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Conventional heating, ventilating, and air conditioning (HVAC) systems have traditionally used the temperature and the humidity ratio as the quantitative indices of comfort in a room. Recently, the carbon dioxide (CO_{2}) concentration has also been recognized as having an important contribution to room comfort. This paper presents the modeling of an augmented HVAC system including CO_{2} concentration, and its control strategies. Because the proposed augmented HVAC system is multiinput multioutput (MIMO) and has no relative degree problem, the dynamic extension algorithm can be employed; then, a feedback linearization technique is applied. A linearquadratic regulator (LQR) is designed to optimize control performance and to stabilize the proposed HVAC system. Simulation results are provided to validate the proposed system model, as well as its linearized control system.
heating ventilating and air conditioning (HVAC) systemmultiinput multioutput (MIMO) systemdynamic extension algorithmfeedback linearization1. Introduction
HVAC systems are automatic systems that control temperature and humidity in buildings, providing people with a comfortable environment. The use of HVAC systems represents more than 50% of the world energy consumption [1,2,3,4]. Thus, balancing occupant comfort and energy efficiency is a main goal of HVAC control strategies.
In most previous studies, HVAC systems have been modeled considering only the temperature and the humidity ratio [5,6,7,8]. A nonlinear HVAC model that includes dynamics of temperature and humidity ratio is proposed in [5], which includes the design of an observer to estimate the thermal and moisture loads. In [6], an adaptive fuzzy output feedback controller is proposed, based on an observer for the HVAC system. In [7,8], a backstepping controller and a decentralized nonlinear adaptive controller are respectively applied to the same model.
Recently, the CO_{2} concentration has been recognized as having an important contribution to room comfort [9,10]. Some researchers have proposed hybrid HVAC systems that represent the temperature and humidity ratio as continuous states and CO_{2} concentration as a discrete state [11,12]. However, because these states are strongly interrelated, it is more appropriate to integrate these continuous and discrete dynamics into a single model that includes temperature, humidity ratio, and CO_{2} concentration as states.
This paper presents a modeling and control strategy for a novel HVAC system that considers temperature, humidity ratio, and CO_{2} concentration. In the process of modeling, the dynamic extension algorithm of [13] is employed to deal with noninteracting control problem and no relative degree problem. After the dynamic extension process, a feedback linearization method can be applied to the proposed HVAC system to convert a bilinear system into a linear system. Linear controllers, pole placement and LQR can be designed for the linearized novel HVAC system to stabilize it and improve its control performance.
This paper is organized as follows: in Section 2, we present the bilinear model for the conventional HVAC system, including valve dynamics. Section 3 presents a novel HVAC system including CO_{2} concentration and its applicability in the feedback linearization method. Also, dynamic extension algorithm is applied for solving the no relative degree and interacting control problems in the MIMO system. In Section 4, we describe the design of linear controllers for the linearized HVAC system, such as pole placement and LQR controllers, to improve the system’s control performance and to verify the effectiveness of the proposed model.
2. Conventional HVAC System with Temperature and Humidity Ratio
As mentioned, conventional HVAC systems control only temperature and humidity. In this paper, we consider the singlezone system shown in Figure 1 as a representative conventional HVAC system. It consists of the following components: a heat exchanger; a chiller, which provides chilled water to the heat exchanger; a circulating air fan; the thermal space; connecting ductwork; dampers; and mixing air components [5]. The conventional HVAC system controls the temperature and humidity ratio as follows [5]:
Fresh air is introduced into the system and is mixed in a 25:75 ratio with recirculated air (position 5) at the flow mixer.
Second, air mixed at the flow mixer (position 1) enters the heat exchanger, where it is conditioned.
Third, the conditioned air is moved out of the heat exchanger; this air is ready to enter the thermal space, and is called supply air (position 2).
Fourth, the supply air enters the thermal space (position 3), where it offsets the sensible (actual heat) and latent (humidity) heat loads acting upon the system.
Finally, the air in the thermal space is drawn through a fan (position 4); 75% of this air is recirculated and the rest is exhausted from the system.
Model of the representative conventional HVAC system.
The control inputs for a conventional HVAC system are the flow rate of air, which is varied using a variablespeed fan (position 2), and the flow rate of water from the chiller to the heat exchanger. However, in our proposed HVAC system, the air recirculation rate (position 4) is added as a new control input, and some modifications are made to the basic operation rules listed above. That is to say, the 75% recirculation rate listed above in the first and last steps becomes a variable quantity, and is used as the third control input.
2.1. Mathematical Modeling of Conventional HVAC System
The conventional HVAC system is a model considering the temperature and the humidity ratio as states. The differential equations describing the dynamic behavior of the HVAC system in Figure 1 can be derived from energy conservation principles and are given by [5]:
The dynamic system given by Equation (1) can be converted into a state variable form for the purposes of control. Let
u1=F,
u2=gpm,
z1=T3,
z2=W3,
z3=T2 and define the following parameters:
α1=60Vs,
α2=60hfgCpVs,
α3=1ρaCpVs,
α4=1ρaVs,
β1=60Vhe,
β2=1ρaCpVhe, and
β3=60hwCpVhe. Then, the dynamic equations given in Equation (1) can be written in the following state variable form.
The conventional HVAC system of Equation (2) is a 2input, 2output MIMO system: its inputs are the volumetric air flow rate and the chilled water flow rate, and its outputs are the temperature and the humidity ratio of the thermal space.
2.2. Adding Valve Dynamics to the Conventional HVAC Model
The Control input signals u=[u1u2]T in the system described in Equation (2) are implemented using liquid valves. The valve dynamics can be modeled as follows in which ψ(s) is the valve inherent characteristic and
u(s) is the flow rate of the liquid which enters the valve [14,15]:
u(s)=1(1+τs)ψ(s)
By considering the characteristic of a linear valve as
ψ(s)=kv(s), the valve transfer function can be written as:
u1=k11+τ1sν1, u2=k21+τ2sν2 where k1k2τ1 and τ2 are the constant gains and the time constants, respectively; ν=[ν1ν2]T is the control signal applied to the actuator; and u=[u1u2]T is the signal that is input to the HVAC system.
An augmented state space model with the new state vector,
z=[z1z2z3u1u2]T=[z1z2z3z4z5]T can be derived as:
{z˙=f(z)+g(z)ν=[a1(z)a2(z)a3(z)a4(z)a5(z)]+[0000k1τ10000k2τ2]νy=[z1z2]T where:
Thus, the system in Equation (5) represents a conventional HVAC system to which valve dynamics have been applied to implement the control signals. This conventional system can regulate only the temperature and the humidity ratio; other factors such as CO_{2} concentration, which affects the health of occupants or workers indoors, cannot be considered.
3. Novel Modeling of HVAC System including CO<sub>2</sub> Concentration
If the recirculated air contains too much CO_{2}, it can affect the health and work efficiency of the building’s occupants. Therefore, CO_{2} concentration should be one of the quantitative indices of room comfort, along with temperature and humidity ratio. In this paper, we propose an HVAC system that continuously controls all three of these indices.
From the mass balance equation, the average CO_{2} concentration
Cs in the room can be represented as [14]:
VsC˙s=Cg+(1−μ)(Ci−Co) where
Cs=Ci−Co,
Cg is the amount of CO_{2} generated in the room;
Ci is the CO_{2} concentration in the inlet air;
Co is the CO_{2} concentration of air leaving the room, and
(1−μ),0≤μ≤1, is the air exchange rate.
3.1. Proposed HVAC System Model
The proposed HVAC system model includes CO_{2} concentration as a state. The differential Equation (6) can be integrated into the dynamic equations in (1). The valve dynamics of
u3 are added to the control input vector
u=[u1u2]T, and the control signal applied to the actuator of
ν3 also can be added to the actuator input vector
ν.
Let
u3=1−μ,
x1=z1,
x2=z2,
x3=z3,
x4=Cs, and let an augmented state vector
x=[x1x2x3x4u1u2u3]T. Then, the whole dynamics can be written in the state variable form as:
where:
The novel HVAC system of (7) is a 3input, 3output MIMO system: its inputs are the volumetric air flow rate, the chilled water flow rate, and the outdoor air flow rate, and its outputs are the temperature, humidity ratio, and CO_{2} concentration of the thermal space. This proposed HVAC system can be linearized using a feedback linearization control method, as shown in Figure 2. Thus, we can finally obtain a linearized HVAC system that can be controlled using linear controllers.
Overall block diagram for controlling the proposed HVAC system.
3.2. Conditions for InputOutput Feedback Linearization
An inputoutput feedback linearization method can be applied to the state space model given in (7) to track the desired temperature, humidity ratio, and CO_{2 }concentration only when the decoupling matrix is nonsingular. However, the decoupling matrix (refer to Appendix A2):
D(x)=[Lg1Lf1x1Lg2Lf1x1Lg3Lf1x1Lg1Lf1x2Lg2Lf1x2Lg3Lf1x2Lg1Lf1x4Lg2Lf1x4Lg3Lf1x4]=[γ1k1τ100γ2k1τ10000γ5k3τ3] is singular; here,
L(∙) represents the Lie derivative and the total relative degree is
6≠n, where
n is the system order [16]. When the invertibility condition is violated, some method is needed to carry out an inputoutput linearization; the dynamic extension algorithm used herein is such a method.
3.2.1. Dynamic Extension Algorithm
Because inputoutput linearization can be achieved only when the decoupling matrix
D(x) is nonsingular, employing the dynamic extension algorithm involves choosing some new inputs that are the derivatives of some of the original system inputs, in such a way that the decoupling matrix
D(x) becomes nonsingular, as shown in Figure 3 [16].
Concept of the dynamic extension algorithm [13].
Let state vector
x=[xv1v3]T=[x1x2x3x4u1u2u3v1v3]T. Then, the state space can be written as:
where:
The decoupling matrix as changed by the dynamic extension (refer to Appendix A3):
is nonsingular. The vector relative degree is {333} and and the total relative degree is equal to the system order
n, which means that there are no internal dynamics [16]. Therefore, we can achieve relative degree and noninteracting control.
3.3. Equivalent Linearization HVAC System
To be able to apply the feedback linearization method conveniently, we can change system (8) into the equivalent system (9) by using Equation (A3):
{x˙′=Ax′+B[F(x′)+G(x′)ζ]y=[y1y2y3]T=[x1x2x4]T where:
x′=[x1x1(1)x1(2)x2x2(1)x2(2)x4x4(1)x4(2)]T;F(x′)=[f1f2f3]T;G(x′)=[g11g12g13g21g22g23g31g32g33];A=diag[[010001000][010001000][010001000]];B=diag[[001][001][001]]; and
C=diag[[100][100][100]].
This equivalent system can apply the feedback linearization law to linearize the HVAC system. By putting proper control gain, the linearized HVAC system can be regulated to maintain the set points of temperature, humidity ratio, and CO_{2} concentration.
4. Control Strategies Using Feedback Linearization Control
The proposed HVAC system is linearized by a feedback linearization method. The linearized HVAC system shown can be controlled by linear controllers such as pole placement and LQR controllers.
Table 1 shows the numerical values of system parameters used in the simulations. The initial state and reference values are given in Table 2.
energies0703599t001_Table 1
Numerical values for system parameters.
Parameter
Value
Unit
ρ
0.074
1b/ft^{3}
V_{he}
60.75
ft^{3}
V_{s}
58464
ft^{3}
C_{p}
0.24
Btu/lb °F
W_{s}
0.0070
lb/lb
M_{o}
166.06
lb/hour
Q_{o}
289,897.52
Btu/hour
T_{o}
85
°F
W_{o}
0.018
1b/1b
C_{o}
400
ppm
τ_{1},τ_{2},τ_{3}
0.008
hour
k_{1},k_{2},k_{3}
5

energies0703599t002_Table 2
Initial state and reference values.
T_{initial}
76 °F
T_{initial}
0.021 lb/lb
T_{initial}
1300ppm
T_{initial}
71 °F
T_{initial}
0.0092 lb/lb
T_{initial}
1200ppm
4.1. Design of Pole Placement Control for Linearized HVAC System
Let the reference signal yd=[y1dy2dy3d]T,
yd(3)=[y1d(3)y2d(3)y3d(3)]T, and matrix
Yd=[y1dy1d(1)y1d(2)y2dy2d(1)y2d(2)y3dy3d(1)y3d(2)]T. Define the tracking errors as
e1=y1−y1d,
e2=y2−y2d and
e3=y3−y3d, and let the error matrix
e=x′−Yd=[e1e1(1)e1(2)e2e2(1)e2(2)e3e3(1)e3(2)]T.
The feedback linearization control law for the proposed HVAC system (9) is designed as:
ζ=G−1(−F+yd(3)−Ke) choosing
K=diag([k11k12k13],[k21k22k23],[k31k32k33]) so that the polynomial
s3+ki1s2+ki2s1+ki3=0,i=1,2,3 has all its roots strictly in the lefthalf complex plane, thereby meeting the desired performance specifications such as those for the transient response of the steadystate error. By substituting Equation (10) into Equation (9), the linearized HVAC system can finally be obtained as follows:
x˙′=Ax′+Bη where
η=yd(3)−Ke.
The gain K_{1} corresponds to the case in which the pole is located at −2, −2, and −4, whereas the gain K_{2} corresponds to the case in which the pole is located at −2, −3, and −5, and the gain K_{3} corresponds to the case in which the pole is located at −5, −6, and −7. According to the pole placement, the control performance is varied. Figure 4 show the system responses in terms of temperature, humidity ratio, and CO_{2} concentration, respectively. Table 3 shows the control performance metrics of settling time, rising time, settling max value, and settling min value for each value of gain.
energies0703599t003_Table 3
Control performance metrics of the pole placement controller.
Temperature
Humidity ratio
CO_{2} concentration
K_{1}
K_{2}
K_{3}
K_{1}
K_{2}
K_{3}
K_{1}
K_{2}
K_{3}
T_{s}
3.2695
2.8431
1.1618
3.3889
2.7312
1.1595
3.3857
2.7270
1.1581
T_{r}
1.8919
1.5184
0.6804
1.8971
1.5124
0.6795
1.8970
1.5119
0.6794
Settling min
70.9962
71.0172
70.9927
0.0092
0.0092
0.0092
1200.1
1200
1199.9
Settling max
71.4929
71.4741
71.4297
0.0104
0.0104
0.0102
1209.9
1210
1208.5
(a) Temperature response for each pole placement; (b) Humidity ratio response for each pole placement; (c) CO_{2} concentration response for each pole placement.
4.2. Design of Linear Quadratic Regulator for Linearized HVAC System
The linearized HVAC system of (11) can be controlled by a linear controller. A linearquadratic regulator (LQR) aims at designing stable controller which can minimize the cost function J represents the performance characteristic requirement as well as the controller input limitation [17]. The cost function is:
J=12∫0∞[x′TQx′+ηTRη]dt where
Q is a positive semidefinite weight matrix and
R is a positive definite weight matrix. The weighting matrices
Q and
R are chosen by the Bryson’s rule (refer to Appendix A5) [18].
The feedback control law that minimizes the values of cost is:
η=−k′e+yd(3) where
e=x′−Yd=[e1e1(1)e1(2)e2e2(1)e2(2)e3e3(1)e3(2)]T;
K′ is given by
K′=R−1BTP; and
P is found by solving the continuous time algebraic Riccati equation:
ATP+PA−PBR−1BTP+Q=0
Table 4 shows the gain values resulting from the LQR process solving algebraic Riccati equation.
The LQR controller was applied to the proposed HVAC system model (11) and the simulation results are shown in Figure 5. The temperature response of the proposed HVAC system is shown in Figure 5(a), and its humidity ratio response and CO_{2} concentration response are shown respectively in Figure 5(b,c).
From these simulation results, we can see that the proposed HVAC system is effective, and that linear controllers are suitable for application to the proposed HVAC system model. Table 5 shows the control performance metrics of settling time, rising time, settling max value, and settling min value for each value of gain.
energies0703599t005_Table 5
Control performance metrics of the LQR controller.
Temperature
Humidity ratio
CO_{2} concentration
K_{1}
K_{2}
K_{3}
K_{1}
K_{2}
K_{3}
K_{1}
K_{2}
K_{3}
T_{s}
1.4886
2.0107
3.2315
1.4982
1.9880
3.2028
1.5049
1.9873
3.1964
T_{r}
0.5279
0.7187
1.3093
0.5277
0.7189
1.3098
0.5277
0.7189
1.3098
Settling min
70.6601
70.7922
70.8702
0.0084
0.0087
0.0089
1193.2
1196
1197.5
Settling max
71.4249
71.3586
71.4777
0.0102
0.0100
0.0103
1208.5
1207.2
1209.6
(a) Temperature response by LQR; (b) Humidity ratio response by LQR; (c) CO_{2} concentration response by LQR.
5. Conclusions
Herein we have presented a novel HVAC system model that considers not only temperature and humidity ratio, but also CO_{2} concentration as the quantitative indices of comfort in a room. In applying an inputoutput feedback linearization method to linearize the HVAC system, problems of singularity, no relative degree, and interacting controls were encountered and a dynamic extension algorithm was used to solve these problems. The key contribution of this report is the addition of a continuous CO_{2} concentration state and corresponding valve dynamics to a conventional HVAC system to allow continuous control of CO_{2} concentration. Two types of linear controllers, pole placement and LQR controllers, were able to regulate the linearized HVAC system at the desired set point. Simulation results validated the proposed HVAC model, demonstrating its effectiveness in maintaining comfortable conditions. In future work, we will conduct further study on developing disturbance observer based controllers or intelligent controllers using fuzzy logic or artificial neural networks for a HVAC system considering parameter uncertainty and disturbance effect.
Author Contributions
A coauthor, JoilIl Park, cooperates on dynamic extension algorithm for achieving relative degree and applying linearization method to the HVAC system in Section 3.2.1. A coauthor, Mignon Park, cooperates on deriving modeling of HVAC system including CO_{2} concentration in Section 3.1. And Jaeho Baek cooperates on the equivalent linearization HVAC system in Section 3.3.
Nomeclaturehw
Enthalpy of liquid water
Wo
Humidity ratio of outdoor air
hfg
Enthalpy of water vapor
Vhe
Volume of heat exchanger
Ws
Humidity ratio of supply air
W3
Humidity ratio of thermal space
Cp
Specific heat of air
To
Temperature of outdoor air
Mo
Moisture load
Qo
Sensible heat load
T2
Temperature of supply air
T3
Temperature of thermal space
Vs
Volume of thermal space
ρ
Air mass density
F
Volumetric air flow rate (ft3/min)
gpm
Chilled water flow rate (gal/min)
μ
Air recirculation rate
Appendix A1
Consider the MIMO system of the form:
x˙=f(x)+G(x)u where X is the n×1 state vector; u is the m×1 control input vector (of components ui);
y is the
m×1 vector of system outputs (of components
yi);
f and
h are smooth vector fields; and
G is a
n×m matrix whose columns are smooth vector fields
gi.
Assume that
ri is the smallest integer for which at least one of the inputs appears in yi(ri); then:
yi(ri)=Lfrihi+∑j=1mLgjLfri−1hiuj with
LgjLfri−1hi(x)≠0 for at least one
j. Applying Equation (A2) for each output
yi yields:
[y1(r1)⋮⋮ym(rm)]=[Lfr1h1(x)⋮⋮Lfrmhm(x)]+D(x)u
Therefore, the decoupling matrix
D(x) is defined as:
D(x)=[Lg1Lfri−1h1⋯LgmLfri−1h1Lg1Lfri−1h2⋯LgmLfri−1h2⋮⋯⋮Lg1Lfri−1hm⋯LgmLfri−1hm] where
1≤i, j≤m.
For example, in our proposed HVAC system case,
m=3 and r=3:
Appendix A2
From the system given in (6):
y=h(x)=[h1(x)h2(x)h3(x)]T=[x1x2x4]T and the output
y1:
Therefore, the relative degree
r1=2 with respect to output
y1. According to Equation (A2):
y1(2)=Lf2h1+Lg1Lf1h1v1+Lg2Lf1h1v2+Lg3Lf1h1v3
Lg1Lf1h1:γ1k1τ1;
Lg2Lf1h1:0;
Lg3Lf1h1: 0.
The above procedure is similar to that applied for output
y2:
Thus, the relative degree
r2=2 with respect to output
y2. By Equation (A2):
y2(2)=Lf2h2+Lg1Lf1h2v1+Lg2Lf1h2v2+Lg3Lf1h2v3
Lg1Lf1h2:γ2k1τ1;
Lg2Lf1h2:0;
Lg3Lf1h2: 0.
In the case of output
y3:
Therefore, the relative degree
r3=2 with respect to output
y3. By Equation (A2):
y3(2)=Lf2h3+Lg1Lf1h3v1+Lg2Lf1h3v2+Lg3Lf1h3v3
Lg1Lf1h3:0;
Lg2Lf1h3:0;
Lg3Lf1h3:γ5k3τ3;
From the above results, the vector relative degree is
{222}, the total relative degree is 6, and the decoupling matrix is:
D(x)=[Lg1Lf1h1Lg2Lf1h1Lg3Lf1h1Lg1Lf1h2Lg2Lf1h2Lg3Lf1h2Lg1Lf1h3Lg2Lf1h3Lg3Lf1h3]=[γ1k1τ100γ2k1τ10000γ5k3τ3].
Appendix A3
From the system given in (7):
y=h(x−)=[h1(x−)h2(x−)h3(x−)]T=[x1x2x4]T and the output
y1:
.
Therefore, the relative degree
r−1=3 with respect to output
y1. According to Equation (A2):
y1(3)=Lf−3h1+Lg−1Lf−2h1ζ1+Lg−2Lf−2h1ζ2+Lg−3Lf−2h1ζ3
Lg−1Lf−2h1:γ1k1τ1
Lg−2Lf−2h1:α1γ4k2u1τ2
Lg−3Lf−2h1:0.
The above procedure is similar to that applied for output
y2:
Thus, the relative degree
r−2=3 with respect to output
y2. By Equation (A2):
y2(3)=Lf−3h2+Lg−1Lf−2h2ζ1+Lg−2Lf−2h2ζ2+Lg−3Lf−2h2ζ3
Lg−1Lf−2h2:γ2k1τ1
Lg−2Lf−2h2:0
Lg−3Lf−2h2:0.
In the case of output
y3:
Therefore, the relative degree
r3=3 with respect to output
y3. By Equation (A.2):
y3(3)=Lf−3h3+Lg−1Lf−2h3ζ1+Lg−2Lf−2h3ζ2+Lg−3Lf−2h3ζ3
Lg−1Lf−2h3:0;
Lg−2Lf−2h3:0;
Lg−3Lf−2h3:γ5k3τ3.
From the above results, the vector relative degree is
{333}, the total relative degree is 9, and the decoupling matrix is:
Appendix A4
From Equation (A2) and our proposed 3input
(ζi) and 3output
(yi) MIMO HVAC system given in (7):
y=h(x)=[h1(x)h2(x)h3(x)]T=[x1x2x4]T
Appendix A5
Bryson’s rule specifies the weighting of the Q and
R matrices where they are selected to be diagonal matrices:
Qii=1maximum acceptance values ofx′i2,i∈{1,2,…,l}Rjj=1maximum acceptance values ofηj2,j∈{1,2,…,m}
This corresponds to the following criteria:
J=12∫0∞[∑i=1lQiix′2+ρ∑j=1mRjjη2]dt where
x′ is state vector;
η is control input, and
ρ=(max state errormax control input)2.
Conflicts of Interest
The authors declare no conflict of interest.
ReferencesHodgeB.K.MullT.E.LevermoreG.HordeskiM.ArguelloSerranoB.VelezReyesM.Nonlinear control of a heating, ventilating, and air conditioning system with thermal load estimationBaekJ.KimE.ParkM.Adaptive Fuzzy Output Feedback Control for the Nonlinear Heating, Ventilating, and Air Conditioning SystemSemsarE.YazdanpanahM.J.LucasC.Nonlinear Control and Disturbance Decoupling of an HVAC System via Feedback Linearization and BackSteppingHuaguangZ.CaiL.Decentralized nonlinear adaptive control of an HVAC systemEmmerichS.J.PersilyA.K.LuX.LuT.ViljanenM.Estimation of space air change rates and CO_{2} generation rates for mechanicallyventilated buildingsChiangM.L.FuL.C.Hybrid System based Adaptive Control for the Nonlinear HVAC SystemChiangM.L.FuL.C.Adaptive control of switched systems with application to HVAC systemIsidoriA.UnderwoodC.P.SemsarKazerooniE.YazdanpanahM.LucasC.Nonlinear control and disturbance decoupling of HVAC systems using feedback linearization and backstepping with load estimationSlotineJ.J.E.LiW.LewisF.L.VrabieD.SyrmosV.L.FranklinG.F.PowellJ.D.EmamiNaeiniA.