T. Kumbasar et al. / International Journal of Approximate Reasoning 54 (2013) 253–272 255
Fig. 1. Illustration of (a) type-2 fuzzy trapezoidal membership function and (b) interval secondary membership function.
Fig. 2. Block diagram of interval type-2 fuzzy logic system.
in which 0 ≤ μA(x, u) ≤ 1. For a continuous universe of discourse, A can be expressed as
A =∫x∈X
∫u∈Jx
μA(x, u)/(x, u), Jx ⊆ [0, 1] (2)
where∫ ∫
denotes union over all admissible x and u. Jx is referred to as the primary membership of x, while μA(x, u) is atype-1 fuzzy set known as the secondary set. The uncertainty in the primarymembership of a type-2 fuzzy set A is defined by
a region named footprint of uncertainty (FOU).WhenμA(x, u) = 1 for∀u ∈ Jx ⊆ [0, 1], an interval type-2 fuzzy set (IT2-FS)
is obtained.Anexampleof a trapezoidal type-2 fuzzy set is given in Fig. 1. It canbedescribed in termsof anuppermembership
functionμA(x) and a lower membership functionμA(x). The primarymembership Jx , and its associated possible secondary
membership functions can be trapezoidal, interval, etc.When the interval secondarymembership function that is illustrated
in Fig. 1(b) is taken, an interval type-2 fuzzy set is obtained [19,29,32].
2.2. Interval type-2 fuzzy logic systems
A fuzzy logic system described with at least one interval type-2 fuzzy set is called an interval type-2 fuzzy logic system
(IT2-FLS). The internal structure of the IT2-FLS is given in Fig. 2.
An IT2-FLS is characterized by IF-THEN rules like type-1 fuzzy logic systems, but its antecedent and/or consequent
fuzzy sets are defined by interval type-2 fuzzy sets. The type-2 fuzzy set outputs are then processed by the type-reducer
which combines the output sets which leads to type-1 fuzzy sets called the type-reduced set. Then, the type-reduced set is
defuzzified to produce crisp output [29,32].
An interval type-2 fuzzy rule can be defined as following:
rj : IF x1 is A1 and x2 is A2 and . . . and xn is An THEN y is Cj, j = 1, 2, . . . ,N (3)
where A1, A2, . . . , An are interval type-2 fuzzy sets on the universe of discourses x1, x2, . . . , xn, while Cj is the consequent
interval set of the jth IF-THEN rule, N is the total number of rules. The firing intervals for each type-2 fuzzy set are:
μA1 = [μA1, μA1]μA2 = [μA2, μA2]...
μAn = [μAn, μAn]
(4)
256 T. Kumbasar et al. / International Journal of Approximate Reasoning 54 (2013) 253–272
whereμAj andμAj are the firing sets of the lower membership function and upper membership function, respectively. The
firing sets inherit the uncertainties of the antecedents and affect the interval consequent sets. The total firing set for each
rule is defined as:
f j =[f j f
j]
(5)
where
f j = μA1 ∗ μA2 ∗ · · · ∗ μAn
fj = μA1 ∗ μA2 ∗ · · · ∗ μAn
Here, the operator ∗ represents the t-norm, which is the product operator.
The consequent part Cj of the IT2-FLS is also an interval set and defined as:
Cj = [cj, cj] (6)
y∗(x) = [yl(x), yr(x)] is the interval output of IT2-FLS. Here, yl and yr can be calculated via a type reductionmethod [29,32].
Then, the type reduced fuzzy set for an IT2-FLS can then be expressed as:
YTR = [yl(x), yr(x)] (7)
where YTR is the type reduced interval set determined by its two end points yl(x) and yr(x). The output of the IT2-FLS can
be obtained by using the average value of yl(x) and yr(x). So the crisp output of IT2-FLS is calculated using
y∗(x) = yr(x)+ yl(x)
2(8)
2.3. Karnik–Mendel type reduction method
The iterativeKarnikandMendelmethod is commonlyused for type reductionanddefuzzification [19].Wu[41]pointedout
two features of the Karnik–Mendel type reductionmethod; namely, novelty and adaptiveness. Noveltymeans that the upper
and lowermembership functions of the same type-2 fuzzy setmayormaynot beused simultaneously in computing the type-
reduced set. Theadaptivenesshas themeaning that theboundsof the type-reduced interval set changeas the inputs change. It
has been stated that the alternativefive approximations [4,12,35,44,45] to theKMalgorithms cannot simultaneously capture
novelty and adaptiveness features [41]. These two features provide the model to handle the uncertainties that may appear
within systems much better.
To be able to calculate the two end points of the type reduced set (yl and yr), first cj and cj are re-ordered such that
c1 ≤ c2 ≤ · · · ≤ cN , c1 ≤ c2 ≤ · · · ≤ cN and the corresponding weights f and f are matched, respectively.
Defining y(R)r and y
(L)l [29] for 1 ≤ L, R ≤ N − 1, as:
y(L)l =
∑Lj=1 f
jcj + ∑N
j=L+1 fjcj∑L
j=1 fj + ∑N
j=L+1 fj
(9)
y(R)r =∑R
j=1 fj cj + ∑N
j=R+1 fjcj∑R
j=1 fj + ∑N
j=R+1 fj
(10)
Then yl(x) is the minimum of all y(L)l (x), and yr(x) is the maximum of all y
(R)r (x), i.e.
yl(x) = min1≤L≤N−1
{yLl (x)} = yL∗(x)l (x) =
∑L∗(x)j=1 f
jcj + ∑N
j=L∗(x)+1 fjcj∑L∗(x)
j=1 fj + ∑N
j=L∗(x)+1 fj
(11)
where
L∗(x) = arg min1≤L≤N−1
{y(L)l (x)} (12)
and
yr(x) = max1≤R≤N−1
{yRr (x)} = y(R∗(x))
r (x) =∑R∗(x)
j=1 f jcj + ∑Nj=R+1 f
jcj∑R∗(x)
j=1 fj + ∑N
j=R∗(x)+1 fj
(13)
T. Kumbasar et al. / International Journal of Approximate Reasoning 54 (2013) 253–272 257
Table 1
Calculation of the two end points of the type reduced set.
Steps The Karnik Mendel algorithm for computing yl The Karnik Mendel algorithm for computing yr
1. Sort cj (j = 1, 2, . . . ,N) in increasing order such that c1 ≤c2 ≤ · · · ≤ cN . Match the corresponding weights f
j, f j (with
their index corresponds to the renumbered cj)
Sort cj (i = 1, 2, . . . ,N) in increasing order such that c1 ≤c2 ≤ · · · ≤ cN . Match the corresponding weights f
j, f j (with
their index corresponds to the renumbered cj)
2. Initialize fj by setting Initialize fj by setting
fj = fj+fj
2, j = 1, 2, . . . ,N fj = fj+fj
2, j = 1, 2, . . . ,N
Compute y =∑N
j=1 fj cj∑N
j=1 fjCompute y =
∑Nj=1 fj c
j∑Nj=1 fj
3. Find the switch point L(1 ≤ L ≤ N − 1) such that Find the switch point R(1 ≤ R ≤ N − 1) such that
cj ≤ y ≤ cj+1 cj ≤ y ≤ cj+1
4. Set fj ={
f j j ≤ L
f j j > LSet fj =
{f j j ≤ R
f j j > R
and compute y′ =∑N
j=1 fj cj∑N
j=1 fjand compute y′ =
∑Nj=1 fj c
j∑Nj=1 fj
5. Check if y = y′ . If not go to step 3 and set y′ = y. If yes stop and
set y = y′Check if y = y′ . If not go to step 3 and sety′ = y. If yes stop and
set y = y′
where
R∗(x) = arg max1≤R≤N−1
{yRr (x)} (14)
The iterative Karnik and Mendel algorithm is given in Table 1 [43].
3. Interval type-2 fuzzy logic system inversion
Consider an IT2-FLS with n inputs (xk ∈ Xk , k = 1, 2, . . . , n) and one output (y ∈ Y). This system can be represented as
a mapping from (x1, x2, . . . , xn) to y as:
f : x1, x2, . . . , xn → y = f (x1, x2, . . . , xn) (15)
where f represents interval type-2 fuzzymapping. Assuming x1 as the manipulated variable, the inversion of IT2-FLS can be
In achieving such a goal, the Karnik–Mendel type reduction method is preferred and considered. Due to the novelty and
the adaptiveness features of this type reduction method, a closed form of the inverse mapping (f−1) cannot be obtained.
To overcome this bottleneck, an iterative method will be presented to find inverse solution set based on an analytical
methodology. Then, the proposed analytical methodology is generalized via the decomposition property. Through this
property, the IT2-FLS can be decomposed into IT2 fuzzy subsystems. The decomposition property can only be applied if and
only if the universe of discourse of each input variable is fully partitioned in the sense of upper and lower fuzzy sets. This
assumption guarantee that each input variable is described with at least two type-2 fuzzy sets. As a consequence, for an IT2
FLS with n inputs, at most 2n rules are activated for any input vector. The objective of this decomposition is to generalize the
proposed inversion methodology.
3.1. Decomposition of an interval type-2 fuzzy logic system
In this subsection, the decomposition property for ordinary (type-1) fuzzy logic systems is generalized to type-2 fuzzy
logic systems. The decomposition property is a simple constructive procedure which is first introduced for type-1 fuzzy
logic systems [47]. Based on this property, a general fuzzy system is decomposed into several fuzzy subsystems [14,47].
The decomposition property will facilitate the inversion methodology significantly which will be explained in detail in the
following subsection.
Let us consider an interval IT2-FLS of n inputs ∀xk ∈ Xk, k = 1, . . . , n, one output y ∈ Y , and consists of fuzzy rules
defined as:
Ri1,i2,...,in : IF x1 is Ai11 and x2 is A
i22 and . . . and xn is Ain
n , THEN y is Ci1,i2,...,in
where Aikk (k = 1, . . . , n) are IT2 FSs representing the inputs (x1, . . . , xn) and Ci1,i2,...,in = [ci1,i2,...,in , ci1,i2,...,in ] is the
consequent interval set. Considering Nk membership functions for describing xk , ik ∈ Ik = {1, 2, . . . ,Nk}, k = 1, . . . , n,the complete rule base is composed of N = ∏
k=1,...,n Nk rules with the corresponding index set is I = I1 × I2 × · · · In.The IT2 FSs of the IT2-FLS must partition the input space such that a crisp value of any xk belongs to A
ikk and A
ik+1
k type-2
fuzzy sets. As a consequence, each input variable is describedwith two IT2-FSs at themost. Thus, it is guaranteed that for any
258 T. Kumbasar et al. / International Journal of Approximate Reasoning 54 (2013) 253–272
Fig. 3. Input membership functions of the IT2-FLS for xk .
Table 2
The rulebase and consequents of the IT2 FLS for the illustrative example-1.
It is a known fact that, the objective of the inverse mapping is to calculate the input signal such that the system output
y∗(x) will become equal to a desired setpoint denoted as Sdes. Since the output uncertainty parameter (γ ) represents the
T. Kumbasar et al. / International Journal of Approximate Reasoning 54 (2013) 253–272 261
uncertainty between the defuzzified output to one of the end points of the type reduced set, the set values for the two end
points (of the type reduced set) are defined as
Sdes_l = Sdes − γ (21)
Sdes_r = Sdes + γ (22)
Here, γ is an unknown parameter and will vary for each input set. It can be noted that the mean of the right and left desired
values will be equal to the desired set point.
In this study, each input domain is defined with interval type-2 trapezoidal membership functions as shown in Fig. 3.
The IT2-FLS uses fully overlapping trapezoidal membership functions in the sense of upper and lower fuzzy sets. The upper
and lower fuzzy membership functions satisfy the following property for ∀xk ∈ Xk given in (23).
μAikk (xk)+ μA
ik+1
k (xk) = 1
μAikk (xk)+ μA
ik+1
k (xk) = 1(23)
It should be noted that alternative interval type-2 sets can be used as long as two linear relations can be written between
two consequent interval type-2 fuzzy membership functions, i.e. Aikk and A
ik+1
k , or between the upper and lower of the same
type-2 fuzzy set, Aikk . For instance, triangular and gaussian type interval type-2 fuzzy sets can be preferred. However, it
should be noted that, the generalization of the presented method to gaussian type-2 fuzzy membership functions can only
be accomplished under some assumptions. As it is known, a Gaussianmembership function is definedwith two parameters,
its center (c) and deviation (σ ), i.e. Gaussian(x, [σ, c]). In order to be able to use this type of membership functions in the
proposed inversion method the following assumptions must be satisfied by the related membership function:
I. The value of the Gaussian membership functions is equal to zero after 3σ , i.e. Gaussian(x = 3σ, [σ, c]) = 0 (three
sigma rule).
II. The absolute difference between two consequent centers (cik and cik+1) of the type-2 fuzzy sets must be greater than
3σik and 3σik+1, i.e. |cik+1
− cik | ≥ 3σik , 3σik+1.
As it has been asserted in the previous sections, for any input the IT2-FLS can be decomposed to IT2 fuzzy subsystems
with 2n rules. Reminding that, since each universe of discourse is fully partitioned in the sense the upper and lower fuzzy
membership functions, only one fuzzy subsystem composed of 2n rules is activated for any input set (x1, . . . , xn). Therefore,the proposed inversion methodology will be explained on a two input (x1, x2) and one output (y) IT2-FLS which consists of
four rules with interval singleton consequents [28]. The rules are defined in the form as:
R1 : IF x1 is A11 and x2 is A1
2, THEN y is C1
R2 : IF x1 is A11 and x2 is A2
2, THEN y is C2
R3 : IF x1 is A21 and x2 is A1
2, THEN y is C3
R4 : IF x1 is A21 and x2 is A2
2, THEN y is C4
where each input domain (x1, x2) is definedwith two interval type-2 trapezoidalmembership functions and the consequent
interval set is Cj = [cj, cj].The total interval firing set for each rule is defined as
f j =[f j f
j]; j = 1, . . . ,N = 2n = 4
where
f 1 = [μA11μA
12], f 1 = [μA1
1μA12]
f 2 = [μA11μA
22], f 2 = [μA1
1μA22]
f 3 = [μA21μA
12], f 3 = [μA2
1μA12]
f 4 = [μA21μA
22], f 4 = [μA2
1μA22]
(24)
For the simplicity, it is assumed that the interval consequent set is inorder such that c1 ≤ c2 ≤ · · · ≤ cN , c1 ≤ c2 ≤ · · · ≤cN and the corresponding weights f
jand f j are matched. Otherwise, the consequents must be reordered similar to the KM
procedure. Now, at any certain time (when an input signal activates the IT2-FLS) desired setpoint for the output (y), i.e. Sdes,
and the consequent interval parameters (cj, cj) are known; while the fuzzified measured variables (μA12, μA
22, μA
22, μA
12)
of x2 are calculated. The only unknown parameters are γ , L∗ and R∗. Since the calculation of L∗ and R∗ values depends on the
262 T. Kumbasar et al. / International Journal of Approximate Reasoning 54 (2013) 253–272
fuzzified variables of the manipulated input x1(μA11, μA
21, μA
21, μA
11) which are unknown, they cannot be predetermined.
Therefore, each combination of L, R ∈ {1, 2, 3} should be tested iteratively. At each iteration, it will be assumed that the
initialized values are the extrema points (L∗, R∗) of (18) and (20), respectively. After initializing L∗, Sdes_l is equated to yl(x)from (9):
Sdes_l =∑L∗
j=1 f jcj+∑Nj=L∗+1
f jcj∑L∗j=1 f j+∑N
j=L∗+1f j
Sdes − γ =∑L∗
j=1 f jcj+∑Nj=L∗+1
f jcj∑L∗j=1 f j+∑N
j=L∗+1f j
0 = −(Sdes − γ )
(L∗∑j=1
f j + N∑j=L∗+1
f j
)+
(L∗∑j=1
f jcj +N∑
j=L∗+1
f jcj
)(25)
Using (24) and after some mathematical manipulations, (25) can be reformulated as follows:
tl(.)μA11 + t2(.)μA
21 + t3(.)μA
21 + t4(.)μA
11 = 0 (26)
where ti, i = 1, . . . , 4 is a function of (Sdes, γ, cj) and a possible combination of the following parameters (μA12, μA
22,
μA22, μA
12). The mathematical derivations are given in Appendix A.
After initializing R∗ and similarly equating Sdes_r to yr(x) from (10), the following equation is obtained.
Sdes_r =∑R∗
j=1 f j cj+∑Nj=R+1 f
jcj∑R∗
j=1 f j+∑Nj=R∗+1
f j
Sdes + γ =∑R∗
j=1 f j cj+∑Nj=R+1 f
jcj∑R∗
j=1 f j+∑Nj=R∗+1
f j
(27)
Using (24) and after some mathematical manipulations:
kl(.)μA11 + k2(.)μA
21 + k3(.)μA
21 + k4(.)μA
11 = 0 (28)
where ki, i = 1, . . . , 4 is a function of (Sdes, γ, cj) and a possible combination of (μA12, μA
22, μA
22, μA
12). The mathematical
derivations are given in Appendix B.
Combing (23), (26) and (28),⎡⎢⎢⎢⎢⎢⎢⎣1 0 1 0
0 1 0 1
t1(.) t2(.) t3(.) t4(.)
k1(.) k2(.) k3(.) k4(.)
⎤⎥⎥⎥⎥⎥⎥⎦︸ ︷︷ ︸
ψ
θ =
⎡⎢⎢⎢⎢⎢⎢⎣1
1
0
0
⎤⎥⎥⎥⎥⎥⎥⎦︸ ︷︷ ︸
b
(29)
where θ =[μA1
1 μA21 μA
21 μA
11
]Tis the solution set.
The solution set can be found via following simple linear matrix equation:
θ = ψ−1b (30)
such that θ ∈ [0, 1] and where ∀θi ∈ θ is function of γ .
Lemma 1. For certain L and R values, the ψ matrix in (29) is invertible if and only if γ > 0 and the universe of discourse each
input variable is fully partitioned in the sense the upper and lower fuzzy membership functions.
Proof.
(i) Since the inputmembership functionsare type-2 fuzzysets, andbecauseof thedefinitionof the IT2FSs themembership
values of the upper and lower membership functions will be always be different μAjk = μA
jk (k = 1, . . . , n, j =
1, . . . ,Nk) and likewise since the output consequents are interval sets cj = cj .
(ii) Since the universe of discourse each input variable is fully partitioned in the sense the upper and lower fuzzy mem-
bership function, the upper and lower fuzzy membership functions satisfy the property given in (23) for ∀xk ∈ Xk .
T. Kumbasar et al. / International Journal of Approximate Reasoning 54 (2013) 253–272 263
Fig. 8. Illustration of partitioned type-2 fuzzy membership functions of the manipulated variable (x1).
(iii) From (i) it can be concluded that for all ti in (26), the following condition satisfies tm = tn ∀m = n, m, n = 1, . . . , 4while for all ki in (28), the following condition satisfies, km = kn ∀m = n, m, n = 1, . . . , 4. Naturally all ti and ki get
distinct values.
(iv) From (i) and since the Karnik–Mendel type reducer computes always the two distinct end points of the type reduced
interval set for γ > 0, the coefficients of (26) and (28) will be distinct, i.e., ti = ki.
Therefore, in the light of (ii), (iii) and (iv) stated above the rank of the ψ matrix will always be in full rank for certain L
and R values. �
The following assertions can bemade on the existence of inverse solution according to the output uncertainty parameter
(γ ) in three cases:
Case 1: If the desired set value is not in the defined universe of discourse then the output uncertainty parameter γ does
not exist then an inverse solution cannot be found.
Case 2: If γ > 0 and there does not exist any γ which satisfies ∀θi(γ ) ∈ [0, 1] then an inverse solution cannot be found
for certain L and R values.
Case 3: If γ > 0 and there exists γ which satisfies ∀θi(γ ) ∈ [0, 1], it must be checked whether the solution set (θ(γ ))lies on the defined type-2 fuzzy membership functions. Then, a candidate inverse solution set can be found for
the certain L and R values.
ForCase 3, themembership functions ofmanipulatedvariable (x1) arepartitioned in three regions as it has been illustrated
in Fig. 8. As a consequence, γ must satisfy at least one of the following equalities.
(a) μA11(γ ) = 1, μA2
1(γ ) = 0, μA11(γ ) ≥ β, μA2
1(γ ) ≤ 1 − β (31a)
(b) μA21(γ )− μA2
1(γ ) = 1 − β, μA11(γ )− μA1
1(γ ) = 1 − β (31b)
(c) μA21(γ ) = 1, μA1
1(γ ) = 0, μA21(γ ) ≥ β, μA1
1(γ ) ≤ 1 − β (31c)
where z1, z2, z3, z4 are the limiting points of each region as shown in Fig. 8.
Moreover, from many possible candidate solutions the correct solution(s) is the one(s) which satisfies the following
lemma.
Lemma 2. If the following conditions are satisfied then the certain L∗ and R∗ are the solutions which minimize/maximize (18)and (20), respectively.
(a) If L∗(x) = arg min1≤L≤N−1
{yL∗(x)l (x)} then y
(L∗(x))l (x) < y
(L(x))l (x) for ∀L (L = L∗).
(b) If R∗(x) = arg max1≤R≤N−1
{yR∗(x)r (x)} then y
(R∗(x))r (x) > y
(R(x))r (x) for ∀R (R = R∗).
Proof. The Karnik–Mendel type reduction algorithm has been proven to converge monotonically and there exits only one
solution set of (L, R)which minimize/maximize (9) and (10), respectively [33,43]. �
If there exists γ ∗ which satisfies conditions of Lemma-2, then defuzzify the solution set θ(γ ∗) and obtain the crisp value
of the manipulated variable (x1). Otherwise, the calculated candidate solution set θ(γ ∗) is not an inverse solution.
While obtaining the inverse solution set from the proposed iterative method based on an analytical methodology, two
important issues must be considered which are the existence of an inverse solution and the possibility of multiple solu-
tions. Reminding that during the proposed inversion procedure, the two end points of the type reduced set are equated as
264 T. Kumbasar et al. / International Journal of Approximate Reasoning 54 (2013) 253–272
23 γ ∗ = 0.6523 Not satisfied No solution γ ∗ = 0.5288 Satisfied x1 = −1.152731 γ ∗ = {} Not satisfied No solution γ ∗ = {} Not satisfied No solution
32 γ ∗ = {} Not satisfied No solution γ ∗ = {} Not satisfied No solution
33 γ ∗ = {} Not satisfied No solution γ ∗ = {} Not satisfied No solution
Table 5
The inversion algorithm for decomposable IT2-FLSs.
Step 1 Decompose the IT2 FLS,∏
k=1,2,...,n (Nk − 1) IT2 fuzzy subsystems.
Determine the possible activated IT2 fuzzy subsystems.
Step 2 Calculate interval the firing strength: f j =[f j f
j].
Sort cj (j = 1, 2, . . . ,N) in increasing order. Match the corresponding weights fj, f j .
Sort cj (i = 1, 2, . . . ,N) in increasing order. Match the corresponding weights fj, f j .
Set L0 = 1 and R0= 1 or L0= N2.4
and R0= N1.7
(Enhanced Karnik Mendel initialization [39]).
Step 3 Equate yl(x) = Sdes − γ , yr(x) = Sdes + γ and calculate ti(.) and ki(.).
Find the candidate inverse solution set θ(γ ) from θ = ψ−1b such that θ(γ ) ∈ [0, 1].Step 4 Check for each partitioned region of the membership functions of manipulated variable via (31a)–(31c) whether the inverse solution set
lies on the defined T2-FS for γ . If not go to Step 2 and increase/decrease L/R value, otherwise calculate the inverse solution set θ(γ ∗).Step 5 Replace the calculated the solution set θ(γ ∗) in (9) and (10) and check the conditions of Lemma-2. If both conditions are satisfied θ(γ ∗)
is an inverse solution, then defuzzify the solution and obtain the crisp value of themanipulated variable. Otherwise increase/decrease L/Rvalue go to Step 2.
Step 6 If there does not exist any solution the desired value cannot be reached. In the case of multiple inverse solutions, a selection is done basisof additional criteria. Apply the inverse solution to the IT2-FLS.
for γ ∗ = 0.6709, L∗ = 1 and R∗ = 3, the conditions in Lemma-2 are satisfied. Therefore θ(γ ∗) is an inverse solution, and
the corresponding crisp value is calculated as x1 = 0.6. In Table 4, the cases of different values of L and R are presented. It
can be clearly seen that only for γ ∗ = 0.6709, L∗ = 1 and R∗ = 3 an inverse solution exists.
In the second case (x2 = 0.4, Sdes = 0.0781), the existence of multiple inverse solutions has been encountered. As it has
been illustrated in Table 4, the proposed inversion procedure generates three inverse solutions for the manipulated variable
(x1). All solutions will force the system output to the desired setpoint. However, in this case an extra selection criterion such
as minimum/maximum energy is needed to choose one of the solutions.
In the third case, if Sdes = 0.8 and x2 = 0.4, an inverse solution does not exist for any (R, L) combination.
3.4. Decomposable interval type-2 fuzzy logic system inversion
In this subsection, the inversion methodology will be generalized for a decomposable interval type-2 fuzzy logic system
with n inputs (x1, . . . , xn), and Nk membership functions for describing the inputs. As it has been illustrated in Section 3.2,
the IT2 FLS can be decomposed of IT2- fuzzy subsystems composed of the 2n rules. Reminding that, since each universe
of discourse is fully partitioned in the sense the upper and lower fuzzy membership functions, only one fuzzy subsystem
composed of 2n rules is activated for ∀(x1, . . . , xn). Therefore, in the proposed inversion methodology each combination
of L, R ∈ {1, 2, . . . , 2n − 1} must be tested iteratively. Moreover, since the number of the possible activated subsystems
will increase, the inversion procedure must be applied for each possible activated decomposed IT2 fuzzy subsystem. As a
consequence, as the number of inputs (n) inputs and membership functions (Nk) increases, the computational complexity
will increase. On the other hand since the inversion is handled analytically, the exactness of this method is guaranteed. The
general inversion procedure is presented in Table 5.
In order to validate the exactness of the proposed generalized inversion strategy, a simulation study is performed on the
presented IT2-FLS presented in example 1. The general scheme of the inversion scheme is illustrated in Fig. 10. The shared
variable x2 is a random number uniformly distributed in [0–2] as shown in Fig. 11(a). The desired setpoint trajectory that
should be tracked by the IT2-FLS is chosen as Sdes = 0.5 + 0.2 ∗ sin(k). According the maximum and minimum energy
criteria two inverse solution sets of x1 are plotted in Fig. 11(b). The exactness is validated by randomly selecting one of the all
possible generated inverse solutions which is plotted in Fig. 12(a). As it can be clearly seen in Fig. 12(b), the output produced
by the fuzzy system converges to the desired setpoint trajectory, i.e. y = Sdes. This confirms that the inversion is exact.
266 T. Kumbasar et al. / International Journal of Approximate Reasoning 54 (2013) 253–272
Fig. 10. Interval type-2 fuzzy logic system inversion scheme.
Fig. 11. Illustrations of (a) calculated inverse solutions (x1) and (b) shared variable x2.
4. Conclusion
In this study, an exact inversion methodology for decomposable interval type-2 fuzzy logic systems is presented. At first,
the handled type-2 fuzzy logic system is decomposed into IT2 fuzzy subsystems based on the generalized decomposition
property for interval type-2 fuzzy logic systems. Then, the analytical formulation of the type-2 fuzzy subsystem output is
tried to be reached for an inverse solution. However, since the type reduced set of the type-2 fuzzy logic subsystem out-
put cannot be presented in a closed form, the switching points of the type reduced set are initialized by setting them to
arbitrary values. After this setting, the analytical formulation of the inverse interval type-2 fuzzy logic subsystem output
is explicitly driven and the candidate inverse solutions are calculated. It has been proven in Lemma-1 that the candidate
inverse solution set only exist under certain conditions, namely the existence of the output uncertainty parameter (γ ) and∀θi(γ ) ∈ [0, 1] for certain L and R values. If these conditions are satisfied, then the certain switching points (L, R) must
be checked whether they are the optimal ones or not for the calculated inverse solution set (θ(γ )). For this purpose, two
conditions (Lemma-2) must be checked which are extracted from the meta-rules of the Karnik–Mendel type reduction
method. If the conditions are satisfied the IT2 fuzzy subsystem can be inverted and an inverse solution set can be calculated.
Finally, the proposed inversion methodology driven for IT2 fuzzy subsystem is extended to the inversion of IT2 FLSs using
the decomposition property. The developed inversion strategy determines either an exact unique solution or exact multi-
ple solutions if there exist. It is important to note here that the developed inversion strategy is based on a pure analytical
method.
T. Kumbasar et al. / International Journal of Approximate Reasoning 54 (2013) 253–272 267
Fig. 12. Illustrations of (a) randomly selected inverse solution (x1) and (b) IT2-FLS output (y) and the tracking error.
The proposed inversionmethod can be easily extended to IT2 FLSs with alternative type reductionmechanisms or mem-
bership functions. Preferring an alternative type-reduction method may significantly reduce the computational complexity
of the inversion method. However, two important criteria must be considered while extending for other types of IT2-FLS
namely, the decomposability property and the handled fuzzy systems must have singleton type consequents.
The effectiveness and the performance of the method have been shown on an example in which the output of an IT2
FLS has been forced to follow a certain desired reference signal in an open loop fashion. Inspecting the tracking error values
between the desired input reference signal and the systemoutput, it can easily be concluded that inversion of interval type-2
fuzzy logic system is exact.
The proposed type-2 fuzzy logic system inversion method can be applied in various areas especially in model-based
control, feedback linearization and disturbance observer structures. Especially in fuzzy internal model control structures
wherer the inverse fuzzy system is implemented as the main controller.
Acknowledgments
This research is supported by a project given to the Scientific Research Project (SPR-BAP 34492) of Institute of Science
and Technology of Istanbul Technical University. All of support is appreciated.
Appendix A. Mathematical manipulations for (26)
In this part, it will be shown that (A.1) can be formulated as (A.3) for each combination of L ∈ {1, 2, 3}.
0 = −(Sdes − γ )
⎛⎝ L∗∑j=1
f j +N∑
j=L∗+1
f j
⎞⎠ +⎛⎝ L∗∑
j=1
f jcj +N∑
j=L∗+1
f jcj
⎞⎠ (A.1)
where the total firing set is:
f 1 = [μA11μA
12], f 1 = [μA1
1μA12]
f 2 = [μA11μA
22], f 2 = [μA1
1μA22]
268 T. Kumbasar et al. / International Journal of Approximate Reasoning 54 (2013) 253–272
f 3 = [μA21μA
12], f 3 = [μA2
1μA12]
f 4 = [μA21μA
22], f 4 = [μA2
1μA22]
(A.2)
0 = tl(.)μA11 + t2(.)μA
21 + t3(.)μA
21 + t4(.)μA
11 (A.3)
where ti, i = 1, . . . , 4 is a function of (Sdes, γ, cj) and a possible combination of the following parameters
(μA12, μA
22, μA
22, μA
12).
(i) For L∗ = 1:
0 = −(Sdes − γ )
(L∗=1∑j=1
f j + N=4∑j=L∗+1
f j
)+
(L∗=1∑j=1
f jcj +N=4∑
j=L∗+1
f jcj
)
0 = −(Sdes − γ )(f 1 + f 2+f 3 + f 4)+ (c1f1 + c2f
Replacing total interval firing set (A.2) in (A.12)
0 = (0)μA11 + (μA1
2 + c2γμA12 − c2μA
12Sdes)μA
21 + (μA2
2 + c4γμA22 − c4μA
22Sdes)μA
21 + · · ·
+ (μA12 + c1γμA
21 − c1μA
21Sdes + μA2
2 + c3μA22 − c3μA
22Sdes)μA
11
(A.13)
Reformulating (A.13)
tl(.)μA11 + t2(.)μA
21 + t3(.)μA
21 + t4(.)μA
11 = 0 (A.14)
where
tl(.) = 0
t2(.) = (μA12 + c2γμA
12 − c2μA
12Sdes)
t3(.) = (μA22 + c4γμA
22 − c4μA
22Sdes)
t4(.) = (μA12 + c1γμA
21 − c1μA
21Sdes + μA2
2 + c3μA22 − c3μA
22Sdes)
(A.15)
As it has been above, (A.1) can be represented in form given in (A.2) for each possible value of L (L ∈ {1, 2, 3}).
Appendix B. Mathematical manipulations for (28)
In this part, it will be shown that (A.16) can be reformulated as (A.18) for each combination of R ∈ {1, 2, 3}.
0 = −(Sdes + γ )
⎛⎝ R∗∑j=1
f j +N∑
j=R∗+1
f j
⎞⎠ +⎛⎝ R∗∑
j=1
f jcj +N∑
j=R+1
fjcj
⎞⎠ (A.16)
f 1 = [μA11μA
12], f 1 = [μA1
1μA12]
f 2 = [μA11μA
22], f 2 = [μA1
1μA22]
f 3 = [μA21μA
12], f 3 = [μA2
1μA12]
f 4 = [μA21μA
22], f 4 = [μA2
1μA22]
(A.17)
0 = kl(.)μA11 + k2(.)μA
21 + k3(.)μA
21 + k4(.)μA
11 (A.18)
(i) For R∗ = 1:
0 = −(Sdes + γ )
(R∗=1∑j=1
f j + N=4∑j=R∗+1
f j
)+
(R∗=1∑j=1
f j cj +N=4∑
j=R+1
f jcj
)
0 = −(Sdes + γ )(f 1 + f 2 + f 3 + f 4)+ (c1f1 + c2f
2 + c3f3 + c4f
4)
0 = f 1(1 − c1(Sdes + γ ))+ f 2(1 − c2(Sdes + γ ))+ f 3(1 − c3(Sdes + γ ))+ f 4(1 − c4(Sdes + γ ))
(A.19)
270 T. Kumbasar et al. / International Journal of Approximate Reasoning 54 (2013) 253–272
Replacing total interval firing set (A.17) in (A.19)
0 = (μA22 − c3μA
22 − c3μA
22Sdes)μA
11 + (μA1
2 − c1γμA12 − c1μA
12Sdes)μA
11 + · · ·
+ (0)μA21 + (μA1
2 − c2γμA12 + γμA2
2 − c4γμA22 − c2μA
12Sdes − c4μA
22Sdes)μA
21
(A.20)
Reformulating (A.20)
kl(.)μA11 + k2(.)μA
21 + k3(.)μA
21 + k4(.)μA
11 = 0 (A.21)
where
kl(.) = (μA22 − c3γμA
22 − c3μA
22Sdes)
k2(.) = (μA12 − c2γμA
12 + γμA2
2 − c4γμA22 − c2μA
12Sdes − c4μA
22Sdes)
k3(.) = (0)
k4(.) = (μA12 − c1γμA
12 − c1μA
12Sdes)
(A.22)
(ii) For R∗ = 2:
0 = −(Sdes + γ )
(R∗=2∑j=1
f j + N=4∑j=R∗+1
f j
)+
(R∗=2∑j=1
f j cj +N=4∑
j=R+1
f jcj
)
0 = −(Sdes + γ )(f 1 + f 2 + f 3 + f 4)+ (c1f1 + c2f
2 + c3f3 + c4f
4)
0 = f 1(1 − c1(Sdes + γ ))+ f 2(1 − c2(Sdes + γ ))+ f 3(1 − c3(Sdes + γ ))+ f 4(1 − c4(Sdes + γ ))
(A.23)
Replacing total interval firing set (A.17) in (A.23)
0 = (μA22 − c3γμA
22 − c3μA
22Sdes)μA
11 + (μA2
2 − c4γμA22 − c4μA
22Sdes)μA
21 + · · ·
+ (μA12 − c2γμA
12 − c2μA
12Sdes)μA
21 + (μA1
2 − c1γμA12 − c1μA
12Sdes)μA
11
(A.24)
Reformulating (A.24)
kl(.)μA11 + k2(.)μA
21 + k3(.)μA
21 + k4(.)μA
11 = 0 (A.25)
where
k1(.) = (μA22 − c3γμA
22 − c3μA
22Sdes)
k2(.) = (μA22 − c4γμA
22 − c4μA
22Sdes)
k3(.) = (μA12 − c2γμA
12 − c2μA
12Sdes)
k4(.) = (μA12 − c1γμA
12 − c1μA
12Sdes)
(A.26)
(iii) For R∗ = 3:
0 = −(Sdes + γ )
(R∗=3∑j=1
f j + N=4∑j=R∗+1
f j
)+
(R∗=3∑j=1
f j cj +N=4∑
j=R+1
f jcj
)
0 = −(Sdes + γ )(f 1 + f 2 + f 3 + f 4)+ (c1f1 + c2f
2 + c3f3 + c4f
4)
0 = f 1(1 − c1(Sdes + γ ))+ f 2(1 − c2(Sdes + γ ))+ f 3(1 − c3(Sdes + γ ))+ f 4(1 − c4(Sdes + γ ))
(A.27)
Replacing total interval firing set (A.17) in (A.27)
0 = (0)μA11 + (μA2
2 − c4γμA22 − c4μA
22Sdes)μA
21 + (μA1
2 − c2γμA12 − c2μA
12Sdes)μA
21 + · · ·
+ (μA12 − c1γμA
12 + μA2
2 − c3γμA22 − c1μA
12Sdes − c3μA
22Sdes)μA
11
(A.28)
Reformulating (A.28)
kl(.)μA11 + k2(.)μA
21 + k3(.)μA
21 + k4(.)μA
11 = 0 (A.29)
T. Kumbasar et al. / International Journal of Approximate Reasoning 54 (2013) 253–272 271
where
kl(.) = (0)
k2(.) = (μA22 − c4γμA
22 − c4μA
22Sdes)
k3(.) = (μA12 − c2γμA
12 − c2μA
12Sdes)
k4(.) = (μA12 − c1γμA
12 + μA2
2 − c3γμA22 − c1μA
12Sdes − c3μA
22Sdes)
(A.30)
As it has been shown, (A.16) can be represented in form given in (A.18) for each possible value of R (R ∈ {1, 2, 3}).
References
[1] K.K. Ahn, D.Q. Truong, Online tuning fuzzy PID controller using robust extended Kalman filter, Journal of Process Control 19 (2009) 1011–1023.[2] R. Babuska, Fuzzy Modeling for Control, Kluwer Academic Publishers, Boston, 1998.
[3] S. Barkati, E.M. Berkouk, M.S. Boucherit, Application of type-2 fuzzy logic controller to an induction motor drive with seven-level diode-clamped inverter
and controlled infeed, Electrical Engineering 90 (2008) 347–359.[4] M. Begian, W. Melek, J. Mendel, Stability analysis of type-2 fuzzy systems, in: Proceedings of the IEEE International Conference on Fuzzy Systems, Hong
Kong, June 2008, pp. 947–953.[5] M. Biglarbegian, W. Melek, J. Mendel, On the robustness of type-1 and interval type-2 fuzzy logic systems in modeling, Information Sciences 181 (2011)
1325–1347.[6] M. Biglarbegian,W.W.Melek, J.M.Mendel, On the stability of interval type-2 TSK fuzzy logic control systems, IEEE Transactions on Systems,Man, Cybernetics:
Part B 4 (3) (2010) 798–818.
[7] R. Boukezzoula, S. Galichet, L. Folloy, Nonlinear internalmodel control: application of inversemodel based fuzzy control, IEEE Transactions on Fuzzy Systems11 (6) (2003) 814–829.
[8] O. Castillo, L. Aguilar, N. Cázarez, S. Cárdenas, Systematic design of a stable type-2 fuzzy logic controller, Applied Soft Computing 8 (3) (2008) 1127–1274.[9] O. Castillo, P. Melin, A review on the design and optimization of interval type-2 fuzzy controllers, Applied Soft Computing 12 (4) (2012) 1267–1278.
[10] A. Celikyilmaz, B. Türksen, Modelling Uncertainty With Fuzzy Logic: With Recent Theory and Applications, Springer, New York, 2009.[11] S. Coupland, R. John, New geometric inference techniques for type-2 fuzzy sets, International Journal of Approximate Reasoning 49 (1) (2008) 198–211.
[12] X. Du, H. Ying, Derivation and analysis of the analytical structures of the interval type-2 fuzzy-PI and PD controllers, IEEE Transactions on Fuzzy Systems 18
(4) (2010) 802–814.[13] I. Eksin, M. Guzelkaya, F. Gürleyen, A new methodology for deriving the rule-base of a fuzzy logic controller with a new internal structure, Engineering
Applications of Artificial Intelligence 14 (5) (2001) 617–628.[14] M.J. Fuente, C. Robles, O. Casado, S. Syafiie, F. Tadeo, Fuzzy control of a neutralization process, Engineering Applications of Artificial Intelligence 19 (2006)
905–914.[15] S. Galichet, R. Boukezzoula, L. Folloy, Explicit analytical and exact inversion of decomposable fuzzy systems with singleton consequents, Fuzzy Sets and
Systems 146 (2004) 421–436.
[16] M. Galluzzo, B. Cosenza, A. Matharu, Control of a nonlinear continuous bioreactor with bifurcation by a type-2 fuzzy logic controller, Computers & ChemicalEngineering 32 (12) (2008) 2986–2993.
[17] M. Guzelkaya, I. Eksin, E. Yesil, Self-tuning of PID-type fuzzy logic controller coefficients via relative rate observer, Engineering Applications of ArtificialIntelligence 16 (2003) 227–236.
[18] H. Hagras, Type-2 FLCs: a new generation of fuzzy controllers, IEEE Computational Intelligence Magazine 2 (1) (2007) 30–43.[19] N.N. Karnik, J.M. Mendel, Q. Liang, Type-2 fuzzy logic systems, IEEE Transactions on Fuzzy Systems 7 (1999) 643–658.
[20] M. Aliasghary, I. Eksin, M. Guzelkaya, T. Kumbasar, Design of an interval type-2 fuzzy logic controller based on conventional PI controller, in: Proceedings of
the International Conference on MED, 2012, pp. 627—632.[21] T.Kumbasar, I. Eksin,M.Guzelkaya, E.Yesil,Adaptive fuzzymodelbased inversecontrollerdesignusingBB–BCoptimization,ExpertSystemsWithApplications
38 (2011) 12356–12364.[22] T. Kumbasar, I. Eksin, M. Guzelkaya, E. Yesil, Type-2 fuzzy model inverse controller design based on BB–BC optimization method, in: The 18th IFAC World
Congress, Milano, Italy, 2011.[23] T. Kumbasar, I. Eksin,M. Guzelkaya, E. Yesil, Type-2 fuzzymodel based controller design for neutralization processes, ISA Transactions 51 (2) (2012) 277–287.
[24] T. Kumbasar, I. Eksin, M. Guzelkaya, E. Yesil, Interval type-2 fuzzy inverse controller design in nonlinear IMC structure, Engineering Applications of Artificial
Intelligence 24 (6) (2011) 996–1005.[25] T. Kumbasar, I. Eksin, M. Guzelkaya, E. Yesil, Big Bang Big Crunch optimization method based fuzzy model inversion, in: MICAI 2008: Advances in Artificial
Intelligence, Lecture Notes in Computer Science, vol. 5317, 2008, pp. 732–740.[26] T. Kumbasar, I. Eksin, M. Guzelkaya, E. Yesil, Adaptive fuzzy internal model control design with bias term compensator, in: IEEE International Conference on
Mechatronics (ICM 2011), April 13–15, 2011.[27] T.Kumbasar, E. Yesil, I. Eksin,M.Guzelkaya, Inverse fuzzymodel controlwithonlineadaptationviabigbang–big crunchoptimization, in: The3rd International,
Symposium on Communications, Control and Signal Processing, ISCCSP, Malta, March 12–14, 2008, pp. 697 – 702.[28] T. Kumbasar, I. Eksin, M. Guzelkaya, E. Yesil, An inversion method for interval type-2 fuzzy logic systems, in: 11th International Conference on Intelligent
Systems Design and Applications – ISDA’2011, Cordoba, Spain, 2011.
[29] Q. Liang, J.M. Mendel, Interval type-2 fuzzy logic systems: theory and design, IEEE Transactions on Fuzzy Systems 8 (5) (2000) 535–550.[30] T.C. Lin, Observer-based robust adaptive interval type-2 fuzzy tracking control of multivariable nonlinear systems, Engineering Applications of Artificial
Intelligence 22 (2010) 420–430.[31] R. Martinez, O. Castillo, L.T. Aguilar, Optimization of interval type-2 fuzzy logic controllers for a perturbed autonomous wheeled mobile robot using genetic
algorithms, Information Sciences 179 (13) (2009) 2158–2174.[32] J.M. Mendel, Advances in type-2 fuzzy sets and systems, Information Sciences 177 (2007) 84–110.
[33] J.M. Mendel, F. Liu, Super-exponential convergence of the Karnik–Mendel algorithms for computing the centroid of an interval type-2 fuzzy set, IEEE
Transactions on Fuzzy Systems 15 (2) (2007) 309–320.[34] S.F. Molaeezadeh, M.H. Moradi, A 2uFunction representation for non-uniform type-2 fuzzy sets: theory and design, International Journal of Approximate
Reasoning, in press.[35] M. Nie, W.W. Tan, Towards an efficient type-reduction method for interval type-2 fuzzy logic systems, in: Proceedings of the IEEE International Conference
on Fuzzy Systems, Hong Kong, June 2008, pp. 1425–1432.[36] S. Oblak, T. Kumbasar, I. Skrjanc, E. Yesil, Inverse-model predictive control based on INFUMO–BB–BC optimization, in: The 10th IFACWorkshop on Adaptation
and Learning in Control and Signal Processing – ALCOSP 2010, Antalya, Turkey.
[37] J.T. Starczewski, Efficient triangular type-2 fuzzy logic systems, International Journal of Approximate Reasoning 50 (5) (2009) 799–811.
272 T. Kumbasar et al. / International Journal of Approximate Reasoning 54 (2013) 253–272
[38] A.R. Varkonyi-Koczy, G. Péceli, T.P. Dobrowiecki, T. Kovácsházy, Iterative fuzzy model inversion, in: IEEE International Conference on Fuzzy Systems, FUZZ-IEEE’98, Anchorage, Alaska, USA. 1998, pp. 561–566.
[39] C.L. Walker, E.A. Walker, Sets with type-2 operations, International Journal of Approximate Reasoning 50 (1) (2009) 63–71.[40] D.Wu,On the fundamental differences between interval type-2 and type-1 fuzzy logic controllers, IEEE Transactions on Fuzzy Systems20 (5) (2012) 832–848.
[41] D. Wu, Approaches for reducing the computational cost of interval type-2 fuzzy logic controllers: overview and comparison, IEEE Transactions on Fuzzy
Systems, in press.[42] D. Wu, J.M. Mendel, Enhanced Karnik–Mendel algorithms, IEEE Transactions on Fuzzy Systems 17 (4) (2008) 923–934.
[43] D. Wu, J.M. Mendel, On the continuity of type-1 and interval type-2 fuzzy logic systems, IEEE Transactions on Fuzzy Systems 19 (1) (2011) 179–192.[44] D. Wu, W.W. Tan, Genetic learning and performance evaluation of internal type-2 fuzzy logic controllers, Engineering Applications of Artificial Intelligence
19 (2006) 829–841.[45] H. Wu, J.M. Mendel, Uncertainty bounds and their use in the design of interval type-2 fuzzy logic systems, IEEE Transactions on Fuzzy Systems 10 (5) (2002)
622–639.
[46] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning – I, Information Science 8 (1975) 199–249.[47] X.-J. Zeng, M.G. Singh, Decomposition property of fuzzy systems and its applications, IEEE Transactions on Fuzzy Systems 4 (2) (1996) 149–165.
