MATRIX-BASED HIERARCHICAL FUZZY SYSTEMS
Santiago Aja-Fern
´
andez and Carlos Alberola-L
´
opez
Laboratorio de Procesado de Imagen
ETSI Telecomunicaci
´
on, Universidad de Valladolid
Campus Miguel Delibes 47011 Valladolid, Spain)
Keywords:
Hierarchical fuzzy systems, FITM, transition matrices, SAM.
Abstract:
A matrix inference method for fuzzy systems is used to deal with hierarchical fuzzy systems (HFSs). A method
to decompose a multiple input fuzzy system into a HFS is presented. This method is based in representing
the structure of a fuzzy system using matrices. An example of such a conversion for a three-input system is
included.
1 INTRODUCTION
In fuzzy systems (FSs) and fuzzy controllers with
multiple inputs a rule-explosion problem arises when
the number of inputs is large. As the number of
rules increases, the rule base is less and less under-
standable to human users. This problem is known as
the curse of dimensionality, which in fuzzy systems
causes some well known problems related with com-
putational and memory usage, real-time performance,
and difficulty to properly define the system. One of
the methods proposed to overcome this problem is the
use of hierarchical fuzzy systems (HFSs); a complex
system with many rules is decomposed into a num-
ber of hierarchically-connected low-dimensional FSs.
These new systems, which turn out to be simpler than
the original, have usually smaller rule bases.
In this paper we will present a new method to carry
out this system decomposition. The method is based
on a matrix procedure known as FITM (Fast Infer-
ence using Transition Matrices) (Aja-Fern
´
andez and
Alberola-L
´
opez, 2004a), which has been proposed
as a methodology to perform inferences in SAM
(Standard Additive Model (Kosko, 1997)) FSs effi-
ciently. It is based on representing each input to the
FS as a vector and the overall fuzzy inference pro-
cedure becomres a simple matrix operative. FITM
is a procedure initially intended for Computing with
The authors acknowledge the CICyT for research grant
TEC2004-06647-C03-01, the FIS for grant PIO-41483, the
Junta de Castilla y Le
´
on for grant VA075A05 and the Euro-
pean Commission for the funds associated to the Network
of Excellence SIMILAR (FP6-507609).
Words (Zadeh, 1996) environments, but it may find
applicability in other fields, such as control and feed-
back systems.
1.1 Hierarchical Fuzzy Systems
It is easy to see that if a system has n input vari-
ables and each variable has m terms (defined over
m fuzzy sets) the system has a complete set of m
n
different rules (Raju et al., 1991). Moreover, as pre-
viously stated, the rule base is less and less under-
standable to human users as the number of rules in-
creases. Originally HFSs were proposed to overcome
this problem. Instead of having one complex system
with many rules, the system was decomposed into a
number of hierarchically-connected low-dimensional
FSs. These new systems, which turn out to be simpler
than the original, have smaller rule bases, a fact that
makes them more transparent to the designer. If the
number of intermediate variables is small enough
2
the HFS has also fewer rules than the original system.
One of the first approaches is due to Raju, Zhou
and Kisner (Raju et al., 1991; Raju and Zhou, 1993).
An adaptive HFS is introduced in order to facilitate
the tuning of the parameters of a controller. Their aim
was to design a series of hierarchical fuzzy processors
with a small number of input variables distributed in
each processor. Yager in (Yager, 1993) suggested an
extension of the basic FS modeling framework. The
purpose of this extension was to allow for a prioriti-
2
A simple criterion to quantify this “enough” can be
found in (Lee et al., 2003).
92
Aja-Fernández S. and Alberola-López C. (2006).
MATRIX-BASED HIERARCHICAL FUZZY SYSTEMS.
In Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics, pages 92-97
DOI: 10.5220/0001206500920097
Copyright
c
SciTePress
zation of the rules by using hierarchical representa-
tion of the rules. This new representation is known as
Hierarchical Prioritized Structure (HPS). In (Hor
´
a
ˇ
cek
and Binder, 1997) Hor
´
a
ˇ
cek and Binder discussed two
structures of hierarchically-organized groups of rules:
series and parallel hierarchies . The aim of this work
is to design a nonlinear controller whose rules were
transparent to the designer. Many other approaches to
overcome the dimensionality problem based on HFS
have been proposed elsewhere (Holve, 2003; Lee
et al., 2003). Additionally, some authors have focused
on theoretical justifications of the approximation ca-
pability of HFS (Wang, 1998; Zeng and Keane, 2005).
In the HFS literature a fuzzy set with two inputs and
one output is known as Fuzzy Logic Unit (FLU). De-
pending of the connection between the FLUs in a hi-
erarchical decomposition of a MISO-FS, we can have
three main types of hierarchical models (Hor
´
a
ˇ
cek and
Binder, 1997; Kikuchi et al., 1998) as shown in Fig. 1
(although one may think of more): Serial HFS, where
the inputs to the FLUs are the outputs of the previous
FLUs and an external input; Parallel HFS, where the
inputs to the FLUs are the outputs of two FLUs of the
previous layer and Hybrid HFS.
FS
FS
FS
. . .
. . .
x
1
x
2
x
3
y
2
y
1
y
N
= y
y
N1
x
N+1
Serial HFS
FS
FS
FS
FS
. . .
. . .
Layer 2 Layer iLayer 1
x
1
x
2
x
3
x
4
x
2N
x
2N1
x
2N2
x
2N3
y
Parallel HFS
FS
FS
FS
FS
FS
. . .
. . .
x
1
x
2
x
3
x
4
x
5
y
x
N
Hybrid HFS
Figure 1: Different hierarchical models.
In the following sections we will show how to deal
with HFSs using transition matrices. This method
was previously introduced in (Aja-Fern
´
andez and
Alberola-L
´
opez, 2004b).
2 BACKGROUND ON FITM
The FITM procedure was originally proposed in (Aja-
Fern
´
andez and Alberola-L
´
opez, 2004a) to perform in-
ferences efficiently in the SAM-FSs (Kosko, 1997);
its result is proportional to the inference output given
by a SAM-FS, and it is totally equivalent to the SAM-
FS in terms of the output centroid. The benefit of
FITM is the considerable reduction of the overall
computational complexity in the inference process,
provided that an initial assumption is held; specifi-
cally, inputs are required to be linear combinations of
the fuzzy sets that the input linguistic variable (LV)
consists of. If this is so, each input is represented as
a vector in the input space, with components equal to
the contribution of each fuzzy set of the input linguis-
tic variable to the current input.
This requirement holds in computing with words
(CWW) (Zadeh, 1996) applications, where the in-
puts to the FS are words, concepts or labels, mod-
eled as fuzzy sets. Additionally, this sort of inputs
can also be found in those problems where the out-
put of one SAM-FS is the input of another one. This
is clearly the case of hierarchical systems. General
inputs and particularly, crisp inputs, can indeed be
dealt with; in this case there would be no computa-
tional savings with FITM, but some benefits can be
obtained from the matrix representation of the sys-
tem. An extension of the procedure to non-linear op-
erators (i.e. generic t-norms and t-conorms) has also
been reported in (Aja-Fern
´
andez and Alberola-L
´
opez,
2005). Following, we describe the method to build a
FITM inference engine when linear operators (SAM)
are used and (crisp) numerical inputs are considered;
attention will be focused on the 2-input single out-
put case (2-ISO). Only conclusions will be described
here. Details can be found in (Aja-Fern
´
andez and
Alberola-L
´
opez, 2004a).
Assume a 2-input single output FS, with inputs X
and Y and output Z. The inputs and the output are
all (crisp) numbers. For each input and the output a
linguistic variable (LV) is defined. Assume that the
first input LV consists of M possible fuzzy sets A
k
defined on the universe U R; the second input Y is
a LV consisting of N possible fuzzy sets B
l
defined
on the universe V R and the output LV consists
of L possible fuzzy sets D
n
defined on the universe
W R.
The whole SAM inference process may be ex-
pressed using transition matrices as
γ =
N
X
l=1
α
l
l
!
β (1)
with
l
an array of transition matrices of the system.
γ is the output vector and α and β are the input vec-
MATRIX-BASED HIERARCHICAL FUZZY SYSTEMS
93
tors defined as
β =
A
1
(x
0
)
A
2
(x
0
)
.
.
.
A
M
(x
0
)
=
β
1
β
2
.
.
.
β
M
(2)
being x
0
the numerical input. α is defined in a sim-
ilar way using sets B
i
and the numerical input y
0
.
In order to build the transition matrices we must de-
fine some intermediate data structures that can be dis-
carded once matrices
l
have been calculated. How
to build these matrices is now addressed.
First step is to create the activation matrix of each
input. As both inputs will be crisp, they will make
use of identity matrices, i.e. R
A
and R
B
, will be the
M × M and the N × N matrices respectively. Next
step is to calculate the set of matrices G
l
, 1 l N ,
that bear the relation between inputs
G
l
= R
A
[R
B
E
l
] (3)
where is the Kronecker tensor product (Golub and
Loan, 1996). E
l
is a column selection vector, i.e., a
column vector with all entries zero but the one at row
l, 1 l N, the value of which is unity. This vector
has the purpose of extracting column l from matrix
R
B
.
The rule base of the system is coded in matrix C.
This is a selection matrix with as many rows as rules
in the rule base and, for row j, all the entries are zero
but the one at column i if the output consequent for
rule j is D
i
.
C =
D
1
D
2
· · · D
L
R
1
R
2
.
.
.
R
M×N
0 0 · · · 1
0 1 · · · 0
. . . . . . . . .
1 0 · · · 0
(4)
As for matrix C, rules in the rule base are ordered
by varying the second antecedent for each value of
the first antecedent (see Appendix I in (Aja-Fern
´
andez
and Alberola-L
´
opez, 2004a)).
Finally, the transition matrices are calculated by
l
= C
T
G
l
l = 1, · · · N (5)
The output centroid, if desired, can be calculated from
the output vector γ by
c
out
=
[c
1
c
2
· · · c
L
]γ
[1 1 · · · 1]γ
=
c
T
γ
1
T
γ
(6)
with c the vector of the output set centroids c
i
(1
i L) and γ as defined in equation (1). one from the
conventional SAM-FS.
For the case of a multiple-input single-output
(MISO) FS the expressions are extended accordingly.
The relation among coefficients is now given by:
γ =
N
1
X
i
1
=1
· · ·
N
F
X
i
F
=1
α
1
i
1
· · · α
F
i
F
F
j=1
i
j
!
β (7)
where α
i
and β are the input vectors and
F
j=1
i
j
are the transition matrices of the system.
3 HIERARCHICAL
DECOMPOSITION OF FS
Consider the serial HFS shown in Fig. 1. A re-
interpretation of this system using transition matrices
and vectors is shown in fig. 2. The different inputs
and outputs to the system can be viewed as vectors
in their respective spaces. As it has been indicated in
section 2, these vectors represent a linear combination
of the sets of each linguistic variable, or, in the case
of crisp inputs, the activation of each set by this input.
Input variables x
i
will be represented by vectors α
i
and output variables y
i
by vectors γ
i
. Each FLU can
be characterized by an array of matrices
i
l
.
α
1
α
2
α
3
α
N+1
γ
1
γ
2
γ
N
γ
N1
1
2
N
Figure 2: A HFS in FITM notation.
i
α
i+1
γ
i
γ
i1
Figure 3: Single FITM block. For i = 1 γ
i1
= α
i
.
A single block of the whole system (a FLU rewrit-
ten in FITM terminology) is showed in 3. For the
i
th
(i 2) block the inputs will be the vectors γ
i1
and α
i+1
, and the output the vector γ
i
. According to
eq. (1) the inference process is performed following
the equation
γ
i
=
X
l
α
i+1
l
i
l
!
γ
i1
(8)
Vector γ
i1
is the output of the previous block and
α
i+1
is an external input. According to eq. (8) we
can write the output of the overall system as
γ =
X
l
N
α
N+1
l
N
N
l
N
!
· · ·
X
l
1
α
2
l
1
1
l
1
!
α
1
(9)
which can be rewritten as
γ =
X
l
N
· · ·
X
l
1
α
N+1
l
N
· · · α
2
l
1
N
l
N
· · ·
1
l
1
!
α
1
=
X
l
N
· · ·
X
l
1
α
N+1
l
N
· · · α
2
l
1
l
N
···l
1
!
α
1
(10)
This is the FITM inference equation for a MISO sys-
ICINCO 2006 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION
94
. . .
α
1
α
2
α
N+1
γ
Figure 4: Multiple Input Single Output fuzzy system.
tem as the one shown in fig. 4, with its transition ma-
trices defined as
l
N
···l
1
=
N
l
N
· · ·
2
l
2
1
l
1
(11)
There is an equivalent relation between the transition
matrices of a MISO FS and the matrices of each FLU
of the derived HFS. Knowing the matrices of the FLU
of the HFS we can easily calculate the matrix of the
overall system. The inverse procedure is also possi-
ble.
3.1 From MISO to HFS. Special
Case: 3ISO
Consider a fuzzy system with 3 inputs and 1 output,
as shown in fig. 4. The imputs will be x
1
(M possi-
ble fuzzy sets, vector α
1
), x
2
(N possible fuzzy sets,
vector α
2
) and (x
3
: P possible fuzzy sets, vector α
3
).
The output is y (Q possible fuzzy sets, vector γ). The
inference can be carried out using an array of transi-
tion matrices
ij
, with i = 1, . . . , N , j = 1, . . . , P
and size Q × M. An equivalent HFS to this system
will be the one shown in fig. 5. Two FLUs replace
the 3-ISO FS. The output of the first block is modeled
by a vector β, defined over a space with K possible
fuzzy sets. The value of K will be discussed later.
Both systems are defined by its transition matrices,
A
i
(Q × K size, i = 1, · · · , P ), and
B
j
(K × M
size, j = 1, · · · , N ).
α
1
α
2
α
3
γ
β
A
B
Figure 5: Equivalent HFS.
There is a relation between the matrices of the 3-
ISO and the hierarchical systems:
A
j
B
i
=
ij
(12)
This relation can be rewritten using block matrices as
A
1
.
.
.
A
P
!
(
B
1
· · ·
B
N
) =
11
· · ·
N1
. . . . . .
1P
· · ·
NP
A
T
[QP ×K]
B
T
[K×M N ]
=
T
[QP ×MN ]
(13)
The solutions of this system are determined by the
rank of matrix
T
. It will also determine the number
of sets in the transition space, as the minimum num-
ber of columns of matrix
A
T
is the rank of matrix
T
.
Specifically K rank(
T
). We choose the equality
to obtain the minimum-size matrices. If a decompo-
sition of matrix
T
according to eq. (13) is feasible,
the hierarchical system determined by matrices
A
i
and
B
j
will be totally equivalent to the original sys-
tem.
3.2 Optimizing the HFS
The decomposition process described in section 3.1
assumes that the inputs are arranged in a particular
order, but this order is somehow arbitrary. A different
arrangement will give rise to a different matrix
T
,
and therefore to a different decomposition in FLUs.
For a 3ISO system, as the one in the previous sec-
tion, there are three possible arrangements; each of
them gives rise to a different system (see fig. 5):
[x
1
, x
2
, x
3
]: x
3
will be the external input to the sec-
ond FLU (this is the one indicated in Fig. 5).
[x
1
, x
3
, x
2
]: x
2
will be the external input to the sec-
ond FLU.
[x
2
, x
3
, x
1
]: x
1
will be the external input to the sec-
ond FLU.
We can also think in other possibilities, such as
[x
2
, x
1
, x
3
], but the final HFS will be the same one
as in the first case. Assuming that the rearranged in-
puts are [x
0
1
, x
0
2
, x
0
3
], with size N
0
1
, N
0
2
and N
0
3
the
total number of rules in the 3ISO system and in the
HFS will be N
3ISO
= N
0
1
× N
0
2
× N
0
3
and N
HF S
=
N
0
1
×N
0
2
+K × N
0
3
, being K = rank(
T
). In order to
have the minimum number of rules, we must choose
the arrangement of the inputs that minimize N
HF S
.
3.3 An Example
Suppose a fuzzy system with 3 inputs x, θ and
˙
θ and
one output y. Each input and the output are defined as
a linguistic variable with 3 possible fuzzy sets: Posi-
tive (P), Zero (Z) and Negative (N). The rule base for
this system is on table 1.
The inference can be carried out using FITM. The
system will be a 3ISO system as the one in fig. 4. The
output will be the centroid of
γ =
X
i
X
j
ij
α
i
2
α
j
3
α
1
being γ the output vector, and α
i
the input vectors.
Accepting that the inputs are crisp values, these vec-
tors are defined as
α
i
= [P (x
i
) Z(x
i
) N (x
i
)]
T
MATRIX-BASED HIERARCHICAL FUZZY SYSTEMS
95
C
T
=
1 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 0
0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0
0 1 1 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1
(14)
Table 1: Rule base for the example.
x θ
˙
θ y
P P P P
P P Z N
P P N N
P Z P N
P Z Z P
P Z N Z
P N P Z
P N Z N
P N N N
Z P P Z
Z P Z N
Z P N N
Z Z P N
Z Z Z P
Z Z N P
Z N P P
Z N Z P
Z N N N
N P P P
N P Z P
N P N N
N Z P P
N Z Z Z
N Z N Z
N N P Z
N N Z N
N N N N
As indicated before, the activation matrices R
x
, R
θ
and R
˙
θ
are the 3 × 3 identity matrix. The selection
matrix C is on eq. (14). The resulting transition ma-
trices are (if an ordering [x, θ,
˙
θ] is assumed)
11
=
1 0 1
0 1 0
0 0 0
21
=
12
=
0 0 1
0 0 0
1 1 0
31
=
23
=
0 1 0
1 0 1
0 0 0
22
=
1 1 0
0 0 1
0 0 0
32
=
13
=
33
=
0 0 0
0 0 0
1 1 1
Once the whole system is defined we can make the
HFS decomposition. To build the
T
matrix in
eq. (13) we have three different ways of ordering the
inputs:[x, θ,
˙
θ], [x,
˙
θ, θ] and [θ,
˙
θ, x]. So, three differ-
ent
T
matrices can be built:
1
T
=
1 0 1 0 0 1 0 1 0
0 1 0 0 0 0 1 0 1
0 0 0 1 1 0 0 0 0
0 0 1 1 1 0 0 0 0
0 0 0 0 0 1 0 0 0
1 1 0 0 0 0 1 1 1
0 0 0 0 1 0 0 0 0
0 0 0 1 0 1 0 0 0
1 1 1 0 0 0 1 1 1
2
T
=
1 0 1 0 0 1 0 0 0
0 1 0 0 0 0 0 0 0
0 0 0 1 1 0 1 1 1
0 0 1 1 1 0 0 1 0
0 0 0 0 0 1 1 0 1
1 1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
1 0 1 0 0 0 0 0 0
0 0 0 1 1 1 1 1 1
3
T
=
1 0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 1 0
0 1 0 1 0 1 1 0 1
0 0 1 0 1 0 0 1 0
1 0 0 0 0 0 0 0 0
0 1 0 1 0 1 1 0 1
1 1 0 1 0 0 0 0 0
0 0 1 0 1 0 0 1 0
0 0 0 0 0 1 1 0 1
the ranks of which are rank(
1
T
) = 6, rank(
2
T
) =
6 and rank(
3
T
) = 5. The number of rules in the
original system is 3 × 3 × 3 = 27. In the HFS we
have 3 × 3 + rank(
T
) × 3 rules. So we choose the
third matrix (the one with smaller rank) to build the
HFS.
According to eq. (13) we may find two matrices
A
T
and
B
T
so that
A
T
B
T
=
T
. One way to do
it is using a LU factorization, and use some further
algebra to eliminate redundant lines. By doing so, we
obtain matrices
A
T
and
B
T
and the transition matri-
ces for each FLU will be
A
1
=
1 0 0 0 0
0 0 1 0 0
0 1 0 0 0
!
A
2
=
0 0 1 1 0
1 0 0 1 0
0 1 0 0 0
!
A
3
=
1 1 0 1 1
0 0 1 1 0
0 0 0 0 1
!
B
1
=
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
B
2
=
0 1 0
1 0 1
0 0 0
0 1 0
0 0 1
B
3
=
0 0 0
1 0 1
0 1 0
0 0 0
1 0 1
The inference using FITM can be done as
γ =
X
i
A
i
α
i
1
!
β
β =
X
j
B
j
α
i
3
α
2
ICINCO 2006 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION
96
The system represented as a HFS is totally equivalent
to the one we departed from.
3.4 General Case
When the number of inputs is greater than 3, the HFS
can be obtained in a very similar way to the 3-ISO
system. Suppose a generic MISO system with N in-
puts x
1
, · · · , x
N
. The system can be represented us-
ing transition matrices
l
2
···l
N
l
2
= 1, · · · , M
2
· · · l
N
= 1, · · · , M
N
being M
i
the dimension of the i-th input. This system
is equivalent to a system with N inputs and another
system with two inputs, as shown in fig. 6.
. . .
. . .
x
1
x
N
y
x
N1
x
2
Figure 6: First step in hierarchical decomposition of a
MISO-FS.
To obtain the transition matrices of each system,
the following equation must be solved
A
l
3
···l
N
B
l
2
=
l
1
···l
N
(15)
Written in matrix form
A
T
B
T
=
T
A
11···1
A
21···1
.
.
.
A
M
3
···M
N
B
1
· · ·
B
M
2
=
11···1
· · ·
M
2
1···1
12···1
· · ·
M
2
2···1
.
.
.
.
.
.
1M
3
···M
N
· · ·
M
2
M
3
···M
N
(16)
Matrices
B
l
2
can be easily obtained from this equa-
tion. The same procedure is applied on the remaining
MISO system, until only two inputs are left.
As in the 3ISO case, the proper election of the or-
dering of the inputs will give a better size of the rule
base of each FLU. This size is related again with the
rank of matrix
T
. When this optimization process
is carried out, the final HFS system can be any of the
types introduced in section 1.1: serial, parallel or hy-
brid.
4 CONCLUSIONS
In this paper we have shown a simple procedure to
convert a large MISO FS into a serial HFS. Such a
conversion leads to a number of smaller rule bases
than the original, which, in turn, may facilitate human
interpretation and/or fine tune the rule bases out of
examples.
REFERENCES
Aja-Fern
´
andez, S. and Alberola-L
´
opez, C. (2004a). Fast
inference in SAM fuzzy systems using transition ma-
trices. IEEE Trans. Fuzzy Systems, 12(2):170–182.
Aja-Fern
´
andez, S. and Alberola-L
´
opez, C. (2004b). Fuzzy
hierarchical systems wih FITM. In Proc. of FUZZ-
IEEE’04, Budapest, Hungary.
Aja-Fern
´
andez, S. and Alberola-L
´
opez, C. (2005). Fast
inference using transition matrices: An extension to
non-linear operators. IEEE Trans. Fuzzy Systems,
13(4):478–490.
Golub, G. and Loan, C. V. (1996). Matrix Computations.
The Johns Hopkins University Press, Baltimore, ME.
Holve, R. (2003). Rule generation for hierarchical fuzzy
systems. In NAFIPS ’97. 1997 Annual Meeting of the
North American Fuzzy Information Processing Soci-
ety, pages 444–449, Syracuse, NY, USA.
Hor
´
a
ˇ
cek, P. and Binder, Z. (1997). Hierarchical fuzzy con-
trollers. Annual Reviews in Control, 21:93–101.
Kikuchi, H., Otake, A., and Nakanishi, S. (1998). Func-
tional completeness of hierarchical fuzzy modeling.
Information Sciences, 110:51–60.
Kosko, B. (1997). Fuzzy Engineering. Prentice-Hall Inter-
national, New Jersey.
Lee, M. L., Chung, H., and Yu, F. (2003). Modeling of
hierarchical fuzzy systems. Fuzzy Sets and Systems,
138:343–361.
Raju, G. and Zhou, J. (1993). Adaptive hierarchical fuzzy
controller. IEEE Trans. on System, Man and Cyber-
netics, 23(4):973–980.
Raju, G., Zhou, J., and Kisner, R. A. (1991). Hierarchical
fuzzy control. Int. J. Control, 54:1201–1216.
Wang, L. X. (1998). Universal approximation by hierarchi-
cal fuzzy systems. Fuzzy Sets and Systems, 93:223–
230.
Yager, R. (1993). On a hierarchical structure for fuzzy mod-
eling and control. IEEE Trans. on System, Man and
Cybernetics, 23(5):1189–1197.
Zadeh, L. A. (1996). Fuzzy logic = computing with words.
IEEE Trans. Fuzzy Systems, 4(2):103–111.
Zeng, X. and Keane, J. (2005). Approximation capabili-
ties of hierarchical fuzzy systems. IEEE Trans. Fuzzy
Systems, 13(5):659–672.
MATRIX-BASED HIERARCHICAL FUZZY SYSTEMS
97