Investigation of ARX Model on Partial Input-Output Data in Heating Process

Nurlaila Ismail Faculty of Electrical Engineering

UiTM Shah Alam, Selangor, Malaysia nrk_my@yahoo.com

Mohd Hezri Fazalul Rahiman Faculty of Electrical Engineering

UiTM Shah Alam, Selangor, Malaysia hezrif@ieee.org

Mohd Nasir Taib Faculty of Electrical Engineering

UiTM Shah Alam, Selangor, Malaysia dr.nasir@ieee.org

Abstract—This paper presents an investigation of ARX model on partial input-output data from heating process in a steam distillation essential oil extraction (SDEOE) system. The study was performed by applying system identification and it was carried out by using four general stages; experimental work to collect input-output data, model structure selection where ARX model is chosen, estimation to obtain the ARX model, and validation. The result from this investigation showed that the ARX model obtained by the partial input output data is sufficient and adequate, i.e. the model behaviour of heating process is closed to the real system performance.

Keywords-ARX; modeling; heating process; essential oil extraction

I. INTRODUCTION In recent years, there has been an increasing

interest in modeling heating process especially in essential oil extraction system where linear models are by far the most common in industrial practice [1]. Heating is an important process in essential oil extraction as it greatly influences the production and quality of the oil [1-4]. Modelling heating process is a need to fulfill some requirements in many essential oil extraction processes such as to explain and understand the observed phenomena of system, to run production safely and efficiently and not limited to aid design the regulators and controller of the system [1, 5].

There are several techniques available for deriving the desired process model [6-8] and one of them is system identification (black-box) approach. It is an experimental approach where the model is estimated based on input-output data from the experiments. This technique is used to estimate the unknown parameters and/or the model structure of observed data by performing some experiments on the system to extract input-output data [5, 9].

II. LINEAR SYSTEM IDENTIFICATION Heating is a linear process and specifically for

linear models, there are some common black-box models such as Finite Impulse Response (FIR), Auto-Regressive with Exogenous Input (ARX) model, Auto-Regressive Moving Average (ARMA) model, Auto-Regressive Moving Average with

Exogenous Input (ARMAX) model, Output-Error (OE) model and Box-Jenkins (BJ) model [6].

Among many linear models, ARX model has been found to be the simplest structure model, easy to find analytical solutions and provide excellent performance [2, 6, 7, 10-13]. The ARX model is widely applied and brings a lot of advantages to the research works. The implementation of generalized predictive control in a floatation plant [14] is done by identified ARX model by using one set of data and another set data for the model validation. Example in [10] is an identification of linear model in electric tube furnace through experimentation where first order ARX model are employed to describe the characteristics of the furnace.

The performance of heating system implementing electrical immersion heater in distillation column is evaluated by using ARX model. The model is found sufficient to describe on heating process behaviour and sufficient for controller design purposes [15].

III. EXPERIMENTAL

A. Pilot plant of SDEOE system The essential oil extraction process pilot plant is

located at Distributed Control System (DCS) Laboratory at Level 5, Faculty of Electrical Engineering, Universiti Teknologi MARA, Shah Alam [16]. The plant applies steam distillation technique to extract the essential oil. It is perturbed by input signal, PRBS with the steam temperature as output signal at one second time interval data collection. The plant is heated by a 1.5kW, 240V and 50Hz immersion heater. There is a temperature sensor located between the packed bed and water level to sense the temperature and send the info to data acquisition (DAQ) system via signal conditioning circuit (SCC). The system temperature is controlled by a programmed system in computer to switch the heater ON and OFF with the help of a solid-state relay (SSR). The pilot plant is equipped by a packed bed to accumulate steam distillation extraction in its distillation column. During heating process, the water is boiled to evaporate the steam and passes through the packed bed. The steam is condenses first in the condenser before it drop into

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a decanter. All the data collection is transferred to the MATLAB version R2006a for ARX modelling approaches. The implementation on heating process in pilot plant of essential oil extraction system is described in Fig.1

Fig. 1 The implementation on heating process in pilot

plant of SDEOE system

B. Data Organization The data collection from pilot plant of essential

oil extraction system also is run at ambient temperature until steady state condition. This data is named data z1. The data z1 is divided into four data sets such as data z2, data z3, data z4 and data z5 to investigate either sufficient or not for each partial data from heating process in producing adequate ARX model. All the data from these experiments is modelled by using MATLAB. The results and relevant analytical studies are shown and discussed in Result and Discussion. Data z1 has the whole sample number of heating process where its output temperature is start at ambient temperature, 23°C until steady state condition temperature, around 93°C. Data z2 is a first part of data z1 where output temperature is start at 23°C to temperature at 53°C. Then, data z3 is a second part of data z1 and the output temperature of heating process is at 53°C to 83°C. Next is data z4 where data z4 is a third part of data z1 and output temperature of heating process is at 83° to 93°C. Lastly is data z5 where data z5 is the last part of data z1 and output temperature of heating process is at steady state condition, around 93°C.

C. Linear ARX Model Fitting The data collection (which is input-output

data) from pilot plant experiment of SDEOE system is depicted in Table I. In general, an interlacing technique is used to preprocess the input-output data from both experiments. The odd numbers (1, 3, 5...) for each data in Table I (data z1(u,y), data z2(u,y), data z3(u,y), data z4(u,y) and

data z5(u,y)) are used to estimate the ARX models. They are called estimation data and their names are ez1, ez2, ez3, ez4 and ez5. The fitted model is named as m_z1, m_z2, m_z3, m_z4 and m_z5. Meanwhile even numbers (2, 4, 6…) for each data is used to validate the ARX model obtained apriori using the estimation data. They are called validation data and their names are vz1, vz2, vz3, vz4 and vz5.

TABLE I. DATA ORGANIZATION FOR PILOT PLANT EXPERIMENT OF SDEOE SYSTEM

Data Temperature Model’s Name

Estimation Data

Validation Data

z1(u,y) from ambient (23°C) to steady state condition (around 93°C)

m_z1 ez1 vz1

z2(u,y) from 23°C to 53°C m_z2 ez2 vz2

z3(u,y) from 53°C to 83°C m_z3 ez3 vz3

z4(u,y) from 83°C to 93°C m_z4 ez4 vz4

z5(u,y) at steady state condition (around 93°C)

m_z5 ez5 vz5

Generally, the ARX model for pilot plant of

EOE system is estimated based on linear regression method. The estimation involves minimization of criteria in the form of loss function. This estimation can be automatically computerised in MATLAB programming, in this work we used MATLAB version R2006b.

The model validation is performed to validate the ARX model estimated from pilot plant experiment of SDEOE system. Generally, the ARX model is validated based model fitting, R2 value, 1-SAP with its residual and correlation tests which are ACF and CCF. The model validation is implemented in two parts. First, the estimated model is validated using the estimation data, and secondly, the model is cross-validated using other validation data, i.e. those not employed for estimation. Similar as model estimation, the validation also is done by using MATLAB programming, version R2006b.

IV. RESULT AND DISCUSSION

A. Data collection The input-output data from heating process

consists of 2400 sample numbers and it is denoted as follows: u (t): Input signal, PRBS (0, 1) and y (t): Output signal, Temperature (ºC). Fig.2 shows the input-output data obtained from the experiment. Data z1 has 2400 sample numbers representing input and output of SDEOE experiment. The data is extracted to form other four groups which mean every data will has 600 sample numbers. These four groups are named data z2, data z3, data z4 and data z5. From Fig.2, it is observed that as sample number is increased, the output temperature (�(�)) also

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increase. Except for data z5, the form plot is different among others as at this time heating process is in its steady state condition.

Fig. 2 Input and output data of heating process (a)

Output signal, temperature (°C) (b) Input signal, PRBS

B. ARX Model The ARX model fitting from pilot plant

experiment of SDEOE system is tabulated in Table II. The ARX models are estimated by estimation data of data z1, data z2, data z3, data z4 and data z5. The modelling results return the value for polynomial A and B in ARX structure.

TABLE II. GENERAL ARX MODEL FROM PILOT PLANT EXPERIMENT OF SDEOE SYSTEM

Model ARX Model

m_z1 �(�) +1.001�(��1) =0.014u( t�1) +e(t)m_z2 �(�) +1.001�(��1) =0.0106u (t�1) +e(t)m_z3 y(�) +1.001�(��1) =0.002043u(t�1) +e(t)m_z4 �(�) +1.002�(��1) =0.02125u(t�1) +e(t)m_z5 � (�) +1.001�(��1) =0.03258u(t�1) +e(t)

From Table II, the result for ARX models fitting from data z1, data z2, data z3, data z4 and data z5 are first order model. They are shown by the model order of ARX model structure; the polynomials order of �(�) and �(�) are one and the error signal is white noise (it is equal to one). It means that the modelling gives ARX model that predicts the output by one recent past output and by one recent past input with the delay response time is one. In other words, for ARX model fitting, temperature output, �(�� of heating process from this experiment is predicted by two regressors which are �(��1) and �(��1). The differ in all models is the values of numerator for input, �(�� i.e. 0.014, 0.0106, 0.002043, 0.02125 and 0.03258. Meanwhile, the values of denominator for output, �(�� in all data (z1, z2, z3 and z5) is similar which is 1.001 except for data z4 where the value is 1.002. From this table, new ARX models are found for heating process in pilot plant experiment in EOE system.

C. Model validation The results for models fitting or R2 are

tabulated in Table III. Generally the results show that the R2 values for all cases are very good (from 93.44% to 99.49%). It can be seen that the values for validation data are different to each other according to their groups; v1, v2, v3, v4 and v5. The values for validation data v1 is above 99%, followed by the 68 values for validation data v2 is above 94%, then the values for validation data v3 is above 97%, next the values for validation data v4 is above 98% and lastly the values for validation data v5 is above 93%.The estimation on data e1 until data e5 and validation on data v1 (in the first row under validation) gives the highest values of percentage among others which is above 99%. It is noted that the validation data v1 is from data set z1 and it is belongs to the whole dynamic of heating process. Its temperature is from ambient temperature until steady state condition temperature. This means that validation on data v1 gives the best validation result for the ARX model estimated from the whole sampled data that is cover the dynamic range of heating process.

The second highest value of percentage is above 98% where the values’ range belongs to the validation on data v4. The validation data v4 is owned to data set z4 and its heating temperature is from 83°C to 93°C. Followed by the third highest percentage where the value is above 97%. This result comes from validation on data v3. The next result is validation on data v2 where the percentage of R2 is above 94%. Finally is the lowest percentage of R2 values among others which is above 93%. This result belongs to the validation on data v5. The validation data v5 is a validation data during steady state condition of heating process and its temperature is 93°C.

Moreover, in every row of validation groups, it is observed that the R2 values is very highly accepted when the validation data is used to validate the model estimated by the estimation data in the similar group i.e. the estimation ARX model by using data e1 is validated by using validation data v1 is the highest percentage of R2 (99.49%) compared to the validated the same model by using validation data v2, v3, v4 and v5. It is justified by the values for validation data v1, validation data v2, validation data v3, validation data v4 and validation data v5 which are 99.83%, 99.77%, 98.44% for estimation data e1, e2, e3, e4 and e5, respectively. Hence, it can be mentioned that validation on data in similar temperature with the estimation data is ideal to be used to validate the ARX model fitted.

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TABLE III. VALIDATION RESULT BASED ON MODEL FITTING OR R2 VALUES IN %

Estimation Validation v1 v2 v3 v4 v5

e1 99.49 94.83 97.73 98.32 94.59 e2 99.49 94.83 97.72 98.30 94.63 e3 99.47 94.79 97.77 98.42 93.89 e4 99.45 94.70 97.73 98.44 93.44 e5 99.46 94.61 97.54 98.09 94.84

Fig.3 and Fig.4 show ARX model validation

results by using 1-SAP and correlation test. This result belongs to ez1vz1. The predicted output by 1-SAP plot in Fig.3 shows good agreement with the measured output, which can be confirmed by the residual plot. Fig.4 shows the correlation tests which are comprises of ACF and CCF. The result from ACF is all the correlation coefficient has been detected outside the confidence limit. The residual has failed the whiteness test. Meanwhile CCF resulted between input signals and residual from temperature output signal, no correlation is detected. All the correlation coefficients lie within the confidence region.

Fig. 3 1-SAP validation results for ez1vz1.

Fig. 4 Correlation tests for ez1vz1

D. Summary The summary of validation results for the pilot

plant experiment of SDEOE system can be summarised as shown in Table IV.

TABLE IV. SUMMARY OF VALIDATION TEST FOR PILOT PLANT EXPERIMENT OF SDEOE SYSTEM

Model

Validation on

estimated data

(R2, %)

Cross-validatio

n (R2, %)*

ACF (whiteness

)

CCF (whiteness

)

m_z1 99.49 96.37 lower lower m_z2 94.83 97.54 lower high m_z3 97.77 96.64 lower high m_z4 98.44 96.33 lower lower m_z5 94.84 97.43 lower lower *average value high – refer to correlation coefficient are within 95% confidence limit lower – refer to some of the correlation coefficient are outside 95% confidence limit

From Table IV, there are five ARX models have

been for pilot plant experiment of SDEOE system. All these estimated ARX are validated by using first, its estimation data and second, other validation data, i.e. cross-validation. The average value is average value of R2 value for cross-validation test. Most of the models show that the validation tests on estimated data is higher than the average value of cross-validation test. For example model m_z1, the validation on estimated data is 99.49% is higher than the average value of cross-validation test, i.e. 96.37%.

The correlation has been detected in the residuals as the entire ACF test shows the whiteness test is lower. For CCF test, some models, i.e., m_z2 and m_z3 show high for CCF whiteness test. It is meaning that no correlation has been detected between input and output residual. However, there are some models i.e. m_z1, m_z4 and m_z5 show lower for whiteness test which meaning there are correlation has been detected between input and residual.

In general, m_z1 perform very well since they are based on the complete data for their respective modelling. The other models also show high accurate validation, indicative of the linearity of the dynamic underlying the heating data. The model m_z5 is more important to perform the best model as it will be used to monitor or control of the heating process of the SDEOE plant. This is because this model represents steady state condition of the temperature for heating process in SDEOE plant.

V. CONCLUSION The experiment is successfully done by using

system identification techniques. The input-output data is successfully estimated by linear regression and the fitting model is validated by model fitting or Multiple Correlation Coefficient (R2), 1-SAP for measured output with its residual and correlation tests (ACF and CCF). The results from both experiments show that all the values for model fitting or R2 values are highly accepted. It is supported by residuals plot that show all the measured output is in good agreement with 1-SAP

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output. In correlation tests (residual analysis), not all the residual in ACF passed the whiteness test. However, most of correlation coefficients in CCF lie within the confidence limit. This indicates that the ARX model is adequate, i.e. the model behaviour of heating process is closed to the real system performance.

ACKNOWLEDGEMENT This work was concluded on the data gathered at

the Faculty of Electrical Engineering , UiTM Shah Alam with the support of RMI UiTM, FKE UiTM, MOSTI Sciencefund 03-01-01-SF0226 and MOSTI NSF scholarship. The authors would like to thank all staff involved.

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