ieee transactions on control systems technology, … · fig. 1. the experimental hvac system in...

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. XX, NO. Y, MONTH 2005 101

MIMO Robust Control for Heating, Ventilating andAir Conditioning (HVAC) Systems

Michael Anderson, Michael Buehner, Peter Young, Member, IEEE, Douglas Hittle, Charles Anderson, Jilin Tu,and David Hodgson

Abstract— Potential improvements in heating, ventilating,and air conditioning (HVAC) system performance are investi-gated through the application of multiple-input, multiple-output(MIMO) robust controllers. This approach differs dramaticallyfrom today’s prevalent method of building HVAC controllersusing multiple single-input, single-output (SISO) control loops.

A simulation model of an experimental HVAC system is used inthe design and simulation testing of controllers. While simulationcan be insightful, the only way to truly verify the performanceprovided by different HVAC controller designs is by actuallyusing them to control an HVAC system. Thus, an experimentalsystem for testing advanced HVAC controllers is built. Theconstruction and modeling of this system is the focus of a separatearticle [2].

While this system is only a portion of an overall HVAC system,it is representative of a typical hot water-to-air heating system.The performance of controllers in regulating the discharge airtemperature and flow rate is verified using the experimentalsystem.

While simple optimal and SISO robust HVAC controllers havebeen designed and their performance experimentally verified, thisproject does so using MIMO robust controllers. The design andtesting of these controllers provides valuable insight into potentialimprovements in performance, as well as constraints, associatedwith applying this control methodology to HVAC systems. Testresults demonstrated that performance gains (reductions indischarge air temperature settle time) in excess of 300% may beachieved. Furthermore, it may be possible for such performancegains to be achieved without significant impact to current HVACsystem architecture (interconnection).

Index Terms— HVAC, MIMO, robust, H∞, control, experi-mental verification, discharge air system, MATLAB, Simulink,Realtime Workshop, Windows Target.

I. INTRODUCTION

M. Anderson, while preparing this article, was a Graduate Student in theElectrical and Computer Engineering Department, Colorado State University(CSU), Fort Collins, Colorado, E-mail: [email protected].

M. Buehner, who edited this article, is a Graduate Student in the Electricaland Computer Engineering Department, Colorado State University (CSU),Fort Collins, Colorado, E-mail: [email protected].

P. Young is an Associate Professor in the Electrical and Computer Engi-neering Department, Colorado State University (CSU), Fort Collins, Colorado,E-mail: [email protected].

D. Hittle is a Professor in the Mechanical Engineering Department,Colorado State University (CSU), Fort Collins, Colorado, E-mail: [email protected].

C. Anderson is an Associate Professor in the Computer Science De-partment, Colorado State University (CSU), Fort Collins, Colorado, E-mail:[email protected].

J. Tu, is a Graduate Student in the Computer Science Department, ColoradoState University (CSU), Fort Collins, Colorado, E-mail: [email protected].

D. Hodgson, is a Graduate Student in the Mechanical Engineering De-partment, Colorado State University (CSU), Fort Collins, Colorado, E-mail:[email protected].

Fig. 1. The Experimental HVAC System

IN commercial heating, ventilating, and air conditioning(HVAC) systems, a central air supply provides air at a

controlled temperature and flow rate for use in heating (orcooling) a space. A heating coil is used in the central airsupply for heating the discharged air. The temperature of thedischarged air is controlled by regulating the rate at whichhot water flows through the heating coil. The flow rate of thedischarged air is regulated to maintain a predetermined staticair pressure within the temperature controlled space. Typically,the space within a building is divided into smaller zones,allowing the temperature within each zone to be maintainedindependently of the others. Each zone contains a reheat coilwhich is used to moderate the final temperature of the airdischarged into the zone.

A characteristic of today’s HVAC systems is the use of acentralized hot water supplies in servicing multiple centralair supplies. Such an HVAC interconnection (architecture)restricts the controller design and potentially impacts theperformance of the resulting system. A full MIMO controllerwould require control of the temperature and flow rates ofboth the air and water flowing through the heating coil.Consequently, independent air and water supplies would berequired for each coil. Such a system would represent a majorshift from current HVAC design paradigms.

A problem with current HVAC controller design is thatindividual HVAC subsystems are treated as uncoupled (or


� ��

PSfrag replacements

External Air

Interface

Mixing Box

ReturnAir

Discharge Air

Boiler

Blower

Var.Freq.Drive

Valve

Heating Coil

T

T

TT

T

T

T

E

E

E

P

P

P

⇒

Outputs

Inputs

Tae

Ade

Cde Cdr

Adr

Tar

Tai

Twi

Two

Fw Tws

Fws

Cw

h Pwh

Avp

Cvp

Tao Fa

Cbs

Fig. 2. Diagram of the Experimental System and Interface Signals

TABLE I

KEY TO MNEMONICS

Cdr Command damper returnCde Command damper externalCwh Command water heaterCvp Command valve positionCbs Command blower speed

Adr Actual damper returnAde Actual damper externalAvp Actual valve positionPwh Power (input) water heater

Tai Temp. of air inputTao Temp. of air outputTws Temp. of water supplyTwi Temp. of water into coilTwo Temp. of water out of coilTae Temp. of air externalTar Temp. of air returnFw Flow rate of water (coil)Fws Flow rate of water (system)Fa Flow rate of air

loosely coupled) systems, controlled using distributed SISOproportional-plus-integral (PI) controllers, when in fact anHVAC system is a truly multivariable system. This articleapproaches HVAC controller design from a multivariable pointof view and is one of the first to implement multivariablecontrollers on a physical HVAC system.

Between today’s HVAC systems employing distributedSISO PI based controllers and a system using a full MIMOrobust controller, a wide assortment of configurations arepossible. To gain insight into potential performance improve-ments, as well as the constraints associated with this breadthof controllers, several diverse designs are implemented andtested, both in simulation and on the experimental HVACsystem.

II. EXPERIMENTAL HVAC SYSTEM

The experimental HVAC system shown in Fig. 1 wasconstructed for verifying the performance of the advancedHVAC controller designs. This system consisted of externaland return air dampers, a variable speed blower and a heatingcoil, which are similar to commercial hot water-to-air heatingsystems. A diagram representing this system is shown in Fig. 2with the mnemonics defined in TABLE I.

The temperature of the discharged air was a function of thetemperature and flow rate of both the air and water flowingthrough the coil. The flow rate of the air was primarily afunction of the speed at which the blower is operating, butwas affected slightly by the position of the return and externaldampers. The dampers were electronically “ganged” together,allowing the return and external (outside) air mix to be varied,in regulating the temperature of the air flowing into the coil.A three-way mixing valve allows the flow rate of the waterthrough the coil to be varied.

PSfrag replacements

uy

wz P

K

∆u∆y∆

Fig. 3. General Feedback Interconnection for µ Synthesis

Controllers for the experimental system were implementedusing MATLAB c© and Simulink c© with the Real-Time Work-shop c© (RTW) and Windows Target c© toolboxes on a Win-dows98 c© based PC fitted with standard data acquisition cards.The design, construction, commissioning and modeling of thissystem is covered in previous works [1], [2].

III. SOME KEY POINTS REGARDING ROBUST CONTROL

While it is not possible to provide a complete tutorialon robust control theory in the space allowed, the followingintroduces a few of the key concepts relative to our approach.Inevitably, the models used in controller design only approx-imate the physical systems they intend to represent. Robustcontrol theory addresses the affects that discrepancies betweenthe model and the physical system (model uncertainty) mayhave on the design and performance of (linear) feedbacksystems. These uncertainties may arise from model approxima-tion, neglected or unmodelled dynamics, unknown parameters,or even sensor and/or actuator imperfections. The techniquesintroduced here are based upon the structured singular value(µ) [10], which is frequently used in dealing with model uncer-tainty. Robust control provides a unified design approach underwhich the concepts of gain margin, phase margin, tracking,disturbance rejection and noise rejection are generalized intoa single framework. Typically, the uncertainties considered inrobust control theory are bounded using norms. The H∞ andH2 norms are frequently applied in the robust controller designprocess [12], as they may be used to bound, respectively, themagnitudes or the energy content of signals. Herein, only theH∞ robust control design methodology is considered.

To facilitate the task of controller synthesis, the generalconfiguration shown in Fig. 3 is adopted. The block labelledP is the generalized plant model, which contains the nominalopen-loop plant model along with the performance and un-certainty weighting functions. The block labelled ∆ is block-diagonal complex uncertainty set (i.e. ∆ ∈ ∆, where ∆ ={∆ : ∆ = block diag(∆1, . . . ,∆n)}). The individual ∆i’s, aredefined to be any stable transfer function with ‖∆i‖∞ ≤ 1,for i = 1, . . . , n, where || · || denotes the H∞ norm. Thisimplies that the Nyquist plot for each ∆i(jω) is entirelycontained within a circle having a radius of one, centeredat the origin. The block labelled K represents the controller.The generalized plant model (P ) has three input vectors:perturbations (u∆), disturbances (w) and control (u) and threeoutput vectors: perturbations y∆, errors (z) and measurements(y).

The application of µ in robust control theory relies heavilyupon a class of linear feedback loops called linear fractional


transformations (LFTs). If P is a 2×2 block-partitioned matrixand ∆ is a matrix such that det(I−P11∆) 6= 0, then the upperLFT is defined FU (P,∆) = P22 + P21∆(I − P11∆)−1P12.This LFT defines the perturbed open-loop transfer functionfrom w to z. Similarly, if K is a matrix such that det(I −P22K) 6= 0, then the lower LFT is defined FL(P,K) =P11 + P12K(I −P22K)−1P21. This LFT defines the nominalclosed-loop transfer function from w to z. LFTs provide ageneral framework for defining a system interconnect. Formore information regarding LFTs see [17], [3].

The set of perturbed plants (Π) to be controlled is definedby the LFT in eqn (1), where σ(∆) refers to the maximumsingular value of ∆. For more information regarding thegeneral configuration see [12].

Π = FU (P,∆) : ∆ ∈ ∆, supω

σ[∆(jω)] < 1 (1)

The robust controller design objective is to find a controller(K) that, under all normalized uncertainties (||∆||∞ < 1),stabilizes the perturbed closed-loop system and satisfies eqn(2).

||FL[FU (P,∆)︸︷︷︸Π

,K]||∞ ≤ 1 (2)

The standard H∞ optimal control problem is to find allstabilizing controllers K that minimize (3).

||FL(P,K)||∞ = supω

σ(FL(P,K)(jω)) (3)

There currently is no method that directly synthesizes anH∞ robust controller; however, for complex perturbations, amethod known as DK-iteration provides a good approxima-tion. DK-iteration is a mix of H∞-synthesis and µ-analysis.An upper bound on µ, denoted as µUB , is given in eqn (4),where D is the set of scaling matrices having the propertyD∆̂ = ∆̂D for every D ∈ D, ∆̂ ∈ ∆.

µUB : µ∆̂(FL(P,K)) ≤ infD∈D

σ(DFL(P,K)D−1) (4)

Here, we can consider a robust controller that minimizes thepeak value of µUB over frequency. This is expressed moreprecisely by eqn (5).

minK

stabilizing

( minD∈D

stable,min−phase

||DFL(P,K)D−1||∞) (5)

The DK-iteration process, which attempts to approxi-mately solve eqn (5), is to alternate between minimizing||DFL(P,K)D−1||∞ with respect to either K or D, whileholding the other constant. In practice, this method oftenconverges to a µ-optimal controller. While µ-synthesis is notthe only way to synthesize a robust controller, it is the methodchosen for this application. In the work presented herein, thetask of controller design, including DK-iteration, was carriedout using the MATLAB µ-Analysis and Synthesis Tool Box.For more information regarding robust control theory see [10],[14], [12], [17], [18].

IV. OVERVIEW OF HVAC CONTROL

Early investigations regarding HVAC control focused ondistributed SISO PI controllers. It has long been recognizedthat the necessarily low gains and tedious and (sometimes)inaccurate tuning of PI-based HVAC controllers contributedto poor performance [11], [9], [6]. Also, SISO controllersare unable to take advantage of the multivariable interactionsin an HVAC system, which makes MIMO controllers moreattractive.

Optimal control has been proposed as a means for MIMObased controller design (e.g., [15], [7], [16]); however, anoptimal controller’s sensitivity to discrepancies between themodel used in designing the controller and that of the physicalplant (model uncertainty) poses a major problem in HVACsystems, where accurate models are not readily available.Furthermore, the characteristics of an HVAC plant change(deteriorate) over time. Nonetheless, several optimal HVACcontroller designs have been simulated and one has been testedon a simple system [7].

The application of robust control theory in controllingHVAC systems has only recently been considered. It hasbeen proposed as a means to account for and compensatefor both the uncertainty associated with the model used incontroller design and the nonlinear nature of HVAC systems[4]. SISO robust controllers have been implemented and theirperformance verified in simulation [13] and on an experimentalHVAC system [8]. Here, MIMO robust controllers for a simpleHVAC application are implemented and tested.

V. MIMO ROBUST HVAC CONTROLLER DESIGN

To simplify the formulation, controller designs for theexperimental HVAC system were restricted to discharge airtemperature and airflow rate control (i.e. command Tao andFa to track reference inputs). The resulting discharge air sys-tem (DAS) is similar to the central air supply in a commercialHVAC system.

The four key control variables in a DAS are the input airand water temperatures (Tai and Twi) and flow rates (Fa andFw) supplied to the heating coil. All four of these variables arecontrolled in a full MIMO DAS. However, in current HVACsystems, a common water supply typically provides hot waterto multiple heating coils (i.e. Twi is held constant). Thus, thewater supply temperature cannot be varied at each DAS.

The design of an H∞ robust controller involves selectingfrequency dependent weighting functions (transfer functions)that define the model uncertainty and the H∞ optimizationcriteria. To facilitate the design of these weighting functions,the experimental system model was arranged in a canonicalform for controller synthesis, which is shown in Fig. 4.

This canonical form isolates the exogenous inputs (w1 andw2) and exogenous outputs (z1 and z2). The exogenous inputsare the weighted external disturbances and commanded inputs(w1) and the weighted sensor noise measurements (w2), aninevitable component of measurement signals. The exogenousoutputs are the weighted error signals to be minimized (z1)and the weighted controller outputs (z2). Equation (6) shows


PSfrag replacements

Wi ∆ Wo

z1 Wperf

Nominal

HVAC ModelWdist w1

w2 Wnoise Σ KPade

Delay

Wcontrol z2

Fig. 4. System Model Configuration for Controller Synthesis

the signals (used in the design of controller KR2) associatedwith the vectors w1, w2, z1, z2, u and y.

w1 =

Tae

Tar

∆Fa

rFa

rTao

rTws

rTai

z1 =

errorFa

errorTai

errorTws

errorTao

u =

Cvp

Cbs

Cdr

Cwh

w2 =

noiseFw

noiseFa

noiseTai

noiseTws

noiseTwi

noiseTwo

noiseTao

noiseTae

noiseTar

z2 =

Cvp

Cbs

Cdr

Cwh

y =

Fw

errorFa

errorTai

errorTws

Twi

Two

errorTao

(7)

Note: vectors w1, w2, z1 and z2 are weighted (optimization criteria),vectors u and y are not weighted (physical signals).

In relation to Fig. 3, the exogenous inputs and outputs arethe (stacked) vectors

w =

[w1

w2

]and z =

[z1

z2

], (8)

respectively, and the generalized plant (P ) contains the nom-inal plant, weighting matrices, and Pade delay. The goal ofµ-synthesis is to find a controller (K) such that

∥∥∥∥w1

w2

∥∥∥∥∞

< 1 ⇒

∥∥∥∥z1

z2

∥∥∥∥∞

< 1, (9)

for all (stable) ∆ ∈ ∆, ||∆||∞ < 1, assuming one exists.Note that the inputs and outputs are normalized (boundedby one in magnitude). This is the standard arrangement forµ-synthesis. Note that this normalization is not restrictive,since the physical signals may be scaled through the use ofweighting functions. For example, if the design requires thatan output signal be less than 0.2 in magnitude (in a givenfrequency range), then the weight associated with that signalwould have a magnitude of 5 (in the given frequency range).This is the crux of the design process, which will be discussedin this section and the next.

As the structure of the controller is embedded in the nominalplant model, many controller architectures may be realizedusing the generalized model shown in Fig. 4. The signals

comprising the vectors w1, w2, z1, z2, u and y depend uponthis structure. The weighting matrices in Fig. 4 are allowed tohave a full structure; however, in the context of this paper, theywill be assumed to be diagonal matrices. This is done to so thateach signal has only one weight describing its uncertainty oroptimization criteria, which provides a more intuitive designprocess. As the design is performed in continuous-time andthe controller implemented in discrete-time, Pade delays areused in the feedback loop to include the effects of sampling.All of the MIMO robust controllers presented in this chapterconform to the system model of Fig. 4.

A. Weight Selection

The selection of weights is a major part of the controllerdesign process. For the experimental system shown in Fig. 4,there were two types of weights that needed to be specified,namely model uncertainty weights (Wi and Wo) and opti-mization criteria weights ( Wdist (disturbance), Wperf (perfor-mance), Wnoise (sensor noise), and Wcontrol (control)). Themodel uncertainty weights describe how accurate the model isat various frequencies. For the MIMO robust controller designspresented herein, additive uncertainty weights were used. Theoptimization criteria weights were used to limit the size ofthe exogenous signals. When choosing these weights, therewas a tradeoff between performance (i.e. good tracking andlarge control signals) and robustness (i.e. poor tracking andsmall control signals). For this application, performance wasrequired at lower frequencies in exchange for robustness athigher frequencies. This meant that most of the weightingfunctions took the form of either lowpass filters (LPFs) orhighpass filters (HPFs).

All of the lowpass and highpass filters used were first orderfilters. The LPFs were designed using the transfer function

WL(s) =gωc

s + ωc

, (10)

where g was the approximate passband gain and ωc was thecutoff frequency in radians per second (rad/sec). The HPFswere designed using the transfer function

WH(s) =g∞(s + ωL)

s + g∞

go

ωL

, (11)

where go was the approximate stopband gain that extends fromDC (zero rad/sec) to ωL (rad/sec), and g∞ was the approximatepassband gain for frequencies above ωH = g∞

go

ωL. Theseweights were designed by specifying the stopband cutofffrequency (ωL). Note that the size of the transition band (ωL

to ωH ) is directly proportional to the ratio g∞

go

.The selection of weighting functions was not purely an

analytic procedure, but rather an iterative process of initialestimation, testing, and revision. The initial weights are cho-sen based upon knowledge of the system or signals (e.g.,noise amplitude and frequency, actuation limits, measuredresponses, etc.), and a controller was synthesized. Based onthe performance in simulation, the weights were tweaked toprovide a good response. Once a controller was workingwell in simulation, it was implemented on the experimental


system. Based on these results, the controller weights wereagain tweaked, simulated in software, and implemented onthe experimental system. To this extend, the weights wereregarded as tuning parameters.

As mentioned earlier, only diagonal weight matrices wereused. The diagonal elements of these weight matrices werethe transfer functions (as in eqns (10) and (11)) that describedthe frequency based weight for a given signal. In the µ-synthesis software used, these weighting matrices were neverexplicitly defined. Instead, they were implicitly defined bythe system interconnect. In the next two subsections, thedesign methodology for the model uncertainty weights andthe optimization criteria weights is addressed. An exemplarMIMO robust controller design is given in Section VIII.

B. Model Uncertainty Weights

The HVAC system contained five subsystems, namely thevalue, the blower, the mixing box, the boiler, and the heatingcoil. A careful inspection of Fig. 2 will reveal these subsys-tems. Additive uncertainty was incorporated around each ofthese five subsystems.

Model uncertainty weights are usually either HPF or con-stant weights (i.e. allpass filters). HPF are generally usedwhen the model is assumed to be very accurate at lowfrequencies (i.e. a small magnitude means less uncertainty)and may not be very accurate at high frequencies (i.e. a largermagnitude means more uncertainty). This generally gives amore accurate uncertainty description, but will increase theorder of the synthesized controller [12]. When the uncertaintydescription of a particular input or output does not appear tohave a great affect on the final µ plot (or for ease of design),constant weights may be used. This method will generallyassume a larger model uncertainty than is necessary at lowerfrequencies, but will result in a synthesized controller of lowerorder.

C. Optimization Criteria Weights

The HVAC controller designs presented herein were de-signed to have performance at “low” frequencies and ro-bustness at “high” frequencies. Due to the slow dynamicsof the HVAC system, “low” frequencies were in the DC to10 mHz range and “high” frequencies were 5 Hz and higher.In between these frequency ranges was the transition regionwere the weights changed from performance specifications torobustness specifications. Choosing how the weights transitionin this region will have an affect on how well the controllerperforms. For the experimental system used, the focus ofthe transition band was in the 0.5 Hz to 2 Hz range. Theoptimization criteria weights were used to manipulate the H∞

controller synthesis under the constraint given in eqn (9). Insome cases, these weights were used to limit the size of thephysical (performance) signals (i.e. control and error signals),and in other cases, they were used to represent the magnitudeof the physical (external) signals (i.e. disturbance and noisesignals). The design of these weights will be discussed in moredetail in the next four sections.

1) Disturbance Weights: In the HVAC setup consideredherein, there were three types of disturbances, namely externaltemperature measurements (Tae and Tar), variations in theflow rate of air (∆Fa), and reference inputs. Since disturbancesoccurred at “lower” frequencies, LPFs were used to representthem. Intuitively, these LPFs provide performance (i.e. thecontroller responds to disturbances) at lower frequencies androbustness (i.e. the magnitude of the scaled disturbance is sosmall the the H∞ optimization ignores the signal) at higherfrequencies.

2) Performance Weights: The performance weights and thedisturbance weights were closely related, since they effectivelyscaled the same signals. The performance weights scaledthe error signals of the tracked reference inputs (and thedisturbance weights scaled the the reference inputs). Sinceperformance was sought at “lower” frequencies (i.e. goodperformance at DC means good tracking in steady state),LPFs were employed. Even more so, the LPFs were stableapproximations to integrators (i.e. first-order strictly propertransfer functions with a pole in the 10µHz to 100µHz range).This was done since the HVAC system was designed to trackstep changes. Over any given frequency range, the amount oftracking obtained (for a given reference input) is determinedfrom the product of a disturbance weight and the associatedperformance weight.

3) Sensor Noise Weights: The sensors were used to providea signal representation of the measured system variables,which inevitably contain noise. These sensors were moreaccurate at lower frequencies and not as accurate at highfrequencies. Therefore, these weights were chosen to be HPFs.In order for the H∞ work properly, these weights must besemi-proper (i.e. they must have a non-zero gain at highfrequencies).

4) Control Weights: Since the actuation rates and rangesof all control devices are finite, the magnitude and fre-quency response of their corresponding control signals mustbe limited. For these weights, a larger magnitude (in a givenfrequency range) will force the H∞ optimization routine tolimit the amount of control allowed (in that frequency range).Therefore, a smaller magnitude is needed at lower frequencieswhere performance is sought (i.e. the controller must be ableto exert control at lower frequencies) and a larger magnitude isneeded at higher frequencies were robustness is sought. Thesecontroller objectives were satisfied using HPFs.

VI. OVERVIEW OF HVAC CONTROLLER DESIGNS

The four controllers considered herein were intended forcontrolling HVAC systems having diverse architectures. Theyrange from a PI based controller (KPI ), for controllingsystems with a shared water supply, to a full MIMO robustcontroller (KR3) requiring separate water supplies for eachDAS. The system architecture and controller features consid-ered are summarized in TABLE II. The four controllers arespecified in the first (left) column. Each “major” row containsdata pertinent to each of the four controller designs, whilethe five right-most columns contain data relative to the keysystem variables: water supply temperature (Tws), flow rate of


TABLE II

SUMMARY OF HVAC SYSTEM AND CONTROLLER ARCHITECTURES

Controller Water Supply Air Supply Temp. AirTws (Twi) Fw Tai Fa Out (Tao)

KPI SISO (PI) ⇒ Tao SISO (PI) Open-loop SISO (PI)Tracks rTws Tracks rTai Tracks rTao

Varied to Manually3 : SISO Control Tao Controlled f(Fw)

KR1 SISO (PI) MIMO ControlledTracks rTws Tracks rFa Tracks rTao

MIMO (3 × 6) Free Variable Free Variable (at low freq.)plus one SISO f(Fw, Tai, F

∗

a )KR2 MIMO Controlled

Tracks rTws Tracks rTai Tracks rFa Tracks rTao

Free Variable (at low freq.) (at low freq.)MIMO (4 × 7) f(Fw, T ∗

ai, F∗

a )KR3 MIMO Controlled

Tracks rTai Tracks rFa Tracks rTao

Free Variable Free Variable (at low freq.)MIMO (4 × 7) f(Fw, Tws, F ∗

a )Key: * dynamic participation only (tracks reference signal in steady-state)

water through the coil (Fw), temperature of air into the coil(Tai), air flow rate (Fa) and temperature of air out (Tao). Themost important variables being regulated, air flow rate (Fa)and output air temperature (Tao), appear in the two right-mostcolumns. For each controller (major-row, including sub-rows),each column contains data pertaining to one of the five keyvariables. The upper sub-row for each controller, identifiesthe controller regulating each of the five variables. This issignificant for the first two controllers, KPI , and KR1, wherethe “overall” DAS controller consists of multiple controllers.The lower sub-row for each controller contains descriptiveinformation regarding the control variable (e.g., if the variabletracks a reference signal, and if so, only at low frequencies, orif it is a “free” control variable). For the output air temperature(right-most) column, the lower sub-row also indicates whatvariables are used in controlling the output air temperature(f(·)).

As indicated in TABLE II, the reference controller KPI

consisted of three SISO (PI) controllers tracking three refer-ence signals. They were used in regulating the water supplytemperature (Tws), input air temperature (Tai) and output airtemperature (Tao). While the air flow rate (Fa) could easilyhave been controlled in a similar manner, it was adjustedmanually (in open loop) by varying the blower speed (Cbs), asair flow rate (Fa) was virtually a direct function of the blowerspeed. The output air temperature (Tao) was controlled by aPI controller which varied the valve position (Cvp) to regulatethe flow rate of water (Fw) through the coil.

The structure of controller KR1, provided a “minimal”MIMO robust controller compatible with a system in whichthe water supply may be shared between multiple DASs. Asshown in TABLE II, the overall controller consisted of a 3×6MIMO controller and a SISO PI controller that was used toregulate the water supply temperature. The MIMO controllertracked two reference signals, namely rFa (at low frequencies)and rTao, and had two free control variables, namely thetemperature of the air flowing into the coil (Tai) and the rateat which the water flows through the coil (Fw). Assuminga constant water supply temperature (Fws), the output airtemperature (Tao) was a function of the water flow rate (Fw),the input air temperature (Tai), and the air flow rate (Fa).

Controller KR2 was a “constrained” MIMO robust con-troller. This controller was similar to KR1 in the sense that thewater flow rate through the coil (Fw) was a “free” control vari-able, and that the output air temperature (Tao) was a function

of the water flow rate (Fw), the input air temperature (Tai), andthe air flow rate (Fa). Unlike KR1, this controller regulatedthe input air temperature to track a reference input (rTai)and incorporated the control of the water supply temperatureinto the MIMO controller. Like KR1, this controller held theshared water supply for all DASs at a constant temperature.This controller would be the most applicable to current HVACsetups were one MIMO controller would be used to regulatethe airflow rate (Fa) and discharge air temperature (Tao) ineach zone while maintaining a constant shared water supplytemperature.

The last controller KR3, was a “full” MIMO robust con-troller and in a system having multiple DASs, would requirean independent (separate) water supply for each DAS. In thiscontroller, the (heater command output controlling the) watersupply temperature was a “free variable” allowing the watersupply temperature to vary. The input air temperature wascontrolled to track the reference input (rTai) during steadystate, but was allowed to vary some during transitions. Thiscontroller was the only MIMO robust controller that useddynamic model uncertainty weights (i.e. HPFs were usedinstead of constant weighting functions). Only a minimalamount of improvement was achieved by using these weights.

In the next two sections, the design of the reference PIcontroller and the KR2 MIMO robust controllers will bediscussed in detail. A similar design procedure was used fordesigning the other MIMO robust controllers, only the designof KR2 will be discussed in detail. For a detailed discussionof the KR1 and KR3 designs, see [1].

VII. REFERENCE PI CONTROLLER DESIGN

Contemporary HVAC controllers utilize multiple PI con-trollers to regulate individual SISO control loops. Since theblower fan already had built-in controller (variable frequencydrive), the airflow rate (Fa) was controlled by commandedblower speed (Cbs). Therefore, only three (SISO) PI con-trollers, namely KTws

PI , KTai

PI , and KTao

PI , were used to controlthe commanded water heater temperature (Cwh), commandedreturn (and hence external) air damper position (Cdr and henceCde), and commanded value position (Cvp), respectively. Thisis illustrated in Fig. 5.

� ��

- - -

PSfrag replacements

External Air

Mixing

box

Return Air

Discharge Air

Boiler

Blower

VariableFreq.Drive

Flow Control Valve

Heating Coil

(Filter)

T

T

T

E

E

EP

P

P

⇒

Cde

Cdr Ta

i

Twi

Tws

Cw

h

Cvp

Tao

Fa Cbs

C−1

dr KTai

PI ΣΣΣ KTws

PI KTao

PI

rTai rTws rTao

Fig. 5. HVAC Controller Based Upon Three SISO PI Controllers


TABLE III

PROPORTIONAL AND INTEGRAL GAINS FOR PI CONTROLLERS

PI Controller Proportional Gain(Kp) Integral Gain(Ki)

(KTaiPI ) 0.20 0.010

(KTaoPI ) 0.24 0.025

(KTwsPI ) 1.80 0.015

Though providing tolerable performance, the PI controlfaces several challenges. First, the subsystems that the theycontrol have gains that vary nonlinearly. Thus, the PI controllermust be tuned at the highest gain state to maintain stabilityover the entire range. This provides a slow response in lowgain states. This makes tuning the PI controller time con-suming and tedious, with optimal performance rarely attained.Nonetheless, considerable effort was put into the design ofthese controllers to get the best possible response on thephysical system. Specifically, these gains were tuned for theexperimental HVAC system using the method outlined in [5,p. 285]. The final controller gains are given in TABLE III.

Windup is another problem common in HVAC systems,which occurs when the controller states become inconsistentwith the control signals (e.g., a linear control drives an actuatorinto saturation). Following such an occurrence, the controlloop may experience severe transients. The PI controller designpresented herein employed an antiwindup technique to avoidthis problem.

VIII. MIMO ROBUST CONTROLLER KR2 WITH

REFERENCED WATER SUPPLY TEMPERATURE CONTROL

DESIGN

The first step in designing the KR2 controller was todetermine the system and controller architectures. ControllerKR2 controlled the water supply temperature (Tws), input airtemperature (Tai), air flow rate (Fa) and the discharge airtemperature (Tao). A system interconnect incorporating thiscontroller is shown in Fig. 6.

The KR2 controller synthesis task required a more detaileddescription than the canonical synthesis model shown inFig. 4. This was accomplished by creating a textual description(defined using sysic, a component of the MATLAB µ-Analysisand Synthesis Toolbox) of the model (given in [1]) thatdefined the overall HVAC system interconnect and associatedweighting functions. The crux of the design is to select thefrequency dependent weights, which are discussed in detail inthe next two subsections.

A. Model Uncertainty Weight Selection

Additive uncertainty was incorporated around each of thefive subsystems. For each subsystem, the number of additiveuncertainty weights was determined by the number of physicalinputs and outputs. The number of inputs and outputs for eachsubsystems determined the size of individual blocks in ∆ =diag(∆V ,∆B ,∆M ,∆H ,∆C). Note that each subsystem hadits own uncertainty block.

� ��

-

-

-

-

PSfrag replacements

External Air Mixing Box

Return Air

Discharge Air

Heating Coil

Boiler

Blower

VariableFreq.Drive

Flow Control Valve

(Filter)

⇒

inv(Cdr)

Cde

Cbs

Cdr

Tai

Twi

Fw Tws

Two

Cw

h

Cvp

Tao

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rFa

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errorTws

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errorTaoKR2

T

T

TT

T

E

E

E P

P

P

Fig. 6. Experimental System Using MIMO Robust Controller KR2

The model uncertainty for each subsystem was split intouncertainty on the inputs (Wi) and uncertainty on the outputs(Wo). While it is possible to lump all of the uncertaintyinto either the input or the output, splitting up uncertaintydescription will give a more balanced H∞ design and a moreintuitive weight selection process.

In some applications, the µ-synthesized controller is verysensitive to model uncertainty weights; however, for the KR2

controller design, it was only important that some uncertaintywas included for each input and output signal. Therefore,constant model uncertainty weights were used. Since theoverall uncertainty for a subsystem is (effectively) determinedby the product of the input and output weights, each of thefive subsystems had the same effective uncertainty description,namely the inputs had either a weight of 0.001 or 0.005 andthe outputs had either a weight of 0.005 or 0.001, respectively.The weights for each subsystem is summarized in TABLE IV.

As mentioned earlier, the matrices Wi and Wo were neverexplicitly defined. Instead, the weights were embedded into thesystem interconnect (using sysic) and only the block struc-ture, which was determined from the physical interconnect,was given to the µ-synthesis software. The particular blockdiagonal structure for KR2 is given in eqn (12).

∆BlkStr =

2 11 21 31 42 4

: blk str for ∆V V alue: blk str for ∆B Blower: blk str for ∆M MixingBox: blk str for ∆H Heater: blk str for ∆C Coil

(12)

This structure should be evident by inspecting TABLE IV.From this, Wi is a matrix whose diagonal elements are theinput weights in TABLE IV, and Wo is a matrix whosediagonal elements are the output weights in TABLE IV.


TABLE IV

MODEL UNCERTAINTY WEIGHTS USED IN THE DESIGN OF KR2

V alve UncertaintywVin

(Cvp) = 0.001 wVout(Fw) = 0.005

wVout(Fws) = 0.005

Blower UncertaintywBin

(Cbs) = 0.001 wBout(Fa) = 0.005

wBin(Cdr) = 0.001

Mixing Box UncertaintywMin

(Cdr) = 0.005wMin

(Tar) = 0.005 wMout(Tai) = 0.001

wMin(Tae) = 0.005

Water Heater (Boiler) UncertaintywHin

(Cwh) = 0.005wHin

(Two) = 0.005 wHout(Tws) = 0.001

wHin(Fw) = 0.005

wHin(Fws) = 0.005

Heating Coil UncertaintywCin

(Fa) = 0.005wCin

(Fw) = 0.005 wCout(Two) = 0.001

wCin(Tai) = 0.005 wCout

(Tao) = 0.001wCin

(Twi) = 0.005

B. Optimization Criteria Weight Selection

The optimization criteria weights were used to scale thenormalized inputs and outputs so that they accurately representthe relative magnitudes that are seen by the HVAC system.When a weight was too small, the H∞ controller synthesissoftware assumed that the signal would not have a largeenough magnitude to affect the overall stability. This oftenresulted in a controller that is over-reactive (e.g. if too muchcontrol authority is given to a particular control signal).Similarly, when a weight was too large, the H∞ controllersynthesis software put too much effort into minimizing thegain of a signal that the physical system would never see.This usually lead to a lack of performance. Ultimately, themagnitudes and frequency ranges of the weights had to betuned until the fastest response (while maintaining disturbancerejection) was obtained on the experimental system. While itis not possible to discuss every minor change made to theweights, the justification for the final weights is given in thenext four sections.

1) Disturbance Weights: For the KR2 controller, the distur-bances (w1) consisted of air temperatures from the surroundingenvironment (Tae and Tar), variations in the flow rate of air(∆Fa) due to wind gusts, and commanded inputs (rTao, rFa,rTai, and rTws). The air temperatures from the environmentgenerally did not change vary rapidly, so the disturbanceweights associated with them had a bandwidth of 3 mHz.Since these disturbances did not affect the output as much asthe reference inputs, a gain of 0.5 was used. The wind gustswere assumed to have an even smaller affect on the outputs, soits magnitude was 0.005 in the passband. This weight coveredwind gusts in the DC to 0.32 Hz (2 rad/sec) range, whichwas enough bandwidth to “cover” the wind gusts seen inexperimental testing. The reference input weights all had acutoff frequency of 0.16 Hz (1 rad/sec), which was reasonable

TABLE V

DISTURBANCE (& REFERENCE) WEIGHTS USED IN THE DESIGN OF KR2

wDist(Tae) = 0.01s+0.02

wDist(Tar) = 0.01s+0.02

wDist(∆Fa) = 0.01s+2

wDist(rFa) = 0.8s+1

wDist(rTao) = 5

s+1wDist(rTws) = 1

s+1

wDist(rTai) = 1

s+1

TABLE VI

PERFORMANCE WEIGHTS USED IN THE DESIGN OF KR2

wPerf (Fa) = 0.1s+0.0002

wPerf (Tao) = 0.005s+0.00025

wPerf (Tai) = 0.001

s+5×10−5wPerf (Tws) = 0.02

s+0.004

since the highest frequency allowed for controlling the valueand the dampers was about 0.1 Hz (see the section on controlweight selection). Since the reference signals were the largestdisturbance signals, they had the largest passband magnitudes.The transfer functions used to describe these weights are givenin TABLE V.

2) Performance Weights: The performance weights(wPerf ) were chosen to specify the performance objectivesthat the controller was to attain. In particular, good steadystate tracking was desired, so these weights had large valuesat DC and then rolled off above 0.1 mHz. The transferfunctions describing these weights are given in TABLE VI.The approximate error allowed in steady state for eachcommanded input was determined by taking the reciprocalof the product of the reference disturbance weight andperformance weight at DC (e.g. wDist(rTao) = 5 andwPerf (Tao) = 20 at DC; therefore, the temperature air outtracking error (eTao) was less than 0.01 in steady state).The resulting tracking errors allowed in steady stage forthe commanded inputs were eFa < 0.0025, eTws < 0.2,eTai < 0.05, and eTao < 0.01. NB: These error bounds arein the units of the physical signals (i.e. flow rate of air is inm3

sand temperature is in ◦C).

3) Sensor Noise Weights: Sensor noise is typically a highfrequency phenomenon (i.e. sensors are assumed to be mostaccurate at frequencies around DC). Therefore, these weightswere chosen to be HPFs. When possible, these weights arechosen based on the sensor product description. While themagnitude of these weights are generally small and the weightsgenerally do not have a large impact on the synthesizedcontroller, they play an important role, namely they guaranteethat the H∞ optimization is well posed. This is due to thefact that that the weights have non-zero magnitude at highfrequencies. For more information regarding this, see [12]. Thetransfer functions describing the sensor noise weights (wSN )are given in TABLE VII.

4) Control Weights: The control weights (wCont) had themost impact on the synthesized controller. The flow controlvalue and the dampers were pneumatically controlled, whichresulted in them responding the slowest to step changes.


TABLE VII

SENSOR NOISE WEIGHTS USED IN THE DESIGN OF KR2

wSN (Fw) = 10s+1×10−6

1×106s+1wSN (Fa) = 0.1s+0.001

0.1429s+1

wSN (Tai) = 0.1s+0.0010.1s+1

wSN (Tws) = 0.05s+0.010.05s+1

wSN (Twi) = 0.05s+0.010.05s+1

wSN (Two) = 0.05s+0.010.05s+1

wSN (Tao) = 0.1s+0.010.1s+1

wSN (Tae) = 0.1s+0.010.1s+1

wSN (Tar) = 0.1s+0.010.1s+1

TABLE VIII

CONTROL WEIGHTS USED IN THE DESIGN OF KR2

wCont(Cvp) = 2s+0.0150.2s+1

wCont(Cbs) = 0.1592s+0.010.01592s+1

wCont(Cwh) = 0.1592s+0.150.001592s+1

wCont(Cdr) = 1.333s+0.020.1333s+1

Therefore, their control inputs (Cvp and Cdr) were allowedto provide large control signals in the DC to 2 mHz rangeand were forced to be negligible above the 1 Hz range. Theseranges were chosen to fit within the actuator limits for thevalue and damper. The blower had a built-in controller that wasable to change the flow rate of air (Fa) with the commandedblower speed (Cbs); however, high frequency changes in onevariable disrupted the other outputs. Therefore, the com-manded blower speed (Cbs) was allowed to have the largestcontrol signals in the DC to 10 mHz range, and was forcedto be negligible above 10 Hz. For the three weights discussedso far, the control weights were in the 0.01 to 0.02 magnituderange. For this system, weight values below 0.01 resulted inoveractive controllers and weight values above 0.02 resulted insluggish controllers (i.e. controllers that had a longer settlingtime). The fourth control variable was the commanded watertemperature in the system (Cwh)). The water heater requiredsmaller control signals, so a larger weight (about ten times thesize of the blower speed weight (wCont(Cbs))) was used. Boththe blower speed weight (wCont(Cbs)) and the water heaterweight (wCont(Cwh)) reached a magnitude of one around the1 Hz range, which may be viewed as a loose upper boundto the fastest controllable dynamic in the HVAC system. Thetransfer functions describing the control weights are given inTABLE VIII.

C. KR2 Structured Uncertainty Description

The robust stability block diagonal structure (∆) associatewith the model uncertain was given in eqn (12). The structurefor the nominal performance block was determined from thenumber of exogenous inputs and outputs defined in eqn (6) (i.e.the number of optimization criteria weights). The nominal per-formance block structure ∆p had 8 inputs (i.e. the exogenousoutputs (z) from the system were perturbation inputs) and 16outputs (i.e. the exogenous inputs (w) into the system wereperturbation outputs), which yielded the nominal performanceblock ∆p ∈ C

16×8. Together, these block structures were usedto create the robust performance block ∆̂ in eqn (13).

∆̂BlkStr =

2 11 21 31 42 416 8

: blk str for ∆V V alue: blk str for ∆B Blower: blk str for ∆M MixingBox: blk str for ∆H Heater: blk str for ∆C Coil: blk str for ∆p (z & w scaling)

(13)

NB: Only the system interconnect (with the individual weightsdefined) and the block structure in eqn (13) were required forµ-synthesis software. For more information on synthesizing µ-optimal robust controllers using the µ-Analysis and Synthesistoolbox, see [3] and [12].

IX. EXPERIMENTAL VERIFICATION OF CONTROLLERS

An experimental system for testing advanced HVAC con-trollers, was built using standard HVAC components. Whileonly a portion of an overall HVAC system, this discharge airsystem (DAS) was representative of a typical hot water to airheating system. The construction and modeling of this systemis the focus of another article [2].

A. Reference PI Controller Verification

A Simulink model in conjunction with MATLAB’s Real-time Workshop Toolbox (RTW) was used to test the perfor-mance of the reference PI controller. This controller was firstsimulated in software and then compiled and downloaded ontothe hardware, where it was used to control the experimentalHVAC system in real-time. After the experimental systemwas running and had reached steady state under closed-loopcontrol, it was subjected to a series of step changes in theoutput air temperature (rTao) and air flow rate (rFa) referenceinputs. The response of the experimental system to these inputsis shown in Fig. 7.

At 308 seconds, the output air temperature reference input(rTao) was increased (stepped) from 39.5◦C to 44.5◦C. Thecontroller KTao

PI responded by increasing the valve position(Cvp) output (increasing the water flow rate (Fw)), whichcaused the temperature of the discharge air (Tao) to rise. After931 seconds elapsed, the reference level was reached. At 1, 800seconds, the air flow rate (rFa using the Cbs output) wasabruptly stepped down from 0.272

m3

s (38.2% of the maximumrate) to 0.151

m3

s (21.4%). In response to the rising temperatureof the output air (Tao), the controller KTao

PI reduced the valveposition command (Cvp) (decreasing the water flow rate (Fw)).However, the output air temperature (Tao) increased 1.65◦Cbefore the water flow (Fw) was reduced and the temperaturereturned to the reference level (rTao) approximately 1, 400seconds after initiation of the air flow rate change. At 3, 400seconds, the air flow rate (Fa) was returned to its originallevel, causing the output air temperature to drop 2◦C beforereturning to the reference level 1, 610 seconds after initiationof the air flow rate change. At 5, 152 seconds, the outputair temperature reference (rTao) returned to its original level(39.5◦C). After roughly 931 seconds elapsed, the output airtemperature (Tao) decreased to this level.

These experimental results demonstrate that an HVACsystem is not an uncoupled dynamical system but rather amultivariable system. Specifically, this is illustrated by the


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Fig. 7. Controller KPI Experimental Test Results

inverse response from increasing the airflow rate (Fa) whichdecreased the output air temperature (Tao). To the controllerKTao

PI is a disturbance, and it is well known that PI controllersare good at disturbance rejection. Therefore, well-tuned PIcontrollers are able to provide acceptable performance incommercial applications. However, the overall performance islimited by the PI controllers inability to make coordinatedcontrol actions. More performance may be achieved by usinga multivariable controller that is able to make coordinatedcontrol actions. The next section illustrates an example MIMO

controller’s ability to make coordinated control actions and theperformance achieved as a result of coordinated control.

B. MIMO Robust Controller with Reference Water SupplyTemperature Control Verification

The controller setup in Fig. 6 was converted to a Simulinkdiagram (with RTW) that was used to test the performance ofcontroller KR2. As with the PI controller, this controller wasfirst simulated in software and then compiled and downloaded


onto the hardware, where it was used to control the experi-mental HVAC system in real-time. Unlike the PI controller,this HVAC system was brought to steady-state in open-loop,and the loop was closed using a (nearly) bumpless transfertechnique. For details, see [1]. The closed-loop system (afterthe bumpless transfer had settled) was subjected to a seriesof step changes. The response of the experimental system tothese inputs is shown in Fig. 8.

From comparing the results shown in Figs. 7 and 8, it is ap-parent that controller KR2 responded faster than the referencecontroller KPI . As a detailed comparison of controller perfor-mance appears later, consider for the moment the experimentalresults for controller KR2, shown in Fig. 8. At 306 seconds,the output air temperature (rTao) reference was increased from39.5◦C to 44.5◦C. The controller responded by increasingboth the valve position (Cvp) and water heater (Cwh) outputs,as well as (after first increasing) momentarily decreasing theblower speed output (Cbs) and briefly increasing the return airdamper position output (Cdr). In response to these actions, thewater flow rate (Fw) increased, the water supply temperature(Tws) was maintained, the air flow rate (Fa) momentarilydecreased, and the input air temperature (Tai) increased briefly.The output air temperature (Tao) reached the specified temper-ature 322 seconds after the step in the reference signal. Sincethe controller KR2 controlled the input air temperature to insteady-state, track a reference input (rTai), only momentarydeviations were allowed. However, these momentary changesin input air temperature (Tai) did aid in reducing the responsetime of the system.

At 800 seconds, the air flow rate reference (rFa) wasreduced from 0.29

m3

s (40.7%) to 0.19m3

s (26.7%). The con-troller responded by rapidly reducing the blower speed output(Cbs) to 90%, then gradually to 100% of the new level. Thisquickly decreased the air flow rate to 0.2

m3

s (28.1%), thengradually decreased to the commanded level in 381 seconds.Simultaneously, while reducing the air flow rate (Fa), thecontroller momentarily reduced the return air damper (Cdr)output and reduced both the water heater (Cwh) and the valveposition (Cvp) outputs. These actions momentarily lowered theinput air temperature, maintained the water supply temperatureand reduced the water flow rate (Fw). At 1, 590 seconds, theair flow rate reference was increased from 0.19

m3

s (26.7%) to0.39

m3

s (54.7%). The control responded by rapidly increasingthe air flow rate (Cbs) output (raising the air flow rate (Fa)to 0.37

m3

s (51.9%)), momentarily increasing the return airdamper (Cdr) output, and increasing both the water heater(Cwh) and valve positions (Cvp). The full air flow rate wasreached in 307 seconds. At 2, 678 seconds, the air flow ratereference (rFa) was reduced to the original level (0.29 m3

s ).The control responded in a similar manner, as it did to thefirst step reduction in air flow rate, reaching the commandedlevel after 283 seconds.

At 3, 380 seconds, the output air temperature reference(rTao) was reduced from 44.5◦C to 39.5◦C. The controlresponded in a manner similar, but opposite from that of thestep increase in the output air temperature reference. Afterroughly 322 seconds, the reference output air temperature was

TABLE IX

COMPARISON OF CONTROLLER PERFORMANCE

Performance Controller unitsMeasurement KPI KR1 KR2 KR3

Rise Time 891 150 178 91 sec.Tao Settle Time 931 275 322 218 sec.

Overshoot 0.0 2.8 2.0 4.0 %Disturbance 96.0 96.4 98.3 96.3 %Rejection

Rise Time NA 199 42 44 sec.Fa Settle Time NA 214 46.7 51 sec.

Overshoot NA 1.1 1.5 2.0 %Disturbance NA 77 83.6 88.7 %Rejection

achieved.At 4, 059 seconds, both the output air temperature (rTao)

and air flow rate (rFa) references were increased. The systemresponded by simultaneously increasing the blower speed(Cbs), valve position (Cvp), and water heater (Cwh) outputswhile momentarily increasing the return air damper output(Cdr). These actions raised both the output air temperature(Tao) and air flow rates (Fa) with the reference levels beingreached in 304 seconds and 387 seconds, respectively. At4, 654 seconds, both the output air (rTao) and air flow rate(rFa)references were reduced with the control responding inthe reverse of the previous process.

The coordinated control actions from this MIMO controllerprovided both a faster settling time and better disturbancerejection then the uncoupled PI controllers. The next sectionprovides a detailed comparison of the PI controller and thethree MIMO controllers. For a more detailed explanation ofthe experiments run using the MIMO controllers KR1 andKR3, see [1]. Also, an overview of the experiment done withcontroller KR3 appears in [2].

X. PERFORMANCE OF DAS CONTROLLERS

Considering the architectural factors influencing the HVACcontroller designs, the performance of the controllers on theexperimental system are now contrasted. The settling times ofthe four DAS controllers in response to step changes in outputair temperature and air flow rate are summarized in TABLE Xand Fig. 9. These results delineate the increased performanceobtained by the coordinated actions of the MIMO controllers.

The reference controller was based upon three (well-tuned)SISO PI controllers that regulated the water supply tem-perature (Tws), input air temperature (Tai) and output airtemperature (Tao), while tracking their respective referenceinputs. This design, using the HVAC industries’ preferredcontroller methodology, provided a reference for comparingthe performance of the MIMO controllers.

For a step change of 5◦C in the discharge air temperature,the system with the SISO PI controllers required 931 secondsto settle at the new level. This was 4.3 times longer than thetime required by the fastest MIMO robust controller (KR3).The performance of the PI controller was also evaluated withregard to how well it tracked the reference input in thepresence of airflow rate disturbances (∆Fa). In response to a0.12 m3

s step change in air flow rate (rFa), the output air tem-perature (Tao) rose/fell approximately 2◦C (96% disturbance


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Fig. 8. Controller KR2 Experimental Test Results

rejection) before recovering. This was very good, since the bestdisturbance rejection obtained by a MIMO robust controller(KR2) was only 2.3% better. In response to a change in theair flow rate (blower speed) command (Cbs)), the air flowrate (Fa) settled to the new value in approximately 5 seconds.However, rapid change in the air flow rate (Fa) disturbedthe output air temperature (Tao). While this condition alsooccurred with the MIMO robust controllers, the referencecontroller (KPI ) was more than five-times slower in returningthe output air temperature (Tao) to the reference level.

Controller KPI ’s “slow” response time was due in large partto the “low” gains associated with the PI controllers. Recallthat these gains were chosen so that the system would remainstable even in the system’s highest gain states. Thus, the PIcontroller was stable throughout the entire operating range ofthe system. The PI based controller also provided the greatestautonomy and allowed the sharing of water supplies amongmultiple DASs.

The first MIMO robust controller (KR1), a 3×6 controller,regulated the air flow rate (Fa) and output air temperature


(Tao) to track reference signals. In this design, the water supplytemperature (Tws) was externally maintained using a separatePI control and the return air damper command was regardedas a free control variable, allowing the input air temperature(Tai) to be varied. Additionally, as the air flow rate trackedonly at low frequency, it was allowed to vary during transitions(to improve performance), but tracked the reference signal atsteady-state.

In regulating the output air temperature (Tao), controllerKR1 (in response to a 5◦C step change) was 3.4 times faster insettling to the reference level than the reference (PI) controllerand did slightly better in rejecting a (0.1 m3

s ) step change inthe air flow rate. In controlling the air flow rate, KR1 had a214 second settling time, which was 4.6 times slower than thatyielded by the fastest MIMO robust controller (KR2).

The second MIMO robust controller (KR2), a 4 × 7 con-troller, regulated the input air temperature (Tai), water supplytemperature (Tws), air flow rate (Fa) and output air tempera-ture (Tao) to track reference levels. The water supply and out-put air temperatures tightly tracked the reference inputs whilethe air flow rate and input air temperatures were allowed tovary in transition, but closely track their references at steady-state. This allowed the controller some freedom for improvingboth the system’s response and disturbance rejection.

In regulating the output air temperature (Tao), controllerKR2, had a slightly longer settling time (322s) than did con-troller KR1 (but was still 2.9 times faster than the reference PIcontroller) and yielded the best disturbance rejection (98.3%)of all the controllers. More importantly, the settling time inresponse to a step change in air flow rate (46.7 seconds) wasa 4.6 times faster than that provided by controller KR1.

The structure of controller KR2 was essentially the same ascontroller KR1, but incorporates control of the water supplytemperature into the MIMO controller. While the performanceof KR2 in response to step changes in air flow rate wasmarkedly better than KR1, this effect was due more tothe design performance (weight) specifications than to ar-chitectural/structural limitations. Specifically, the performanceweight for KR1 required twice as much performance (in theperformance weight) at DC. This resulted in controller thatallowed less overshoot and took longer settle. These resultsare apparent by comparing KR1 and KR2 in TABLE X.

The third and final MIMO robust controller design (KR3),also a 4×7 controller, provided the best overall performance ofthe controller designs. As did controller KR2, this controllerregulated the input air temperature (Tai), air flow rate (Fa)and output air temperature (Tao) to track reference levels.However, within this controller, the water heater control output(Cwh) was left as a free control variable, allowing the watersupply temperature to be varied. The input and output airtemperatures tightly tracked their reference inputs1 while theair flow rate was allowed to vary in transition, but closelytracked its reference at steady-state. This allowed the controlleryet more freedom for improving both the system’s response

1The requirement for tracking the input air temperature over “all operating”frequencies was achieved by selecting a larger disturbance weight. Specifi-cally, the disturbance weight used for wDist(Tao) in TABLE V was alsoused for the wDist(Tai) disturbance weight.

PSfrag replacements

⇐ Common watersupplies

Independent ⇒ watersupplies

4T

ao

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Fig. 9. Comparison of Controller Settling Times

and disturbance rejection. In addition, this controller designused dynamic weights that assumed less model uncertainty atlower frequencies than the other two MIMO robust designs,allowed for a more aggressive controller.

In regulating the output air temperature, KR3 provided asettling time of 218 seconds (in response to a 5◦C step inoutput air temperature), the fastest of all of the controllers and4.3 times faster than the reference PI controller. Additionally,the controller provided a 96.3% disturbance rejection (to a 0.1m3

s step change in the air flow rate). In tracking a step changein the air flow rate, KR3 yielded a settle time of 51 seconds.It also provided a 88.7% disturbance rejection (in response toa 5◦C step in output air temperature.

In Fig. 9, the settling times of the four controllers areplotted above the controller designations which appear on thehorizontal axis. The left and right vertical axes provide thetime scales for the output air temperature (black bars) andair flow rate (gray bars), respectively. The sequence of thecontrollers is related to the architecture (interconnection) of theHVAC system they were intended to control. From left to right,they progress from sharing water supplies (with other DASs)to requiring independent water supplies (for each DAS).

From the preceding, controller KR3 clearly out-performedthe other controllers. It also required independently controlled(separate) water supplies for each DAS being controlled anddiffered the most architecturally from today’s systems.

A. Factors Affecting Performance

The performance improvements provided by MIMO robustHVAC controllers (over the PI reference controller) were dueto several factors. First, the MIMO controllers designed andtested were designed to operate over a limited operating range(since they were linear controllers) and not the entire operatingrange of the system, where as the PI controllers were designedto be stable over the entire operating range (to conform withcurrent industry standards). Secondly, since the controlledvariables were interrelated (e.g., discharge air temperature(Tao), is a function of – Fw, Fa, Twi, Tai), the MIMO robustcontrollers was able to make coordinated control actions


PSfrag replacementsSystem

Architecture

Robust

ControllerDesign

Performance

OperatingRange

ModelUncertainty

Fig. 10. Interrelationship of Controller Design Objectives/Parameters

using all of the control variables. Such coordinated actionprovided performance improvements that were not possibleusing using multiple (uncoordinated) SISO controllers. Finally,robust control design methods provided a controller which wasinsensitive to (limited) variations between the actual HVACplant and the (nominal) model used during the MIMO designprocess.

In order to fully make the MIMO controllers applicablein commercial applications, a gain-scheduled robust controllerwould need to be designed to cover the entire operating range.This task would be an extension to the work presented hereand beyond the scope of this article. In this case, the gain-scheduled MIMO robust controller should be compared againsta gain-scheduled PI to fairly compare the improvement gainedfrom the MIMO controller.

As shown in Fig. 10, there are four interrelated factorsaffecting the robust performance provided by a control design.The architecture of the controller must be consistent withthat of the HVAC system. Together with the quality (andobjectives) of the design implementation, the overall systemarchitecture impacts the performance of the system. A linearcontroller, controlling the nonlinear HVAC process has a lim-ited operating range in which it may achieve its performanceobjectives. In a robust controller (on a nonlinear system), the“true” model uncertainty (that not used in covering nonlinear-ities) allowed for in the design (providing robustness) and thecontrollers range of operation are interrelated. In general, thegreater the (nonlinear) operating range, the less “true” modeluncertainty is available. As more and more of the uncertaintyis utilized in “covering” nonlinearities, there is less left tocover “true” uncertainties. The less uncertainty available forcovering “true” model uncertainty, the less robust the resultingcontroller design. Similarly, the operating range is affectedby the implementation (constraints) of the controller design.As shown in Fig. 10, all of these factors impact the robustperformance of the closed-loop HVAC system. The design ofHVAC controllers requires consideration of these interrelatedfactors while seeking a suitable compromise between them.

XI. CONCLUSIONS

The “best” response time of all the HVAC designs consid-ered in this project was provided by controller KR3, a fullMIMO robust controller. Response, however, is not the solecriteria for determining the “best” HVAC controller design.The efficiency of the overall system is perhaps the most

important factor considered. Increased efficiency tends toreduce the cost associated with operating the HVAC system.However, the economics associated with an HVAC system aredetermined by the initial cost of building the system, as well asthe operating costs (efficiency). While increased performancemay increase efficiency, the associated cost of doing so maynot always be economically justified. The various HVACcontroller designs considered by this project provide differentperformance merits. Many of these merits are closely relatedto the architecture of the HVAC system for which they weredesigned to control.

The PI controller, consisting of separate SISO (PI) con-trollers, is well suited to today’s HVAC systems, which havewater supplies supporting multiple DASs. The system responseis “slow,” but the efficiency afforded by sharing air and watersupplies is economically appealing.

Controller KR3, a “full” MIMO robust design, respondedthe fastest of all the controllers (a step in Tao settles in 218second). However, this design requires independent (separate)water supplies for each DAS controlled and it differs the mostarchitecturally from today’s systems. The cost associated withbuilding a system with separate water supplies for each DASmay be economically hard to justify.

While more constrained, controller KR2 provides much ofthe performance (response) of controller KR3 (a step in Tao

settles in 322 second). This controller design is compatiblewith HVAC systems where the DASs share a common watersupply. It offers a good compromise between performance andthe economics of a shared water supply.

The controllers KR2 and KR1 operated with an externallycontrolled water supply (boiler) and provided a significantincrease in performance over the reference system. The perfor-mance differences (in tracking reference airflow inputs (rFa))exhibited by these controllers were due primarily to differencesin the (performance) weighting functions used in their design,not differences in their architecture. These two controllerswould be the most practical for current HVAC architectures.

The experimental results demonstrate that the application ofrobust MIMO controls to an HVAC system offers a dramatic(more than 3 times) improvement in performance over currentPI based HVAC controllers. Furthermore, increased perfor-mance may be realized without the increased cost associatedwith separate water supplies (for each DAS controlled), thatare required with a “full MIMO” controller.

This article was not intended to address the various PIdesigns (i.e. only a well tuned industry standard PI controllerwas used), but rather exemplify the design and implementa-tion of MIMO robust controllers for various HVAC systemarchitectures (one of which represents a major shift in thecurrent design paradigm) and to demonstrate the improvementthat would be expected on a commercial HVAC system. Thisarticle (based on the research in [1]) is one of the first toapproach HVAC controller design from a multivariable pointof view.

ACKNOWLEDGMENTS

The authors would like to thank the National ScienceFoundation for providing funding for this project under awards


9804757, 9732986, and 0245291.

REFERENCES

[1] Michael L. Anderson, MIMO Robust Control for Heating, Ventilatingand Air Conditioning (HVAC) Systems, Masters Thesis, Colorado StateUniversity, Fort Collins, CO, Spring 2001.

[2] Michael L. Anderson, Michael R. Buehner, Peter M. Young, DouglasC. Hittle, Charles W. Anderson, Jilin Tu, and David Hodgson, AnExperimental System for Testing Advanced Heating, Ventilating and AirConditioning Controllers, Submitted to International Journal of Heating,Ventilating, Air-Conditioning and Refrigerating Research, Summer 2005.

[3] Gary J. Balas, John C. Doyle, Keith Glover, Andy Packard and RoySmith, µ-Analysis and Synthesis TOOLBOX User’s Guide, The MathWorks, Inc., Natick, Mass. 01760-1500, 1996.

[4] Y.H. Chen, K. M. Lee and W. J. Wepfer, Adaptive Robust Control SchemeApplied to a Single-Zone HVAC System, Proceedings of the AmericanControl Conference, pp. 1076-1081,May 23-25 1990.

[5] Roger W. Haines and Douglas C. Hittle, Control Systems for Heating,Ventilating, and Air Conditioning, 5th ed. Chapman & Hall, New York,NY, 1993.

[6] Douglas C. Hittle, Dynamic Response and Tuning, ASHRAE Journal,pp. 40-43, Sept. 1997.

[7] Masato Kasahara, Tadahiko Matsuba, Yukihiro Hashimoto, Itaru Mura-sawa, Akiomi Kimbara, Kazuyuki Kamimura and Shigeru Kurosu, Op-timal Preview Control for HVAC System, Proceedings of the 1998ASHRAE Winter Meeting, pp. 502-513, 1998.

[8] Masato Kasahara, Tadahiko Matsuba, Yoshiaki Kuzuu, Takanori Ya-mazaki, Yukihiro Hashimoto, Kazuyuki Kamimura and Shigeru Kurosu,Design and Tuning of Robust PID Controller for HVAC Systems,ASHRAE Transactions, vol. 105, no. 2, pp. 154-166, Jun 18-23 1999.

[9] Clay G. Nesler and William F. Stoecker, Selecting the Proportionaland Integral Constants in the Direct Digital Control of Discharge AirTemperature, ASHRAE Transactions, vol. 90, part 2B, pp. 834-845, 1984.

[10] Andy Packard and John C. Doyle, The Complex Structured SingularValue, Automatica, vol. 29, no. 1, pp. 71-109, 1993.

[11] G. Shaavit and S. G. Brandt, The Dynamic Performance of A DischargeAir Temperature System With A P-I Controller, Honeywell Inc., Com-mercial Division, Arlington Heights, Illinois, 1982.

[12] Sigurd Skogestad and Ian Postlethwaite, Multivariable Feedback Con-trol, John Wiley and Sons Ltd., West Sussex, England, 1996.

[13] C. P. Underwood, Robust Control of HVAC Plant II: Controller Design,Building Services Engineering Research and Technology, vol. 21, no. 1,pp. 63-71, 2000.

[14] Peter M. Young, Controller Design with Real Parameteric Uncertainty,International Journal of Control, vol. 65, pp 469-509, 1996.

[15] M. Zaheer-Uddin, Optimal, Sub-Optimal and Adaptive Control Methodsfor the Design of Temperature Controllers for Intelligent Buildings,Building and Environment, vol. 28 no. 3, pp. 311-322, July 1993.

[16] G. R. Zheng and M. Zaheer-Uddin, Discharge Air System: Modellingand Optimal Control, International Journal of Energy Research, vol. 23,no. 8, pp. 727-738, June 1999.

[17] Kemin Zhou, John C. Doyle and Keith Glover, Robust and OptimalControl, Prentice Hall, Upper Saddle River, NJ 07458, 1996.

[18] Kemin Zhou and John C. Doyle, Essentials of Robust Control, PrenticeHall, Upper Saddle River, NJ 07458, 1998.

PLACEPHOTOHERE

Michael Lee Anderson recieved his B.S. and M.S.degrees in Electrical Engineering from ColoradoState University in 1998 and 2001, respectively. Hisgraduate studies focused on control systems, withhis thesis topic being MIMO Robust Control forHeating, Ventilating and Air Conditioning (HVAC)Systems. From 1993 to 1997 he was the Engineeringand Projects Manager for ABB Flexible Automation,Welding Systems Division. From 1978 through 1992he worked for ESAB Automation Inc. as an Electri-cal Engineer and Engineering Manager.

PLACEPHOTOHERE

Michael Ryan Buehner received his B.S. and M.S.degrees in Electrical Engineering from ColoradoState University in 2002 and 2004, respectively.His masters research was in DSP programmingand communication systems. Currently, Michael ispursuing a Ph.D. at Colorado State University incontrol theory.

PLACEPHOTOHERE

Peter Michael Young was born in Wakefield, Eng-land, on September 21, 1962. He received the B.A.degree from Oxford University in 1985, the M.S.degree from the University of Florida in 1988, andthe Ph.D. from California Institute of Technology in1993. From 1985 to 1986 he worked as an ExecutiveEngineer at British Telecom Research Laboratories,and from 1993 to 1995 he was a PostdoctoralAssociate at Massachusetts Institute of Technology.Currently Dr. Young is an Assistant Professor atColorado State University. His research interests

include the the development of advanced analysis and design techniquesfor large-scale uncertain MIMO systems, subject to both multiple uncertainparameters, and multiple dynamic uncertainties. He is also interested in thedevelopment of synthesis tools for robust controller design, and a numberof specific application areas, which include structural vibration suppression,control of HVAC systems, power systems, hard disk drives, and hapticinterfaces.

PLACEPHOTOHERE

Dr. Doug Hittle received his PhD degree from theUniversity of Illinois in 1981. From 1969 to 1973he was a designer of heating and air-conditioningsystems at an Air Force base. He then joined theUS Army Construction Engineering Research Labo-ratory. His primary areas of interest were buildingenergy analysis, and heating and air-conditioningcontrol. He was project team leader for the devel-opment of the Building Loads Analysis and SystemThermodynamics (BLAST) program and carried outa research program on control systems. In 1986 Dr.

Hittle joined the faculty at Purdue University and carried out research at theHerrick Laboratory. In 1989 he became Director, Solar Energy ApplicationsLaboratory, at Colorado State University.

Dr. Hittle is a Fellow in ASHRAE and has received its DistinguishedService Award. In ASHRAE, he is a member of the Load Calculation technicalcommittee, past chair of the Energy Calculations technical committee, andcurrently a member of the Handbook committee. He is co-author of a bookentitled ”Control Systems for Heating, Ventilating and Air-conditioning.”

PLACEPHOTOHERE

Charles Anderson received the B.S. degree incomputer science in 1978 from the University ofNebraska, and the M.S. and Ph.D. degrees in com-puter science in 1982 and 1986, respectively, fromthe University of Massachusetts, Amherst. From1986 through 1990, he was a Senior Member ofTechnical Staff at GTE Labs in Waltham, MA. In1991, he joined the faculty of the Department ofComputer Science at Colorado State University inFort Collins, CO, where he is currently an associateprofessor. His research interests are in reinforcement

learning and neural networks for signal processing and control. Currentprojects include robust reinforcement learning, EEG pattern recognition, semi-automated segmentation of medical images, and atmosphere model inversion.Additional information can be found at http://www.cs.colostate.edu/ anderson.


PLACEPHOTOHERE

Jilin Tu received his B.S and M.S. degrees in En-gineering from Huazhong University of Science andTechnology, P.R. China in 1994 and 1997, respec-tively. He currently is investigating reinforcementlearning in pursuit of an M.S. degree in ComputerScience, at Colorado State University. From 1997to 1999 Jilin worked as a research member, atthe Institute for Pattern Recognition and ArtificialIntelligence, Huazhong University of Science andTechnology, P.R.China.

PLACEPHOTOHERE

David Hodgson received his B.S. in MechanicalEngineering from Rensselaer Polytechnic Institutein 1994. Following his under graduate studies, heworked as a Manufacturing Engineer for LutronElectronics in Coopersburg, PA and St. Kitts. Davidis currently pursuing an M.S. degree in the solar ther-mal field and plans to pursue a Ph.D in MechanicalEngineering, studying control theory.

ieee transactions on control systems technology, … · fig. 1. the experimental hvac system in...

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