Singular Perturbation Method For Smart Building ...koushik/mypapers/AJC2017.pdf · Santosh K. Gupta, Koushik Kar, Sandipan Mishra, John T. Wen ABSTRACT In this work we propose a framework

Asian Journal of Control, Vol. 00, No. 0, pp. 1–15, Month 0000Published online in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/asjc.0000

Singular Perturbation Method For Smart Building Temperature Control

Using Occupant Feedback

Santosh K. Gupta, Koushik Kar, Sandipan Mishra, John T. Wen

ABSTRACT

In this work we propose a framework for incorporating occupantfeedback towards temperature control of multi-occupant spaces, and analyzeit using singular perturbation theory. Such a system would typically haveto accommodate occupants with different temperature preferences, andincorporate that with thermal correlation among multiple zones to obtainoptimal control for minimization of occupant discomfort and energy cost. Incurrent practice, an acceptable temperature set point for the occupancy level ofthe zone is estimated, and the control law is designed to maintain temperatureat the corresponding set point irrespective of the changes in occupancy andthe preferences of multiple occupants. Proposed algorithm incorporates activeoccupant feedback to minimize aggregate user discomfort and total energycost. Occupant binary feedback in the form of hot/cold or thermal comfortpreference input is used by the control algorithm. The control algorithm alsotakes the energy cost into account, trading it off optimally with the aggregateoccupant discomfort. For convergence to the optimal, sufficient separationbetween the occupant feedback frequency and the temperature dynamics ofsystem is necessary; in absence of which, the occupant feedback provideddo not correctly reflect the effect of current control input value on occupantdiscomfort. Under sufficient time scale separation, using singular perturbationtheory, we establish the stability condition of the system and show convergenceof the proposed solution to the desired temperature that minimizes energycost plus occupant discomfort. The occupants are only assumed to be rationalin that they choose their own comfort range to minimize individual thermaldiscomfort. Optimization for a multi-zone building also takes into account thethermal correlation among different zones. Simulation study with parametersbased on our test facility, and experimental study conducted in the samebuilding demonstrates performance of the algorithm under different occupancyand ambient conditions.

Key Words: Smart building temperature control, singular perturbationmethod, occupant comfort feedback, human-centered controlalgorithm.

Santosh K. Gupta, Koushik Kar and John T. Wen arewith the Department of Electrical, Computer & SystemsEngineering, Rensselaer Polytechnic Institute (RPI), Troy, NY,12180 USA e-mail: ([email protected]). Sandipan Mishra iswith the Department of Mechanical, Aerospace & NuclearEngineering, RPI.

I. INTRODUCTION

As a multitude of electronic gadgets make theirway into our day-to-day lives, technological innovationsare redefining the notion of human comfort, requiringit to be more personalized and adaptive. At thesame time, changes in our general living style and

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2 Asian Journal of Control, Vol. 00, No. 0, pp. 1–15, Month 0000

expectations are accompanied with a surge in ourindividual energy demands. This calls for the design ofsmart systems that can achieve desired human comfortlevels without putting additional pressure on energyresources. Energy usage in buildings, both residentialand commercial, account for one of the major source ofenergy consumption both within the US and worldwide.Data suggests that nearly 40% of the total energyconsumption in US, and 20% of the total energyconsumption worldwide, is attributed to the residentialand commercial building usage [1]. More than 75%of current electricity consumption is also due to theusage in buildings. Hence, global energy efficiencycannot be obtained without devising ways for energyefficient operation of buildings. Heating, ventilation,and air conditioning (HVAC) system is one of the majorenergy consumers in buildings. Numerous design andsolution approaches have been proposed for the controland efficiency of HVAC systems. The approaches takenso far can be broadly classified into those focusing onoptimal energy usage through variable electricity rates[2], [3], [4], [5], active and passive thermal energystorage [4], [5], and more recently model predictivecontrol (MPC) approach exploiting information throughweather forecast [6], [7], [8], [9], [10], [11]. Thebuilding and its occupants combined are a single entityfor which the HVAC control needs to be optimized.More recent work, based on this philosophy, havefocused on using occupant feedback at binary/multiplelevels towards determining the direction of temperatureadjustment and perform thermal management usingaverage user vote [12], [13], [14].

A smart building energy control system that seeksand takes into account the comfort level feedback ofits occupants to determine the HVAC control input islikely to be more efficient. This can however be achallenging task in large commercial buildings such asschools, libraries, offices etc. which have a very diversecollection of occupants. Achieving energy efficiencyin a building taking into account the comfort levelsof its occupants needs a new and unique approachtowards problem solution. Our work is loosely related toThermovote [12] and a more recent work [13] that seekuser feedback at binary/multiple levels, and determinesthe direction of temperature adjustment based on theaverage user vote. Some prior work [15] have triedto link the cost of building environment control tothe occupant efficiency, health and satisfaction. Otherrecent work [16], [17] have evaluated thermal complaintbehavior using one-class classifier. In a recent work[18], Jazizadeh et al. implemented a complementarycontrol strategy for the HVAC control based on a

fuzzy predictive model to learn occupant comfortprofiles. Zhao et al. used a simulation study, against abaseline rule-based algorithm, to tie occupant subjectivethermal comfort feedback with MPC control algorithmfor the active HVAC system [19]. Vellei et al. [20]conducted a six week study to establish the feasibilityof enhanced occupant and energy savings based oncomfort feedback. Chen et al. [21] developed an MPCalgorithm based on the Actual Mean Vote (AMV)approach developed in [22], in order to achieve abalance between thermal comfort and energy savings.

In this work, unlike most of the existing studies, wepose the effective temperature control of the buildingformally as an optimization problem that takes intoaccount both the temperature dynamics as a functionof the energy input and the discomfort function ofthe building occupants [23]. We prove that our controlsolution converges to the optimal trade off pointbetween energy cost and occupant discomfort. Nosuch guarantees are known to hold for the existingapproaches proposed in the related studies. To furthercontrast it with our recent work [24], [25], the approachin this work is much simpler for the occupants asthey just have to share their upper and lower limittemperatures (or provide binary input) without havingto worry about pricing or penalty feedback fromthe building system. The collaborative (collaborationbetween occupants, declaring thermal comfort range inself-interest) and coordination (the thermostat controlis set in a coordinated manner, taking into accountthe multi-zone thermal correlations) aspect of ourproposed framework also distinguishes it more othersmart thermostat solutions such as those by Google-Nest [26] and [27]. In one of our related experimentalwork [28] we demonstrated energy savings on top of alearning thermostat.

In this paper we present a user feedback drivencontrol mechanism to optimally trade off the overallenergy cost with the aggregate occupant discomfort.Major contributions of our work can be summarizedas follows. (i) A novel algorithm is developed andanalyzed using gradient optimization and singularperturbation theory. Simple feedback of the formof ‘heat up’ or ‘cool down’ is generated fromthe occupant’s device (such as desktop or smartphone/tablet) based on the occupant’s preferences,which is incorporated into the control algorithm. Theoccupants can share their comfort range limit points proactively and also provide instantaneous binary feedbackas ’hot or cold’ if needed. (ii) We consider a multi-zone building, and optimize its thermal environmenttaking into account the thermal correlations between

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S. K. Gupta et al.: Singular Perturbation Method For Bldg. Temp. Cont. 3

its different zones. (iii) For convergence of our controlalgorithm to the optimal, sufficient separation betweenthe occupant feedback frequency and the dynamics ofthe system is necessary. We use singular perturbationtheory to analyze the system, with temperatureevolution on a faster time scale and occupant feedbackon a relatively slower time scale. With such timescale separation, we establish the stability conditionunder which the proposed control algorithm achievesconvergence to a desired temperature that minimizesthe sum of total energy cost and the aggregateoccupant discomfort. (iv) We run simulations (usingparameters of our smart building testbed) as well asconduct experimental study in the testbed to establishviability and evaluate the performance of our proposedalgorithm. The experimental study is conducted in ourWatervliet test facility but it uses occupant comfortpreference range based on our related experimentalwork [28], as the experimental facility is not equippedto host occupants in real-time.

A high level work flow of the proposed algorithmis presented in Figure 1. The flow starts with the

Fig. 1. High-level process flow diagram of the proposed algorithm.

occupant comfort perception based on which occupantscan provide their upper and lower limit temperaturesusing a smart phone application. This enables theoccupants to declare their comfort pro actively. Basedon the limits the system generates hot/cold feedback.At any point if an occupant wants to change theirprevious limits they can just provide binary inputaround the limit boundaries and the system incorporatesthat. Feedback corresponding to each of the occupant,based on their zone is consolidated and the control inputis adequately determined. This is then used to update theHVAC system of the building accordingly. Note that theproposed algorithm is intended for any multi-occupantbuilding, whether residential or commercial. The keycontribution of our work around collaboration among

multiple occupants and coordination among multiplethermal zones makes it applicable to all multi-occupantspaces.

The rest of the paper is structured as follows. InSection II we develop the problem and propose thesingular perturbation based solution. In Section III weestablish the condition for stability of our proposedsolution. In Section IV we present results correspondingto the simulation and experimental study based on ourtest bed, and finally conclude in Section V.

II. SINGULAR PERTURBATION APPROACH

In this section, we first describe our optimizationobjective (Section 2.1) and occupant discomfortmodeling (Section 2.2). We then present the widelypopular RC building model used to express our controllaw (Section 2.3), and finally introduce our singularperturbation based solution in Section 2.4.

2.1. Objective Function

Our overall minimization objective (overall cost)is the sum of two terms: (i) energy cost (i.e., costof heating/cooling), and (ii) aggregate discomfort costof the occupants. For the development of objectivefunction consider a building with m zones. The energycost E(u) is assumed to be a convex function of thecontrol input vector u. For the sake of definiteness weexpress E(u) in the following quadratic form:

E(u) =1

2uT�u, (1)

where � is a positive definite matrix, referred to asenergy cost parameter in our analysis, and u 2 Rm isthe vector of heating (cooling) inputs into differentzones of the building. Note that our framework andanalysis also extends to other convex energy costs (heatcost could have been taken as the 1-norm or 2-normof the heat input as well). Next, let Sj denote the setof all occupants in zone j, and ⇢ =

Pmj=1 |Sj | be the

total number of occupants in the building. Also let Gs

denote the (convex) discomfort function of occupants in zone j. Then the aggregate occupant discomfortcost is expressed as

Pmj=1

Ps2Sj

Gs(yj), where yjdenotes the jth element of temperature vector y, or thetemperature of zone j. Our minimization objective canthen be expressed as:

U(u, y) =1

2uT�u+ �

mX

j=1

X

s2Sj

Gs(yj). (2)



In (2), � is a scalar constant that defines the relativeweight provided to the aggregate occupant discomfort,as compared to the energy cost and is referred to asuser weight parameter in our analysis. The matrix � andscalar � together determine the weight of the energycost relative to the aggregate discomfort cost of theoccupants. It can be based on the actual energy cost,or can alternately be explicitly defined by the buildingoperator (home owner) based on the building type torelatively weigh the aggregate occupant comfort andoverall energy cost.

2.2. Occupant Discomfort ModelingOccupant thermal comfort modeling has been

extensively researched and can be summarized intothe following three major approaches: (1) the chamberstudy model, based on mapping thermal comfort fromenvironmental and personal factors to a 7-level comfortvalue scale, viz. the Predicted Mean Vote - PredictedPercent Dissatisfied (PMV-PPD) [29], [30]; (2) humanbody physiology based models such as Gagge’s core toskin model [31], Stolwijk’s comfort model for multi-human segments [32], and Zhang et al.’s sensation onhuman body segments [33]; and (3) adaptive comfortmodels developed in field study, viz. Humphreys [34]and [35]. Recent work based on thermal complaintbehavior using one-class classifier [36], [37] havealso been presented. However, existing work mainlyfocus on average thermal comfort models insteadof individual comfort modeling. Such group comfortmodels only capture average behavior and are notparticularly useful in maximizing aggregate comfortfor multi-occupant spaces, with individual thermalpreferences differing from each other.

In the current work, therefore, we take intoconsideration discomfort functions of the occupantsindividually - modeled as convex quadratic functions oftemperature variation based on the PMV-PPD model.Our model captures the difference across occupantsin their comfortable temperature range. For simulationand experimental study we adopt occupant discomfortfunction of the form:

Gs(yj) =

8><

>:

(yj � yUs )2 if yj > yUs ,

0 if yLs yj yUs ,

(yj � yLs )2 if yj < yLs ,

(3)

where yUs and yLs are the upper and lower limittemperatures respectively of the user s located in zonej.

Note that these values are not explicitly indicatedby the user, but conveyed implicitly to the system

through user feedbacks. In our simulation we assumethat a user is required to submit feedback in a simplebinary form, indicating whether he/she is feeling hotor cold in the current setting. The lack of feedbackfrom the user is assumed to be an indication that theuser is currently comfortable. Based on the receiveduser feedback the system constantly estimates thecomfortable range (discomfort function) for each userand controls the temperature accordingly. We alsorecognize the fact that user feedback around theboundaries of its comfortable temperature range can besomewhat imprecise. This is taken into account in oursimulation by associating a probability distribution withthe user feedback in a � range around the comfort rangeboundaries. More precisely, if yUs is the upper limit ofthe comfort range of user s in zone j, the feedbackof the user changes from 0 (no feedback, or the userfeels comfortable) to 1 (user feels hot) linearly as yjvaries in the range [yUs �, y

Us + �]. A similar probability

distribution is also associated for user feedback in therange [yLs �, y

Ls + �], where yLs is the lower limit of

the users comfort range. For this simulation study wehave used the value of � as 1. Note that this practicalconsideration would also account for slight changes inthe upper and lower limit of comfortable temperaturewith the users mood and attire. This is depicted inFigure 2.

Fig. 2. Algorithm for estimating thermal comfort limits from occupantbinary feedback.

2.3. Building RC Model

Several building modeling strategies have beenproposed in past, which include the finite elementmethod based model [38], lumped mass and energytransfer model [39], graph theoretic model based onelectrical circuit analogy [40], [41], [42], [43] andmore recently thermal networks [44]. The electrical



analogy approach to modeling multiple interconnectedzones reduces the heat transfer model to an equivalentelectrical circuit network. The model can be furthermodified to include building occupancy, room andheating equipment dynamics [43], [45]. In this paperwe take this electrical circuit analogy approach,and combine it with occupant discomfort feedbackmodeling.

A building is modeled as a collection ofinterconnected zones, with temperature dynamicsevolving according to a lumped heat transfer model.In the lumped heat transfer model, a single zone ismodeled as a thermal capacitor and a wall is modeledas an RC network. This results in the standard lumped3R2C wall model [41]. The heat flow and thermalcapacitance model can be written for all the thermalcapacitors in the system, with Ti as the temperature ofthe ith capacitor. Consider the system to have n thermalcapacitors and l thermal resistors. Taking the ambienttemperature (T1) into account, we can write the overallheat transfer model of the system with m zones as [46]:

CT = �DR�1DTT +B0T1 +Bu, (4)

where T 2 Rn is the temperature vector (representingthe temperature of the thermal capacitors in the 3R2Cmodel), u 2 Rm (as mentioned previously) is the vectorof heating (cooling) inputs into different zones of thebuilding, and B 2 Rn⇥m is the corresponding inputmatrix. Also, note that (T, u) are functions of time(T (t), u(t)) and accordingly T = dT

dt . Note that positivevalues of u correspond to heating the system whilenegative values of u correspond to cooling. In the aboveequation, C 2 Rn⇥n consists of the wall capacitanceand is a diagonal positive definite matrix; R 2 Rl⇥l

consists of the thermal resistors in the system and isa diagonal positive definite matrix as well. Also, D2 Rn⇥l is the incidence matrix, mapping the systemcapacitance to the resistors, and is of full row rank, andB0 = �DR�1dT0 2Rn is a column vector with non-zeroelements denoting the thermal conductance of nodesconnected to the ambient.

In our experimental facility all the zones areactuated, which in turn implies that B is of full rowrank. Also, since matrix D is of full row rank theproduct DR�1DT is a positive definite matrix. Thevector of zone temperatures, denoted by y (which is afunction of T ) can be expressed as,

y = BTT. (5)

For simulation and experimental runs we obtainthe model parameters from our Watervliet, NY basedindoor test facility.

2.4. Singular Perturbation Solution

Using equilibrium condition (setting T = 0 in (4))we obtain:

T = h(u) = (DR�1DT )�1(B0T1 +Bu). (6)

Define, J(u) = U(u, h(u)), (7)

i.e., J(u) is obtained by plugging in T = h(u) from (6)into (2). Note that energy cost term in (2) is strictlyconvex in u; and the aggregate occupant term is convexin T , and therefore convex in u when T is set toh(u), since h(u) is affine in u. This implies that J(u)is strictly convex in u. Therefore J(u) has a uniqueoptimal solution u⇤. Define

T ⇤ = h(u⇤), (8)

which is also unique by definition.With the goal of driving the system to (u⇤, T ⇤), we

propose the control input u be updated once every �time units as

uk+1 = uk � ⌘��uk + �Y ⇤F (y)

�, (9)

where k is the iteration or the time step and ⌘ is a scalarthat can be loosely interpreted as the “feedback gain”of the system. This feedback gain (step size) parametercontrols the size of change in u in each step, and isdiscussed in more detail at a later stage. Furthermore,Y 2 Rm⇥m in the above is the Jacobian obtained using(5) and the equilibrium condition (6), expressed as

Y = (@y

@u) = BT (DR�1DT )�1B. (10)

Also, ⇤ 2 Rm⇥⇢ is the zone-occupant matrix thatindicates which occupants are present in a zone (⇤js =1 if s 2 Sj , and 0 otherwise), and F (y) 2 R⇢⇥1 is the“marginal discomfort” vector of the occupants, obtainedby taking partial derivative of the occupant discomfortfunctions with respect to y. This marginal discomfortcan possibly result from the zonal temperatures beingoutside of the occupant’s preferred comfort range. Themarginal discomfort vector is obtained by taking partialderivative of the occupant discomfort functions withrespect to the zonal temperature vector y.

In other words, the sth element of F (y), wheres 2 Sj , is obtained as

Fs(yj) =dGs(yj)

dyj, s 2 Sj . (11)



Comparing (9) with (2) provides the motivation ofour control algorithm: roughly speaking, (9) updatesu in the gradient direction of U(u, T ), while takinginto account the relationship between T and u atequilibrium, as given by (6). In other words, it attemptsto update u in the direction of �rJ(u), where J(u)is defined by (7). In this interpretation, ⌘ representsthe constant “step size” associated with the gradientdescent. Note however that using (2) - (7), rJ(u) isexpressed as:

rJ(u) = �u+ �Y ⇤F (BTh(u)). (12)

From (12) we note that update of u in the gradientdirection of J(u) requires user discomfort feedbackat y = BTh(u), the equilibrated zone temperaturescorresponding to u. In practice, however, a user s 2Sj will provide a comfort feedback at the currenttemperature it experiences, yj = [BTT ]j (different ingeneral from the equilibrated temperature [BTh(u)]j),which is what we incorporate into our control algorithmas stated in (9). This implies that our control algorithmas described in (9) does not exactly move u inthe gradient direction (�rJ(u)). The effect of thisdifference (error) can be analyzed using singular

perturbation theory [47], [48], which in our caserequires (for convergence to optimality) that theoccupant feedback be collected after long intervals(i.e. � is large), allowing the temperature T to settledown close to h(u) before the next occupant feedbackcollection.

Towards developing a singular perturbation modelof our system, we first consider a continuousapproximation to the evolution of the control input u:

u ⇡ uk+1 � uk

�= � ⌘

�

⇣�u+ �Y ⇤F (y)

⌘. (13)

Note that time step � is the interval at which userfeedback is solicited and the control input u is updated.A larger � implies a slower evolution of u. We nextexpress the system evolution in the time scale of theevolution of u (slower time scale as compared to thetime scale at which T evolves).

Define ✏ = 1� as the perturbation parameter; then

⌧ = t� = ✏t is the slower time scale. Then

d⌧

dt= ✏ =) u =

du

dt= ✏

�dud⌧

�; T =

dT

dt= ✏

�dTd⌧

�.

(14)Using the fact that y = BTT , control input equation(13) can now be expressed in terms of ⌧ as follows:

du

d⌧= �⌘

⇣�u+ �Y ⇤F (BTT )

⌘. (15)

Similarly, equation (4) modeling the temperatureevolution of the building can now be expressed as:

C✏dT

d⌧= �DR�1DTT +B0T1 +Bu. (16)

Equations (15) and (16) represent a singularly perturbedsystem. Note, � " =) ✏ # leading to steady statecondition for temperature evolution. In the next sectionwe establish the global asymptotic stability of oursystem as given by equations (15) and (16).

III. STABILITY ANALYSIS

The system evolution is governed by the set ofequations (15) and (16). In equation (16) the coefficientDR�1DT is positive definite which makes the unforcedsystem (with u = 0) exponentially stable. We usesingular perturbation analysis [48] to establish thecondition for stability of the system.

Theorem 1 There exists an ✏⇤ > 0 such that (u⇤, T ⇤)is a globally asymptotically stable equilibrium of the

system given by (15) and (16) for all ✏ < ✏⇤.

Proof: We first introduce Lyapunov functions V (u) andW (u, T ) that will be used in our stability analysis:

V (u) = J(u)� J(u⇤), and (17)

W (u, T ) = (T � h(u))TP (T � h(u)), (18)

where P is a symmetric positive definite matrix (theexact choice of matrix P will be determined at a laterstage). We now define a combined Lyapunov functionL(u, T ):

L(u, T ) = (1� ↵)V (u) + ↵W (u, T ), (19)

where ↵ satisfies 0 < ↵ < 1.We start by evaluating the conditions to establish

stability using Theorem 2.1 and Corollary 2.1 fromchapter 7 of [48]. We propose the following comparisonfunctions for the analysis:

(u) = krJ(u)k, and (20)

�(T � h(u)) = kT � h(u)k. (21)

We assume that the user discomfort function Gs(yj) hasbounded second derivative i.e. there exists a < 1:

d2Gs(yj)

d2yj , 8yj , s 2 Sj . (22)



Note that from equation (11) above:

F0

s(yj) =dFs(yj)

dyj=

d2Gs(yj)

d2yj , s 2 Sj . (23)

We can now use the Mean Value Theorem to assert:

Fs(yj)� Fs

�[BTh(u)]j

�= F

0

s(yj)�yj � [BTh(u)]j

�,

for some yj 2 (yj , [BTh(u)]j), s 2 Sj .

(24)

Applying Cauchy-Schwartz inequality on (24), for s 2Sj :

kFs(yj)� Fs

�[BTh(u)]j

�k kF

0

s(y)kkyj � [BTh(u)]jk.(25)

Define = kBT k. Then from (5), (23) and (25)we have:

kF (y)� F (BTh(u))k k(T � h(u))k. (26)

Now we evaluate the conditions in Assumption 2.3,chapter 7 of [48] on V (u) to obtain:

@V

@u

⇣dud⌧

��h(u)

⌘= �⌘(rJ(u))T (rJ(u)) �⌘ 2(u),

(27)

and@V

@u

⇣dud⌧

��T� du

d⌧

��h(u)

⌘

= �⌘�(rJ(u))TY ⇤�F (y)� F (BTh(u)

�

⌘�kY kk⇤k (u)�(T � h(u)). (28)

The above expressions are obtained by taking partialderivative of V (u) as defined in (17) and substitutingdud⌧ from (15). Next, evaluating conditions fromAssumption 2.2, chapter 7 of [48] on W (u, T ) yields(29) and (30):

@W

@T

⇣✏dT

d⌧

⌘= �2(T � h(u))TPC�1A(T � h(u))

�2�min�2(T � h(u)),

(29)

where A = DR�1DT and �min is the minimumeigenvalue of the symmetric part of the matrix(PC�1A), assumed to be positive definite. Note that thesymmetric positive definite matrix P must be chosensuch that (PC�1A) is positive definite. One choicewould be P = C since A is positive definite. Anotherchoice is P = I; it is reasonable to assume that thesymmetric part of the matrix (C�1A) has positive

eigenvalues, as we have verified to hold for the dataset used in our simulation study presented in the nextsection.

@W

@u

⇣dud⌧

��T

⌘= 2⌘(T � h(u))TPA�1BrJ(u)

+ 2⌘�(T � h(u))TPA�1BY ⇤�F (BTT )� F (BTh(u)

�

2⌘kPkkA�1kkBk (u)�(T � h(u))

+ 2⌘�kPkkA�1kkBkkY kk⇤k�2(T � h(u)).

(30)

Equation (29) is obtained using (16) and the definitionof W (u, T ) in (18). Equation (30) is obtained using(18) as well as (15), (12) and (26). Given the Lyapunovfunctions V (u) and W (u, T ) satisfy the conditions(27) - (30) and are radially unbounded by definition,Theorem 2.1 with Corollary 2.1 of [48] states thatfor every ↵, L(u, T ) as given by equation (19), is aLyapunov function for ✏ < ✏⇤. For our system ✏⇤ can beobtained in terms of the conditions derived in equations(27) - (30):

✏⇤ =�min

2⌘�kPkkA�1kkBkkY kk⇤k . (31)

Note that L(u, T ) is minimized uniquely at (T ⇤, u⇤).It follows therefore that (T ⇤, u⇤) is a globallyasymptotically stable equilibrium point for all ✏ < ✏⇤,or all � > �⇤.⇤

Note, that in our simulation study we observedthat our algorithm converges even when the occupantfeedback is provided and incorporated at fast timescales (much faster than that suggested by our time-scale bound in Theorem 1). However, it would behard to technically establish that the system movesin an acute angle with the gradient direction withoutassuming sufficient time-scale separation.

IV. SIMULATION AND EXPERIMENT

In this section we first briefly present the layout ofour test bed facility located in Watervliet, NY in section4.1. The simulation study of different scenarios usingthe model parameters from our test facility is presentedin section 4.2. Finally we present our experimentalstudy results in section 4.3 conducted on the same testfacility.

4.1. Testbed LayoutWe consider our six zone physical testbed of

an intelligent building located in Watervliet, NY for



simulation and experimental study of our proposedalgorithm and validating performance of the same.Figure 3 represents the dimensions of the facility asgenerated using the Building Resistance-CapacitanceModeling (BRCM) toolbox [49]. BRCM toolbox isused to generate the RC model of the six zone testfacility, mapping it to 31 building elements resulting ina total of 93 capacitive elements.

Fig. 3. Test bed with the building elements as generated by the BRCMToolbox.

Each zone of the testbed (except for zone 2, whichis the hallway) is actuated with thermo electric coolers.Real time temperature sensing is enabled through J-typethermocouples spread across the test facility. Sensordata is acquired through wireless communication inreal time to a central server, which also runs thecontrol loops to operate the coolers and achieve thedesired ambient condition. Further details of the test bedlayout, instrumentation, and software architecture canbe referred to in [50].

The simulated occupancy of the building isrepresented in Figure 4. Zones 1 and 6 are occupiedby two occupants each and the other zones 3, 4 and 5have one occupant each. Occupants U1 and U2 are inZone 1, U3 in Zone 3, U4 in Zone 4, U5 in Zone 5, andfinally U6 and U7 in Zone 6. All the occupants havetheir own specific temperature preference as depicted inFigure 5. Occupancy mapping with the correspondingpreference range is also presented in Table 1. Notethat the occupants U1 and U2 co-located in Zone 1

Fig. 4. Layout of the Watervliet test facility with the simulatedoccupancy pattern of each zone.

have no common range of comfort preferences, whereasthe occupants U6 and U7 co-located in Zone 6 havean overlapping region of comfort preference. Thisdistribution enables us to capture all possible scenariosin terms of conflicting and common preferences amongco-located occupants.

Fig. 5. Preferred temperature comfort range of each occupant in �

C. U6, U7 in Zone 6 have an overlapping comfort preference,whereasU1, U2 in Zone 1 have no common range.

4.2. Simulation Study

The control input u in our study refers to thepower input to the system (in watts). For simulationpurpose we consider both heating and cooling actuationin each zone with a typical power rating of 1000 W.The experimental setup has a limitation of coolinginput only, limited to 600 W in the larger zones and400 W in the smaller ones. The system dynamics andthe control algorithm are simulated using MATLABand SimuLink. Using the occupancy distribution as



Table 1. Occupant preferred temperature range with occupancymapping.

Location Occupant Lower Limit(� C)

Upper Limit(� C)

Zone 1 U1 24 24.5Zone 1 U2 26 27Zone 3 U3 20 23Zone 4 U4 23 25Zone 5 U5 22 24Zone 6 U6 18 21Zone 6 U7 19 22

per Figure 4, occupant preference as per Table 1 andthe model parameters for the Jacobian generated usingBRCM toolbox, we first simulate temperature dynamicsfor a 48 hour period with a fixed ambient condition of30�C. Figure 6 represents the temperature dynamics forthe 48 hour period simulation run.

Time of Day9am 2pm 7pm 12am 5am 10am 3pm 8pm 1am 6am 11am

Tem

perature

(◦C)

14

16

18

20

22

24

26

28

30

32

Zone 1Zone 3Zone 4Zone 5Zone 6Ambient

Fig. 6. Temperature dynamics for a 48 hour period simulationwith ambient condition higher than the occupant preferences,occupancy preference as per Table 1.

The temperature of zone 6 settles at 21�C, whichis acceptable to both the occupants U6 and U7, andsimultaneously energy optimal being closer to theambient temperature. Note that anything between 19�Cto 21�C would have been comfortable for both U6and U7 based on their comfort preferences, with 21�Cbeing optimal for the given ambient condition. Zone1 settles around 25.5�C, which tends to minimizethe aggregate discomfort of both users U1 and U2and simultaneously minimizes energy consumptionconsidering the thermal correlation among all the zones.Similarly, the temperature of other zones also settleat a point to minimize aggregate occupant discomfort

and the energy cost. In this simulation we also setthe initial condition of each zone to the ambienttemperature as that represents the extreme scenario,thus providing a good performance evaluation of theproposed algorithm. In most practical condition theinitial zonal temperatures would be much closer to thecorresponding desired temperatures.

To demonstrate the energy saving in this scenario,we compare it to the prevalent set point based methodof temperature control in buildings. With the givenoccupant preferences, set point method would considerthe mid point of each occupants comfort range asthe zonal set point. In case of multiple occupantsan average of the occupant set point can be used.Using this approach the corresponding zonal set pointsare presented in Table 2. Compared to the set pointbased approach, our algorithm achieves energy optimaltemperature for the zones and in this particular caseresults in an energy saving of 12.1%. In Table 3 wealso show the final zonal temperature when using setpoint versus the proposed optimal approach. Note thatthis for the case with ambient temperature at 30�C. Theproposed algorithm becomes more effective and energyefficient with varying ambient conditions and changesin occupancy as demonstrated in the later cases.

Table 2. Temperature set point for each zone when using the setpoint based method of building temperature control.

Location Occupant setpoints (� C)

Zonal setpoints (� C)

Zone 1 24.25, 26.5 25.4Zone 3 21.5 21.5Zone 4 24 24Zone 5 23 23Zone 6 20.5, 19.5 20

Table 3. Temperature set point for each zone when using the setpoint based method of building temperature control versus theproposed algorithm.

Location Set point approach(� C)

Optimal approach(� C)

Zone 1 25.4 25.5Zone 3 21.5 23Zone 4 24 25Zone 5 23 24Zone 6 20 21

Next we simulate another case with fixed ambienttemperature at 15�C. The corresponding temperature



dynamics are represented in Figure 7. Note that inthis case the temperature of Zone 6 settles at 19�C,compared to 21�C in Figure 6 as 19�C is more energyoptimal for the ambient condition of 15�C.


Tem

perature

(◦C)

14

16

18

20

22

24

26

28

30

32Zone 1Zone 3Zone 4Zone 5Zone 6Ambient

Fig. 7. Temperature dynamics for a 48 hour period simulationwith ambient condition lower than the occupant preferences,occupancy as per Table 1.

Fixed ambient temperature is an over simplifica-tion and does not represent true variations in ambientconditions. Next, we consider a sinusoidal variation ofthe ambient with T1 taking the form: T1 = 20�C +5�Csin(2⇡t/t0), with t0 = 24 hr. The correspondingtemperature dynamics are presented in Figure 8. Thecontrol algorithm lets the zonal temperature varywith ambient till it hits the comfort limit, at whichpoint appropriate heating/cooling input is applied. Thisapproach is more optimal than maintaining a fixedset point as it harnesses the ambient variation withoutthe need to constantly re-adjust the fixed set point.For our particular setup we observed a relative energysaving of 5.3%. Note that the relative energy savingsobtained would depend on the ambient condition.Harsher ambient condition that creates load on theHVAC system would lead to higher percentage ofrelative energy saving using our algorithm compared toset point based approach.

Another simplification that has been applied to ourstudy so far has been the un-interrupted occupancy ofthe occupants throughout the period of 48 hours. Theoccupants would be moving around and in case of nonoccupancy no occupant feedback would be generated.In practice, this occupancy information can be obtainedusing motion sensors or blue tooth low energy (BLE)beacons that can detect the presence of smart phone. Wenext consider a typical work environment occupancyschedule of the occupants entering their respective


Tem

perature

(◦C)

14

16

18

20

22

24

26

28

30

32Zone 1Zone 3Zone 4Zone 5Zone 6Ambient

Fig. 8. Temperature dynamics for a 48 hour period simulation whereambient condition follows a sinusoidal variation with period of24 hours, occupancy as per Table 1.

zones at 9 am in the morning and departing at 5 pmin the evening. Further, we also consider lunch timeand simulate non occupancy during 12 to 1 pm. In theevening after 5 pm we increase the heat cost factor �.For the lunch break heat cost factor remains unchangedbut due to non-occupancy no occupant feedback isgenerated. The results are presented in Figure 9.


Tem

perature

(◦C)

14

16

18

20

22

24

26

28

30

32

Zone 1Zone 3Zone 4Zone 5Zone 6Ambient

Fig. 9. Temperature dynamics corresponding to 48 hour simulation ofoccupancy schedule along with sinusoidally varying ambient,occupancy as per Table 1. Energy cost is increased between 5pm to 9 am with no occupant feedback during lunch time.

Due to increase in energy cost factor after 5pm, the zonal temperatures tend to follow ambientvariation resulting in immense energy savings. Thekink in Zone 1 and Zone 6 temperatures during lunchtime is attributed to thermal correlation as no occupantfeedback is generated resulting in temperature variationas per prevailing thermal conditions. Compared to a



set point based approach (with predetermined energysaving set point after 5pm) this approach can result inenergy savings of 6.1%. Note that if the set point basedapproach does not implement energy saving mode after5pm then the relative saving through our approachwould be even higher. Compared to scenario in Figure 8additional energy saving can be attributed to efficiencyduring lunch break.

4.3. Experimental StudyAlthough the RC model parameters for the

simulation study has been obtained using the testfacility information, we conduct an experimental testrun on the facility to further validate our modelparameters and the proposed algorithm. The mainpurpose of these experimental run is to establish that thealgorithm can be used in a real building to achieve thezonal temperatures that optimizes the group comfort ofthe occupants and the energy cost, based on occupantfeedback mechanism. Currently we only have coolingcapacity in the test bed for each zone, with powerlimited to 600 W in the larger zones (1 and 6) and 400W in the smaller ones (3,4, and 5) [50]. The occupancysimulation for experimental study is as per Figure 4.Owing to the cooling power limitation of the test bed wechange the occupant preferences slightly for occupantsU3, U6 and U7 so as to reduce the maximum differencebetween the ambient temperature and desired zonetemperature. The occupancy mapping with occupantcomfort preference for experimental study is listedin Table 4. The control algorithm and the real-timetemperature data collection was implemented using NILabView software.

Table 4. Occupant comfortable temperature range with occu-pancy mapping for experimental study.

Location Occupant Lower Limit(� C)

Upper Limit(� C)

Zone 1 U1 24 24.5Zone 1 U2 26 27Zone 3 U3 22.5 24.5Zone 4 U4 23 25Zone 5 U5 22 24Zone 6 U6 21 24Zone 6 U7 22 25

The temperature dynamics corresponding to a 10hour experimental study is presented in Figure 10.The ambient is being maintained at 28.5�C (notethat ambient is maintained inside the building using

standard set point based thermostat). The buildingis pre-cooled to 26�C prior to occupant’s arrival at9am. This is in line with the current practice of pre-cooling/pre-heating in most of the commercial building.At 9am once the occupants arrive the temperature ofeach zone settles as per occupant comfort preferencesin Table 4. Zone 1 with users U1 and U2 havingconflicting preferences settles at 25.5�C. Zone 3 settlesat 24.5�C, Zone 4 at 25�C, and Zones 5 and 6 at 24�C.Note that these zonal temperatures correspond to energyoptimal set points given the preference of correspondingoccupants.

Between 12pm to 1pm the occupants take alunch break. During this time no occupant feedback isgenerated resulting in the zonal temperatures increasingunder ambient effect till they hit the pre-cool limit of26�C. Note that Zone 6 being larger and having settledat the lowest relative temperature of 24�C, has muchslower dynamics and takes the longest to reach closeto 26�C during lunch hour. As the occupants get backat 1pm the zonal temperatures once again settle inaccordance to their respective preferences.

At 5pm (post eight hours of work) the energy costfactor � is doubled with the occupants still in theirrespective zones. This results in the zonal temperaturessettling at a relatively higher point slightly out of theoccupant comfort range to compensate for the rise inenergy cost. At 6pm the energy cost factor � is doubledonce again, with the occupants still in their respectivezones, resulting in further discomfort to the occupantsto compensate for the additional rise in energy cost. At7pm when all the occupants leave their respective zones,the zonal temperatures are once again maintained at thepre-cooling limit of 26�C.

The energy cost parameter � can be tuned toreal time energy pricing coming directly from a utilityservice company. Thus our model facilitates very easyadaptation to energy price signals through adaptingthe � parameter appropriately. The experimental studyin Figure 10 demonstrated a preliminary case ofthe same, as the zonal temperatures moved towardsambient after 5pm due to increase in energy cost.The overall evolution and convergence of our systemis dependent on the values of three major modelparameters, feedback gain (step size) parameter (⌘),user feedback weight parameter (�), and the energy costparameter (�). User weight parameter (�), signifies theweight given to the penalty associated with the buildingoccupant discomfort. A higher � would result in a highvalue of control input for a given user comfort feedbackvector. This in turn would cause the temperature tochange sharply as a result of occupant discomfort, but



Time of Day9am 10am 11am 12pm 1pm 2pm 3pm 4pm 5pm 6pm 7pm

Tem

perature

(◦C)

20

22

24

26

28

30

32

Zone 1

Zone 3

Zone 4

Zone 5

Zone 6

Ambient

Fig. 10. Temperature dynamics for a typical work day from 9am to 7pm for the experimental study at our test facility. The zonal temperatures adjustto occupants preferences when they walk in at 9am. During lunch break there is no occupant feedback generated. Energy cost is doubled at 5pmand then again at 6pm, and finally the occupants leave at 7pm.

may also result in large overshoots as well as higherenergy costs. Note that the final relative weight ofenergy cost and occupant discomfort is determined bythe ratio of the energy cost parameter (�) and userweight parameter (�), and could as well be incorporatedthrough appropriate scaling of just one parameter. Thefeedback gain (step size) parameter controls the sizeof change in u in each step. Small values of ⌘ wouldlead to a very slow convergence of the system to thedesired temperature. With a high value of ⌘ we can hitthe desired range of temperature much faster, but at thecost of high temperature overshoots resulting in muchhigher total energy input. Also, a sufficiently high valueof ⌘ could violate the time scale separation assumptionthat is needed for stability. In our study, ⌘ was tunedto obtain a reasonable trade-off. The parameter valuesused for the simulation and experimental study ispresented in Table 5.

The cooling power input (in Watts) and thecumulative energy input (in Joules) to the buildingduring the experimental duration is presented in Figure11. The power consumption peaks at 9am when theoccupants first get into their respective zones, and at1pm when the occupants return to their respective zonesfrom lunch break. During the lunch break the powerconsumption is minimal as is also evident from therelatively flat cumulative energy curve during that hour.The cumulative energy curve gets relatively flatter asthe energy cost factor � is increased at 5 pm and at

Table 5. Parameter values used for the simulation andexperimental study.

Parameter Symbol ValueProbability range � 1�CTime step � 10 minFeedback gain ⌘ 0.1User weight parame-ter

� 10

Energy cost parame-ter

� 1

6 pm, owing to lower energy consumption or in otherwords the building starts to operate in energy savingmode beyond the regular work hours.

V. CONCLUSION

In this work we have demonstrated that thebuilding temperature and energy usage can becontrolled successfully and efficiently through dynamicfeedback from the occupants based on their comfortlevels. Under the reasonable assumption that occupantfeedback is provided at a slower time scale as comparedto the building temperature dynamics, our analysisshows that the proposed control algorithm results in thedesired (optimal) trade-off between energy usage andoccupant discomfort. The simulation and experimental



Time of Day9am 10am 11am 12pm 1pm 2pm 3pm 4pm 5pm 6pm 7pm

Coo

lingInput(W

)

0

1000

2000

Total

Energy

Con

sumption

(J)

×106

0

5

10

Fig. 11. Power input and cumulative energy consumption during the experimental study from 9am to 7pm at our test facility. Peak power consumptionis at 9am and 1pm when the occupants get in after a break. Note the relatively flat cumulative energy curve during lunch break and on increasingenergy cost beyond 5pm.

studies presented further establish that with effectivetuning of a few parameters, the control algorithmcan be effectively combined with a demand responsemechanism that reacts to energy price signals.

While we have relied on the RC model of abuilding to develop and analyze our optimal controlsolution, it is worth noting that the control law describedin (9) is reasonably ‘model independent’, allowing forpractical implementation. The only term in (9) thatdepends on the building model is the Jacobian matrixY given by (10). In practice the matrix Y wouldbe sparse owing to low coupling between zones farapart in a building, and could be easily estimatedthrough measurements. Results from various scenariosconsidered under the simulation and experimental studyof this work seem promising. Further implementationon a full scale building with real occupants is anongoing effort.

REFERENCES1. L. Lombard, J. Ortiz, C. Pout, “A review

on buildings energy consumption information,”Energy and Buildings, vol. 40, pp. 394-398, 2008..

2. H. Sane, C. Haugstetter, S. Bortoff, Building hvaccontrol systems - role of controls and optimization,In Proc. of ACC, 2006.

3. J. Braun, Reducing energy costs and peak electricaldemand through optimal control of building thermalstorage, ASHRAE transactions, 1990.

4. G. Henze, Energy and cost minimal controlof active and passive building thermal storageinventory, Journal of Solar Energy Engineering, vol.127, 2005.

5. G. Henze, C. Felsmann, G. Knabe, Evaluation ofoptimal control for active and passive buildingthermal storage, International Journal of ThermalSciences, vol. 40, 2004.

6. Y. Ma, G. Anderson, F. Borrelli, A distributedpredictive control approach to building temperatureregulation, In Proc. of ACC, 2011.

7. A. Kelman, Y. Ma, F. Borrelli, Analysis of localoptima in predictive control for energy efficientbuildings, In Proc. of 50th IEEE CDC-ECC, 2011.

8. Y. Ma, F. Borrelli, Fast stochastic predictive controlfor building temperature regulation, In Proc. ofACC Montreal, Canada, 2012.

9. N. Gatsis, G. Giannakis, Residential demandresponse with interruptible tasks: Duality andalgorithms, In Proc. of 50th IEEE CDC-ECC, 2011.

10. S. Prvaraa, J. Sirokyb, L. Ferkla, J. Ciglera, Modelpredictive control of a building heating system: Thefirst experience, Energy and Buildings, vol. 43, pp.564-572, 2011.

11. I. Hazyuka, C. Ghiausa, D. Penhouetc, Opti-mal temperature control of intermittently heatedbuildings using Model Predictive Control: Part IBuilding modeling, Building and Environment, vol.51, pp.379-387, 2012.



12. V. L. Erickson, A. E. Cerpa, Thermovote:participatory sensing for efficient building hvacconditioning, BuildSys ’12 Proceedings of theFourth ACM Workshop on Embedded Sensing Sys-tems for Energy-Efficiency in Buildings Toronto,ON, Canada, 2012.

13. S. Purdon, B. Kusy, R. Jurdak, G. Challen, Model-free hvac control using occupant feedback, IEEE38th Conf. on LCN Workshops) Sydney, NSW,2013.

14. N. Klingensmith, J. Bomber, and S. Banerjee,Hot, cold and in between: Enabling fine-grainedenvironmental control in homes for efficiency andcomfort, in Proc. 5th Int. Conf. Future Energy Syst.,Cambridge, U.K., 2014, pp. 123132.

15. O. Seppanen, W. J. Fisk, A Model to Estimatethe Cost-Effectiveness of Improving Office Workthrough Indoor Environmental Control, ASHRAETransactions, vol. 111, 2005.

16. Q. C. Zhao, Z. J. Cheng, F. L. Wang, Y. Jiang,J. L. Ding, Experimental study of group thermalcomfort model, IEEE International Conference onAutomation Science and Engineering, pp. 1075-1078, Taipei, Taiwan, Aug. 18-22, 2014.

17. Q. C. Zhao, Y. Zhao, F. L. Wang, Y. Jiang, F. Zhang,Preliminary study of learning individual thermalcomplaint behavior using one-class classifierfor indoor environment control, Building andEnvironment, Vol. 72, pp. 201-211, Feb. 2014.

18. F. Jazizadeh, et al., User-led decentralized thermalcomfort driven HVAC operations for improvedefficiency in office buildings, Energy Buildings,Vol. 70, pp. 398-410, 2014.

19. J. Zhao, et al., Occupant-oriented mixed-mode EnergyPlus predictive controlsimulation, Energy Buildings (2015),http://dx.doi.org/10.1016/j.enbuild.2015.09.027.

20. M. Vellei, S. Natarajan, B. Biri, J. Padget,I. Walker, The Effect of Real-Time Context-Aware Feedback on Thermal Comfort and HeatingEnergy Use: A Case Study, in 31th InternationalPLEA Conference: ARCHITECTURE IN (R)EVOLUTION 2015

21. X. Chen, Q. Wang, J. Srebric, Model predictivecontrol for indoor thermal comfort and energyoptimization using occupant feedback, Energy andBuildings, Vol. 102, pp. 357-369, 2015.

22. X. Chen, Q. Wang, J. Srebric, M. O. Fadeyi,Data-driven state-space modeling of indoor thermalsensation using occupant feedback, in AmericanControl Conference (ACC), 2014, pp. 1710-1715.

23. S. K. Gupta, K. Kar, S. Mishra, J. T. Wen, Buildingtemperature control with active occupant feedback,

19th World Congress, The International Federationof Automatic Control, Cape Town, South Africa,Aug 24-29, 2014.

24. S. K. Gupta, K. Kar, S. Mishra and J. T. Wen, Dis-tributed Consensus Algorithms for CollaborativeTemperature Control in Smart Buildings, In Proc.of American Control Conference (ACC), Chicago,July 1-3, 2015.

25. S. K. Gupta, K. Kar, S. Mishra, J. T. Wen, Collab-orative Energy and Thermal Comfort ManagementThrough Distributed Consensus Algorithms, IEEETransactions on Automation Science and Engineer-ing, vol. 12, pp. 1285-1296, 2015.

26. www.nest.com.27. http://lyric.honeywell.com.28. S. K. Gupta et al., BEES: real-time occupant feed-

back and environmental learning framework forcollaborative thermal management in multi-zone,multi-occupant buildings, Energy and Buildings,vol. 125, pp. 142-152, 2016.

29. P. O. Fanger, Thermal comfort: Analysis andapplications in environmental engineering, DanishTechnical Press, 1970.

30. J. V. Hoof, Forty years of Fanger’s model of thermalcomfort: comfort for all?, Indoor Air, Vol. 18, Issue3, pp. 182-201, 2008.

31. A. P. Gagge, Y. Nishi, An effective temperaturescale based on a simple model of human physiolog-ical regulatory response, ASHRAE Transactions,Vol. 77, pp. 247-263, 1971.

32. J. A. J. Stolwijk, Mathematical model of thermoregulation, Annals of The New York Academy ofSciences, Vol. 355, pp. 98-106, 1980.

33. H. Zhang, E. Arensa, C. Huizengaa, T. Hanb,Thermal sensation and comfort models for non-uniform and transient environments, part III:Whole body sensation and comfort, Building andEnvironment, Vol. 45, Issue 2, pp. 399-410, 2010.

34. M. A. Humphreys, J. F. Nicol, I. A. Raja, Fieldstudies of thermal comfort and the progress ofthe adaptive model, Advances in Building EnergyResearch, Vol. 1, pp. 55-88, 2007.

35. R. J. Dear de, G. S. Brager, Towards an AdaptiveModel of Thermal Comfort and Preference,ASHRAE Transactions, Vol. 104, Issue 1, pp. 145-167, 1998.

36. Q. C. Zhao, Z. J. Cheng, F. L. Wang, Y. Jiang,J. L. Ding, Experimental study of group thermalcomfort model, IEEE International Conference onAutomation Science and Engineering, pp. 1075-1078, Taipei, Taiwan, Aug. 18-22, 2014.

37. Q. C. Zhao, Y. Zhao, F. L. Wang, Y. Jiang, F. Zhang,Preliminary study of learning individual thermal



complaint behavior using one-class classifierfor indoor environment control, Building andEnvironment, Vol. 72, pp. 201-211, Feb. 2014.

38. B. Mebee, Computational approaches to improvingroom heating and cooling for energy efficiencyin buildings, PhD thesis, Virginia State andPolytechnic Institute, 2011.

39. P. Riederer, D. Marchio, J. Visier, A. Husaunndee,R. Lahrech, Room thermal modeling adapted tothe test of hvac control systems, Building andEnvironment, vol. 37, 2002.

40. H. Boyer, J. Chabriat, B. Grondin-Perez, C.Tourrand, J. Brau, Thermal building simulation andcomputer generation of nodal models, Building andEnvironment, vol. 31, 1996.

41. G. Fraisse, C. Viardot, O. Lafabrie, G. Achard,Development of simplified and accurate buildingmodel based on electrical analogy, Energy andBuildings, vol. 34, pp. 1017-1031, 2002.

42. B. Xu, L. Fu, H. Di, Dynamic simulation of spaceheating systems with radiators controlled by trvs inbuildings, Energy and Buildings, vol. 40, 2008.

43. A. Athienitis, M. Chandrashekhar, H. Sullivan,Modelling and analysis of thermal networksthrough subnetworks for multizone passive solarbuildings, Applied Mathematical Modelling, vol. 9,1985.

44. C. Ghiaus, Causality issue in the heat balancemethod for calculating the design heating andcooling load, Energy, vol. 50, pp. 292-301, 2013.

45. V. Chandan, Modeling and control of hydronicbuilding hvac systems, Masters thesis, UIUC, 2010.

46. S. Mukherjee, S. Mishra, J. Wen, Buildingtemperature control: A passivity-based approach,IEEE 51st Annual Conference on Decision andControl, 2012.

47. H. Khalil, Nonlinear Systems (Chp. 11), 3rdEdition, Prentice-Hall, 2002.

48. P. Kokotovic, H. Khalil, J. OReilly, SingularPerturbation Methods In Control: Analysis andDesign, London: Academic Press, 1986.

49. D. Sturzenegger, D. Gyalistras, V. Semeraro,M. Morari and R. S. Smith, BRCM MatlabToolbox: Model Generation for Model PredictiveBuilding Control, Proceedings of American ControlConference, pp. 1063-1069, Portland, OR, June 4-6, 2014.

50. M. Minakais, et. al, Design and Instrumentationof an Intelligent Building Testbed, IEEE CASE,Gothenberg, Sweden, August 24-28, 2015.


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Singular Perturbation Method For Smart Building ...koushik/mypapers/AJC2017.pdf · Santosh K. Gupta, Koushik Kar, Sandipan Mishra, John T. Wen ABSTRACT In this work we propose a framework