Fitting of parameters for a temperature control model by means of continuous derivative-free optimization: a case study in a broiler house

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Fitting of parameters for a

temperature ontrol model by means

of ontinuous derivative-free

optimization: a ase study in a broiler

house Bruno H. Cervelin

1

, Dante Conti

2

, Denise T. Dets h

3

, Maria A. Diniz-Ehrhardt

4

, José Mario Martínez

5

CompAMaVol.5,No.1,pp.117-139,2017-A eptedJuly11,2017

Abstra t

Intensivebroilerprodu tionrequiresofa urate ontrolsystemsaimedtomaintain ideal onditions inside the fa ilities. The a hievement of an appropriate environ-ment guarantees good performan e and sustainability of the produ tion. Control and monitoring of temperature is a key fa tor during the produ tion y le. In ountrieswithtropi alandsubtropi al limate,su hasBrazil,highvalues of tem-peratures an ae t negatively the broiler produ tion. Based on a temperature ontrol modeldevelopedbythe authors,thisresear h isfo usedonthe determina-tion and tting of the intrinsi parameters of the model. Conse utive exe utions

1

DepartmentofAppliedMathemati s,InstituteofMathemati s,Statisti sand S ien-ti Computing, UniversityofCampinas,SP,Brazil(b ervelingmail. om).

2

DepartmentofAppliedMathemati s,InstituteofMathemati s,Statisti sand S ien-ti Computing, UniversityofCampinas,SP,Brazil( ontiime.uni amp.br).

3

DepartmentofEngineeringandExa tS ien es,FederalUniversityofParaná(Palotina Se tor),Brazil(denise.dets hufpr.br).

4

DepartmentofAppliedMathemati s,InstituteofMathemati s,Statisti sand S ien-ti Computing, UniversityofCampinas,SP,Brazil( hetiime.uni amp.br).

5

DepartmentofAppliedMathemati s,InstituteofMathemati s,Statisti sand S ien-ti Computing, UniversityofCampinas,SP,Brazil(martinezime.uni amp.br).

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ofthe modeland hanges inthefa ilities suggest adapting parameters onstantly undertheperspe tive of real-time systems. Four strategiesof derivative-free opti-mization were applied to adjust the parameters of the model. Experiments were ondu ted with data olle ted from a pilot farmin South-eastern Brazil. Results demonstratedthatthepro essofupdatingparametersneedstobeimplementedon thetemperature ontrol model. BOBYQAmethodresultedto bethebeststrategy to be taken into onsideration fortheimprovement of thesystem.

Keywords: Parameter optimization, broiler produ tion, ontrol of temperature, derivative-free methods, real-time.

1 Introdu tion

Livesto k produ tion requires ontrol, monitoring and surveillan e of oper-ating onditions in order toguarantee good performan e, produ tivity, sus-tainabilityand animal omfort. Basi ally, these operating onditions are re-latedtothermalvariables: temperature,relativehumidity,airvelo ity/wind speed amongst other fa tors and their intera tion with automated or semi-automated devi es inside the fa ilities(ventilation systems and ontrollers). Temperature is one of the key variables to be kept under ontrol during a rearingpro ess [1℄. Forintensive broilerprodu tion( hi ken meat), temper-atureinside the fa ilities(broilerhouses) isa ru ialfa tor that needs tobe ontrolled and monitored almost inreal-time[2,3℄.

Brazilisoneofthe topthreebroilerprodu ersintheworld togetherwith United States of Ameri a and China. In ountries where limate is tropi al and subtropi al,su hasBrazil,variationsoftemperatureae ts the rearing pro ess by putting at risk the thermal omfort of the animals. Dis omfort produ esheat stressand high ratesof mortality [4℄; in onsequen e, produ -tion is ae ted negatively in terms of weight onversion, feed e ien y and animal welfare [5℄. Therefore, an e ient ontrol of thermal onditions is ne essary to maximizethe produ tionand guarantee its sustainability.

Ane ient ontrolofthermal onditionsis ommonlysupported by ven-tilationsystemsand ontrollersinsidethefa ilities. Thea hievementofideal onditionsisasso iatedtotheintera tionwithautomated ontrollersand op-eratingpoli ies whi h respond tothe urrent onditions (thermal variables) by swit hing on oro devi es (exhaust fans, ooling pads and humidiers). The hallenge is to guarantee the most omfortable mi ro limate inside the

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approa hes dealwith quantitative methodsindata-driven models whi h use Computation FluidDynami s (CFD), Statisti s, Data Mining, Arti ial In-telligen e andAppliedMathemati sinordertounderstand, ontroland sup-port a urately the thermal onditions at the fa ilities by intera ting with automated devi es. All these approa hes are alled as pre ision livesto k farming (PLF). Some interesting resear hes an be found in [610℄.

Re ently,authorsofthis resear hdevelopedatemperature ontrolmodel aimed to support thermal onditions inside broilerhouses. The model om-bines applied mathemati s, optimizationand some empiri al onsiderations in order to equilibrate a ura y with fast exe ution for real-time pra ti e. The developed model, des ribed in details in [11℄, uses a one-dimensional representation of the broiler house, with left and right walls, in whi h the temperature is propagated by a diusion pro ess, subje t to boundary on-ditionsgivenby externaltemperatureandinitial onditionsprovided by sev-eral sensors pla ed along the house. The ventilation system (Exhaust fans) is modeled as heat sour es that ontribute to balan e the temperature in-side the fa ilities. This modelwould require of the determination/ tting of threeparametersasso iatedtoitforits orre tuse asasupportingtool. The tting pro ess is ne essary when hanges or stru tural modi ations of the broilerhouseareeviden ed. These hangesandthereal-timeapproa h ould produ esigni antvariationsofthe oe ients,sothe updateofparameters be omesa key task to be implemented.

In this resear h, the determination and tting of the parameters asso i-ated to the temperature ontrol model[11℄ is managed by using ontinuous derivative-free optimization. This family of optimization methods tries to a hievethe optimal value of anobje tivefun tion withoutevaluating or ap-proximating its derivatives. They are used, for example, when a bla k-box obje tive fun tion is present, i.e., when the a tual equation of the obje tive fun tion is not available,so itsderivatives are alsonot available. Parameter optimization problems usually tin this family of problems.

Somerelatedproblemsare itedinAudetandOrban[12℄. Theyproposed anobje tivefun tiontooptimizetheparametersof atrustregionmethodin termsofthepro essingtimeofthemethod. Cervelin[13℄proposedsome vari-ationsoftheobje tivefun tionandoptimizedtheparametersofaderiv ative-free method in relation to the number of fun tion evaluation performed by it. Thesetwoworksfo us moreinthe optimizationpro essthanthe appli a-tion of the te hniques in real-world problems. Wild [14℄ used derivative-free

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physi s, and Mukherjee [15℄ enun iates some derivative-free te hniques that are implemented for metal utting pro esses.

Under these premises, the paper is aimed to des ribe how ontinuous derivative-freeoptimizationisappliedtothetemperature ontrolmodel. Pa-rametersasso iatedtothemodel orrespondtothediusion oe ientinside thehouse,the diusion oe ientinthewallsand theee t ofea hexhaust fanonthe variationofthe temperatureinone unitoftime. Fittingthese o-e ientstoa tualdata isataskthat may bea omplishedby minimization algorithms. The temperaturemodel is part of a ontrolmodel and, sin e it an be modied in order to improve the ontrol model e ien y, the merit fun tion that measures the quality of the approximation an hange. So, even if itis not impossible to ompute the derivatives of the merit fun tion, thene essity of hangingthe modelstru ture ledus tothe useof derivative-free methods. These methods allow taking advantage of enough exibility and satisfa tory speed of exe ution forreal-timesituations.

The stru ture of the paper is presented as follows: Se tion 2 des ribes a summarizationofthetemperature ontrolmodelobtainedfrom[11℄together withthe ontrolpro essand itsrelationwith derivative-freete hniques, Se -tion 3givesa basi des ription with ow harts of the ontinuous derivative-free optimization te hniques used in the resear h. In Se tion 4 the ase study for a Brazilian broiler house is detailed with its orrespondingresults anddis ussion. Finally,thepaperis losedinSe tion5withthe on lusions.

2 The temperature ontrol model

2.1 Basi s of the model

The BroilerHouse is represented asasegment

[0, L]

, where

L

represents the lengthofthehouse,the segments

[−a, 0]

and

[L, L + a]

representtheleft-wall and the right-wall respe tively. Thus,

a

may be thought as the thi kness of ea h wall. The ontrol devi es have the property of de reasing the internal temperature (

T

)

∂x

2 (x, t)

if

x ∈ [L, L + a],

T (x, 0)

given for all

x ∈ [−a, L + a],

T (−a, t) = T (L + a, t)

given forall

t ≥ 0,

∂

2 _T

∂x

2 (0, t) =

∂

2 _T

j

(x, t) = α − 0.05jp

3 .

So, in the absen e of onne ted exhaust fans, the internal temperature in reases

α

degrees pertimeunitbut thea tivationofea hfande reases the temperature

0.05p

3

degrees per time unit.

The one-dimensional PDE model des ribed has three parameters that need to be tted to real data before (or during) the exe ution in broiler houses. As mentioned before, the three parameters orrespond to the diu-sion oe ient insidethe house,the diusion oe ientinthe wallsand the ee t of ea hexhaust fan onthe variationof the temperaturein one unit of time. Fittingoftheseparameters isperformedby derivative-freealgorithms.

2.2 The Control Pro ess and Fitting of Parameters

ThePDE des ribedinSe tion2.1predi tsthetemperatureinsidethe broiler house using predi tions of the external one. To do so, the PDE uses some internal parameters (diusion oe ients and fans ee ts). These parame-ters were previously estimated fora parti ular broilerhouse, but, sin e they depend on several other parameters that are not onsidered in the model (quantity and size of the birds, broilerhouse wallmaterial,the maintenan e oftheexhaustfansandotherspe i parametersofthehouse),theyshouldbe adjusted forea hbroilerhouse. Andmore, sin esome ofthe non- onsidered

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duringthe simulation. The parametersare updated if the predi tedinternal temperature isvery dierent fromthe observed one.

During the rearing pro ess, a set of sensors are strategi ally positioned insidethe broilerhouse. These sensorsprovidethetemperatureinrealtime. Human operator or an automated operator ( ontroller) reads these urrent temperatures. In addition, an external sensor olle ts the outside tempera-tureand its fore astedvalues for anext period. Thesevaluesare also trans-mitted and read by the operator. A ording to the values of the registered temperatures the operator de idesto swit hon oro the orre tnumberof exhaust fans in order to approximate the internal temperature to the ideal one whi h depends basi ally on the infrastru ture of the fa ility, age and breed of the animal. For that purpose, the operator fore asts the internal temperature for a period of (say) one hour and, based on this predi tion, de ideshow topro eed with the exhaust fans.

Tomakethe de ision,the operator ompares the resultsof onne ting or dis onne tingevery ombinationof the exhaust fans along the period under onsiderationand hoosesthe ombinationthat,a ordingtothepredi tion, produ esthebestproleofinternaltemperaturesintermsofanimal omfort. The pro ess of hoosing the onguration of onne ted fans is a ombi-natorialoptimization problemwhere a omfort fun tion,represented by the dieren ebetween the a hieved temperature and the ideal one isoptimized. In turn, the evaluation of the omfort-like fun tion involves the experien e of the operator (if human), the onsolidatedadvi e in some operationsheet (alsoree ting humanexperien e) orthe exe utionof apredi tionmodel, in the ase that the ontrolis automati .

In thetemperature ontrolmodel, thepredi tion isgiven by the solution ofthe one-dimensional PDE system briey des ribed inSe tion2.1. As said above,the parameters of the PDE may be modiedduring the pro ess, and for appli ability of the temperature ontrol model, it must be done in real-time.

In fa t,the wholesystem involvespermanent olle tionof data and par-alleltting of the parameters to updated data. This self- orre ting s heme shouldimprovethe su ess ofthe operationasfar astimegoesonina single broiler house. However, stru tural modi ations of the broiler house may produ e signi ant variations of the tted oe ients. Therefore, it is im-portant to implement e ient and reasonably fast tting algorithms to the parametersof the temperature ontrol model.

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fun tiongivestheperforman eofthe method. In thisspe i ase, itwillbe represented bythe

2

-normofthe dieren eofthe predi tedtemperatureand theobservedoneatseveralinstantsoftimes. Usuallythiskindofproblemhas a bla k-box obje tive fun tion, i.e., the equation that denes the obje tive fun tion is not known. Sin e there is no a ess to itsexpression, then there is no a ess to its derivatives, so derivative-free methods an be used to optimize the fun tion.

It also supposed that fun tion

m

depends on some parameters

p

of the methodand thesetof problemsusedtondtheoptimalparametersis alled trainingset. In this resear h,the measure fun tion

m

is the errorperformed by the rst order PDE when solving the problems on the trainingset. The main parameters of the PDE modelare the diusion of temperature in the broiler house(

p

1

), the diusiontemperatureinthe walls (

p

2

), and the ee t ofea hexhaustfanonthevariationoftemperatureinonetimeunit(

p

3

). For instan e, if the predi ted variationof temperaturea ording tothe diusion modelatthe position

x

fromtime

t

to

t + 1

intheabsen e of exhaustfansis

α

,thepredi ted variationwithone onne tedexhaust fanwillbe

α − 0.05p

3

. The optimization problem whi h tries to nd the values of parameters that best ts the data is:

minimize m(p) ≡

kT

S

(p) − T

r

k

2

2 n

such that 0 ≤ p

i

≤ 40 i = 1, 2, 3,

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where

n

is the number of simulated temperatures,

p

is the ve tor with the parameters inwhi hwe areinterested,

T

S

(p)

isave tor withsimulated tem-peratures for dierent instants of time using the parameters

p

and

T

r

is a ve tor with the observed temperatures at the same times. Note that the upperlimit

40

for

p

3

meansthat ea h exhaustfande reases thetemperature 2 degrees Celsiusperunit of time.

Three main strategies were sele ted to solve the problem: the rst one uses BOBYQA [1618℄ that optimizes the obje tive fun tion using a trust region model based onquadrati interpolation; the se ond one uses Pattern Sear h [19℄ whi h tries to nd the optimal point moving through a positive generating set of dire tions, and nally the third strategy uses SID-PSM [20℄ whi h ombines the Pattern Sear h approa h with trust regions based on simplex derivatives. These strategies are able to deal with multidimen-sional derivative-free problems, as it happens in this resear h. An optional

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Sear h[21℄. Although,thislastmethodisappli ableonlyforone-dimensional problems,it wasused by optimizingthe parametersin asequentialway,i.e., ea h parameter ata time.

Results found by ea h of the four strategies will help to determine the nal de ision about the type of minimization method that should be used inpra ti alsituations fromnow on. This de ision is madeby establishing a omparison/ben hmarkingpro ess within the whole set of strategies.

3 Basi ba kground of ontinuous derivative-free

optimization te hniques

3.1 BOBYQA

BOBYQA (Bound Optimization BY Quadrati Approximation) was pre-sented in [18℄. Some theoreti al properties are des ribed in [16,17℄. As its namestates,ittriestooptimizeaproblemapproximatingtheobje tive fun -tion by quadrati models, whi h are built using interpolation points. These points are the ones where the obje tive fun tion was already evaluated at previous iterations.

In ea h iteration we build a model, optimize it in a trust region and verify if the minimizer of the model de reases the value of the obje tive fun tion. If it happens, we a ept this point as the new approximation to theminimizerandupdatethepointsoftheinterpolationandthetrustregion size; otherwise, we update the points of the interpolation and de rease the trust regionsize.

Figure1 shows the BOBYQA pro ess.

3.2 Pattern sear h

Pattern Sear h [19℄ is a derivative-free optimization method that tries to ndthe optimalpointof anobje tivefun tion

f

k

+ αd

i

)

where

d

i

isave torof

D

and

α

isthestep parameter of the method. If this fun tion value is less than

f (x

k

)

, then set

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Fig. 1: Flow hart of the BOBYQA method.

x

k+1

= x

k

+ αd

i

anditispossibletoupdatethestepparameterin reasing its size, otherwise, another dire tion in

D

should be used. If all the dire tions were used and there is no improvement in the fun tion value, then the step parameter sizeshould bede reased.

Usually, the stop riteria of this method is the step parameter, if it is smaller thana value atiteration

k

, then

x

k

isanapproximation ofthe mini-mizer of

f

.

Figure2 shows a ow hart des ribing the method.

3.3 SID-PSM

The SID-PSM (Simplex Derivativesin Pattern Sear h Method) methodwas presented at [20℄. It is a ombination of Pattern Sear h with trust region method.

Inea hiterationthe pointsinwhi h thefun tion isevaluatedarestored. Then, given anapproximation

x

k

fortheminimizerof

f

andapositive span-ning set

D

, the method an be des ribed in three steps:

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Fig. 2: Flow hart of the pattern sear h method.

imatethe obje tive fun tion;

2. Ifthereisaset ofpointswith goodgeometry,webuildamodel

m

,and minimize it. If its minimizer de reases the obje tive fun tion value, wea ept this point asthe new approximationfor the minimizerof

f

. Else, goto step 3.

3. In the last step we perform the Pattern Sear h. Also, one an try to improvethe behavior of the PatternSear hby addingsome dire tions using datafrom the model.

As in the Pattern Sear h method, if the iteration is su essful we an in rease the step size parameter;otherwise, wemust de rease it.

Figure3 des ribes the ow hart of the method.

3.4 Golden Se tion

TheGoldenSe tionSear hmethodisanone-dimensionalderivative-free op-timizationmethod[21℄. Thismethodshrinksaninitialinterval

[a, b]

,inwhi h itis known to have a lo al minimizerof the obje tive fun tion

f

.

.

The reason to use the golden ratio is that the point not used in the update pro ess will be the

c

or

d

of the next iteration,so we must evaluate the obje tivefun tion inonlyonepointatea h iteration. Thestop riterion for this methodis usually used as the distan e between

a

and

b

.

As mentioned before, this method is used to optimize one-dimensional problems. Here, our target refers to multidimensional problems. However, the method was used as an alternative strategy. In order to apply this method,weoptimizedea hvariableatatime. Also,itisimportanttonoti e that there is no mathemati al guarantee that this strategy will onverge to

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4 Case study

4.1 Material & Methods

Experimentswere arriedoutinapilotfarmsituatednearthe ityofCabreuva, StateofSão Paulo,South-easternBrazil. Thebroilerhousehasthefollowing spe i ations:

150

m of length,

15

m of width and

2.5

mof height.

24, 000

birds (Cobb breed) in average are rearing there in a produ tion y le with anestimated duration of

42

days. The fa ility has ventilation system whi h ontains

9

exhaustfans,anautomati ontrollerandotherdevi eswhi hhelp onthe environmental ontrolinside the house.

Four temperature/relative humidity sensors are strategi ally pla ed in thefa ility,three ofthemtoregistertheinside temperatureand oneofthem to register the outside temperature, respe tively. These sensors transmit the temperature values at ea h

5

minutes whi h are re orded in a simple datasheet of type sv or txt.

For this resear h a total of

10803

valid values were used to perform the experiments. The datasheet has

3

olumns labeled as: Number of exhaust fans (number of fans), average of temperature in Celsius degrees from the three inside sensors (int temp

◦

_C

) and outside temperature in Celsius de-grees(exttemp

◦

6

rows.

Tab. 1: A Datasheet sample. Number of fans int. temp.

◦

_C

ext. temp.

◦

_C

0.57

29.57

23.12

0.23

29.26

22.12

0.13

29.49

21.80

0.13

29.74

21.50

0.12

29.89

21.50

0.14

30.11

21.86

For instan e, in the rst line, Table 1presents the average internal tem-perature

29.57 ◦

_C

,theexternaltemperature

23.12 ◦

_C

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fan was turnedon

57%

of the time (

2

minutes,

51

12.5

days).

Threesets ofparameters were estimated orresponding tothe three peri-odsof the bird's life.

Therefore, the four strategies of derivative-free optimization are imple-mented in the three datasets. The pro ess began by giving trial values of the parameters

obs

(60)

and

T

obs

(120)

, respe tively. We pro eed inthe same way in the interval times

[120, 180]

,

[180, 240]

, ...,

[17940, 18000]

.

For these al ulations we used the trial parameters, the data given by external temperatures, and the exhaust fan information given in

A

. The obje tive fun tion value is the sum of squares of the dieren es between the predi ted temperatures

T

model

(t)

and the observed temperature

T

obs

(t)

, divided by the number of predi ted temperatures, in this ase

300

. In a similar way we omputed the obje tive fun tions that orrespond to the se ond and third part of the data.

Even though the temperatures for every

5

minutes were available, only the temperatures at ea h hour were used at the obje tive fun tion. This hoi e was made onsidering that the PDE model is proposed as a tool to predi t the internaltemperatures fora reasonablylarge timeinterval. If the parametersare optimizedusingthe predi tionsforevery

5

minutes,they an benot properly ttedfor predi tions of larger time intervals.

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of simulation, that is, to predi t the internal temperature at instant

60

, we use the real internal temperature at instant

0

, to predi t the temperature at instant

120

we use the real one at instant

60

, and so on. Finally, we estimated another set of parameters, orresponding to the whole broilers' life. The orresponding data set wasstored atmatrix

D

.

PatternSear hwasexe utedbyusingtheroutinepatternsear h(default parameters,ex eptfortoleran e)inMatLab. SID-PSMversion2.0 (default parameters,ex ept fortoleran e and the step of the in remental parameter, inthis ase value of

2

)in MatLabfor the SID-PSM method. BOBYQA was exe uted in FORTRAN (default parameters, ex ept for the toleran e) and GoldenSe tionwasimplemented inMatLab bysettingthe distan e between onse utive points to dene the stopping riterion. For the whole set of strategies, theparameter that denes the stopping riterionwasset to

10 −8

.

4.2 Results and dis ussion

Results of the optimization for ea h training set and ea h method are pre-sented in Tables 2 to 5. The improvement with respe t to the initial ap-proximation given by

p

1 = 1, p

2 = 0.5, p

3 = 1

, is also shown. The initial approximation has been obtained from[11℄ by trialand error.

Tab. 2: Optimization of

m(p)

using

A

. This table shows the fun tion opti-mal value, the optimal parameters obtained, the improvement with respe t tothe initialapproximation(

%

)and the number of fun tion evaluationsperformed by ea h method.

Method

m(p)

p

1 p

2 p

3 %

fevals Initialapprox.

1.50

1.00

0.50

B

. This table shows the fun tion opti-mal value, the optimal parameters obtained, the improvement with respe t to the initialapproximation(

%

) and the number offun tion evaluationsperformed by ea h method.

Method

m(p)

p

1 p

2 p

3 %

fevals Initial approx.

2.13

1.00

0.50

1.00

0.0 −

Pattern Sear h

1.06 18.76 40.00 0.34 50.2

1140

SID-PSM

1.06 18.76 40.00 0.34 50.2

716

BOBYQA

1.06 18.76 40.00 0.34 50.2

184

Golden Se tion

1.09 16.67 40.00 0.47 48.7

144 m(p)

using

C

. This table shows the fun tion opti-malvalue, the optimal parameters obtained, the improvement with respe t to the initialapproximation(

%

) and the number offun tion evaluationsperformed by ea h method.

Method

m(p)

p

1 p

2 p

3 %

fevals Initial approx.

4.00

1.00

0.50

1.00

0.0 −

Pattern Sear h

1.99 13.07 40.00 0.31 50.2

1088

SID-PSM

1.99 13.07 40.00 0.31 50.2

577

BOBYQA

1.99 13.07 40.00 0.31 50.2

107

Golden Se tion

2.02 11.20 40.00 0.39 59.7

144

2. Allthe methodsfound thesamesolutions,ex eptGoldenSe tion, that obtained poorer approximations.

3. BOBYQA was the most e ient method sin e it always obtained the lowest fun tionalvaluesand the smallestnumber of fun tional evalua-tions.

4. Theoptimalparameter

p

2

alwaysrea heditsupperbound. Thismeans that, inthe best ase, the modelruns essentially withoutwalls. This is a limitationofthe one-dimensional formulation.

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m(p)

using

D

. This table shows the fun tion opti-mal value, the optimal parameters obtained, the improvement with respe t tothe initialapproximation(

%

)and the number of fun tion evaluationsperformed by ea h method.

Method

m(p)

p

1 p

2 p

3 %

fevals Initialapprox.

2.54

1.00

0.50

B

are shown.

Theseprolesshowthat

m

,asafun tionof

p

2

,tends asymptoti allytoa horizontalline and the un onstrained optimization problemhas no (global) minimizer. Also, it an be seen that

m

is well-behaved and seems to have nostationarypointsother than thesolution ofthe problem. Althoughthese graphi s were plotted using the xed variables as the optimal values found using SID-PSM, similar behavior is obtained when they are xed as the optimalvalues found by Pattern Sear h or BOBYQA.

5 Con lusions

The real-time features of the temperature ontrol model make it ne essary tosele tthe fastestmethodforparameterestimationandtting. Corre tion of the parameters (

p

1

,

p

2

and

p

3

) should o ur during the operation of the

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(a) PatternSear h. (b)SID-PSM.

( )BOBYQA. (d) GoldenSe tion.

Fig. 4: Graphi s of the fun tion value per iteration for dierent methods using the fourth trainingset.

with the growing, rearing and dynami s pro ess of the broilers by itself. As was mentioned before, the ontrol of thermalenvironment inside the broiler houses is a omplex problem. A strategy related to impose xed param-eters for onse utive exe ution in real-time of the model is unfeasible and unreliable. So, self-optimization and in orporation of the estimation/tting software into the ontrol model is ru ial for the ee tive and a urate re-sponse of the system.

In this sense and after being evaluated the results of the four strategies with derivative-free methods, Powell's software BOBYQA seems to be the most adequatetoolforthistypeofparameterestimation. Thereasonforthe superiority of BOBYQA is that, as shown in gures 5 and 6, the obje tive

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points, other than the solutions of the problem. BOBYQA pro eeds inter-polating quadrati models, so its omparative performan e improves when the fun tion is smooth and presumably unimodal. The derivatives of the obje tivefun tion with respe t tothe parametersare omputable,either by hand- al ulation or by automati dierentiation. However, the human ost of these tasks are rather dis ouraging, espe ially at a level of development inwhi h one hangesfrequentlythe stru ture ofthe model. The friendliness of derivative-free software makes it preferable to more sophisti ated alter-natives. Moreover, the ommer ial feasibility of temperature ontrol model imposes that our own solver for real time optimization must be developed. Thus, the present resear h allowed dete ting the best type of software and adjustmentsthat should be implemented.

Inadditiontothede isiononthebestsolverforparameterestimation,the present numeri al study provided the understanding of the stru ture of the problem. The fa tthat the best wall diusionparameter is innity revealed that something is inadequate in the formulation of the PDE model. This is notsurprisingbe ausethe PDEmodel omesfromaradi alone-dimensional simpli ation of a Fluid Me hani s problem whose solution in real time is impossible,atleastsubje ttothebudgetrestri tionsofthis proje tinterms of omputing time. Sin e 3D models are ertainly unaordable, it an be onje turedthat 2Dmodelsshould bedeveloped. Ourpresent feelingisthat models based on a parallel plane to the oor ould ree t adequately the diusionthrough dierent typesof walls.

A knowledgements

ThisworkwassupportedbyPRONEX-CNPq/FAPERJ E-26/111.449/2010-APQ1, FAPESP (grants2010/10133-0,Cepid-Cemeai 2011-51305-02,

2013/03447-6,2013/05475-7,and 2013/07375-0),and CNPq.

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(a) The average squared error as fun tionof

p

1

hasauniqueminimizer at

p

1 = 4.39

.

(b) The average squared error as fun tion of

p

1

in a smaller interval,

p

1

varyingfrom

3

to

3 p

3 = 1.19

.

(f) The average squared error as fun tion of

p

3 p

3

varyingfrom

0

to

2

.

Fig.5: Proles using Matrix

A

, these graphi shave two variables set as the optimum value found by SID-PSM and the other one is free. They show how the average error behaves when we hange just one of the

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(a) The average squared error as fun tionof

p

1 p

1 = 18.76

.

(b) The average squared error as fun tion of

p

1 p

1

varyingfrom

18

to

3 p

3 = 0.34

.

(f) The average squared error as fun tion of

p

3 p

3

varyingfrom

0

to

1

.

Fig. 6: Proles using Matrix

B

, these graphi s have two variables set as the optimum value found by SID-PSM and the other one is free. They show how the average error behaves when we hange just one of the