3-D Fuzzy Logic Controller for Spatially Distributed Dynamic

7/30/2019 3-D Fuzzy Logic Controller for Spatially Distributed Dynamic

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AbstractThree-dimensional fuzzy logic controller (3-D

FLC) is a new fuzzy logic controller for spatially distributed

dynamic systems. The goal of this tutorial is to wipe of the magic

behind the FLC. This tutorial focuses on building an intuition

for how and why 3-D FLC works. Additionally, recent

development on 3-D FLC is presented. The hope is that by

addressing both aspects, readers of all levels will be able to gain

a better understanding of 3-D FLC as well as the when, the how

and the why of applying the techniques.

I. INTRODUCTIONhree-dimensional fuzzy logic control (3-D FLC) [1] is a

novel fuzzy control method developed for spatially

distributed dynamic systems. 3-D FLC uses a new fuzzy set

three-dimensional (3-D) fuzzy set, which is composed of the

traditional two-dimensional fuzzy set and a third dimension

for the spatial information, and carries out a 3-D inference

engine. Therefore, it has the inherent capability to express and

deal with spatial information. Through emulating the

behaviors that human operators or experts control the

spatially-distributed field from the point of view of overall

space domain, satisfactory nonlinear controller can be

developed. Moreover, the design of 3-D FLC doesnt require

accurate mathematical models of the systems. Thereby, 3-D

FLC has a great potential to a wide range of engineering

applications for spatially-distributed processes.The goal of this tutorial is to provide both an intuitive feel

for 3-D FLC, and a thorough discussion of this topic. We

begin with a simple example and provide an intuitive

explanation of the goal of 3-D FLC. Then, we will see how

and why 3-D FLC works. This understanding will lead us to a

prescription for how to apply 3-D FLC in the real world. We

will also give some recently developed results of 3-D FLC

such as mathematical explanation of its structure, stability

issue, and its extension to multiple control sources.

II. BACKGROUNDMost of physical processes or systems have the nature of

spatial distribution. Examples can be given from civil life to

industrial process, e.g. the temperature regulation in a

building, chemical reactor, semiconductor manufacturing,

Manuscript received February 1, 2009. This work was supported by the

National Science Foundation of China under Grant 60804033.Xian-xia Zhang and Ye Jiang are with Shanghai Key Laboratory of Power

Station Automation Technology, School of Mechatronics and Automation,

Shanghai University, Shanghai, China. (email: [email protected])Han-xiong Li is with Department of MEEM, City University of Hong

Kong , Hong Kong, China. (email: [email protected])

solar power plan, etc. Such process or system is usually called

spatially distributed dynamic system, since its states,

controls, and outputs depend on the spatial position as well as

the time [2]. To control such system, it usually involves

spatially distributed sensors and control sources. Moreover,

the control goal is usually to track a spatially distributed

profile. It is completely different to the traditional lumped

system, which ignores the spatial characteristics.

Take for example a problem of reheating a steel slab [2]

shown in Fig. 1. The steel slab is reheated by thermal

radiation for rolling in a furnace. For proper rolling

characteristics, it is necessary for the slab to have a specified

temperature distribution along z coordinates. Thus, our

problem is to control the heat flux to the surface of the slab in

such a way as to approach that specified temperature

distribution.

Fig. 1 Radiant heating of a steel slab

This is a standard problem in physics where the

temperature along the z coordinates varies with time t. In

other words, the underlying dynamics of the steel slab can be

expressed as a function ofz and t. Mathematically, it can be

represented by partial differential equations.

For such system, if we choose model-based control

method, then we will make more efforts to build an accurate

model before designing the controller. Therefore, we prefer to

concentrate on model-free method, such as FLC, which can

directly employ experts control knowledge and experience.

However, the traditional fuzzy set only has two dimensions,

both of which cannot express spatial information. Based on

the fuzzy set, the traditional FLC loses the capability to

handle spatial information and then limits its application for

spatially-distributed systems.

Thus, we will seek a new fuzzy set and a new fuzzy control

strategy to deal with the spatial information. 3-D FLC is such

a fuzzy logic controller that it can express spatial information

via using a new fuzzy set called as 3-D fuzzy set and carry out

a 3-D fuzzy control strategy to handle spatial information. In

the following section, we will have a good understanding of

its working principle.

3-D Fuzzy Logic Controller for Spatially Distributed Dynamic

Systems: a Tutorial

Xian-xia Zhang, Ye Jiang, and Han-xiong Li, SeniorMember, IEEE

T

FUZZ-IEEE 2009, Korea, August 20-24, 2009

978-1-4244-3597-5/09/$25.00 2009 IEEE 854


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III. FUNDAMENTALS OF 3-DFLCThe goal of 3-D FLC is to control a spatially distributed

dynamic system via utilizing humans control knowledge and

experience. 3-D fuzzy set and 3-D fuzzy control strategy are

two important concepts. With a good understanding of them,

we can easily penetrate the structure and design of 3-D FLC.

A. 3-D fuzzy setA 3-D fuzzy set A defined on the universe of discourse

X for primary variable x and the universe of discourse

Z for spatial variable z is characterized by a membership

function (MF) ( , )A

x z which takes on values in the interval

[0, 1]. The 3-D fuzzy set A in X Z may be represented as a

set of ordered pairs of the generic elements ( ,x z) and its

grade of MF:

{(( , ), ( , )) , }A

A x z x z x X z Z= , 0 ( , ) 1A

x z . When

X and Z are continuous, A is commonly written as

( , ) /( , )AZ X

A x z x z= . When Xand Zare discrete, A is

commonly written as ( , ) / ( , )AZ XA x z x z=

.For example, the temperature in a space domain can beviewed as high or low. We can find the MF description in

Fig. 2, where high temperature in the entire space domainwith a MF expression is described for clearness. Assumed

that we use a slice vertical to cut the 3-D MF in the z

coordinate, we can find the cross section is a traditional fuzzy

set. In that sense, the 3-D fuzzy set can be regarded as theassembly of infinite (in continuous space domain) or finite (indiscrete space domain) traditional fuzzy sets, thus, the design

method for traditional fuzzy sets may be also applied to the

3-D fuzzy set. Particularly, in practical application finitepoint sensors are usually used for the measurement. The

space dimension in the 3-D fuzzy set is usually discrete andcomposed of those effect measurement points.

Fig. 2 3-D fuzzy set

3-D fuzzy set is the extension of the traditional fuzzy set by

adding a space dimension to express spatial information. If

the space dimension is wiped off, 3-D fuzzy set will be

directly degenerated to a traditional fuzzy set. 3-D fuzzy setincreases the capability of traditional fuzzy set to express the

spatial information.

B. 3-D fuzzy control strategyOn the basis of the 3-D fuzzy set, 3-D fuzzy control

strategy emulates humans intuition and knowledge to control

a spatially distributed domain from the point of view of entire

space domain, as shown in Fig. 3. All the effectivemeasurement from the space will be taken as one inputvariable, which is called spatial input variable since it

contains spatial information. Using the 3-D fuzzy set, spatial

information is able to easily embed into fuzzy set and then is

implicated in a 3-D rule base. For example, if temperatureerror and error change from the entire space domain are taken

as the spatial inputs, then the 3-D fuzzy control strategy is

able to be realized by the following rule base: If ( ) is and ( ) is Then isl l l l R e z E r z F u K (1)

where ( )e z and ( )r z denote the temperature error and error

change from the entire space domain respectively, lE andlF denote 3-D fuzzy sets, lK is the traditional fuzzy set for

control action u , 1, ,l M= " , M denote the number of

rules.

From Fig. 3, we can find that 3-D fuzzy control strategy

aims to manipulate the overall behavior of the space domain.

Fig. 3 3-D fuzzy control strategy

C. Structure of 3-D FLC3-D FLC has a similar basic structure to the traditional FLC

as shown in Fig.4.It consists of fuzzification, rule inference,

and defuzzification. Due to its spatial nature, the first two

parts involve inherent 3-D feature. In that sense, fuzzification

and rule inference can be called as 3-D fuzzification and 3-Drule inference. In particular, 3-D rule inference is composed

of three parts: spatial information fusion, dimension

reduction, and traditional inference. Its purpose is to processspatial information and to realize two main functions: one isfor overall behavior capture from spatial domain (realized by

spatial information fusion and dimension reduction) and the

other is for traditional fuzzy inference (realized by traditional


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inference). In detail, the spatial information fusion operationis to fuse information at each spatial point and ultimatelyform a spatial membership distribution for each fired 3-D

rule, the dimension reduction operation is to compress the

3-D spatial distribution information into 2-D information for

each fired rule, and the traditional inference is to execute thetraditional inference operation and transform the 2-D fuzzy

information into traditional fuzzy output.

Fig. 4 Structure of 3-D FLC

For example, as for the heating system in section II, wedesign a 3-D FLC and describe the computation procedure of

each part. Firstly, select the detailed component or

computation unit for each part. 3-D fuzzification has two categories: singleton and

non-singleton, which are the extensions of traditional

fuzzification by adding the space dimension. Here, for

simplicity and convenience, we choose singleton 3-D

fuzzification.

3-D MF is more complex than 2-D MF. To simplify the

design of 3-D MF, 3-D MF can be viewed as the spatial

assembly of 2-D MFs, thus, the shapes of 2-D MFs will

determine the shape of 3-D MF. For instance, if 2-D MF at

each spatial point is selected as triangular MF, then the

resultant 3-D MF is a spatial triangular MF. Here, we choose

spatial triangular MF for spatial inputs ( )e z and ( )r z . 3-D rule base is greatly simpler than the traditional

multivariable rule base. 3-D rule base represents 3-D fuzzy

control strategy. Here, for simplicity and convenience, we

choose linear rule base, whose rule can be expressed as

( , )( , ) : ( ) ( )

l l l l l l i j f i jR i j If e z is A and r z is B Then u is V (2)

whereli

A andlj

B are 3-D fuzzy sets, ( , )l lf i j is a linear

function of li and lj , ( , )l lf i jV is 2-D singleton fuzzy set,

li and lj are integer, representing the partition of 3-D fuzzy set

on its domain. For simplicity, let ( , ) ( )l l l l f i j i j H= + , H is

the center-center distance of adjacent fuzzy set , then( , )l lf i j

V

is nonzero only at ( )l li j H+ . Let us see an example. Suppose

that there are five identical input 3-D fuzzy sets for ( )e z and

( )r z : negative medium (nm), negative small (ns), zero (z),

positive small (ps), and positive medium (pz). They are

represented as 2 1 0 1 2, , , , }{A A A A A and

2 1 0 1 2, , , , }{B B B B B . Then, output fuzzy sets are represented

as 4 3 2 1 0 1 2 3 4, , , , , , , , }{V V V V V V V V V .

In 3-D rule inference, min and max are used for thet-norm and t-conorm fuzzy operation in spatial information

fusion respectively, weighted aggregation approach is used

for the dimension reduction, and min is used for the t-norm

in the traditional inference.

The defuzzification is chosen as Center-of sets type.

In detail, we give the computation formula according to theabove design.

3-D fuzzification

1{ , , }pZ z z= " , ( )i ie z e E = \ and ( )i ir z r R= \

denote the inputs from the sensing location iz z= , E and

R denote universe of discourse.

( )

( )

( ( ), ) /( ( ), )

( ( ), ) /( ( ), )

Ez Ezz Z e z E

Rz Rzz Z r z R

C e z z e z z

C r z z r z z

=

=

(3)

where ( ( ), ) 1Ez e z z = , ( ( ), ) 1Rz r z z = ,when ( ) ( )ie z e z = ,

( ) ( )ir z r z = ,and iz z= ( 1, , p" ); ( ( ), ) 0Ez e z z = ,

( ( ), ) 0Rz r z z = , when ( ) ( )ie z e z , ( ) ( )ir z r z ,and iz z .

Combining the two fuzzifications, we have

( ) ( )

( ) ( )

( ( ), ( ), ) /( ( ), ( ), )

( ( ), ) ( ( ), ) /( ( ), ( ), )

XX Cz Z e z E r z R

Ez Rzz Z e z E r z R

C e z r z z e z r z z

e z z r z z e z r z z

=

=

(4)

where is t-norm operation.

3-D rule inference

Using the rule (2), the lth fired rule can formulate the

following fuzzy relation

: 1,2, ,l l l l R A B V l N = " (5)

where N denotes the number of fired rules; for simplicity,lR , lA , lB , and lV represent ( , )

l lR i j ,

liA ,

ljB , and ( , )l lf i jV

in (2) respectively.

i) Spatial information fusionMathematically, it is realized by an extended sup-star

composition operation on the input set and the antecedent set

and results in a spatial fuzzy set lW

( )l l lXW C A B= D (6)

The MF of lW is given as

( )

( ) , ( )

( )

( )

( ) ( ( ), ( ), )

sup [ ( ( ), ( ), ) ( ( ), ( ), )]

{sup [ ( ( ), ) ( ( ), )]}

{sup [ ( ( ), ) ( ( ), )]}

l l lX

l lX

l

l

W C A B

e z E r z R C A B

e z E Ez A

r z R Rz B

z e z r z z

e z r z z e z r z z

e z z e z z

r z z r z z

=

=

=

D

(7)

Since singleton fuzzification and min t-norm are used, (7)can be written as

1 1 1( , )

( , )

l l l

z e r

l l l

zp ep rp

T

T

=

=

# (8)

where leI andl

rI are the membership grades of the inputs

Ie and Ir at the sensing location Iz z= with respect to the

antecedent sets of the lth fired rule; ( , )T is t-norm, min is

chosen as the t-norm here.


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ii) Dimension reduction

The dimension reduction operation is realized by using aweighted aggregation approach for the compression of spatial

information, assumed that the weights at the same sensing

location for all fired rules are same and have a linear

relationship with the spatial membership grades at thatsensing location.

In this operation, let I ( 1, ,I p= " ) denote the weight of

the spatial membership grade at the sensing location Iz z= ,

where I \ and 1 0p

II

=> . Then, the spatial

information at each fired rule is compressed by the weightedaggregation operation, which results in a firing strength for

each fired rule as follows

1

1 1 1

1 1 2 2

1 1 2 2

1 1 2 2

l

N

z z p zpR

l l l

z z p zpR

N N N

z z p zpR

= + + +

= + + + = + + +

"

#

"

#

"

(9)

where lR

denotes the firing strength of the lth fired rulel

R .

iii) Traditional inference

The Mamdani implication is employed for the lth fired

rule as follows

( , ),( ) ( , ( ))l

f i jl li j VRl l

u T u = uu U

Due to singleton fuzzy sets designed for the output

variable, (9) becomes

,

,

( ) ( ,1) for ( )

( ) ( ,0) 0 for with ( )

i j l l R Rl ll l

i j u l l Rll l

u T u i j H

u T u U u i j H

= = = +

= = +

DefuzzificationFinally, after a center-of-sets for defuzzification, a crisp

out is given by

1 11 1[( ) ( ) ] ( )N NN NR R R R

u i j H i j H = + + + + + +" "

Generally speaking, once MF and rule base are designed,

the 3-D FLC is determined. To reduce the complexity to tune

the controller, we can add scaling factors to the inputs and

output. Many cases have proved that once MF and rule base

are properly designed, satisfactory control performance could

be achieved by only tuning scaling factors.

IV. RECENT DEVELOPMENTS ON 3-DFLCSince the first journal paper [1] about 3-D FLC was

published, much research has been carried out. At present,

some results of 3-D FLC have been achieved. In this section,we will present some important results.

A. Mathematical explanation of its spatial structureUsing the method of rule base decomposition [3], an

analytically mathematical expression of 3-D FLC designed insection II can be derived as follows [4]

1 1 1 1 1( ) / 2 ( )p p p p pu H s s c H k k

= + + + + +" "

(10)

where:

I I I eI I dI Is e r k e k r += = +

eIk and dIk represent the scaling gains of actual error Ie

and

Ir respectively.

1

1 1 2 2(1 2 ) (1 2 ) (1 2 )p p

= + + + + + +"

1I I Ik i j= + + 1, ,I p= "

c is the center-center distance of adjacent triangular fuzzy set.

Ii and Ij are integers, with ( Ii c , Ij c ) representing the

location coordinates of inference cell ( , )I IQ i j where the

scaled input pair Ie , Irfrom the sensing location Iz=

appears in the rule base plane (shown in Fig. 5).

I is a membership grade. Its value varies in different

sub-region where the scaled input pair ( ,I Ie r) falls into the

inference cell ( , )I IQ i j in the rule base.

Fig. 5 Decomposing rule base into many inference cells

The 3-D FLC presents two different structures as follows.

1) Sliding mode structure

The final output of a PD-type 3-D FLC is given by

( ) ( )

sat( ( ))

u u u

u eq

U k u k H s Kc c k H

k H s c u

= = +

= +(11)

where:

1 1 2 2 p

s s s s = + + +"

I I Ikcs s

=

1 1 2 2 p pK k k k = + + +"

1 1 1(1 2 ) (1 2 )p p pk k = + + + +"

1 p = + +"

s s Kc= is a switching function.

eq uu k H = is an equivalent control term.

uk is the scaling factor of output.


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sgn( )sat( ( ))

( )

s s cs c

s c s c

=

, sat( ( ))s c is actually

the continuous approximation of the function sgn( )s by

introducing a boundary layer.

The 3-D FLC presents a global sliding mode structure [4]

over the space domain, as shown in Fig. 6, in whicheq

u plays

an equivalent control term and sat( ( ))uk H s c plays a

switching control term. At each sensing locationi

z z=

( 1, ,i p= " ), there is a local sliding surface i is k c

= , where

i is k c

= is called a pseudo sliding surface when 0ik , and a

real sliding surface when 0ik = . Through the aggregation of

all the local sliding surfaces over the space domain, a global

sliding surface s Kc = is formulated, where s Kc = is a

pseudo sliding surface when 0K and a real one when

0K= .

Fig. 6 Sliding mode structure of 3-D FLC

2) Spatial aggregation of multiple traditional FLCs(10) can be transformed into the following form

1 1( ( ) / )

p p I

I I I I I FLCI Iu Hk H e r c u

= == + + = (12)

where:

( ) /IFLC I I I I Iu Hk H e r c = + +

( 0.5)I I Ie e i c = +

( 0.5)I I Ir r j c = +

1 1(1 2 ) ( (1 2 ) (1 2 ))I I I p p = + + + + +"

1 1[ (1 2 ) (1 2 )]I I p p = + + + +"

3-D FLC can be regarded as a spatial equivalent structure

[5][6] shown in Fig. 7. At each sensing location Iz= , one

traditional FLC works, which behaves like a global

two-dimensional multilevel ( I IHk ) relay plus a local PI/PD

controller ( ( ) /I I IH e r c + ). In the entire space domain,

multiple traditional FLCs aggregated by spatially-coupling

parameters I and I ( 1, ,I p= " ) constitute 3-D FLC.

Fig. 7 Spatial equivalent structure of 3-D FLC

B. Stability issueSystem stability is considered from two different points of

view: Lyapunov stability and BIBO stability.

The spatially-distributed nonlinear system is representedby . Assumed that enough sensors deployed in the space

domain are used to extract the spatial information of the

spatially-distributed nonlinear system , the input-output

nonlinear dynamic relationship of at finite spatial points

where sensors are located can be represented by .

The input-output relation of the system can be expressedapproximately in terms of ordinary differential equations as

follows.

( )

1 1 1 1

( )

( , )

( , )

n

n

p p p p

y f y d bU

y f y d b U

= +

= +

# (13)

where ( , )i iy y z t= denotes the output measurement value at

the sensing location iz z= ; y denotes a vector associated

with { }1 2, ..., py y y ; id denotes the disturbance (e.g.

unmeasured spatial information) toi

y ; ( , )i i

df y denotes

unknown function of y and id ; p denotes the number of

spatial points;i

b indicates the strength of the source U acting

on spatial pointi

z z= , 0ib , 1, ,i p= " .

1) Lyapunov stability [4]

For the nonlinear system (13) with 2n = , two assumptions

are posed:

Assumption 1 The upper bound of /i ei di di i ir k k y f F

+

( 1, ,i p= " , 0iF > ) exists.

Assumption 2 The upper bound ofeq

u exists and satisfies

i i eq iF b u F + < ( 0iF > ).

The control given by (10) is employed. If Assumption 1 and 2

hold, the scaling factors of the error and error in change

satisfy the inequation (14), and the parameters of 3-D fuzzy

logic controller in (10) satisfy the equation (15), then the


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global stability of the closed-loop fuzzy control system is

guaranteed.

1 1 max max

1 1 max max

, ,

, ,

e ep p

d dp p

k L e k L e

k L r k L r

< , L Nc= ( 2 1N+ denotes the number of fuzzy

sets),maxi

e andmaxi

r denote the maximum of the absolute

value of ie and ir

respectively.

2) BIBO stability [5] [6]

If the system has a bounded norm (gain) < and

the parameters of the 3-D FLC c satisfy

max max 1uk Hk p c < (16)

where:

max 1 1max{ , , }e d ep dpk k k k k = + +"

max 1max{ , , }p = "

1 2( )I I p = + + +" and 0I > ( 1, ,I p= " )

then the nonlinear 3-D fuzzy control system is to be globally

BIBO stable.

C. Extension to multiple control sourcesThe spatially distributed system with multiple control

sources shows more complex spatial distribution than the onewith one control source. For such system, we can consider to

employ advanced control strategy. In most of spatiallydistributed systems, sources usually have local influence on

the space domain. Using this spatial feature, the system withmultiple control sources can be decomposed into multiple

subsystems with one control source. Consequently, the

aforementioned 3-D FLC can be employed as the controller

for each subsystem. Since interaction exists amongsubsystems and it decreases as the distance of subsystems

increase, suitable coordination actions should be added.

Under this strategy, a local coordination type 3-D FLC [6] [7]

is proposed. For each subsystem, one local coordination type3-D FLC is used. Its output is composed of two parts: main

output and coordinated output. The main output is equivalent

to the output of a standard 3-D FLC, and the coordinated

output is the influence output from spatially adjacent

subsystems.

V. CONCLUSIONSIn this tutorial, basic working principle of 3-D FLC is

provided. 3-D fuzzy set and 3-D fuzzy control strategy aretwo very important concepts to understand 3-D FLC.

Through a simple example, a detailed design and computation

procedure of 3-D FLC is given. Recent development on 3-D

FLC is described. Some important results includingmathematical explanation of its structure, stability issue, and

its extension to multiple control sources are given.

REFERENCES

[1] H.X. Li, X.X. Zhang, and S.Y. Li, A Three-dimensional FuzzyControl Methodology for a Class of Distributed Parameter System,

IEEE Trans. Fuzzy Syst, vol.15, no.3, pp. 470-481, 2007.

[2] W.H. Ray,Advanced process control. New York: McGraw-Hill, 1981[3] H. Ying, Fuzzy Control and Modeling: Analytical Foundations and

Applications. New York: IEEE Press, 2000.

[4] X.X. Zhang, H.X. Li, and S.Y. Li, Analytical Study and StabilityDesign of Three-Dimensional Fuzzy Logic Controller for Spatially

Distributed Dynamic Systems,IEEE Trans. Fuzzy Syst, vol.16, no.6,

pp. 1613-1625, 2008.

[5] X.X. Zhang, S.Y. Li, and H.X. Li, Structure Analysis ofThree-dimensional Fuzzy Two-term Controller and BIBO Stability ofits Control System, Mathematics and Computers in Simulation, 2nd

revised, 2008.[6] X.X. Zhang, Design and Analysis of 3-Domain Fuzzy Logic

Controller for Spatially Distributed Systems, PhD dissertation,

Shanghai Jiao Tong University, Shanghai, China, 2008.[7] X.X. Zhang, S.Y. Li, and H.X. Li, Novel fuzzy control for spatially

distributed systems based on decomposition and coordination

strategy, Control and Decision (in Chinese), vol. 23, no. 6, pp.709-713, 2008.


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3-D Fuzzy Logic Controller for Spatially Distributed Dynamic