Fuzzy Sets and Systemsauthor.uthm.edu.my/uthm/www/content/lessons/4519/ACSE-Worksho… · Web...

Computational Intelligence Approaches: Fuzzy Systems

http://www.infm.ulst.ac.uk/~siddique

Computational Intelligence Approaches to Systems Modelling and Control

Dr. M.N.H. Siddique email: [email protected],

mailto:[email protected]



Computational Intelligence Approaches

Keep it simple:As simple as possible,

But not simpler.

- Albert EinsteinIntroduction to CI

Much is unknown about intelligence, and much will remain beyond human comprehension.

Fundamental nature of intelligence is only poorly understood

According to a dictionary, intelligence means the ability to comprehend, reason and learn.



A demanding definition of intelligent system -An intelligent system is capable to comprehend with or without much a priori information about the process, reason and learn about the process

Other aspects of intelligence that describes human intelligence are creativity, skills, consciousness, emotion and intuition.

Highly complex mathematical description can seriously inhibit the ability to develop system models.

Required to cope with significant un-modelled and unanticipated changes in the plant,

Dissatisfaction with conventional modelling techniques is growing with increasing complexity.

Necessitates the use of more human expertise and knowledge.



Intelligent techniques are aimed at processes that are

ill-defined, complex, nonlinear, time varying and stochastic.

Central to intelligent systems is the construction of the process model.

Many real world processes are not amenable to mathematical modelling because

i. processes are too complex to represent mathematicallyii. process models are difficult and expensive to evaluate

iii. uncertainties in process operationiv. process is nonlinear, distributed, incomplete and stochastic in nature



Paradigms of CI The area of computational intelligence is in fact interdisciplinary Combine and extend theories and methods from other disciplines Inter-relationship between these disciplines is illustrated as

Figure 1: Periphery of computational intelligent



Approaches to CI

According to Engelbrecht there are five basic approaches to computational intelligence. These are

i. Fuzzy logicii. Neural networks

iii. Evolutionary computingiv. Swarm intelligencev. Learning theory

vi. Probabilistic methods

We will cover three basic methodologies: -Fuzzy logic-Neural networks-Evolutionary computing

Other methodologies such as Swarm intelligence, learning theory and probabilistic methods will be addressed in later date.



Introduction

Two justification for fuzzy systems

(1) Real world too complicated- Fuzziness introduced to obtain a reasonable model

(2) Dissatisfaction growing with conventional approaches- Human knowledge becoming more and more important

Simple Example

Figure 2: A real world situation.



A fuzzy system can combine the two

Figure 3: Example of a practical system.



Notational Conventions

A notational convention of fuzzy sets for a discrete and finite universe of discourse X is:

(1) (2)

represents the degree of membership of x in A.

Where ‘+’ does not mean arithmetic addition or logical OR.

Membership Functions



A membership function (MF) characterizes the fuzziness of a fuzzy set A in X, which associate each point in X a real number in the interval [0,1] with the value of at x representing the grade of membership.

Choice of membership function is usually problem dependent and very often determined by heuristically and subjectively.

Triangular MF Triangular MF: specified by three parameters , Figure 5(a).

(3) Parameters with determine the coordinates of the three corners of

the underlying triangular MF.



Figure 4(a): Triangular MF.

Trapezoidal MF Trapezoidal MF: specified by four parameters , Figure 5(b).

(4) Parameters with determine the coordinates of the four corners

of the underlying trapezoidal MF.



Figure 4(b): Trapezoidal MF.

Gaussian MF Gaussian MF: specified by two parameters , Figure 5(c).

(5) Parameters and represent the centre and width of the Gaussian MF

respectively.



Figure 4(c): Gaussian function. Figure 4(d): Gaussian MF.

Bell-shaped MF Bell-shaped MF: specified by three parameters , Figure 5(d).

(6) Parameters and represent the centre and width of the bell-shaped MF. Parameter a, usually positive, controls slope of the MF at crossover point.



Figure 4(e): Bell-shaped function Figure 4(f): Bell-shaped MF

Gaussian and bell shaped MFs achieve smoothness - unable to specify asymmetric MFs, which are important in certain applications.

Sigmoidal MF, which is either open left or right, can be used in such applications.



Heuristic selection of parameters of membership functions are widely used and practised in fuzzy systems applications.

Different approaches to construct membership functions, such as

- heuristic selection, - clustering approach, - C-means clustering approach, - adaptive vector quantization, and - self-organising map.



Linguistic Variables and Hedges

Linguistic variable is a variable whose values are words or sentences, which serve as names of fuzzy subsets of a universe of discourse.

For example, Speed is a linguistic variable if its values are slow, not slow, very slow, not very slow etc.

Specifically, a linguistic variable is characterised by a quintuple of primary terms, connectives, hedges, and markers.



For example: linguistic variable with hedges

Figure 5: Linguistic variables and hedges.



Purpose of the hedges is to generate a larger set of values for a linguistic variable from a small collection of primary terms.

Hedges are realised on primary terms through the processes

intensification or concentration, dilation, and fuzzification.

Example:Operator VERY is concentration (or intensification) operator

Very (Small) = Small2

Very(Very (u)) = ( Very (u))2 = (u2)2 = u4



Linguistic hedges very and more or less – shown in Figure 6.

Figure 6: Dilation and Concentration

For example, more or less very nice can be represented as

More or less (Very (Nice)) = More or less (Nice2) = (Nice2)1/2 = Nice



Linguistic variable can be a word or sentence

Such natural language expressions are fuzzy

For example, Slow OR Medium, Medium AND Fast.

Figure 7: Three fuzzy sets for speed – Slow, Medium and Fast



Figure 8: Expression for “Slow OR Medium”.

Figure 9: Expression for “Medium AND Fast”.



Some widely used Hedges

Hedge Meaning

About, around, near roughly Approximates a scalar

Above, more than Restricts a fuzzy region

Almost, definitely, precisely Contrast intensification

Below, less than Restricts a fuzzy region

Generally, usually Contrast diffusion

Neighbouring, close to Approximate narrowly

Not Negation or complement

Quite, rather, somewhat Dilute a fuzzy region

Very, extremely Intensify a fuzzy region



Linguistic variable can be used with more than one hedge.

For example

- Almost very fast but generally below 100 km/hr- Close to 100m but not very high (but AND)- Not more than about zero Not (more than (about zero))

Linguistic variable and hedges allow constructing mathematical models for expression in natural language.

These models can than be used to write computer programs and process rules and simulate real world process and behaviour.



Fuzzy If-Then Rules

If-part of the rule ‘x is A1’ is called the antecedent or premise. Then-part of the rule ‘y is B2’ is called the consequent.

Example: Speed and pressure of a steam engine can be expressed with the following linguistic

conditional statementIf Speed is Slow Then Pressure should be High.

Graphically, this statement looks like

Figure 10: If-Then rule.



Rule Forms

In general, three forms of rule exist for any linguistic variables.

Assignment statemente.g. x is not large AND not very small

Conditional statement e.g. IF x is very big THEN y is medium

Unconditional statement e.g. set pressure high



Compound Rules

Linguistic statement expressed by a human might involve compound rule structures.

Any compound rule structure may be decomposed and reduced to a number of simple Canonical rules.

Two forms of Canonical forms

Conjunctive antecedent Disjunctive antecedent



Conjunctive antecedents

Multiple conjunctive antecedents of the form

IF x is A1 AND x is A2 … AND x is An THEN y is Bs

Can be rewritten in compact form as

IF x is As THEN y is Bs

As = A1 A2 A3 … An

As expressed by means of membership functions based on fuzzy intersection operation



Disjunctive antecedents

Multiple disjunctive antecedents of the form

IF x is A1 OR x is A2 … OR x is An THEN y is Bs

Can be rewritten in compact form as

IF x is As THEN y is Bs

As = A1 A2 A3 … An

As expressed by means of membership function based on fuzzy union operation



Aggregation of rules

Aggregated output y is found by fuzzy intersection of the entire individual rule consequent , where .

y = y1 AND y2 AND … AND yr

ory = y1 y2 … yr

y defined by the membership function

for

For disjunctive system of rules where at least one rule must be satisfied, the rules are connected by OR connectives.

Aggregated output y is found by fuzzy union of all the individual rule consequent , where .



y = y1 OR y2 OR … OR yr or

y = y1 y2 … yr

y defined by the membership function

for

Fuzzification



Process that allows converting a numeric value (or crisp value) into a fuzzy input is called fuzzification.

Two ways to do fuzzification:

Aui is a fuzzy singleton

Aui is a fuzzy set such that

Singleton fuzzification is generally used in implementation where there is no noise.



Figure 11: Fuzzification in different MFs

Defuzzification

Defuzzification is the reverse process of fuzzification.



Mathematically, the defuzzification of a fuzzy set is the process of conversion of a fuzzy quantity into a crisp value, i.e. rounding off from its location in the unit hypercube to the nearest vertex.

Fuzzy control engineers have many different ways of defuzzifying; fuzzy reasoning, however, can usually use quite simple methods.

There are many defuzzification methods available in the literature.

Decision as to which method to use is a fairly technical matter; but the experts should know that these method are available and should try to see which works best.

Max-membership method

The max-membership method is both simple and quick method.



This method takes the peak value of each fuzzy set and builds the weighted sum of these peak values.

This method is given by the algebraic expression as

for all

Shown in Figure 12(a).

12(a): Max-membership defuzzification

Centre of Gravity method

In the literature also referred to as centre of area and centroid method. In the continuous case it is given by the expression as



For a discrete universe with m quantisation levels in the output

This method determines the centre of the area below the combined membership function. Shown in Figure 12(b).

12(b): Centre of gravity defuzzification Weighted average method

Only valid for symmetrical output membership functions. It is given by the algebraic expression as

where denotes an algebraic sum. Shown in Figure 12(c).



12(c): Weighted average defuzzification For example, the two functions shown in Figure 12(c) would result in the following

defuzzified value:

Mean-max membership

If the mean or average of all local maxima in X is taken, a single defuzzified output is generated by this mean of maximum approach through

where xj = max (x) and N is the number of times the membership function reaches the maximum support value. Shown in Figure

12(d).



12(d): Mean-max defuzzification The defuzzified output is then

Centre of sums



This process involves the algebraic sum of individual output fuzzy sets, say C1 and C2, instead of their union.

One drawback of this method is that the intersecting areas are added twice. The defuzzified value x* is given in discrete case by the following

expression:

Shown in Figure 12(e).

12(e): Centre of sums defuzzification

Inference Mechanism

Inference is the process of formulating a nonlinear mapping from a given input space to output space.



The mapping then provides a basis from which decisions can be taken. The process of fuzzy inference involves all the membership functions, fuzzy logic

operators and if-then rules. There are three types of fuzzy inference, which have been widely employed in

various applications. The differences between these three fuzzy inferences, also called fuzzy models, lie

in the consequents of their fuzzy rules, aggregations and defuzzification procedures. These fuzzy inferences are

(1) Mamdani fuzzy inference(2) Sugeno fuzzy inference (3) Tsukamoto fuzzy inference

Mamdani Fuzzy Inference

Mamdani type fuzzy modelling was proposed as the first attempt to control a steam engine and boiler by a set of linguistic control rules by Mamdani and Assilian in 1974.

Max-min is the most common rule of composition.



In max-min rule of composition the inferred output of each rule is a fuzzy set chosen from the minimum firing strength.

In max-product rule of composition the inferred output of each rule is a fuzzy set scaled down by its firing strength via algebraic product.

A typical rule in Mamdani-type fuzzy model with two-input single-output has the form

If x is A and y is B then z is C

In Mamdani’s fuzzy model crisp values are used as inputs and defuzzification is used to convert a fuzzy set to a crisp value.



Figure 13: Two-input single-output Mamdani fuzzy model.

Sugeno Fuzzy Inference

The Sugeno fuzzy model, also known as the TSK fuzzy model, was proposed by Takagi, Sugeno and Kang (Takagi and Sugeno, 1985; Sugeno and Kang, 1988) in an



effort to develop a systematic approach to generate fuzzy rules from a given input-output data set.

A typical fuzzy rule in Sugeno fuzzy model has the form

If x is A and y is B then z=f(x,y)

where A and B are fuzzy sets in the antecedent part, while z=f(x,y) is crisp function in the consequent part.

Usually f(x,y) is polynomial in the input variables x and y but it can be any function as long as it can appropriately describe the output of the model within the fuzzy region specified by the antecedent of the rule.



Figure 14: Two-input single-output Sugeno fuzzy model.



Features of Sugeno-type Inference Systems

When f(x,y) is a first-order polynomial, the resulting fuzzy inference system is called a first-order Sugeno fuzzy model which was proposed by Sugeno and Takagi.

When f(x,y) is a constant, it is zero-order Sugeno fuzzy model, which can be considered as a special case of Mamdani fuzzy model, in which the consequent of each rule is specified by a fuzzy singleton or by a pre-defuzzified consequent or a special case of Tsukamoto fuzzy model in which the consequent of each rule is specified by an MF of a step function.

Output of zero-order Sugeno model is a smooth function of its input variables as long as neighbouring MFs in the antecedent have enough overlap.

Overlap of MFs in the consequent of a Mamdani model does not have a decisive effect on the smoothness.

Overlap of the antecedent MFs that determines the smoothness of the resulting input-output behaviour.



Overall output of a Sugeno fuzzy model is obtained via weighted average of the crisp output, thus avoiding the time consuming process of defuzzification required by Mamdani model.

Zero-order Sugeno fuzzy model is functionally equivalent to a radial basis function network under certain minor constraints which was shown by Jang and Sun.

In practice, weighted average operator is sometimes replaced with weighted sum operator to reduce computation further, especially in the training of a fuzzy inference system.

Simplification could lead to the loss of MF linguistic meaning unless the sum of firing strengths is close to unity.

Tsukamoto Fuzzy Inference



In the Tsukamoto fuzzy model, the consequent of each fuzzy if-then rule is represented by a fuzzy set with a monotonic MF.

As a result, the inferred output of each rule is defined as crisp value included by the rule’s firing strength.

The overall output is taken as the weighted average of each rule’s output.

Since each rule infers a crisp output, the Tsukamoto fuzzy model aggregates each rule’s output by the method of weighted average and thus avoids the time-consuming process of defuzzification.



Figure 15: Two-input single-output Tsukamoto fuzzy model.

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