Online Model Learning of Buildings Using …downloads.hindawi.com/journals/jcse/2017/3035892.pdfJournalofControlScienceandEngineering 3 knownasPILCO.Inthisframework,therobotcanlearn

Research ArticleOnline Model Learning of Buildings Using Stochastic HybridSystems Based on Gaussian Processes

Hamzah Abdel-Aziz and Xenofon Koutsoukos

Department of Electrical Engineering and Computer Science Institute for Software Integrated Systems Vanderbilt UniversityNashville TN 37235 USA

Correspondence should be addressed to Hamzah Abdel-Aziz hamzahabdelazizvanderbiltedu

Received 17 March 2017 Revised 12 June 2017 Accepted 2 July 2017 Published 23 August 2017

Academic Editor Tushar Jain

Copyright copy 2017 Hamzah Abdel-Aziz and Xenofon Koutsoukos This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

Dynamical models are essential for model-based control methodologies which allow smart buildings to operate autonomouslyin an energy and cost efficient manner However buildings have complex thermal dynamics which are affected externally by theenvironment and internally by thermal loads such as equipment and occupancyMoreover the physical parameters of buildingsmaychange over time as the buildings age or due to changes in the buildingsrsquo configuration or structure In this paper we introduce anonlinemodel learningmethodology to identify a nonparametric dynamicalmodel for buildings when the thermal load is latent (iethe thermal load cannot bemeasured)The proposedmodel is based on stochastic hybrid systems where the discrete state describesthe level of the thermal load and the continuous dynamics represented by Gaussian processes describe the thermal dynamics ofthe air temperature We demonstrate the evaluation of the proposed model using two-zone and five-zone buildings The data forboth experiments are generated using the EnergyPlus software Experimental results show that the proposed model estimates thethermal load level correctly and predicts the thermal behavior with good performance

1 Introduction

Heating ventilation and air conditioning (HVAC) in build-ings is a major source of energy consumption Annualreports show that it is the highest cause of energy consump-tion in residential buildings Moreover HVAC along withmiscellaneous electric loads accounts for the highest twoenergy consumption sources in commercial buildings [1]Therefore there is a high demand for developing advancedHVAC control methodologies that can reduce the energyconsumption of HVAC units without compromising usersrsquocomfort Many of these advanced control methodologies aremodel-based and rely on accurate thermal models [2]

Developing accurate dynamical models is a challengingtask because of the complex behavior of buildings Thermaldynamics of buildings are stochastic nonlinear dynamicswhich are affected externally by the environment (egambient temperature) and internally by interactions betweenadjacent thermal zones Additionally the thermal dynamics

of buildings behave differently based on the thermal load(eg occupancy) which may not be measured Developinga unified parametric model for buildings is usually notpractical because thermal dynamics differ from one buildingto another since each building has different constructionmaterials size layout and location Moreover the modelparameters may change over time as the buildings age or dueto changes in the configuration or structure Such challengescan be addressed using nonparametric modeling approachesbased on online model learning

In this paper we introduce a novel modeling frame-work to learn multizone data-driven dynamical models forbuildings in an online fashion The nonparametric nature ofthe model supports creating unified data-driven models forbuildings Furthermore we do not assume that the thermalload can be measured and we estimate it as a latent discretevariable The proposed approach incorporates the effects ofthe thermal load and as a result it increases the modelaccuracy Additionally the online learning adapts the model

HindawiJournal of Control Science and EngineeringVolume 2017 Article ID 3035892 18 pageshttpsdoiorg10115520173035892

2 Journal of Control Science and Engineering

to the time variability during the system operation We alsoinclude the ambient temperature prediction to incorporatethe environment effects in the model

The main contributions of this work are as follows

(1) We introduce a nonparametric SHS model basedon Gaussian processes which can be used to modelcomplex coupled discrete and continuous dynamicsrequired to develop data-driven models of smartbuildings

(2) We develop an online clustering-based learningmethodology for SHS when the discrete dynamicscannot be measured In the proposed methodologywe use the 119870-means clustering algorithm to identifythe discrete states of the system and to segment thedata into each corresponding discrete state Theneach segment is used to build a distinct GP modelto abstract the continuous dynamics Learning thesedistinct models is more efficient and mitigates thecomputation limitations of using a single GP modelwith every variable as inputs

(3) We utilize the data-driven modeling approach toprovide an accurate thermal model for multizonebuildings when the buildingsrsquo thermal load is latentWe evaluate the performance and the efficiency of theproposed model is evaluated using datasets of two-zone and five-zone buildings In both experiments weshow themodel accuracy of estimating the level of thethermal load and predicting the zone air temperature

The organization of this paper is as follows Section 2presents the related work of modeling thermal dynamics andmodel learning of dynamical systems using GPs Section 3illustrates a brief theoretical background of GPs The pro-posed model is defined in Section 4 along with the modellearning problem The proposed model learning approach ispresented in Section 5 Finally the experimental results arediscussed in Section 6 with the evaluation

2 Related Work

21 Thermal Modeling of Buildings Developing thermalmodels for buildings has been investigated extensively in theliterature Many studies proposed linear models for single-zone buildings [3 4] while other studies use lumped thermalmodels to approximate multizone buildings as a single-zone model [5 6] Although these methods are useful theytypically result in very simple models On the other handmultizone models have been investigated to identify thethermal dynamics of each zone in multizone buildings [7ndash9] These models are more accurate and they can be used foradvanced control design (eg model predictive control [8])

Most of the above-mentioned work relies on parametricmodels to represent the thermal behavior of buildingsHowever parametric models require the buildingrsquos detailedstructure andor the thermal equations to be known apriori Also they typically use linear dynamics and are notvery accurate Recently nonparametric models have drawnattention to construct accurate nonlinear thermal models

A nonparametric thermal model based on recurrent neuralnetwork (RNN) architecture has been developed to learna compact thermal model for a single thermal zone [10]Models based on RNNs are very useful to represent thenonlinearity of thermal dynamics however they typicallydo not consider the stochasticity in the thermal dynamicsMoreover thesemodels require a large set of the training datato learn the model with the desired quality To mitigate theselimitations a nonparametric probabilistic approach basedon Gaussian processes (GPs) has been developed to learnthermal models in an online adaptive learning framework[11] Both the RNN and the GP models assume that thethermal load of buildings (eg occupancy heating rate fromlight and equipment) can be measured and they used itas a model input However this assumption may not bevalid in many real scenarios Another modeling approachis developed to learn the effect of the thermal load when itcannot be measured [12] This model is based on using agray box parametric model for the thermal dynamics andcombining it with a latent force model based on Gaussianprocesses The latent force model improves the parametricmodel accuracy by learning the dynamical effects of the latentthermal load

Existing nonparametric modeling approaches for build-ings assume that the thermal load can be measured andused as a model input This assumption may limit the use ofthese approaches in many systems therefore we propose anonparametric SHS model for multizone buildings when thethermal load is latentThe proposedmodel estimates the levelof the thermal load and learns a distinct model for each levelin order to improve the model efficiency and accuracy

22 Model Learning of Dynamical Systems Using GPs In thispaper the proposedmodeling approach is based on Gaussianprocesses [13] Gaussian processes have shown a great successinmodelingmanymodern systems because of their attractivefeatures [14] GPs are nonparametric probabilistic modelswhich can express the uncertainty in the system dynamicsand the model confidence through the model predictivevariance Furthermore models based on GPs are very simplesince they have few hyperparameters and relatively requiresmall datasets to learn the modelThese features allow GPs tobuild efficient robust and adaptive dynamical models GPshave been used recently to develop many time-series modelsfor short-term andmultistep forecasting [15 16] For instanceGPs are used to build forecast models for the ambienttemperature and carbon intensity These models are used todevelop an adaptive heating control algorithm thatminimizesthe energy cost and carbon emissions [4] Also GPs are usedto develop a load forecasting model for power systems in[17] and a short-term traffic volume prediction model witha high performance [18] In addition to time-series modelsGPs are used to identify state-space models with a controlinput in order to support robust filtering smoothing andprediction of the system state [19] For instance in roboticsapplication GPs have been used to build a model-baseddata-efficient learning framework for control policy search

Journal of Control Science and Engineering 3

known as PILCO In this framework the robot can learnthe system dynamics and the control policy efficiently in anonline fashion [20]

Gaussian processes have been used widely to modelstochastic continuous systemsHowever they typically do notconsider systems with coupled discretecontinuous dynamics(eg SHS) In this paper we utilize GP models to representthe continuous dynamics of SHS Furthermore we considersystems with latent discrete dynamics This allows us toefficiently integrate the state-space modeling techniques forstochastic continuous systems into the proposed data-drivenSHS modeling framework

23 System Identification of SHS Stochastic hybrid systems(SHS) are dynamical systems that integrate continuous anddiscrete dynamics Moreover the continuous andor the dis-crete dynamics exhibit a probabilistic behavior [21] Systemidentification of hybrid systems (HS) has been investigatedin the literature substantially to develop system identificationmethods for various classes of HS Typically these methodsaim to estimate the system model parameters for a givenmodel complexity [22] For HS with unknown model com-plexity a kernel-based approach is developed to identify apopular class ofHS known as piecewise affine systems whereGPs are used tomodel the impulse response of each submodelof the HS [23]

System identification of SHS has an additional level ofcomplexity because of the presence of uncertainty in themodel behavior along with the coupled continuousdiscretedynamics For a given model structure many methods havebeen developed to learn the model parameters (ie param-eter identification) such as simulation-based approachesSimulation-based approaches use the simulated trajectoriesto identify the model parameters based on randomizedoptimization techniques such as Markov chain Monte Carlo(MCMC) [24] Most of these approaches assume a knownmodel structurewith unknownparameters However findingthe model structure is very crucial in many complex systemsAlternatively online model learning with reachability analy-sis framework based on Gaussian processes is developed tolearn nonparametric models for SHS with deterministic andknown discrete dynamics [25]

Recent nonparametric models for SHS are either lim-ited to deterministic discrete dynamics or developed foroffline learning In this paper we propose a nonparametricclustering-based modeling framework based on GPs whichutilizes sensory data to learn SHS in an online fashion

3 Background

31 Gaussian Process Model Gaussian processes (GPs) arenonparametric probabilistic models that require only high-level knowledge about the system behavior and use theobserved data to model the behavior of the underlyingsystem [13] Generally GPs build aGaussian distribution overfunctions by which a function index variable is mapped toan infinite-dimensional function space GPs are identifiedby mean and covariance functions The mean functionrepresents the expected value before observing any data and

the covariance function (also called kernel) identifies theexpected correlation between the observed data For instancethe mean function119898(x) and the covariance function 119896(x x1015840)are defined as

119898(x) = E [119891 (x)] 119896 (x x1015840) = E [(119891 (x) minus 119898 (x)) (119891 (x1015840) minus 119898 (x1015840))] (1)

The function modeled by the GP can be written as

119891 (x) sim GP (119898 (x) 119896 (x x1015840)) (2)

We typically use a zero mean function for simplicity andsquared exponential (SE) covariance kernel for its expressive-ness combined with a noise kernel Therefore the mean andcovariance functions can be expressed as

119898(x) = 0119896 (x x1015840) = 1205902119891 exp [minus12 (x minus x1015840)119879Λminus1 (x minus x1015840)]

+ 120575xx10158401205902120596(3)

where 120590119891 is the kernel signal variance Λ fl diag ([11989721 1198972119863])is the characteristic length-scales matrix 120575 is the Kroneckerdelta and 120590120596 is the noise variance The above GP modelbuilds a probability distribution over the functions 119901(119891(x))bymapping 119899-samplesX of a continuous variable x to a vectorof random variable f with a Gaussian joint distribution suchthat

119901 (y) sim N (0K (XX)) (4)

where K is nD-by-nD covariance matrix generated by (3)We are interested in the GP posterior distribution given

some test inputs and observations (training data) We definethe set of test inputs where we want to predict the functionvalue as Xlowast After observing data D and according to (4)the joint distribution of the known y and the unknown ylowastfunction values is

[ yylowast] sim N(0 [ 119870 (XX) 119870 (XXlowast)119870 (XlowastX) 119870 (XlowastXlowast)]) (5)

Therefore the posterior distribution 119901(ylowast | XlowastX y) isalso a conditional Gaussian distribution with a mean and acovariance given by

E [ylowast | yXXlowast] = K119879lowast120573var [ylowast | yXXlowast] = Klowastlowast minus K119879lowast (K + 1205902120596I)minus1 Klowast (6)

where Klowast = 119896(XXlowast) Klowastlowast fl 119896(XlowastXlowast) K fl 119896(XX) and120573 fl (K + 1205902120596I)minus1y

The hyperparameters of the above GP model are definedas Θ fl (120590119891 11989721 1198972119863 120590120596) The training data (D = (x119894 119910119894) |119894 = 1 119899) are used to identify the model hyperparameterswhich best represent the training data The learning process


can be expressed as an optimization problem where the opti-mal hyperparameters (Θ) maximize the marginal likelihoodsuch that

Θ = argmaxΘ

log119901 (y | ΘD) (7)

where the marginal likelihood is given by

119901 (y | ΘD) = minus12y119879Ky minus 12 log (|K|) minus 1198992 log (2120587) (8)

Optimization algorithms based on conjugate gradients havebeen developed to optimize the hyperparameters [13 26]Also the popular quasi-Newton optimization method fornonlinear functions has been used to learn the GPsrsquo hyper-parameters effectively

4 Stochastic Hybrid Systems

We introduce in this section a nonparametric stochastichybrid systems (SHS) model based on GPs To formalize theSHS model we define Q as the set of discrete states anddenote the continuous state space by R119863 for each discretestate 119902 isin Q with dimension 119863 The hybrid state space isdefined as S fl Q times R119863 For each discrete mode 119902 isin Qits corresponding continuous dynamics evolve according to astochastic process modeled by a GP model The discrete statemay also change based on a stochastic process Furthermorewe consider systems with two types of inputs (1) controlinput and (2) external uncontrolled input (disturbance) fromthe environmentThe control input usually affects the systemdynamics based on a control policy (120587(S) S rarr U) whichmaps the hybrid state space (S) into the control input space(U) On the other hand the external uncontrolled input (V isinV) affects the systemdynamics and represents the interactionwith the environment Therefore we propose to model theexternal input as a time-series disturbance model (119864 N rarrV) The model is formalized by the following definition

Definition 1 (nonparametric SHS) A nonparametric SHS isdefined as a tupleH = (Q 119883 InitUV 119860 120575)

(i) Q = 1199021 1199022 119902119898 for some 119898 isin N represents thediscrete state space

(ii) 119883 is a set of continuous variables in the Euclideanspace R119863

(iii) Init B(S) rarr [0 1] is an initial probability measureon the Borel spaceB(S) where S fl Q timesR119863

(iv) U sub R119864 for some 119864 isin N represents the control inputspace

(v) V sub R119865 for some 119865 isin N represents the externaluncontrolled input space

(vi) A assigns to each discrete state 119902 isin Q a randomfunction (119909119896+1 = 119891119902(119909119896 119906119896 V119896)) modeled by a GPwhich represents the evolution of the continuous stategiven the predecessor continuous state 119909119896 isin R119863 acontrol input 119906119896 isin U and an external uncontrolledinput V119896 isin V

E(k)S(k) u(k) v(k)k

Controlpolicy

Time-seriesmodel

SHS model

P12

P21

120587(s(k))

q = 1

x sim GP1

q = i

x sim GPi

q = 2

x sim GP2

Figure 1 Nonparametric SHS components and interconnections

A graphical representation of the model is shown inFigure 1

The goal of model learning is to identify the discretestate space (ie Q fl 1199021 1199022 119902119898 and the number of thediscrete states 119898) and the continuous dynamics (ie GP119902(sdot)for all 119902 isin Q) from a given dataset (D) The data consistsof the continuous state the control input and the externaluncontrolled input Formally we can define the training dataas

D fl (x119896 y119896) 119896 = 119879119904 119879119890 (9)

where y119896 is the successor continuous state (ie y119896 = x119896+1)x119896 is defined as (x119896 119906119896 V119896) and [119879119904 119879119890] is the time periodat which the data have been collected For many complexsystems such as buildings achieving the model learningobjective is a challenging task because the sensory data donot necessarily include information about the discrete stateexplicitly For instance learning the level of the thermal loadin buildings is a major challenge because the thermal loadcannot be measured in many real scenarios Moreover thesystem dynamics vary over time due to the variability of thesystem parameters

In summary model learning of SHS encompasses thefollowing challenges

(1) Identifying the discrete state space from data causesa major challenge because the training data may nothave explicit information about the discrete state

(2) The hybrid dynamics add a level of complexity to thelearning algorithm because identifying the continu-ous dynamics requires segmenting the data for eachcorresponding discrete state so that a distinct GPmodel can be learned using each data segment


Feature extractionClustering

(eg K-means)Data segmentation Learning a GP model for each

Initialization (oine phase)

New sensor measurement Feature extraction Classification to estimate q

(eg nearest centroid)

Predict the system response Prediction and model update (online phase)

Update training data

windowing

Update the classifier byreclustering the data() based on

relearn GPqusing GPq

data segment q(12 m)

Training data

Update q and

Figure 2 Clustering-based online model learning for SHS

(3) As mentioned earlier the system dynamics dependon an external uncontrolled input V Therefore anappropriate time-series model119864(119896) should be learnedand integrated into the SHS modeling framework

(4) Because of the variability of the system parametersthe learning algorithmmust adapt the model to thesechanges and therefore themodel learning needs to beperformed in an online fashion

5 Online Clustering-Based ModelLearning for SHS

In this section we present a novel online learning approachwhich can be used to learn a nonparametric SHS model ofcomplex systems with latent discrete dynamics Moreoverthe approach is used for online learning in order to adaptto system changes The proposed learning approach consistsof two phases offline phase and online phase as depicted inFigure 2 In the offline phase we initialize the SHS model byidentifying its discrete space first We determine the numberof the discrete states heuristically using Silhouette analysismethod and then we identify the discrete state of each datapoint by clustering the dataThe clustering is also used to labeland segment the training data to the corresponding discretestates This allows us to learn the continuous dynamics foreach discrete state using a distinct GP model In the onlinephase we estimate the discrete state in order to determinethe GP model used to predict the continuous dynamicsThen we update themodel accordinglyThis section providesa detailed discussion of the major steps in the proposedapproach

51 Initialization (Offline Phase)

511 Feature Extraction Feature extraction is a techniqueused to transform the training data into a set of featuresA set of features (usually referred to as a feature vector)contains the useful data needed for the clustering stage wherethe irrelevant information is discarded Feature extractionis needed to distinctively filter the training data Generallythe feature vector can be computed based on time-domainfeatures (eg mean root mean square) or frequency-domainanalysis (eg transfer function Fourier transforms) In thispaper the feature vector is computed based on time-domainfeatures such as mean and rate of change because of thesimplicity and the efficiency of these features

512 Data Clustering for Discrete Space Identification Thegoal of data clustering is to estimate the discrete state ofeach data point Various clustering algorithms can be used tolearn the discrete state such as 119870-means Gaussian mixturemodel (GMM) and hierarchical clustering algorithms Thechoice of the appropriate algorithm depends on the natureof the application and the collected data In this section wedescribe the 119870-means clustering algorithm which calculatesthe optimal centroids (C) for119870 clusters so that Cminimizesthe following potential function

C = argminC

sum119892(119909)isin120594

min119888isinC

1003817100381710038171003817119892 (x) minus 11988810038171003817100381710038172 (10)

where 120594 isin R119899times119889 is the feature matrix extracted from thetraining data (D) with size 119899 and feature dimension 119889 119892(x) isinR119889 is the feature vector for the data point x isin D and C isinR119870times119889 represents the optimal119870 centroids


The clustering-based learning approach considers thedata that lie close to each other to probably belong to the samediscrete stateHowever it requires the number of clusters (iethe number of discrete states 119898 isin N) to be known a prioriWe identify the number of discrete states119898 using a heuristicalgorithm known as Silhouette analysis method [27] TheSilhouette analysis method determines the best number ofclusters () which results in the best clustering consistencySilhouette analysis evaluates the clustering consistency for agiven119870 by calculating a scoring coefficient for each clustereddata point The Silhouette scoring coefficient has a range of[minus1 1] where scores near +1 are assigned to data points thatlie far from the neighboring clusters On the other handscores near 0 are assigned to data points that lie very closeto the boundary between their cluster and a neighboring oneNegative Silhouette scores are assigned to data points whichmight have been allocated to the wrong cluster Thereforeclusters with a higher average Silhouette score have betterconsistency than clusters with a lower average Silhouettescore Formally the Silhouette score 119904(119894) for a given data point119894 can be obtained using the following formula

119904 (119894) = 119887 (119894) minus 119886 (119894)max 119886 (119894) 119887 (119894) (11)

where 119886(119894) is the average distance from the data point 119894 and theother points in its cluster and 119887(119894) is theminimumminimizedover clusters average distance between the data point 119894 andother points in a different cluster

We use Silhouette analysismethod to identify the numberof discrete states such that the average Silhouette score ismaximized 119898 = = argmax

119870

119904 (119894) 119870 isin 119866 (12)

where 119866 is a finite set of potential values of119898

513 Data Segmentation and Learning GP Models for theContinuous Dynamics As shown in the previous subsectiondata clustering enables us to identify the discrete states(1199021 1199022 119902119898) and to label each data point in D withthe corresponding discrete state The labeled data is used tosegment the training data into119898 datasets

D119902 fl (x y)119894 119902 = argmin1199021015840isinQ

10038171003817100381710038171003817119892 (x119896) minus 1198881199021015840100381710038171003817100381710038172 forall (x y)119894isin D forall119902 isin Q

(13)

where 1198881199021015840 is the cluster centroid of the discrete mode 1199021015840and 119892(x) is the feature vector of the data point x We useeach data segment D119902 to learn a distinct GP model inorder to learn the continuous dynamics for the discretestate 119902 isin Q As discussed in Section 3 learning a GPmodel is an optimization process that calculates the optimalhyperparameters Θ119902 of the GP in order to maximize the loglikelihood functionΘ119902 = argmax

Θ119902

log119901 (y | Θ119902D119902) (14)

The offline phase learns the SHS model using the initialtraining dataset and then the model is updated onlinewhenever a newmeasurement is received in order to improvethe model quality and to adapt to the variation in theunderlying system

52 Prediction and Model Update (Online Phase)

521 Classification for State EstimationPrediction Duringthe system operation at each time step 119896 we classify the newmeasured data point (x119896) to estimate the current discrete state119902119896 The new data point is classified using a nearest centroidclassifierThis classifier assigns to the new data point the labelof the class with the nearest centroidThe discrete state can beestimated

119902119896 = argmin119897

1003817100381710038171003817119892 (x119896) minus 11988811989710038171003817100381710038172 (15)

where 119888119897 is the centroid of class 119897522 Online Learning of SHS with Windowing Onlinelearning is very useful in order to adapt the model to thevariability of the system parameters however it requires aproper data selection method to update the training datasetWe use a moving window method to update the modeldataset (ie D119902 for all 119902 isin Q) based on first-in-first-out(FIFO) policy where the new data point is inserted and theearliest one is dequeued The dataset (D119902) for each discretestate 119902 is updated independently and then it is used toupdate its corresponding (GP119902) model and to reoptimizeits hyperparameters (Θ119902) We also use the moving windowmethod to update the training dataset for the clustering (ieD) and then we update the classifier centroids accordinglyBoth the classifier centroids and the GP associated with thecurrent discrete mode are updated by repeating the learningprocess of those models using the updated datasets

We maintain a fixed window size for all the119898 datasets ofthe GPmodels and use a different window size of the trainingdataset for the clustering algorithm The sizes of the trainingdatasets affect the model learning running complexity anddetermine the forgetting weight of the model Typically thereis a trade-off between selecting large and small dataset sizeOnline algorithms with large dataset size suffer from highrunning complexity and they adapt the updated models tothe system variability in a slow pace (ie low forgettingweight of old information) However these algorithms canlearn models with high quality On the other hand onlinealgorithms with small dataset size have efficient runningcomplexity and the updated models adapt to the systemvariability faster However these algorithms may miss usefulinformation from the discarded old data especially in modelswith large input space Therefore we consider the datasetssizes as configuration parameters because they depend onthe nature of the modeled system and the requirementsand the constraints of the system application The updatedmodels adapt to the variation in the systems so that themodel accuracy is improved and reflects the behavior of theunderlying system


523 Prediction of the System Response We predict thecontinuous state of the underlying systemusing theGPmodelcorresponding to the estimated discrete state (ie GP119902119896)Hence the predictive distribution of the continuous state canbe formalized as

119901 (x119896+1 | x119896) = 119891119902119896 (x119896) sim GP119902119896 (119898 (x) 119896 (x x)) (16)

where x119896 is the tuples (x119896 119906119896 V119896) The prediction and thelearning steps in the online phase are repeated iteratively eachtime when new data measurements arrive

The quality of the learned model is evaluated by measur-ing the prediction performanceThe prediction performancerepresents the agreement between themodel and the physicalprocess output This comparison can be measured quantita-tively using the rootmean square error (RMSE) and themeanrelative square error (MRSE) metrics The RMSE and MRSEare defined as

RMSE = radic 1119873119873sum119896=1

1198902119896

MRSE = radic sum119873119896=1 1198902119896sum119873119896=1 1199102119896 (17)

where 119910119896 is the process output and 119890119896 = 119910119896 minus 119910119896 is theprediction error of the 119896th time step

Multistep Predictionwithin aDiscreteModeMultistep predic-tion of the system response is required in many verificationand optimization algorithms For the proposed SHS modelmultistep prediction can be achieved by propagating thepredictive distribution in (16) using the GP model of thecurrent estimated mode (ie GP119902) However this propaga-tion faces a major challenge where it requires to predict thesystem response at an uncertain input defined by a Gaussiandistribution (ie 119901(Xlowast) sim N(120583lowastΣlowast)) The GP posterior ofthis predictive distribution can be obtained by

119901 (ylowast) = ∬119901 (ylowast | Xlowast) 119901 (Xlowast) 119889ylowast119889Xlowast (18)

However the posterior distribution shown in (18) is ana-lytically intractable [16] To overcome this limitation weapproximate the posterior distribution as Gaussian distribu-tion using a linearization methodology [16 20] where theapproximated posterior (119901(ylowast) sim N(120583119910Σ119910)) can be obtainedby

120583119910 = E [ylowast | 120583lowast] Σ119910 = var [ylowast | 120583lowast] + VΣlowastV

119879 (19)

whereE[ylowast | 120583lowast] and var[ylowast | 120583lowast] is themean and covarianceof the GP posterior calculated at the mean 120583lowast of the inputdistribution as in (6) and V is defined by

V = 120597120583119910120597120583lowast = 120573119879 120597119896 (X120583lowast)120597120583lowast (20)

53 Learning Time-Series Model for the Uncontrolled InputAs mentioned earlier the proposed approach identifies atime-series model 119864(119896) for the uncontrolled input V usingsingle Gaussian processes model The time-series modelis learned online and independently of the SHS modellearning algorithm Therefore the time-series model 119864(119896) ofan observed time-dependent variable V119896 is modeled as

V119896 = 119891 (119896) sim GP (119898 (119896) 119870 (119896 1198961015840)) (21)

We also use the moving window method to learn the abovetime-series model online

6 Evaluation

In this section we evaluate the performance of the proposedframework and its efficacy to learn thermal models forbuildings when the applied thermal load (eg occupancy)cannot be measured Thermal models are used to capturethe dynamics of the buildingsrsquo air temperature which areessential for developing many advanced control method-ologies [2] In this case we use the proposed SHS tobuild a thermal model for buildings such that the zoneair temperature represents the model continuous state theHVAC heatingcooling rate represents the control input theambient temperature represents the uncontrolled input andthe thermal load represents the latent discrete state

We evaluate the thermal model learning for (1) a two-zone data center building and (2) a five-zone office build-ing The actual behaviors for both buildings are generatedusing simulations based on the EnergyPlus software [28]EnergyPlus is an open-source cross-platform building energysimulator engine funded by the US Department of Energy(DOE) and Building Technologies Office (BTO) and man-aged by the National Renewable Energy Laboratory (NREL)EnergyPlus is used by engineers architects and researchersfor high fidelity simulation of buildings EnergyPlus requirestwo inputs (1) the ambient temperature and the environ-ment data and (2) the building description The buildingdescription defines its structure and layout the constructionmaterials the thermal zones with their dimensions andarea the HVAC system the control strategies and more Italso defines the building thermal loads with their schedulessuch as occupancy lights and electrical equipment Thesedetailed descriptions are used to construct several models(eg airflow network model pollution model and on-sitepower model) by which EnergyPlus simulates the buildingbehavior

We use EnergyPlus to represent the system responseand use the data collected from EnergyPlus simulation (thebuilding state and the control input values) to learn andto evaluate the proposed model learning framework Wealso use the weather data used by EnergyPlus to learna time-series model in order to forecast the uncontrolledinput At each time step 119896 we collect the system state (thezone air temperatures) and the control input (the appliedheatingcooling rate from theHVACunit)The collected dataare then used to learnupdate the model online as describedearlier Moreover we compare the model prediction against


Weather data Building description

SHS modeling framework

HVAC system

Time-series model E(k)

Thermostat

SHS model Thermal zones

Prediction evaluation

xk

xk+1

k

xk+1

uk

Figure 3 Block diagram of the experiment setup

the system response (ie zone air temperatures for the nexttime step 119896 + 1) in order to evaluate the model learningperformance A general block diagram of the experimentsetup is shown in Figure 3

The proposed framework has been implemented usingMATLAB and Statistics and Machine Learning ToolboxRelease 2016a [29] Gaussian processes models are learnedusing a quasi-Newton optimization algorithm in order tooptimize the model hyperparameters The following sub-sections discuss the experiment results for a two-zone datacenter building and a five-zone office building

61 Two-Zone Data Center Building In this experiment weuse a dataset of a two-zone data center building to learn andto evaluate the building thermal model using the proposedSHS modeling framework This dataset is synthetic datagenerated by EnergyPlus and is used to evaluate buildingsrsquomodeling in [10] The data center building consists of twozones the west zone and the east zone as shown in Figure 4The dataset contains hourly data for one-year simulationEach data point consists of zone air temperature ambienttemperature thermal load heating rate from IT equipmentlights and electrical equipment andHVAC unit cooling rateThe heating rate from the building thermal load dependson the building activities (eg how utilized or idle the ITequipment is) Many models in the literature assume that thethermal load can be measured and therefore can be used asa model input (eg [10]) however this is very expensive inreal data centers Therefore we consider the thermal load asa latent discrete state of the system and we use our proposedapproach to estimate it from the measurable thermal data(ie zone temperature and HVAC unit cooling rate)

To formalize the model we define the SHS model asfollows the continuous state (x isin R2) represents the airtemperature for both zones the discrete state 119902 represents thethermal load level the uncontrolled input (V isin R) represents

Westzone

Eastzone

Figure 4 Two-zone data center building

Table 1 The average Silhouette scores for different numbers ofclusters

119898 Average Silhouette scoreWest zone East zone

2 070 0553 071 0654 062 0605 065 065

the ambient temperature and the control input (119906 isin R)represents the HVAC cooling rate Therefore the predictivedistribution of the zone air temperature can be representedas

x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (22)

In the offline phase of the model learning approach weused the first four weeks of data (ie 672 data points) toinitialize the model and to learn its discrete states usingthe 119870-means clustering algorithm Data clustering starts byextracting the time-domain features from the data In thisexperiment the features are the average cooling rate (ie119906(119896)) and the zone air temperature difference (ie Δ119909 = 119909119896 minus119909119896minus1) (note that the values of these features are normalizedin order to unify their scale) Then the number of discretestates is estimated using the Silhouette analysis method Thecalculated average Silhouette scores of this experiment areshown in Table 1 Based on this analysis the number ofthe discrete states is three (ie 119898 = 3) Since the discretestate represents the thermal load level we identified threethermal load levels which correspond to low medium andhigh heat gained from the thermal loads Figure 5 depicts thethree clusters of the training data for each zone Finally wesegment the data into three datasets and use them to learnthree distinct GP models for each zone

As explained earlier the ambient temperature is consid-ered as an uncontrolled input and thus a time-series GPmodel is used to forecast its value for the next hour We alsoupdate the forecast model every hour when we receive a newmeasurement Figure 6 shows the forecasting results of theambient temperature for three days

In the online phase we predict the zone temperature forthe next hour using the learned SHS model and the pre-dicted ambient temperature Further we update the trainingdatasets every hour when we receive a new data point using


1

05

0

minus05

minus1

minus15

minus2

uwest

1

2

3

West features

0 2minus1minus2 1

Δxwest

(a)

1

05

0

minus05

minus1

minus15

minus2

East features

Δxeast

0 2minus1minus2 1

ueast

1

2

3

(b)

Figure 5 Clustering of the training data using119870-means algorithm

Real dataPredicted mean95 prediction limits

690 700 710 720 730 740 750680Time

0

5

10

15

20

Am

bien

t tem

pera

ture

(C)

Figure 6 Prediction of ambient temperature using a time-series GPmodel

themovingwindowmethodwith a size of one-week data (ie168 points)The updated datasets are then used to relearn thecorresponding models We run the predictionlearning stepsiteratively for almost 11 months and use the results to evaluatethe proposed SHS model learning approach The results forthe first three days of the discrete mode estimation (iethermal load level) and the west zone temperature predictionare shown in Figures 7 and 8 respectively The depictedresults present the performance of the online model learningapproach to predict the continuous state accurately and toestimate the discrete state successfully As shown in Figure 8we compare the model prediction with a full GP model Thefull GPmodel assumes that the thermal load can bemeasuredand the thermal model is defined as

x119896+1 = 119891 (x119896 119906119896 V119896 119897119896) (23)

Table 2 Performance metrics for the prediction of the zone airtemperature with a comparison between the SHSmodel and the fullGP model

West zone East zoneFull GP SHS Full GP SHS

RMSE 005 006 007 006MRSE 0008 0009 001 0009Max 06 103 06 08

where 119897119896 is the total heating rate from the thermal load Asshown in the results the proposed SHS model and the fullGPmodel have a similar performance However the learningof the full GP is computationally more expensive because ofthe large dimension of the input data On the other hand SHSmodel segments the data into smaller distinctmodels for eachestimated level of the thermal load

We evaluated the prediction performance of the contin-uous state using the root mean square error (RMSE) and themean relative square error (MRSE) metrics defined in (17)The error metrics statistics for the SHS model and the fullGP model are shown in Table 2 The results show that ourSHS model predicted the zonesrsquo temperature with a goodperformance Also the performance of the SHS model issimilar to that of the full GP model which assumes that thethermal load is known and can be measured

To evaluate the improvement of the online learningapproach we compared the RMSE metric for the proposedonline learned model against another offline model which islearned once offline only In this experiment we considered


1

15

2

25

3

Estim

ated

stat

e

10 30 40 50 6020 70Time

7030 40 50 602010Time

354

455

556

Tota

l the

rmal

load

(W) 104

Figure 7 Discrete state estimation of the west zone versus the actualtotal thermal load

the changing of the building climate due to seasons changingas an example to represent the variability in the systemFigure 9 shows the RMSE statistics for bothmodels calculatedfor each week of the year The results indicate that onlinelearning allows our model to adapt to the variation in thesystem with a good performance On the other hand theoffline model failed to adapt to these variations

611 Evaluating Different Clustering Algorithms There aremany clustering algorithmswhich can be used to estimate themode of the systemThe choice of such algorithm depends onthe nature of the data and the application In this experimentwe studied three different clustering algorithms which are119870-means Gaussian mixture model (GMM) and hierarchicalclustering algorithms Figure 10 shows the clustering ofthe training data using each algorithm Both 119870-means andGMM algorithms require the number of the clusters to begiven a priori where we used Silhouette analysis methodto estimate it Hierarchical algorithm on the other handhas the advantage of determining the number of the clustersbased on its stopping criteria (eg cluster inconsistency) butit has many design parameters and suffers from run timecomplexity 119874(1198992 log(119899)) for large-scale data as indicated inFigure 11

Figure 12 shows the boxplot of the heating rate causedby the thermal load (ie light occupancy and equipment)for each estimated discrete mode (ie cluster) Despite thedifferences between these algorithms the results show thatthey identify the discrete state with distinctive levels of thethermal load GMM estimated the thermal load levels withthe lowest accuracy (ie low high) Hierarchical algorithmhas the highest accuracy (ie low medium-low medium-high and high) of the thermal load levels

612 Multistep Prediction within the Same Mode Asdescribed earlier multistep prediction requires propagating

Table 3 Performance metrics for multistep prediction of both SHSmodel and unimodal GP model

SHS model Unimodal GPWest East West East

RMSE 023 03 056 03MRSE 0009 0013 0023 0013LD 51 6 27 257

the uncertainty of the predictive distribution for each timestep In addition the control policy of the HVAC is requiredto predict the control sequence of the HVAC unit (iecooling rate) To do so we used GP to learn the controlpolicy of the HVAC as a function of the hybrid systemstate (ie zone air temperature and discrete mode) and theambient temperature that is

u119896+1 sim 119891 (s119896 V119896) s119896 = (119902119896 x119896) (24)

Furthermore we use the RMSE and MRSE metrics as aperformance measure in order to evaluate the predictedmean We also use another performance metric known aslog predictive density (LD) to evaluate the prediction perfor-mance of the predicted mean and the predicted variance aswell [14] LD is defined as

LD = 12 log (2120587) + 12119873119873sum119894=1

log (1205902119894 ) + 11989021198941205902119894 (25)

where 120590119894 is the prediction variance in 119894th step and 119890119894 is theerror between the system output and the predicted meanFigure 13 shows the multistep prediction of the zone airtemperature for both west and east zones using the proposedSHS model We also implemented the multistep using atypical single GP model which does not estimate the discretemode of the buildings (ie thermal load level) Figure 14shows the multistep prediction of the zone air temperaturefor both west and east zones using a typical unimodal GPmodel Table 3 shows the performance metrics statistics forthe multistep prediction within the same mode of the SHSand the unimodal GP

The performance metric shows that the SHS modelprovides amore accurate prediction than the unimodal GP asfar as the systemdoes not change its discretemodeMoreoverthe variance of the unimodal GP model is falsely optimistic

62 Five-Zone Office Building In this experiment we evalu-ate the scalability of the framework using a five-zone officebuilding dataset The dataset is a synthetic dataset generatedby EnergyPlus for a single story office building The officebuilding is a rectangular building with five zones and a returnplenum The building has four windows in each facade andthere are two interior glazed doors between the west and thecore zone and between the east and the core zone as shownin Figure 15

The main thermal sources for all the five zones are theHVAC unit heating and cooling supply air the office lightsand equipment and the office occupancy The dataset mea-sures the building thermal behavior hourly for one year using


95 prediction limitsPredicted meanReal data

22

225

23

235

24Te

mpe

ratu

re (C

)

690 700 710 720 730 740 750680Time

(a)

22

225

23

235

Tem

pera

ture

(C)

690 700 710 720 730 740 750680Time


(b)

Figure 8 The predicted temperature of the west zone using (a) the proposed SHS model and (b) the full GP model

Offline modelOnline model

01

02

03

04

05

06

07

08

09

1

11

RMSE

10 15 20 25 30 35 40 45 505Week number

Figure 9 Error metric (RMSE) of the prediction of the zone air temperature for both the online and the offline learned SHS model

weather data of Chicago ILThemeasurements consist of theambient temperature zone air temperatures coolingheatingrate from the HVAC unit and heating rate from the thermalload (lights occupancy and office equipment) aggregatedand averaged for every hour

The thermal model of the building is represented bythe following SHS model the continuous state (x isin R5)represents the zone air temperatures The discrete state 119902represents the latent thermal load The uncontrolled input(V isin R) represents the ambient temperature Lastly thecontrol input (119906 isin R) represents the HVAC heatingcoolingrate Therefore the predictive distribution of the zone airtemperatures is given by

x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (26)

Like the two-zone building we also use the first fourweeks of data (ie 672 data points) to initialize themodel and

learn its discrete space (ie 119898) and use the heatingcoolingrate (ie 119906(119896)) and the zone air temperature difference (ieΔ119909 = 119909119896 minus 119909119896minus1) as time-domain features for the 119870-meansclustering algorithm We use the extracted feature to clusterthe training data and to identify the discrete states based onthe Silhouette analysis The estimated number of the discretestates of the south the east the west and the core zones is twoand for the north zone is three Figure 16 shows the clusteringof the training dataset for the south and the north zones andFigure 17 shows the clustering results of the west the coreand the east zones

In the online phase we use the learned model to predictthe zone air temperatures for the next hour Also we updatethe training data and its corresponding models every hourwhen we receive new data points We run the predic-tionlearning steps iteratively for the rest of the data (about11 months) The results for the discrete mode estimation (ie


minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(a)

minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(b)

minus1 0 1 2minus2

Δxwest

minus2

minus1

0

1

uwest

(c)

Figure 10 Clustering of the training data for the west zone using (a)119870-means algorithm (b) GMMalgorithm and (c) hierarchical algorithm

Table 4 Prediction error statistics per zone

South zone East zone North zone West zone Core zoneFull GP SHS Full GP SHS Full GP SHS Full GP SHS Full GP SHS

RMSE 032 028 025 036 028 029 028 027 022 011MRSE 011 005 008 006 010 005 01 005 06 002Max 35 63 55 101 121 45 67 45 25 32

K-meansGMMHierarchical

2000 4000 6000 8000 100000Number of training data

0

02

04

06

08

1

12

14

16

18

Tim

e (s)

Figure 11 Clustering time for different data sizes of 119870-meansalgorithm GMM algorithm and hierarchical clustering algorithm

thermal load level) and zone air temperature prediction ofthe north zone are shown in Figures 18 and 19 respectivelyTypically office environments have two modes busy andidle because people tend to go to their offices during thebusiness hours and then the building becomes almost idleduring the nights and holidaysThemodel learning approachestimated this office behavior successfully as shown in theresults

To evaluate the prediction performance we use the rootmean square error (RMSE) and the mean relative squareerror (MRSE) metrics defined in (17) The prediction errorevaluation statistics are shown in Table 4 These results showthat our SHS model predicted the zone air temperatures witha good performance where the average prediction error is lessthan or equal to 6 Further the proposed model shows asimilar performance to the full GPmodel which assumes thatthe thermal load is known and can be measured

63 Computation of Efficiency and Scalability of the LearningAlgorithm Despite the attractive features and properties ofGPs the running time becomes a major factor in the GP


2 31Cluster number

4

5

6

7

Hea

ting

rate

(wat

t)

104

(a)

21Cluster number

3

4

5

6

7

Hea

ting

rate

(wat

t)

104

(b)

2 3 41Cluster number

4

5

6

Hea

ting

rate

(wat

t)

104

(c)

Figure 12 Average heating rate caused by west zonersquos thermal load for each discrete mode (in watt) estimated by (a)119870-means algorithm (b)GMM algorithm and (c) hierarchical algorithm

Discrete mode changed

4 6 8 10 12 14 162Prediction time step


21522

22523

235

East

tem

pera

ture

21

22

23

24

25

Wes

t tem

pera

ture


Figure 13 Multistep prediction of zone air temperature for both zones using the proposed SHS model


21

22

23

24

25

Wes

t tem

pera

ture

21215

22225

23235

East

tem

pera

ture





Figure 14 Multistep prediction of zone air temperature for both zones using a typical unimodal GP model

Core zone

North zone

South zone

Eastzone

Westzone

Figure 15 Five-zone office building layout

performance when the dimension of the training data islarge Learning a GP model requires the inversion of the119899 times 119899 covariance matrix where 119899 is the size of the data Thematrix inversion has a complexity of 119874(1198993) Therefore GPlearning becomes more computationally expensive when thedimension of the model (ie number of regressors) andorthe training dataset increase To address this limitation atypical solution is to divide or distribute the computationWe mitigate this limitation by approximating the thermalload as a discrete state Therefore we divide the trainingdata to learn a distinct GP model independently for eachmode Further in the context of buildings modeling themodel dimension can be decreased by using the adjacentzones only in the regressors regardless of the total numberof zones in the buildings (ie 119909119894119896+1 = 119891119902119896(x119894119896 119906119896 V119896) 119902119896 isin Q where x119894119896 is the zone air temperature of theadjacent zones of zone 119894) This approximation is accept-able because the zone air temperature is conditionallyindependent of other zones given the adjacent zonesrsquo airtemperature

To evaluate the model scalability we used EnergyPlusto generate data for five different buildings with a differentnumber of zones For each building we use the vector of zoneair temperatures as the continuous state variable thereforethe model dimension increases as the number of zonesincreasesWeuse a fixed training data size for all the buildingsto learn aGPmodel of the thermal dynamics Figure 20 showsthe variation of the learning time of the GP model versus thenumber of zones in the building

We also evaluated the performance and the efficiency ofthe model learning with respect to the training data size Todo so we learn both the SHS and the full GP model for thenorth zone in the five-zone building online every hour forone week using different dataset sizes Figure 21 compares themodel learning time and the prediction error between the fullGPmodel and the proposed SHSmodelThe results show thatbothmodels have a similar prediction performance howeverthe proposed SHSmodel is faster than the full GPmodelTheSHS is faster because the training data are segmented intotwo smaller datasets for each level of the thermal load andthen used to learn two distinct GPmodelsThis segmentationdistributes the computation of the GPs in the SHS modelOn the contrary the full GP uses a single GP with all dataat once Further the SHS model approximates the thermalload as a discrete state instead of an input which reducesthe model dimension Despite these approximations theSHS prediction performance is almost similar to the full GPperformance

7 Conclusion

In this work we proposed a nonparametric SHS modelingframework with a clustering-based online learning approachThe proposedmodel can be used to build a thermalmodel for


1

2

3

South zone

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

us

0 5minus5

Δxs

(a)

North zone

0 2minus2

Δxn

minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

un

1

2

(b)

Figure 16 119870-means clustering of the south and the north zones

1

2

minus2000

minus1000

0

1000

2000

3000

0 5 10minus5

Δxw

uw

(a)

0 5 10minus5

Δxe

minus2000

minus1000

0

1000

2000

3000

ue

1

2

(b)

minus2000

0

2000

4000

6000

8000

uc

0 1minus1

Δxc

1

2

(c)

Figure 17 119870-means clustering of the (a) west zone (b) east zone and (c) core zone

buildings when the thermal load is latent (ie the thermalload cannot be measured) We evaluated the performanceof our model by learning thermal models for buildingsOnline learning is useful to adapt ourmodel to nonstationary

variations and to improve the model efficiency because thetraining data in the offline phase is not necessarily completeDespite these advantages online learning dictates real-timeconstraints Experimental results demonstrate the efficiency


40Time

60 80 100 120 140 1600 200

5

10Th

erm

al lo

ad (W

)

4020 60 80 100 120 140 1600Time

1

2

Estim

ated

stat

e

106

Figure 18 North zone discrete state estimation versus the actual total thermal load


161820222426

Tem

pera

ture

(C)

710 720 730 740 750 760 770700Time

Figure 19 The predicted zone air temperature of the north zone

0

100

200

300

400

500

Aver

age l

earn

ing

time (

ms)

2 3 4 5 6 7 8 91

Number of zones

Figure 20 Average learning time of one GP model for different buildings with different numbers of zones

of the proposed model learning approach The learningprocess runs online with an adequate computation time Theaverage learning time for one GP model was 140ms and

368ms for the two-zone building and the five-zone buildingrespectively Further the average prediction time was 1ms forboth buildings


5 10 15 20 25 30 350Training data size (days)

0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)

Figure 21 Evaluating model learning efficiency and performance of the north zone in the five-zone building using different training datasetsizes with a comparison between the full GPmodel and the SHSmodel (a) Average learning time (b) Prediction error statistics using RMSE

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work is supported in part by the National Science Foun-dation under Award CNS-1238959 and by AFOSR DDDASFA9550-13-1-0227

References

[1] US Energy Information Administration ldquoThe annual energyoutlook 2015 with projection to 2040rdquo Tech Rep 2015

[2] S Prıvara J Cigler Z Vana F Oldewurtel C Sagerschnig andE Zacekova ldquoBuilding modeling as a crucial part for buildingpredictive controlrdquo Energy and Buildings vol 56 pp 8ndash22 2013

[3] A Aswani N Master J Taneja et al ldquoIdentifying models ofHVAC systems using semiparametric regressionrdquo in Proceed-ings of the 2012 American Control Conference (ACC rsquo12) pp3675ndash3680 IEEE 2012

[4] A Rogers S Maleki S Ghosh and J Nicholas R ldquoAdaptivehome heating control through gaussian process prediction andmathematical programmingrdquo in Proceedings of the in SecondInternational Workshop on Agent Technology for Energy Systems(ATES rsquo11) pp 71ndash78 event Dates 2011

[5] Y Ma F Borrelli B Hencey B Coffey S Bengea and P HavesldquoModel predictive control for the operation of building coolingsystemsrdquo IEEE Transactions on Control Systems Technology vol20 no 3 pp 796ndash803 2012

[6] F Oldewurtel A Parisio C N Jones et al ldquoEnergy efficientbuilding climate control using Stochastic Model PredictiveControl and weather predictionsrdquo in Proceedings of the 2010American Control Conference (ACC rsquo10) pp 5100ndash5105 IEEEJuly 2010

[7] Q Hu F Oldewurtel M Balandat E Vrettos D Zhou andC J Tomlin ldquoBuilding model identification during regularoperation - Empirical results and challengesrdquo in Proceedings of

the 2016 American Control Conference ACC 2016 pp 605ndash610July 2016

[8] D Sturzenegger D Gyalistras M Morari and R S SmithldquoSemi-automated modular modeling of buildings for modelpredictive controlrdquo in Proceedings of the 4th ACMWorkshop onEmbedded Systems for Energy Efficiency in Buildings BuildSys2012 pp 99ndash106 November 2012

[9] B Sun P B Luh Q-S Jia Z Jiang F Wang and C SongldquoBuilding energy management integrated control of activeand passive heating cooling lighting shading and ventilationsystemsrdquo IEEE Transactions on Automation Science and Engi-neering vol 10 no 3 pp 588ndash602 2013

[10] H Zhao D Quach S Wang et al ldquoLearning based compactthermal modeling for energy-efficient smart building man-agementrdquo in Proceedings of the 34th IEEEACM InternationalConference on Computer-Aided Design ICCAD 2015 pp 450ndash456 November 2015

[11] F Massa Gray and M Schmidt ldquoThermal building modellingusing Gaussian processesrdquo Energy and Buildings vol 119 pp119ndash128 2016

[12] S Ghosh S Reece A Rogers S Roberts A Malibari andN R Jennings ldquoModeling the thermal dynamics of buildingsA latent-force-model-based approachrdquo ACM Transactions onIntelligent Systems and Technology vol 6 no 1 Article ID2629674 2015

[13] C Rasmussen and C Williams Gaussian Process for MachineLearning The MIT Press Boston Mass USA 2006

[14] J Kocijan ldquoGaussian processmodels for systems identificationrdquoin Proceedings of the 9th Int PhDWorkshop on Sys and Cont pp8ndash15 2008

[15] S Roberts M Osborne M Ebden S Reece N Gibson andS Aigrain ldquoGaussian processes for time-series modellingrdquoPhilosophical Transactions of the Royal Society of London SeriesA Mathematical Physical and Engineering Sciences vol 371 no1984 20110550 25 pages 2013

[16] A Girard C E Rasmussen J Q Candela and R Murray-Smith ldquoGaussian process priors with uncertain inputs applica-tion tomultiple-step ahead time series forecastingrdquoAdvances inNeural Information Processing Systems pp 545ndash552 2003


[17] H Mori and M Ohmi ldquoProbabilistic short-term load fore-casting with Gaussian processesrdquo in Proceedings of the 13thInternational Conference on Intelligent Systems Application toPower Systems ISAPrsquo05 pp 452ndash457 November 2005

[18] Y Xie K Zhao Y Sun and D Chen ldquoGaussian processes forshort-term traffic volume forecastingrdquo Transportation ResearchRecord no 2165 pp 69ndash78 2010

[19] R Turner M P Deisenroth and C E Rasmussen ldquoState-spaceinference and learning with Gaussian processesrdquo Journal ofMachine Learning Research pp 868ndash875 2010

[20] M P Deisenroth D Fox and C E Rasmussen ldquoGaussianprocesses for data-efficient learning in robotics and controlrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 37 no 2 pp 408ndash423 2015

[21] L M Bujorianu Stochastic Reachability Analysis of HybridSystems Communications and Control Engineering SeriesSpringer Science amp Business Media 2012

[22] S Paoletti A L Juloski G Ferrari-Trecate and R VidalldquoIdentification of hybrid systems a tutorialrdquo European Journalof Control vol 13 no 2-3 pp 242ndash260 2007

[23] G Pillonetto ldquoA new kernel-based approach to hybrid systemidentificationrdquo Automatica A Journal of IFAC the InternationalFederation of Automatic Control vol 70 pp 21ndash31 2016

[24] K Koutroumpas E Cinquemani P Kouretas and J LygerosldquoParameter identification for stochastic hybrid systems usingrandomized optimization A case study on subtilin productionby bacillus subtilisrdquo Nonlinear Analysis Hybrid Systems vol 2no 3 pp 786ndash802 2008

[25] H Abdel-Aziz and X Koutsoukos ldquoLearning and reachabil-ity analysis for stochastic hybrid systems using mixtures ofGaussian processesrdquo in Proceedings of the 24th MediterraneanConference on Control and Automation MED 2016 pp 332ndash337June 2016

[26] D J MacKay Information Theory Inference and LearningAlgorithms Cambridge University Press New York NY USA2003

[27] L Kaufman and P J Rousseeuw Finding Groups in Data AnIntroduction to Cluster Analysis vol 344 John Wiley amp SonsNew York NY USA 1990

[28] D B Crawley L K Lawrie F C Winkelmann et al ldquoEner-gyPlus creating a new-generation building energy simulationprogramrdquo Energy and Buildings vol 33 no 4 pp 319ndash331 2001

[29] ldquoMATLAB and Statistics Toolbox Release 2016ardquo The Math-Works Inc Natick MA USA 2016

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Submit your manuscripts athttpswwwhindawicom

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Navigation and Observation



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to the time variability during the system operation We alsoinclude the ambient temperature prediction to incorporatethe environment effects in the model

The main contributions of this work are as follows

(1) We introduce a nonparametric SHS model basedon Gaussian processes which can be used to modelcomplex coupled discrete and continuous dynamicsrequired to develop data-driven models of smartbuildings

(2) We develop an online clustering-based learningmethodology for SHS when the discrete dynamicscannot be measured In the proposed methodologywe use the 119870-means clustering algorithm to identifythe discrete states of the system and to segment thedata into each corresponding discrete state Theneach segment is used to build a distinct GP modelto abstract the continuous dynamics Learning thesedistinct models is more efficient and mitigates thecomputation limitations of using a single GP modelwith every variable as inputs

(3) We utilize the data-driven modeling approach toprovide an accurate thermal model for multizonebuildings when the buildingsrsquo thermal load is latentWe evaluate the performance and the efficiency of theproposed model is evaluated using datasets of two-zone and five-zone buildings In both experiments weshow themodel accuracy of estimating the level of thethermal load and predicting the zone air temperature

The organization of this paper is as follows Section 2presents the related work of modeling thermal dynamics andmodel learning of dynamical systems using GPs Section 3illustrates a brief theoretical background of GPs The pro-posed model is defined in Section 4 along with the modellearning problem The proposed model learning approach ispresented in Section 5 Finally the experimental results arediscussed in Section 6 with the evaluation

2 Related Work

21 Thermal Modeling of Buildings Developing thermalmodels for buildings has been investigated extensively in theliterature Many studies proposed linear models for single-zone buildings [3 4] while other studies use lumped thermalmodels to approximate multizone buildings as a single-zone model [5 6] Although these methods are useful theytypically result in very simple models On the other handmultizone models have been investigated to identify thethermal dynamics of each zone in multizone buildings [7ndash9] These models are more accurate and they can be used foradvanced control design (eg model predictive control [8])

Most of the above-mentioned work relies on parametricmodels to represent the thermal behavior of buildingsHowever parametric models require the buildingrsquos detailedstructure andor the thermal equations to be known apriori Also they typically use linear dynamics and are notvery accurate Recently nonparametric models have drawnattention to construct accurate nonlinear thermal models

A nonparametric thermal model based on recurrent neuralnetwork (RNN) architecture has been developed to learna compact thermal model for a single thermal zone [10]Models based on RNNs are very useful to represent thenonlinearity of thermal dynamics however they typicallydo not consider the stochasticity in the thermal dynamicsMoreover thesemodels require a large set of the training datato learn the model with the desired quality To mitigate theselimitations a nonparametric probabilistic approach basedon Gaussian processes (GPs) has been developed to learnthermal models in an online adaptive learning framework[11] Both the RNN and the GP models assume that thethermal load of buildings (eg occupancy heating rate fromlight and equipment) can be measured and they used itas a model input However this assumption may not bevalid in many real scenarios Another modeling approachis developed to learn the effect of the thermal load when itcannot be measured [12] This model is based on using agray box parametric model for the thermal dynamics andcombining it with a latent force model based on Gaussianprocesses The latent force model improves the parametricmodel accuracy by learning the dynamical effects of the latentthermal load

Existing nonparametric modeling approaches for build-ings assume that the thermal load can be measured andused as a model input This assumption may limit the use ofthese approaches in many systems therefore we propose anonparametric SHS model for multizone buildings when thethermal load is latentThe proposedmodel estimates the levelof the thermal load and learns a distinct model for each levelin order to improve the model efficiency and accuracy

22 Model Learning of Dynamical Systems Using GPs In thispaper the proposedmodeling approach is based on Gaussianprocesses [13] Gaussian processes have shown a great successinmodelingmanymodern systems because of their attractivefeatures [14] GPs are nonparametric probabilistic modelswhich can express the uncertainty in the system dynamicsand the model confidence through the model predictivevariance Furthermore models based on GPs are very simplesince they have few hyperparameters and relatively requiresmall datasets to learn the modelThese features allow GPs tobuild efficient robust and adaptive dynamical models GPshave been used recently to develop many time-series modelsfor short-term andmultistep forecasting [15 16] For instanceGPs are used to build forecast models for the ambienttemperature and carbon intensity These models are used todevelop an adaptive heating control algorithm thatminimizesthe energy cost and carbon emissions [4] Also GPs are usedto develop a load forecasting model for power systems in[17] and a short-term traffic volume prediction model witha high performance [18] In addition to time-series modelsGPs are used to identify state-space models with a controlinput in order to support robust filtering smoothing andprediction of the system state [19] For instance in roboticsapplication GPs have been used to build a model-baseddata-efficient learning framework for control policy search







3 Background





119891 (x) sim GP (119898 (x) 119896 (x x1015840)) (2)



+ 120575xx10158401205902120596(3)


119901 (y) sim N (0K (XX)) (4)










Θ = argmaxΘ

log119901 (y | ΘD) (7)













E(k)S(k) u(k) v(k)k

Controlpolicy

Time-seriesmodel

SHS model

P12

P21

120587(s(k))

q = 1

x sim GP1

q = i

x sim GPi

q = 2

x sim GP2




D fl (x119896 y119896) 119896 = 119879119904 119879119890 (9)













windowing




Training data

Update q and









C = argminC

sum119892(119909)isin120594

min119888isinC

1003817100381710038171003817119892 (x) minus 11988810038171003817100381710038172 (10)




119904 (119894) = 119887 (119894) minus 119886 (119894)max 119886 (119894) 119887 (119894) (11)



119870

119904 (119894) 119870 isin 119866 (12)





(13)


Θ119902

log119901 (y | Θ119902D119902) (14)




119902119896 = argmin119897

1003817100381710038171003817119892 (x119896) minus 11988811989710038171003817100381710038172 (15)





119901 (x119896+1 | x119896) = 119891119902119896 (x119896) sim GP119902119896 (119898 (x) 119896 (x x)) (16)



RMSE = radic 1119873119873sum119896=1

1198902119896







119879 (19)




V119896 = 119891 (119896) sim GP (119898 (119896) 119870 (119896 1198961015840)) (21)


6 Evaluation







HVAC system


Thermostat



xk

xk+1

k

xk+1

uk






Westzone

Eastzone




2 070 0553 071 0654 062 0605 065 065


x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (22)





1

05

0

minus05

minus1

minus15

minus2

uwest

1

2

3

West features

0 2minus1minus2 1

Δxwest

(a)

1

05

0

minus05

minus1

minus15

minus2

East features

Δxeast

0 2minus1minus2 1

ueast

1

2

3

(b)



690 700 710 720 730 740 750680Time

0

5

10

15

20

Am

bien

t tem

pera

ture

(C)



x119896+1 = 119891 (x119896 119906119896 V119896 119897119896) (23)



RMSE 005 006 007 006MRSE 0008 0009 001 0009Max 06 103 06 08





1

15

2

25

3

Estim

ated

stat

e

10 30 40 50 6020 70Time

7030 40 50 602010Time

354

455

556

Tota

l the

rmal

load

(W) 104








RMSE 023 03 056 03MRSE 0009 0013 0023 0013LD 51 6 27 257


u119896+1 sim 119891 (s119896 V119896) s119896 = (119902119896 x119896) (24)


LD = 12 log (2120587) + 12119873119873sum119894=1

log (1205902119894 ) + 11989021198941205902119894 (25)







22

225

23

235

24Te

mpe

ratu

re (C

)

690 700 710 720 730 740 750680Time

(a)

22

225

23

235

Tem

pera

ture

(C)

690 700 710 720 730 740 750680Time


(b)



01

02

03

04

05

06

07

08

09

1

11

RMSE

10 15 20 25 30 35 40 45 505Week number




x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (26)





minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(a)

minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(b)

minus1 0 1 2minus2

Δxwest

minus2

minus1

0

1

uwest

(c)




RMSE 032 028 025 036 028 029 028 027 022 011MRSE 011 005 008 006 010 005 01 005 06 002Max 35 63 55 101 121 45 67 45 25 32



0

02

04

06

08

1

12

14

16

18

Tim

e (s)






2 31Cluster number

4

5

6

7

Hea

ting

rate

(wat

t)

104

(a)

21Cluster number

3

4

5

6

7

Hea

ting

rate

(wat

t)

104

(b)


4

5

6

Hea

ting

rate

(wat

t)

104

(c)





21522

22523

235

East

tem

pera

ture

21

22

23

24

25

Wes

t tem

pera

ture




21

22

23

24

25

Wes

t tem

pera

ture

21215

22225

23235

East

tem

pera

ture






Core zone

North zone

South zone

Eastzone

Westzone





7 Conclusion



1

2

3

South zone

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

us

0 5minus5

Δxs

(a)

North zone

0 2minus2

Δxn

minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

un

1

2

(b)


1

2

minus2000

minus1000

0

1000

2000

3000

0 5 10minus5

Δxw

uw

(a)

0 5 10minus5

Δxe

minus2000

minus1000

0

1000

2000

3000

ue

1

2

(b)

minus2000

0

2000

4000

6000

8000

uc

0 1minus1

Δxc

1

2

(c)





40Time

60 80 100 120 140 1600 200

5

10Th

erm

al lo

ad (W

)

4020 60 80 100 120 140 1600Time

1

2

Estim

ated

stat

e

106



161820222426

Tem

pera

ture

(C)

710 720 730 740 750 760 770700Time


0

100

200

300

400

500

Aver

age l

earn

ing

time (

ms)

2 3 4 5 6 7 8 91

Number of zones






0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















3 Background





119891 (x) sim GP (119898 (x) 119896 (x x1015840)) (2)



+ 120575xx10158401205902120596(3)


119901 (y) sim N (0K (XX)) (4)










Θ = argmaxΘ

log119901 (y | ΘD) (7)













E(k)S(k) u(k) v(k)k

Controlpolicy

Time-seriesmodel

SHS model

P12

P21

120587(s(k))

q = 1

x sim GP1

q = i

x sim GPi

q = 2

x sim GP2




D fl (x119896 y119896) 119896 = 119879119904 119879119890 (9)













windowing




Training data

Update q and









C = argminC

sum119892(119909)isin120594

min119888isinC

1003817100381710038171003817119892 (x) minus 11988810038171003817100381710038172 (10)




119904 (119894) = 119887 (119894) minus 119886 (119894)max 119886 (119894) 119887 (119894) (11)



119870

119904 (119894) 119870 isin 119866 (12)





(13)


Θ119902

log119901 (y | Θ119902D119902) (14)




119902119896 = argmin119897

1003817100381710038171003817119892 (x119896) minus 11988811989710038171003817100381710038172 (15)





119901 (x119896+1 | x119896) = 119891119902119896 (x119896) sim GP119902119896 (119898 (x) 119896 (x x)) (16)



RMSE = radic 1119873119873sum119896=1

1198902119896







119879 (19)




V119896 = 119891 (119896) sim GP (119898 (119896) 119870 (119896 1198961015840)) (21)


6 Evaluation







HVAC system


Thermostat



xk

xk+1

k

xk+1

uk






Westzone

Eastzone




2 070 0553 071 0654 062 0605 065 065


x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (22)





1

05

0

minus05

minus1

minus15

minus2

uwest

1

2

3

West features

0 2minus1minus2 1

Δxwest

(a)

1

05

0

minus05

minus1

minus15

minus2

East features

Δxeast

0 2minus1minus2 1

ueast

1

2

3

(b)



690 700 710 720 730 740 750680Time

0

5

10

15

20

Am

bien

t tem

pera

ture

(C)



x119896+1 = 119891 (x119896 119906119896 V119896 119897119896) (23)



RMSE 005 006 007 006MRSE 0008 0009 001 0009Max 06 103 06 08





1

15

2

25

3

Estim

ated

stat

e

10 30 40 50 6020 70Time

7030 40 50 602010Time

354

455

556

Tota

l the

rmal

load

(W) 104








RMSE 023 03 056 03MRSE 0009 0013 0023 0013LD 51 6 27 257


u119896+1 sim 119891 (s119896 V119896) s119896 = (119902119896 x119896) (24)


LD = 12 log (2120587) + 12119873119873sum119894=1

log (1205902119894 ) + 11989021198941205902119894 (25)







22

225

23

235

24Te

mpe

ratu

re (C

)

690 700 710 720 730 740 750680Time

(a)

22

225

23

235

Tem

pera

ture

(C)

690 700 710 720 730 740 750680Time


(b)



01

02

03

04

05

06

07

08

09

1

11

RMSE

10 15 20 25 30 35 40 45 505Week number




x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (26)





minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(a)

minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(b)

minus1 0 1 2minus2

Δxwest

minus2

minus1

0

1

uwest

(c)




RMSE 032 028 025 036 028 029 028 027 022 011MRSE 011 005 008 006 010 005 01 005 06 002Max 35 63 55 101 121 45 67 45 25 32



0

02

04

06

08

1

12

14

16

18

Tim

e (s)






2 31Cluster number

4

5

6

7

Hea

ting

rate

(wat

t)

104

(a)

21Cluster number

3

4

5

6

7

Hea

ting

rate

(wat

t)

104

(b)


4

5

6

Hea

ting

rate

(wat

t)

104

(c)





21522

22523

235

East

tem

pera

ture

21

22

23

24

25

Wes

t tem

pera

ture




21

22

23

24

25

Wes

t tem

pera

ture

21215

22225

23235

East

tem

pera

ture






Core zone

North zone

South zone

Eastzone

Westzone





7 Conclusion



1

2

3

South zone

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

us

0 5minus5

Δxs

(a)

North zone

0 2minus2

Δxn

minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

un

1

2

(b)


1

2

minus2000

minus1000

0

1000

2000

3000

0 5 10minus5

Δxw

uw

(a)

0 5 10minus5

Δxe

minus2000

minus1000

0

1000

2000

3000

ue

1

2

(b)

minus2000

0

2000

4000

6000

8000

uc

0 1minus1

Δxc

1

2

(c)





40Time

60 80 100 120 140 1600 200

5

10Th

erm

al lo

ad (W

)

4020 60 80 100 120 140 1600Time

1

2

Estim

ated

stat

e

106



161820222426

Tem

pera

ture

(C)

710 720 730 740 750 760 770700Time


0

100

200

300

400

500

Aver

age l

earn

ing

time (

ms)

2 3 4 5 6 7 8 91

Number of zones






0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Θ = argmaxΘ

log119901 (y | ΘD) (7)













E(k)S(k) u(k) v(k)k

Controlpolicy

Time-seriesmodel

SHS model

P12

P21

120587(s(k))

q = 1

x sim GP1

q = i

x sim GPi

q = 2

x sim GP2




D fl (x119896 y119896) 119896 = 119879119904 119879119890 (9)













windowing




Training data

Update q and









C = argminC

sum119892(119909)isin120594

min119888isinC

1003817100381710038171003817119892 (x) minus 11988810038171003817100381710038172 (10)




119904 (119894) = 119887 (119894) minus 119886 (119894)max 119886 (119894) 119887 (119894) (11)



119870

119904 (119894) 119870 isin 119866 (12)





(13)


Θ119902

log119901 (y | Θ119902D119902) (14)




119902119896 = argmin119897

1003817100381710038171003817119892 (x119896) minus 11988811989710038171003817100381710038172 (15)





119901 (x119896+1 | x119896) = 119891119902119896 (x119896) sim GP119902119896 (119898 (x) 119896 (x x)) (16)



RMSE = radic 1119873119873sum119896=1

1198902119896







119879 (19)




V119896 = 119891 (119896) sim GP (119898 (119896) 119870 (119896 1198961015840)) (21)


6 Evaluation







HVAC system


Thermostat



xk

xk+1

k

xk+1

uk






Westzone

Eastzone




2 070 0553 071 0654 062 0605 065 065


x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (22)





1

05

0

minus05

minus1

minus15

minus2

uwest

1

2

3

West features

0 2minus1minus2 1

Δxwest

(a)

1

05

0

minus05

minus1

minus15

minus2

East features

Δxeast

0 2minus1minus2 1

ueast

1

2

3

(b)



690 700 710 720 730 740 750680Time

0

5

10

15

20

Am

bien

t tem

pera

ture

(C)



x119896+1 = 119891 (x119896 119906119896 V119896 119897119896) (23)



RMSE 005 006 007 006MRSE 0008 0009 001 0009Max 06 103 06 08





1

15

2

25

3

Estim

ated

stat

e

10 30 40 50 6020 70Time

7030 40 50 602010Time

354

455

556

Tota

l the

rmal

load

(W) 104








RMSE 023 03 056 03MRSE 0009 0013 0023 0013LD 51 6 27 257


u119896+1 sim 119891 (s119896 V119896) s119896 = (119902119896 x119896) (24)


LD = 12 log (2120587) + 12119873119873sum119894=1

log (1205902119894 ) + 11989021198941205902119894 (25)







22

225

23

235

24Te

mpe

ratu

re (C

)

690 700 710 720 730 740 750680Time

(a)

22

225

23

235

Tem

pera

ture

(C)

690 700 710 720 730 740 750680Time


(b)



01

02

03

04

05

06

07

08

09

1

11

RMSE

10 15 20 25 30 35 40 45 505Week number




x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (26)





minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(a)

minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(b)

minus1 0 1 2minus2

Δxwest

minus2

minus1

0

1

uwest

(c)




RMSE 032 028 025 036 028 029 028 027 022 011MRSE 011 005 008 006 010 005 01 005 06 002Max 35 63 55 101 121 45 67 45 25 32



0

02

04

06

08

1

12

14

16

18

Tim

e (s)






2 31Cluster number

4

5

6

7

Hea

ting

rate

(wat

t)

104

(a)

21Cluster number

3

4

5

6

7

Hea

ting

rate

(wat

t)

104

(b)


4

5

6

Hea

ting

rate

(wat

t)

104

(c)





21522

22523

235

East

tem

pera

ture

21

22

23

24

25

Wes

t tem

pera

ture




21

22

23

24

25

Wes

t tem

pera

ture

21215

22225

23235

East

tem

pera

ture






Core zone

North zone

South zone

Eastzone

Westzone





7 Conclusion



1

2

3

South zone

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

us

0 5minus5

Δxs

(a)

North zone

0 2minus2

Δxn

minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

un

1

2

(b)


1

2

minus2000

minus1000

0

1000

2000

3000

0 5 10minus5

Δxw

uw

(a)

0 5 10minus5

Δxe

minus2000

minus1000

0

1000

2000

3000

ue

1

2

(b)

minus2000

0

2000

4000

6000

8000

uc

0 1minus1

Δxc

1

2

(c)





40Time

60 80 100 120 140 1600 200

5

10Th

erm

al lo

ad (W

)

4020 60 80 100 120 140 1600Time

1

2

Estim

ated

stat

e

106



161820222426

Tem

pera

ture

(C)

710 720 730 740 750 760 770700Time


0

100

200

300

400

500

Aver

age l

earn

ing

time (

ms)

2 3 4 5 6 7 8 91

Number of zones






0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















windowing




Training data

Update q and









C = argminC

sum119892(119909)isin120594

min119888isinC

1003817100381710038171003817119892 (x) minus 11988810038171003817100381710038172 (10)




119904 (119894) = 119887 (119894) minus 119886 (119894)max 119886 (119894) 119887 (119894) (11)



119870

119904 (119894) 119870 isin 119866 (12)





(13)


Θ119902

log119901 (y | Θ119902D119902) (14)




119902119896 = argmin119897

1003817100381710038171003817119892 (x119896) minus 11988811989710038171003817100381710038172 (15)





119901 (x119896+1 | x119896) = 119891119902119896 (x119896) sim GP119902119896 (119898 (x) 119896 (x x)) (16)



RMSE = radic 1119873119873sum119896=1

1198902119896







119879 (19)




V119896 = 119891 (119896) sim GP (119898 (119896) 119870 (119896 1198961015840)) (21)


6 Evaluation







HVAC system


Thermostat



xk

xk+1

k

xk+1

uk






Westzone

Eastzone




2 070 0553 071 0654 062 0605 065 065


x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (22)





1

05

0

minus05

minus1

minus15

minus2

uwest

1

2

3

West features

0 2minus1minus2 1

Δxwest

(a)

1

05

0

minus05

minus1

minus15

minus2

East features

Δxeast

0 2minus1minus2 1

ueast

1

2

3

(b)



690 700 710 720 730 740 750680Time

0

5

10

15

20

Am

bien

t tem

pera

ture

(C)



x119896+1 = 119891 (x119896 119906119896 V119896 119897119896) (23)



RMSE 005 006 007 006MRSE 0008 0009 001 0009Max 06 103 06 08





1

15

2

25

3

Estim

ated

stat

e

10 30 40 50 6020 70Time

7030 40 50 602010Time

354

455

556

Tota

l the

rmal

load

(W) 104








RMSE 023 03 056 03MRSE 0009 0013 0023 0013LD 51 6 27 257


u119896+1 sim 119891 (s119896 V119896) s119896 = (119902119896 x119896) (24)


LD = 12 log (2120587) + 12119873119873sum119894=1

log (1205902119894 ) + 11989021198941205902119894 (25)







22

225

23

235

24Te

mpe

ratu

re (C

)

690 700 710 720 730 740 750680Time

(a)

22

225

23

235

Tem

pera

ture

(C)

690 700 710 720 730 740 750680Time


(b)



01

02

03

04

05

06

07

08

09

1

11

RMSE

10 15 20 25 30 35 40 45 505Week number




x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (26)





minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(a)

minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(b)

minus1 0 1 2minus2

Δxwest

minus2

minus1

0

1

uwest

(c)




RMSE 032 028 025 036 028 029 028 027 022 011MRSE 011 005 008 006 010 005 01 005 06 002Max 35 63 55 101 121 45 67 45 25 32



0

02

04

06

08

1

12

14

16

18

Tim

e (s)






2 31Cluster number

4

5

6

7

Hea

ting

rate

(wat

t)

104

(a)

21Cluster number

3

4

5

6

7

Hea

ting

rate

(wat

t)

104

(b)


4

5

6

Hea

ting

rate

(wat

t)

104

(c)





21522

22523

235

East

tem

pera

ture

21

22

23

24

25

Wes

t tem

pera

ture




21

22

23

24

25

Wes

t tem

pera

ture

21215

22225

23235

East

tem

pera

ture






Core zone

North zone

South zone

Eastzone

Westzone





7 Conclusion



1

2

3

South zone

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

us

0 5minus5

Δxs

(a)

North zone

0 2minus2

Δxn

minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

un

1

2

(b)


1

2

minus2000

minus1000

0

1000

2000

3000

0 5 10minus5

Δxw

uw

(a)

0 5 10minus5

Δxe

minus2000

minus1000

0

1000

2000

3000

ue

1

2

(b)

minus2000

0

2000

4000

6000

8000

uc

0 1minus1

Δxc

1

2

(c)





40Time

60 80 100 120 140 1600 200

5

10Th

erm

al lo

ad (W

)

4020 60 80 100 120 140 1600Time

1

2

Estim

ated

stat

e

106



161820222426

Tem

pera

ture

(C)

710 720 730 740 750 760 770700Time


0

100

200

300

400

500

Aver

age l

earn

ing

time (

ms)

2 3 4 5 6 7 8 91

Number of zones






0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119904 (119894) = 119887 (119894) minus 119886 (119894)max 119886 (119894) 119887 (119894) (11)



119870

119904 (119894) 119870 isin 119866 (12)





(13)


Θ119902

log119901 (y | Θ119902D119902) (14)




119902119896 = argmin119897

1003817100381710038171003817119892 (x119896) minus 11988811989710038171003817100381710038172 (15)





119901 (x119896+1 | x119896) = 119891119902119896 (x119896) sim GP119902119896 (119898 (x) 119896 (x x)) (16)



RMSE = radic 1119873119873sum119896=1

1198902119896







119879 (19)




V119896 = 119891 (119896) sim GP (119898 (119896) 119870 (119896 1198961015840)) (21)


6 Evaluation







HVAC system


Thermostat



xk

xk+1

k

xk+1

uk






Westzone

Eastzone




2 070 0553 071 0654 062 0605 065 065


x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (22)





1

05

0

minus05

minus1

minus15

minus2

uwest

1

2

3

West features

0 2minus1minus2 1

Δxwest

(a)

1

05

0

minus05

minus1

minus15

minus2

East features

Δxeast

0 2minus1minus2 1

ueast

1

2

3

(b)



690 700 710 720 730 740 750680Time

0

5

10

15

20

Am

bien

t tem

pera

ture

(C)



x119896+1 = 119891 (x119896 119906119896 V119896 119897119896) (23)



RMSE 005 006 007 006MRSE 0008 0009 001 0009Max 06 103 06 08





1

15

2

25

3

Estim

ated

stat

e

10 30 40 50 6020 70Time

7030 40 50 602010Time

354

455

556

Tota

l the

rmal

load

(W) 104








RMSE 023 03 056 03MRSE 0009 0013 0023 0013LD 51 6 27 257


u119896+1 sim 119891 (s119896 V119896) s119896 = (119902119896 x119896) (24)


LD = 12 log (2120587) + 12119873119873sum119894=1

log (1205902119894 ) + 11989021198941205902119894 (25)







22

225

23

235

24Te

mpe

ratu

re (C

)

690 700 710 720 730 740 750680Time

(a)

22

225

23

235

Tem

pera

ture

(C)

690 700 710 720 730 740 750680Time


(b)



01

02

03

04

05

06

07

08

09

1

11

RMSE

10 15 20 25 30 35 40 45 505Week number




x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (26)





minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(a)

minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(b)

minus1 0 1 2minus2

Δxwest

minus2

minus1

0

1

uwest

(c)




RMSE 032 028 025 036 028 029 028 027 022 011MRSE 011 005 008 006 010 005 01 005 06 002Max 35 63 55 101 121 45 67 45 25 32



0

02

04

06

08

1

12

14

16

18

Tim

e (s)






2 31Cluster number

4

5

6

7

Hea

ting

rate

(wat

t)

104

(a)

21Cluster number

3

4

5

6

7

Hea

ting

rate

(wat

t)

104

(b)


4

5

6

Hea

ting

rate

(wat

t)

104

(c)





21522

22523

235

East

tem

pera

ture

21

22

23

24

25

Wes

t tem

pera

ture




21

22

23

24

25

Wes

t tem

pera

ture

21215

22225

23235

East

tem

pera

ture






Core zone

North zone

South zone

Eastzone

Westzone





7 Conclusion



1

2

3

South zone

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

us

0 5minus5

Δxs

(a)

North zone

0 2minus2

Δxn

minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

un

1

2

(b)


1

2

minus2000

minus1000

0

1000

2000

3000

0 5 10minus5

Δxw

uw

(a)

0 5 10minus5

Δxe

minus2000

minus1000

0

1000

2000

3000

ue

1

2

(b)

minus2000

0

2000

4000

6000

8000

uc

0 1minus1

Δxc

1

2

(c)





40Time

60 80 100 120 140 1600 200

5

10Th

erm

al lo

ad (W

)

4020 60 80 100 120 140 1600Time

1

2

Estim

ated

stat

e

106



161820222426

Tem

pera

ture

(C)

710 720 730 740 750 760 770700Time


0

100

200

300

400

500

Aver

age l

earn

ing

time (

ms)

2 3 4 5 6 7 8 91

Number of zones






0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119901 (x119896+1 | x119896) = 119891119902119896 (x119896) sim GP119902119896 (119898 (x) 119896 (x x)) (16)



RMSE = radic 1119873119873sum119896=1

1198902119896







119879 (19)




V119896 = 119891 (119896) sim GP (119898 (119896) 119870 (119896 1198961015840)) (21)


6 Evaluation







HVAC system


Thermostat



xk

xk+1

k

xk+1

uk






Westzone

Eastzone




2 070 0553 071 0654 062 0605 065 065


x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (22)





1

05

0

minus05

minus1

minus15

minus2

uwest

1

2

3

West features

0 2minus1minus2 1

Δxwest

(a)

1

05

0

minus05

minus1

minus15

minus2

East features

Δxeast

0 2minus1minus2 1

ueast

1

2

3

(b)



690 700 710 720 730 740 750680Time

0

5

10

15

20

Am

bien

t tem

pera

ture

(C)



x119896+1 = 119891 (x119896 119906119896 V119896 119897119896) (23)



RMSE 005 006 007 006MRSE 0008 0009 001 0009Max 06 103 06 08





1

15

2

25

3

Estim

ated

stat

e

10 30 40 50 6020 70Time

7030 40 50 602010Time

354

455

556

Tota

l the

rmal

load

(W) 104








RMSE 023 03 056 03MRSE 0009 0013 0023 0013LD 51 6 27 257


u119896+1 sim 119891 (s119896 V119896) s119896 = (119902119896 x119896) (24)


LD = 12 log (2120587) + 12119873119873sum119894=1

log (1205902119894 ) + 11989021198941205902119894 (25)







22

225

23

235

24Te

mpe

ratu

re (C

)

690 700 710 720 730 740 750680Time

(a)

22

225

23

235

Tem

pera

ture

(C)

690 700 710 720 730 740 750680Time


(b)



01

02

03

04

05

06

07

08

09

1

11

RMSE

10 15 20 25 30 35 40 45 505Week number




x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (26)





minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(a)

minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(b)

minus1 0 1 2minus2

Δxwest

minus2

minus1

0

1

uwest

(c)




RMSE 032 028 025 036 028 029 028 027 022 011MRSE 011 005 008 006 010 005 01 005 06 002Max 35 63 55 101 121 45 67 45 25 32



0

02

04

06

08

1

12

14

16

18

Tim

e (s)






2 31Cluster number

4

5

6

7

Hea

ting

rate

(wat

t)

104

(a)

21Cluster number

3

4

5

6

7

Hea

ting

rate

(wat

t)

104

(b)


4

5

6

Hea

ting

rate

(wat

t)

104

(c)





21522

22523

235

East

tem

pera

ture

21

22

23

24

25

Wes

t tem

pera

ture




21

22

23

24

25

Wes

t tem

pera

ture

21215

22225

23235

East

tem

pera

ture






Core zone

North zone

South zone

Eastzone

Westzone





7 Conclusion



1

2

3

South zone

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

us

0 5minus5

Δxs

(a)

North zone

0 2minus2

Δxn

minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

un

1

2

(b)


1

2

minus2000

minus1000

0

1000

2000

3000

0 5 10minus5

Δxw

uw

(a)

0 5 10minus5

Δxe

minus2000

minus1000

0

1000

2000

3000

ue

1

2

(b)

minus2000

0

2000

4000

6000

8000

uc

0 1minus1

Δxc

1

2

(c)





40Time

60 80 100 120 140 1600 200

5

10Th

erm

al lo

ad (W

)

4020 60 80 100 120 140 1600Time

1

2

Estim

ated

stat

e

106



161820222426

Tem

pera

ture

(C)

710 720 730 740 750 760 770700Time


0

100

200

300

400

500

Aver

age l

earn

ing

time (

ms)

2 3 4 5 6 7 8 91

Number of zones






0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












HVAC system


Thermostat



xk

xk+1

k

xk+1

uk






Westzone

Eastzone




2 070 0553 071 0654 062 0605 065 065


x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (22)





1

05

0

minus05

minus1

minus15

minus2

uwest

1

2

3

West features

0 2minus1minus2 1

Δxwest

(a)

1

05

0

minus05

minus1

minus15

minus2

East features

Δxeast

0 2minus1minus2 1

ueast

1

2

3

(b)



690 700 710 720 730 740 750680Time

0

5

10

15

20

Am

bien

t tem

pera

ture

(C)



x119896+1 = 119891 (x119896 119906119896 V119896 119897119896) (23)



RMSE 005 006 007 006MRSE 0008 0009 001 0009Max 06 103 06 08





1

15

2

25

3

Estim

ated

stat

e

10 30 40 50 6020 70Time

7030 40 50 602010Time

354

455

556

Tota

l the

rmal

load

(W) 104








RMSE 023 03 056 03MRSE 0009 0013 0023 0013LD 51 6 27 257


u119896+1 sim 119891 (s119896 V119896) s119896 = (119902119896 x119896) (24)


LD = 12 log (2120587) + 12119873119873sum119894=1

log (1205902119894 ) + 11989021198941205902119894 (25)







22

225

23

235

24Te

mpe

ratu

re (C

)

690 700 710 720 730 740 750680Time

(a)

22

225

23

235

Tem

pera

ture

(C)

690 700 710 720 730 740 750680Time


(b)



01

02

03

04

05

06

07

08

09

1

11

RMSE

10 15 20 25 30 35 40 45 505Week number




x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (26)





minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(a)

minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(b)

minus1 0 1 2minus2

Δxwest

minus2

minus1

0

1

uwest

(c)




RMSE 032 028 025 036 028 029 028 027 022 011MRSE 011 005 008 006 010 005 01 005 06 002Max 35 63 55 101 121 45 67 45 25 32



0

02

04

06

08

1

12

14

16

18

Tim

e (s)






2 31Cluster number

4

5

6

7

Hea

ting

rate

(wat

t)

104

(a)

21Cluster number

3

4

5

6

7

Hea

ting

rate

(wat

t)

104

(b)


4

5

6

Hea

ting

rate

(wat

t)

104

(c)





21522

22523

235

East

tem

pera

ture

21

22

23

24

25

Wes

t tem

pera

ture




21

22

23

24

25

Wes

t tem

pera

ture

21215

22225

23235

East

tem

pera

ture






Core zone

North zone

South zone

Eastzone

Westzone





7 Conclusion



1

2

3

South zone

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

us

0 5minus5

Δxs

(a)

North zone

0 2minus2

Δxn

minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

un

1

2

(b)


1

2

minus2000

minus1000

0

1000

2000

3000

0 5 10minus5

Δxw

uw

(a)

0 5 10minus5

Δxe

minus2000

minus1000

0

1000

2000

3000

ue

1

2

(b)

minus2000

0

2000

4000

6000

8000

uc

0 1minus1

Δxc

1

2

(c)





40Time

60 80 100 120 140 1600 200

5

10Th

erm

al lo

ad (W

)

4020 60 80 100 120 140 1600Time

1

2

Estim

ated

stat

e

106



161820222426

Tem

pera

ture

(C)

710 720 730 740 750 760 770700Time


0

100

200

300

400

500

Aver

age l

earn

ing

time (

ms)

2 3 4 5 6 7 8 91

Number of zones






0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

05

0

minus05

minus1

minus15

minus2

uwest

1

2

3

West features

0 2minus1minus2 1

Δxwest

(a)

1

05

0

minus05

minus1

minus15

minus2

East features

Δxeast

0 2minus1minus2 1

ueast

1

2

3

(b)



690 700 710 720 730 740 750680Time

0

5

10

15

20

Am

bien

t tem

pera

ture

(C)



x119896+1 = 119891 (x119896 119906119896 V119896 119897119896) (23)



RMSE 005 006 007 006MRSE 0008 0009 001 0009Max 06 103 06 08





1

15

2

25

3

Estim

ated

stat

e

10 30 40 50 6020 70Time

7030 40 50 602010Time

354

455

556

Tota

l the

rmal

load

(W) 104








RMSE 023 03 056 03MRSE 0009 0013 0023 0013LD 51 6 27 257


u119896+1 sim 119891 (s119896 V119896) s119896 = (119902119896 x119896) (24)


LD = 12 log (2120587) + 12119873119873sum119894=1

log (1205902119894 ) + 11989021198941205902119894 (25)







22

225

23

235

24Te

mpe

ratu

re (C

)

690 700 710 720 730 740 750680Time

(a)

22

225

23

235

Tem

pera

ture

(C)

690 700 710 720 730 740 750680Time


(b)



01

02

03

04

05

06

07

08

09

1

11

RMSE

10 15 20 25 30 35 40 45 505Week number




x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (26)





minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(a)

minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(b)

minus1 0 1 2minus2

Δxwest

minus2

minus1

0

1

uwest

(c)




RMSE 032 028 025 036 028 029 028 027 022 011MRSE 011 005 008 006 010 005 01 005 06 002Max 35 63 55 101 121 45 67 45 25 32



0

02

04

06

08

1

12

14

16

18

Tim

e (s)






2 31Cluster number

4

5

6

7

Hea

ting

rate

(wat

t)

104

(a)

21Cluster number

3

4

5

6

7

Hea

ting

rate

(wat

t)

104

(b)


4

5

6

Hea

ting

rate

(wat

t)

104

(c)





21522

22523

235

East

tem

pera

ture

21

22

23

24

25

Wes

t tem

pera

ture




21

22

23

24

25

Wes

t tem

pera

ture

21215

22225

23235

East

tem

pera

ture






Core zone

North zone

South zone

Eastzone

Westzone





7 Conclusion



1

2

3

South zone

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

us

0 5minus5

Δxs

(a)

North zone

0 2minus2

Δxn

minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

un

1

2

(b)


1

2

minus2000

minus1000

0

1000

2000

3000

0 5 10minus5

Δxw

uw

(a)

0 5 10minus5

Δxe

minus2000

minus1000

0

1000

2000

3000

ue

1

2

(b)

minus2000

0

2000

4000

6000

8000

uc

0 1minus1

Δxc

1

2

(c)





40Time

60 80 100 120 140 1600 200

5

10Th

erm

al lo

ad (W

)

4020 60 80 100 120 140 1600Time

1

2

Estim

ated

stat

e

106



161820222426

Tem

pera

ture

(C)

710 720 730 740 750 760 770700Time


0

100

200

300

400

500

Aver

age l

earn

ing

time (

ms)

2 3 4 5 6 7 8 91

Number of zones






0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

15

2

25

3

Estim

ated

stat

e

10 30 40 50 6020 70Time

7030 40 50 602010Time

354

455

556

Tota

l the

rmal

load

(W) 104








RMSE 023 03 056 03MRSE 0009 0013 0023 0013LD 51 6 27 257


u119896+1 sim 119891 (s119896 V119896) s119896 = (119902119896 x119896) (24)


LD = 12 log (2120587) + 12119873119873sum119894=1

log (1205902119894 ) + 11989021198941205902119894 (25)







22

225

23

235

24Te

mpe

ratu

re (C

)

690 700 710 720 730 740 750680Time

(a)

22

225

23

235

Tem

pera

ture

(C)

690 700 710 720 730 740 750680Time


(b)



01

02

03

04

05

06

07

08

09

1

11

RMSE

10 15 20 25 30 35 40 45 505Week number




x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (26)





minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(a)

minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(b)

minus1 0 1 2minus2

Δxwest

minus2

minus1

0

1

uwest

(c)




RMSE 032 028 025 036 028 029 028 027 022 011MRSE 011 005 008 006 010 005 01 005 06 002Max 35 63 55 101 121 45 67 45 25 32



0

02

04

06

08

1

12

14

16

18

Tim

e (s)






2 31Cluster number

4

5

6

7

Hea

ting

rate

(wat

t)

104

(a)

21Cluster number

3

4

5

6

7

Hea

ting

rate

(wat

t)

104

(b)


4

5

6

Hea

ting

rate

(wat

t)

104

(c)





21522

22523

235

East

tem

pera

ture

21

22

23

24

25

Wes

t tem

pera

ture




21

22

23

24

25

Wes

t tem

pera

ture

21215

22225

23235

East

tem

pera

ture






Core zone

North zone

South zone

Eastzone

Westzone





7 Conclusion



1

2

3

South zone

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

us

0 5minus5

Δxs

(a)

North zone

0 2minus2

Δxn

minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

un

1

2

(b)


1

2

minus2000

minus1000

0

1000

2000

3000

0 5 10minus5

Δxw

uw

(a)

0 5 10minus5

Δxe

minus2000

minus1000

0

1000

2000

3000

ue

1

2

(b)

minus2000

0

2000

4000

6000

8000

uc

0 1minus1

Δxc

1

2

(c)





40Time

60 80 100 120 140 1600 200

5

10Th

erm

al lo

ad (W

)

4020 60 80 100 120 140 1600Time

1

2

Estim

ated

stat

e

106



161820222426

Tem

pera

ture

(C)

710 720 730 740 750 760 770700Time


0

100

200

300

400

500

Aver

age l

earn

ing

time (

ms)

2 3 4 5 6 7 8 91

Number of zones






0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











22

225

23

235

24Te

mpe

ratu

re (C

)

690 700 710 720 730 740 750680Time

(a)

22

225

23

235

Tem

pera

ture

(C)

690 700 710 720 730 740 750680Time


(b)



01

02

03

04

05

06

07

08

09

1

11

RMSE

10 15 20 25 30 35 40 45 505Week number




x119896+1 sim 119891119902119896 (x119896 119906119896 V119896) 119902119896 isin Q (26)





minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(a)

minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(b)

minus1 0 1 2minus2

Δxwest

minus2

minus1

0

1

uwest

(c)




RMSE 032 028 025 036 028 029 028 027 022 011MRSE 011 005 008 006 010 005 01 005 06 002Max 35 63 55 101 121 45 67 45 25 32



0

02

04

06

08

1

12

14

16

18

Tim

e (s)






2 31Cluster number

4

5

6

7

Hea

ting

rate

(wat

t)

104

(a)

21Cluster number

3

4

5

6

7

Hea

ting

rate

(wat

t)

104

(b)


4

5

6

Hea

ting

rate

(wat

t)

104

(c)





21522

22523

235

East

tem

pera

ture

21

22

23

24

25

Wes

t tem

pera

ture




21

22

23

24

25

Wes

t tem

pera

ture

21215

22225

23235

East

tem

pera

ture






Core zone

North zone

South zone

Eastzone

Westzone





7 Conclusion



1

2

3

South zone

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

us

0 5minus5

Δxs

(a)

North zone

0 2minus2

Δxn

minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

un

1

2

(b)


1

2

minus2000

minus1000

0

1000

2000

3000

0 5 10minus5

Δxw

uw

(a)

0 5 10minus5

Δxe

minus2000

minus1000

0

1000

2000

3000

ue

1

2

(b)

minus2000

0

2000

4000

6000

8000

uc

0 1minus1

Δxc

1

2

(c)





40Time

60 80 100 120 140 1600 200

5

10Th

erm

al lo

ad (W

)

4020 60 80 100 120 140 1600Time

1

2

Estim

ated

stat

e

106



161820222426

Tem

pera

ture

(C)

710 720 730 740 750 760 770700Time


0

100

200

300

400

500

Aver

age l

earn

ing

time (

ms)

2 3 4 5 6 7 8 91

Number of zones






0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(a)

minus2

minus1

0

1

uwest

minus1 0 1 2minus2

Δxwest

(b)

minus1 0 1 2minus2

Δxwest

minus2

minus1

0

1

uwest

(c)




RMSE 032 028 025 036 028 029 028 027 022 011MRSE 011 005 008 006 010 005 01 005 06 002Max 35 63 55 101 121 45 67 45 25 32



0

02

04

06

08

1

12

14

16

18

Tim

e (s)






2 31Cluster number

4

5

6

7

Hea

ting

rate

(wat

t)

104

(a)

21Cluster number

3

4

5

6

7

Hea

ting

rate

(wat

t)

104

(b)


4

5

6

Hea

ting

rate

(wat

t)

104

(c)





21522

22523

235

East

tem

pera

ture

21

22

23

24

25

Wes

t tem

pera

ture




21

22

23

24

25

Wes

t tem

pera

ture

21215

22225

23235

East

tem

pera

ture






Core zone

North zone

South zone

Eastzone

Westzone





7 Conclusion



1

2

3

South zone

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

us

0 5minus5

Δxs

(a)

North zone

0 2minus2

Δxn

minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

un

1

2

(b)


1

2

minus2000

minus1000

0

1000

2000

3000

0 5 10minus5

Δxw

uw

(a)

0 5 10minus5

Δxe

minus2000

minus1000

0

1000

2000

3000

ue

1

2

(b)

minus2000

0

2000

4000

6000

8000

uc

0 1minus1

Δxc

1

2

(c)





40Time

60 80 100 120 140 1600 200

5

10Th

erm

al lo

ad (W

)

4020 60 80 100 120 140 1600Time

1

2

Estim

ated

stat

e

106



161820222426

Tem

pera

ture

(C)

710 720 730 740 750 760 770700Time


0

100

200

300

400

500

Aver

age l

earn

ing

time (

ms)

2 3 4 5 6 7 8 91

Number of zones






0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2 31Cluster number

4

5

6

7

Hea

ting

rate

(wat

t)

104

(a)

21Cluster number

3

4

5

6

7

Hea

ting

rate

(wat

t)

104

(b)


4

5

6

Hea

ting

rate

(wat

t)

104

(c)





21522

22523

235

East

tem

pera

ture

21

22

23

24

25

Wes

t tem

pera

ture




21

22

23

24

25

Wes

t tem

pera

ture

21215

22225

23235

East

tem

pera

ture






Core zone

North zone

South zone

Eastzone

Westzone





7 Conclusion



1

2

3

South zone

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

us

0 5minus5

Δxs

(a)

North zone

0 2minus2

Δxn

minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

un

1

2

(b)


1

2

minus2000

minus1000

0

1000

2000

3000

0 5 10minus5

Δxw

uw

(a)

0 5 10minus5

Δxe

minus2000

minus1000

0

1000

2000

3000

ue

1

2

(b)

minus2000

0

2000

4000

6000

8000

uc

0 1minus1

Δxc

1

2

(c)





40Time

60 80 100 120 140 1600 200

5

10Th

erm

al lo

ad (W

)

4020 60 80 100 120 140 1600Time

1

2

Estim

ated

stat

e

106



161820222426

Tem

pera

ture

(C)

710 720 730 740 750 760 770700Time


0

100

200

300

400

500

Aver

age l

earn

ing

time (

ms)

2 3 4 5 6 7 8 91

Number of zones






0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










21

22

23

24

25

Wes

t tem

pera

ture

21215

22225

23235

East

tem

pera

ture






Core zone

North zone

South zone

Eastzone

Westzone





7 Conclusion



1

2

3

South zone

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

us

0 5minus5

Δxs

(a)

North zone

0 2minus2

Δxn

minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

un

1

2

(b)


1

2

minus2000

minus1000

0

1000

2000

3000

0 5 10minus5

Δxw

uw

(a)

0 5 10minus5

Δxe

minus2000

minus1000

0

1000

2000

3000

ue

1

2

(b)

minus2000

0

2000

4000

6000

8000

uc

0 1minus1

Δxc

1

2

(c)





40Time

60 80 100 120 140 1600 200

5

10Th

erm

al lo

ad (W

)

4020 60 80 100 120 140 1600Time

1

2

Estim

ated

stat

e

106



161820222426

Tem

pera

ture

(C)

710 720 730 740 750 760 770700Time


0

100

200

300

400

500

Aver

age l

earn

ing

time (

ms)

2 3 4 5 6 7 8 91

Number of zones






0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

2

3

South zone

minus3000

minus2000

minus1000

0

1000

2000

3000

4000

us

0 5minus5

Δxs

(a)

North zone

0 2minus2

Δxn

minus1000

minus900

minus800

minus700

minus600

minus500

minus400

minus300

minus200

un

1

2

(b)


1

2

minus2000

minus1000

0

1000

2000

3000

0 5 10minus5

Δxw

uw

(a)

0 5 10minus5

Δxe

minus2000

minus1000

0

1000

2000

3000

ue

1

2

(b)

minus2000

0

2000

4000

6000

8000

uc

0 1minus1

Δxc

1

2

(c)





40Time

60 80 100 120 140 1600 200

5

10Th

erm

al lo

ad (W

)

4020 60 80 100 120 140 1600Time

1

2

Estim

ated

stat

e

106



161820222426

Tem

pera

ture

(C)

710 720 730 740 750 760 770700Time


0

100

200

300

400

500

Aver

age l

earn

ing

time (

ms)

2 3 4 5 6 7 8 91

Number of zones






0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










40Time

60 80 100 120 140 1600 200

5

10Th

erm

al lo

ad (W

)

4020 60 80 100 120 140 1600Time

1

2

Estim

ated

stat

e

106



161820222426

Tem

pera

ture

(C)

710 720 730 740 750 760 770700Time


0

100

200

300

400

500

Aver

age l

earn

ing

time (

ms)

2 3 4 5 6 7 8 91

Number of zones






0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

1

2

3

4

5

6Av

erag

e lea

rnin

g tim

e (s)

Full GPSHS model

(a)

Full GPSHS model

0010203040506070809

1

RMSE


(b)




Acknowledgments


References
































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation























RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Online Model Learning of Buildings Using …downloads.hindawi.com/journals/jcse/2017/3035892.pdfJournalofControlScienceandEngineering 3 knownasPILCO.Inthisframework,therobotcanlearn