Mamdani & Takagi-Sugeno Controllers · Mamdani style inference: The Good News: This method is...

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FUZZY CONTROL:Mamdani & Takagi-Sugeno Controllers

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Khurshid Ahmad, Professor of Computer Science,

Department of Computer ScienceTrinity College,

Dublin-2, IRELANDNov 15-16th, 2011

https://www.cs.tcd.ie/Khurshid.Ahmad/Teaching/Teaching.html

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FUZZY CONTROL

Control Theory?

•The term control is generally defined as a mechanism used to guide or regulate the operation of a machine, apparatus or constellations of machines and apparatus.

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FUZZY CONTROL

Control Theory?

An Input/Output Relationship

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4

FUZZY CONTROL

Control Theory?

An Input/Output Relationship IDENTIFIED using Fuzzy Logic

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FUZZY CONTROL

Control Theory?

An Input/Output Relationship IDENTIFIED using Fuzzy Logic

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FUZZY CONTROL

Control Theory?•Typically, rules contain membership functions for both antecedents and consequent.

•Argument is that the consequent membership function can be simplified – this argument is based on a heuristic that operators in a control environment divide the variable space (say, error, change in error and change in control) into PARTITIONS;

• Within each partition the output variable is a simple, often linear function of the input variables and not membership functions

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7

FUZZY CONTROL

Control Theory?•Typically, rules contain membership functions for both antecedents and consequent.

Mamdani Controller

If e(k) is positive(e) and ∆e(k) is positive(∆e)

then ∆u(k) is positive (∆u)

Takagi-Sugeno Controllers:

If e(k) is positive(e) and ∆e(k) is positive(∆e) then ∆u(k) =ααααe(k)+ß ∆e(k)+δ;

• αααα, ß and δ are obtained from empirical observations by relating the behaviour of the errors and change in errors over a fixed range of changes in control

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FUZZY CONTROL FUZZY CONTROLLERS – Takagi-Sugeno Controllers

According to Yager and Filev, ‘a known disadvantage of the linguistic modules is that they do not contain in an explicit form the objective knowledge about the system if such knowledge cannot be expressed and/or incorporated into fuzzy set framework' (1994:192).

Typically, such knowledge is available often: for example in physical systems this kind of knowledge is available in the form of general conditions imposed on the system through conservation laws, including energy mass or momentum balance, or through limitations imposed on the values of physical constants.

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Tomohiro Takagi and Michio Sugeno recognised two important points:

1. Complex technological processes may be described in

terms of interacting, yet simpler sub processes. This is

the mathematical equivalent of fitting a piece-wise

linear equation to a complex curve.

2. The output variable(s) of a complex physical system,

e.g. complex in the sense it can take a number of input

variables to produce one or more output variable, can

be related to the system's input variable in a linear

manner provided the output space can be subdivided

into a number of distinct regions.

Takagi, T., & Sugeno, M. (1985). ‘Fuzzy Identification of Systems and its Applications to Modeling and Control’. IEEE Transactions on Systems, Man and Cybernetics. Volume No. SMC-15 (No.1) pp 116-132.

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Takagi-Sugeno fuzzy models have been widely used to identify the structures and parameters of unknown or partially known plants, and to control nonlinear systems.

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Mamdani style inference:

The Good News: This method is regarded widely ‘for capturing expert knowledge’ and facilitates an intuitively-plausible description of knowledge;

The Bad News: This method involves the computation of a two-dimensional shape by summing, or more accurately integrating across a continuously varying function. The computation can be expensive.

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Mamdani style inference:The Bad News: This method involves the computation of a two-dimensional shape by summing, or more accurately integrating across a continuously varying function. The computation can be expensive.

For every rule we have to find the membership functions for the linguistic variables in the antecedents and the consequents;

For every rule we have to compute, during the inference, composition and defuzzification process the membership functions for the consequents;

Given the non-linear relationship between the inputs and the output, it is not easy to identify the membership functions for the linguistic variables in the consequent

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Literature on conventional control systems has suggested that a complex non-linear system can be described as a collection of subsystems that were combined based on a logical (Boolean) switching system function.

In realistic situations such disjoint (crisp) decomposition is impossible, due to the inherent lack of natural region boundaries in the system, and also due to the fragmentary nature of available knowledge about the system.

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Takagi and Sugeno (1985) have argued that in order to develop a generic and simple mathematical tool for computing fuzzy implications one needs to look at a fuzzy partition of fuzzy input space.

In each fuzzy subspace a linear input-output relation is formed. The output of fuzzy reasoning is given by the values inferred by some implications that were applied to an input.

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Takagi and Sugeno have described a fuzzy implication Ras:

R: if (x1 is µA(x1),… xk is µA(xk))then y = g(x1, …, xk)

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A solution for the coefficients of the consequent in TSK Systems

)(

][*)(

)(:

1

101

1011

x

xppxy

NCOMPOSITIO

xppyTHENxisxRule

µµ

µ

+=

+=

There are two unknowns: p0 and p1. So we need two simultaneous equations for two values of x, say x1

and x2, and two values of y – y1 and y2

21

2110

21

211

2102

1101

;

xx

yyyp

xx

yyp

xppy

xppy

−−−=

−−=

+=+=

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Consider a two rule system:

][*)(ˆ][*)(ˆ

)()(

][*)(][*)()(ˆ

)()(

][*)(][*)(

)(:

)(:

1202211011

21

12022110111

21

1202211011

120222

110111

xppxxppxy

xx

xppxxppxx

xx

xppxxppxy

NCOMPOSITIO

xppyTHENxisxRule

xppyTHENxisxRule

+++=∴+

+++=

++++=

+=

+=

µµµµµµµ

µµµµ

µµ

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There are 4 unknowns p01, p11, p02, p12, so we need 4 equations. And, these can be obtained from 4 observations comprising 4 diffierent values of and 4

1144211441024201414

1111211111021201111

4321

1202211011

**)(ˆ**)(ˆ*)(ˆ*)(ˆ

....................................................

....................................................

**)(ˆ**)(ˆ*)(ˆ*)(ˆ

,,,,

][*)(ˆ][*)(ˆ

pxxpxxpxpxy

pxxpxxpxpxy

avaluehavewillxeachforandxxxxxofvaluesFour

xppxxppxy

µµµµ

µµµµ

µµ

+++=

+++==

+++=∴

7

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There are 4 unknowns p01, p11, p02, p12, so we need 4 equations. And, these can be obtained from 4 observations comprising 4 diffierent values of and 4

+++

+

=

+++==

+++=∴

21

11

02

01

442441424241

332331323231

222121222221

112111121211

4

3

2

1

1111211111021201111

4321

1202211011

*)(ˆ*)(ˆ)(ˆ)(ˆ)(ˆ

*)(ˆ*)(ˆ)(ˆ)(ˆ)(ˆ

*)(ˆ*)(ˆ)(ˆ)(ˆ)(ˆ

*)(ˆ*)(ˆ)(ˆ)(ˆ)(ˆ

**)(ˆ**)(ˆ*)(ˆ*)(ˆ

,,,,

][*)(ˆ][*)(ˆ

p

p

p

p

xxxxxxx

xxxxxxx

xxxxxxx

xxxxxxx

y

y

y

y

pxxpxxpxpxy

avaluehavewillxeachforandxxxxxofvaluesFour

xppxxppxy

µµµµµµµµµµµµµµµ

µµµµµµµµµ

µµ

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There are 4 unknowns p01, p11, p02, p12, so we need 4 equations.

[ ] [ ] [ ][ ] [ ] [ ]YXXXP

PXY

p

p

p

p

xxxxxx

xxxxxx

xxxxxx

xxxxxx

y

y

y

y

TT

T

1

21

11

02

01

4424414241

3323313231

2221212221

1121111211

4

3

2

1

][

*)(ˆ*)(ˆ)(ˆ)(ˆ

*)(ˆ*)(ˆ)(ˆ)(ˆ

*)(ˆ*)(ˆ)(ˆ)(ˆ

*)(ˆ*)(ˆ)(ˆ)(ˆ

−=

=

=

µµµµµµµµµµµµ

µµµµ

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Takagi and Sugeno have (a) generalised the method to an n-rule, m-parameter system; and (b) claim that ‘this method of identification enables us to obtain just the same parameters as the original system, if we have a sufficient number of noiseless output data for identification’ (Takagi and Sugeno, 1985:119).

[ ] [ ] [ ][ ] [ ] [ ]YXXXP

PXY

p

p

p

p

xxxxxx

xxxxxx

xxxxxx

xxxxxx

y

y

y

y

TT

T

1

21

11

02

01

4424414241

3323313231

2221212221

1121111211

4

3

2

1

][

*)(ˆ*)(ˆ)(ˆ)(ˆ

*)(ˆ*)(ˆ)(ˆ)(ˆ

*)(ˆ*)(ˆ)(ˆ)(ˆ

*)(ˆ*)(ˆ)(ˆ)(ˆ

−=

=

=

µµµµµµµµµµµµ

µµµµ

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In order to determine the values of the parameters p in the consequents, one solves the LINEAR system of algebraic equations and tries to minimize the difference between the

ACTUAL output of the system (Y) and the simulation

[X]T[P] :

[ ] [ ] [ ] ε≤− PXY T

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In order to determine the values of the parameters p in the consequents, one solves the LINEAR system of algebraic equations and tries to minimize the difference between the

ACTUAL output of the system (Y) and the simulation

[X]T[P] :

[ ] [ ] [ ] ε≤− PXY T

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Takagi and Sugeno have described a fuzzy implication R as:

R: if (x1 is µA(x1),… xk is µA(xk)) then y = g(x1, …, xk), where:

A zero order Takagi-Sugeno Model will be given asR: if (x1 is µA(x1),… xk is µA(xk)) then y = k

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FUZZY LOGIC & FUZZY SYSTEMS

Knowledge Representation & Reasoning: The Air-conditioner Example

Consider the problem of controlling an air-conditioner (again). The rules that are used to control the air-conditioner can be expressed as a cross product:

CONTROL = TEMP ×SPEED

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The rules can be expressed as a cross product of two term sets: Temperature and Speed.

CONTROL = TEMP × SPEED

Where the set of linguistic values of the term sets is given as

TEMP = COLD + COOL + PLEASANT + WARM + HOTSPEED = MINIMAL + SLOW + MEDIUM + FAST + BLAST

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A Mamdani ControllerRecall that the rules governing the air-conditioner are as follows:RULE#1: IF TEMPis COLD THEN SPEEDis MINIMAL

RULE#2: IF TEMPis COOL THEN SPEEDis SLOW

RULE#3: IF TEMPis PLEASENT THEN SPEEDis MEDIUM

RULE#4: IF TEMPis WARM THEN SPEEDis FAST

RULE#5: IF TEMPis HOT THEN SPEEDis BLAST

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A Zero-order Takagi-Sugeno ControllerRecall that the rules governing the air-conditioner are as follows:RULE#1: IF TEMPis COLD THEN SPEED=k1

RULE#2: IF TEMPis COOL THEN SPEED= k2

RULE#3: IF TEMPis PLEASENT THEN SPEED=k3

RULE#4: IF TEMPis WARM THEN SPEED=k4

RULE#5: IF TEMPis HOT THEN SPEED=k5

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A First-order Takagi-Sugeno ControllerRecall that the rules governing the air-conditioner are as follows:RULE#1: IF TEMPis COLD THENSPEED=j1+k1*T

RULE#2: IF TEMPis COOL THENSPEED= j2+k2*TRULE#3: IF TEMPis PLEASENT THENSPEED=k3RULE#4: IF TEMPis WARM THENSPEED= j4+ k4 T

RULE#5: IF TEMPis HOT THENSPEED=k5

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FUZZY LOGIC & FUZZY SYSTEMS Knowledge Representation & Reasoning: The Air-conditioner Example

Temperature Fuzzy Sets

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

Temperature Degrees C

Tru

th V

alu

e ColdCoolPleasentWarmHot

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The analytically expressed membership for the reference fuzzy subsets for the temperatureare:

301)(

3025115.2

)(''

5.275.225.55

)(

5.225.175.35

)('''

;205.1785.2

)(

5.171565.2

)(''

;5.175.125.35

)(

5.1205.12

)(''

;100110

)(''

≥=

≤≤−=

≤≤−−=

≤≤−=

≤≤+−=

≤≤−=

≤≤+−=

≤≤=

≤≤+−=

TT

TT

THOT

TT

T

TT

TWARM

TT

T

TT

TPLEASENT

TT

T

TT

TCOOL

TT

TCOLD

HOT

HOT

WARM

WARM

PLEA

PLEA

COOL

COOL

COLD

µ

µ

µ

µ

µ

µ

µ

µ

µ

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Speed Fuzzy Sets

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90 100

Speed

Tru

th V

alue

MINIMALSLOWMEDIUMFASTBLAST

For FLC of Mamdani type

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The zero-order speed control just takes one SINGLETON value at fixed values of the velocity; for all other values the membership function is defined as zero

1001)(''

701)(''

501)('

301)(''

;01)(''

==========

VVBLAST

VVFAST

VVMEDIUM

VVSLOW

VVMINIMAL

BLAST

FAST

MED

SLOW

MINIMAL

µµµµµ

12

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Let the temperature be 5 degrees centigrade:Fuzzification: 5 degrees means that it can be COOL and COLD;Inference: Rules 1 and 2 will fire:

Composition:The temperature is ‘COLD’ with a truth value of µ COLD=0.5 � the SPEED will be k1The temperature is ‘COOL’ with a truth value of µCOOL =0.5 � the SPEED will be k2‘DEFUZZIFICATION’: CONTROL speed is(µ COLD*k1+ µCOOL *k2)/(µ COLD+ µCOOL)=(0.5*0+0.5*30)/(0.5+0.5)=15 RPM

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Zero Order Takagi Sugeno Controller

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100 120 140

Speed

Mem

ber

ship

Fu

nct

ion

MINIMAL

SLOW

MEDIUM

FAST

BLAST

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FUZZIFICATION: Consider that the temperature is 16oC and we want our knowledge base to compute the speed. The fuzzification of the the crisp temperature gives the following membership for the Temperature fuzzy set:

µCOLD µCOOL µPLEASENT µWAR

M

µHOT

Temp=16 0 0.3 0.4 0 0

Fire Rule (#)

yes/no

(#1)

no

(#2) yes

(#3)

yes

(#4)

no

(#5)

no

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INFERENCE: Consider that the temperature is 16oC and we want our knowledge base to compute the speed. Rule #2 & 3 are firing and are essentially the fuzzy patches made out of the cross products of

COOL x SLOWPLEASANT x MEDIUM

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COMPOSITION: The COOL and PLEASANT sets have an output of 0.3 and 0.4 respectively. The singleton values for SLOW and MEDIUM have to be given an alpha-level cut for these output values respectively:

Zero-order Takagi-Sugeno Model

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 20 40 60 80 100 120

Speed

Mem

ber

ship

Fu

nct

ion

(alp

ha-

cut)

MINIMAL

SLOW

MEDIUM

FAST

BLAST

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DEFUZZIFICATION: The problem of finding a single, crisp value is no longer a problem for a Takagi-Sugeno controller. All we need is the weighted average of the singleton values of SLOW & MEDIUM.

Recall the Centre of Area computation for the Mamdani controller

)()(

1.

.

.....1

.....1

yyy

output

xxoutput

xx

n

n

µµ

η ΣΣ

=

14

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DEFUZZIFICATION: For Takagi-Sugeno, the computation for η is restricted to the singleton values of the SPEED linguistic variable – we do not need to sum over all values of the variable y:

(.)(.)

(.)*(.)*

)(

)(

__

__

.....1

.....1

'''

''

__________

'

__________

.

.

µµµµ

µµ

η

MEDIUM

x

SLOW

x

MEDIUM

x

SLOW

x

output

xx

output

xx

MEDIUMSLOW

y

yy

n

n

+

+=

Σ=

Σ

Σ

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DEFUZZIFICATION: For Takagi-Sugeno, the computation for η is restricted to the singleton values of the SPEED linguistic variable – we do not need to sum over all values of the variable y:

41.42857)4.03.0(

)50*4.030*3.0(

(.)(.)

(.)*(.)*

__

__

'''

''

__________

'

__________

=++=

+

+=

Σµµ

µµη MEDIUM

x

SLOW

x

MEDIUM

x

SLOW

xMEDIUMSLOW

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DEFUZZIFICATION: Recall the case of the Mamdani equivalent of the fuzzy air-conditioner – where we had fuzzy sets for the linguistic variables SLOW and MEDIUM: The ‘Centre of Area’ (COA) computations involved a weighted sum over all values of speed between 12.5 and 57.5 RPM: in the Takagi-Sugeno case we only had to consider values for speeds 30RPM and 50 RPM.

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DEFUZZIFICATION: Recall the case of the Mamdani equivalent of the fuzzy air-conditioner – where we had fuzzy sets for the linguistic variables SLOW and MEDIUM: The ‘Centre of Area’ (COA) computations involved a weighted sum over all values of speed between 12.5 and 57.5 RPM: in the Takagi-Sugeno case we only had to consider values for speeds 30RPM and 50 RPM.

SPEED SLOW MEDIUM OUTPUT OF 2 RULES WEIGHTED SPEED

12.5 0.125 0 0.125 1.5625

15 0.25 0 0.25 3.75

17.5 0.3 0 0.3 5.25

20 0.3 0 0.3 6

22.5 0.3 0 0.3 6.75

25 0.3 0 0.3 7.5

27.5 0.3 0 0.3 8.25

30 0.3 0 0.3 9

32.5 0.3 0 0.3 9.75

35 0.3 0 0.3 10.5

37.5 0.3 0 0.3 11.25

40 0.3 0 0.3 12

42.5 0.3 0.25 0.3 12.75

45 0.25 0.4 0.4 18

47.5 0.125 0.4 0.4 19

50 0 0.4 0.4 20

52.5 0 0.4 0.4 21

55 0 0.4 0.4 22

57.5 0 0.25 0.25 14.375

SUM 5.925 218.6875

The speed is

36.91 RPM

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DEFUZZIFICATION:

For Mean of Maxima for the Mamdani controller, we had to have an alpha-level cut of 0.4, and the summation ran between 45-57.5 RPM, leading to a speed of 50 RPM. We get the same result for Takagi-Sugeno controllers:

η= (0.4*50)/0.4=50 RPM

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DEFUZZIFICATION: Comparing the results of two model identification exercises – Mamdani and Takagi-Sugeno- we get the following results:

Controller

Takagi-Sugeno

(RPM)

Mamdani

(RPM)

Centre of Area

41.43 36.91

Mean of Maxima

50.00 50.00

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DEFUZZIFICATION: Comparing the results of two model identification exercises – Mamdani and Takagi-Sugeno- we get the following results:

Mamdani Controller and the use of COA is the

best result ErrorController Takagi-Sugeno Mamdani

(RPM) (RPM)

Centre of Area 12% 0%Mean of Maxima 35% 35%

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A formal derivationConsider a domain where all fuzzy sets are associated withlinear membership functions.

Let us denote the membership function of a fuzzy set A asµµµµA(x), x ∈∈∈∈X. All the fuzzy sets are associated with linearmembership functions. Thus, a membership function ischaracterised by two parameters giving the greatest grade1 and the least grade 0.

The truth value of a proposition “x is mA and y is mB” isexpressed as

BABA µµµyµx Λ=isandis

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FUZZY CONTROL FUZZY CONTROLLERS- An example

e(k)

N Z P

∆∆∆∆e(k) N N N Z

Z N Z P

P Z P P

A worked exampleConsider an FLC of Mamdani type:

which expresses rules like:

Rule 1: If e(k) is negative AND ∆e(k) is negative then ∆u(k) is negativeALSO

Rule 9: If e(k) is positive AND ∆e(k) is positive then ∆u(k) is positive

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e(k)

N Z P

∆∆∆∆e(k) N N N Z

Z N Z P

P Z P P

A worked exampleConsider an FLC of Mamdani type:

The nine rules express the dependence of (change) in the value of control output on the error (the difference between expected and output values) and the change in error).

This dependence will capture some very complex non-linear, and linear

relationships between e and ∆e and ∆u.

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e(k)

N Z P

∆∆∆∆e(k) N αααα1 αααα2 αααα3

Z αααα4 αααα5 αααα6

P αααα7 αααα8 αααα9

A zero-order Takagi-Sugeno Controller

which expresses rules like:Rule 1:If e(k) is negative AND ∆e(k) is negative then ∆u(k) =αααα1

Rule 9:If e(k) is positive AND ∆e(k) is positive then ∆u(k) = αααα9

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e(k)

N Z P

∆∆∆∆e(k) N αααα1e+ß1 ∆∆∆∆e +δ1 αααα2e+ß2 ∆∆∆∆e+ δ2 αααα3e+ß3 ∆∆∆∆e+ δ3

Z αααα4e+ß4 ∆∆∆∆e +δ4 αααα5e+ß5 ∆∆∆∆e+ δ5 αααα6e+ß6 ∆∆∆∆e+ δ6

P αααα7e+ß7 ∆∆∆∆e+ δ7 αααα8e+ß8 ∆∆∆∆e+ δ8 αααα9e+ß9 ∆∆∆∆e+ δ9

which expresses rules like:

Rule 1:If e(k) is negative AND ∆e(k) is negative then ∆u(k) =αααα1e+ß1 ∆∆∆∆e+δ1

Rule 9:If e(k) is positive AND ∆e(k) is positive then ∆u(k) =αααα9e+ß9 ∆∆∆∆e+δ9

A first-order Takagi-Sugeno Controller

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52


Takagi and Sugeno have described a fuzzy implication R is of the format:

R: if (x1 is µA(x1),… xk is µA(xk)) then y = g(x1, …, xk), where:

y Variable of the consequence whose value is inferred

x1, …, xkVariables of the premise that appear also in the part of the consequence

mA(x1),….. mA(x1)

Fuzzy sets with linear membership functions representing a fuzzy subspace in which the implication R can be applied for reasoning.

f Logical function connects the propositions in the premise.

g Function that implies the value of y when x1,…. xk

satisfies the premise.

53


In the premise if µA(xi) is equal to Xi for some i where Xi is the universe of discourse of xi, this term is omitted; xi is unconditioned; otherwise xi

is regarded as conditioned.

54


In the premise if µA(xi) is equal to Xi for some i where Xi is the universe of discourse of xi, this term is omitted; xi is unconditioned. The following example will help in clarifying the argumentation related to 'conditioned' and 'unconditioned' terms in a given implication:

R: if x1 is small and x2 is big then y = x1 + x2 + 2x3.

The above implication comprises two conditioned premises, x1 and x2, and one unconditioned premise, x3.

The implication suggests that if x1 is small and x2 is big, then the value of y would depend upon and be equal to the sum of x1, x2, and 2x3., where x3 is unconditioned in the premise.

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55


Typically, for a Takagi-Sugeno controller, an implication iswritten as:

R: if x1 is µ1 and … and xk is µk

then y = p0 + p1x1 + … + pkxk.

The assumption here is that only ‘and’ connectives are used in the antecedants or premises of the rules. And, that the relationship between the output and inputs is strictly a LINEAR (weighted average) relationship. (The weights here

are p0 ,p1….. pk).

56


Reasoning AlgorithmRecall the arguments related to the membership functions of the union and intersection of fuzzy sets. The intersection of sets A and B is given as:

µA∩B = min (µA, µB)We started this discussion by noting that we will explore the problems of multivariable control (MultipleInputSingleOutput ). Usually, the rule base in a fuzzy control system comprises a number of rules; in the case of multivariable control the relevant rules have to be tested for what they imply.

57


Consider a system with n implications (rules); the variable of consequence, y, will have to be notated for each of these implications, leading to yi variables of consequence. There are three stages of computations in Takagi-Sugeno controllers:FUZZIFICATION: Fuzzify the input. For all input variables compute the implication for each of the rules;INFERENCE or CONSEQUENCES: For each implication compute the consequence for a rule which fires. Compute the output y for the rule by using the linear relationship between the inputs and the output (y = p0 + p1x1 + … + pkxk.). AGGREGATE (& DEFUZZIFICATION): The final output y is inferred from n-implications and given as an average of all individual implications yi with weights |y= yi |:

y = (Σ |y= yi | * yi )/ Σ |y= yi |

where |y= yi | stands for the truth value of a given proposition.

20

58


Consider the following fuzzy implications (or rules) R1,R2, R3 used in the design of a Takagi-Sugeno controller:

R1 �� If x1 is small1 & x2 is small2 then y(1) =x1+x2

R2�� If x1 is big1 then y(2) = 2x1

R3�� If x2 is big2 then y(3) =3x2

where y (i) refers to the consequent variable for each rule labelled Ri andx1andx2 refer to the input variables that appear in premise of the rules.

59


x Small1 Small2 Big1 Big20 1 1 0 01 0.938 0.875 0 02 0.875 0.75 0 03 0.813 0.625 0 0.1254 0.75 0.5 0 0.255 0.688 0.375 0 0.3756 0.625 0.25 0 0.57 0.563 0.125 0 0.6258 0.5 0 0 0.759 0.438 0 0 0.875

10 0.375 0 0 111 0.313 0 0.1 112 0.25 0 0.2 113 0.188 0 0.3 114 0.125 0 0.4 115 0.063 0 0.5 116 0 0 0.6 117 0 0 0.7 118 0 0 0.8 119 0 0 0.9 120 0 0 1 1

The membership function for small1, small2, big1 and big2 are given as follows

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Takagi-Sugeno Example pp 117

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20

Input x

Mem

bers

hip

Func

tion

Small1

Small2

Big1

Big2

The membership function for small1, small2, big1 and big2 are given as follows

21

61

FUZZY CONTROL FUZZY CONTROLLERS – Takagi-Sugeno Controllers An example

Let us compute the FINAL OUTPUT y for the following values:

x1 = 12 & x2 = 5 using Takagi and Sugeno’s formula:

y = (Σ |y= yi | * yi )/ Σ |y= yi |

where |y= yi | stands for the truth value of a given proposition.

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FUZZUFICATION: We have the following values of the membership functions for the two values x1 = 12 & x2 = 5:

Small1 Small2 Big1 Big212 0.25 0 0.2 15 0.6875 0.375 0 0.375

x 1 =

x 2 =

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INFERENCE & CONSEQUENCE :x1 = 12 & x2 = 5

Rule Premise 1 Premise 2 Consequenc

e

Truth ValueMin (Premise 1 & Premise2)

R1 Small1 ( x1 )

= 0.25

Small2 ( x2 )

= 0.375

y(1)= x1 + x2

=12+5

Min(0.25, 0.375)=0.25

R2 Big1 ( x1 )

= 0.2

y(2)= 2x1

= 24

0.2

R3 Big2 ( x2 )

= 0.375

y(3)= 3x2

= 15

0.375

22

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AGGREGATION (&DEFUZZIFICATION) :x1 = 12 & x2 = 5

y = (Σ |y= yi | * yi )/ Σ |y= yi |Using a Centre of Area computation for y we get:

8.17375.02.025.0

15*375.024*2.017*25.0

*

3,1

)(3,1

)()(

=++

++=

=

==

∑

∑

=

=

y

yy

yyy

y

i

ii

ii

65

FUZZY CONTROL FUZZY CONTROLLERS

Key difference between a Mamdani-type fuzzy system and the Takagi-Sugeno-Kang System?

A zero-order Sugeno fuzzy model can be viewed as a special case of the Mamdani fuzzy inference system in which each rule is specified by fuzzy singleton or a pre-defuzzified consequent. In Sugeno’s model, each rule has a crisp output, the overall input is obtained by a weighted average – this avoids the time-consuming process of defuzzification required in a Mandani model. The weighted average operator is replaced by a weighted sum to reduce computation further. (Jang, Sun, Mizutani (1997:82)).

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Takagi, T., & Sugeno, M. (1985). ‘Fuzzy Identification of Systems and its Applications to Modeling and Control’. IEEE Transactions on Systems, Man and Cybernetics. Volume No. SMC-15 (No.1) pp 116-132.

X

XX

X

X

23

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Takagi and Sugeno (1985) have argued that in order to develop a generic and simple mathematical tool for computing fuzzy implications one needs to look at a fuzzy partition of fuzzy input space. In

each fuzzy subspace a linear input-output relation is formed. The output of fuzzy reasoning is given by the values inferred by some implications that were applied to an input.

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Takagi-Sugeno model is an approximation of a Mamdani controller – in that the Takagi-Sugeno model ignores the fuzziness of linguistic variables in the consequent, but accounts for the fuzziness of variables in the antecedants; whereas Mamdani controller takes into the fuzziness of variables appearing both in the antecdents and the consequent.

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Model Identification: Given a choice between two models, say Mamdani and Takagi-Sugeno, we have to first identify why to chose a fuzzy logic system (will a crisp description not work as it is simpler to compute) and second whether to use an elaborate model (say Mamdani) rather than an approximation to the model (say, Takagi-Sugeno).

The choice can be based on the relative performance of the two models

Mamdani & Takagi-Sugeno Controllers · Mamdani style inference: The Good News: This method is...

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