Type-2 Fuzzy Sets Based Modeling of Nonlinear System ... - CiteSeerX

INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.IJCC.US), VOL. 6, NO. 1, MARCH 2008

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Type-2 Fuzzy Sets Based Modeling of Nonlinear System Using Choquet Integral and Fuzzy Difference Equation Madhusudan Singh, Smriti Srivastava, M. Hanmandlu and J.R.P. Gupta

Abstract— In this paper, the Choquet Integral is used to replace the consequent part of a fuzzy rule to make the resulting fuzzy system non-additive and type-2 fuzzy sets are used in the antecedent part of the rule to account for varying uncertainties and vagueness. As the stability of the model is also highly dependent on the learning of the system we use Lyapunov Stability (LS) in combination with fuzzy difference (FD). FD gives the range of variation of parameters having the lower and the upper bounds within which the system is stable. The underlying fuzzy model is investigated for identification of nonlinear systems. The identification of fuzzy model involving the Choquet integral and type-2 fuzzy sets is developed with the strength of the rules and fuzzy measures as the input information to be determined. The fuzzy measures aggregated to yield the non additive fuzzy model are constructed from the fuzzy densities c 2008 Yang’s Scientific Research of type-2 fuzzy sets. Copyright ° Institute, LLC. All rights reserved. Index Terms— Choquet integral, λ-measure, fuzzy density, type-2 fuzzy sets, Lyapunov stability, fuzzy difference equation.

I. I NTRODUCTION

W

E PRESENT here a new class of fuzzy logic system (FLS), i.e., the non-additive fuzzy system involving the Choquet Integral and type-2 fuzzy sets. In this FLS, type2 fuzzy sets are used in antecedent part and Choquet Integral is used in consequent part. The concept of type-2 fuzzy set was introduced by Zadeh as an extension of the concept of an ordinary fuzzy set (henceforth called a type-1 fuzzy set). Such fuzzy sets whose membership grades themselves are type-1 fuzzy sets are very useful in circumstances where it is difficult to determine an exact membership function for a fuzzy set [35]. Quite often, the knowledge used to construct rules in a fuzzy logic system (FLS) is uncertain. This uncertainty leads to rules having uncertain antecedent and/or consequent membership functions. In fuzzy logic (FL), we may view computing the defuzzified output by the various techniques as analogous to computing the mean of a probability density function. But Manuscript received June 14, 2007; revised November 06, 2007. Madhusudan Singh, Smriti Srivastava and J.R.P. Gupta, Department of Instrumentation and Control Engineering, N.S.I.T., Sector–3, Dwarka, New Delhi-110058, India. M. Hanmandlu, Department of Electrical Engineering I.I.T. Delhi, Hauz Khas, New Delhi-110016, India. Corresponding Author- Dr. Smriti Srivastava Postal Address: Department of Instrumentation & Control Engineering N.S.I.T., Sector-3 Dwarka New Delhi-110075, India. Email: [email protected], [email protected] Publisher Item Identifier S 1542-5908(08)10101-4/$20.00 c Copyright °2008 Yang’s Scientific Research Institute, LLC. All rights reserved. The online version posted on July 09, 2008 at http://www.YangSky.com/ijcc/ijcc61.htm

in practical statistical-based designs, we need to capture the probabilistic distributions. Type-2 FLS provides a measure of dispersion of the membership functions which is essential to the design of systems that include linguistic and/or numerical uncertainties. Just as one can go beyond second-order moments in probabilistic modeling, we can also use higher than type-2 sets in fuzzy modeling; but, as we move on to higher types, the complexity of the system increases rapidly. The learning of the premise and consequent parameters of the fuzzy membership function is accomplished by Lyapunov Stability (LS) in combination with Fuzzy Difference (FD). In general, the learning methods like gradient descent, back propagation etc. have the problem of local minima. Hence, using the concept of LS this problem is eliminated allowing us to build a stable system with the global minima. For a stable system, the region of stability is also an important aspect. Using the FD in combination with LS solves this problem. FD gives the upper and lower bounds of the solution, which define the stability region for the system. If the parameters exceed this stability limit by means of perturbation or external noise etc., the stability of the system deteriorates. Moreover, the region of stability provides the limit of tolerance, which a stable system can sustain. In other words, it defines the robustness of the system. The paper is organized as follows: In section II, a brief description of Choquet fuzzy integral is given. Section III, describes the Modeling of a system. Section IV describes the learning approach based on gradient descent (GD) law together with LS and FD. Simulated results and comparison of our work with the existing works is discussed in Section V. Finally Section VI is relegated to conclusions of the proposed work. II. T HE C HOQUET F UZZY I NTEGRAL It aggregates the overlapping input information from the input sets using the fuzzy measures to yield the output. The fuzzy measures provide an alternative computational scheme for aggregating the information. Let X = {x1 , x2 , . . . , xn } be a finite set and A, B ⊂ X with A ∩ B = φ. A set function g : Ω → [0, 1], where Ω is a sigma-algebra of subsets of X then we can define the fuzzy measure of A and B as [2]: g (A ∪ B) = g(A) + g(B) + λg(A)g(B)

(1)

for a fixed value of λ > −1. A probability, belief or plausibility measure can be devised by choosing the value of

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INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.IJCC.US), VOL. 6, NO. 1, MARCH 2008

λ as zero, negative and positive. The value of λ can be found from g(X) = 1, which amounts to solving, λ+1=

n Y

(1 + λgi )

(2)

i=1

The definition of λ-measure is highly constrained by the uniqueness of the parameter λ computed by solving an (n−1) degree polynomial in λ, which is expensive and restrictive. If h(x1 ), h(x2 ), h(x3 ), . . . , h(xn ) constitute the information provided by the input sources x1 , x2 , . . . , xn and g is a λmeasure then the Choquet fuzzy integral is constructed as: Z h(·) ◦ g(·) (3)

as it combines fuzzy measures from subsequent fuzzy sets and the resulting system is called non-additive fuzzy system unlike the function in TS model where each coefficient corresponds to its own fuzzy set and there is no concept of fuzzy measure in this model. This means that the fuzzy sets are independent and the resulting system is called additive fuzzy system as the output is directly related to these fuzzy sets not to the fuzzy measures. The integral type function is suitable to most real life applications where the information from several sources needs to be combined and it has the following form:

x

For a finite set X, the Choquet fuzzy integral is computed as: n X

¡ ¢ h(xi ) gAi − gA(i+1)

(4)

i=1

where, gAi is a λ fuzzy measure and Ai = {xi , xi+1 , . . . , xn } then gAi can be computed recursively as gAn = g({xn }) = gn gAi = gi + gA(i+1) + λgi gA(i+1) ,

for 1 ≤ i < n.

yˆk (j) = f k (xj ) = G0k (j) + G1k (j)x1 (j) + ... + Gnk (j)xn (j) n n X X ¡ ¢ = xi (j)Gik (j) = xi (j) gAik (j) − gA(i+1)k (j)(8) i=0

Here, gAik (j) = gik (j) + gA(i+1)k (j) + λgik (j)gA(i+1)k (j) and gAnk (j) = gnk (j).

(5) (6)

Since this work involves overlapping fuzzy sets, it is important to represent them by means of fuzzy measures, which according to Sugeno [14] are defined so as to replace the additivity axiom of classical measures with weaker axioms of monotonicity and continuity [5]. Therefore, a new class of fuzzy measures called q-measure is introduced in [2] by generalizing λ-measure.

This paper makes an attempt to improve the modeling capabilities of FLS. In practice, dynamic system modeling is based on some a priori knowledge and input-output data of the system. Here we will not seek a priori knowledge but only the input-output data. Modeling consists of determining the structure identification of a model, which in turn consists of determining a suitable number and shape of fuzzy partitioning of input-output space, since the number of fuzzy partitions gives the number of rules and the shape of fuzzy partition determines the membership function parameters. Note that type-2 non-additive fuzzy rules are characterized by type-2 fuzzy sets in the antecedent part and the Choquet integral in the consequent part. A rule of the following form underlies the model

where yl (j) the lower bound output is defined as: yl (j)

= = =

if x1 (j) is A˜k1 and x2 (j) is A˜k2 and x3 (j) is A˜k3 . . . , and xn (j) is A˜kn then yˆk (j) = f k (xj ) (7)

where, k varies from 1 to r, r being the number of rules, n is the total number of inputs and j is the j th training sample. In the above rule, the function f k (xj ) contains fuzzy densities and fuzzy measures, which represent information of the corresponding fuzzy sets. This function is of integral type

r X

k k βjl f (xj ) =

k=1 Ã r X

k βjl

k=1

k βjl

n X

Gik (j)xi (j)

i=0

! n X ¡ ¢ gAik (j) − gA(i+1)k (j) xi (j) i=0

k=1 Ã r X

n X

k=1

i=0

k βjl

r X

! ¡ ¢ gik (j) 1 + λgA(i+1)k (j) xi (j)

and yu (j) the upper bound output is defined as:

yu (j)

= = =

:

(9)

gAik (j) is a fuzzy measure and gik (j) is the fuzzy density, which are calculated from the LS+FD learning approach and j is the j th training sample. The defuzzified output of the model is: y(j) = [yl (j), yu (j)]

III. F UZZY M ODELING OF N ONLINEAR S YSTEM

Rk

i=0

r X

k k βju f (xj ) =

k=1 Ã r X

r X k=1

k βju

n X

Gik (j)xi (j)

i=0

! n X ¡ ¢ k βju gAik (j) − gA(i+1)k (j) xi (j) i=0

k=1 Ã r X

n X

k=1

i=0

k βju

! ¡ ¢ gik (j) 1 + λgA(i+1)k (j) xi (j) (10)

Similar meaning exist for all the parameters involving ‘l’ and ‘u’ in the subscripts place. Then type reduced output is: y(j) =

1 [yl (j) + yu (j)] 2

(11)

SINGH, SRIVASTAVA, HANMANDLU & GUPTA, TYPE-2 FUZZY SETS BASED MODELING OF NONLINEAR SYSTEM

where

3

B. Parameter update formula by GD µkl

k βjl = P r

m=1

µku

(xj )

µm l (xj )

k , and βju = P r

m=1

(xj )

.

(12)

µm u (xj )

We apply the gradient learning to update the parameters alki , cki , lki and gik by

auki ,

∆alki (j) = ∆auki (j) =

Here, µkl (xj ) =

n h Y

³ ¯ ¯lki ´i exp − ¯alki (xi (j) − cki )¯

∆cki (j) =

i=1

∆lki (j) =

and

∆gik (j) = µku (xj ) =

n h ³ ´i Y l exp − |auki (xi (j) − cki )| ki i=1

are the generalized Gaussian function and approximates the symmetric triangular to trapezoidal membership functions depending on the value of exponential power, lki ; the indices i, k indicate ith input xi and k th rule respectively; cki is the central value of the fuzzy set for ith premise variable xi corresponding to k th rule; and a1ki represents the width of the fuzzy set. Determination of the number of fuzzy rules and initialization of parameters are accomplished by the fuzzy curve method [1]. Here, we recall that the range (lower and upper) of the width parameter in the Gaussian membership £ ¤ function denotes the uncertainty as aki ∈ alki , auki . To begin with we assume generally ±a% of deviation in the parameters to compute their lower and upper bounds so as to learn them later. But the value of a% varies for different applications. The choice of ‘a’ is also important as initialization of parameters always affects the convergence of the parameters. Hence, we use the fuzzy difference equation to solve this problem by finding the upper and the lower bounds of the parameters such that the system is stable.

∂J , ∂alki ∂J −η u , ∂aki ∂J −η , ∂cki ∂J −η , ∂lki ∂J −η . ∂gik (j)

−η

(14)

where, η is the learning rate > 0. In Eq. (14), the parameters are learned by GD learning law. GD method has the problem of yielding the local minima. We will attempt to modify this method so as to make the system globally stable. To achieve this, let us choose Wki (alki , auki , cki ,lki and gik ) as the generalized weight and rewrite the update Eq. (14) for the j th sample as: Wki (j + 1) = Wki (j) + ∆Wki (j)

(15)

Next, we will convert Eq. (15) in to the general form by introducing additional learning parameters, which are estimated using the Lyapunov function. C. Lyapunov Stability Based Learning Rewriting Eq. (15) in the generalized form as: Wki (j + 1) = α1 Wki (j) + α2 ∆Wki (j)

(16)

where, α1 and α2 are constants. In view of Eqs. (13) and (14), we can write ∆Wki (j) in Eq. (16) as: ∂J ∂J ∂e(j) ∂y(j) = −η ∂Wki ∂e(j) ∂y(j) ∂Wki ∂y(j) = ηe(j) . (17) ∂Wki

∆Wki (j) = −η IV. T HE L EARNING A PPROACH A. Gradient Descent based learning Algorithm: Fine-tuning of rules can be achieved by minimizing the objective function J. It is a function of normalized mean square error with respect to the parameters alki , auki , cki , lki and gik . The following form is assumed for J:

Note that Eq. (17) is still gradient descent based having the problem of local minima. We are therefore concerned with eliminating this drawback by taking recourse to Lyapunov stability. As we know, ∂e(j) ∂y(j) =− ∂Wki ∂Wki

(18)

We choose a Lyapunov function as: J=

1 2

M X

e2 (j)

(13)

j=1

V (j) =

i 1h 2 2 (e(j)) + (Wki (j)) 2

Such that V (j) > 0, then where, e (j) = yd (j) − y (j) and yd is the desired output and y is the actual output.

∆V (j) = V (j + 1) − V (j) ≤ 0

(19)

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∆V (j)

=

1h 2 2 2 (e(j + 1)) − (e(j)) + (Wki (j + 1)) 2 i 2 − (Wki (j))

1 [(e(j + 1) − e(j)) (e(j + 1) + e(j)) 2 + (Wki (j + 1) − Wki (j)) × (Wki (j + 1) + Wki (j))] 1 [∆e(j) (2e(j) + ∆e(j)) = 2 +∆Wki (j) (2Wki (j) + ∆Wki (j))] =

where, z is a positive value. Now, Eq. (25) can be written as: "Ã µ ¶2 ! 1 ∂y(j) 2 1+ (∆Wki (j)) 2 ∂Wki (j)

e(j + 1) − e(j) = ∆e(j) and Wki (j + 1) − Wki (j) = ∆Wki (j), the above becomes, 1h 2 2 ∆V (j) = (∆e(j)) + (∆Wki (j)) + 2e(j)∆e(j) 2 +2Wki (j)∆Wki (j)] " à µ ¶2 ! 1 ∆e(j) 2 = (∆Wki (j)) 1 + 2 ∆Wki (j) µ µ ¶¶¸ ∆e(j) +2∆Wki (j) Wki (j) + e(j) ∆Wki (j) (20) For a very small change, we can write: " à µ ¶2 ! ∂e(j) 1 2 (∆Wki (j)) 1 + 2 ∂Wki (j) µ µ ¶¶¸ ∂e(j) +2∆Wki (j) Wki (j) + e(j) ∂Wki (j)

µ µ ¶¶ ¸ ∂y(j) +2 Wki (j) − e(j) ∆Wki (j) + z = 0 (26) ∂Wki (j) Note that Eq. (26) is in quadratic form, i.e. aˆ x2 +bˆ x +c = 0 2 and for deriving (ˆ x − d) = 0, where d is the solution, we must ensure that b2 − 4ac = 0. Hence, taking, à µ ¶2 ! ∂y(j) a = 0.5 1 + ∂Wki (j) , µ µ ¶¶ ∂y(j) b = Wki (j) − e(j) ∂Wki (j) and c = 0.5z, then, b2 − 4ac = 0, µ µ ¶¶2 à µ ¶2 ! ∂y(j) ∂y(j) Wki (j) − e(j) − 1+ z=0 ∂Wki (j) ∂Wki (j)

(21)

Hence, ³

³ ´´2 ∂y(j) Wki (j) − e(j) ∂W ki (j) µ z= , ³ ´2 ¶ ∂y(j) 1 + ∂W ki (j)

Using Eq. (18) we can rewrite (21) as: " à ¶2 ! µ 1 ∂y(j) 2 = (∆Wki (j)) 1 + 2 ∂Wki (j) µ µ ¶¶¸ ∂y(j) +2∆Wki (j) Wki (j) − e(j) ∂Wki (j) (22) From the condition of Lyapunov stability: ∆V ≤ 0 assures the stability; hence, " à µ ¶2 ! ∂y(j) 1 2 (∆Wki (j)) 1 + ∆V (j) = 2 ∂Wki (j) µ µ ¶¶¸ ∂y(j) +2∆Wki (j) Wki (j) − e(j) ∂Wki (j) ≤ 0 (23) Theorem: Lyapunov function, i 1h 2 2 (e(j)) + (Wki (j)) > 0 V (j) = 2

(24)

Proof: Let, from Eq. (23): " à µ ¶2 ! 1 ∂y(j) 2 ∆V (j) = (∆Wki (j)) 1 + 2 ∂Wki (j) µ µ ¶¶¸ 1 ∂y(j) +2∆Wki (j) Wki (j) − e(j) = − z (25) ∂Wki (j) 2

Since,

=

and ∆V (j) ≤ 0 if and only if ³ ³ ´´ ∂y(j) − Wki (j) − e(j) ∂W ki (j) µ ∆Wki (j) = ³ ´2 ¶ ∂y(j) 1 + ∂W ki (j)

(27)

which is a positive value. Hence, ∆Wki (j) = −b 2a . Also, ³ ³ ´´ ∂y(j) − Wki (j) − e(j) ∂W (j) ki µ ∆Wki (j) = (28) ³ ´2 ¶ ∂y(j) 1 + ∂Wki (j) If we use Eq. (28) then our system is Lyapunov stable thus proving our statement in Eq. (24). Incorporating Eq. (28) into Eq. (16) we get: ³ ³ ´´ ∂y(j) Wki (j) − e(j) ∂W ki (j) µ Wki (j + 1) = α1 Wki (j) − α2 ³ ´2 ¶ ∂y(j) 1 + ∂W ki (j) Wki (j + 1) = k1 Wki (j) + k2 ,

(29)

SINGH, SRIVASTAVA, HANMANDLU & GUPTA, TYPE-2 FUZZY SETS BASED MODELING OF NONLINEAR SYSTEM

where

 k1 =  α1 − µ and

h i L U lki (j) = 0.5 (lki ) (j, α) + (lki ) (j, α) h i L U gik (j) = 0.5 (gik ) (j, α) + (gik ) (j, α)



 ³ 1+

α2 ∂y(j) ∂Wki (j)

 ´2 ¶  

D. Fuzzy Difference Equation We now consider Eq. (29) as the fuzzy difference:

where, ¡

¢ k L βjl α

¡ k ¢L µl α (xj ) = P r ¡ ¢L µkl α (xj ) m=1

and ¡ k ¢L µl α (xj ) = µ ¯ n ³ ´¯(lki )L (j,α) ¶¸ Y· ¯¡ ¢L ¯ L exp − ¯ alki (j, α) xi (j) − (cki ) (j, α) ¯ ,

(30)

Here we require that the updated value of the weight at j th iteration must depend on its value at (j-1)th iteration assuming W0 = Wki (j − 1). This will assure us in speeding up the learning rather than choosing the initial values. While finding the solutions of Eq. (30) we treat k1 and k2 as constant at that instant. Let Wki (j) be the solution of Eq. (30). As our motive is to calculate the parameters which give rise to the robust system, we chooseW0 , k1 and k2 to be fuzzy; hence we model this uncertainty by invoking the triangular fuzzy numbers for ¯ is defined W0 , k1 and k2 . The triangular fuzzy number N by the three values that satisfy the condition:a1 < a2 < a3 , where a1 , a3 form the base of triangle and a2 is the vertex. Let the uncertainty of W0 be defined by the fuzzy num¯ N = (W 1 /W 2 /W 3 ), and k1 is given by K ¯1 = ber, W N N N ¯ 2 = (k21 /k22 /k23 ). Then α-cuts (k11 /k12 /k13 ) and k2 by K of Wki (j) are: · ¸ −k21 (α) k21 (α) j L 1 Wki (j, α) = + WN (α) + (1 + k12 (α)) k12 (α) k12 (α) (31) · ¸ −k22 (α) k22 (α) j 2 = + WN (α) + (1 + k11 (α)) k11 (α) k11 (α) (32) Eqs. (31) and (32) give the lower and upper bound of the parameter Wki (j) respectively. As the uncertainty is represented by the width parameter of the membership function, we calculate the lower and upper width parameter as: ¡ ¢L alki (j) = alki (j, α) U Wki (j, α)

and

i=0 ´ ´ ³ ¢L ¡ × 1 + λ gA(i+1)k (j, α) xi (j) (34)

k=1

Now Eq. (29) is the update equation entrusted with the Lyapunov stability. In order to find the upper bound and lower bounds of the weights, we use the fuzzy difference equation.

W0 = Wki (j − 1)

(33)

We can now find the lower and upper bounds of the output as: Ã r n X ¡ k ¢L X L L Y (j) = βjl α (gik ) (j, α)

³ ´ ∂y(j) α2 e(j) ∂W ki (j) k2 = µ ³ ´2 ¶ . ∂y(j) 1 + ∂W ki (j)

Wki (j + 1) = k1 Wki (j) + k2 ,

5

U

auki (j) = (auki ) (j, α).

By doing so we need not deviate from the initial parameters by ±a% only as it is always different for different applications. It has to be got through the learning of the system and gives an idea about how much perturbation the system can withstand yet being stable. By this way the robustness of the system can also be ascertained. The other parameters are obtained from: h i L U cki (j) = 0.5 (cki ) (j, α) + (cki ) (j, α) ,

i=1

Y U (j) = ! Ã r n ³ ´ X ¡ k ¢U X ¡ ¢U U βju α (gik ) (j, α) 1 + λ gA(i+1)k (j, α) xi (j) k=1

i=0

(35) where,

¡ k ¢U ¡ k ¢U µu α (xj ) βju α = P r U (µku )α (xj ) m=1

and

³

´U µku (xj ) = α · µ ¯ n ³ ´¯(lki )U (j,α) ¶¸ Y ¯ ¯ exp − ¯(auki )U (j, α) xi (j) − (cki )U (j, α) ¯

i=1

Here, Eqs. (34) and (35) give the lower and the upper bounds of the output. Using these equations we can find the maximum robustness of the system, i.e. system remains stable in the presence of the maximum variation of the parameters. The global stability of the system is guaranteed as we are imposing the Lyapunov stability criterion on the learning. V. R ESULTS OF S IMULATION For the purpose of simulation we have taken up an example of a chemical plant. Example 1: We have taken an example from [11]. In this example, we model the role of an operator’s control of a chemical plant. The main operation of the plant is the production of a polymer by the polymerization of some monomers. Since the start up of the plant is very complicated, it requires manual operation. The structure of human operation is shown in Fig. 1. In this plant the inputs denoted by xi and output by yare given as follows: x1 :Monomer concentration x2 :Change in monomer concentration x3 :Monomer flow rate

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x4 :Local temperatures x5 :Local temperatures y :Monomer flow rate

Fig. 1.

Structure of plant operation in example 1.

There are a total of 70 input-output pairs. Using the method of Fuzzy Curve [1], five fuzzy curves c1 (x1 ), c2 (x2 ), c3 (x3 ), c4 (x4 ) and c5 (x5 ) are plotted for the 70 input-output data. Fig. 2 shows the curves. Hence the minimum numbers of rules needed is 4. Also, the parameters of the membership functions are initialized using fuzzy curve as explained in [1]. Fig. 3 shows the final membership functions for different inputs. Fig. 4 shows the final output of the system with the corresponding lower and upper bounds. Table I shows the performance index (J) at the last iteration. The proposed type2 (T2) non-additive model using the Choquet Integral (CI) with LS and FD learning (T2+CI+LS+FD) is compared with (i) the same model for type-1 fuzzy sets (T1+CI+LS+FD) (ii) the same model for type-2 sets but with GD learning algorithm (T2+CI + GD) and (iii) the same model with type-1 fuzzy sets but with GD learning (T1+CI + GD), and (iv) by T-S model. Table I shows that the proposed system gives the minimum PI and also assures the robustness and global stability of a nonlinear system.

Fig. 3.

Plot of final membership functions for example 1.

Fig. 4.

Plot of plant output for example 1.

x4 (t) = y(t-4), x5 (t) = u(t-1), x6 (t) = u(t-2), x7 (t) = u(t3), x8 (t) = u(t-4), x9 (t) = u(t-5), x10 (t) = u(t-6) }. This makes it as ten inputs and single output system with 290 input-output data points. Using the method of Fuzzy Curve [1], ten fuzzy curves c1 (x1 ) to c10 (x10 ) are plotted for input-output data. Fig. 5 shows the curves. Minimum numbers of rules needed is 2 for example 2. Fig. 6 shows the final membership functions after learning for different inputs. Fig. 7 shows the final output of the system with the corresponding lower and upper bounds. Table II shows the performance index (J) at the last iteration. Proposed type-2 (T2) non additive model using the Choquet Integral (CI) with LS and FD learning (T2+CI+LS+FD) is compared with other schemes and superiority of the T2+CI+LS+FD scheme is well Fig. 2.

Plot of fuzzy curves for example 1.

Example 2: As a second example of identification of a well-known Box-Jenkins data set is taken from [11]. There are originally 296 data points {y(t), Where output is CO2 concentration and input is gas flow rate. Here, yd (t) is predict on { x1 (t) = y(t-1), x2 (t) = y(t-2), x3(t) = y(t-3),

TABLE I Performance Index (J) T2+CI+LS+FD T1+CI+LS+FD T2+CI + GD T1+CI + GD T-S Model

Example 1 7.6238 × 105 7.9731 × 105 8.3685 × 105 8.4832 × 105 8.7639 × 105

SINGH, SRIVASTAVA, HANMANDLU & GUPTA, TYPE-2 FUZZY SETS BASED MODELING OF NONLINEAR SYSTEM

7

justified in table II.

Fig. 7.

Plot of plant output for example 2. TABLE II

Fig. 5.

Plot of fuzzy curves for example 2.

Performance Index (J) T2+CI+LS+FD T1+CI+LS+FD T2+CI + GD T1+CI + GD T-S Model

Example 2 3.4565 × 102 3.7214 × 102 5.7532 × 102 6.1437 × 102 6.4396 × 102

R EFERENCES

Fig. 6.

Plot of final membership functions for example 2.

VI. C ONCLUSIONS Present work opens the new direction in the field of type2 fuzzy systems for including the overlapping information from the input fuzzy sets by way of modeling the type-2 non-additive fuzzy set system with Choquet Integral in the consequent part. The global stability of system is assured as system parameters are learned by using the Lyapunov stability criterion. The use of fuzzy difference equation takes care of the uncertainty in the antecedent part by determining the lower and the upper bounds of the parameters such that the system remains within the bounds. Next the lower and upper bounds of the output are utilized to compute the maximum deviation in the parameters. Finally, the simulation and results show the superiority of the proposed type-2 non-additive fuzzy model over other models.

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