Full Parameterization Process for Singleton Fuzzy Logic Controllers: A Computing Algorithm

Full Parameterization Process for Singleton Fuzzy Logic Controllers: A Computing Algorithm

Osama A. Awad Rami A. Maher

College of Information Engineering. Dept. of Electrical Engineering Nahrain University Isra University

Baghdad, Iraq Amman, Jordan

AbstractAlthough fuzzy systems demonstrate their ability to solve different kinds of problems in various applications, there is an increasing interest on developing solid mathematical implementations suitable for control applications such as that used in fuzzy logic controllers (FLC). It is well known that, wide range of parameters is needed to be specified before the construction of a fuzzy system. To simplify in a systematic way the design and construction of a general fuzzy system, and without loss for generality a full parameterization process for a singleton type FLC is proposed in this paper. The presented methodology is very helpful in developing a universal computing algorithm for a standard fuzzy like PID controllers. An illustrative example shows the simplicity of applying the new paradigm.

Keywordsparameterization;fuzzy logic controller(FLC); singletone FLC ; PID

1. INTRODUCTION

   Since Zadeh introduced the basics of fuzzy sets [1] in 1965, and the fuzzy logic concepts [2] in 1968; fuzzy logic has been successfully applied to a wide range of applications in various fields. Mamdani and Assilian [3] first applied the fuzzy logic control in to the control field, and since then fuzzy logic controllers have attracted a great deal of interest among many researchers. Later on, fuzzy logic controller is proven to be an effected way in control engineering applications.

   There are mainly two types of a ruled base fuzzy system. One is the Mamdani type FLC [4], and the other is the Takagi-Sugeno (TS) [5]. Structure for the both types are the same, the only difference is related to the definition of the output in the consequent field of the rule base. TS type uses a crisp values for the output in the rule base, where it is a fuzzy linguistic in the case of Mamdani type.

   Another type gaining a wider acceptance in control and industrial applications, which is called a singleton fuzzy controller [6] will be adopted and focused on by this paper. Although it defines a singleton membership function over the output, it is actually uses a constant real value called a singleton of the rule output, representing the position of the trivial output MF. With this type several activation, accumulation and defuzzification methods yield identical results [7].

   As the field of fuzzy computing is an active research field, many methodologies are developed for constructing and computing the FLC. The designer of a fuzzy controller for certain control application is faced with the many design choices that the fuzzy set theory provides. Fundamental comparisons and suggestions are found in the literature, and they are well presented by [8-11,16-18]. These computing approaches are not unique; it is mainly due the lack of having a general good mathematical formulation for the fuzzy system construction algorithm.

   A solution for this problem may be solved if good parameterization process is developed. The parameterization of a fuzzy system is insufficiently addressed in the literature. This work is a trial to solve this problem, and mainly devoted to present a fuzzy system for control applications, whether used in construction of the FLC, or in fuzzy system modeling or used in design and tuning of the FLC itself.

2. PROPLEM FORMULATION

   Fuzzy logic system (of which FLC is a special application) is a natural extension of fuzzy set theory to relations between fuzzy sets and rules. A FLC is characterized by four modules: [fuzzifier, inference engine, knowledge base, and defuzzifier]. A schematic representation of FLC is presented in Fig. 1.

   Fig. 1. Basic structure of a FLC.

   The parameters of an FLC can be classified into four categories [12]: logical, structural, connective, and operational as can be shown in Table I.

   1. Parameterization Process of the FLC

      In the following, we will discuss the suggested FLC parameterization methodology specified for a singleton type

      fuzzy system, from different aspects according to the classification of parameters summarized by Table I.

   2. Parameterization of Input Membership Functions

      Consider the fuzzy system as shown in Fig. 2. which is a

      systems employ a normalized fuzzy sets to specify the entire partition of a fuzzy variables), a great number of reduction in the number of parameters defining the input MF can be introduced. Without loss of generality, this implementation is adopted in the work. Hence (3) will be reduced to:

      simplified form of Fig. 1 with input x and output u ,

      where x nx is the fuzzy system input variables,

      b11b12b13

      in

      b b b

      (4)

      u nu is the fuzzy system output variables, and { n ,

      x

      21 22 23

      nu } are the dimension of input and output variables.

      TABLE I. PARAMETER CLASSIFICATION OF A FLC

   3. Parameterization of Membership Degree

      In FLC computing, only the degree of membership is further processed. Often information is lost during this procedure, although it is not required that the MFs are

      normalized (i.e. their sum is equal to one for all x ), this

      property is called fuzzy partition and often is employed because it makes the interpretation and computation easier [7]. The degree of membership is obtained for the current input vector by:

      CLASS	PARAMETERS
      LOGICAL	REASONING MECHANISM, FUZZY OPERATORS, MEMBERSHIP FUNCTIONS TYPE, DEFUZZIFICZTION METHOD
      STRUCTURAL	RELEVANT VARIABLES, NUMBER OF MEMBERSHIP FUNCTIONS, NUMBER OF RULES
      CONNECTIVE	ANTECEDENT PART OF THE RULE, CONSEQUENT PART OF THE RULE, RULE WEIGHTS
      OPERATIONAL	MEMBERSHIP FUNCTION VALUES

      and,

      x

      i

      mxi

      (5)

      (x ; (i) , m ),i 1,2,3……, n .

      (6)

      xi i in xi x

      where

      (.;.) is a generalized MF producing degrees of

      membership for all fuzzy sets related to the input

      xi and

      x

      xi

      i

      [ 1, 2 ,….., k ,……m ] (7)

      x

      is a vector of dimension m , each element represents the

      i

      Fig. 2. A simplified input-output fuzzy block.

      degree of membership for fuzzy subset

      (k )

      associated with

      The input membership functions are parameterized by:

      input xi and evaluated using the input parameter

      (i )

      in

      at a

      in

      nx ni

      i

      where: ni c max( mx ;i 1,2,……nx )

      (1)

      (2)

      given point in the range of the relative input.

      Example 2:

      and c represents the standard number of parameters that

      Consider the previous example; membership degree can

      define a certain type of MF,

      m is the number of fuzzy sets

      x

      i

      be evaluated for input

      x1 at a measured value cx1 by using

      assigned for input xi . The i th row of in includes all the

      (7) as follows:

      (cx ; (1) , m

      3)

      parameters that characterize the MFs which are related to the

      i th input (shape, no. of sets, width of the fuzzy set, spacing and overlapping between the sets).

      x1 T 1 in x

      x

      1

      12 3

      (8)

      Example 1:

      For a fuzzy system with two inputs, each defined with three

      where the index T is put as an indicator targeting for using

      the triangular function in calculating the membership degree.

      1 T (cx1;[a11b11c11]), k 1

      triangular MFs, then nx 2 ; and c 3.

      m 3;

      x

      1

      m 3

      x

      2

      2 T (cx1;[a12b12c12 ]), k 2

      3 T (cx1;[a13b13c13 ]), k 3

      (9)

      A triangular MF is normally defined by three points (a, b, c).

      The membership degree of each fuzzy set

      (k 1,2,3) for a

      Therefore in

      will be defined by

      in

      29

      a11b11c11a12b12c12a13b13c13

      (3)

      triangular MF is evaluated by:

      in a b c a b c a b c

      21 21 21 22 22 22 23 23 23

      As the triangular MF is characterized by its core (most fuzzy

      0

      cx a

      1 1k

      ; cx1

      ; a

      a1k

      cx b

      ( j ) T (

      x

      1

      , x2

      ,….,

      x

      i

      ,….

      xnx

      ( j )

      ;

      rules

      ), j 1,2,..nR (1

      3)

      b a

      1k 1

      1k

      where (13) operates the T-norm or T-conorm between the

      k (cx1

      ) 1k

      c

      1k

      • cx

      (10)

      elements of the vectors defined byrules . This operation

      1k 1

      ; b cx c

      represents the aggregation stage of the inference engine.

      c b

      1k 1 1k

      0

      1k

      1k

      ; c1k

      cx1

      Example 3

      Consider the same previous example, for which the premise

      at least one value of the vector x is not zero, otherwise the

      part of the complete RB will be constructed as follows:

      1

      value of the input x1 is not represented by any of the MFs

      T

      1

      3

      x

      2

      3

      1

      2

      3

      1

      2

      1 1 1 2 2 2 3 3 3 x1

      (14)

      (fuzzy sets) defined over the relative UOD.

      For an acceptable (50%) overlapping between MFs, it

      rules

      2

      is sufficient to parameterize the inputs by their cores only as shown in Fig. 3.

   4. Parameterization of Rule Base Premise

   Without loss of generality, completeness of the rule

   where the dimension of the RB according to (12) is nR 9 .

   Then, to compute the dof for rule j 6 , we will proceed as follows:

   1st -Access (14) given j equal six, then get the index for the

   MF defined for each input variable

   k 2

   for

   input

   x1 , and

   k 3

   for input

   x2 .

   2nd-Apply the T-norm on the degree of memberships

   x

   x

   ( and

   1 2

   ) as given by (6), choosing product operator

   Fig. 3. Triangular input MF parameters represented by their cores.

   (or minimum) to represent the T-norm, then (13) gives the following:

   x

   6 T (

   1

   , x2

   (6)

   ;

   )

   rules

   (15)

   2

   base (RB) will be assumed, by taking all the possibilities encountered by the predefined MFs over the input variables.

   (6)

   (x1 ) 3

   (x2 )

   (16)

   This approach, which is adopted by the presented methodology, will cover both reduced and full RB representation by letting the certainty of each undefined rule to be zero, and set one otherwise.

   Also, most of FLCs used in control application are assumed to have fuzzy propositions connected with fuzzy and connectives only. Thus the structure of the rules premise is defined by:

   Hence, certainty of the 6th rule is evaluated by the product of the 2nd MF value (defined by mf2 ) for input 1 (calculated at the measurement value of input x1 ) and the 3rd MF value (defined by mf3 ) for input 2 (calculated at the measurement value of input x2 ).

   rules

   nR nx

   (11 )

   1. Parameterization of the Rule Base Output

      After the dof has been calculated for all the rules, and, for

      where nR denotes the total number of rules in the RB, and

      nx represents the number of inputs used in constructing the

      the inference engine process to be completed, it is required now to consider the outputs of the RB.

      Now, consider the output parameters vector by defining:

      FLC. Noting that

      nR is defined by:

      out

      nR nu

      (17)

      x

      n

      nR mx

      (12)

      where nu represents the number of controller outputs, nR

      i

      i1

      a row of rules , connects the index of the input fuzzy sets for

      represents the dimension of the rule base. The i th row within out is defined by:

      each input variable defined by each rule, and hence reflects

      (i)

      [h h h

      ]; j 1,2,….n

      . (18)

      the degree of membership function to be taken into

      out

      i1 ij

      inu u

      xi

      considerations for evaluating the premise certainty (truth

      where

      hij

      is a constant real value representing the place of

      value) of the specified rule, defined some times in the

      the singleton MF selected from a number of m

      u

      j

      fuzzy sets

      literature as degree of fulfillment (dof ) or firing strength.

      Accordingly, the firing strength of the rule is performed using the generalized T-norm or T-conorm function given by:

      defined over the output u j at rule i .

      Since the output membership value is always one at the core and zero elsewhere, hence the outputs in the RB are always defined by their singletons which are represented by:

      Example 4

      hij

      ck

      ; k =1,2,3 .

      m

      u

      j

      (19)

      variable y is observed, and given by:

      e(k) y* (k) y(k)

      (25)

      For a singleton FLC structure with one output ( nu

      1)

      The main control objective is to keep the error signal as small as possible. Also, the rate of change in the error signal

      u

      defined over five MFs ( m

      1

      5 ), then the RB consequence

      is given by:

      e(k) e(k) e(k 1)

      (26)

      91

      of the previous example could be represented by the following parameterization vector:

      with those two inputs { e(k), e(k) } the FLC can perform

      out

      [ p1

      p1

      p1 . . . .

      h81

      h ] T (20)

      the PD or PI type control depending on whether the output is

      and, for five singleton MFs defined by: [ c1 c2 c3 c4 c5 ]

      taken to be the pure control signal u(k) or the change in the control signal u(k) . Different PID like FLC structures can

      then

      out

      could be set as follows:

      be generated using the above concepts in various forms.

      out

      [ c1 c2 c3

      c2 c3

      c4 c3

      c c ] T (21)

      H. Parameterization of the Fuzzy Operators

      5

      4

      for an arbitrary values of the output singleton MF cores, as given by [ 1 0.7 0 0.7 1 ]

      then, the output RB parameterization vector could be set by:

      There are multiple choices for representing the premise fuzzy conjunction and , and fuzzy disjunction or operators. A common choice is the (min-max) composition [13] and the

      out

      [ 1 0.7 0

      0.7 0

      0.7 0

      0.7

      1] T (22)

      (product-sum) composition [14]. It is found that representing the T-norm by any operator other than the product operator will introduce un-adjustable nonlinearities [11]. In this work

   2. Parameterization of the FLC Output

      As a final stage for calculating the crisp output out of the FLC, a defuzzificztion stage is mandatory for this purpose. Many defuzzificztion formulas are developed [11], each is suitable for a certain application. For control applications it is found tht using the center of area (COA), and a well known version named the center of gravity (COG) are highly recommended [8].

      The output U of the FLC is evaluated by:

      T

      product operator is used to implement both and conjunction and the T-implication.

      I. Parameterization of the scaling factors

      Although scaling factors is not part of the parameters for a fuzzy system, but practically, is highly acceptable to be part of the FLC structure. Those scaling factors related to the inputs g X ( ge , ge ) and the outputs gU ( gu , gu ) are

      playing an important role in tuning the fuzzy controller and for normalization the input and output UODs.

      n

      U out

      R

      (23)

      UODx

      [,] g x

      1/

      i1

      (i )

      1

      x

      UOD

      2

      1

      x

      [ , ] g

      2

      1/

      where and

      out

      are defined by (13) and (18)

      UOD

      u

      1

      [ , ] g

      u

      1

      1/

      (27)

      respectively.

      The generalized form of the defuzzified output can be written in the following form:

      Actually they are part of the pre-processing and post- processing stages.

      U D(,out )

      (24)

3. FLC COMPUTING ALGORITHM

   where D can be any defuzzification formula applied to evaluate the crisp output U .

   1. Parameterization of Input and Output Variables

   In closed loop systems as shown in Fig. 4., there are several signals which should be taken into consideration when the control signal is calculated. The error signal between the set point y and the measurement output

   Back to Table 1, a complete set of FLC parameters are defined and connected by systematic mathematical formulation suitable for the computing algorithm, and summarized as follows:

   1. The inference mechanism uses the singleton fuzzy system as given by the general RB structure:

      IF (rules ) THEN (out ) (28)

   2. Relevant inputs are { e1 , e1 } and outputs are

      { u1 , u1 }.

   3. Number of MFs defined for each input and output is given

      x

      u

      by m and m .

      i j

      Fig. 4. Typical FLC in a closed loop control system.

   4. Input MFs characteristics

      in are defined by (3), and

      output MFs are defined by singletons (19).

   5. The fuzzy operators are defined to use the product

      operator for calculating the rule premise certainty as given by (13) and for calculating the implication as given by numerator part of (23).

   6. Number of rules is calculated by (12).

   7. Antecedents of the rules are defined by rules (11).

   1. FLC construction

      We will show in a systematic way how the presented parameterization methodology is applied for computing the control action u using (29) as follows:

      1) The FLC relevant inputs are chosen to be x1 e y

      1. Consequents of the rules are defined by out

         (18).

         and

         x2 e y

         with u as the relevant only output,

      2. Membership values are calculated by (7).

      3. Firing strength of the rule (dof) is defined by (14).

      4. Defuzzifications are performed using (24).

      Based on the parameterization process proposed above for

      hence nx 2; and nu 1.

      2 2

      1. The active UODs for e and e are chosen to be [ , ]

         and [ , ] respectively, and for the output u1 is

         a singleton FLC, the output of the FLC is given by: 4 4

         U F(X ,in ,rules ,out , g X , gU )

         (29)

         chosen to be [ 20,20 ] (i.e. the control signal boundaries).

         Fig. 5. shows the main components constructing the FLC

         generated according to the presented parameterization methodology.

      2. The input variables are to be partitioned into five triangular MFs satisfying symmetricity feature with 50% overlapping. While the output variable is chosen to have nine equidistant singleton MFs.

         Hence:

         x

         x

         m 5; m

         1 2

         5; m 9

         u

         1

         And according to (2) for a triangular MF it gives: c 3

         Fig. 5. Components of a FLC computing algorithm.

         and ni

         15 .

4. ILUSTRATIVE EXAMPLE (INVERTED PENDULUM PROBLEM)

Consider the problem of balancing an inverted pendulum

4)in

Fig. 6. Inverted pendulum on a cart.

is evaluated by (1), and its reduced version (4) gives:

on a cart [15] as shown in Fig. 6. for which the dynamic equation is given by:

1

in 1

0.5 0

0.5 0

0.5 1

1

0.5

u 0.05 y 2 sin( y)

with normalization scaling factors evaluated by (27) as:

9.8sin( y) cos( y)

1.1

g x 2 / ; g x 4 / ; gu

1/ 20

y

4 0.1

(30) 1 2 1

0.5 3

1.1

cos 2 ( y)

1. rules reflects the indices of the MFs involved in the construction of the rule premise, which is identified by

   the two inputs

   x1 and

   x2 respectively.

   nR 25 is

   where y is the angle of the pendulum with respect to the vertical line and y is the pendulum angular velocity and u

   is the applied control force.

   calculated by (13), and RB premise is generated by (12) as:

   1111122222333334444455555T

   rules 1234512345123451234512345

   To evaluate the presented methodology, a PD like fuzzy

   controller is to be constructed for the non-linear control system given by (30), using the developed singleton FLC computing algorithm. We will discuss the regulator problem,

   i.e. keeping the inverted pendulum balanced in a vertical

2. The consequence out of the rule base is constructed according to (18), using the summation formula defined by passino [8], gives the following:

   position with reference to different initial positions of

   y(0) .

   out [ 1 .75 .5 .25 0 | .75 .5 .25 0 -.25|

   .5 .25 0 -.25 -.5| .25 0 -.25 -.5 -.75|

   0 -.25 -.5 -.75 – 1 ]T

   1

   where the output MFs are assumed to have the same partitions of the input MFs, that is identified by:

   { mf1 , mf2 , mf3 , mf4 , mf5 }= {NL, NS, Z, PS, PL}

   If normalization is used, it will be in the form of:

   {-1,-0.5, 0, 0.5, 1}

   The rule base table generated by and could

   the pendulum to its vertical position from different initial conditions bounded by y(0) [42.97,42.97] .

   VI. CONCLUSION

   A full parameterization process for a singleton fuzzy system is developed. It presents a systematic methodology for developing a singleton fuzzy logic controller for control applications. The assumptions made by the parameterization process is highly simplified the FLC computing algorithm. A well-known inverted pendulum problem is chosen for

   be summarized as shown by Table II.

   rules

   out

   evaluating the capability of the

3. The firing strength i is accomplished using (13), taking into consideration the choice of a triangular form MFs, hence (10) is used to evaluate the membership value at the given measurement points

   TABLE II.

   RULE-BASE TABLE FOR THE INVERTED PUNDULUM PROBLEM

   Fig. 7. Inverted pendulum for different initial conditions using the proposed singleton FLC.

   cx1

   and

   cx2

   for x1

   and x2

   respectively, for the

   proposed approach. The obtained result shows the effectiveness of the developed approach in terms of simplicity and transparency in setting the solution.

   givenin .

4. After calculating the vectors andout , the COG formula as given by (23) is used for defuzzification, which will calculate the final crisp value for the control action u1 at instants k based on a the measurement

Although, the proposed methodology is tested for constructing the FLC, a current research now is initiated for developing a structural design methodology using soft- computing controller based on the presented parameterization process. Both FLC structure and FLC parameters are to be designed and tuned in one single phase

simultaneously.

values cx1 and cx2 , knowing that cx1 e(k) and

cx2 e(k) .

1. Simulation and results

Complete set of programs are written to simulate the presented computing algorithm using MATLAB 7.3. The differential equation for the inverted pendulum problem is solved by using the fourth order Runge-Kutta algorithm. The sampling time is taken to be 0.02 seconds.

The closed loop system output which is identified by the angular position y is examined for different initial

conditions of

y(0) . The designed FLC shows a high

stability regulation for the inverted pendulum system over the entire UOD defined for the controller input variables. The behavior of the controlled system is summarized by Fig. 7. It is clearly shown that the effective margin of stability for the proposed simple FLC controller is capable to control back

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