Matrix representation and implementationof fuzzy system

Zhinong Miao 1,2 Xiangyu Zhao2Center of Intelligent Control and Development Southwest 2Department of engineering of electronic information,

Jiaotong University. Panzhihua University.Chengdu 610031, Sichuan. P.R. China Panzhihua 617000,Sichuan. P.R. China

miao2]5(lvahoo. com. cn miao2]5(lvahoo. com. cn

Abstract - fuzzy logic is wildly used in many fields in the fuzzy system. This paper attempt to construct a matrixrecent years. It is also the theoretic base of fuzzy control. 'A representation and implementation method based on thenovel matrix representation and implementation method is state space method which have achieved great success inprompted in this paper. The new method employes the modern control theory.concepts of state space which achieved great success in themodern control theory and uses matrix to represent fuzzy the restfi ap iorganized follows: s onelmodels including the fuzzification, inference mechanism, rule gie mth icrepresentation metho fo aifuzzy mdbase and defuzzification. Some new combining operators for and matrix representation of fuzzy system is discussed infuzzy logic inference are also defined in this paper. To show section III. The implementation of fuzzy logic based on thethe correctness and efficiency of the new method, a nonlinear matrix representation is prompted in section IV. Somesystem is discussed employing the new methods. valuable property is listed in the conclusion.

Index Terms -fuzzy system, matrix representation, fuzzy II FuzzY MODEL REPRESENTATIONlogic implementation

As with all modelling problems, the first step is toI. INTRODUCTION identify the relevant quantities whose interaction the

model will specify. These quantities can be classified intoAs a solution to control complex systems in their input, output, and state variables. Let U, Y andE denote

whole operation range, fuzzy controllers constitute a good the input, output, and state spaces, respectively. Tooffer. Since the first application of fuzzy theory to simplify the discussion, we assume that all these spaces areautomatic, fuzzy control has gradually been constituted as subsets of Euclidean space of (possibly) differenta powerful technique of control [2] [3]. Fuzzy controllers dimensions. Because we are interested in dynamicalare non-linear controllers that provide a formalmrethodologyefr crolepresenting,h maiatpvin andol

systems we also need to specify a set T c R of times ofmethodology for representing, manipulating anditrs;tpcly rR otnostm

implementing a human's heuristic knowledge about how interest; typically, T R or R for continuous timeto control a system. They could be viewed as artificial systems and T = {kI|c > 0 and k E Z or N} fordecision-makers that operate in a closed-loop system in discrete time systems. Given these sets, a generalreal time [4]. There are many issues discussing the dynamical system is defined in as a quintuplemethodology for design and analysing of fuzzy control D = (u, Z, y, s, r) where is the state space, U is a setsystem, including discussing the rule base construction, of input functions uO T U, andy is a set of outputfuzzy modelling, and adaptive fuzzy control [5].

According to them, a problem is how to represent and functions y(): T -* Y. The dynamics are encoded by thecompute process that are imprecisely described or are state transition function Scontrolled by humans without recourse to mathematic s T x T x E x U -* Emodels, algorithms or a deep understanding of thephysical process involved. Another problem is how to get (t1,to, x0, u) - x1 s(t1,to, x0, u)ready adaptive techniques, which permit to have that produces the value of the state XI at time t4 givenintelligent control systems, that is, system is involvinglearning or adaptation in response to changes in process the value of the state at time to and the input for allparameters. FL is a powerful tool for knowledge times. The map is only defined for t, > to . Finally, r is therepresentation in computational intelligence. On the other

hand, adaptive contr, lread-out function.hand, adaptive control, learning and self-organization can r*ExUxT Ybe considered in a lot of cases as optimization or search rZxUxT-process. (xt,u(t),t) -* y(t) =r(xt,u(t),t)

The representation of fuzzy logic or fuzzy controller isnot in a precisely way till now. The problem iS also an thtpoue h uptfnto ttm hobstacle for systematic design and performance analyses of value of the state and input at time t.To keep the

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definition consistent two axioms are imposed on the state control theory for reference. Some important concept istransition function. defined as following:

For fuzzy system, the dynamic system can be described State: state of a dynamic system is a minimized set ofas the same form. variables of the system that it can determine the future

In this paper, we restrict our attention to discrete performance at the time t > to if the variables at timetime models and, in particular, models whose time stampstake values in the set T {kT k ~ N} for someT > o = to and the input at the period t . to are determined.

State variables: state variables for a dynamic system isWithout loss of generality, we assume = ]. a minimized set of variables that determining the system

So a fuzzy dynamical system is a quintuple state. State variables must efficient to express the systemD= (UF, EF, yF , IR, RO) where state which means that the state variables can determine

zF F ocl ocNthe only system behaviour at any time. And it also must beE is the fuzzy state space, E

aX *... X j ,that is, necessary which means that the state variables are the

-xlF- - P1 - minimized set of variables that can be used to representF

C EF F F iai ~~~~~~the system state.for every xF E EF X xi. , X' = ' E IA iState vector: if representing dynamic system behaviour

-XN- paiX needs n variables; the vector X which takes the

withO.p.1i id,...,N and j~1. ..ai; n variables as its sub variables, the vector X is the statewithO<p' <],ic1, -,N and JE], ,OCaI;UF i a vector of the dynamic system. As the state at time °

iS a set of fuzzy input functionst

F T UF C jbiX .jbm and the input u(t) at the period - are determined,YF is a set of fuzzy output functions system state X(t) at any time in t 2 t is determined.F _T yF 1 i c

y T*' c' X ..X I State space: if xi x2 is the state variablesIR {IRI, * IRn } is a set of inference rules for a dynamic system, the n dimensions space is the state

J ZiEF XUF XT I" space. Any state of the system can be represented as a* UF T jal point in the state space.IR: F XUF XT o F The state variables form the state vector for describe

IR produces the value of the state at the next time instant, the object.given the value of the state and the input at the current X = (xi,x2, *.*. x")time instant;RO ={RO,.* * RO}* III MATRIX REPRESENTATION OF THE FUZZY SYSTEM

iS a set of read-out mapsREF X UF x

a s

Ir u

In a typical fuzzy system, there are models forR01 x T Ic controller and controlled objects. The formal approachRO:ZF XUF XT yF discussed in last section prompted a model representationRO produces the value of the output at the current time frame. To make it convenient representation and easy to

given the value of the state and input, carry out the online calculate, parameterized model

Fuzzy state space is a new concept for fuzzy system representation method should be employed. Matrixintrduce fro modrn cntro therem.Thebasic representation method is discussed in this section based onintroduced from modern control theorem. The bsc the formal approach.

problem for fuzzy state space is the choice of state the formal approach,variable. ~~~~~~~~~~~According to the formal approach, the fuzzy model

variable. includes the input space, output space, state space, theConventional fuzzy controller always uses the error ifrnerls edu ucin.S aaeeieinference rules, readout functions. So a parameterized

e and the change of error as it's input variables as representation method should be able to describe all theshowed in figure 1. It is based on human's intuited model components.knowledge about the controlled object and the controller Model component representationuses the input to deduce an output signal to drive the For a fuzzy model, no matter it is a controller or acontrolled object. But it is not sufficient to describe the controlled object, there may be multiple inputs. Thedynamic property of controlled object. universe of discourse of every input variable is a sub input

In such kind of system which is common in practice, space and the Cartesian product constructs the inputa*ie ro ,h hneo ro eane h nu vector. The input space is formed by the possible value of

signal U can not determine the next state of system. So we the input vector.have to use some concept of state variables in modern

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Assume an input vector as U =(ul, u22*., ul) , it is State vector is represented as

formed with I input variables. Every input variable u1 is Pi Pkxdescribed by a series of linguistic value which is given byhuman expert or experienced operator. The linguistic X Xvalues would extend to the whole universe of discourse of X P1 P1 Pk, Pk,the input variables. Each linguistic value is correspondingto a fuzzy set which is defined in the universe of discourse.Meanwhile, proper membership function is defined to each P1 * kXfuzzy set. The membership of the input variable can becalculated and then be represented as the input space in a pmatrix form. That is for each input variable U1: Output variable is represented as yi

P1 Pki

uiF . And output vector is represented as

|k7 j 0 Pi Pkywhere k1 is the index of the fuzzy set which is defined inthe universe of discourse corresponding to the input Y pi pi Pk, Pkivariable U1; P, j = 1, ..., k is the membership that the

input variable U1 belongs to the k1 th fuzzy set. .PThe input space U can be constructed by the input Representation of inference rules and read-out function

variables. For the convenient of representation, define a Conventional inference rule base always employee a state-variable ku as: evaluating rules. The prevailing form is as

ku= max(k,...,k) RULE1 IF x IS A1 AND y IS B1 THEN z IS C1RULE2: IF x IS A2 AND y IS B2 THEN z IS C2then the input space U can be represented as a Ixkumatrix. For the input variables which is k < ku, the

RULEn IF x IS An AND y IS Bn THEN z IS Cnmembership value of pk can be defined equal to the The rules describe the mapping of input variables to output

membership value I' variables. The new method is different to the conventionalmembership value Pki.I method that:

Then The new method takes the system state into account whileP' Pku conventional method doesn't.

Rule base in conventional fuzzy system is therepresentation of intuited knowledge of human expert orexperienced operators. They can not evaluate or describe

U = P1I pi Pki Pk, the potential influence of system state precisely especiallyfor a complicated system with many state variables.

/ / In a complicated system, there may be many statePi Pku variables. There would be large number of rules if all the

State space and output space representation is analogous to system state be included in the inference rule base. Thethe representation of input space. Thy are: controlled table for conventional fuzzy control system is

_ always carried out by off-line calculate. It is impossible forpli large scale fuzzy state variables.

State variable is represented as Practically, system state is of great importance fordescribing a complicated system. Great deal of information

PkL would lose if all the system state is not be included in therule base. This is the main reason for low accuracy andslow response for conventional fuzzy control system.New method doesn't maps the input variables to outputvariables directly but employee the inference rules (IR)and read-out functions (RO) instead. The inference rules(IR) maps the current system state to the system state in

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next instance given a input signal and the read-out x Ffunction (RO) maps the current system state to system I

response (output) given a input signal.Comparing with modern control theory, the inference rules xF(IR) and read-out function (RO) is analogous to the state n

transform function and output function. In modern control u F

theory, the system can be described as:

X(t) f(x, u, t) FY(t) g(x, U, t) UI

the combination ofthe state space and input space actsinference rules JR = {1R1," ,IR} is analogous to the as the premises of the inference rules while the conclusion

state transform function X(t) f(x, u, t) and read-out of the inference rules is state space. For convenient tofunction describe human's intuited knowledge and can be

Y(t)=g(X,aU,ut)ut understand by human, the inference rules should beY(t) g(x,u,t). ~~~~~~~~represented as:

So the inference rules IR should in the form as: - . 1 -- 1if uk is u aidXk is XF tlif Xk+l isXthe premises of the rule is the combination of the input

F Fspace U and the system state space x , the conclusion k).is the new state space x'F The rule can also be n -1 2represented as:kTXk XUk Jk+R-

or

1 |r Xd9 'ki AX t - -- kgF F F

A.44A x AAAi A4omI= A 4A A Xi

~Iq 'XJtik ~ I F F Ft t AxJQ 0 ,0 T A~~~~~xJtk+ltr X? X.1 JThe state spaceX is a n x kx two dimensions matrix, The matrix of inference rules can be divided into two

parts. The part above the line describe the premises of theit describe the membership for the state vector based on inference rules, it is the combination of state space andthe universe of discourse. It also can be viewed as: input space. It is represented as a combination of the fuzzy

'xiF sets which are defined in the universe of discourse of inputand state space. Each column describes one passible

K. 2 combination of the linguistic value of input vector andkx F state vector and then the matrix describe all the passible

combination of the linguistic value that the input and stateBy the same way, the input vector U is a I x ku two vector may have. As discussed before, there are n state

dimensions matrix. It describes the membership for the variables in the state space and kx fuzzy sets are definedinput vector based on the defined universe of discourse. It in the universe of discourse for every state variable. Also,can also be represented as:canualsobe represented as:

there are I input variables in the input space and ku fuzzysets are defined in the universe of discourse for every input

K; 2 variable. So the number of combinations of the linguistic

tF ) value in the input space and state space is kxn x ku' andThe combination of state space and input space can be then the number of inference rules in the IR is

done simply by connection of the two single column also X xk /matrixes as: The part below the line in the matrix describes the

conclusion of the inference rules. As discussed in therepresentation of the fuzzy model, the inference result isstate space and each rule (each column in the matrix)

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deduces a linguistic value of state vector, which is a point 1 I

in the state space. z

Read-out function describes the map of current state tosystem response given a input. That is F = '

Xk XUk oRO=YkZ Pi Pi Pk, Pk,so the representation of RO is analogue to the

representation of IR. Pn nI1 Pkz

IV FuzzY LOGIC IMPLEMENTATION BASED ON MATRIX The calculation for fuzzification is to calculate theREPRESENTATION value membership using the predefined membership

function given the real value of vectors. It can beThe representation method discussed in the last represented as:

section describes the fuzzification process, input space, i _ )state space, inference rules, read out function, and Pk kZidefuzzification process as a series of matrix. The matrixesdescribe the static attributes of the fuzzy model. To DEFUZZIFICATIONdescribe the dynamic property of the fuzzy system, thelingistric matrix coambinmroperthod shofuldyb tem toe Defuzzification is the process to transfer fuzzy value tolinguistic matrix combine method should be defined to

ra au ftevco.fzysse uttk hcarry out the fuzzy inference process and implement the real value of the vector, fuzzy system must take thedynamic mapping of input space state space and the defuzzification process to acquire the output signal anddynamic mapn o nutsae,saepcrealize the interface to other fuzzy system or hybridoutput space.

The main process of fuzzy inference includes: systemFfuzzification of input vector and state vector, combination ziof input vector, state vector and inference rules, ZF ,combination of input vector, state vector and read-outfunction, defuzzification of output variables. The calculate LZsFjmethods based on the linguistic matrix would be discussed process can be represented as:as follows: F

ZF o DF -*z

FUZZIFICATION where z is the representation of a fuzzy vector. It isFuzzification is the process that transfers the real value of the form as the matrix representation which is

vectors to predefined membership functions to acquire the discussed in last section.linguistic value of the vectors. ' .1 PkzF: R _> In

F.(x) ZF pIpPi I Z =pi Pi Pk7 Pk7

F,(x)~~~~~~~~~~Pk k

J DF is the defuzzification matrix which is discussedgiven a vector z ,the process of fuzzification in last section of the form as:

f--1F)-_zs_ i(Yl )for the vector z is as: DF

zOF FS fm (Ym.)where z is the representation of fuzzificated vectorspace . it can be represented as a state space matrix where f[(IF) is the defuzzification function for thebased on the discussion in the last section. fuzzy value y1F.

The calculate process is to calculate the real valueusing the defuzzification function given a fuzzy vector.

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COMBINATIONOF THE INPUTSPACEAND STATE SPACE where YJ is the fuzzy value of output space which is

The premises of inference rules are the combination of deduced by the i th rule in OR; i = n x kU' indexall the input variables in conventional fuzzy system. Thatmeans the premise of inference is the input space. The of the rule in OR .

new prompt method does not map the input variables tooutput directly but map the input space and state space to CONCLUSIONnew state space by inference rules and to output space by The prompted matrix method introduces state spaceread-out function. Then the premise of inference rules concept which achieved great success in modern controland read-out function is the combination of input space system into fuzzy system representation andand state space. X x U. implementation and the new method makes representation

The combination method has been discussed in the and implementation of fuzzy system clearly and precisely.last section. It can be done simply by connect the two It constructs a solid base for fuzzy system design andspace to form a new single column matrix where the analyses. We applied the new method in a fuzzy systemitems is the state variables or input variables. design for a gas burning boiler and acquire a considerable

system response. What more we get by the new method isINFERENCE RULES that it makes the design process much easy compared with

The inference rules IR is defined as a set of inference traditional fuzzy system design process.rule in the last section as:

JR - {1R1,... , IRn } ACKNOWLEDGMENT

Jj:IF UF T I" This work is partly supported by the Department ofElectronic Engineering and Information Panzhihua

JR: ZF X UF x T - ZF University and Department of Mathematics Southwest

JR map the state space to new state space in the next Jiaotong University. It is also a part of work supported byinstant given a input vector. It carry out the fuzzy national nature science fund. Grant number (60474022).implication calculate. There are many kinds of implication REFERENCESoperators which have been discussed in past researchworks. [1] "Editorial: Fuzzy models-What are they and why," IEEETrans.

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X-Xl V VvX1V VXk Anchorage, AK, May 1998, pp. 188-1193.x =XI .. V i... V kx,xkul

where Xi is the fuzzy value of new state space which is

deduced by the i th inference rule; i = I ... kXnx ku' is

the index of inference rule in the JR .Read-out function OR can be represented and

implemented in the same way as inference rules JR:1I ~~~kxnxku

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