identification of dynamic systems

1. IDENTIFICATION OF BENCHMARK DATA USING
ARX AND BOX JENKIN MODELS
1 1 2
A. Moufid , D. Filahi , A. Naitali
(1) Student at Mohammed V University, ENSET, Rabat, Morocco
(2) Mohammed V University, ENSET, Rabat, Morocco, (IEEE Member)
Abstract - This paper is a comparison between two estimated models, that’s ARX and
Box-Jenkins using benchmark data, the first step is to choose the best parameters of every
one the second is to choose the best model that describe perfectly the system.
Keywords : Benchmark, system identification, ARX, Box-Jenkins
1. INTRODUCTION
The term benchmarking means to compare with a
reference, so we have to compare the capacity to
identify and modeling benchmark data ,that present a
raw material to test deferent methods of identification
and modeling.
The strategy is to consider the system as a black box,
so we use only the data of input and output.
So we change the order of the models, and apply it to
benchmark’s data that divided on tree ranges :
Data estimation [1 : 60000]
Data validation [61000 : 120000]
Data test [121000 : 180000]
2. PRESENTATION OF THE
BENCHMARK
The term benchmark mean a model. so it’s a result of
medialization for nonlinear system. the objective is to
compare different method of identification nonlinear
process but not really create a competition. so with a
few words the goal is to more understand the
compabilities of modelling strategies.
3. THE SYSTEM USED IN SIGNAL
ACQUISITION OF INPUTS AND
OUTPUTS
To profit more information about inputs and outputs
we have to load and save the form of signals in order
to analyses them so the system used in acquisition is
the card HPE1433A.
The HPE1433A is a system of acquisition data
realized by HP company so we see in the user guide
of the card principally :
The method of running and making measurement
installation of software libraries.
To communicate with the hardware it is necessary to
use the interface c-language host interface library.
Visa is the input output standard upon which all the
VXI plug and play software component are based.

2. 4. PRESENTATION OF THE
APPROCHE IDENTIFICATION
In this part we present tow methods, the first one is
ARX model the second is the Box-Jenkins.
A. The identification using the ARX
Auto Regressive model with eXternal inputs noted
ARX it suppose the existence of an input of control
and a model that describe noises with an average equal
zero.
Also the model include a temporal delay that
represented in the following equation
)()()()()( 11
tetuqBtyqA  
.
y(t) output of processes, u(t) command applied to the
system e(t)noise, A and B are polynomials with q-1
respectively presents degrees na and nb.
Structure of Nonlinear ARX Models
This block diagram represents the structure of a
nonlinear ARX model in a simulation scenario:
The class : of the arx model is considered as linear
system.
Estimation algorithm for nonlinear ARX models: the
ARX model is based on the ls algorithm named last
square.
B. The identification using the Box-Jenkins
In time series analysis, the Box-Jenkins method
applies autoregressive moving
average ARMA or ARIMA models to find the best fit
of a time-series model to past values of a time series.
So we have to choose the parameters [nb nc nd nf nk]:
That compose a vector of matrix contain commands
and delays of model Box-Jenkins.
The matrix will be composed by numbers strictly
positive
Mathematically we express the Box-Jenkins model by
the following relationship.
Algorithm of Box-Jenkins method
fig.1: Algorithm of Box-Jenkins method
5. RESULT AND DISCUTION
In the figure fig.2 we subdivide benchmarks data into
tree ranges.
The green color means data estimation ;
Red color means data validation ;
Blue color means data test.
fig.2: Presentation of Data
The figures fig.3 and fig.4 present the results of
identification in term of correlation’s percentage (Fit)
.applied to estimation and validation data using the
ARX model with different parameters.

3. fig.3: Data Fit
Remark : we note that the quality of model depend
to parameters na ,nb and nk .
So we have the max fit using values (na,nb,nk)=(6,6,3)
That present a fit 76,32 and 76,36 respectively in
estimation and validation .
Box-Jenkins So we initiate the vector by [nb, nc, nd,
nf, nk]= [2 2 2 2 1]
And varying parameters as it shown in figure
fig.4: varying parameters
We remark that the parameters that present the best
fit is [9 9 7 4 1].
In the other hand that concern the phase of validation.
We change the data from data estimation to data
validation.
Similarly we initiate with [2 2 2 2 1].
Increasing different parameters the results are
organized in the figure.
fig.5: Increasing different parameters
As it shown the parameters that give the best fit is [9
8 7 6 1].
6. CONCLUSION
The best conclusion can’t be better than the remark of
Mr. Leonard that to choose between two models the fit
is not the principle information to analyze .but we have
to look and analyze the flexibility of model .that
described mathematically by the complexity C, in this
paper both models ARX and box Jenkins are given
similar values of fit but we must make on
consideration that the fit and the complexity evolve
contradictory .it’s the art of identification.
In our case the arx model have presented same fit with
low order equal six, compared with Box-Jenkins that
going to the order ten.
As a future researchers we have a desire to more
discover this science by developing new techniques
and algorithm that must a good subject of research.
REFERENCES
[1] J. Schoukens , J. Suykens , L. Ljung : WIENER-
HAMMERSTEIN BENCHMARK.
[2] L. Ljung. System Identification Toolbox: Getting
Started Guide.
Http:www.mathworks.com/help/pdf_doc/ident/ident_
gs.pdf, 2013.