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TEMPERATURECONTROLUSINGAMICROCONTROLLERANINTERDISCIPLINARYUNDERGRADUATEENGINEERINGDESIGNPROJECTJAMESSMCDONALDDEPARTMENTOFENGINEERINGSCIENCETRINITYUNIVERSITYSANANTONIO,TX78212ABSTRACTTHISPAPERDESCRIBESANINTERDISCIPLINARYDESIGNPROJECTWHICHWASDONEUNDERTHEAUTHORSSUPERVISIONBYAGROUPOFFOURSENIORSTUDENTSINTHEDEPARTMENTOFENGINEERINGSCIENCEATTRINITYUNIVERSITYTHEOBJECTIVEOFTHEPROJECTWASTODEVELOPATEMPERATURECONTROLSYSTEMFORANAIRFILLEDCHAMBERTHESYSTEMWASTOALLOWENTRYOFADESIREDCHAMBERTEMPERATUREINAPRESCRIBEDRANGEANDTOEXHIBITOVERSHOOTANDSTEADYSTATETEMPERATUREERROROFLESSTHAN1DEGREEKELVININTHEACTUALCHAMBERTEMPERATURESTEPRESPONSETHEDETAILSOFTHEDESIGNDEVELOPEDBYTHISGROUPOFSTUDENTS,BASEDONAMOTOROLAMC68HC05FAMILYMICROCONTROLLER,AREDESCRIBEDTHEPEDAGOGICALVALUEOFTHEPROBLEMISALSODISCUSSEDTHROUGHADESCRIPTIONOFSOMEOFTHEKEYSTEPSINTHEDESIGNPROCESSITISSHOWNTHATTHESOLUTIONREQUIRESBROADKNOWLEDGEDRAWNFROMSEVERALENGINEERINGDISCIPLINESINCLUDINGELECTRICAL,MECHANICAL,ANDCONTROLSYSTEMSENGINEERING1INTRODUCTIONTHEDESIGNPROJECTWHICHISTHESUBJECTOFTHISPAPERORIGINATEDFROMAREALWORLDAPPLICATIONAPROTOTYPEOFAMICROSCOPESLIDEDRYERHADBEENDEVELOPEDAROUNDANOMEGATMMODELCN390TEMPERATURECONTROLLER,ANDTHEOBJECTIVEWASTODEVELOPACUSTOMTEMPERATURECONTROLSYSTEMTOREPLACETHEOMEGASYSTEMTHEMOTIVATIONWASTHATACUSTOMCONTROLLERTARGETEDSPECIFICALLYFORTHEAPPLICATIONSHOULDBEABLETOACHIEVETHESAMEFUNCTIONALITYATAMUCHLOWERCOST,ASTHEOMEGASYSTEMISUNNECESSARILYVERSATILEANDEQUIPPEDTOHANDLEAWIDEVARIETYOFAPPLICATIONSTHEMECHANICALLAYOUTOFTHESLIDEDRYERPROTOTYPEISSHOWNINFIGURE1THEMAINELEMENTOFTHEDRYERISALARGE,INSULATED,AIRFILLEDCHAMBERINWHICHMICROSCOPESLIDES,EACHWITHATISSUESAMPLEENCASEDINPARAFFIN,CANBESETONCADDIESINORDERTHATTHEPARAFFINMAINTAINTHEPROPERCONSISTENCY,THETEMPERATUREINTHESLIDECHAMBERMUSTBEMAINTAINEDATADESIREDCONSTANTTEMPERATUREASECONDCHAMBERTHEELECTRONICSENCLOSUREHOUSESARESISTIVEHEATERANDTHETEMPERATURECONTROLLER,ANDAFANMOUNTEDONTHEENDOFTHEDRYERBLOWSAIRACROSSTHEHEATER,CARRYINGHEATINTOTHESLIDECHAMBERTHISDESIGNPROJECTWASCARRIEDOUTDURINGACADEMICYEAR199697BYFOURSTUDENTSUNDERTHEAUTHORSSUPERVISIONASASENIORDESIGNPROJECTINTHEDEPARTMENTOFENGINEERINGSCIENCEATTRINITYUNIVERSITYTHEPURPOSEOFTHISPAPERISTODESCRIBETHEPROBLEMANDTHESTUDENTSSOLUTIONINSOMEDETAIL,ANDTODISCUSSSOMEOFTHEPEDAGOGICALOPPORTUNITIESOFFEREDBYANINTERDISCIPLINARYDESIGNPROJECTOFTHISTYPETHESTUDENTSOWNREPORTWASPRESENTEDATTHE1997NATIONALCONFERENCEONUNDERGRADUATERESEARCH1SECTION2GIVESAMOREDETAILEDSTATEMENTOFTHEPROBLEM,INCLUDINGPERFORMANCESPECIFICATIONS,ANDSECTION3DESCRIBESTHESTUDENTSDESIGNSECTION4MAKESUPTHEBULKOFTHEPAPER,ANDDISCUSSESINSOMEDETAILSEVERALASPECTSOFTHEDESIGNPROCESSWHICHOFFERUNIQUEPEDAGOGICALOPPORTUNITIESFINALLY,SECTION5OFFERSSOMECONCLUSIONS2PROBLEMSTATEMENTTHEBASICIDEAOFTHEPROJECTISTOREPLACETHERELEVANTPARTSOFTHEFUNCTIONALITYOFANOMEGACN390TEMPERATURECONTROLLERUSINGACUSTOMDESIGNEDSYSTEMTHEAPPLICATIONDICTATESTHATTEMPERATURESETTINGSAREUSUALLYKEPTCONSTANTFORLONGPERIODSOFTIME,BUTITSNONETHELESSIMPORTANTTHATSTEPCHANGESBETRACKEDINA“REASONABLE”MANNERTHUSTHEMAINREQUIREMENTSBOILDOWNTOALLOWINGACHAMBERTEMPERATURESETPOINTTOBEENTERED,DISPLAYINGBOTHSETPOINTANDACTUALTEMPERATURES,ANDTRACKINGSTEPCHANGESINSETPOINTTEMPERATUREWITHACCEPTABLERISETIME,STEADYSTATEERROR,ANDOVERSHOOTALTHOUGHNOTEXPLICITLYAPARTOFTHESPECIFICATIONSINTABLE1,ITWASCLEARTHATTHECUSTOMERDESIREDDIGITALDISPLAYSOFSETPOINTANDACTUALTEMPERATURES,ANDTHATSETPOINTTEMPERATUREENTRYSHOULDBEDIGITALASWELLASOPPOSEDTO,SAY,THROUGHAPOTENTIOMETERSETTING3SYSTEMDESIGNTHEREQUIREMENTSFORDIGITALTEMPERATUREDISPLAYSANDSETPOINTENTRYALONEAREENOUGHTODICTATETHATAMICROCONTROLLERBASEDDESIGNISLIKELYTHEMOSTAPPROPRIATEFIGURE2SHOWSABLOCKDIAGRAMOFTHESTUDENTSDESIGNTHEMICROCONTROLLER,AMOTOROLAMC68HC705B166805FORSHORT,ISTHEHEARTOFTHESYSTEMITACCEPTSINPUTSFROMASIMPLEFOURKEYKEYPADWHICHALLOWSPECIFICATIONOFTHESETPOINTTEMPERATURE,ANDITDISPLAYSBOTHSETPOINTANDMEASUREDCHAMBERTEMPERATURESUSINGTWODIGITSEVENSEGMENTLEDDISPLAYSCONTROLLEDBYADISPLAYDRIVERALLTHESEINPUTSANDOUTPUTSAREACCOMMODATEDBYPARALLELPORTSONTHE6805CHAMBERTEMPERATUREISSENSEDUSINGAPRECALIBRATEDTHERMISTORANDINPUTVIAONEOFTHE6805SANALOGTODIGITALINPUTSFINALLY,APULSEWIDTHMODULATIONPWMOUTPUTONTHE6805ISUSEDTODRIVEARELAYWHICHSWITCHESLINEPOWERTOTHERESISTIVEHEATEROFFANDONFIGURE3SHOWSAMOREDETAILEDSCHEMATICOFTHEELECTRONICSANDTHEIRINTERFACINGTOTHE6805THEKEYPAD,ASTORM3K041103,HASFOURKEYSWHICHAREINTERFACEDTOPINSPA0PA3OFPORTA,CONFIGUREDASINPUTSONEKEYFUNCTIONSASAMODESWITCHTWOMODESARESUPPORTEDSETMODEANDRUNMODEINSETMODETWOOFTHEOTHERKEYSAREUSEDTOSPECIFYTHESETPOINTTEMPERATUREONEINCREMENTSITANDONEDECREMENTSTHEFOURTHKEYISUNUSEDATPRESENTTHELEDDISPLAYSAREDRIVENBYAHARRISSEMICONDUCTORICM7212DISPLAYDRIVERINTERFACEDTOPINSPB0PB6OFPORTB,CONFIGUREDASOUTPUTSTHETEMPERATURESENSINGTHERMISTORDRIVES,THROUGHAVOLTAGEDIVIDER,PINAN0ONEOFEIGHTANALOGINPUTSFINALLY,PINPLMAONEOFTWOPWMOUTPUTSDRIVESTHEHEATERRELAYSOFTWAREONTHE6805IMPLEMENTSTHETEMPERATURECONTROLALGORITHM,MAINTAINSTHETEMPERATUREDISPLAYS,ANDALTERSTHESETPOINTINRESPONSETOKEYPADINPUTSBECAUSEITISNOTCOMPLETEATTHISWRITING,SOFTWAREWILLNOTBEDISCUSSEDINDETAILINTHISPAPERTHECONTROLALGORITHMINPARTICULARHASNOTBEENDETERMINED,BUTITISLIKELYTOBEASIMPLEPROPORTIONALCONTROLLERANDCERTAINLYNOTMORECOMPLEXTHANAPIDSOMECONTROLDESIGNISSUESWILLBEDISCUSSEDINSECTION4,HOWEVER4THEDESIGNPROCESSALTHOUGHESSENTIALLYTHEPROJECTISJUSTTOBUILDATHERMOSTAT,ITPRESENTSMANYNICEPEDAGOGICALOPPORTUNITIESTHEKNOWLEDGEANDEXPERIENCEBASEOFASENIORENGINEERINGUNDERGRADUATEAREJUSTENOUGHTOBRINGHIMORHERTOTHEBRINKOFASOLUTIONTOVARIOUSASPECTSOFTHEPROBLEMYET,INEACHCASE,REALWORLDCONSIDERATIONSCOMPLICATETHESITUATIONSIGNIFICANTLYFORTUNATELYTHESECOMPLICATIONSARENOTINSURMOUNTABLE,ANDTHERESULTISAVERYBENEFICIALDESIGNEXPERIENCETHEREMAINDEROFTHISSECTIONLOOKSATAFEWASPECTSOFTHEPROBLEMWHICHPRESENTTHETYPEOFLEARNINGOPPORTUNITYJUSTDESCRIBEDSECTION41DISCUSSESSOMEOFTHEFEATURESOFASIMPLIFIEDMATHEMATICALMODELOFTHETHERMALPROPERTIESOFTHESYSTEMANDHOWITCANBEEASILYVALIDATEDEXPERIMENTALLYSECTION42DESCRIBESHOWREALISTICCONTROLALGORITHMDESIGNSCANBEARRIVEDATUSINGINTRODUCTORYCONCEPTSINCONTROLDESIGNSECTION43POINTSOUTSOMEIMPORTANTDEFICIENCIESOFSUCHASIMPLIFIEDMODELING/CONTROLDESIGNPROCESSANDHOWTHEYCANBEOVERCOMETHROUGHSIMULATIONFINALLY,SECTION44GIVESANOVERVIEWOFSOMEOFTHEMICROCONTROLLERRELATEDDESIGNISSUESWHICHARISEANDLEARNINGOPPORTUNITIESOFFERED41MATHEMATICALMODELLUMPEDELEMENTTHERMALSYSTEMSAREDESCRIBEDINALMOSTANYINTRODUCTORYLINEARCONTROLSYSTEMSTEXT,ANDJUSTTHISSORTOFMODELISAPPLICABLETOTHESLIDEDRYERPROBLEMFIGURE4SHOWSASECONDORDERLUMPEDELEMENTTHERMALMODELOFTHESLIDEDRYERTHESTATEVARIABLESARETHETEMPERATURESTAOFTHEAIRINTHEBOXANDTBOFTHEBOXITSELFTHEINPUTSTOTHESYSTEMARETHEPOWEROUTPUTQTOFTHEHEATERANDTHEAMBIENTTEMPERATURETMAANDMBARETHEMASSESOFTHEAIRANDTHEBOX,RESPECTIVELY,ANDCAANDCBTHEIRSPECIFICHEATS1AND2AREHEATTRANSFERCOEFFICIENTSFROMTHEAIRTOTHEBOXANDFROMTHEBOXTOTHEEXTERNALWORLD,RESPECTIVELYITSNOTHARDTOSHOWTHATTHELINEARIZEDSTATEEQUATIONSCORRESPONDINGTOFIGURE4ARETAKINGLAPLACETRANSFORMSOF1AND2ANDSOLVINGFORTAS,WHICHISTHEOUTPUTOFINTEREST,GIVESTHEFOLLOWINGOPENLOOPMODELOFTHETHERMALSYSTEMWHEREKISACONSTANTANDDSISASECONDORDERPOLYNOMIALK,TZ,ANDTHECOEFFICIENTSOFDSAREFUNCTIONSOFTHEVARIOUSPARAMETERSAPPEARINGIN1AND2OFCOURSETHEVARIOUSPARAMETERSIN1AND2ARECOMPLETELYUNKNOWN,BUTITSNOTHARDTOSHOWTHAT,REGARDLESSOFTHEIRVALUES,DSHASTWOREALZEROSTHEREFORETHEMAINTRANSFERFUNCTIONOFINTERESTWHICHISTHEONEFROMQS,SINCEWELLASSUMECONSTANTAMBIENTTEMPERATURECANBEWRITTENMOREOVER,ITSNOTTOOHARDTOSHOWTHAT1TP11TZ1TP2,IE,THATTHEZEROLIESBETWEENTHETWOPOLESBOTHOFTHESEAREEXCELLENTEXERCISESFORTHESTUDENT,ANDTHERESULTISTHEOPENLOOPPOLEZERODIAGRAMOFFIGURE5OBTAININGACOMPLETETHERMALMODEL,THEN,ISREDUCEDTOIDENTIFYINGTHECONSTANTKANDTHETHREEUNKNOWNTIMECONSTANTSIN3FOURUNKNOWNPARAMETERSISQUITEAFEW,BUTSIMPLEEXPERIMENTSSHOWTHAT1TP1_1TZ1TP2SOTHATTZTP2_0AREGOODAPPROXIMATIONSTHUSTHEOPENLOOPSYSTEMISESSENTIALLYFIRSTORDERANDCANTHEREFOREBEWRITTENWHERETHESUBSCRIPTP1HASBEENDROPPEDSIMPLEOPENLOOPSTEPRESPONSEEXPERIMENTSSHOWTHAT,FORAWIDERANGEOFINITIALTEMPERATURESANDHEATINPUTS,K_014_WANDT_295S142CONTROLSYSTEMDESIGNUSINGTHEFIRSTORDERMODELOF4FORTHEOPENLOOPTRANSFERFUNCTIONGAQSANDASSUMINGFORTHEMOMENTTHATLINEARCONTROLOFTHEHEATERPOWEROUTPUTQTISPOSSIBLE,THEBLOCKDIAGRAMOFFIGURE6REPRESENTSTHECLOSEDLOOPSYSTEMTDSISTHEDESIRED,ORSETPOINT,TEMPERATURE,CSISTHECOMPENSATORTRANSFERFUNCTION,ANDQSISTHEHEATEROUTPUTINWATTSGIVENTHISSIMPLESITUATION,INTRODUCTORYLINEARCONTROLDESIGNTOOLSSUCHASTHEROOTLOCUSMETHODCANBEUSEDTOARRIVEATACSWHICHMEETSTHESTEPRESPONSEREQUIREMENTSONRISETIME,STEADYSTATEERROR,ANDOVERSHOOTSPECIFIEDINTABLE1THEUPSHOT,OFCOURSE,ISTHATAPROPORTIONALCONTROLLERWITHSUFFICIENTGAINCANMEETALLSPECIFICATIONSOVERSHOOTISIMPOSSIBLE,ANDINCREASINGGAINSDECREASESBOTHSTEADYSTATEERRORANDRISETIMEUNFORTUNATELY,SUFFICIENTGAINTOMEETTHESPECIFICATIONSMAYREQUIRELARGERHEATOUTPUTSTHANTHEHEATERISCAPABLEOFPRODUCINGTHISWASINDEEDTHECASEFORTHISSYSTEM,ANDTHERESULTISTHATTHERISETIMESPECIFICATIONCANNOTBEMETITISQUITEREVEALINGTOTHESTUDENTHOWUSEFULSUCHANOVERSIMPLIFIEDMODEL,CAREFULLYARRIVEDAT,CANBEINDETERMININGOVERALLPERFORMANCELIMITATIONS43SIMULATIONMODELGROSSPERFORMANCEANDITSLIMITATIONSCANBEDETERMINEDUSINGTHESIMPLIFIEDMODELOFFIGURE6,BUTTHEREAREANUMBEROFOTHERASPECTSOFTHECLOSEDLOOPSYSTEMWHOSEEFFECTSONPERFORMANCEARENOTSOSIMPLYMODELEDCHIEFAMONGTHESEAREQUANTIZATIONERRORINANALOGTODIGITALCONVERSIONOFTHEMEASUREDTEMPERATUREANDTHEUSEOFPWMTOCONTROLTHEHEATERBOTHOFTHESEARENONLINEARANDTIMEVARYINGEFFECTS,ANDTHEONLYPRACTICALWAYTOSTUDYTHEMISTHROUGHSIMULATIONOREXPERIMENT,OFCOURSEFIGURE7SHOWSASIMULINKTMBLOCKDIAGRAMOFTHECLOSEDLOOPSYSTEMWHICHINCORPORATESTHESEEFFECTSA/DCONVERTERQUANTIZATIONANDSATURATIONAREMODELEDUSINGSTANDARDSIMULINKQUANTIZERANDSATURATIONBLOCKSMODELINGPWMISMORECOMPLICATEDANDREQUIRESACUSTOMSFUNCTIONTOREPRESENTITTHISSIMULATIONMODELHASPROVENPARTICULARLYUSEFULINGAUGINGTHEEFFECTSOFVARYINGTHEBASICPWMPARAMETERSANDHENCESELECTINGTHEMAPPROPRIATELYIE,THELONGERTHEPERIOD,THELARGERTHETEMPERATUREERRORPWMINTRODUCESONTHEOTHERHAND,ALONGPERIODISDESIRABLETOAVOIDEXCESSIVERELAY“CHATTER,”AMONGOTHERTHINGSPWMISOFTENDIFFICULTFORSTUDENTSTOGRASP,ANDTHESIMULATIONMODELALLOWSANEXPLORATIONOFITSOPERATIONANDEFFECTSWHICHISQUITE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EMC14489,1990MOTOROLASEMICONDUCTORAPPLICATIONNOTEAN4313MOTOROLA,INC,PHOENIX,AZ,HC05MCULEDDRIVETECHNIQUESUSINGTHEMC68HC705J1A,1995MOTOROLASEMICONDUCTORAPPLICATIONNOTEAN12384MOTOROLA,INC,PHOENIX,AZ,HC05MCUKEYPADDECODINGTECHNIQUESUSINGTHEMC68HC705J1A,1995MOTOROLASEMICONDUCTORAPPLICATIONNOTEAN12395MOTOROLA,INC,PHOENIX,AZ,RAPIDINTEGRATEDDEVELOPMENTENVIRONMENTUSERSMANUAL,1993RAPIDWASDEVELOPEDBYPEMICROCOMPUTERSYSTEMS,INC单片机温度控制一个跨学科的本科生工程设计项目JAMESSMCDONALD工程科学系三一大学德克萨斯州圣安东尼奥市78212摘要本文所描述的是作者领导由四个三一大学高年级学生组成的团队进行的一个跨学科工程项目的设计。该项目的目标是设计一个气室内温度控制系统。该系统的要求是当实际气室的温度阶跃响应时，规定范围内的温度进入气室后，稳定时的温度误差和超调量必须少于一个绝对温度。本组学生开发设计是基于摩托罗拉MC68HC05系列单片机。该问题的教学价值也通过某些步骤的关键描述在本文说明。研究结果表明，解决该方案需要具有广泛的工程学科知识，包括相关电子、机械和控制系统工程的知识。1引言该设计项目来自一个实际应用问题，一个关于显微镜载玻片干燥剂温控器欧米茄CN390温度控制器，而这个设计的目标是研发一个自定义的通用温度控制系统取代欧米茄系统、一个以更低的成本实现相同功能的自定义控制器，就像欧米茄系统一样，并不需要能够全方位的处理各种问题。该载玻片干燥机的机械布局如图1所示。干燥机的主体是一个足够大的绝缘充气室，里面依次存放着薄纸包着的石蜡。为了使石蜡保持适当稳定性，载玻片气室的温度必须维持稳定。第二个气筒（电子围绕元件）设有一个电阻加热器、一个温度控制器以及一个安装在干燥机上的风扇，是为了把风吹过加热器，把热量带到载玻片气室。图11载玻片干燥机的机械布局自199697学年来，本文作者带领四位三一大学工程科学系的高年级学生开展此项目的研究。本文的目的说明了提出一些问题并详细阐述学生的一些解决方案，而且讨论了这种类型的跨学科设计项目在教学方面应用的问题。这份学生报告曾经在1997年全国本科毕业生研讨会上提出过并讨论过。第2节给出该设计的更多详细情况，包括性能规格。第3节具体学生的设计。第4节是论文的主体，讨论该设计在教学应用方面的实施问题。最后，第5节全文总结。2问题阐述该项目基本的思想是设计一个自定义温度控制系统来取代相关的欧米茄CN390温度控制器。温度时通常保持在一个稳定的常数，但重要的是阶跃变化可以被“合理”的跟踪。因此主要要求如下可以对空气室的温度进行设定，同时显示设定值和实际温度，以及在设定温度值情况下，可接受范围内的跟踪阶跃变化，稳态误差，超调量。设定温度接口设定温度显示室内温度显示范围精度准确度60991C1C室内温度阶梯响应范围（稳定状态）精度（稳定状态）最大超调设定时间（到1）60991C1C120S表1精确的规格说明尽管表1部分说明并不明确，但是它清楚的反映了人们对数字显示器在设定值和实际温度的要求和温度应该通过数值输入来设定（而不是，通过电位器设置）。3系统设计根据微控设计，数字温度显示和单点输入的要求可能是最合适的。图2为学生的设计框图。图22温度控制器硬件结构图摩托罗拉MC68HC705B16（简称6805），是系统的核心。它通过一个简单的4键小键盘对温度进行设定，同时使用两个显示驱动控制7段LED数码管来显示定值和气室温度的测量值。所有这些，输入和输出信号与6805的并行口相连。气室的温度值使用预校准热敏电阻测量，并通过6805的数模转换输入。最后，6085的脉冲宽度调制（PWM）输出用来驱动一个继电器，以控制线性电阻加热器的闭合和断开。图3更详细的显示了6805的接口和电子器件。使用暴风3K041103型号四键键盘，通过PA0PA3端口进行数据输入。其中一个重要的功能是进行模式切换。两种模式固定模式和运行模式。在固定模式下，其他两个键用于设定温度，一个增加，一个减少，第四个按键暂无作用。LED显示屏由哈里斯半导体ICM7212进行驱动，通过PB0PB6端口与芯片相连，作为输出。热敏电阻由电压分频器驱动，通过AN0针脚（八个模拟输入端口中的一个）相连。最后，PLMA针脚（两个PWM输出端口中的一个）驱动加热继电器。图23单片机原理图图3单片机原理图是关于用软件实现温度控制算法、保持温度显示以及改变键盘输入响应，这将不会在本文详细讨论，因为这并不是本文的重点，也没有编译完成。软件部分还没有确定控制算法，但很可能是一个简单的比例控制，比PID算法简单。一些控制设计的问题将在第四节讨论。4设计过程虽然该项目的本质是建立一个恒温器，但它有许多很好的契机可以供教学借鉴。高级工程本科教育的知识只是能够让学生们具有解决问题的能力。然而，很多情况下，实际情况却和理论有些不同。不过，这些不是问题，参与这个项目的设计，将获得很多设计方面的宝贵经验。本节的其余部分着眼于其他的几个方面41节讨论系统的一些特征，简化系统热性能的数学模型，以及一些简单理论的证明。42节介绍确定实际控制算法。43节指出控制设计程序的一些不足，并通过模拟环境，指出怎样克服问题。44节给出单片机的一些设计相关概述，以及出现问题和值得借鉴之处。41数学模型集总元件热系统符合线性控制，适用于载玻片干燥机的问题。图4显示了二阶集总元件热量模型的载玻片干燥机。状态变量是温度，TA是箱内空气的温度，TB是箱子本身的温度。该系统输入功率等于QT的热量和环境温度T的和。MA，MB分别对应空气和箱子的质量。CA和CB则分别是其对应热量。M1和M2分别是空气与箱子间以及箱子与外界间的传热系数。图41集总元件热模型由图4可以推出（线性）状态方程拉普拉斯变换1和2等式，并整理TAS。有趣的是，可以推出一个开环的热系统方程。其中K是一个常数，DS是一个二阶的多项式。K,TZ,以及系数DS和在1和2等式中出现的系数功能相近。当然，在1和2等式中各种参数在未知的情况下，不难证明DS与其他参数的值无关，具有两个零点。因此传递函数可以写成（我们假设环境温度为常数）此外,可以推出1/TP11/TZ1/TP2,即，零点在两极之间。开环零极点如图5所示。图42GAQS的零极点为了获取完整的热模型，从3式中除去常数K和3个未知的时间常数。四个未知参数并不少，但由简单的实验表明，1/TP11/TZ，1/TP2统基本上是一阶函数，且TZ,TP2近似为0。因此，开环系可以写成下标P1已经被去掉了过初始温度和热量值大范围内的设置，简单的开环阶跃响应实验结果表明，K014O/W，295S。42控制系