institute of intelligent power electronics – ipe page1 a dynamical fuzzy system with linguistic...

Institute of Intelligent Power Electronics – IPEPage1

A Dynamical Fuzzy System with Linguistic Information Feedback

Xiao-Zhi Gao and Seppo J. Ovaska

Institute of Intelligent Power Electronics

Department of Electrical and Communications Engineering

Helsinki University of Technology, Finland


Outline Introduction Basic Fuzzy Systems Conventional Dynamical Fuzzy Systems Fuzzy Systems with Linguistic Information

Feedback Simulation Results Conclusions and Remarks


Introduction Fuzzy logic theory has found successful

applications in industrial engineering Most fuzzy systems applied in practice are

static– static input-output mappings– no internal dynamics

A new dynamical fuzzy model with linguistic information feedback is proposed– suitable for dynamical system modeling, control,

filtering, time series prediction, etc.


Basic Fuzzy Systems

Feedforward Stucture (Mamdani Type)

IF x is A AND (OR) y is B THEN z is C


Conventional Dynamical Fuzzy Systems

Classical fuzzy systems lack necessary internal dynamics– can only realize static mappings

Feedback is needed to introduce dynamics Two kinds of conventional recurrent fuzzy

systems– Globally feedback fuzzy systems– Locally feedback fuzzy systems

Crisp information feedback


Globally Feedback Fuzzy Systems

Fuzzification Fuzzy Inference Defuzzification

1Z

Output and Crisp Feedback


Locally Feedback Fuzzy Systems

[Lee2000]

Internal Memory Units

Fuzzy Input Membership FunctionsCrisp

Output


Crisp Information Feedback

Defuzzification: Fuzzy->Nonfuzzy Conversion

Unavoidable Information Lost


Dynamical Fuzzy System with Linguistic Information Feedback

Inference Output (Membership Function) is fed back

Mamdani Type

Fuzzification Fuzzy Inference Defuzzification

1Z

)(),(),( kkk )()( yF

k)()1()( yk

)()1( yfk )()1( yF

k


Feedback ParametersTitle:demo_membership.epsCreator:MATLAB, The Mathworks, Inc.Preview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.

)()( )()( kyy Fkk

)()()( )()( yky Fkk

)(

)()( )()(kF

kk yy

)()( yFk


Diagram of Fuzzy Information Feedback Scheme

Linguistic Information Feedback

Feedback is controlled by

Inference Output Before Feedback

Fuzzy Feedback Information

Final System Output

Aggregation (max)

-1 0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1 0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1 0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1 0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Previous System Output

)()( yFk

)()1()( yk

)()1( yfk )()1( yF

k

)(),(),( kkk

,,


Linguistic Information Feedback for Individual Fuzzy Rules

Fuzzification

Fuzzy Rule 1

Defuzzification

1Z

)(),(),( 111 kkk

1Z

)(),(),( 222 kkk

Fuzzy Rule m

)(),(),( kkk mmm

Fuzzy Rule 2

1Z


High-Order Linguistic Information Feedback

Fuzzification Fuzzy Rule i

1Z

)(),(),( )1()1()1( kkk iii )()( yFk)()1(

)( yk

)()1( yfk )()1( yF

k

1Z

)(),(),( )2()2()2( kkk iii )()2()1( yk

)(),(),( )()()( kkk ni

ni

ni )()(

)( ynnk

)()1( yFk

)()( yFnk


Learning Algorithms of Feedback Parameters

Feedback parameters have a nonlinear relationship with system output

It is difficult to derive an explicit learning algorithm

Some general-purpose algorithms can be applied to optimize feedback parameters– genetic algorithms (GA)

)(),(max)( )1()()1( yyy f

kk

F

k

)(DEFUZZ )1(

*

)1( yz F

kk nonlinear operators


Advantages of Linguistic Information Feedback

1. Rich fuzzy inference output is fed back without any information transformation and loss

2. Local feedback connections can store temporal patterns– Suitable for dynamical system identification

3. Training of feedback coefficients leads to an equivalent update of output membership functions– Benefit of adaptation


Simulations

A simple dynamical fuzzy system with linguistic information feedback– single-input-single-output– two inference rules

» IF X is Small THEN Y is Small» IF X is Large THEN Y is Large

max-min and sum-product composition COA defuzzification Step input ( )5.20 x

1,8.0,5.1


Input and Output Fuzzy Membership FunctionsTitle:input_output_membership_functions.epsCreator:MATLAB, The Mathworks, Inc.Preview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.


Step Responses with First-Order Fuzzy Feedback

Solid line: max-min composition. Dotted line: sum-product composition

Title:step_response_1.epsCreator:MATLAB, The Mathworks, Inc.Preview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.


Step Response with Second-Order Fuzzy Feedback

Title:step_response_2.epsCreator:MATLAB, The Mathworks, Inc.Preview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.


Time Sequence Prediction I

1 2 3 40-1.5

-1

-0.5

0

0.5

1

1.5

Time in Samples

x(k) 10101


Fuzzy Predictor with Linguistic Information Feedback

Four fuzzy rules are constructed

– IF x(k) is [-1] THEN x(k+1) is [0]– IF x(k) is [0] THEN x(k+1) is [1]– IF x(k) is [1] THEN x(k+1) is [0]– IF x(k) is [0] THEN x(k+1) is [-1]

Rule 2 and Rule 4 are conflicting

Linguistic information feedback can correct


Input Membership Functions of Fuzzy Predictor

Title:input_membership.epsCreator:MATLAB, The Mathworks, Inc.Preview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.


Evolution of GA-Based Feedback Parameters Optimization

Title:fitness_evolution.epsCreator:MATLAB, The Mathworks, Inc.Preview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.


Prediction Outputs of Fuzzy Predictors

1 2 3 40-1.5

-1

-0.5

0

0.5

1

1.5

Time in Samples

Pre

dic

tion

Ou

tpu

t Dotted line: static fuzzy predictor. Solid line: dynamical fuzzy predictor


Time Sequence Prediction II

1x

2x

1 2

3

4 5

11

1

1

0

1354321


Output Membership Functions of Fuzzy Predictor

Title:output_membership.epsCreator:MATLAB, The Mathworks, Inc.Preview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.


Prediction Outputs of Fuzzy Predictors

1 2 3 40-1.5

1

2

3

4

5

Time in Samples

Pre

dic

tion

Ou

tpu

t

5 6

Dotted line: static fuzzy predictor. Solid line: dynamical fuzzy predictor


Conclusions A new dynamical fuzzy system with linguistic

information feedback is proposed Dynamical properties of our fuzzy model are shown Present paper is a starting point for our future work

under this topic– more simulations are needed– extension to Sugeno type fuzzy sytems– extension to feedforward structure– extension to premise part– applications in dynamical system identification

institute of intelligent power electronics – ipe page1 a dynamical fuzzy system with linguistic...

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