Deep Neural Network System Identification for Servomechanism System
MOHAMED. A. SHAMSELDIN
Mechanical Engineering Department, Future University in Egypt, Cairo, EGYPT
Abstract: This paper presents a systematic technique for designing the input signal to identify the one-stage
servomechanism system. Sources of nonlinearities such as friction and backlash consider an obstacle to
obtaining an accurate model. Also, most such systems suffer from a lack of system parameters data. So, this
study establishes a model using the black-box modeling approach; simulations are performed based on real-
time data collected by LabVIEW software and processed using MATLAB System Identification toolbox. The
input signal for the servomechanism system driver is a pseudo-random binary sequence that considers violent
excitation in the frequency interval and the output signal is the corresponding stage speed measured by rotary
encoder. The candidate models were obtained using linear least squares, nonlinear least squares, and Deep
Neural Network (DNN). The validation results proved that the identified model based on DNN has the smallest
mean square errors compared to other candidate models.
Keywords: Deep Neural Network (DNN), Harmony Search (HS), Servomechanism
Received: June 28, 2021. Revised: March 19, 2022. Accepted: April 16, 2022. Published: May 11, 2022.
1. Introduction
The newest growth of machine tools is to realize high-
speed spindle and feed drives, which causes improvement
in the performance and reduces the machining cycle times.
Besides, the design of feed drives with satisfying dynamic
response and high performance now is considered
necessary in several industrial applications.
Most of the real engineering systems are nonlinear
systems but, the nonlinearity characteristics change
according to the complexity degree of the system. It is
known that the mechanical systems have high nonlinear
behavior and parameters uncertainty because of the
friction phenomenon and the backlash of gear trains and
mechanical parts. For example, the one-stage
servomechanism system which considers a base unit of
CNC machines. So, the traditional mathematical model
cannot achieve accurate presentation for this type of
system. Experimental identification is a well-recognized
methodology to obtain a precise process model, often
intended for control but also for other purposes. The nature
of the input signal for the identification has a great effect
on the accuracy of the model. But, in many cases, the input
signal cannot be easily selected with respect to plant
behavior constrictions. Pseudo-Random Binary Sequences
(PRBS) are often used as violent excitation signal for
system identification, it has a finite length that can be
synthesized frequently with simple generators while
presenting favorable spectra for identification.
The family of candidate models for system
identification can be classified along with several different
aspects. The linear and nonlinear theory has been well
developed and investigated to real applications through
recent years. The linear structure is used to simplify the
analysis where the parameters are constant and do not
vary throughout a simulation, such as autoregressive with
exogenous inputs (ARX) model. In contrast, a non-linear
model presents dependent parameters that are permitted to
vary throughout the period of a simulation run, and its use
becomes necessary where interdependencies between
parameters cannot be considered insignificant.
Artificial neural networks (ANN) trained by back-
propagating error derivatives have the possibility to
develop much better models of data that has nonlinear
behavior. In recent researches, improvements in both
computer hardware and machine learning algorithms have
led to more effective approaches for training deep neural
networks (DNNs) that contain several layers of non-linear
hidden units and a very large output layer known as the
deep learning algorithms.
Lately, deep learning has been become attractive and
has significant attention from a wide range of engineering
applications. Compare to the traditional neural networks,
the vital features of deep learning are to have more hidden
layers and neurons and to improve learning performance.
Using these features, complex and large problems that
could not be solved with traditional neural networks can
be resolved by deep learning algorithms. Therefore, deep
learning has been subjected to various applications
including pattern recognition and classification problems;
for example, handwritten digit recognition [4], speech
recognition [3], human action recognition [5], and so on.
However, to the best knowledge of the authors, no result
has been published in the system identification and
automatic control field. Thus, this paper focuses on
presenting the applying possibility of deep learning in
system identification areas.
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125
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This paper investigates several candidate models for one
stage servomechanism system based on linear and
nonlinear least squares and deep neural networks. The
actual input/output data is collected by NI Data
Acquisition (DAQ) Card 6009 using LABVIEW software.
The input signal is a Pseudo-Random Binary Sequences
(PRBS) while the output signal is the corresponding stage
speed measured by the optical encoder. Comparison
between the candidate models is implemented to select the
best between them.
The paper is ordered as follows: Section II illustrates
the system modeling. The system identification techniques
is involved in Section III. Section IV shows the validation
results. Lately, Section V presents the conclusion.
2. Mathematical Model
It is well-known that several servomechanism system
parameters cannot be determined easily. Consequently,
most of the researchers neglect those parameters which
will cause errors in the design of their own controllers. The
structure of the lead-screw-driven stage is shown in Fig.1.
The stage is stayed by linear guides on a bed. The driving
mechanism contains a DC motor, coupling, screw shaft,
and nut. The DC motor rotates the screw shaft while the
nut changes the movement from rotation to linear motion
and drives the stage. The stage speed and position are
measured through a rotary encoder coupled with a screw
shaft.
Fig. 1. The structure of the lead screw drive system.
A four-inertia model of a lead screw driving system is
shown in Fig. 2. This model considers each component as
a lumped constant, and the four moments of inertia of the
components are connected by stiffness and damping
parameters. The four inertia parameters are the mass of the
screw shaft (), the mass of the stage (), the moment
of inertia of the rotor (), and the moment of inertia of the
screw shaft (), The moment of inertia of the rotor and
that of the screw shaft (which are connected by the
torsional stiffness of the coupling) is rotated by the motor
torque, where and represent a motor angle and
screw-shaft angle respectively. The motion of the screw
shaft in the x-direction is taken into account, and the mass
of the screw shaft has stayed on the axial stiffness of the
screw shaft and its bearing, where represents the gap
between the screw shaft and nut (backlash). The
parameters details in the four-inertia model are defined in
Table I.
Fig. 2. Four-inertia model of lead screw driven stage.
The linear motion of the stage is generated by rotating
the screw shaft through the nut, and stands for
the position at which the nut is mounted. The is a
transformation factor of the lead screw. The mass of the
stage and the nut mounting position are linked through the
axial stiffness of the nut, and denotes the stage position.
The overall system parameters are verified in table 1.
Table 1. Model Parameters
No.
Parameter
Unit
1
kg·m2
2
kg·m2
3
kg
4
kg
5
N·m /
rad
6

N / m
7
N / m
8
N·m /
(rad / s)
9
N·m /
(rad / s)
10
N / (m /
s)
11
m / rad
12
N·m /
(rad / s)
13

N / (m /
s)
14
N / (m /
s)
15
rad
16
rad
17
m
18
m
19
N.m
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The model differential equations can be described as
follows:
󰇘 󰇗 󰇗 󰇗 󰇛 󰇜 (1)
󰇘 󰇗  󰇗 󰇗 󰇗 󰇛
󰇜 󰇗 󰇗 󰇛 󰇜 (2)
󰇘  󰇗   󰇗 󰇗 󰇗
󰇛 󰇜 (3)
󰇘 󰇗 󰇗 󰇗 󰇗 󰇛
󰇜 (4)
3. System Identification
&ROOHFWLQJ,QSXW2XWSXW'DWD
This section demonstrates the main parts of the one-stage
servomechanism system. Also, it shows the open-loop
performance of the servomechanism system. Moreover, it
demonstrates the system identification techniques and the
validation results. Fig. 3 illustrates the main components
of one stage servomechanism experimental setup which
consists of a DC Motor Electro-Mechanical Module. The
stroke of the stage ranges from 0 to 9 Inches. The DC
motor has a nominal speed of 1800 rev/min, and an
armature voltage of 90 V dc. The Optical Encoder is an
add-on that provides position feedback signals (100 pulses
per revolution). Two magnetic limit switches detect when
the sliding block reaches the start or end position. The DC
Motor Drive controls the DC Motor Electro-Mechanical
Module. This versatile drive also allows an external signal
to control the motor speed. A data acquisition card (DAQ)
NI USB-6009 is connected to the computer that is used to
perform the control algorithms.
Fig. 3. The one Stage Servomechanism Experimental Setup.
The main idea of the program has been designed to
make the NI DAQ 6009 generates an analog output signal
(-5V to 5 V) to the linear amplifier. Also, the analog output
signal from the optical encoder has been collected at the
same time. The speed of the DC motor will fluctuate when
the generated signal change continuously. The positive
signal will cause the DC motor speed to fluctuate in the
forward direction, while the DC motor will fluctuate in the
reverse direction through the negative voltage ranges. The
shaft of the optical encoder is coupled with the lead screw
shaft to measure the speed and position of the stage as
shown in Fig. 4.
Fig. 4. The block diagram of experimental setup servomechanism
system.
Fig. 5 demonstrates The PRBS output signal of the
NI DAQ card and input to the DC motor driver while Fig.
6 shows the corresponding linear speed of the stage
measured by the optical encoder.
Also, Fig. 7 displays the actual position of one stage
servomechanism through the experiment.
It can be noted that the stage position increases in
positive ranges of the input signal entering to DC motor
driver while the actual stage position decreases in negative
ranges of the input signal. The input /output data will be
collected and stored in an excel sheet file and then this data
will be used to develop identified model for the
experimental setup.
Fig. 5. The PRBS output signal of NI DAQ card and input to DC motor
driver.
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Fig. 6. The output linear speed of the servomechanism table.
Fig. 7. The corresponding position of the servomechanism table.
The general linear transfer function can be expressed as
follows:
󰇛󰇜
󰇛󰇜 
 (5)
In equation (5),  are the estimated
parameters of the approximate transfer function (T.F). The
y(s) stands for linear stage speed while u(s) is the input
voltage to DC motor drive.
The accuracy of the transfer function improves
significantly when the system degree is increased.
However, often there is a restriction that increasing order
cannot make the model accuracy sufficiently [25].
Therefore, it is necessary to explicitly add the
nonlinearities into the system.
As the servomechanism systems suffer from uncertainty
and complex nonlinear dynamics, the nonlinear ARX
(NLARX) model structure has been developed to model
such systems [26].
The NLARX model consists of several complicated
nonlinear functions which can model the backlash and
friction in servomechanism systems significantly.
An NLARX model can be understood as an extension
of a linear model. A linear SISO ARX model has this
structure [27].
󰇛󰇜 󰇛 󰇜 󰇛 󰇜 󰇛 󰇜
󰇛󰇜 󰇛 󰇜 󰇛  󰇜
󰇛󰇜(6)
where u is input, y is output,  is the number of past
output terms and  is the number of past input terms.
This structure can be extended to create a nonlinear
form where instead of the weighted sum that represents a
linear mapping, the NLARX model has a more flexible
nonlinear mapping function [28].
󰇛󰇜 󰇛󰇛 󰇜 󰇛 󰇜 󰇛 󰇜 󰇛󰇜 󰇛
󰇜 󰇛 󰇜 󰇜 (7)
where f is a nonlinear function (to simulate the behavior
of friction and backlash existing in servomechanism
systems). Inputs to f are model regressors as shown in the
following Fig. 8.
Fig. 8. The NLARX model structure [28].
The obtained input /output data was used to develop
a set of linear and nonlinear identified models to select the
best between them. This task can be carried using the
MATLAB System Identification Toolbox. The input u is
the input voltage to the DC motor driver. The output y
contains the corresponding output linear velocity of the
stage. The sampling interval is 50 milliseconds. Two
models had been obtained, the first identified model is a
linear second-order system while the second identified
model is an NLARX model.
The linear second-order system has the following
specifications.
󰇛󰇜
󰇛󰇜 
 (8)
By matching with equation (1), the estimated
parameters of the linear approximation transfer function
are     
 
On the other hand, the obtained discrete-time ARX
model has the following structure.
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󰇛󰇜󰇛󰇜 󰇛󰇜󰇛󰇜 󰇛󰇜 (9)
where 󰇛󰇜   
󰇛󰇜  
By corresponding with equation (2), the polynomial
orders are  and  . Also, the used nonlinear
function is a wavelet with 25 units.
Artificial Neural Networks (ANN) had been proposed
in 1943. The ANN has been used to solve different types
of problems such as robotic processing, object recognition,
speech and handwriting recognition, and even real-time
sign-language translation. Despite the intuition, that
deeper architectures would yield better results than the
then more commonly used shallow ones, experimental
tests with deep networks had found similar or even worse
results when compared to networks with only one or two
layers. The Latest deep architectures use several modules
that are trained separately and stacked together so that the
output of the first one is the input of the next one. There
are two main types of ANN, the first type is the shallow
neural network and the second type is the deep neural
network. Deep learning has several layers of hidden units
and it also permits many more factors to be used before
over-fitting occurs. Thus, for deep learning, a deep
architecture is used as shown in Fig. 9.
Fig. 9. Deep Neural Network.
The neural network structure in this study contains three
layers the first layer is the input layer which receives the
input PRBS signal to the DC motor driver. The second
layer is the hidden layer that contains several hidden
neurons and receives data from the input layer. The third
layer is the output layer which presents the corresponding
linear stage speed.
󰇛 󰇜 Input vector applied to the
layer, the whole of the hidden neuron input is:



(10)
󰇛 󰇜 (11)
Such as 
is the input vector applied to the layer, and  is the
weights of 󰇛󰇜 input neuron connection, and
represent
the bias of hidden layer neurons.
The neurons of the hidden layer can be written as
follows:
󰇛

󰇜 (12)
The output as:
󰇛

󰇜 (13)
With
: bias of neurons output layer.
Dynamic neural networks are good at time-series
prediction. For instance, the NARX networks can be
developed in open-loop form, closed-loop form, and
open/closed-loop form. Using the Neural Network Time
Series App in MATLAB. The first step, open the Neural
Network Start GUI with this command: start. There are
many selections such as Input-output and curve fitting,
pattern recognition and classification, clustering, and
dynamic time series as shown in Fig. 10.
Fig. 10. Neural Network start GUI.
The second step, select Time Series App to open the
Neural Network Time Series App. The NARX model will
provide better predictions than this input-output model,
because it uses the additional information contained in the
previous values of 󰇛󰇜 so, the first type will be chosen as
displayed in Fig. 11. This form of prediction is called
nonlinear autoregressive with exogenous (external) input,
or NARX.
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Fig. 11. Neural Network time series GUI.
The toolbox then provides a dialog box to define the
network architecture. By default, the toolbox proceeds
with a single hidden layer with a sigmoid activation
function and a linear activation function for the output
layer as shown in Fig. 12. This network also uses tapped
delay lines to store previous values of the x(t) and y(t)
sequences. Note that the output of the NARX network,
y(t), is fed back to the input of the network (through
delays), since y(t) is a function of y(t – 1), y(t – 2), ..., y(t
– d). For this study, a hidden layer with 10 neurons was
decided based on a literature study and multiple iterations.
Fig. 12. Neural Network time series Architecture
Select a training algorithm from the following window
as shown in Fig. 13, then click Train. Levenberg-
Marquardt (trainlm) is recommended for most problems,
Levenberg-Marquardt Algorithm that trains a neural
network 10 to 100 times faster than the usual gradient
descent backpropagation method. It always calculates the
approximate Hessian matrix, which has dimensions n-by-
n.
Fig. 13.Neural Network time series training algorithm.
The toolbox by default divides the data randomly into
three categories: Training (70%), Validation (15%), and
Testing (15%). The training data, as the name suggests, is
used to train the network through an iterative process while
the weights are adjusted during backpropagation based on
the error calculated after each epoch. The validation data
is used to determine if the network has achieved the
required generalization and decides if the training can be
stopped and overfitting is avoided. The testing data is used
as an independent measure to evaluate the prediction
performance of the trained neural network. It provides a
measure of the mean squared error and R-value to assess
how well the network has trained. The network is now
ready to be trained.
Several procedures exist for estimating the network
parameters. In neural network literature, the algorithms are
called learning or teaching algorithms, in system
identification, they belong to parameter estimation
algorithms. The most well-known is back-propagation and
Levenberg-Marquardt algorithms. Backpropagation is a
gradient-based algorithm, which has many variants.
Levenberg-Marquardt is usually more efficient but needs
more capable computer memory. This algorithm does not
require the calculation of the Hessien, it only approximates
it.
(14)
The jacobian is calculated by
(15)
Where J is the jacobien matrix, which contains the
first derivatives of the network errors with respect to the
weights and biases, and (e) is a vector of network errors
is given by:
󰇛
󰇜
 (16)
Where
is the desired output.
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The jacobian matrix can be computed through a
standard backpropagation technique that is much less
complex than computing the hessian matrix. The adjust
weights as:
 󰇟 󰇠 (17)
Where 󰇛 󰇜 
When the scalar is zero, this is just Newton’s
method, using an approximate Hessian matrix, when is
large, this becomes gradient descent with a small step size.
Newton’s method is faster and more accurate near an error
minimum, so the aim is to shift towards Newton’s method
as quickly as possible. Thus, is decreased after each
successful step (reduction in performance function) and is
increased only when a tentative step would increase the
performance function. In this way, the performance
function will always be reduced at each iteration of the
algorithm.
If 󰇛󰇜
󰇛󰇜 󰇛 󰇜 (18)
If  󰇛󰇜
󰇛󰇜 󰇛 󰇜 (19)
Algorithm of Levenberg-Marquadt:
Stage 1: to initialize the weights and skews by small
value random as well as the Parameters: m =
0,001(default value), m - dec= 0,1 (default value), m - Inc
= 10 (default value),
Stage 2: To present the vector of input and desired output.
Stage 3: To calculate the output of the network by using
the expression (12) & (13).
Stage 4: To calculate the error of output (16)
Stage 5: To calculate the error in the hidden layers
Stage 6: To calculate the gradient of the error compared
to the weights
Stage 7: To calculate Hessien approximated by using the
expression (15).
Stage 8: To adjust the weights according to the
expression (17).
Stage 9: One test: by using the expression (18) & (19).
Stage 10: If the condition the error or the iteration count
is reached or m reached m-max, outward journey at stage
11 if not outward journey at stage 2.
Stage 11: End
Fig. 14 demonstrates the linear speed of the actual
experimental setup and the candidate models. It is noted
that identified model based on DNN can simulate the
behavior of the actual experimental setup compared to
other identified models.
Fig. 14. The linear speed of one stage table servomechanism for actual
experimental setup and identified models.
Table 2 demonstrates the mean square error of candidate
models. It can be noted that identified model based on
NARX neural network model has the minimum error
compared to other identified models.
Table 2. Mean square error of each identified model.
MSE
Estimation to fit
data
Model type
0.0499
79.624%
Identified linear model
0.00096
91.5 %
Identified NAR Model
0.00071
93.3%
Identified NAR neural
network model
4. CRQFOXVLRQ
This paper shows the practical design steps of an identified
model for one stage servomechanism drive system. The
influence of nonlinear dead-zone caused by friction,
backlash, and external disturbance is an obstacle to
developing a mathematical model. So, we resort to
constructing a nonlinear identified model by collecting the
experimental input/output data and entering it into the
MATLAB system identification toolbox. The obtained
identified models were validated by comparing them with
the actual input /output data. Also, a comparison between
the candidate-identified models was implemented to
investigate the validity of each model. The experimental
and validation results provide that the identified model
based on NARX deep neural network has the best
performance with respect to other identified models.
DESIGN, CONSTRUCTION, MAINTENANCE
DOI: 10.37394/232022.2022.2.18
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BIBLIOGRAPHY OF AUTHORS
Dr. Mohamed.A. Shamseldin born in
Cairo, Egypt, on October 1, 1987. He
received the B.Sc. degree in
mechatronics engineering in 2010
from faculty of engineering at
Helwan, Helwan University, Cairo,
Egypt. On December 2012, he
received his work in faculty of
engineering and technology at Future
University in Egypt as an instructor in Mechatronics Engineering
Department. He obtained M.Sc. in system automation and
engineering management (2012 to 2016) from, Helwan
University, Cairo, Egypt. In 2020, he obtained the Ph.D. in
Mechatronics Engineering from faculty of engineering, Helwan
University, Cairo, Egypt. Mohamed was visiting staff at
University of Central Lancashire, Preston, UK. His research
activity includes studying Artificial Intelligent techniques,
electrical machines speed control and robot control.
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DESIGN, CONSTRUCTION, MAINTENANCE
DOI: 10.37394/232022.2022.2.18
Mohamed. A. Shamseldin
E-ISSN: 2732-9984
132
Volume 2, 2022