Structure of Zero- and First-order TS Fuzzy Models

The TS model is a combination of a logical and a mathematical model. This model is also formed by logical rules; it consists of a fuzzy antecedent and a mathematical function as consequent part. The antecedents of fuzzy rules partition the input space into a number of fuzzy regions, while the consequent functions describe the system behavior within a given region:

Rj: Ifz1 isA1,j and . . . andzn isAn,j then

y=fj(q1, . . . , qm) (3.7) where z = [z1, . . . , zn]^T is the n-dimensional vector of the antecedent variables, z∈x, q= [q1, . . . , qm]^T is the m-dimensional vector of the consequent variables q ∈ x, where x denotes the set of all inputs of the y = f(x) model. Ai,j(zi) denotes the antecedent fuzzy set for theith input. The antecedents of fuzzy rules partition the input space into a number of fuzzy regions, while thefj(q) consequent functions describe the system behavior within a given region.

The spirit of fuzzy inference systems resembles that of a ‘divide and conquer’

concept – the antecedent of fuzzy rules partition the input-space into a number of local fuzzy regions, while the consequents describe the behavior within a given region via various constituents [131].

With respect to the antecedent membership functions and the structure of the rules three typical ways of partitioning of the input space can be obtained.

Figure 3.4 illustrates these partitions in a two-dimensional input space.

Figure 3.4: Various methods for partitioning the input space.

• Grid partitionThe conjunctive antecedent divides the antecedent space into a lattice of axis-orthogonal hyperboxes. In this case the number of rules needed to cover the entire domain is an exponential function of the input space dimension and of the fuzzy sets used in each variable. This partition method is often chosen in designing a fuzzy system which usually involves a few input variables. This partition strategy needs only a small number of membership functions for each input. However, it encounters problems when we have moderately large number of inputs. For instance, a fuzzy model with ten inputs and two membership functions on each input would result in 2¹⁰= 1024 fuzzy if-then rules. This problem, usually referred to as thecurse

of dimensionality, can be alleviated by other partition strategies introduced below.

• Tree partitionThis partition relieves the problem of the exponential increase of the number of rules. However, more membership functions for each input are needed to define these fuzzy regions and these membership functions are not interpretable as they do not bear clear linguistic meaning such as “small”.

• Scatter partition By covering a subset of the whole input space that char-acterizes a region of possible occurrence of the input vectors, the scatter partition can be obtained that can also limit the number of rules to a rea-sonable amount. In this case the use of multivariate membership functions is the most general one, as there is no restriction on the shape of fuzzy sets.

The boundaries between the fuzzy sets can be arbitrarily curved and opaque to the axes. This multidimensional membership function based fuzzy model is described in [1].

Usually, thefj consequent function is a polynomial in the input variables, but it can be any arbitrary chosen function that can appropriately describe the output of the system within the region specified by the antecedent of the rule. Whenfj(q) is a first-order polynomial,

fj(q) =θ⁰_j+θ¹_jq1+· · ·+θ^m_j qm= m l=0

θ^l_jql,whereq0= 1 (3.8) the resulting fuzzy inference system is calledfirst-order Takagi–Sugeno or simply Takagi–Sugeno fuzzy model. If fj(q) is a constant (fuzzy singleton) fj = θ⁰_j we have a zero-order Takagi–Sugeno orSingleton fuzzy model, which is a special case of the linguistic fuzzy inference systems [104] and the TS fuzzy model (m= 0).

Using fuzzy inference based upon product-sum-gravity at a given input [288], the final output of the fuzzy model,y, is inferred by taking the weighted average of the consequent functions as it is depicted in Figure 3.5:

y=

Nr

j=1

βj(z)fj(q)

Nr

j=1

βj(z)

(3.9)

where the weight, 0 ≤ βj(z) ≤ 1, represents the overall truth value (degree of fulfillment) of theith rule calculated based on the degrees of membership

βj(z) =

&n i=1

Ai,j(zi). (3.10)

Figure 3.5 shows the fuzzy reasoning procedure for a TS fuzzy model.

Figure 3.5: Inference method of the Takagi–Sugeno fuzzy model.

Example 3.1 (SISO TS fuzzy model). This example presents a simple Takagi–

Sugeno fuzzy model that has one input variable. The y =f(x) model consists of three rules with local linear models on the consequent parts

Rj : Ifz1 isA1,j theny=θ⁰_j+θ¹_jq1

wherex1=x, z1=x1, q1=x1, j= 1, . . . ,3.

As Figure 3.6 shows, when the operating regimes of these local models are de-fined by fuzzy sets instead of crisp ones, the resulted model behaves as a smooth nonlinear function.

As Figure 3.7 shows, the membership functions are arranged by Ruspini-type partition keeping the sum of the membership degrees equal to one

il=Ml

il=1

Al,il(zl) = 1l= 1, . . . , n (3.11) whereMlrepresents the number of the fuzzy sets on thelth input domain. Hence, the triangular membership functions are defined by:

Al,il(zl) = zl−al,ij−1

al,il−al,il−1

, al,il−1≤zl< al,il

Al,il(zl) = al,il+1−zl

al,il+1−al,il

, al,il≤zl< al,il+1 (3.12)

−100 −5 0 5 10 0.2

0.4 0.6 0.8 1 1.2

z1

Membership Grades

small medium large

(a) Antecedent MFs for Crisp Rules

−100 −5 0 5 10

2 4 6 8

x

y

(b) Overall I/O Curve for Crisp Rules

−100 −5 0 5 10

0.2 0.4 0.6 0.8 1 1.2

z1

Membership Grades

small medium large

(c) Antecedent MFs for Fuzzy Rules

−100 −5 0 5 10

2 4 6 8

x

y

(d) Overall I/O Curve for Fuzzy Rules

Figure 3.6: Example of a TS fuzzy model.

Figure 3.7: Ruspini parameterization of triangular membership functions.

whereal,ilcores of the adjacent fuzzy sets determine the support (sup_l,il=al,il+1− al,il−1) of a set.

al,il= core(Al,il(zl)) ={zl|Al,il(zl) = 1} . (3.13)

Example 3.2 (Ruspini partition of two-dimensional input space ). In multivariable case, the presented model obtains grid-type axes-parallel partition of the input space that helps to obtain an easily interpretable rule base. Figure 3.8 illustrates such partitioning of a two-dimensional input space whenn= 2,M1= 5, and M2= 5.

Figure 3.8: Partition of the input domain (whenn= 2,M1= 5,M2= 5).

In the following, the mathematical formalism of TS fuzzy models for nonlinear regression is described. Consider the identification of an unknown nonlinear system

y=f(x) (3.14)

based on some available input-output datax_k = [x1,k, . . . , xn,k]^T and yk, respec-tively. The indexk= 1, . . . , N denotes the individual data samples.

While it may be difficult to find a model to describe the unknown system glob-ally, it is often possible to construct local linear models around selected operating points. The modelling framework that is based on combining local models valid

in predefined operating regions is calledoperating regime-based modelling[197]. In this framework, the model is generally given by:

ˆ y=

c i=1

φi(x)

a^T_i x+bi

(3.15) whereφi(x) is the validity function for theith operating regime andθi= [a^T_i bi]^T is the parameter vector of the corresponding local linear model. The operating regimes can also be represented by fuzzy sets in which case the Takagi–Sugeno fuzzy model is obtained [257]:

Ri: IfxisA_i(x)thenyˆ=a^T_i x+bi, [wi] i= 1, . . . , c . (3.16) Here,Ai(x) is a multivariable membership function,ai andbi are parameters of the local linear model, andwi∈[0,1] is the weight of the rule. The value ofwi is usually chosen by the designer of the fuzzy system to represent the belief in the accuracy of theith rule. When such knowledge is not availablewi= 1,∀iis used.

The antecedent proposition “xisA_i(x)” can be expressed as a logical combi-nation of propositions with univariate fuzzy sets defined for the individual com-ponents ofx, usually in the following conjunctive form:

Ri: Ifx1 isAi,1(x1)and . . . andxn isAi,n(xn)thenyˆ=a^T_i x+bi, [wi]. (3.17) Thedegree of fulfillmentof the rule is then calculated as the product of the indi-vidual membership degrees and the rule’s weight:

βi(x) =wiA_i(x) =wi

&n j=1

Ai,j(xj). (3.18) The rules are aggregated by using the fuzzy-mean formula

ˆ y=

c i=1

βi(x)

a^T_ix+bi

c

i=1

βi(x)

. (3.19)