https://doi.org/10.31449/inf.v43i2.2303 Informatica 43 (2019) 291–298 291
  
Evolving Neural Network CMAC and its Applications 
Oleg Rudenko, Oleksandr Bessonov and Oleksandr Dorokhov 
Kharkiv National University of Economics, Nauka Ave 9a, 61166 Kharkiv, Ukraine 
E-mail: aleks.dorokhov@meta.ua 
  
Keywords: neural network, training, nonlinear object, identification, adaptive control 
Received: April 21, 2018 
The conventional neural network (NN) CMAC (Cerebellar Model Articulation Controller) can be 
applied in many real-world applications thanks to its high learning speed and good generalization 
capability. In this paper, it is proposed to utilize a neuro-evolutional approach to adjust CMAC 
parameters and construct mathematical models of nonlinear objects in the presence of the Gaussian 
noise. The general structure of the evolving NN CMAC (ECMAC) is considered. The paper 
demonstrates that the evolving NN CMAC can be used effectively for the identification of nonlinear 
dynamical systems. The simulation of the proposed approach for various nonlinear objects is performed. 
The results proved the effectiveness of the developed methods. 
Povzetek: Razvit je postopek za evolucijsko iskanje najbolj prilagojene CMAC (Cerebellar Model 
Articulation Controller) nevronske mreže za probleme z Gaussovim šumom. 
1 Introduction 
Using a mathematical model of the cerebellar cortex 
developed by D. Marr [1] in 1975 J. Albus proposed a 
model describing the motion control processes that occur 
in the cerebellum, which was subsequently implemented 
in the neural network controller for controlling the robot 
- arm, which he called CMAC - Cerebellar Model 
Articulation Controller [2, 3]. Ease of implementation 
and a good network of approximating properties have 
ensured its wide usage not only in the tasks of controlling 
the robotic arm in real time, but also to solve many other 
practical problems [4-14]. 
However, it should be noted that in designing a 
network CMAC a number of difficulties in the selection 
of parameters such as the number of levels and the 
quantization levels, the shape of the receptive field, the 
type of applied information hashing algorithm and 
training. These parameters have a significant impact on 
the accuracy and speed of CMAC network, and 
therefore, the determination of the optimal values of 
these parameters is an important practical problem.  In 
this article, for eliminating the drawbacks of traditional 
methods of synthesis and functioning ANN CMAC we 
provide the use of a new class of networks - evolving 
ANN (EANN) in which, in addition to traditional 
learning it is used another fundamental form of 
adaptation - evolution, realized by applying the 
evolutionary computation [15-18]. 
The use of two forms of adaptation in EANN - 
evolution, and training, allowing to change the network 
structure, its parameters and learning algorithms without 
external intervention, make the network data most 
suitable for work in non-stationary conditions and 
uncertainty about the properties of the object under study 
and the conditions of its functioning. 
The main advantage of using evolutionary 
algorithms (EA) as learning algorithms is that many 
ANN parameters can be encoded in the genome and 
determined in parallel. Moreover, unlike most 
optimization algorithms designed to solve a problem, EA 
operate with a multitude of solutions - the population, 
which allows reaching a global minimum, without 
getting stuck in the local ones. In this case, information 
about each individual of the population is encoded in a 
chromosome (genotype), and the solution (phenotype) is 
obtained after evolution (selection, crossing, mutation) 
by decoding. 
Among EAs that are stochastic and include 
evolutionary programming, evolutionary strategies, 
genetic algorithms, genetic programming, in particular, 
programming with gene expression, genetic algorithms 
(GA) are the most common [19,20]. GA abstract the 
fundamental processes of Darwinian evolution: natural 
selection and genetic changes due to recombination and 
mutation. 
2 Neural network CMAC 
The modification of the network proposed by Albus is 
shown in Figure 1. The network consists of the input, 
hidden and output layers, labeled L1, L2, L3, 
respectively, and uses two basic conversions: 
    S: X  A,                      (1) 
           P: A  y,                             (2) 
where X - N-dimensional space of continuous input 
signals; A - n-dimensional space associations; y - a one-
dimensional output. 

292 Informatica 43 (2019) 291–298 O. Rudenko et al.  
Converting S  A, in turn, consists of two 
transformations: 
           X  M   (3) 
           M  A,   (4) 
where M - the space of binary variables. 
The principle of the network operation as an 
associative memory is as follows. Approximated function 
y = f (x) is given to a limited number of points (argument 
values) x constituting N-dimensional space of the input 
signals. This space is divided into subspaces M formed 
the input signals x(i) ( M , 1 i = ). 
A number of subspaces M impacts the accuracy of 
the network and number of utilized memory cells. 
Therefore, on the one hand, side it should be big enough 
to ensure good approximation capabilities of the network 
and on the other hand side, it should be not too big to 
save some memory. In constructing the cerebellum 
model Albus proceeded from the fact that the appearance 
of the excitation signal activates its a certain area of the 
cerebellum, or receptive field, characterized by a 
parameter ρ. 
Therefore, storage of values of y(i) (network output 
signal) corresponding to x(i)  ( M , 1 i = ), used ρ memory 
cells, the number of which is constant for all vectors of 
the input signals on the network. At receipt of the input 
signal x(i) a signal y(i) appears at network output, which 
is the sum of ρ addressable cells content. 
Associative CMAC properties manifest themselves 
in the form of used addressing, which is based on a 
special coding input information and called hash coding 
or hashing [21-23]. 
3 Encoding information in CMAC 
Information coding in the network means that to each N-
dimensional input vector x(i) an n-dimensional 
association vector a(i), is assigned and stored in virtual 
memory. 
Elements of a(i) can take the values from the interval 
[0, 1] (in the papers cited above it is assumed that these 
elements take the values 0 or 1). Thus only ρ << n 
elements of the vector have non-zero values, i.e. only ρ 
memory elements are active. 
A continuous plurality of input signals by sampling 
(at the level of quantization) is converted into discrete. 
Thus to represent the i-th input signal components R i 
quantization levels used with the appropriate 
quantization step r i ( ). It should be noted that the 
accuracy of the system identification depends 
substantially on the size of the quantization step, and loss 
of stability is possible in digital automated control 
systems with an incorrect choice of this parameter. 
 
Each stage is characterized by a corresponding 
association matrix A i , only one element of 
which is different from zero. 
Construction associations vector as follows. For a 
given total number of input signals association matrix A i 
of each quantization stage ( ) are formed. The 
columns of these matrixes form association vectors a i (
). 
The dimension of these vectors, n, equal to the sum 
of all elements of the matrices A i ( ) and can be 
calculated by the formula: 
N , 1 i =
( )  , 1 = i
 , 1 i =
 , 1 i =
 , 1 i =
 
Figure 1: Albus network. 

Evolving Neural Network CMAC and... Informatica 43 (2019) 291–298 293 
















+
−
=
N
1
1 R
n


,                          (5) 
where R - the number of used levels for quantizing input 
signals; N - the dimension of the input vector; - means 
rounded to the nearest whole number. 
Since all ρ matrices A i (
 , 1 i =
) have only one non-
zero element, from the n components of the vector a(i) 
only ρ are non-zero. 
The quantization region is arranged in such a way, 
that any of them relating to the adjacent stages have not 
more than (ρ - 1)-th connection. This corresponds to a 
restriction on the maximum total number of cells equal to 
(ρ - 1) used for storing two different vectors of the input 
signals in which their recognition is still possible. 
4 Selecting the basic functions of 
neurons 
Selecting the basic functions of neurons in L1 layer 
significantly affects the approximating properties of 
CMAC network. 
As already noted, the traditional CMAC performs 
piecewise constant approximation, that is a consequence 
of the usage of neurons with a rectangular activation 
function. 
When choosing rectangular basis functions 
computational cost will be minimal. Also, CMAC 
networks widely use B-splines as basis functions.  
B-splines undoubted advantage is the possibility of 
recurrent calculating in both the splines in accordance 
with the formula [24-27]: 
 
) ( ) ( ) (
, 1
1
1 , 1
1
,
x B
x
x B
x
x B
j n
n j j
j
j n
n j j
n j
j n −
+ −
− −
− −
−








−
−
+








−
−
=
 

 

 (6) 
 
and their derivatives  -order: 
 
()
()
, 1, 1
1
( 1)
( ) ( )
( 1)
jn
n j n j
j j n
x
n
B x B x
n



  
−
−−
−−
 −
−
=+

− − −


 
    
()
1,
1
( 1)
()
( 1)
j
nj
j j n
x
n
Bx
n


  
−
−+
 −
−
+

− − −


     (7) 
 
Here:  
 )






=
−
otherwise; , 0
; ,  x if , 1
) (
1
, 0
j j
j
x B
 
 
 
 )






=
−
otherwise; , 0
; ,  x if , 1
) (
1
, 0
) 0 (
j j
j
x B
 
 
 








−
−








−
=
+ −
−
−
− −
− −
−
1
, 1
) 1 (
1
1 , 1
) 1 (
,
) (
) ( ) (
) (
n j j
j n
n j j
j n
j n
x B x B
x B
   
 

     (8) 
 
where
j
 - j-th spline’s node (center of the quantization 
field). 
Thus, after determining the active slot ] , (
1 j j
 
−
 for 
the zero order B-spline, these expressions can be used to 
obtain the values of all nonzero B-spline of higher order 
and, if appropriate, their derivatives.  
Note that traditional СМАС uses zero order B-
spline. Selection of the first order B-spline leads to the 
triangular membership function, and selection of the 
fourth-order B-spline leads to membership function 
similar to the Gaussian. 
The CMAC network also uses Gaussian activation 
function of the form: 
 










−
− =
2
2
)
exp ) (

i
j i
x Ф
μ (x
j             (9) 
As a basis one can use trigonometric functions, for 
example, cosine: 
 
( )





+ − 








−
=
otherwise,                         0
];
2
r
,
2
r
(  х if 
r
cos
) (
j j
j
j



 


i i i j
j i
x
x Ф
(10) 
 
where - i-th center of the quantization field; r j - 
quantization step of j-th component of the input signal. 
However, it should be noted that although the most 
commonly used Gaussian membership functions also 
allow a very simple calculation of derivatives and have 
the property of a local activation, it is difficult to allocate 
clearly enough their  activation boundary, which is often 
important for the implementation of the network that 
used, for example, scaling basis functions. 
In order to eliminate this disadvantage, one can use a 
modified Gaussian function that has the following form: 
 












− −
−
−
=
otherwise.                         0
); , (  х if 
) )( (
4 / ) (
exp
) (
2 1
2 1
2
1
 
 
 
x x
x Ф
i
2
   (11) 
 
As seen from the expression (11), the function is 
strictly defined in the range (λ 1, λ 2), which simplifies the 
process of scaling basis functions when the network 
parameters such as R and ρ are changing. 
5 Network training 
Defining the network parameters, i.e. in the general case 
defining a vector ( ) k  , that includes all network 
parameters (weights, parameters of basis functions, etc.) 
is accomplished by training with the teacher. 
The training criterion can be presented as follows: 
  ( )

=
=
k
i
i e k e F
1
) ( ) (  ,                       (12) 
where ( ) ) (i e  - some loss function. 
  •
i


294 Informatica 43 (2019) 291–298 O. Rudenko et al.  
Gradient network training algorithm has the 
following form: 
( ( ))
ˆˆ
( ) ( 1) ( )
j
F e k
k k k   


= − +

,          (13) 
or 
( )
()
ˆˆ
( ) ( 1) ( ) ( )
j
ek
k k k e k    


 = − +

     (14) 
where 0   – parameter that affects the training speed 
and which can be selected differently for different 
network parameters. 
Training of traditional CMAC that uses 
( )
2
( ) 0,5 ( ) e i e i  = and rectangular basis functions, 
occurs on each step after the presentation of training 
pairs )} ( ), ( { k y k x , where ) (k y – function’s value, 
that corresponds ) (k x , and consists in the correction of 
only those of its  weights that correspond to the single 
components of the association vector for a given vector 
) (k x .  
In this case, the training algorithm for all i , j , for 
which 1 ) ( ) ( = = k a k a
j i
, is the following: 
1
1
( 1) ( ) ( ) ( ) ,
n
j j i
i
w k w k y k w k 

=

+ = + −




 (15) 
where ] 1 , 0 (   – parameter that affects the speed of 
training. 
When membership functions with a form other than 
rectangular are used, this algorithm can be written as 
follows: 
, ) ( ) (
) ( ) (
) ( ) ( ) ( ) (
) ( ) ( ) 1 (
2








−
+ = + k a x
k a x
k w x k a k y
k k w k w
T




(16) 
where ) (k  - in general case a variable parameter. 
Multistep network training algorithms were 
considered in [7,8]. 
6 Evolving ANN CMAC 
During switching from ANN to EANN for all types of 
networks the common evolutionary procedure 
(initialization population, an estimation of the 
population, selection, cross-breeding, mutations) is used. 
Differences are only in the method of encoding the 
structure and parameters of a particular form of ANN in 
the chromosome. 
At the beginning of EA functioning, a population 
0
P 
that consisting of N individuals (ANN):
 
N
H H H P , ,... ,
2 1 0
=
 
is randomly initialized. The 
proper choice of the N’s value is very important as this 
parameter significantly affects the speed of the algorithm 
and its selection is critical for real-time systems.  
Each individual in the population at the same time 
gets its own unique description, encoded in the 
chromosome
 
Lj j j j
h h h H ,... ,
2 1
=
, 
which consists 
of L gene, wherein  
max min
w w h
ij
 - i-th value of j-
gene chromosome (
min
w - the minimum and 
max
w - 
maximum allowable values, respectively). 
Figure 1 shows an example ECMAC chromosome’s 
format and the correspondence between genes and 
network parameters stored in the chromosome. It should 
be noted that chromosome length depends on the 
dimensionality of the problem and the maximum amount 
of memory. 
As seen from the drawing, it consists of a 
chromosome gene in which information about 
corresponding network parameters is stored. At the 
beginning of the chromosome, there are genes that 
contain information about the parameters of the noise 
and they are active only in case of the noisy 
measurements. Next gene’s block encodes the number of 
levels and the quantization steps, the shape of the 
receptive field of neurons and type of algorithm that is 
used for hashing information. 
Due to the large amount of the BF that can be used in 
CMAC, there is a special gene in its chromosome BF, 
that is responsible for coding the type of the used 
functions. There is also a gene H in the chromosome, that 
encodes a type of the hashing algorithm (If its value is set 
to 0 then hashing is not used). 
Then, in the chromosome, there is a group of genes 
encoding weighting parameters directly relevant to the 
associative neurons. During the initialization phase, 
initial values are assigned to all these parameters by 
using a random number generator. 
Since during evolution mutation may occur in the 
parameters affecting the amount of used associative 
neurons, the length of the chromosome can vary. The use 
of variable length chromosomes occur individuals with 
specific genetic code segments (introns) which are not 
used for coding characteristics [28-29]. 
Typically, introns are used in the EA: 
- as noncoding bits that are uniformly added to the 
genetic code (in this case, introns only fill the space 
between the active genes of the chromosome); 
- as the nonfunctional parts of the genetic code, i.e., 
parts of the decision which do not actually do anything, 
thus not affect the fitness of the chromosome (this 
usually occurs in the genetic programming and in the 
chromosomes, which are subject to the cycle of 
development after birth); 
-as posterior useless part of the chromosome, which 
do not participate in the calculation of its fitness (usually 
it manifests itself in some types of competitive-trained 
neural networks, in which only neurons-winners in 
contrast to other neurons that are a posteriori useless 
affect network performance results). 
Introns appear in other types of neural networks, 
where they are called potentially useful waste. 
The control of introns amount in the population 
carried out by: 
- the use of special operators, that alter the length of 
the chromosome and add or remove introns 

Evolving Neural Network CMAC and... Informatica 43 (2019) 291–298 295 
(experimental results show that some number of introns 
improves overall properties EA); 
- the use of the selection operator: depending on the 
number of introns, the value of individual fitness 
function will increase or decrease. 
Once the initial population is formed, the fitness of 
each individual part in it evaluates by some defined 
fitness function. 
Conventionally, as such a function the quadratic one 
is used: 
( )
2
1
*
) ( ˆ ) (
1
) (

=
− =
M
i
i i
x y x y
M
x F
,           (17) 
where 
) (
*
k y
 - the desired network response; 
) ( ˆ k y
 - 
real output signal; M – sample size. 
The next step is the selection of individuals, the 
chromosomes of which are involved in the formation of 
the new generation, and subsequent hybridization.  
The task of crossing operator (crossover) is the 
transfer of genetic information from the parent 
individuals to their offspring.  
After completion of the operator’s work, any gene of 
any individual in the new population may mutate, i.e. 
change its value. 
Since chromosome uses hybrid coding, during the 
mutations various operations must be performed for 
different encoding methods.  
For example, in the case of the gene that is 
responsible for neuron’s activation and uses binary 
encoding, inverse mutation should be used.  
For coding the BF and weighting parameters, that 
uses real values, different types of mutations may be 
used. 
Thus ECMAC algorithm can be represented as 
follows: 
- create an initial population (initialization of each 
individual chromosome, estimation of the initial 
population);  
- the stages of evolution - the construction of a new 
generation (selection of candidates for mating /breeding, 
hybridization, i.e. causing by each pair of selected 
candidates some new individuals, mutation, evaluation of 
the new population); 
- check the completion criterion, if not satisfied - go 
back to the stages of evolution. 
7 First Modeling Experiment 
It is considered the problem of nonlinear dynamic object 
identification that is described by the equation: 
22
10 ( ) 50 ( 1)
( ) max ; ;
u k y k
y k e e
− − −

=

 
( )
22
5 ( ) ( 1)
; 1.25 ( ),
u k y k
ek 
− + −

+


         (18) 
where u(k) - the input signal, that is a stationary random 
sequence with the random uniform distribution in the 
interval [-1, 1]. 
During the study of this object, the population of 
CMAC networks consisting of 100 individuals was used.  
The population evolved during 500 epochs. All 
configurable network parameters (including R and ρ) 
were determined by EA.  
It should be noted that the values of R and ρ 
determine the amount of memory that is used for storing 
network parameters and significantly affect the accuracy 
of the approximation.  
Graphs of the network-winner fitness function and 
the required amount of memory to store its parameters 
are shown in Figures 2 and 3. 
 
Figure 2: The network-winner fitness function. 
 
Figure 3: Required amount of memory. 
Results of the stationary object (18) identification are 
shown in Figures 4 and 5. 
So, Figure 4 shows the surface itself, according to 
the equation described, and Figure 5 - the surface 
recovered by ECMAC with ξ (k) = 0.  
The winning network comprises 814 weighting 
parameters with R = 186 and ρ = 95, and rectangular 
activation function was chosen. 
Figure 6 shows the results of the object (18) 
identification in the presence of the random noise ξ (k) 
that is normally distributed in the interval [-0.3, 0.3].  
In this case, the winning network used cosine 
activation functions (10). 
) (k y
) (k u ) 1 ( − k y
k
) (k y
) (k y
) (k y
k
k

296 Informatica 43 (2019) 291–298 O. Rudenko et al.  
8 Second Modeling Experiment 
We solved the problem of the identification of a dynamic 
object described by the equation: 
 
22
( 1) ( 2) ( 3) ( 4)( ( 3) 1)
()
1 ( ) ( 2)
y k y k y k y k y k
yk
+u k + y k
− − − − − −
=+
−
 
22
()
1 ( ) ( 2)
uk
+u k + y k
+
−
          (19) 
To solve this problem, we used a population of 
evolving CMAC networks comprising 250 individuals.  
After reaching the required accuracy of 
identification, to assess the quality of the resulting 
model, to the object and the winner network the same 
control actions were given:   
a) ( )=sin 200 cos 400 ; u k ( k / )+ ( k / )  
b) ( )= 1 5 0 001 u k . + . k − . 
The experimental results for the cases a) and b) are 
shown in Figure 7 and 8.  
The solid line shows the output signal of the object 
and the dashed - neural network model output.  
 
 
Figure 7: The experimental results for the a)-case. 
 
Figure 8: The experimental results for the b)-case. 
As can be seen from the simulation results, the 
accuracy of identification of a multidimensional object 
(19) via evolving networks CMAC is sufficiently high 
and error – is inessential. 
In Figure 8 there are some minor oscillations that 
appeared due to rounding off calculations. 
9 Third Modeling Experiment 
It is considered the problem of controlling a 
multidimensional nonlinear dynamic object described by 
the following equations: 
 
 
Figure 4: The surface itself, according to the equation 
described. 
 
Figure 5: The surface recovered by ECMAC. 
 
Figure 6: The results of the object (13) identification. 
u(k) 
y(k-1) 
u(k) 
y(k-1) 
y(k) 

Evolving Neural Network CMAC and... Informatica 43 (2019) 291–298 297 
). 1 (
) 1 ( 1
) 1 ( ) 1 (
) (
) 1 (
) 1 ( 1
) 1 (
) (
2
2
2
2 1
2
1
2
2
1
1
− +
−
− −
=
− +
−
−
=
k u
k +y
k y k y
k y
; k u
k +y
k y
k y
      (20) 
Uncorrelated random sequences with a uniform 
distribution law in the interval [–1, 1] were used as inputs 
of the network during the training.  
Training was carried out on the basis of the 
presentation of a network of 10000 training pairs and the 
following parameters of the CMAC neural network: 
activation functions - trigonometric; R=100; ρ=40.  
The required memory capacity for such parameters is 
5818 memory cells. The size of the population was 300 
individuals.  
The required values of the output signals were set in 
the following way: 
 



= +
=
=
=
. 1000 , 501 ), 500 / sin( ) 300 / sin(
; 500 , 1 , 5 . 0
) (
); 100 / sin( ) (
*
2
*
1
k k k
k
k y
k k y
 

(21) 
The results of the neural control are shown in 
Figures 9 and 10.  
 
Figure 9: The experimental results for the y1(k) 
In all these figures, the dotted line shows the 
required output signal ) k ( y
*
i
, solid line – real ) k ( y ˆ
i
, 
and the line with the circles - the corresponding change 
of the control signal u i(k) (i=1,2). 
 
Figure 10: The experimental results for the y2(k). 
10 Conclusions 
The results showed that the evolving neural network 
CMAC is quite effective and convenient in solving 
practical problems (identification of nonlinear objects, 
control, etc). 
Substantial savings of required memory in 
combination with evolutionary training algorithms make 
it particularly attractive for implementation in real 
complex dynamic object control systems in the presence 
of noisy measurements. 
An additional advantage of the evolutionary 
approach to CMAC network training is the solution of 
the problem of choice the receptive field form of the 
associative neurons that is affecting the method and the 
accuracy of the approximation of the studied functions.  
In the case ECMAC this problem is solved 
automatically.  
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