1

Introduction to Type-2 FuzzySets and Systems

Jerry M. MendelMing Hsieh Department of Electrical Engineering

University of Southern Californiamendel@sipi.usc.edu

Outline

I. Type-2 fuzzy sets, especially intervaltype-2 fuzzy sets

II. Type-1 fuzzy logic systemsIII.Type-2 fuzzy logic systemsIV.Conclusions

2

I. Type-2 Fuzzy Sets Especially

Interval Type-2 Fuzzy Sets

What is a T2 FS and How isit Different From a T1 FS?

T1 FS: crisp grades of membership

3

What is a T2 FS and How isit Different From a T1 FS?

T1 FS: crisp grades of membership

T2 FS: fuzzy grades of membership, a fuzzy-fuzzy set

!A

New T2 notation

Where Does a T2 FS ComeFrom?

Consider a FS as a model for a word Words mean different things to different

people So, we need a FS model that can

capture the uncertainties of a word A T2 FS can do this Let’s see how

4

Collect Data froma Group of Subjects

“ On a scale of 0–10 locate the end points of an intervalfor some eye contact”

l r0 x

10

Average of left-end (l) andright-end (r) of intervals

Collect Data froma Group of Subjects

“ On a scale of 0–10 locate the end points of an intervalfor some eye contact”

5

Create a Multitude of T1 FSs Choose the shape of the MF, as we do for T1

FSs, e.g. symmetric triangles Create lots of such triangles that let us cover

the two intervals of uncertainty

Fill-er-in and Some NewTerms

UMF: Upper membership function (MF) LMF: Lower MF Shaded region: Footprint of uncertainty (FOU)

6

Weighting the FOU

Non-uniform secondary MF: General T2 FS Uniform secondary MF: Interval T2 FS

More Terms

7

Interval T2 FSs

Rest of tutorial focuses exclusively on IT2 FSs Computations using general T2 FSs are very costly Many computations using IT2 FSs involve only interval

arithmetic All details of how to use IT2 FSs in a fuzzy logic system

have been worked out Software available Lots of applications have already occurred

Other FOUs

Gaussian

Piece-wiselinear

+ Many others

8

Important Representationsof an IT2 FS: 1

Vertical Slice Representation—Veryuseful and widely used for computation

!A = Vertical Slices (x)!x"X

"

Important Representationsof an IT2 FS: 2 Wavy Slice Representation—Very

useful and widely used for theoreticaldevelopments

!A = Embedded T1 FS( j)!j

"

9

Important Representationsof an IT2 FS: 3

Wavy Slice: Also known as “Mendel-John Representation Theorem (RT)” Importance: All operations involving IT2

FSs can be obtained using T1 FSmathematics

Important Representationsof an IT2 FS: 3

Wavy Slice: Also known as “Mendel-John Representation Theorem (RT)” Importance: All operations involving IT2

FSs can be obtained using T1 FSmathematics

Interpretation of the tworepresentations: Both are coveringtheorems, i.e., they cover the FOU

10

II. Type-1 Fuzzy Logic Systems

Type-1 Fuzzy Logic System

Also called a Fuzzy Logic Controller, FuzzySystem, Fuzzy Rule-Based System, FuzzyExpert System,…

11

Provided by experts or extracted fromnumerical data

Expressed as a collection of IF-THENstatements

e.g., IF pressure is low and temperature ishigh, THEN turn the valve a bit to the right

Rules

Antecedent

Consequent

TI FLS Inference for OneRule

Rule: IF x1 is F

1 and x

2 is F

2, THEN y is G

µB(y)

12

TI FLS Inference-to-Outputfor Two Fired Rules

µB(y)

III.Type-2 Fuzzy Logic Systems

13

Interval Type-2 FLS

• Rules don’t change, only the antecedent and consequent FS models change

• Novel Output Processing: Going from a T2 fuzzy output set to a crisp output—type-reduction + defuzzification

Interpretation for an IT2 FLS

A T2 FLS is a collection of T1 FLSs

14

Use T1 FS Mathematics

Make use of the RT

IF x is !F, THEN y is !G

Use T1 FS Mathematics

B(y) ! µB(i, j ) (y)j=1

nG

!i=1

nF

! = µ"B(y),...,µ "B (y){ } "y#Yd

15

IT2 FLS Inference forOne Rule

Rule: IF x

1 is !F

1 and x

2 is !F

2, THEN y is !G

IT2 FLS Inference to Outputfor Two Fired Rules

16

Output Processing

• Defuzzification is trivial once type-reduction has been performed

Type-Reduction (TR): 1 TR methods are “extended” versions of T1

defuzzification methods that satisfy a: Fundamental design requirement: T2 FLS

results must reduce to T1 FLS results when alluncertainties disappear

17

Type-Reduction (TR): 1 TR methods are “extended” versions of T1

defuzzification methods that satisfy a: Fundamental design requirement: T2 FLS results

must reduce to T1 FLS results when all uncertaintiesdisappear

Different kinds of TR, just as there are different kindsof T1 defuzzification: Centroid Height Modified height Center-of sets

Type-Reduction:2

For an IT2 FS, TR leads to an interval-valued FS

TR involves computing the centroid ofa T2 FS, or a generalized centroid

No closed-form formulas Iterative algorithms are used

18

Centroid Type-Reduction: 1

Discretize y and u Create all of the embedded T1 FSs Compute the COG of each embedded T1 FS There will be COGs

Union of two fired-ruleoutput sets

nB

C!B= {ce1,...,cenB } = {yl ,..., yr}! [yl , yr ] as nB !"

nB

Centroid Type-Reduction: 2

Computing this way is impractical because: Each embedded T1 FS would have to be enumerated For large value of nB this would require an astronomical

number of computations Instead, only the two endpoints of are computed,

namely yl and yr This is done using two iterative algorithms, that are

called “KM Algorithms” (Karnik-Mendel) Regardless of what TR method one chooses, KM

Algorithms must be used to compute the interval end-points of that set

C!B

C!B

19

KM Algorithm for yl

yl = min!"i#[LMF (

!B|yi ),UMF (!B|yi )]

yi"ii=1

N

$ "ii=1

N

$

1. Initialize each !i as [LMF( !B | yi )+UMF( !B | yi )] / 2

2. Compute "c = yi!ii=1

N

# !ii=1

N

#

3. Find k (k = 1,...,N $1) such that yk % "c % yk+1.

4. Set !i =UMF( !B | yi ) i % k

LMF( !B | yi ) i & k +1

'()

5. Compute ""c * "c in Step 2 using !i from Step 4.

6. If ""c = "c STOP, set yl = ""c and call k L. Otherwise go to Step 7.

7. Set "c = ""c and GO TO Step 3.

KM Algorithm for yr

yr = max!"i#[LMF (

!B|yi ),UMF (!B|yi )]

yi"ii=1

N

$ "ii=1

N

$

1. Initialize each !i as [LMF( !B | yi )+UMF( !B | yi )] / 2

2. Compute "c = yi!ii=1

N

# !ii=1

N

#

3. Find k (k = 1,...,N $1) such that yk % "c % yk+1.

4. Set !i =LMF( !B | yi ) i % k

UMF( !B | yi ) i & k +1

'()

5. Compute ""c * "c in Step 2 using !i from Step 4.

6. If ""c = "c STOP, set yr = ""c and call k R. Otherwise go to Step 7.

7. Set "c = ""c and GO TO Step 3.

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About KM Algorithms: 1

Each KM Algorithm locates a switchpoint on the FOU, L for yl and R for yr

About KM Algorithms: 2 Formulas for yl and yr are:

yl = yl (L) =

yiUMF(!B | yi )

i=1

L

! + yiLMF(!B | yi )

i=L+1

N

!

UMF( !B | yi )i=1

L

! + LMF( !B | yi )i=L+1

N

!

yr = yr (R) =

yiLMF(!B | yi )

i=1

R

! + yiUMF(!B | yi )

i=R+1

N

!

LMF( !B | yi )i=1

R

! + UMF( !B | yi )i=R+1

N

!

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Properties of KMAlgorithms: 1

Convergence is monotonic Convergence is super-exponentially fast

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

yr

Linear convergenceSuper-exponentialconvergence

Iterations

Properties of KMAlgorithms: 2

In the limit, as the discretizations of yand u approach 0, L→yl and R→yr

Computation time can be greatlyreduced by using the “Enhanced” KMAlgorithms Better initialization Better organization of the steps Doesn’t recalculate everything from one iteration

to the next

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COS Type-Reduction Computing the union of fired-rule output sets is time-

consuming In Center-of-sets TR:

Centroids of all rule consequents are pre-computed usingKM Algorithms

Firing interval for each fired rule is used with the centroid ofits respective consequent to compute a GeneralizedCentroid–GC (also called an Interval Weighted Average)

Two KM Algorithms are used to compute the GC

GC(x) =

Fi(x)CGi

i=1

M

!

Fi(x)

i=1

M

!= [yl (x), yr (x)]

Computations in IT2 FLSthat Use COS TR

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Real-Time Applications Type-reduction may be a computational

bottleneck for a real-time application ofan IT2 FLS, because is uses theiterative KM Algorithms for itscomputations.

TR can be replaced using minimaxuncertainty bounds—Wu-Mendeluncertainty bounds

WM Uncertainty Bounds: 1

yl and yr are bounded

24

WM Uncertainty Bounds: 1

yl !yl + yl

2yr!yr+ y

r

2

yl and yr are bounded

WM Uncertainty Bounds: 2

To compute the WM UncertaintyBounds for COS TR: Compute COGs of four boundary T1 FLSs

using closed-form formulas (Two firing interval end-points) X (Two

consequent set centroid end-points) = 4 Compute the four uncertainty bounds using

closed-form formulas

25

WM Uncertainty Bounds: 3yli and yr

i : left- and right-end points of centroid of

ith consequent IT2 FS

Boundary T1 FLSs:

! {LMFs, left} : yl(0) (x) = f

iyli

i=1

M

! fi

i=1

M

!

! {LMFs, right} : yr(M ) (x) = f

iyri

i=1

M

! fi

i=1

M

!

! {UMFs, left} : yl(M ) (x) = f

iyli

i=1

M

! fi

i=1

M

!

! {UMFs, right} : yr(()) (x) = f

iyri

i=1

M

! fi

i=1

M

!

WM Uncertainty Bounds: 4yl (x) = min yl

(0)(x), yl

(M )(x){ } y

r(x) = max yr

())(x), yr

(M )(x){ }

yl(x) = yl (x)!

( fi ! f

i)

i=1

M

"

fi

i=1

M

" fi

i=1

M

"#

fiyli ! yl

1( )i=1

M

" fi

i=1

M

" ylM ! yl

i( )

fiyli ! yl

1( )i=1

M

" + fi

i=1

M

" ylM ! yl

i( )

$

%

&&&&

'

(

))))

yr (x) = yr(x)+

( fi ! f

i)

i=1

M

"

fi

i=1

M

" fi

i=1

M

"#

fiyri ! yr

1( )i=1

M

" fi

i=1

M

" yrM ! yr

i( )

fiyri ! yr

1( )i=1

M

" + fi

i=1

M

" yrM ! yr

i( )

$

%

&&&&

'

(

))))

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Computations in IT2 FLSthat Use Uncertainty Bounds

IV. Conclusions

27

Conclusions: 1

IT2 FLSs are simple to understand All computations can be developed

using T1-mathematics IT2 FLSs can be implemented in two

ways: Use type- reduction Use minimax uncertainty bounds

Conclusions: 2Educational perspective of T2 FSs and FLSs

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References

Go to http://en.wikipedia.org/wiki/Type-2_Fuzzy_Sets_and_Systems

It has a section entitled “FurtherReading” that will guide you to anappropriate article or book

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1

Practical and Applications Aspects of Type-2 FLCs

Professor Hani Hagras

The the University of Essex, England, UK

The 2009 IEEE International Conference on Fuzzy Systems, Jeju Island, Korea,

August 2009

2

Tutorial Outline and Learning Objectives

Sources of uncertainties in real world environments.

Type-1 and type-2 FLCs.

Overview of the interval type-2 FLC and its various components.

Hierarchical type-2 FLCs.

Type-2 FLC design.

Overcoming the computational overheads in type-2 FLCs.

Sample type-2 FLC applications.

Future directions in type-2 FLCs.

Conclusions

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3

Uncertainties and Real World Applications

The majority of real world applications face high

levels of uncertainties and imprecision that are

difficult to predict.

It is also very difficult to name and quantity all the

various sources causing these uncertainties.

These uncertainties cause problems for modelling,

decision support and the control for the associated

applications.

Hence, there have been various approaches for

uncertainty handling.

4

Possible Sources of Uncertainty and Imprecision- I

There are many sources of uncertainty facing any control system in dynamic real world unstructured environments and real world applications; some sources of these uncertainties are as follows:

– Uncertainties in the inputs of the system due to:

The sensors measurements being affected by high noise levels from various sources such a electromagnetic and radio frequency interference, vibration induced triboelectric cable charges, etc.

The input sensors being affected by the conditions of observation (i.e. their characteristics can be changed by the environmental conditions such as wind, sunshine, humidity, rain, etc.).

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5

Possible Sources of Uncertainty and Imprecision- II

Other sources of Uncertainties include:

– Uncertainties in control outputs which can result from the change of the actuators characteristics due to wear and tear or due to environmental changes.

– Linguistic uncertainties as words mean different things to different people.

– Uncertainties associated with the change in the operation conditions due to varying load and environment conditions.

6

Fuzzy Logic Control (FLC)

Fuzzy control is regarded as the most widely used application of fuzzy logic.

A Fuzzy Logic Controller (FLC) is credited with being an adequate methodology for designing robust controllers that are able to deliver a satisfactory performance in face of uncertainty and imprecision.

FLCs have been applied with great success to many real world applications.

In these applications, the FLCs have given satisfactory performances similar (or even better) to the human operators and have successfully outperformed the traditional control systems.

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Type-1 FLC-I

The vast majority of the FLCs that have been used

so far were based on the traditional type-1 FLCs.

However, type-1 FLCs cannot fully handle or

accommodate for the high levels of linguistic and

numerical uncertainties.

This is because type-1 FLCs use precise type-1

fuzzy sets and membership functions.

8

Type-1 FLC-II

For example, for a house environment a “Low” temperature could be associated with the following triangular type-1 fuzzy membership function

However, the centre of this triangular membership function and its end points vary according the following:

– The user of the system, where different users will have different preferences.

– Even for the same user, his preference will vary according to the season of year, his mode, country, context and the room in the house where “Low” temperature in the kitchen will be different to “Low” temperature in the living room.

Low Temperature

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Type-1 FLC- III

Hence, type-1 FLCs cannot handle the high levels of linguistic and numerical uncertainties available in the real world environments.

It is also obvious that the designed type-1 fuzzy sets can be sub-optimal under specific environment and operation conditions; however, because of the environment changes and the associated uncertainties, the chosen type-1 fuzzy sets might not be appropriate anymore.

This can cause degradation in the FLC performance, which can result in poor control and inefficiency and we might end up wasting time in frequently redesigning or tuning the type-1 FLC so that it can deal with the various uncertainties.

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Type-2 Fuzzy Sets- I

In a type-1 fuzzy set the membership grade for each element is a crisp number in [0,1]

A type-2 fuzzy set is characterised by a three dimensional membership function and a Footprint of Uncertainty (FOU).

Hence, the membership value (or membership grade); for each element of this set is a fuzzy set in [0,1].

FOUPrimary Membership

Secondary Membership

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11

Type-2 Fuzy Sets-II

The new third-dimension of type-2 fuzzy sets and the footprint of uncertainty provide additional degrees of freedom that make it possible to directly model and handle uncertainties.

Hence type-2 FLCs that use type-2 fuzzy sets in either their inputs or outputs have the potential to provide a suitable framework to handle the uncertainties in real world environments.

It is worth noting that a type-2 fuzzy set embeds a huge number of type-1 fuzzy sets.

12

Type-2 Fuzzy Sets-III

A type-2 fuzzy set is bounded from below by a

Lower Membership Function.

The type-2 fuzzy set is bounded from above by an

Upper Membership Function.

The area between the lower membership function

and the upper membership is entitled the Footprint

of Uncertainty.

Lower Membership

FunctionUpper

Membership Function

F

Footprint of Uncertainty

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13

Interval Type-2 Fuzzy Sets

When the secondary membership functions are interval sets, and, if this is true for x X, we have the case of an interval type-2 membership function which characterizes the interval type-2 fuzzy sets .

Interval secondary membership functions reflect a uniform uncertainty at the primary memberships of x.

Since all the memberships in an interval type-1 set are unity, in the sequel, an interval type-1 set is represented just by its domain interval, which can be represented by its left and right end-points.

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Why type-2 FLCs-I

Using type-2 fuzzy sets to represent the inputs/outputs of a FLC has many advantages when compared to the type-1 fuzzy sets:– As the type-2 fuzzy sets membership functions are fuzzy

and contain a FOU, they can model and handle the linguistic and numerical uncertainties associated with the inputs and outputs of the FLC.

– Using type-2 fuzzy sets to represent the FLC inputs and outputs will result in the reduction of the FLC rule base when compared to using type-1 fuzzy sets. This is because the uncertainty represented in the FOU in

type-2 fuzzy sets lets us cover the same range as type-1 fuzzy sets with a smaller number of labels.

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15

Why type-2 FLCs-II

Each input and output will be represented by a large number of type-1 fuzzy sets, which are embedded in the type-2 fuzzy sets. – The use of such a large number of type-1 fuzzy sets to describe

the input and output variables allows for a detailed description of the analytical control surface as the addition of the extra levels of classification give a much smoother control surface and response.

– In addition, the type-2 FLC can be thought of as a collection of many different embedded type-1 FLCs.

It has been recently shown that the extra degrees of freedom provided by the FOU enables a type-2 FLC to produce outputs that cannot be achieved by type-1 FLCs with the same number of membership functions – Where a type-2 fuzzy set may give rise to an equivalent type-1

membership grade that is negative or larger than unity [Wu and Tan 2005].

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Interval Type-2 FLCs

The general type-2 FLC is computationally intensive

and the computation simplifies a lot when using

interval type-2 FLC (using interval type-2 fuzzy sets).

This enables type-2 FLCs to operate in real time.

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The structure of Interval Type-2 FLC

Crisp Inputs

x

Fuzzifier

Rule Base

Inference

Engine

Defuzzifier

Crisp Outputs

Y

Type-2 input

fuzzy sets

Type-2 output

fuzzy sets

Type-reducer

Type-1 reduced Fuzzy

sets

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Type-2 FLC Operation-Fuzzification

The fuzzifier maps a crisp input vector with p inputs

x=(x1,…,xp)T X1 x X2… x Xp X into input fuzzy sets,

these fuzzy sets can, in general, be type-2 fuzzy input sets.

However singleton fuzzification is widely used as it is fast to

compute and thus suitable for the real time operation.

cm

Membership Function

Near

RFS

x1’

1F

1

1( ')F

x

1 1( ')F x

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Type-2 FLC Operation- Rule Base

The rules will remain the same as in the type-1 FLC.

However, the antecedents and the consequents will

be represented by type-2 fuzzy sets.

Consider a type-2 FLC having p inputs x1 X1,….,

xp Xp and c outputs y1 Y1,…., Yc Yc . The ith

rule for this Multi Input Multi Output FLC can be

written as follows:

Type-2 FLC Operation: The Inference Engine-I

The fuzzy inference engine gives a mapping from the input type-2 sets to the output type-2 sets.

In the inference engine, multiple antecedents in the rules are connected using the Meet operation.

The membership grades in the input sets are combined with those in the output sets using the extended sup-star composition.

Multiple rules are combined using the Join operation.

Just as the sup-star composition is the backbone computation for type-1 FLC, the extended sup-star composition is the backbone computation for a type-2 FLC.

The extended sup-star composition can be obtained simply by extending the type-1 sup-star composition by replacing type-1 membership functions by type-2 membership functions, the sup operation with Join operation and the t-norm operation with the Meet operation.

See [Mendel 2001a] for more details.

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Type-2 FLC Operation: The Inference Engine-II

The result of the input and antecedent operations,

which are contained in the firing set is an interval

type-1 set, as follows:

Type-2 FLC Operation: Type-Reduction-I

Type-reduction takes us from the type-2 output sets of the inference engine to a type-1 set that is called the “the type-reduced set” .

These type-reduced sets are then deuzzified to obtain crisp outputs.

As we are dealing with interval sets, the type-reduced set for the kth output will also be an interval set and has the following structure:

Hence the type-reduced set is identified by its left most point ylk and its right most point yrk where k= (1,….c) and c is the total number of outputs for the FLC.

The center of sets type reduction, is widely used as it has reasonable computational complexity that lies between the computationally expensive centroid type-reduction and the simple height and modified height type-reductions which have problems when only one rule fires.

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Type-2 FLC Operation: Type-Reduction-II

The calculation of the type-reduced sets is divided into two stages;

– The first stage is the calculation of centroids of the type-2 interval consequent sets of the ith rule which is conducted ahead of time and before starting the control cycle of the FLC.

– The second stage happens each control cycle to calculate the type-reduced sets which are then defuzzified to produce the crisp outputs to the actuators.

– In the following slides we will describe these two stages.

Type-Reduction- Calculating the

Centroids of the Rule Consequents- I

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Type-Reduction- Calculating the

Centroids of the Rule Consequents-II

To compute the centroid of each output fuzzy set we will use

the procedure developed by Karnik and Mendel.

In this procedure for the kth output we will discretise each

output fuzzy set into Z points, y1,…..yz where z=(1…Z).

yz

1

0

Lz

Rz

hz=(Lz +Rz)/2

Type-Reduction- Calculating the

Centroids of the Rule Consequents-III

Let hz= (Lz+Rz )/2 and z =(Rz- Lz)/2.

To calculate we will apply the iterative procedure

proposed by Karnik and Mendel.

t

rky

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Type-Reduction- Calculating the

Centroids of the Rule Consequents-IV

1. Without loss of generality assume that yz are arranged in an ascending order; i.e. y1 y2… yZ .Then,

2. Initialize z by setting z =hz for z= 1, …., Z and then compute y’= y(h1 , ….hz) using

3. Find e (1 e Z-1) such that ye y’ye+1.

4. Set wz=hz -z , for ze and wz=hz +z , for e z+1 and compute y”= y(h1-1, …. he-e ,he+1+e+1,…, hZ+Z ) using the above Equation.

5. Check if y”= y’ . If yes stop, y” is the maximum value of y(1 , ….Z) and it is .

6. If no go to step 5, Set y’ equal y”. Go to step 2.

The value of can be obtained using a similar procedure to the one used above with one change

– In step 3, set wz=hz +z , for z e and wz=hz -z , for z e+1 and compute y”= y(h1+1 , …. he+e , he+1-e+1,…, hZ-Z ) using the above Equation .

t

rky

t

lky

t

lky

Type-Reduction: Calculating the Type-Reduced Set-I

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Type-Reduction: Calculating the Type-Reduced Set-II

The procedure used to compute yrk can be written as follows:

Without loss of generality, assume that the pre-computed are arranged in an

ascending order; i.e. . Then,

1. Compute yr in by initially setting

where and let yr’yr

2. Find R (1 R M-1) such that

3. Compute yr in the above Equation using

4. If yr” yr’ then go to step 5. If yr”=yr’ then stop and set yr”yr

5. Set yr’ equal to yr” and return to step 2.

i

ry i

ry

Type-Reduction: Calculating the Type-Reduced Set- III

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Defuzzification

From the type-reduction stage we have for each

output a type-reduced set.

Each type-reduced set is an interval type-1 set

determined by its left most point yl and right most

point yr.

We defuzzify the interval set by using the average of

ylk and yrk

hence the defuzzified crisp output for each output k

is:

Type-2 HFLC-II

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Advantages of Type-2 HFLCs

It simplifies the design of the agent controller and reduces the number of rules to be determined compared to the single rule based systems.

The computation time for this hierarchical structure is small.

The FLC can achieve multiple goals, whose priorities may change with time as the behaviour preferences outputs and the context truth values (weight) are dynamic taking into account the situation of FLC.

The type-2 fuzzy coordination provides a smooth transition between behaviours with a consequent smooth output response which allows more than one behaviour to be active to differing degrees thereby avoiding the drawbacks of on-off switching schema (i.e. dealing with situations where several criteria need to be taken into account).

This hierarchical structure offers a flexible structure where new behaviours can be added or modified easily. The system is capable of performing very different tasks using identical behaviours by changing only the context rules and co-ordination parameters to satisfy a different high level objective.

Type-2 FLC Design- Neuro Fuzzy

Systems (Applied to Marine Diesel Engines)

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GA Learning of Type-2 FLCs-Applied to Robots

Heuristic Approaches- Applied

to Intelligent Buildings

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Overcoming the Computational Overheads in a Type-2 FLC-I

The higher order computation and especially type-

reduction causes computation bottlenecks that can

hinder the real time performance of the type-2 FLC.

However, recent work has presented methods that can

speed the type-2 FLC performance. This will open the

way to enable the widespread of type-2 FLCs in a

similar manner to type-1 FLCs.

The first approach to speed a type-2 FLC is by

overcoming the type-reduction by applying the Wu-

Mendel uncertainty bounds method.

Overcoming the Computational Overheads in a Type-2 FLC: Wu-

Mendel Uncertainty Bounds Method -I

• The Wu-Mendel Uncertainty Bounds method provides mathematical formulas for the inner and outer bound sets which can be used to approximate the type-reduced set.

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Overcoming the Computational Overheads in a Type-2 FLC: Wu-

Mendel Uncertainty Bounds Method -II

The inner bound sets could be written as follows:

–

The outer bounds sets could be written as follows:

Wu-Mendel Uncertainty Bounds Method– Results (Marine Engine)-I

It has been shown that the Wu-Mendel method

produces very similar control results to those

obtained using the KM type-reduction.

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Wu-Mendel Uncertainty Bounds Method–Results (Marine Engine)-II

Also the Wu-Mendel method can result in much faster responses than using

the KM type-reduction method.

Such computational savings will result in a faster control response.

This faster response will map to a better real-time response that will impact

not only our ability to meet hard real-time deadlines but also to free the

processor for executing other high priority components such as alarms and

monitoring whilst still having the superior type-2 FLC control response

Parallel Hardware Implementations of the type-2 FLC

KM CP

WMCP

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Parallel Hardware Implementations of the type-2 FLC- Results

Co-Processors support up to 64 fired rules thus a much larger rule base.

The complete embedded type-2 FLC is now 10 times faster than a

conventional PID controller

The co-processors provide a computational reduction of approximately

100% compared to the sequential implementation

The WM co-processor required an average of 45 % less clock cycles than

the KM co-processor

Type-2 FLC Applications: Industrial

Control- Marine Diesel Engines- I

Marine diesel engines are huge engines which due to their vast sizes and large power outputs require accurate and robust speed control/governing.

Accurate speed control of marine diesel engines is of critical importance as significant deviations from the speed set point could be detrimental and damaging to the engine and the respective loads.

Moreover, for applications such as power generation sets, the engine speed in rpm must be stable multiples of the generated base frequency i.e. 50Hz frequency would require the engine to operate at 1000 rpm, 1500 rpm etc. Hence, significant speed deviation can cause the generation of incorrect frequencies resulting in loss of synchronisation between the generator and associated power grid.

Robustness in speed control is required to overcome and recover quickly from the inherent instabilities and disturbances associated with the fast and dynamic changes of the environment, load and operation conditions.

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Type-2 FLC Applications: Industrial

Control- Marine Diesel Engines- II

There are many sources of uncertainties facing the speed controller of marine diesel engine including:

– Inputs Uncertainties: Sensor measurements are affected by high noise levels i.e. electromagnetic and radio frequency interference, vibration induced triboelectric cable charges

– Outputs Uncertainties: Actuator wear and tear where worn linkage between actuator and fuel pump can result in excessive friction and backlash causing instability.

– Uncertainties in Operation and load conditions: Varying loads, weather and sea conditions, wind strength, hull fouling, vessel displacement (dependant on cargo).

E.g. As a result of wind strength, hull fouling alone ship resistance (force working against ship propulsion) can increase by as much as 50-100% of the total ship resistance in calm weather.

– Designers differ in their choice for the control parameters

Due to these uncertainties the current control strategies that employ type-1 FLCs or PID controllers require continuous tuning to deal with the various faced uncertainties.

Type-2 FLC Applications: Industrial

Control- Marine Diesel Engines- III

There have been various developments of this type-2

FLC which resulted in a commercial embedded Neuro

type-2 FLC which can deliver control responses that

are 10 times faster than the employed commercial PID

controller.

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Other Type-2 FLC Industrial Applications: Buck DC-DC Converter

DC-DC converters are power electronic systems that convert one level of electrical voltage into another level by switching action.

The DC-DC converters are used extensively in consumer electronic devices.

The DC-DC converters are an intriguing subject from the control point of view due to their intrinsic nonlinearity.

The control technique for DC-DC converters must cope with their wide input voltage and load variations to ensure stability in any operating condition while providing fast transient response.

It is required to control the duty cycle so that the output voltage can supply a fixed voltage in the presence of the input voltage uncertainty and load variations.

It has been shown that the performance of the type-2 FLC is better than its type-1 counterpart where the rise time response of type-2 FLC is faster than that of type-1 FLC with no overshoot in the type-2 FLC controlled system.

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Other Type-2 FLC Industrial Applications: The Coupled-Tank

Liquid-Level Control

A GAwas used to evolve a type-2 FLC to control a liquid-level process.

The controlled process is the coupled tank apparatus.

The volumetric flow rate of the pumps in the coupled-tank apparatus used to produce the results is nonlinear, and the system has non-zero transport delay.

It was observed that both the type-1 and the type-2 FLCs are able to attenuate the oscillations when the modelling uncertainties are small.

The liquid level in the tank will eventually reach the desired set-point, though the settling time is shorter when type-2 FLC is employed.

When the modelling uncertainties are larger, the type-1 FLC will give rise to persistent oscillations while the type-2 FLC has the ability to eliminate these oscillations and the liquid level reaches its desired height at steady state.

Type-2 FLC Applications: Ambient Intelligent Environments- I

Ambient Intelligent Environments (AIEs) consist of a multitude of interconnected embedded systems which form a pervasive infrastructure that surround the user.

Intelligent agents are embedded into AIEs to form an intelligent “presence” allowing the AIE to recognize the user and program itself to their needs by learning from their behaviours in a non intrusive way.

AIEs face large amounts of uncertainty from various sources including:

– Uncertainties arising due to change of environmental conditions (such as the external light level, temperature, time of day (morning, evening...etc)) over a considerable long period of time due to seasonal variations.

– Uncertainties caused by the humans occupying these environments as their behaviours, moods and activities are dynamic, unpredictable and non-deterministic and change with time and season There is also the fact that different words mean different things in different times of the year and the values associated with a term such as „warm‟ in reference to temperature can vary from winter to summer.

– Uncertainties in the agent‟s controller outputs due to the change of actuators characteristics as a result of the wear and tear that occurs over time.

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Type-2 FLC Applications: Ambient

Intelligent Environments- II

Unique experiments were conducted with various users during an extended period (spanning the course of the year) inside the Essex intelligent Dormitory (iDorm).

It was possible to demonstrate how the type-2 agent can adapt in a life-long learning mode and handle the faced short- and long-term uncertainties.

The type-2 FLC based agents were compared with type-1 FLC based agents in their ability to model the user‟s behaviour while handling the long-term uncertainties.

The results showed that the type-2 FLC was better able to model the user behaviour and handle the short- and long-term uncertainties while using fewer rules.

It was also shown the type-2 agent was able to outperform the type-1 agent performance while achieving a 60% increase in processing speed as a result of attaining a 50% reduction in the size of the rule base, thus reducing memory usage.

In addition, the online experiments showed that the type-2 agents were able to adjust the environment in a more satisfactory way to the user when compared to the type-1 agents.

Type-2 FLC Applications: Robotics- I

There are many sources of uncertainty facing the robots operating in real world environments including:

– Uncertainties in the robot inputs where the sensors measurements are typically noisy and are affected by the conditions of observation (i.e. their characteristics are changed by the environmental conditions such as wind, sunshine, humidity, rain, etc.).

– Uncertainties in control actions which translates to uncertainties in the output membership functions of the FLC. Such uncertainties can result from the change of the actuators characteristics which can be due to the inconsistency of the terrain or the change of environmental changes or robot physical properties.

– Linguistic uncertainties as the meaning of words that are used in the antecedents and consequents linguistic labels can be uncertain as words mean different things to different designers.

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Type-2 FLC Applications: Robotics- II

Other Type-2 Fuzzy Systems Applications

In all the following applications the type-2 Fuzzy Logic Systems (FLSs) have given excellent responses when handling the faced uncertainties.

– Type-2 Fuzzy Adaptive Filters (FAF) in the equalization of nonlinear time-varying channels,

– Multi-category classification of ground vehicles based on their acoustic emissions,

– MPEG variable bit rate video modelling and classification.

– Connection admission control in ATM Networks.

– Estimating the network lifetimes for wireless sensor networks

– Forecasting sensed signal strength in wireless sensors

– Diagnosis of diseases.

– Representation of the perceptions of lung scan images in order to predict pulmonary emboli using type-2 fuzzy relations.

– Representation of the imprecise perceptions of nurses in the patient assessment process.

– Management of clinical cases of difficult or uncertain diagnosis.

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Notes from the Type-2 FLC Applications-I

When the level of uncertainty is low, the type-2 FLC will give

similar control responses to those achieved using type-1

FLCs.

It is advisable to use type-2 FLCs when there are high or

unpredictable levels of uncertainty. This is the case with the

majority of most real world applications.

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Notes from the Type-2 FLC Applications-II

The type-2 FLC embeds a large number of type-1 FLCs which allows the type-2

FLC to better handle the uncertainties while producing a smooth control surface.

Hence, a type-2 FLC with a smaller rule base can outperform a type-1 FLCs with

larger rule bases.

As the type-1 FLC increases its rule base, it approaches the smooth performance

of the type-2 FLC which uses a much smaller rule base.

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Notes from the Type-2 FLC Applications-III

Type-2 FLCs have been perceived to be

computationally expensive techniques for

uncertainty management in real world applications.

However, recent advances has resulted in speeding

the type-2 FLCs and producing hardware solutions

that enables a parallel type-2 FLC to be faster by 10

times than the used commercial controllers.

This opens the way to the widespread of type-2

FLCs in various real world applications.

Future Directions in Type-2 FLCs

After the success achieved in interval type-2 FLCs and their applications to many real world problems, there is now a strong direction towards investigating the power of general type-2 FLCs.

Encouraging results have been achieved in DMU in the robotics domain and a general type-2 FLC hardware chip has been also produced.

Also new theories are emerging which employs alpha cuts/z-slices which enables the theories developed for interval type-2 FLCs to be used in general type-2 FLCs.

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Conclusions

Type-2 FLCs offer a very powerful paradigm for

uncertainty handling in real world applications and

environments where there are large amounts of uncertainty

that are difficult to predict.

With the recent advances that allowed to speed the

performance of type-2 FLCs, it is now envisaged to see the

widespread of type-2 FLCs to many challenging

applications.

Work has started in investigating the power of general type-

2 FLCs following the great success of interval type-2 FLCs.

7/31/2009

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Generalised type-2 fuzzy sets

and future research directionsBob John and Simon Coupland

Centre for Computational Intelligence

www.cci.dmu.ac.uk

Generalised type-2 fuzzy sets

Generalised type-2 fuzzy sets allow for a third

dimension to capture higher order uncertainty

They lie between interval type-2 and type-1 where t1 is

crisp and it2 provides no extra information

They present design issues for implementation

More parameters

Choice of secondaries

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An example of ‘tallness’ Somebody who is 5ft 9inches (1.75m) – how ‘tall’ are

they?

Type-1 – 0.8

IT2 – [0.75,0.85]

GT2 – about 0.8 e.g.

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Why are they not used much (yet)?

A perception that they are computationally intractable

Defuzzification seen as a problem

Type reduction difficult because of the number of

embedded sets

Difficult to visualise/conceptualise

More difficult to understand the mathematics

More difficult to code

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Should you consider using them?

Yes!

The world is highly uncertain and current uncertainty

models not good at modelling human decision making

for example

Conceptually the ability to model uncertainty in a third

dimension should improve performance in certain

cases

We have used generalised type-2 in control and it

works….

So, what about defuzzification?

Geometric type-2 fuzzy systems

Geometric Type-1 and Type-2 Fuzzy Logic Systems,

IEEE T-FS, Vol. 15:1, 3-15, 2007 – Coupland and John

A Fast Geometric Method for Defuzzification of Type-2

Fuzzy Sets. Coupland, S.and John, R., IEEE

Transactions on Fuzzy Systems. 16(4):929-941, 2008.

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Geometric Type-2 Systems Model a GT2 FS as a surface in 3D:

Geometric Type-2 Systems Logical operators can be defined using geometric

properties rather than via the extension principle

Centroid defuzzication is fast

Sub-second inference on Mamdani FLC on PIC

technology

Enabled first control applications of GT2

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Sampling method

Sample the embedded fuzzy sets in type reduction

Produces surprisingly good results

A Novel Sampling Method for Type-2 Defuzzification

Sarah Greenfield, Robert John and Simon Coupland

Proc. UKCI 2005, pages 120 - 127

Sampling Method The centroid of a general type-2 fuzzy set as derived

using the extension principle contains massive

redundancy

The redundancy stems from larger numbers of

embedded sets using in the calculation give similar

answers

The sampling method calculates the centroid using a

sample of these embedded sets to give an approximate

answer very quickly

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Role of alpha planes and z-slices

Alpha planes/z-slices allow generalised type-2 to be represented by interval valued fuzzy sets

Allows interval valued operations to be used thus reducing computation

zSlices — towards bridging the gap between interval and general type-2 fuzzy logic. Wagner, C. Hagras, H. Fuzzy Systems, 2008. FUZZ-IEEE 2008. 489-497

Alpha-Plane Representation for Type-2 Fuzzy Sets: Theory and Applications. Mendel, J. M.; Liu, F.; Zhai, D. Accepted for IEEE Transactions on Fuzzy Systems.

Alpha Planes and z-slices Deconstruct a general type-2 fuzzy set into planes cut

along the Z axis:

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Alpha Planes and z-slices Process each plane/slice independently:

Implication example

Alpha Planes and z-slices Process each plane/slice independently:

Implication example

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Alpha Planes and z-slices Reconstruct a general type-2 fuzzy set from the

processed planes:

Summary of generalised Current uncertainty models not good at tackling difficult

problems

The third dimension offers an opportunity, particularly in

modelling human decision making

We can now implement generalised type-2 (so there is

no excuse!)

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Future Research

Directions

Current research Current research growing (www.type2fuzzylogic.org)

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Current Research Still relatively small numbers of people

Circa 560 papers on www.type2fuzzylogic.org of which

~50% are journal

But some quality work….

Won TFS award for 2004 and 2007

Best student paper at Fuzz-IEEE twice (2005 and 2006)

Future research Lots to do…

Theoretical and applied

Plenty of scope for tackling hard applications

Many theoretical results need proving for type-2

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Some research questions Optimisation of type-2 systems

Machine Learning for type-2 systems

When to use type-2?

Interval?

Generalised?

New representations for generalised type-2 fuzzy

systems?

Computing with Words

More research questions What do we mean by uncertainty of a type-2 fuzzy set?

Apply in real applications

Modelling human decision making

Control

…..

7/31/2009

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Generalised type-2 fuzzy sets

and future research directionsBob John and Simon Coupland

Centre for Computational Intelligence

www.cci.dmu.ac.uk