compute

the rule strength.

2) Clipping the output membership function at the rule strength.Combining Outputs into an Output Distribution

The outputs of all of the fuzzy rules must now be combined to obtain one fuzzy output

distribution. This is usually, but not always, done by using the fuzzy "or the output membership


4/9

functions on the right hand side of the figure above are combined using the fuzzy "or" to obtain

the output distribution shown on the lower right corner of the above figure.

Defuzzification of Output Distribution

In many instances, it is desired to come up with a single crisp output from a FIS. For example, if

one was trying to classify a letter drawn by hand on a drawing tablet, ultimately the FIS would

have to come up with a crisp number to tell the computer which letter was drawn. This crisp

number is obtained in a process known as defuzzification. There are two common techniques for

defuzzifying:

Center of mass - This technique takes the output distribution found in section and finds its

center of mass to come up with one crisp number. This is computed as follows:

z is the center of mass and

uc is the membership in class c at value zj.

Defuzzification Using the Center of Mass

Mean of maximum - This technique takes the output distribution and finds its mean of maxima

to come up with one crisp number. This is computed as follows:


5/9

z is the mean of maximum,

zj is the point at which the membership function is maximum,

l is the number of times the output distribution reaches the maximum level.

Defuzzification Using the Mean of Maximum

Fuzzy Inputs

It fuzzifies the two inputs by finding the intersection of the crisp input value with the

input membership function. It uses the minimum operator to compute the fuzzy "and" for

combining the two fuzzified inputs to obtain rule strength. It clips the output membership

function at the rule strength. Finally, it uses the maximum operator to compute the fuzzy "or" for

combining the outputs of the two rules.

This can be used to model inaccuracies in the measurement. For example, we may be measuring

the output of a pressure sensor. Even with the exact same pressure applied, the sensor is

measured to have slightly different voltages. The fuzzy input membership function models this

uncertainty. The input fuzzy function is combined with the rule input membership function by

using the fuzzy "and" as shown in the figure.


6/9

A two Input, two rule Mamdani FIS with a fuzzy input

Advantages of the Mamdani Method

It is intuitive. It has widespread acceptance. It is well suited to human input.


7/9

SUGENO-TYPE INFERENCING

In general, Sugeno-type systems can be used to model any inference system in which the output

membership functions are either linear or constant. Sugeno, or Takagi-Sugeno-Kang method offuzzy inference first introduced in 1985. It is similar to the Mamdani method in many respects.

In fact the first two parts of the fuzzy inference process, fuzzifying the inputs and applying the

fuzzy operator, are exactly the same. The main difference between Mamdani-type of fuzzy

inference and Sugeno-type is that the output membership functions are only linear or constant for

Sugeno-type fuzzy inference.

A typical fuzzy rule in azero-order Sugeno fuzzy modelhas the form

ifx isA andy isB thenz= k

where A and B are fuzzy sets in the antecedent, while k is a crisply defined constant in the

consequent. When the output of each rule is a constant like this, the similarity with Mamdani's

method is striking. The only distinctions are the fact that all output membership functions are

singleton spikes, and the implication and aggregation methods are fixed and can not be edited.

The implication method is simply multiplication, and the aggregation operator just includes all of

the singletons.

A typical rule in a Sugeno fuzzy model has the following form:

If Input 1 = x and Input 2 =y, then Output isz = ax + by + c

For a zero-order Sugeno model, the output levelzis a constant (a=b =0).

The output levelzi of each rule is weighted by the firing strength wi of the rule.

For example, for an AND rule with Input 1 = x and Input 2 =y, the firing strength is

whereF1,2 (.) are the membership functions for Inputs 1 and 2.


8/9

The final output of the system is the weighted average of all rule outputs, computed as

whereNis the number of rules.

A Sugeno rule operates as shown in the following diagram.


9/9

Advantages of the Sugeno Method

It is computationally efficient. It works well with linear techniques (e.g., PID control). It works well with optimization and adaptive techniques. It has guaranteed continuity of the output surface. It is well suited to mathematical analysis.

Comparison of Sugeno and Mamdani Methods

Because it is a more compact and computationally efficient representation than a Mamdani

system, the Sugeno system lends itself to the use of adaptive techniques for constructing fuzzy

models. These adaptive techniques can be used to customize the membership functions so that

the fuzzy system best models the data.

mamdani and sugeno

Documents