Journal of Intelligent Learning Systems and Applications, 2010, 2: 12-18
doi:10.4236/jilsa.2010.21002 Published Online February 2010 (
Copyright © 2010 SciRes JILSA
Evolutive Neural Net Fuzzy Filtering: Basic
Juan C. García Infante1, J. Jesús Medel Juárez2, Juan C. Sánchez García1
1Professional School of Mechanical and Electrical Engineering, National Polytechnic Institute, Col. San Francisco Culhuacan Del.
Coyoacan, México D.F.; 2Centre of Computing Research, National Polytechnic Institute, Vallejo, México D. F.
Received October 29th, 2009; accepted January 20th, 2010.
The paper describes the operation principles of the evolutive neuro fuzzy filtering (ENFF) properties, which based on
back propagation fuzzy neural net, this filter adaptively choose and emit a decision according with the reference signal
changes of an external reference process, in order to actualize the best correct new conditions updating a process. This
neural net fuzzy filter mechanism selects the best parameter values into the knowledge base (KB), to update the filter
weights giving a good enough answers in accordance with the reference signal in natural sense. The filter architecture
includes a decision making stage using an inference into its structure to deduce the filter decisions in accordance with
the previous and actual filter answer in order to updates the new decision with respect to the new reference system con-
ditions. The filtering process states require that bound into its own time limit as real time system, considering the Ny-
quist and Shannon criteria. The characterization of the membership functions builds the knowledge base in probabilis-
tic sense with respect to the rules set inference to describe the reference system and deduce the new filter decision, per-
forming the ENFF answers. Moreover, the paper describes schematically the neural net architecture and the deci-
sion-making stages in order to integrate them into the filter architecture as intelligent system. The results expressed in
formal sense use the concepts into the paper references with a simulation of the ENFF into a Kalman filter structure
using the Matlab© tool.
Keywords: Digital Filters, Neuro Fuzzy Systems, Evolutive Systems, Real Time
1. Introduction
The development of new intelligent systems requires me-
chanism that deduces its own external environment. Hav-
ing a characterization of its answers in order to discover
dynamically the changes of an external process; inter-
preting the operation levels to select the best response
updating its answers weights, in order to follow the best
approximation condition in accordance to the external pr-
ocess changes and taking a decision about the actual con-
dition to correct or update a system parameters.
These kinds of systems related to artificial intelligent
mechanisms, which have some problems to develop cap-
acities with high evolutive changing processes.
The new systems should use into its architecture evo-
lutive tools in order to get its own perception giving the
best decision answers, using this to solve complex prob-
lems, actualizing and adapting its perceptions and answ-
ers in accordance with a reference dynamical system.
The use of learning techniques based on searching the
optimal solutions represents a classical alternative man-
ner to obtain knowledge. The evolutive systems into the
neural net fuzzy digital filtering is an option for to obtain
different decision answer levels, in accordance with a
reference model dynamics, adapting its answers to the
possible changes by selecting the best values in order to
get the necessary convergence conditions, which should
has the best operation each time.
An evolutive system with artificial neural network pro-
perties is a computational model imitating natural proc-
esses as the biological systems. Constituting in this case
the processing elements called neurons, all of them con-
nected forming the neural net structure and a decision
stage to loop the process with the best answer [1,2].
The evolutive filtering is constructing with different
kind of decision answers, considering a set of character-
istics limited by the reference operation region. The goal
of this kind of filter is to characterize and infer a system
that has uncertainties in its operation, describing the na-
tural process, with a rule set; needing a feedback law in
order to follow the basic properties in accordance with a
desired input signal, adjusting its parameters to give a
Evolutive Neural Net Fuzzy Filtering: Basic Description13
correct solution dynamically. According to this, the filter
chooses the best decision instruction to update the sys-
The neural net filter stage [3] works as a parallel fuzzy
neurons in loop form, which has an iterative searching
methodology used for evolutive algorithms and based on
the back propagation (BP) algorithm since its parameters
are updated dynamically ([4–6]) at each iteration by de-
grees [7]. This process refers to a back propagation pa-
rameter adaptation [8], using supervised learning by the
knowledge base according with the error e(k) ([9,10]) de-
scribed by the difference between the desired response
y(k) and the actual signal [11]1.
With this perspective, the paper integrates the ENFF
concept with its real time restrictions [9], using statistical
methods in order to characterize the Kalman filter inter-
nal structure to give answers with respect to the operation
levels in natural way making a specific decision in order
to follow the natural reference model ([12,13]).
2. Filtering Stages
The criterion described as the error minimization e(k)[3]
describes the difference between the desired input y(k)
and the actual output filter, allowing to find the co-
rresponding membership function. Which is the best ap-
proximation from the signal
ˆ to y(k) in order to ad-
just the parameters of the filter and get the correct answer
describing the reference system path in order to take the
decision into the decision stage. The error value e(k) sh-
ould be close to γ with a limited interval [0, ] and is
described as }, Zk:)(inf{ ike (i represents an in-
dex, with k interval [14,15])2 Below, there is the stages
description of the evolutive filtering, the first three stages
are the inference mechanism; the stages d and e are the
filtering process and the decision stage is the instruction
Input Inference: In this stage, the natural desired
signal y(k) from the reference system to the input filter
has a description in metric sense, is the first step of the
inference mechanism [5].
Rule base: Dynamical rank intervals with respect to
the input of the filter use the logical binary connector
known as IF, representing the second step of the in-
ference mechanism.
Output inference: The expert action with respect to
the rule base known as consequence uses the logical
binary connector THEN to find the correct weight an-
swer as
ˆ called membership function is the third
step of the inference mechanism.
Filter algorithm: This is the filter stage, which ta-
kes the digital answer, converted in a natural response.
This is the closest distance value to the desired signal,
based on the predefined knowledge base.
Natural feedback: the filtering process takes the co-
rrect linguistic value and feedbacks the filter parame-
ters, updating its operation according to the natural
evolution of the reference system considering the error
differences between
ky and
ˆ dynamically and
using a metric rank of the error functional
kJ .3
Decision stage: Finally, the filter decision mecha-
nism takes the best answer option according with the
values of
ˆ and
ˆky to actualize, the decisi-
on described as
Each decision stage is in accordance with the actual
and previous values of
ˆ described as
ˆky do-
ing a comparison between these values in order to select
the corresponding instruction answer to the reference
system, all the instruction answers are previously defined
using the logic connectors IF-AND-THEN the filter se-
lects the corresponding instruction to actualize the refer-
ence system conditions.
Figure 1, shows the filter architecture interacting with
a reference process to describe its operation and take a
decision instruction.
3. Filtering Description
The evolutive neural net fuzzy filter (see: Figure 1) re-
quires a knowledge base (KB); because, it has all the pos-
sible responses in accordance with the reference system or
process in which the filter is interacting. The evolutive by
neural net fuzzy filter has its state variables bounded by a
1 is a fuzzy value.
)(:)( kykyke 
2Inf is the greatest lower bound error value into the error set.
3The functional
kJ describes the convergence relation among the
real observed and its estimation; symbolically in recursive form:
)1(kJ )()1(
)( kek Figure 1. Evolutive neural net fuzzy filter process
kJ 
Copyright © 2010 SciRes JILSA
Evolutive Neural Net Fuzzy Filtering: Basic Description
rank of values described by fuzzy sets (for example, the
temperature classified by the fuzzy logic as high, me-
dium or low ([3,8,17,18]).
This kind of filter adaptively modifies its response
according to the dynamic changes at the filter de-
sired input y(k) from the reference process alteration (pr-
oduced by neural excitation). It requires that the error
loop inference (comparison between desired input y(k)
and actual output changes the filtering responses
using the knowledge base (KB) with real time conditions
(see: [4,6,9]).
The filter classifies its different operation levels by
membership functions, based on the error minimum fun-
ction Jmin to give a corresponding near specific answer for
each kind of desired input y(k) limited by the error dis-
tribution function, which is also limited by γ. The classi-
fication of the reference system responses realized by the
filter is by the error distribution function considering di-
fferent levels of response (membership functions), each
level delimited by specific metrics bounded all into the
error distribution region. This classification represents th-
e characterization of the reference process operation in
interaction with the filter ([14,19]).
Using this kind of filters avoids the initial instability of
the error convergence because it has all information re-
quired and limited by an interval into a distribution re-
gion as supervised learning technique.
To do a classification of the membership functions the
filter uses an error criterion function using metric inter-
vals to delimit its operational levels. The membership
function set established inside a distribution function co-
rresponding to the error criterion; this classification with
several membership functions is in accordance to each
error interval (as operation linguistic levels).
Respect to Figure 1, is an input natural linguis-
tic value from the reference process interacting with the
filter. is the control area, which include all the fil-
tering processes corresponding to one iteration of a pr-
ocess set, in order to obtain an output response
ˆ. The
desired signal described as
ky is the fuzzy natural re-
ference variable given by the reference system.
ke is
a fuzzy value where the set
Z i,0: ii
0,}inf{ i
and the1,}
sup{ , [5]
with *
that must be the minimum value between y(k)
and ; both values should be almost equal; in other
words, is the difference between , and
ky ,
being this the criterion distance in order to select the
membership function for each case1.
4. Fuzzy Neural Net Structure
The ENFF architecture is develop in order to work as
neural net using back propagation properties ([11,19]).
This characterizes the operational linguistic levels of the
desired variable set {y(k)}, expressing as a basic estima-
tion result5, in accordance with the inference rules
with respect to the error value criterion selecting the cor-
responding membership function.
The neural net conditions are [1]:
The desired input y(k) represents fuzzy labels, and
each one has different operation levels.
The input inference recognizes the variable type (A,
B or C).
The weights required for the internal adaptive ad-
justments into the filter architecture, to get a correct
response described as membership functions, rene-
wing the values of )k(a
ˆparameter using the
knowledge base.
The connection into the filter according to the error
differences1 has a minimum operation criterion, se-
lecting the membership function based on the mini-
mum cost of the filter response to update its wei-
It uses supervised learning into the knowledge base,
therefore, the neuron has previously all the infor-
mation included inN
The decision stage compares the previous output
and the actual value of the filter to select a instruc-
tion decision [7].
Globally, according to Figure 2, the filter has four
neural layers:
The first layer with
Zpnnp ,,
input neurons rep-
resenting a set of desired inputs {y(k)}, a single hidden
layer with p processing units in which the inference me-
chanism operates in order to find the respective mem-
bership function; the output{}, that is the an-
swers set of the neural process (see: Figure 2). The filter
has a decision stage layer as evolutive condition in order
to give control instructions for an external process ac-
cording with the previous filter answers.
The neurons architecture are simple processors, which
get information from the first input layer from the outside
world as desired signal
}{ ky to the next neurons into
the hidden filter layer.
The neural net structure proceed the result of their
previous processing step to nodes in the next higher lay-
ers in order to obtain the nearest desired signal value of
ky at the filter output
ˆ. In consequence the deci-
sion stage in the top-layer communicate their decision
results to the outside world as , which its instruc-
tions bounded inside the external reference process op-
eration in order to give the best control value.
Subsequently, using rules the fuzzy system performs
the classification in both stages: filtering and decision; its
Copyright © 2010 SciRes JILSA
Evolutive Neural Net Fuzzy Filtering: Basic Description15
rules are generated automatically by the evolutive neural
net fuzzy filtering evaluating features which are difficult
to extract or to evaluate directly delimited by the filtering
The neural net represents a set of services, activating a
specific neuron means the use of specific service with
different levels of response. The filter requires a variable
value according to the inference classification of the de-
sired signal set, according to the changes of the error
rank e(k) per iteration, one neuron is activating; next it-
eration the filter could renew the neuron or only change
its value by degrees.
The value of y(k) defines a neuron to activate or to se-
lect, in order to use its own set of operation levels
(membership functions by degrees) to give the corre-
sponding value of , and gives a correct natural
answer [2].
5. Rule Base Strategy
This kind of filtering has operational properties defined
by the rules base to learn, recall, associate and compares
the new information delimited before according to the
error variance limits predefined. The fuzzy rules condi-
tions established as logical connectors (IF-THEN) con-
stitute the rules base with respect to the error intervals
and its respective response described as membership fun-
The rules generated constitute the inference of the de-
sired signal5 as the logical connector IF, and the
logical connector THEN selecting automatically the pa-
rameter weighs of according to the knowledge ba-
se ([18,20]):
The KB has an automatic classification of the filter
conditions: having the knowledge of the filtering opera-
tion levels. In addition, selects the corresponding mem-
bership function (value of ) of the knowledge base,
adapting its responses weighs to give a correct answ-
er , near to.
A rules base filter characterized by a set of desired
signals at the filter input, classified by intervals
delimited previously into the error difference in order to
select the corresponding membership function, which has
the corresponding value giving a correct response
seeking the closest distance to:
}{ ky
kN kykyT 1
In addition, mathematically correspondence of
and , expressed with respect to the second probabil-
ity moment, has the infimum value, described as:
NyyJJ )
Figure 2. Evolutive neural filter layers (L-low, M-medium,
The rules set constitute a simple filter operation, based
on the error set {e(k)} as limit indicator of the corre-
sponding membership function set to adapt the parameter
The decision stage described as selects the best in-
struction answer to control a process according with the
filter answer, and its previous value, with the decision
inference mechanism by fuzzy rules.
For the decision stage, which is the finally inference of
the evolutive neural net fuzzy filter, using a set of fuzzy
rules to get the correct control instruction from its own
knowledge base (KB). First, consider the value level and
its previous value as the actual environment and select
the corresponding condition or strategy from the knowl-
edge base to loop an external process as.
To describe the decision stage mechanism, we use fu-
zzy rules to infer and select the corresponding value with
If-then rules; the filtering system selects and gives a spe-
cific instruction answer as in order to update and improve
the control instruction performance minimizing the error
process, in accordance with the fuzzy rule structure will
If value is____ and value is____ Then value is ____.
This filter (ENFF) has two knowledge bases, one for
to select the parameter (as in Figure 1) and the second
used for the decision stage to select the instruction an-
swer (as in Figure 3).
The Figure 3 illustrates the decision stage mechanism
as fuzzy rules as:
(2) Figure 3. Decision stage inference
Copyright © 2010 SciRes JILSA
Evolutive Neural Net Fuzzy Filtering: Basic Description
6. Real Time Scheme
DEFINITION 1 The real time properties of an evolutive
neural net fuzzy digital filter, which is an adaptive filter
performed according to ([6] and [10]):
To take and to emit signals with fuzzy information
into intervals limited in accordance to the reference
process response according to the stability criteria
To take and to emit information through semi-open
time intervals, synchronized with the process evolu-
tion time, described in a relative way by semi-open
time intervals, considering the criteria into refer-
ences [15,21,22].
A membership functions group forms a control area,
according to the properties considered in a) and b)
points, respectively ([18,20,23]) .
A set of fuzzy rules builds the knowledge base ac-
cording to the fuzzy desired signal from the refer-
ence model)(ky , obtaining a specific filter answer
The adaptation algorithm updates the filter coeffi-
cients according to the selected membership func-
tion respectively to the established error criterion
symbolically expressed as*
The decision stage mechanism selects the best in-
struction answer as )k(ds , in order to perform the
external process operation at each time.
The knowledge base has all information that the filter
requires to adjust its gains in optimal form and gives an
answer accomplishing the convergence range, inside of
the time interval (indexed with) in agreement wi-
th the Nyquist sense, without loss of the stability pro-
perties [6,20,23]):
ky is a measurable variable classified in metric
ranks as degrees in linguistic sense (described into
a state space variable bounded symbolically into
linguistic natural expressions as high, medium or
low values),
kT is the control area described in pairs formed
ˆ and
ky limited in time interval (has a
velocity change bounded in the sense exposed by
ke is the fuzzy value defined by the difference
ˆ and
ky , which is bounded by the
set Zi
, with ,}inf{ *
i such
that 0
, ,}sup{ *
i 1
, means that
ˆ is approximately equal to
ky metrically
speaking as linguistic variables, both are the same
natural value.
kds is the instruction answer of the filter acc-
ording with the output of the filter
ˆ and its
previous value in fuzzy sense, to selects the corre-
sponding candidate function classified in probabil-
istic sense into a knowledge base to update an ex-
ternal process into a new condition.
6.1 Local and Global Description
DEFINITION 2 An ENFF in local and global temporal
sense has quality of response according to the conver-
gence criterion1 with respect to the real time conditions
Global characteristics: The convergence intervals de-
fined by
,0 with measures up to zero through er-
ror functional J(k) considering and the convergence rela-
tion1, temporally parameterized to the membership func-
tion according to the linguistic variables values ([3,24]),
without loss that /e(k)/<1 in agreement to [6].
According to the evolutive neural net fuzzy concepts,
the global characteristics specified in stochastic sense
according to [15], where
min{ k(see:
(2)), with
kk JJ }
{ and
without loss that its natural evolution described by [22]:
maxmin f5.0
7. Simulations
For the simulation of the ENFF, in this case we use the
Kalman filter structure to integrate the filter stages, with
a transition matrix described by the knowledge base in
accordance to the error functional criterion [5]. The evo-
lution times into a soft system integrated in a PC with
AMD Sempron processor 3100+ with k intervals, having
a mean evolution time of 0.004 sec ± 0.0002 sec.
The basic reference system in discrete state space ex-
pressed by the first order difference, as:
In accordance with the system (4) the output is:
4Sup is the least upper bound of a partially ordered set
5The desired signal commonly has the basic and explicit description as
, where is known as stability pa-where: {x(k)}= is the set of internal states, {a(k)}= is the
parameters sequence, {w(k)}= is the noise set system
perturbation, {y(k)}= is the set of desired signal from the
system reference, v(k)= is the output noises.
)()1()()( kkykaky
rameter (see: [12] and [10]),
is the perturbation output noise,
)(ky is the desired reference signal.
Copyright © 2010 SciRes JILSA
Evolutive Neural Net Fuzzy Filtering: Basic Description17
There are different operation levels described in prob-
ability sense in order to match with the functional error1
limit, in accordance with the desired signal y(k) and the
Kalman filter answer. The filter operation bounds
into the second probability moment, establishing the fil-
ter classification as linguistic natural variables expressed
as low, medium and high levels.
According to the parameter selected by the rule
strategy, the Figure 4 shows the output answer
ˆ of
the filter with respect to the desire signal y(k) at the input
Figure 5 shows the decision stage process to obtain the
instruction answer as according with the
values and its previous stage described as:
The evolution time is less than the reference process
proposed as 0.092 sec., satisfying the condition described
in (3).
The Figure 6 shows the decision stage as linguistic
levels according with the filtering process in order to
give different decision answers level described as DA#:
The global convergence time of the filter is 0.08 sec,
which is less than the evolution condition of the system,
known as system maximum evolution time, oscillating
around 0.092 sec.
Figure 7 shows the functional described as J(k) with
respect to the filter:
8. Conclusions
The paper was about the analysis of the evolutive neural
net fuzzy filtering and its real time conditions, in order
show the applicability conditions into dynamical systems.
The paper describes the adaptive inference mechanism
that classifies and deduces the filter answers by the error
value as limit, in order to search the adaptive weights and
update its parameters to give a correct response dynami-
cally as a natural linguistic answer.
Figure 4.Estimation in accordance with
ˆ value
and reference
Figure 5. Decision answer
kds with respect to
ˆ and
Figure 6. Decision answer as linguistic levels (DA)
Figure 7. Parameter of convergence illustratively by the
functional J(k)
The paper establishes how to construct and character-
ize the membership functions of the knowledge base in a
probabilistic manner with the decision rules set, making a
Copyright © 2010 SciRes JILSA
Evolutive Neural Net Fuzzy Filtering: Basic Description
Copyright © 2010 SciRes JILSA
description of the real time conditions that the evolutive
neural net fuzzy filter (ENFF) has to perform, which
architecture works as a neural net. The filter has a final
decision stage in accordance with the filter operation
making a fuzzy inference between the actual and previ-
ous filter answers in order to select the corresponding
decision answer value.
The results are in formal sense and described using the
definitions considered in the papers referenced here. Fi-
nally, this work showed a simulation of the ENFF opera-
tion using the Matlab tool and the Kalman filter struc-
tureto integrate the evolutive properties, considering five
decision answers into the decision stage, having an accu-
rate filtering time response with respect to the reference
system in accordance with the real time properties pro-
posed in this work.
9. Acknowledgments
The authors thank the sponsors for the support of this
work: Conacyt and National Polytechnic Institute (IPN).
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