REC 2006 - Georgia Tech University Savannah Feb 22-24 Reduction in Space Complexity And Error Detection/Correction of a Fuzzy controller F. Vainstein Georgia

REC 2006 - Georgia Tech University SREC 2006 - Georgia Tech University Savannah Feb 22-24avannah Feb 22-24

Reduction in Space Complexity And Error Detection/Correction of a Fuzzy controller

F. VainsteinGeorgia Institute of Technology

[email protected]

E. MartePontificia universidad Católica Madre y Maestra

[email protected]

V. OsoriaUniversidad Tecnológica de Santiago

[email protected]

R. RomeroInstituto Tecnologico de Santo [email protected]

mailto:[email protected]





AgendaAgenda

IntroductionIntroductionFuzzification, Rule Base and DefuzzificationFuzzification, Rule Base and DefuzzificationAddress Generation Address Generation Variable Radix Numbers (Multi Radix) Variable Radix Numbers (Multi Radix) Reduction in space complexity Reduction in space complexity Data compression and error correcting codes in Rule Base Data compression and error correcting codes in Rule Base Conclusion Conclusion


IntroductionIntroduction

Fuzzy controllers prove to be very useful for practical applications, Fuzzy controllers prove to be very useful for practical applications, especially in the cases when there is no appropriate mathematical especially in the cases when there is no appropriate mathematical model of behavior of the controlled object. Control signal is model of behavior of the controlled object. Control signal is computed by fuzzy controller with the use of rule base table. computed by fuzzy controller with the use of rule base table.

Fuzzy sets and some basic ideas pertaining to their theory were first Fuzzy sets and some basic ideas pertaining to their theory were first introduced in 1965 by Lofti A. Zadeh, a Professor of Electrical introduced in 1965 by Lofti A. Zadeh, a Professor of Electrical Engineering at the University of California a Berkeley. The Engineering at the University of California a Berkeley. The development of fuzzy set theory and fuzzy logic experimented development of fuzzy set theory and fuzzy logic experimented changes since their introduction. Therefore the “Fuzzy Boom” (since changes since their introduction. Therefore the “Fuzzy Boom” (since 1989) has been characterized by a rapid increase in successful 1989) has been characterized by a rapid increase in successful industrial applications that have netted impressive revenues. industrial applications that have netted impressive revenues.


Braunstingl (1995) developed a wall-following robot that used a fuzzy logic Braunstingl (1995) developed a wall-following robot that used a fuzzy logic controller and local navigation strategy to determine its movement. The controller and local navigation strategy to determine its movement. The fuzzy logic controller uses the input variables to control the firing of 33 rules.fuzzy logic controller uses the input variables to control the firing of 33 rules.

A fuzzy system developed by Surmann (1995) controls the navigation of an A fuzzy system developed by Surmann (1995) controls the navigation of an autonomous mobile robot. The entire system has about 180 fuzzy rules that autonomous mobile robot. The entire system has about 180 fuzzy rules that associate 30 fuzzy inputs with 11 outputs. Potentially, the number of fuzzy associate 30 fuzzy inputs with 11 outputs. Potentially, the number of fuzzy rules can be very large.rules can be very large.


The contribution of this paper is to tackle the space complexity of the The contribution of this paper is to tackle the space complexity of the system by decreasing the number of addresses lines used to store If-Then system by decreasing the number of addresses lines used to store If-Then rules. Also the incorporation of error correcting codes in the memory used to rules. Also the incorporation of error correcting codes in the memory used to store If-Then rules without substantial increasing space complexity as well store If-Then rules without substantial increasing space complexity as well as application of signatures analysis for error detection/location in a fuzzy as application of signatures analysis for error detection/location in a fuzzy controller. controller.


Fuzzification, Rule Base and DefuzzificationFuzzification, Rule Base and Defuzzification

FuzzySensor 0 Attributes

Values of Membership Functions

0.00.60.30.0

Sensor k 0.0 Crisp Output

0.00.00.70.50.00.00.0

Fuzzification

Fuzzification Addressgenerator

0r

kr

Rule Base(1,2,..r)

Computational Block

Fig. 1 Basic Fuzzy Controller Block diagram

Fuzzification Fuzzification

Rule BaseRule Base

Computational blockComputational block

DefuzzificationDefuzzification

REC 2006 - Savannah, Georgia Feb 22-24

Address Generation

Sensor 0(Position)

NM 0.0 To ComputationalNS 0.6 BlockZE 0.3PS 0.0

Sensor 1 PM 0.0 3(Velocity)

3

NM 0.0NS 0.0ZE 0.7PS 0.5

PM 0.0

Fuzzification

Fuzzification Addressgenerator

50 r

51 r

Fig. 2 Example of Address Generation

NM NS ZE PS PMNM 1 4 3 2 1NS 5 2 1 3 2ZE 1 3 2 1 4PS 5 5 3 2 1PM 1 2 5 1 2

Position

to computationalblock

Rule Base

velo

city

Fig. 3 Bidimensional Table for Decision

Take of the fuzzy Controller


Number of Lines and Space Complexity

Rule Base

02log r

kr2log

r2log

Fig. 4 Straight forward Address Generation

k

iirN

020 ][log

Example: Let k=9,

5... 910 rrr

Using (1) we obtain

300 N

.

(1)

number ofaddress lines

space complexit

y

exponential

Fig. 5 Space Complexity Vs Address Lines


Variable Radix Numbers (Multi Radix)

),...,( 0aaa kr (2)

},...,{ 0 krrr

}1,...,0{ 00 ra

}1,...,0{ 11 ra

……………….

}1,...,0{ kk ra

110102010 ...... kkr rrrarraraaa

The Multy Radix number ra has the valueBy Definition Multy Radix number can be written as follow:

Example: Let’s assume that

7,2,5 210 rrr

Then the multi radix number 4123 it then has the decimal value

10303ra


There exists natural one-to-one correspondence between the set of multi radix numbers and the set of fuzzy rules. It is demonstrated on Fig. 6 and Fig. 7.

Sensor 0

01

Sensor 1 2

01234

0 1 2 3 4

0

0 0

1

01 4

02 3

03 5

0 4 1

1

1 0 3

11 4

12 4

13 1

142

2

2 0 2

211

22 1

231

242

Fig. 6 Set of fuzzy Rules

Fig. 7 One-to-One Mapping


Multi Radix to Binary ConverterMulti Radix to Binary Converter

3

3

2

3

Convertor

+

0w

1w

2w

3w

0a

1a

2a

3a

9][log 32102 rrrr

Fig. 8 Multi Radix to Binary Converter

Denote by

kk rrrwrrwrw ...,...,, 1010201

.

Then

kkr wawawaaa ...22110

(4)

0 1 2 35, 5, 3, 6r r r r


Reduction in space complexityReduction in space complexity Sensor 0

Sensor k

N

Fuzzification

Fuzzification

Addressgenerator in

noncompact binary

0r

kr

Multiradix to Binary

Converter

Rule Base

][log 02 r

][log 2 kr

Fig. 9 Rule Base with Multi radix to Binary Converter

0202102 ][log...][log]...[log NrrrrrN kk (5)


Example:

Let 5..,9 90 rrk Then the number of initial addresses lines

300 N

.

The number of addresses lines after the Multi Radix to Binary Converter is equal to 24][log2 krN

Reduction in space complexity (cont.)Reduction in space complexity (cont.)

Data compression and error correcting codes in Rule BaseData compression and error correcting codes in Rule Base

Example:Suppose we have 2 Sensors 50 rr , 5r

Normally, for this case it will take 15 bits for representing a single row as shown in Fig.10.

Fig. 10 Single Row 2 Sensors Representation

1 4 3 0 1

Fig. 11 Center Row

To convert it to binary we need 12]15[log 52 bits. We saved 3 bits.

These bits can be used for error correction

1 4 3 0 1 001 100 011 000 001 15 Bits

Example:Suppose that we have 3 sensors, ;5210 rrr

5rIn this case we have a number with 25 digits. The biggest possible number

255 1To convert it to binary we need 59 bits. Therefore we saved 75-59=16 bits, see Fig. 12.

2 1 1 2 4 3 5 3 1 2

25 Positions25*3=75 Bits

…...…….

Fig. 12 3 Sensors representation

Data compression and error correcting codes in Rule Base Data compression and error correcting codes in Rule Base (cont.)(cont.)


Rule Base with data compression and Error Rule Base with data compression and Error Detection/correctionDetection/correction

N

Addressgenerator

(noncompact binary)

Variable Radix to Binary

Converter

Rule Base with error correction

code

][log 02 r

][log 2 kr

0N

Decoder

…Code Word Code Word Code Word

Decode

check bits(16)

Fig. 13 Rule Base block Diagram with data compression and Error Detection/Correction


Testing of a Fuzzy Controller by signature Analysis of test Testing of a Fuzzy Controller by signature Analysis of test

ResponseResponse

Test MPatters N Test

Responses

Variable Radix to Binary

Converter

Rule Base with error correction

code

][log 02 r

][log 2 kr

0N

Decoder

Fig. 14 Testing by signature analysis

MN ZZf 220:

(6)

)(xfS xf (7)

)(xfS xf

(8)

If the signatures fS and fS

are the same, the test is passed.

MZ2 Can be considered as a field FGF m )2(

FZx xN ,0

2


Conclusion and future workConclusion and future work

In this paper we introduced a new representation of numbers – Variable Radix Number system. In this paper we introduced a new representation of numbers – Variable Radix Number system. Using a Variable Radix Number system we decreased the number of addresses lines in a ROM Using a Variable Radix Number system we decreased the number of addresses lines in a ROM that is used to store If-Then rules, thus reducing the space complexity of a fuzzy controller. Also that is used to store If-Then rules, thus reducing the space complexity of a fuzzy controller. Also we incorporated error correcting codes in the memory used to store If-Then rules without we incorporated error correcting codes in the memory used to store If-Then rules without substantial increasing space complexity.substantial increasing space complexity.

Documents

REC 2006 - Georgia Tech University Savannah Feb 22-24 Reduction in Space Complexity And Error Detection/Correction of a Fuzzy controller F. Vainstein Georgia