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Topic 5 - Fuzzy Logic Control

Topic 5Topic 5

FUZZY LOGIC CONTROLFUZZY LOGIC CONTROL

Instructional Objectives1. State the factors involved in a fuzzy logic controller design.

2. Sketch and describe the structure of fuzzy logic controllers.

3. Define fuzzification and defuzzification strategies.

4. Explain the mechanism involved in fuzzy reasoning.

5. Describe what are linguistic terms.

6. Explain how data base and rule base are used in fuzzy logic controllers.7. Sketch and explain Mandani's and Larsens inferences.

8. Define mean of maxima and center of area methods.

9. Describe how fuzzy control rules are derived.

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CONTENTS

5.1 Introduction

5.2 Basic Structure of Fuzzy Logic Controllers

5.3 Fuzzy Logic Controller Design Issues5.4 Design of Fuzzy Control Rules

5.5 Detail Examples on Fuzzy Logic Control

Self-Test Questions

REFERENCES

1. Using Fuzzy Logic by J. Yan, M. Ryan & J. Power( pages 45 - 61 )

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5.1 INTRODUCTION

Digital control is more flexible due to the wide variety of algorithms

that may be used. Digital systems are generally better for coping with

varying environments arising from load disturbances, process non-

linearities, changes of plant parameters and so on..The algorithm used may be based on proportional-integral-derivative

(PID), model reference adaptive control and fuzzy logic. The PID

control is effective for a fixed control environment, but is unable to

cope with a varying control system non-linearity. The fuzzy logic

approach uses inputs, outputs and control response that are specified in

terms similar to that used by a human expert. Complex mathematical

models of the system under control are not required..

Fuzzy logic controllers have been applied to cement process plant,

robot control, motor control, automatic train operation, crane control,

servo loop control and many more.

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Theprincipal factors in designing a fuzzy logic controller are:

The actual inputs and outputs and their universes of discourse i.e.

the range of values which each may take.

The scale factors of the input-output variables.

The fuzzy membership functions to be used in setting up the fuzzy

values for each input and output variable.

The fuzzy control rule base.

The key issue in the design are the construction of the membership

functions and the fuzzy control rules. A fuzzy logic controller may

also have self-organizing or learning capabilities.

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5.2 BASIC STRUCTURE OF FUZZYLOGIC CONTROLLERS

FLC PlantR UE Y+

-

The fuzzy logic controller (FLC) is normally incorporated in a

closed-loop control system.

The main elements of the FLC are a fuzzification unit, a fuzzy

logic reasoning unit, a knowledge base and a defuzzificationunit

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Fuzzy Reasoning

Mechanism

Fuzzification

Unit

Defuzzification

Unit

Rule Base Data Base

Input E Output U

The fuzzy knowledge base contains 2 main types of information:

A data base defining the membership functions of the fuzzy sets used

as values for each system variables

A rule base that maps fuzzy values of the input to fuzzy values of the

outputs.

Measured fromcontrolled process

Used by FLC tocontrol the process

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Fuzzification strategy involves

Acquiringcrisp values of input values.Mappingcrisp values of the inputs into corresponding universes

of discourse.

Convertingthe mapped data into fuzzy singletons or into suitable

linguistic terms.

Defuzzification involves weighting and combininga number of fuzzy

sets resulting from the the fuzzy inference process that will give

single crisp value for each output.

The rule basepart consists of a number of fuzzy rules which

expresses the control relationship such as.

Rule k: IF x is PB THEN y is NB.where x is an input variable, y is an output variable, PB is one of the fuzzy

sets defined for x in X and NB is one of the fuzzy sets defined for y in Y.

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The fuzzy logic reasoning module uses fuzzy logic in a mannersimilar to some aspects of human decision making. It performs

fuzzy inference to arrive at the fuzzy control actions by evaluating

the knowledge base for the fuzzified inputs.

The 3 basic operations involve are:.

Determination of degree of matchbetween fuzzy input data and

the defined fuzzy sets for each system input variable.

Carry out fire strength calculations for each rule based on thedegree of match and the connectives used in the antecedent part.

Derive the control outputsbased on the calculated fire strength

and the defined fuzzy sets for each output variable in the

consequent part.

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5.3 FUZZY LOGIC CONTROLLER

DESIGN ISSUES

System Variable and Fuzzy Parameters

The number of input and output variables determines the complexity

of the fuzzy system starting with single-input-single-output system

(SISO), MISO and MIMO.

When a FLC is designed to replace a conventional PID controllerin

a MISO system, 3 input variables are normally used:

State error, e

Sum of the state error, e Derivative of state error, deA vector E = [ e e de ]t is commonly used. The output variable isa control signal u.

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Linguistic terms are usually used to define each system variable inthe fuzzy sets such as PB (positive big), PM (positive medium), PS

(positive small), ZE (Zero), NS ( negative small), NM ( negative

medium), NB (negative big), etc.

Selection of membership functions based on the range and shape

for a variable is somewhat a subjective design choice.

Symmetrically distribute the fuzzy sets across the defined

universe of discourse. Use an odd number of fuzzy sets for each variable.

Overlap adjacent fuzzy sets to ensure no crisp value fails to

correspond to any set.

Use triangular or trapezoidal membership functions as theserequire less computation.

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Fuzzification Strategies

In process control, the observed data is usually crisp and

fuzzification is required to map the observed range of crisp inputs to

corresponding fuzzy values for the system input variables. The

mapped data are further converted into suitable linguistic terms aslabels of the fuzzy sets defined for system input variables.

EXAMPLE : Voltage Reading From A Digital Voltmeter

voltage

Low voltage

membership

Crisp Reading

1

0.5

Fuzzy singleton

voltage

0.4

Medium

voltage

membership

Fuzzy Reading

1

Fuzzy number

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Knowledge Base

Data Base Rule Base+

Data Base

Defines the universe of discourse for each variable, determining the

number of fuzzy sets and designing the membership functions.

Sampling & Quantization resolution

Input variable

Discretization of

universe of discourse

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Discretization of universe of discourse is important such that the

performance of a FLC will be less sensitive to small deviations in

values of the system variables. Quantization during analog-to-

digital conversion must be selected at an appropriate quantization

level known as resolution.

EXAMPLE

A universe of discourse is discretized into 7 levels with 5 fuzzy sets.

Level no. Range NB NS ZE PS PB-3 x < -12 1.0 0.3 0.0 0.0 0.0

-2 -12 < x < -8 0.3 1.0 0.3 0.0 0.0

-1 -8 < x < -4 0.0 0.7 0.7 0.0 0.0

0 -4 < x < +4 0.0 0.3 1.0 0.3 0.01 +4 < x < +8 0.0 0.0 0.7 0.7 0.0

2 +8 < x < +12 0.0 0.0 0.3 1.0 0.3

3 +12 < x 0.0 0.0 0.0 0.3 1.0

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Rule Base

A collection of fuzzy controlbased on the control goals and control

policy given by the experts. Each fuzzy rule is a set-level implication

representing expert knowledge. A typical MISO system will have the

following rule base:.

Rule 1: IF x1 is A11 AND AND xm is A1m THEN y is B1.

Rule 2: IF x1 is A21 AND AND xm is A2m THEN y is B2.

Rule n: IF x1 is An1 AND AND xm is Anm THEN y is Bn..

( Aij and Bi are fuzzy sets such as PB, PM, PS, ZE, NS, NB )

A fuzzy control algorithm should be able to infer a proper controlaction for any input in the universe of discourse referred to as

completeness.

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Number of Control Rules

If the number of fuzzy sets or predicates for each input variable is m and

the number of system input variables is n, then there are mn different

rules required for a completeness in a conventional system. FLC rule

base typically uses a small numberof rules to attain completeness.

EXAMPLE

A FLC has been designed to improve the autofocus function in video camcorders.

The fuzzy logic control system uses 7 input variables to be judged and producesone output, the focus. How many rules might be used if 3 predicates were utilized?

Number of rules = 37 or 3 x 3 x 3 x 3 x 3 x 3 x 3 = 2187.

If the number of predicates increase to 7, we will have 77

or 823,543 rules.In this case, only 20 rules have been used in the FLC system and the

system performance is greatly enhanced.

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Reasoning Techniques

The 2 main fuzzy inference methods commonly used in industrial

FLC are:.

The point-valued MAX-MIN fuzzy inference.

The point-valued MAX-DOT fuzzy inference.Due to the nature of industrial process control, the input data is

most often crisp in nature. Hence, fuzzification involves treating

these as fuzzy singletons that are used with either MAX-MIN or

MAX-DOT.

Taking a fuzzy control rule base with 2 rules:

Rule 1: IF x is A1 AND y is B1 THEN z is C1.

Rule 2: IF x is A2 AND y is B2 THEN z is C2.The fire strength of the ith rule for inputs x0 and y0 of the rule base are:

1 = A1(x0) B1(y0) and 2 = A2(x0) B2(y0)

is the min operator.

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Mamdanis Inference: MAX-MIN

The membership of the inferred consequence C is given by:

C(w) = ( 1 C1(w) ) ( 2 C2(w) )

Input values x0 and y0 are regarded as fuzzy singletons.

= min

= max

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Larsens Inference: MAX-DOT

The membership of the inferred consequence C is given by:

C(w) = ( 1 C1(w) ) ( 2 C2(w) )

Input values x0 and y0 are regarded as fuzzy singletons.

= product

= max

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Defuzzification Strategies

Theprocess of mappingfrom a space of inferred fuzzy control actions

to a space of crisp ( non-fuzzy ) control actions that best represents

the possibility distribution of the inferred control action..

In real-time implementation of fuzzy logic control, the commondefuzzification strategies used are the mean of maximum (MOM)

and the center of area (COA).

Mean of Maximum

Generates a control action that represents the mean value of all

local actions whose membership functions reach the maximum.

zMOM = zj / m = z dz / dzm

j=1

z = support value at which the

membership value reaches

maximum value.

m = number of support values.

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Center of Area

EXAMPLE

z

A

Smallest of Max.

Largest of Max.

Mean of Max. (MOM)

Center of Area (COA)

Generates the center of gravity of the possibility distribution of the

control action.

zCOA

= A

(z) z dz / A

(z) dz z = amount of control output.

A(z) = membership value inthe output fuzzy set A.= A(z) . z / A(z)

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5.4 DESIGN OF FUZY CONTROL RULES

Four principal methods have been employed to derive fuzzy control

rules based on the following:.

An Experts Experience or Control Engineering Knowledge.

It is a heuristic approach aimed at capturing the experts rules of

thumb. Its is an iterative process, with the fuzzy parameters of the

initial system being tuned and adjusted until satisfactory

performance is achieved. Fuzzy control rules provide a naturalframeworkfor capturing expert knowledge.

.

Modeling The Operators Control Actions.

It is a deterministic approach where modeling the operators

control actions is carried out by observing the human controllers

actions over a period of time and expressing them in terms of the

operational input-output data. The structure and control parameters

can be determined that satisfied the control objective.

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Fuzzy Modeling.

It is a qualitative modelingscheme by which the behavior of the

system to be controlled is qualitatively described using fuzzy

quantities or languages. Based on the fuzzy model model, wecan generate a set of fuzzy control rules to obtain optimal

performance of the system.

Self Organizing or Learning.Fuzzy control systems can be built which simulate human

leaningand have the ability to create fuzzy control rules and to

modify them based on system performance. Such schemes

include neural networks, self-tuning and self-organization.

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EXAMPLE 1

FLC PlantE

CE

CI

YYd

-

Objectives

Design a set of

fuzzy control

rules to reduce

the overshoot

and the rise

time.

5.5 DETAIL EXAMPLES ON FUZZY LOGIC CONTROL

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Assume that the fuzzy subsets for each input and output are represented by

the terms Negative, Zero and Positive i.e. [ N, Z, P ]. Then 2 fuzzy control

rules can be formulated as:.

Type 1 - Shorten the rise time of the system.

IF E is Positive AND CE is negative THEN CI is Positive..

Type 2 - Decrease the overshoot of the system.

IF E is Negative AND CE is negative THEN CI is Negative.

Rule No. E CE CI References Functions

1 P Z P points a, e, i shortens rise time2 Z N N points b, f, j reduce overshoot

3 N Z N points c, g, k reduce overshoot

4 Z P P points d, h, l reduce oscillation

5 Z Z Z set point braking system

6 P N P ranges A, E shorten rise time7 N N N ranges B, F, J reduce overshoot

8 N P N ranges C, G reduce overshoot

9 P P P ranges D, H reduce oscillation

10 P N Z range I braking system

11 N P Z range K braking system

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Finer fuzzy partitions for each input/output variable will result inbetter control performance such as using the following set

[ NB, NM, NS, ZE, PS, PM, PB ] for the fuzzy variable [ E,CE, CI ].

Theprinciples to construct the fuzzy control rules for a FLC in

process control can be generalized as follows:.

If the output has the desired value and the change of error is zero,

keep the output of the FLC constant..

If the output diverges from the desired value, the control actiondepends on the sign and value of the error and its change.

A matrix can be used to express this control process. It is used to

approximate a human operators thinking and actions. Projecting thematrix on an axis perpendicular to the main diagonal, we will have

the control form and policy.

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CE

-B -M -S -Z +Z +S +M +B+B +Z +S -M -B -B -B -B -B

+M +S -Z -S -M -M -M -B -B

+S +M +S -Z -S -S -S -M -B

+Z +M +M +S +Z -Z -S -M -M

E -Z +M +M +S +Z -Z -S -M -M-S +B +M +S +S +S +Z -S -S

-M +B +B +M +M +M +S +Z +Z

-B +B +B +B +B +B +M +Z +Z

-B -M -S -Z+Z+S+M+B+B

+M

+S

-Z

+Z

-S-M

-B

BM

SZ

SMB

Positive

Negativ

e

Fuzzy Control

Matrix

Fuzzy Control

Resolution

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EXAMPLE 2

FLC PlantE

CE

UYYd

-

N Z P

P Z P P

Z N Z P

N N N Z

CE

EControl Rules

Note: P - positive, N - negative, Z = zero

Find the output U given that E = 0.5

and CE = 0.25 using COA method.1

E

PZN

0 1-1

1

CE

PZN

0 0.5-0.5

Membership

Functions

1

U

0 5-5

N Z P

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1

E

PZN

0 1-1

1

CE

PZN

0 0.5-0.5

1

U

0 5-5

N Z P

E=0.5 CE=0.25

0.5 0.5

At E = 0.5,

the fire strength for Z is 0.5

the fire strength for P is 0.5.

At CE = 0.25,

the fire strength for Z is 0.5

the fire strength for P is 0.5.

Thus, there are 4 active rules:.

Rule 1: IF E is Z AND CE is Z THEN U is Z.

Rule 2: IF E is Z AND CE is P THEN U is P.

Rule 3: IF E is P AND CE is Z THEN U is P.

Rule 4: IF E is P AND CE is P THEN U is P.

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Outcome of the rules:.Rule 1: .. U is Z. output = 0V.Rule 2: .. U is P. output = 5V.Rule 3: .. U is P. output = 5V.Rule 4: .. U is P. output = 5V.

Defuzzification using COA method:

Ucrisp = [ (0 x 0.5) + (5 x 0.5 ) + (5 x 0.5) + (5 x 0.5) ]

0.5 + 0.5 + 0.5 + 0.5

= 3.75.

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EXAMPLE 3

A moving crane is often

required in a factory workshop

to carry loads of various sizes

and weights. The load willswing when the motion of the

crane is accelerated or

decelerated. A fuzzy control

can be developed to enablehigh speed crane travel with

minimal load swing.

Position sensors are used to detect the swing angle. The swingangular velocity is computed by differentiating the swing angleand the previous swing angle.

i i C l

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The swing angle and swing angular velocity are taken as inputs

to the fuzzy logic controller (FLC). The inferred output controlsignal from the FLC is the change of the crane travel speed, CV.

Fuzzy Set Fuzzy Set

Fuzzy Set CV

T i 5 F L i C l

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Using the human operators knowledge and intuition about the crane

operation, a fuzzy knowledge base is constructed. Some of the fuzzy

control rules are listed below:

Rule 1: IF is NB AND is N THEN CV is PS.Rule 2: IF is NB AND is P THEN CV is PB.Rule 3: IF is NS AND is N THEN CV is ZE.Rule 4: IF is NS AND is P THEN CV is PS.Rule 5: IF is ZE AND is Z THEN CV is ZE.Rule 6: IF is PB AND is N THEN CV is NS.Rule 7: IF is PB AND is P THEN CV is NB.Rule 8: IF is NS AND is N THEN CV is ZE.Rule 9: IF is NS AND is P THEN CV is NS.

Rule 10: IF is ZE AND is P THEN CV is NS.Rule 11: IF is ZE AND is N THEN CV is PS.

T i 5 F L i C t l

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The crane swing control is realized by the control of the cranespeed that is implemented using the procedure below:

Crane is traveling

Load swing detection

Measure i

Fuzzy inference

Inferred CVModify crane speedV = V + CV

Compute = ii-1

T i 5 F L i C t l

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EXAMPLE 4

A process control of inflow

valve V1 and outflow valve

V2 is used to measure the

liquid level L inside the tankthat is kept constant at 50%

level. The fuzzy variables

are level and valve positions

as shown.

The fuzzy rules used are:

Rule 1: IF the level is normal AND valve 2 is open THEN open valve 1.

Rule 2: IF the level is high AND valve 2 is half-open THEN close valve 1.

Rule 3: IF the level is low THEN open valve 1.

Rule 4: IF the level is low OR valve 2 is open THEN half open valve 1.

Topic 5 Fuzzy Logic Control

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Evaluation of

Rules 1, 2 & 4

for level of 70%

and valve 2position at 60%.


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Continue

All activated rules are aggregated by using the max operator to

form a max-min inference and defuzzified using the COG orCOA method yielding a 50% valve position for V1.

How would the result change if max-product inference is used ?


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SELF-TEST QUESTIONS

Question 1Which one is not a main element in

a fuzzy logic controller structure ?

(a) Knowledge base.(b) Fuzzification unit

(c) Defuzzification unit

(d) PID rules.

Question 2How many fuzzy rules are there if

the number of fuzzy sets for each

input variable is 3 and the number

of system input variables 4 ?

(a) 81

(b) 64

(c) 48

(d) 12


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Question 3

Fill in the blanks with the most appropriate answers.

The key issues in the design of a fuzzy logic controller are

construction___________________ and ________rules.

The fuzzy knowledge base consist of 2 main types of information

which are ____________ and ___________.

The process of acquiring crisp values and mapping this crisp values

into corresponding universe of discourse is called ______________.

Linguistic terms are used to define each system variable in the fuzzy

sets. Name some of these terms. Answer: _____________________.

The 2 main fuzzy inference used in industrial FLC are ___________

and ___________ inferences.


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Question 4

What are the 2 most popular method of defuzzification ?

(a) _____________ (b) ____________

Question 5

List down 4 ways that fuzzy control rules can be design ?

(a) _____________________________

(b) _____________________________

(c) _____________________________

(d) _____________________________


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Question 6Apply the 2 most popular defuzzification methods to the following:

z

z

z

z

(a)

(b)

(c)

(d)

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