Multivariable control of indoor air temperature and humidity in a direct

Multivariable control of indoor air temperature and humidity in a direct expansion (DX)air conditioning (A/C)systemQi Qi,Shiming Deng *Department of Building Services Engineering,The Hong Kong Polytechnic University,Kowloon,Hong Kong SAR,Chinaa r t i c l e i n f oArticle history:Received 26June 2008Received in revised form 10October 2008Accepted 4November 2008Keywords:Temperature and humidity Multivariable control Air conditioning system Direct expansiona b s t r a c tControlling indoor humidity at an appropriate level is very important since this affects occupants’thermal comfort and indoor air quality (IAQ).The paper presents an investigation on developing a multi-input multi-output (MIMO)control strategy for simultaneously controlling the indoor air temperature and humidity by varying the speeds of both compressor and supply fan in an experimental DX A/C system.The MIMO-based controller was designed based on the linearized dynamic model of the experimental DX A/C system.The Linear Quadratic Gaussian (LQG)technique was used in designing the MIMO-based controller.The controllability tests with respect to both the disturbance rejection capability and the command following capability were carried out to assess the control performance of MIMO controller.The results of disturbance rejection capability test showed that the MIMO control strategy can effectively maintain the indoor air temperature and humidity to their respective settings after an unmeasured heat load disturbance was imposed by simultaneously varying speeds of both the compressor and the supply fan of the DX A/C system.Furthermore,in the command following capability test for indoor air temperature,the test results showed that the indoor air temperature can be controlled to its new setting while indoor humidity remained unchanged.Similar test results were also observed in the command following capability test for indoor humidity.Therefore,the MIMO controller developed can effectively control indoor air temperature and humidity simultaneously by varying compressor speed and supply fan speed of the DX A/C pared to the previous related studies using conventional on–off control method or single-input single-output (SISO)control strategy,which can only effectively control either air temperature or relative humidity,the MIMO controller can simultaneously control the indoor air temperature and humidity with adequate control sensitivity and accuracy.The application of MIMO control strategy developed can be extended to other HVAC systems in the future to improve their operating performance and energy efficiency.Ó2008Elsevier Ltd.All rights reserved.1.IntroductionControlling indoor humidity at an appropriate level is very important since this directly affects occupants’thermal comfort and indoor air quality (IAQ)[1].Traditionally,reheating has been the principal method for indoor humidity control in central HVAC systems.This method is inherently costly and energy inefficient since it uses a great deal of energy to overcool the air,and then more energy to reheat the air to a suitable supply temperature.However,reheating is uncommon for DX A/C systems and controlling indoor humidity at an appropriate level using a DX A/C system is difficult.This is because the cooling coil in a DX A/C system must perform both cooling and dehumidification simulta-neously.Most DX A/C units are currently equipped with a single-speed compressor and supply fans,relying on on–off cycling compressor as a low-cost approach to maintain only indoor air dry-bulb temperature.This results in either space overcooling or an uncontrolled equilibrium indoor relative humidity (RH)level.The recent advancement of low-cost variable-speed drive (VSD)technology offers tremendous opportunities for improving indoor thermal control and energy efficiency for air conditioning.When the speed of a supply fan decreases,a lower airflow rate will decrease the sensible heat ratio (SHR)and improve the latent heat transfer in an air cooling and dehumidifying coil.This has been proved by the previous studies which demonstrated that lowering the airflow rate across an evaporator coil can improve moisture removal and thus enhance the control of indoor humidity [2,3].However,this would also impact on air temperature,so that the air exiting the coil will be not only drier but also cooler.On the other hand,for a DX A/C system when its compressor speed increases,its output cooling capacity will increase but its evaporating tempera-ture will decrease,hence its ability to remove moisture from air is*Corresponding author.Tel.:þ852********;fax:þ852********.E-mail address:besmd@.hk (S.Deng).Contents lists available at ScienceDirectBuilding and Environmentjournal homepage:/locate/buildenv0360-1323/$–see front matter Ó2008Elsevier Ltd.All rights reserved.doi:10.1016/j.buildenv.2008.11.001Building and Environment 44(2009)1659–1667improved.However,the changes in compressor speed would also affect the sensible heat transfer as its output cooling capacity changes.This will affect the indoor air temperature.Therefore,varying either supply fan speed or compressor speed would influ-ence both indoor air temperature and humidity although at difference magnitudes.Therefore it is possible to simultaneously control indoor air temperature and humidity by varying speeds of both compressor and supply fan for DX A/C system.Various control strategies have been developed for air condi-tioning systems.A fuzzy model-based predictive control scheme for controlling the supply air temperature was developed by Thompson and Dexter [4].Experimental results showed that the control performance of supply air temperature was satisfactory at both high and low airflow rates.Huang and Lam [5]developed an adaptive learning algorithm based on genetic algorithms (GA)for tuning proportional–integral–derivative (PID)controllers to achieve optimal control performance.Krakow et al.[6]proposed a PID control method to control indoor air temperature by varying compressor speed,and humidity by varying supply fan speed,respectively.Air conditioning systems have been traditionally controlled using simple single-input single-output techniques.Two control loops have been treated as two independent SISO systems [6],with their cross-coupling effects being often ignored.However,MIMO control strategies are able to take into account the coupling effects of multiple variables.They are more effective in simultaneously achieving multiple control objectives,such as:temperature,humidity,capacity,efficiency,etc.The performance of conventional SISO control strategies is inherently inferior to that of multi-input multi-output control strategies [7].He et al.[8]considered that there were strong cross-coupling effects among various operating parameters in a vapor compres-sion refrigeration cycle,such as evaporating temperature,condensing temperature,degree of superheat,etc.Therefore,in order to improve the transient behavior of the cycle in terms of maintaining a desired degree of superheat and evaporating temperature,an MIMO feedback controller was designed based on a low order lumped-parameter dynamic model [7].Controllability test results showed that the MIMO controller could effectively improve the operating performance and energy efficiency of the vapor compression refrigeration cycle.Rasmussen and Alleyne [9]presented the application of a multivariable adaptive control strategy to a general automotive air conditioning system.The possibility of coordinated modulations of both the expansion valve opening and the airflow rate passing through the evaporator as a control strategy was explored.It was demonstrated that the MIMO-based adaptive approach was very successful in achieving multiple objectives by regulating both degree of superheat and evaporating pressure.Furthermore,an MIMO controller for air temperature in,and ventilation to multiple zones in a building with a model predictive control (MPC)strategy was developed by Yuan and Perez [10].On the other hand,a direct digital control (DDC)-based control algorithm developed by Li and Deng [11,12]used space sensible heat ratio (SHR)as a controlled variable to simul-taneously control space air temperature and relative humidity.However,the control algorithm was not MIMO based.The experimental investigation by Krakow et al.[6]suggested that indoor air temperature and relative humidity may be controlled by separately varying compressor speed and supply fan speed.However,such a control strategy remained fundamentally inadequate.The transient performance of the two decoupled feedback loops (i.e.,the control loop for temperature by varying compressor speed and that for RH by varying supply fan speed)was inherently poor,due to the strong cross-coupling between the two feedback loops.Both air temperature and relative humidity were affected by compressor speed and supply fan speed.It can therefore be seen that artificially decoupling the control of two strongly coupled indoor air parameters,i.e.,air temperature and RH,would only yield poor control performance.Therefore an MIMO control strategy has been developed and is reported in this paper.The strategy considered the coupling effect of indoor air temperature and humidity for simultaneously controlling indoor air temperature and humidity by varying compressor speed and supply fan speed.It helped eliminate the artificial decoupling of the two coupled control variables,so that the control accuracy and sensitivity were improved.The organization of the paper is as follows.Section 2describes briefly an experimental DX A/C system and its dynamic model.The design of the MIMO controller based on the dynamic model afteritsQ.Qi,S.Deng /Building and Environment 44(2009)1659–16671660linearization is presented in Section3.The results of controllabilitytests of the MIMO controller are reported in Section4,and discussions presented in Section5.2.The experimental DX A/C system and its dynamic modelThe experimental DX A/C system was mainly composed of two parts,i.e.,a DX refrigeration plant(refrigerant side)and an air-distribution sub-system(air side).Its simplified schematic diagram is shown in Fig.1.The major components in the DX refrigeration plant included a variable-speed rotor compressor,an electronic expansion valve(EEV),a high-efficiency tube-louver-finned DX evaporator and an air-cooled tube-plate-finned condenser.The evaporator was placed inside the supply air duct on the air side to work as a DX air cooling coil.The nominal output cooling capacity from the DX refrigeration plant was9.9kW,operating with an evaporating temperature of7.2 C,a condensing temperature of 54.4 C,an ambient temperature of35 C and the wind speed of 2m sÀ1.The workingfluid of the plant was refrigerant R22,with a total charge of5.3kg.The air side included an air-distribution ductwork,a variable-speed centrifugal supply fan,and a conditioned space.The space was measured at 6.8m(L)Â3.9m(W)Â2.9m(H).Inside the space,there were sensible heat and moisture load generating units (LGUs),simulating the cooling load in the space.The experimental DX A/C system was fully instrumented for measuring and moni-toring all system operating parameters including temperatures and flow rates of both air and refrigerant,and pressures.The dynamic mathematical model for the DX A/C system was mainly derived from the energy and mass conservation principles. The following assumptions were made in developing the mathe-matical model:(1)perfect air mixing inside all heat exchangers and the thermal space,and no fresh air intake to the system;(2)two regions on the air side of the DX evaporator,i.e.,dry-cooling region and wet-cooling region;(3)negligible thermal losses in air ducts.In the DX A/C system to be modeled,the temperature and moisture content of the air leaving the DX cooling coil,as shown in Fig.1,were T1and W1,respectively.With the air perfect mixing assumption,air temperature,T2,and air moisture content,W2, leaving the conditioned space can be regarded as being equal to those in the conditioned thermal space.Based on the energy conservation principle,the sensible energy balance equation for the conditioned space was C p r V d T2¼C p r fðT1ÀT2ÞþQ loadþQ spl(1) where V was the volume of the conditioned space,Q load the space sensible load,Q spl heat gain of the supply fan,f is the air volumetric flow rate.The relationship between the air volumetricflow rate and supply fan speed S fan is as follows:f¼k fan S fan(2) The heat gain of supply fan increased with the airflow rate as follows:Q spl¼k spl f(3) On the other hand,the moisture mass balance inside the conditioned space wasr V d W2d t¼r fðW1ÀW2ÞþM(4)At the air side of the evaporator,with the assumption of no fresh air, the temperature and moisture content of the air entering evapo-rator were T2and W2,respectively.The air temperature decreased along the evaporator wall and was equal to T3at the end of dry-cooling region.Applying the energy balance principle in the dry-cooling region on the air side yielded:C p r V h1d T3d t¼C p r fðT2ÀT3Þþa1A1T wÀT2þT32(5)On the other hand,the coupled air cooling and dehumidification mainly took place in the wet-cooling region.Therefore the energy balance in the wet-cooling region can be written in enthalpy form:r V h2d h1¼r fðh3Àh1Þþa2A2T wÀT3þT1(6)The relationship between air enthalpy,temperature and mois-ture content wash¼C p Tþh fg W(7) where h fg was the latent heat of vaporization of water.Substitute Eq.(7)into Eq.(6)to obtain:C p r V h2d T1d tþr V h2h fg d W1d t¼C p r fðT3ÀT1Þþr fh fgðW2ÀW1Þþa2A2T wÀT3þT1(8)On the other hand,the refrigerant massflow rate can be determined by[13]:M ref¼sV comv s1À0:015hðP c=P eÞ1=bÀ1i(9)where s was the compressor speed and v s was the specific volume of superheated refrigerant,which can be obtained from measured pressure and temperature of refrigerant at compressor suction using the R22State Equations[14].Therefore the energy balance equation for the evaporator wall can be written asÀC p r VÁwd T wd t¼a1A1T2þT32ÀT wþa2A2T3þT12ÀT wÀM refðh r2Àh r1Þ(10)Variable speedcompressorCondenserFig.1.Simplified schematic diagram of the experimental DX A/C system.Q.Qi,S.Deng/Building and Environment44(2009)1659–16671661The supply air leaving the DX evaporator was assumed to be at 95%saturated.The relationship between air moisture content and temperature can be derived by plotting and curving fitting:d W 1d t Àð2Â0:0198T 1þ0:085Þd T1d t ¼0(11)Eqs.(1),(4),(5),(8),(10)and (11)formed the dynamic model of the experimental DX A/C system,which can be written as follows:The dynamic mathematical model of DX A/C system was linearized and written in state-space representation which was suitable for designing multivariable control,as follows:&_x¼Ax þBu y ¼Cx(13)where the state variables x ¼½d T 1;d T 2;d T 3;d T w ;d W 1;d W 2 T ,input variables u ¼½d f ;d s T ,and the output variables y ¼½d T 2;d W 2 T ,which were the dynamic deviations from their set points,respec-tively.A ,B and C were the coefficient matrices.The above dynamic mathematical model for the experimental DX A/C system has been experimentally validated and was reported elsewhere in detail [15].3.Design of the MIMO controllerThe MIMO-based controller was designed based on the linear-ized dynamic model of the experimental DX A/C system,i.e.,Eq.(13).A schematic diagram of the MIMO control system is shown in Fig.2.The main difference between an MIMO control system and an SISO control system was that the decoupled SISO control system only used one feedback signal to generate a single control input to a controlled object.For example,indoor air temperature was used to generate a signal to control compressor speed.However,the MIMO control system used two feedback signals,i.e.,indoor air temperature and moisture content,to generate two signals to control both compressor speed and supply fan speed simultaneously.Fig.3shows the block diagram of MIMO feedback control system.The MIMO controller was designed based on the technique of Linear Quadratic Gaussian (LQG)[16].An LQG multivariable controller was an observer-based compensator that used the Kal-man filter to optimally estimate unmeasured state variables basedon measured controlled variables,and then used the optimal full state feedback to generate a control law.The optimal gain matrix,K 1,and Kalman filter gain matrix,L ,were determined based on the system matrices in the dynamic model of the DX A/C system and the weighting matrices for linear quadratic regulator and weighting matrices for linear quadratic estimator.r was the reference set points of indoor air temperature and moisture content,and y the actual air temperature and moisture content inside the conditioned space.The optimal gain matrix,K 1,was calculated such that the feed-back control law,u ¼ÀK 1x ,minimized the cost function which compromised output errors and control efforts:J ¼Zðy 0Qy þu 0Ru Þd t(14)where Q and R are the weighting matrices.To obtain Kalman filter gain matrix,L ,the linear quadratic estimator design was considered.For the system with process noise and measurement noise [13]:&_x¼Ax þBu þw y ¼Cx þv(15)where w was process noise and v measurement noise,its optimal observer was given by_b x ¼A b x þBu þL y ÀC b x(16)where b x was an estimate of the state variables x .The observer gain was computed fromL ¼XC 0R À1(17)wherePwas the solution of an Algebraic Riccati Equation [13]:AXþXA 0þQ 0ÀXC 0R À10CX¼0(18)The weighting matrices Q 0and R 0can be suitably selected to obtainthe desire performance of the system.In this paper,Q 0¼1001!and R 0¼ 11!After the gain matrices K 1and L were obtained,the transfer function of the observer-based compensator LQG can be expressed asK ðs Þ¼K 1ðsI ÀA þBK 1þLC ÞÀ1L(19)In order to track the reference,an integrator should be included in the controller.The integrator variable was written as follows:_h¼r Ày (20)Therefore,the augmented state equationsbecomeFig.2.The schematic diagram of the MIMO control system.Q.Qi,S.Deng /Building and Environment 44(2009)1659–16671662_x a¼_x_h !¼A0ÀC0!xh!þB00I!ur!¼A a x aþB a u a(21)y¼½C0 xh!¼C a x a;x h!¼x a(22)The gain matrix,K2,can be selected to obtain the desire performance of the controlled system.In this paper,K2¼À0:0068750:00165À0:000405À0:000162!After obtaining the gain matrix K2,the control law can be expressed as follows:u¼ÀK1b xÀK2h¼À½K1K2 b xh!¼ÀK a x a(23)The transfer function of the LQG with integrators compensator can be expressed asKðsÞ¼K aðsIÀAþBK aþLCÞÀ1L(24) 4.Controllability test resultsThe MIMO controller was implemented in the experimental DX A/ C system.With the computerized instrumentation,all the measured data can be recorded for subsequent analysis,and the control inputs to the DX A/C system can be generated based on the feedback signals of measured indoor air temperature and moisture content.For the MIMO controller,its control performances were evalu-ated based on the following criteria:(1)Disturbance rejection:the output variables,i.e.,indoor airtemperature and moisture were to be maintained at their respective set points through simultaneously varying compressor speed and supply fan speed when the space was subjected to disturbances of space sensible and latent heat load changes.(2)Command following:the output variables can track changes oftheir set points with stability.When the set point of air temperature or moisture content was changed,the MIMO controller could react so that indoor air temperature or moisture content can be maintained at their respective new set point.For all controllability tests,indoor humidity control was via controlling indoor air wet-bulb temperature.Air moisture content can be evaluated by the properties of moist air after knowing air dry-bulb and wet-bulb temperatures.The MIMO controller was designed around an operating point of an indoor air dry-bulb temperature of24 C and a wet-bulb temperature of17 C(or an equivalent of air moisture content of9.34g/(kg dry air)or indoor relative humidity of50%).The compressor speed and supply fan speed were all initially set at3955rpm and2180rpm,both being 50%of their maximum speeds,respectively.The condenser cooling airflow rate was maintained constant at3100m3/h,with its inlet temperaturefixed at35 C.The refrigerant degree of superheat was maintained constant at6 C by using the EEV.In order to protect the compressor and supply fan and to avoid the speeds from changing sharply,each step change of both compressor speed and supply fan speed was set at not more than5%of their respective maximum speeds.The time interval between step changes was30s in all tests.4.1.Disturbance rejection capability testFor the test,indoor air settings were24 C(dry-bulb tempera-ture)and17 C(wet-bulb temperature)(or an equivalent of indoor air moisture content9.34g/kg dry air).These settings were expected to be maintained after the disturbances of both sensible load and latent heat loads were introduced.The MIMO controller would be enabled when either the deviation between the measured indoor air dry-bulb temperature and its setting or the deviation between the measured indoor air wet-bulb temperature and its setting was greater thanÆ0.5 C as a result of the disturbances.Figs.4and5present the controllability test results of distur-bance rejection capability of the MIMO controller.The variation profile of heat loads imposed over the entire test period is shown in Fig.4a.At t¼350s,the disturbances were introduced,with the sensible and latent heat loads increased from5.0kW to5.6kW,and from2.0kW to2.3kW,respectively.Fig.4b shows the variation profile of indoor air temperatures.During thefirst350s of the test, indoor air temperatures were maintained at their respective set points,i.e.,24 C dry-bulb and17 C wet-bulb before the distur-bances were introduced.Afterward,both air dry-bulb and wet-bulb temperatures increased gradually.At t¼1360s when indoor air dry-bulb temperature went over24.5 C,the MIMO controller was enabled.Fig.4c shows the variation profiles of compressor speed and supply fan speed.Following the action of the MIMO controller, both speeds increased.Towards the end of the test,the speeds of compressor and supply fan stabilized at around4750rpm and 2380rpm,respectively.The variation profiles of indoor air moisture content and relative humidity are presented in Fig.5a and b, respectively.As seen from Fig.4b,the MIMO controller can well control the indoor air dry-and wet-bulb temperatures to their respective set points,achieving a satisfactory performance in termsIntegratorFig.3.Block diagram of MIMO feedback control system.Q.Qi,S.Deng/Building and Environment44(2009)1659–16671663of disturbance rejection capability.At the end of test,the deviation between indoor air dry-bulb temperature and its setting was less than 0.1 C.mand following capability testsFor the command following capability tests,it was expected that when there were changes in set points,the MIMO controller would react so that the new settings can be tracked and maintained.Fig.6shows the controllability test results of the MIMO controller with respect to its command following capability,following a change in the set point of indoor air dry-bulb temper-ature.When the set point was lowered from 24 C to 23.5 C,with the setting of indoor air moisture content remaining unchanged at 9.34g/(kg dry air),the set point of the indoor air wet-bulb temperature had to be changed correspondingly from 17 C to 16.8 C.During the first 520s of the test,the DX A/C system was stable,and both indoor air temperature and moisture content were20003000400050006000S p e e d (r p m )Time (s)161820222426T e m p e r a t u r e (0C )1234567T h e r m a l l o a d s (k W )Fig.4.The variation profiles of some measured operating parameters in disturbance rejection capability test.05001000150020002500300035004000464748495051525354bR e l a t i v e h u m i d i t y (%)Time (s)8.08.59.09.510.010.511.0aM o i s t u r e c o n t e n t (g /k g d r y a i r )Fig.5.The variation profiles of measured moisture content and RH in disturbance rejection capability test.Q.Qi,S.Deng /Building and Environment 44(2009)1659–16671664maintained at their respective set points.At t ¼500s,the change of indoor air temperature setting from 24 C to 23.5 C was intro-duced,then MIMO controller was immediately enabled,as shown in Fig.6b.From Fig.6a,it can be seen that the MIMO controller can well control the indoor air dry-bulb temperature and wet-bulb temperature to their respective set points.From Fig.6c,it can be observed that indoor air moisture content can be maintained at its set point after the set point of indoor air temperature was altered.Fig.7illustrates the controllability test results of the MIMO controller,also with respect to its command following capability,following a change in indoor air moisture content setting which was achieved via a change in indoor air wet-bulb temperature.The set point of wet-bulb temperature was changed from 17 C to 16.6 C,or equivalently moisture content setting from 9.34g/(kg dry air)to 8.86g/(kg dry air),while the indoor air dry-bulb temperature setting was maintained at 24 C.The variation profiles of both indoor air dry-bulb and wet-bulb temperatures are shown in Fig.7a,and that of moisture content in Fig.7c.As seen from these figures,after the set point was changed,the MIMO controller responded by simultaneously varying speeds of both compressor and supply fan (Fig.7b),so that required new set point was approached in less than 1200s,and well maintained afterward.Therefore,the test results shown in Figs.6and 7suggested that the performance of the MIMO controller in terms of its command following capability was also satisfactory.5.DiscussionsSimultaneous control of indoor air temperature and humidity in a space served by a DX A/C system would require simultaneously varying the speeds of both supply fan and compressor of the DX A/C system.However,the changes in supply fan speed and compressor speed would have different levels of influence on indoor air temperature and humidity.When space sensible and latent heat loads were increased,the control of air dry-bulb temperature required the speeds of both supply fan and compressor to increase.However,the control of humidity required the speed of supply fan to decrease,but meanwhile required compressor speed to increase to output more cooling capacity for more moisture removal.Thus speed control would be in conflict if the coupling effect was not considered,and the result of control performance would be inherently poor.These can only be resolved by applying multivar-iable control strategy.From Fig.4,showing the results of disturbance rejection capa-bility test,when the MIMO controller was enabled,both supply fan speed and compressor speed were collaboratively increased to remove more sensible heat load because of a greater output cooling capacity from the DX cooling coil.Afterward,the speed of supply fan was decreased for removing more moisture from return air.Finally the indoor air temperature and humidity stabilized at around their respective settings.On the other hand,in Fig.6showing the results of command following controllability test after a change of indoor air temperature setting,it can be seen that indoor air moisture content rose above its set point before decreasing under the set point (Fig.6c).This was because the speed of supply fan was initially increased causing the sensible heat ratio of the DX cooling coil to decrease and consequently less moisture was removed even under an increased compressor speed.This also suggested that varying supply fan speed would influence SHR more than varying compressor speed,which agreed well with the results in previous related studies [2,3].Furthermore,similar phenomena may also be observed in Fig.7.8.08.59.09.510.010.5M o i s t u r e c o n t e n t (g /k g d r y a i r )Time (s)100020003000400050006000S p e e d (r p m )161820222426T e m p e r a t u r e (0C )Fig.6.The variation profiles of measured operating parameters in command following test with a change of air temperature setting.Q.Qi,S.Deng /Building and Environment 44(2009)1659–16671665。