Modeling a production system by parametric identification approach

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Modeling a production system by parametric identification approach

Conference Paper · March 2015

DOI: 10.1109/ICoCS.2014.7060882

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Modeling a production system based on parametric identification approach

RACHAD Sofia

, FOURAIJI Hicham

PMMAT Laboratory

Faculty of Sciences Ain Chok, Hassan II University Casablanca, Morocco

1

Sofia.rachad@gmail.com

2

Hicham.fouraiji@gmail.com

BENSASSI Bahloul PMMAT Laboratory

Faculty of Sciences Ain Chok, Hassan II University Casablanca, Morocco

b.bensassi@fsac.ac.ma

Abstract— This paper refers to possibility of utilization a new method for modeling an industrial production system using the identification approach. The production system studied is considered as a black box, which does not require any knowledge of the internal parameters. Two structures of linear parametric models are outlined and compared i.e. Autoregressive with Exogenous (ARX) model and Autoregressive Moving Average with Exogenous (ARMAX) model. The best linear model is ARMAX, due to its higher percentage of best fit.

Keywords- Production system; system identification; ARX model; ARMAX model.

I. I NTRODUCTION

Manufacturing systems are becoming more complex to model, design and control, in a competitive evolving market.

Indeed, the study of a production system proves a very difficult task to perform, which requires very often to have a specific model that accurately describes its dynamic behavior.

These models can be derived directly from physical laws that govern the behavior of the system, but it is often impossible to obtain a priori knowledge complete and accurate of all model parameters. In this case, we use an only estimate behavioral model based on observed input-output production system, also known as modeling "black box".

The modeling approach that we are referring to here, is the identification approach. This is the approach that allows finding a mathematical model belonging to a class of

"universal" model, based on actual measurements taken from the studied system. [11]

In our context we are interested in identifying production systems based on parametric mathematical models. Before going further, let us focus on what offers literature in modeling production systems field. It is interesting to see what concepts are being developed now and of course what techniques or modeling approaches have been discussed. (K.Labadi, 2005) [9] presented a modeling and performance analysis of logistics systems based on a new model of stochastic Petri nets. This model is suitable for modeling flow evolves in discrete amounts (lots of different sizes). It also allows taking into account more specific activities such as customer orders, supply inventory, production and delivery batch model.

(N.Samata, 2011) [16], proposed a model of the global supply chain using Petri nets with variable speed. He transposed the concepts developed on the traffic towards the supply chain of production type. He also proposed a modular approach for modeling the different actors in the supply chain in the supply chain, based on Petri nets formalism with variable speed.

(F.Petitjean, 2008) [3], proposed a methodology for modeling global supply chain from an audit of the company. Then by using the UML model, he created a platform for simulation, and he also proposed the control principles for integrated supply chain. (B.Rohee, 2007) [1], used hybrid Petri Nets to propose an approach of offline simulation that supports multiple constraints (change control, time ENTERED developments, friction ....). Originality of his work lies in the fact that he proposed a simulation of continuous production and the study of interactions between the continuous model and the discrete data exchanged with the control part. This allows simulating and controlling the system without using the actual operative part. (H.Sarir, 2013) [5], proposed to model and to control the current stocks in the production lines by macroscopic analogy with the model for controlling a hydraulic tank. He used the automatic control concepts for controlling the currents stocks .

He also presented a production line model by using behavioral identification with transfer functions , he used the PEM algorithm for the construction of models and the simulation was performed on the Graphic user interface ( IDENT ) of MATLAB © [7] . (K. Tamani, 2008) [10], in turn, proposed a controlling products flow approach where he decomposed the system studied as elementary production modules. He proposed thereafter, controlling the flow through each module and production supervision based on fuzzy logic.

Based on this literature review, we observe that production systems represent a major focus for the Scientific Research, and there are various methods to model them. We also note that the parametric identification approach is rarely used for modeling manufacturing systems.

In this paper, we present a new method for modeling a

production system using the approach of identification.

II. DESCRIPTION OF PRODUCTION SYSTEM

In order to simplify, we propose to study the case of a production line consisting of three single-

(Mi), and with single-machine operation. We assumed that the flow of material is continuous; The system is in a operating state without failure. The production rate is identical with a negligible transfer time between machines.

Figure 1 illustrates the studied system of transformation line Type:

Thereafter, as shown in Figure 2, the production system is represented as a “black box”. Where the input u(t) and output y(t) have been extracted from the ERP of the company.

III. M ODELING METHOD

In this paper, the modeling method used is the identification approach of dynamic systems.

The identification approach, as shown in Fig. 3, is a processes that enable find a mathematical model belonging to a class of universal models. This model, exposed to test signals (input) gives a response (output) closest to the real system. [1]

The first step is to acquire the input-output data under experimental protocol; the number of measured points (data input-output) must be significant to complete the test: The applied signal input must be in high frequency. Thereafter, a model structure among existing universal models is selected.

We will be interested in the following on parametric models. In this case, we will compare the two parametric models ARX and ARMAX.

In the third step, we estimate the model parameters from experimental data using a specified estimation algorithm or a specific criterion to be minimized. In the end, the model is validated only after obtaining an appropriate model to represent the system.

Figure 1: Transformation Line

M2

u(t)

Figure 2: Black box modeling of the production

M1 M2

u(t) u(t)

Black box

CTION SYSTEM

In order to simplify, we propose to study the case of a -product machines machine operation. We assumed that the flow of material is continuous; The system is in a operating state without failure. The production rate is identical with a Figure 1 illustrates the studied system of transformation line

Thereafter, as shown in Figure 2, the production system is represented as a “black box”. Where the input u(t) and output y(t) have been extracted from the ERP of the company.

ODELING METHOD

In this paper, the modeling method used is the identification The identification approach, as shown in Fig. 3, is a set of processes that enable find a mathematical model belonging to a class of universal models. This model, exposed to test signals (input) gives a response (output) closest to the real system. [1]

output data under an experimental protocol; the number of measured points (data output) must be significant to complete the test: The applied signal input must be in high frequency. Thereafter, a model structure among existing universal models is selected.

interested in the following on parametric models. In this case, we will compare the two parametric models ARX estimate the model parameters from estimation algorithm or a ion to be minimized. In the end, the model is validated only after obtaining an appropriate model to represent

A. Collecting data

In this work, the input and output data of the production line are measured, collected and sent to a system identification process, as shown in Fig. 4.

In this work, the data is divided into two parts. The first part (blue) is used to estimate the system model and the second (red), is used to validate the model.

M3

y(t)

Figure 2: Black box modeling of the production system

M3

y(t) y(t)

Choose model structure Collecting data

Parameter model estimation

Validate model

Model accepted

Application of selected model Yes

Figure 3: The system identification loop [1]

Model Production line

System Identification Input

Figure 4: modeling approach by collecting

In this work, the input and output data of the production line are measured, collected and sent to a system identification In this work, the data is divided into two parts. The first used to estimate the system model and the second (red), is used to validate the model.

Choose model structure Collecting data

Parameter model estimation

Validate model

Model accepted

No

Application of selected model

Figure 3: The system identification loop [1]

Model Production line

System Identification Output

Figure 4: modeling approach by collecting data

Figure 5 shows the input and output data extracted from ERP of the company. The collected data 250 for each input- output sample are used for modeling purpose.

B. Sélection of the model structure

The principle of identification based on parametric models is to extract a mathematical model from measured data. The model must be capable of measuring the output of the system whatever the instant t, if the initial conditions of the system are known. For this, we can use the input values at present and previous times (u(t), u(t-1), ...) and the previous values of the output (y(t-1), y(t-2), ...) in the case of a regression model.

There are various kinds of parametric models (ARX, ARMAX, OE, BJ ...) [1].

The difference lies mainly in the modeling of the perturbation e(t) and the presence or not of an external input. In the following, we are mainly interested in the structure of the ARX model (Autoregressive with external input) and ARMAX (autoregressive moving average with external input).

The structure of the ARX model can be written as follows:

( ) ( ) = ( ) ( − ) + ( ) (1) Figure 6 describe the structure of the ARX model as below:

The ARMAX structure is similar to ARX structure, but with an additional term, which represents the moving average error.

The ARMAX model can be written in the following compact form:

( ) ( ) = ( ) ( − ) + ( ) ( ) (2) The signal flow of ARMAX model structure can be representing as:

Whether it’s the ARX or ARMAX model, A(q), B(q) and C(q) are polynomials to be estimated. For the ARX model, the polynomial C(q) = 1 .

These polynomials represent the overall system dynamics, where A(q), B(q) and C(q) are defined by:

( ) = 1 + + ⋯ +

( ) = + ⋯ +

( ) = 1 + + ⋯ +

u(t) and y(t) are respectively the input and the output of the system. Referring to our work, u and y are shown in figure 5, t is time unity and q represents the delay operator and e(t) is a white noise signal.

Variables a , b and c are model parameters to be estimated, with i = 1, … na, j = 1, … nb l = 1, … nc. nk is the pure delay model. The minimum value of nk is supposed to be equal to 1.

C. Parameter estimation

Parameter estimation is a step in which mathematical optimization is involved; it comes to choosing an estimating algorithm or a criterion to be minimized.

The main parameter estimation goal is to find the optimum values of parameters described in (3).

In this work, the estimated model is done by using the prediction error method (PEM). The prediction error is given by:

( , ) = ( ) − ( | ) (4) 1

u

e

y

Figure 6: The ARX model structure

Figure 7: The ARMAX model structure u

e

y

(3)

Figure 5: Input-Output Data

Where y(t)the observed output is y(t|θ output and θ is the parameter vector defined by:

= , … ,

, … , , …

Fig. 8 shows the principle of the parameters estimation method based on the prediction error method.

The measurement of the prediction error is often represented by a function, known also as the loss function. It can be written as:

( ) = 1

2 ²( , )

Minimization technique employed is the least squares algorithm (LSA).

D. Model validation

Model validation is often done by considering the highest best fit.

The model fit is often measured by the coefficient of determination R , it shows the cloand can be expressed by:

= 1 −

( )²

0 ≤ ≤ 1

Where is error between measured and desired output at time t. The R is usually represented in percentage. High percentage of R , means that the model is good.

IV. R ESULTS AND DISCUSSION

Among the existing types of linear parametric models, we chose to study and compare the ARX and ARMAX model structure to perform production system modeling. The results are shown below for ARX and ARMAX based on the best fit.

Figure 8: Principle of system identification based on the prediction error

u(t) y(t)

ŷ(t) Production

system

Model

Criterion

θ) is the predicted output and θ is the parameter vector defined by:

, (5) Fig. 8 shows the principle of the parameters estimation method based on the prediction error method.

The measurement of the prediction error is often represented by a function, known also as the loss function. It

(6)

Minimization technique employed is the least squares

ften done by considering the highest The model fit is often measured by the coefficient of

, it shows the cloand can be expressed by:

1 (7) is error between measured and desired output at

is usually represented in percentage. High , means that the model is good.

AND DISCUSSION

Among the existing types of linear parametric models, we mpare the ARX and ARMAX model structure to perform production system modeling. The results are shown below for ARX and ARMAX based on the best fit.

From the results of the two linear models, the most accurate model for this system is the ARMAX

highest percentage of best fit. The summary of model properties are shown in table 1

V. C ONCLUSION

The originality of this work resides in the projection of methods inspired from automatic field systems of production.

We proposed a “black box" modeling of a production line from the observed inputs and outputs.

To estimate the parameters of the studied system, we opted for auto-regressive parametric models.

The ARX and ARMAX models parameters were estimated using MATLAB and its System Identification Toolbox

Following the comparative study was conduct

ARX and ARMAX models, tests showed that the best linear model is the ARMAX 2211 because of its lower order and higher percentage of best fit.

model

Linear Polynomial

ARX

1 2 2 ARMAX

1 2 2 Figure 8: Principle of system identification based on the prediction error

Figure 9: Measured output based on ARX model fit and ARMAX model fit

TABLE I. S

- +

ARX

From the results of the two linear models, the most accurate model for this system is the ARMAX model as it exhibits the highest percentage of best fit. The summary of model

ONCLUSION

The originality of this work resides in the projection of methods inspired from automatic field systems of production.

We proposed a “black box" modeling of a production line and outputs.

To estimate the parameters of the studied system, we opted regressive parametric models.

The ARX and ARMAX models parameters were estimated using MATLAB and its System Identification Toolbox

Following the comparative study was conducted for the ARX and ARMAX models, tests showed that the best linear model is the ARMAX 2211 because of its lower order and

Linear Polynomial Parameters

Best fit R² (%)

1 - 1 22,43%

1 - 1 43,09%

2 - 1 90,62%

1 1 1 44,62%

2 1 1 92,43%

2 2 1 89,95%

Figure 9: Measured output based on ARX model fit and ARMAX model fit

UMMARY OF LINEAR MODEL PROPERTIES ARX

ARMAX

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