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Abstract

odels with external input (ARMAX), could be used in order to predict the

1. Introduction

researches and theoretical studies have demonstrated that apotential saving is possible by improving the handling of energy

can be reduced between 20 and 30% and simultaneously solvesome comfort problems [3].

varying loads, could operate more efficiently and keep bettercomfort conditions than conventional control schemes. Theadaptable control techniques are necessary in HVAC applica-tions because the characteristics of the process change

16establishing the correct or optimal operation of the HVACsystems in commercial buildings. The energetic consumptionAn important goal of the intelligent buildings is to improveusers' comfort and security with a global reduction in theconsumption of energy [1]. The inside air quality and thermalcomfort should be assured by means of an appropriate heatingand ventilation of the spaces. The consumption of energy byheating, ventilating and air-conditioning (HVAC) equipment inindustrial and commercial buildings constitutes 50% of theworld energy consumption [2]. The high energetic consumptionin the HVAC systems is related to the use of inefficient controloperation sequences and to failures in the system. Experimental

Engineers and architects have influenced the energy use ofthe new buildings through the design of the envelope, theHVAC systems selection and the operation sequences specifi-cation; however, once the building has been finished, theconsumption of energy is decided mainly by its control,maintenance and by its occupants.

An important factor affecting the efficiency of the HVACsystems is that most of them are set to operate at design thermalloads while actual thermal loads, which affect the system, aretime-varying. Therefore, control schemes that consider time-interior air temperature of a building. In particular, the obtained results in the classrooms of the Universidad Autnoma de Quertaro, U.A.Q.,Mxico, are shown. Outside air temperature, global solar radiation flux, outside air relative humidity and air velocity were used as the inputvariables. The obtained results showed that the ARX models give a better prediction of the temperature than the ARMAX models, obtaining thebest results with the ARX (2,3,0) with a coefficient of determination of 0.9457 and ARX (2,2,1) with a coefficient of determination of 0.9056. 2006 Elsevier B.V. All rights reserved.

Keywords: Intelligent buildings; Comfort; Autoregressive modelsmodels with external input (ARX) and autoregressive moving average mOne of the main problems of the intelligent buildings is to give comfort to its occupants and to increase the user's performance at a low cost.The excessive demand of electric energy due to heating, ventilating, and air-conditioning (HVAC) systems require temperature forecast andcontrol to make maximum reduction of the electrical energy. The objective of this paper is to investigate in what extent linear autoregressiveModelling temperature in intautoregres

G.J. Ros-Moreno , M. TrejoV.M. Hernndez-Guz

Faculty of Engineering, Universidad Autnoma de Quertaro,

Accepted 20

Automation in Construction Corresponding author. Fax: +52 442 192 12 00+1+6064+#.E-mail address: riosg@uaq.mx (G.J. Ros-Moreno).

0926-5805/$ - see front matter 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.autcon.2006.11.003igent buildings by means ofe models

erea, R. Castaeda-Miranda,n, G. Herrera-Ruiz

ro de las Campanas s/n, CP 76010, Quertaro, Qro., Mxico

ember 2006

(2007) 713722www.elsevier.com/locate/autconcontinuously by effect of the variations in the climate andoccupation of the building. An adaptable control systemdepends on a dynamic model that describes the parameters

which draw the behavior of a system; the more accurate thepattern is to the real characteristics of the process, the moreefficient control actions can be accomplished.

The parameters' adjustment of a model used to represent asystem is called System Identification. The parameters' estima-tion of the continuous systems in the time is a relevant topic thathas several applications which range from control and signal

averaged parameters, different from Fanger's concept, pointingout the differences of each one and giving their definition. Thestudy showed that there are differences between the heat andenergy dissipation model of the human body at the walking stateand the model of quiet seating postures. However, he mentionsthat this does not influence thermal comfort concepts, becauseseveral heat and energy dissipation models can be applied to thecalculation equations in order to estimate the three indexes ofthe thermal comfort.

The purpose of this work is to determine the appropriatestructure of the (ARX and ARMAX) models to predict thetemperature in the interior of a building (classrooms). It alsopretends to study the external climate variables that should beincluded in these models to achieve the best estimation. Thedynamic model of autoregressive structure can become apractical alternative for the implementation of an adaptablecontroller that will allow to increase the building occupants'comfort and at the same time, to reduce the electric energyconsumption of the HVAC systems.

714 G.J. Ros-Moreno et al. / Automation iprocessing to astrophysics and economy [4]. This is due to thefact that the majority of the systems or physical phenomenonsare from continuous nature in time. With the arriving of thedigital computers, research for the control and identification ofthese continuous systems and processes in time has beenconcentrated in the discretized models with samples of inputsand outputs of the fundamental systems. A discrete model of thecontinuous climatic system in time of a building can be obtainedin several ways; one of them is by an autoregressive relationshipbetween the discrete output y(t) and the input u(t). Twostructures for autoregressive models commonly used in theestimation of systems are the autoregressive models withexternal input (ARX) and autoregressive moving averagemodel with external input (ARMAX).

The methods of identification are often classified as grey-box(semi-physical) or black-box models (non-physical). The grey-box models have a structure that is based partly on physical,chemical or biological laws (like deterministic models) andpartly on empiricism. In the black-box models previousknowledge of the system is not needed, this could be anadvantage if the information of the system dynamics is limited,but it implies the problem of selecting an appropriate structurefor the pattern. Another advantage is the possibility of obtaininga vague model with a relatively small group of measurements.The pattern can be improved by introducing new data.

In literature, numerous works have tried to respond to thedemand of thermal comfort and energy saving. The develop-ment of a dynamic model is a fundamental part of the proposals.The work presented by Loveday [5] points toward the utilityand the potential of using the models of temperature to handleresources in intelligent buildings. Some studies related to thecontrol methods of the HVAC systems based on models of thesystem are presented in [611]. Another way in which themodels can be used is by applying them in different strategies ofdetecting, locating and predicting the presence of defects whichcause incorrect operations. In [3] an autoregressive algorithmwas used for the development of a detection dynamic model anda failure diagnosis of a HVAC system. Recent studies focusedon the comfort evaluation in classrooms can be found in Krger[12] who evaluated classrooms in Brazil, highlighting fourFig. 1. Blocks diagram of the pattern of black box used to predict thetemperature of the inside air in classrooms.different study areas of comfort: thermal, luminic, acoustic andergonomic. The results obtained in different evaluations showeda strong interdependence among the first three areas. Forevaluating thermal comfort, he considered solar orientation andbuilding materials on walls and roofs. Hwang [13] presented themethodology of the ASHRAE standard 55 for the study ofthermal comfort in classrooms in Taiwan. In order to obtain thedata, surveys and field experiments in 26 air-conditionedclassrooms and 10 ventilated in a natural way were applied. Theresults demonstrated that air temperature, air movement andaverage radiant temperature have a significant influence, buthumidity does not make any statistical significance. Hanqing[14] carried out a study based on Fanger's thermal comfortconcept [15]. He proposed several concepts by using computingresults obtained from a large eddy simulation (LES). The authorestablished three indexes of the thermal comfort based on time

Fig. 2. Typical school building viewed from the south.

n Construction 16 (2007) 713722This paper is organized as follows: in Section 2 the theo-retical considerations for the identification of the autoregressivesystems are presented. Section 3 describes the materials and

w (

G.J. Ros-Moreno et al. / Automation inmethods for the field measurements and also presents theaccuracy measurements to explain the variation of thetemperature data. In Section 4 a discussion of the obtainedresults of the selected models ARX and ARMAX is carried out.Finally, the conclusions and the future works are presented inSection 5.

2. Theoretical considerations

ASHRAE STANDARD, claims that to determine a comfortzone it is necessary to take into account the operativetemperature for given values of humidity, air speed, metabolicrate, and clothing insulation. The comfort zone is defined interms of a range of operative temperatures that provideacceptable thermal environmental conditions, or in terms ofthe combination of air temperature and mean radiant temper-ature that people find thermally acceptable [16]. The excess ofheat, coming from the atmosphere or generated by their own

Fig. 3. Typical plan viemetabolism, should be eliminated in order to maintain aconstant temperature in the body and to assure thermal comfort.By sure, the sensation of thermal comfort is given by the

Fig. 4. Distribution of the sensors in the interior of the classroom 19. Superiorview, plane XY.climatic conditions, the heat production of the humanmetabolism and the transfer of heat with the environment.

In a closed atmosphere there are six main elements thatdetermine the perception of the environmental quality from thethermal point of view: Temperature of the air (T), solar radiation(Ra), wind speed (Vw), relative humidity (Rh), average of theinstantaneous air velocity over an interval of time (Va) [13,16]and occupation (Oc).

The identification of the system is simplified by omittingthe Oc and Va variables to calculate the model of the clas-sroom building. On one hand, the difficulty of being able toaccurately measure the dissipation of energy of the humanbody of the Oc individuals inside the interior of the clas-srooms is presented. On the other, due to the fact that thebehavior of the classroom becomes highly non-linear duringHVAC ventilation operation inclusion of the Va effect in themodels is discarded.

Based on this, the proposed mathematical model to predict

dimensions in meters).

715Construction 16 (2007) 713722the temperature inside of the intelligent building of class-rooms in this research is based on the analysis of the followinginput variables: outside air temperature (To), global solarradiation flux (Ra), wind speed (Vw), outside air relativehumidity (Rho); being the inside air temperature the outputvariable (Ti) (Fig. 1).

Although the climate characteristics are continuous vari-ables, they are measured and registered in discrete form. In thediscrete domain, the dynamic system of the building can bemodeled by means of linear autoregressive relations betweenthe discrete output y(t) and the discrete input u(t), such as theautoregressive models with external input (ARX) and auto-regressive moving average models with external input(ARMAX). Assuming a single inputsingle output system,the following expression can be used to describe thisrelationship [4]:

yt a1yt1 : : :anaytna b1ut1 : : : bnbutnb et c1et1 : : : cncetnc

1

For a multivariable system [4] in which the number of inputsis given by nu and the number of outputs by ny, A(z) and B(z)are ny by ny and nu by nu matrices, respectively, whichelements are the polynomials in the shift operator zm (with mas any natural number). The entries aij (z) and bij (z) of thematrices A(z) and B(z), respectively, can then be expressed as:

aijz dij a1ij z1 : : : anaij znaij 10

and,

bijz b1ij znkij : : : bnkij znkijnbij1 11

Where ij represents Kronecker symbol.According to this, it is clear that the ARX structure has a

system which can be defined by means of the number of thepoles na the number of zeros nb1 and the time delay nk. Thedefinition of the ARMAX structure additionally requires theorder of the measured error nc to be known. The matricesA(z),B(z) and the vector C(z) are determined by means of off-lineparameter identification methods.

n in Construction 16 (2007) 713722where: y=output signal, u= input signal, t=discrete time,ana=model parameters, bnb=model parameters, cnc=modelparameters, e=error, na=number of poles (positive integer),nb=number of zeros increased by one (positive integer) andnc=order of the measured error (positive integer).

The determination of all or some values is obtained bymeans of the estimation procedures, that is, the coefficientsenter as parameters to be determined. will be named as thevector that has all the parameters to estimate. The descriptionof the model is the following:

yt; Gz;ut Hz;et: 2

Because the white noise term enters as a direct error to thedifference equations, the model of Eq. (1) is also known asthe model or structure of error equation. In this case theparameters of adjustment will be:

a1 a2 : : : ana b1 b2 : : : bnb T 3

For ARMAX, the Eq. (1) is often represented as:

Fig. 5. Distribution of the sensors inside of the classroom 19. Lateral right view.

716 G.J. Ros-Moreno et al. / AutomatioAzyt Bzutnk Czet 4

And for ARX as:

Azyt Bzutnk et 5Where the matrices A(z) and B(z) and the vector C(z) aregiven by:

Az : 1 a1z1 : : : anazna 6

Bz : 1 b1z1 : : : bnbznb 7

Cz : 1 c1z1 : : : cncznc 8and z1 is the backward shift operator

z1ut ut1: 9Fig. 6. Original data versus results of the (a) autoregressive moving average

model with external input and the (b) autoregressive model with external inputand the difference between both models for the first period of sampling; ,__,measured;......, estimated.

Table 1Model structures and typical parameter identification results for the period models giving the inside air temperature (Ti) as a function of outside air temperature (To), global solar radiation flux (Ra), wind speed (Vw),outside air relative humidity (Rho), and perturbation (e) in the discrete time (t) domain, using the z-transform operator (z

1); a1 to a5, b11 to b43, y, c1 to c3 are regressive coefficients as defined in the model structure

Tit Tot Rat Vwt Rhot1 a1z1 a2z2 a3z3 a4z4 a5z5 b11z1 b12z2 b13z3b21z1 b22z2 b23z3b31z1 b32z2 b33z3b41z1 b42z2 b43z3

2664

3775

1 c1z1 c2z2 c3z31 a1z1 a2z2 a3z3 a4z4 a5z5 et

a1 a2 a3 a4 a5 b11 b12 b13 b21 b22 b23 b31 b32 b33 b41 b42 b43 c1 c2 c3

First period of samplingARMAX4,1,3,0 1.88 1.84 1.32 0.39 0 5.87104 0 0 2.36103 0 0 0.03 0 0 0.02 0 0 0.48 0.77 0.024,1,2,0 1.83 1.76 1.25 0.37 0 6.33104 0 0 2.50103 0 0 0.03 0 0 0.03 0 0 0.43 0.74 0

ARX1,3,0 0.98 0 0 0 0 7.14104 3.67103 3.93103 0.02 0.02 3.23104 0.51 0.28 0.21 0.10 0.12 0.22 0 0 03,3,0 1.28 0.26 0.03 0 0 7.33104 2.98103 3.38103 0.02 0.03 0.01 0.47 0.44 0.02 0.02 0.15 0.17 0 0 02,3,0 1.29 0.30 0 0 0 7.54104 2.90103 3.35103 0.02 0.03 0.01 0.47 0.44 0.02 0.03 0.15 0.18 0 0 0

Second period of samplingARMAX5,3,1,1 2.14 1.34 0.09 0.12 0.02 5.56104 6.17104 9.17105 0.02 0.04 0.02 0.11 0.17 0.06 0.05 0.03 0.01 0.73 0 05,3,2,1 2.42 1.96 0.50 0.08 0.05 5.31104 6.63104 1.57104 0.02 0.04 0.02 0.10 0.16 0.06 0.06 0.08 0.01 1.01 0.22 0

ARX2,2,1 1.50 0.51 0 0 0 0 6.71104 4.57104 1.56104 1.15103 0 0.32 0.32 0 0.01 0.02 0 0 0 03,3,1 1.41 0.30 0.11 0 0 0 6.20104 3.47104 2.33103 7.93103 7.13103 0.29 0.24 0.04 3.07103 0.05 0.04 0 0 03,2,1 1.41 0.30 0.11 0 0 0 6.28104 4.37104 1.79103 2.89104 0 0.32 0.31 0 4.73103 0.02 0 0 0 0

ARMAX: autoregressive moving average model with external input; ARX: autoregressive model with external input.

717G.J.

Ros-M

orenoet

al./Autom

ationin

Construction

16(2007)

713722

According to Ljung [4] once the data have been recorded, thefirst two thirds or so of the data record were used in order todetermine the model coefficients and the remaining data forvalidation.

Even when the measures were done during a continuousperiod of time, the data of the first 24 days was used to get thecoefficients of the model. Once these coefficients werecalculated, a prediction of the inside temperature for the next12 days was done by comparing the predicted temperature ofthe model and the real temperature taken in the classroom.These relation is shown in Fig. 6(a and b).

The approximated temperate of the room was taken from theaverage temperature of the 12 sensors for each sample. Theaverage value calculated corresponds to the inside temperature,which was compared to the output data of models ARX andARMAX. Thus, from only one calculated model, the insidetemperature was calculated.

3.2. Measures of accuracy

The coefficient of determination (R2) is the measure ofcorrelation between the observations and predictions [18].

n in Construction 16 (2007) 7137223. Materials and methods

3.1. Field measurements

The procedure was carried out in the graduate building of theEngineering school at the Quertaro State University; thebuilding is shown in Fig. 2. This construction is equipped witha saving and analysis digital system of the consumption andquality of electrical energy called MONITO UAQ [17], whichallows a continuous monitoring of the power consumption andelectrical energy parameters in order to execute control actionsthat prevent failures and enable energy savings. The structureof the building has five classrooms, a meeting room and acontrol room. The rooms chosen for this study were consideredto be identical.

A schematic plan view of one of these units shows therelative position of the rooms, doors, windows and overhangs,in addition to the overall dimensions (Fig. 3). Four classroomshave an area of 48 m2; one classroom has an area of 40 m2, andthe meeting room an area of 56 m2. A control room, where therecording and monitoring of the variables were done, has anarea of 16 m2. The height of all of them was 3 m.

The building has a 12 cm thick concrete layer on top. Theconcrete layer was armed and has a resistance of 300 kg/cm2.Lateral walls were built with 14 cm wide30 cm long coredbricks. With the purpose of providing a better illumination, thedesign included two 4 m long2 m high windows on the northside and one on the south side. The walls dividing the rooms aremade of concrete blocks lined up with mortar with finishedinteriors and exteriors. It has a 0.5 mm and 0.7 cm thick vinylpainting. The top is covered by water proof Protexa layer andred sand.

To measure the climate variables inside classroom 19, twelvesensors of temperature were installed and distributed in theinterior of the room. Sensors were numbered from 0 to 11. Ninesensors were distributed in plane XY, placed horizontally at adistance of 1.5 m on the X axis, and placed on the Y axis at adistance of 2 m (Fig. 4). Three sensors were placed in the superiorpart of Z axis at a distance from floor to roof of 2.9 m (Fig. 5).

At the outside part of the building a resistance temperaturedetector (RTD) of the Omega Engineering Inc. was placed inorder to measure temperature in C. A capacitive sensor wasused to measure the relative humidity (%). Wind speed (m/s)was measured by an anemometer, and to measure the globalsolar radiation (W/m2) a pyranometer sensor was utilized. Thesecomponents were installed at the southwest part of thebuilding at 4.5 m from the floor. All sensors were connected tothe TUNA SCCII v4 system, developed in the BiotronicsLaboratory of the Quertaro State University. This system wasconfigured to sample data every 5 min.

In order to be able to obtain the coefficients for themathematical model several measures were done to predicttemperature in the classrooms of the intelligent building. Thisresearch is based on the analysis of the input variables, which

718 G.J. Ros-Moreno et al. / Automatioare: To, Ra, Vw, Rho, being the inside air temperature the outputvariable (Ti). The measures of the foregoing variables were doneby sampling every 5 min during a period of 36 days.Fig. 7. Original data versus results of the (a) autoregressive moving average

model with external input and the (b) autoregressive model with external inputand the difference between both models for the first period of sampling; ,__,measured;......, estimated.

Some measures of variances are the percent standard error of theprediction (% SEP) [19], the coefficient of efficiency (E) [20]and the average relative variance (ARV) [21]. These estimatorsare not biased by the variation range of its elements. These areused to determine how the model is able to explain the totalvariance of the data. The percent standard error of the predictionis defined as:

%SEP 100yk

XNk1

ykk2

N

vuuuut 12

the estimated ones for the ARMAX (4,1,2,0) model obtainingan estimate of 82.22% and for the ARX (2,3,0) an estimate ofthe 87.71%. It can be observed that the models which give abetter prediction of the temperature are the ARX models withregard to the ARMAX models.

The assessment corresponding to the second period ofsampling can be observed in Fig. 7 showing the accuracy of theARMAX and ARX models. The simulated results are comparedwith the real data for the temperature of the interior air. Thedifferences between the measured temperature and the estimat-ed one in the interior of the classrooms between both models areshown in Fig. 7(a and b). The ARMAX (5,3,2,1) model showsan estimate of 82.27% and the ARX (3,2,1) an estimate of87.33%. Based on the results of Figs. 6 and 7(a and b) themodels that give a better prediction of the temperature are the

G.J. Ros-Moreno et al. / Automation inWhere yk is the observed output k of the pattern; k is theestimate output for the pattern; N is the total number of thegeneralization patterns and yk is the mean value of the observedoutputs of the prediction set.

The coefficient of the efficiency (E) and the average relativevariance (ARV) are expressed by:

E SobsSSobs

; ARV SSobs

13

Sobs XNk1

ykyk2; S XNk1

ykk2 14

Where Sobs is the measure of variability of the observedvalues from their means and S is the measure of the associationbetween the predicted and observed values. For a perfect match,R2 and E should be close to 1.0 and the values of %SEP and(ARV) close to 0.

4. Result and discussion

4.1. Selection of the ARX and ARMAX models

Table 1 shows the coefficients of the selected models for thefirst and second period of sampling. It can be observed that allthe selected models included the four variables of the externalclimate. It was found that in most cases, the values of the outside

Table 2Results of the validation of the ARMAX and ARX models corresponding to thefirst sampling period

Model Validation

R2 E % SEP ARV

ARMAX4,1,3,0 0.9082 0.9998 0.0062 2.2344104

4,1,2,0 0.9107 0.9997 0.7320 3.0476104

ARX1,3,0 0.9434 0.9996 0.85 4.1690104

3,3,0 0.9455 0.9986 1.54 1.410210342,3,0 0.9457 0.9993 1.22 7.442810

ARMAX: autoregressive moving average model with external input;ARX: autoregressive model with external input.climate data which were older than 20 min did not improve themodel performance. In most cases, outside climate data olderthan 15 min were not considered. Consequently, the resultingmodels were rather compact. This should be considered as anadvantage since this allows their use in climate controlapplications.

The coefficients of the ARMAX and ARX models of Table 1of the first period with regard to the coefficients of the ARMAXand ARX models of the second period show a minimumvariation. It is observed that most of the coefficients do notdiffer and they present a similarity according to the two periods.But the coefficients of the ARMAX models of the first period ofsampling with regard to the coefficients of the ARX models ofthe first and second periods show a greater variation amongthem. In a similar way the coefficients of the ARMAX modelsof the second period of sampling with regard to the coefficientsof the ARX models of the second and first period of samplingshow a bigger variation.

In order to assess the accuracy of the ARMAX and ARXmodels, the simulated results are compared with the originaldata for the inside air temperature. Fig. 6(a and b) show anexample for both first and second period models. The differencebetween the measured and the estimated inside air temperaturesis shown in these figures. The measured data are shown versus

Table 3Results of the validation of the ARMAX and ARX models corresponding to thesecond sampling period

Model Validation

R2 E % SEP ARV

ARMAX5,3,1,1 0.8861 0.8525 16.97 0.14755,3,2,1 0.8906 0.8640 15.34 0.1360

ARX2,2,1 0.9056 0.9325 10.81 0.06753,3,1 0.9093 0.9363 10.50 0.06373,2,1 0.9096 0.9391 10.27 0.0609

ARMAX: autoregressive moving average model with external input;ARX: autoregressive model with external input.

719Construction 16 (2007) 713722ARX models with regard to the ARMAX models.Based on the results in Table 1 and on Figs. 6 and 7(a and b)

it is observed that in both periods of sampling, the models

Fig. 10. Scatterplot of the observed versus estimated temperature variation for

720 G.J. Ros-Moreno et al. / Automation in Construction 16 (2007) 713722presented similar behaviors, but the ARX models are morecompact and give a better estimate for the internal temperatureof the air with regard to the ARMAX model. It can be alsoappreciated that the best models for the ARMAX of the first andsecond period are the models (4,1,2,0) and (5,3,2,1) and thatthey have an estimate of 82.22 and 82.27% respectively. Thebest models of the first and second period for the ARX are(2,3,0) and (3,2,1) and they have an estimate of the 87.71 and87.33% respectively.

The simulated results on the real data of the lineal regressionfor the 5 selected models from a total of 21 are shown in Table 2.It can be observed that the coefficient of determination is closeto 1 (that is 0.9082) and that the percent standard error of theprediction is close to 0 (that is 0.015), while the coefficient of

Fig. 8. Scatterplot of the observed versus estimated temperature variation for theARMAX (4,1,2,0) model corresponding to the first sampling period.efficiency is superior to 0.9992 and the average relative varianceis very close to 0 (that is 0 .00074). When the results of theautoregression were compared, it could be observed that most of

Fig. 9. Scatterplot of the observed versus estimated temperature variation for theARMAX (5,3,2,1) model corresponding to the second sampling period.the models gave good results although the performances of theARMAX models were not as good as those of the ARX models.

Table 3 shows the results of the lineal regression for the 5selected models from a total of 36. It is observed that they had acoefficient of determination close to 1 (that is 0.8861) and apercent standard error of the prediction close to 0 (that is0.1697), while the coefficient of efficiency is superior to 0.8525and the average relative variance is very close to 0 (that is0.1475). Comparing the results of autoregression it wasobserved that the models gave good results although theperformances of the ARMAX models again were not as goodthan those of the ARX models.

Comparing the results of Tables 2 and 3, it is observed thatthe structures of the patterns of a bigger order, and in

the ARX (2,3,0) model corresponding to the first sampling period.consequence more complex showed a lower performance.Finally, these clearly illustrate that the ARX models showed abetter performance than the ARMAX models.

Fig. 11. Scatterplot of the observed versus estimated temperature variation forthe ARX (3,2,1) model corresponding to the first sampling period.

constrained predictive control case study: an environmental test chamber,

on inThe Scatterplot and the linear regression between theobserved data and the estimated ones are shown in Fig. 8.The characteristics of the temperature variation provided thelower deviations around the regression line. It is observed thatthe best ARMAX model is the (4,1,2,0) where a (R2 =0.9107)coefficient of determination is presented, a percent standarderror of prediction of (% SEP=0.7320), a coefficient ofefficiency of (E=0.9997), besides average relative variance of(AVR=3.0476104).

The linear regression for the ARMAX (5,3,2,1) model inFig. 9, presents a (R2 =0.8906) coefficient of determination, astandard error of prediction of (% SEP=15.34), a coefficient ofefficiency of (E=0.8640), besides average relative variance of(AVR=0.1360), showing a lower correlation in comparisonwith the ARMAX (4,1,2,0) model.

The ARMAX models present good linear regressionsregarding to the measured data versus the observed ones, butthey do not improve the performances of the model ARX. TheARX (2,3,0), Fig. 10, has the lowest dispersion around themodels shown in Table 2, presenting the best coefficient ofdetermination (R2 =0.9457), one of the lower percent standarderror of prediction (% SEP=1.22) and the best coefficient ofefficiency of (E=0.9993) besides an average relative varianceof (AVR=7.4428104) which is the lowest.

In Fig. 11 a Scatterplot with good results can be appreciated.It shows a low dispersion of the data, given by the ARX (3,2,1)model, improving this way any of the performances of theARMAX models with a (R2 =0.9096) coefficient of determi-nation, a percent standard error of prediction of (%SEP=10.27), a coefficient of efficiency of (E=0.9391) and anaverage relative variance of (AVR=0.0609), shown in Table 3.

It is observed that the ARMAX models give goodpredictions of the interior temperature of the classrooms, butthese results are improved by the ARX models, due to a goodadjustment, the models should present a coefficient ofdetermination and a coefficient of efficiency close to 1; themain characteristic that is observed in the results of the ARXmodels according to the data that are given in Tables 2 and 3.Likewise, they should present a percent standard error ofprediction and the average relative variance close to zero, acharacteristic that the ARX models also present in general; thisexplains the capacity of the ARX models to give a betterprediction.

Figs. 811 show the dispersion diagram of the temperaturedata. It can be considered that the models that will be includedare the ARX models as identification models of the buildingfor the prediction of the internal temperature. Besides, theypresent coefficients of smaller order and they obtain a lowerdispersion; such case was observed with the model of Fig. 10,ARX (2,3,0).

5. Conclusion and future work

This work studied how to apply the linear autoregressive

G.J. Ros-Moreno et al. / AutomatiARX and ARMAX models as an approach to the dynamicbehavior of the temperature of the air inside the classrooms. Forthis, measurements of temperature, global solar radiation flux,Automatica 27 (4) (1991) 611626.[10] W.J. Graham, A.L. Dexter, Practical experience of self-tuning temper-

ature control in an office building, Proc. Int. Symp. Resent Advances inControl and Operation of Building HVAC Systems, Trondheim, 1984,pp. 124133.

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[17] M. Trejo Perea, J.G. Ros Moreno, E. Rivas, V. Rauch, Savings andwind speed and outside air relative humidity were carried out asinput variables to the system, and different structures of ARXand ARMAX models were intended. By means of program-ming, the acting indexes were calculated for each one of thestructures; selecting those models with better prediction of thereal conditions of interior temperature. Based on the selectedmodels, the influence of the input variables is discussed in theaccuracy of the models. The best prediction results wereobtained by the structures of ARX (2,3,0) model with acoefficient of determination of 0.9457 and the ARX (3,2,1) witha coefficient of determination of 0.9096.

The information presented in this work will serve as acomplement for a strategy of intelligent control in HVACequipments, where, based on the model of the controlledbuilding will determine the best control sequence in order to getthe desired reference with a minimum expense of energy.

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722 G.J. Ros-Moreno et al. / Automation in Construction 16 (2007) 713722

Modelling temperature in intelligent buildings by means of autoregressive modelsIntroductionTheoretical considerationsMaterials and methodsField measurementsMeasures of accuracy

Result and discussionSelection of the ARX and ARMAX models

Conclusion and future workReferences