Hybrid System Based Adaptive Control for the Nonlinear HVAC System
Ming-Li Chiang and Li-Chen Fu
Abstract— An adaptive controller is designed for the nonlin-ear MIMO heating, ventilating, and air conditioning (HVAC) system. The designed controller has the ability to adapt to the slowly time varying load change of thermal space and maintains good tracking performance for temperature and humidity ratio. Moreover, we integrate a supervisory switching logic into this system to adjust the CO2 concentration of thermal space and the whole closed loop system is modelled as a hybrid system. The obtained control system shows robustness and effectiveness and simulation results are provided to illustrate the control performance.
I. INTRODUCTION
Occupant comfort and energy efficiency are two primary goals of control strategies for the heating, ventilating, and air conditioning (HVAC) systems. As indicated in [1], HVAC systems for buildings are major consumers of electrical energy through the world. Temperature, humidity ratio and
CO2concentration are the quantitative indices of comfort in
a room. To control such systems efficiently and effectively with dynamic interactions and disturbances so as to conserve energy while maintaining the desired thermal comfort level requires more than a conventional methodology. Classical HVAC control techniques such as the ON/OFF controllers (thermostats) and the proportional-integral-derivative (PID) controllers are still very popular because of their low cost. However in the long run, these controllers are expensive since they operate at a very low-energy efficiency. With increasing complexity of modern HVAC systems, how to control and optimize the operation with guaranteed per-formance, stability and reliability becomes a challenging issue. In fact, the air conditioning process is highly non-linear and the interaction between temperature and humidity is significant. A nonlinear HVAC model which includes dynamics of temperature and humidity ratio is proposed in [2] where an observer to estimate the thermal load and moisture load is designed. The controller with load estimator can achieve the desired performance while the value of loads are not the designed one. In that paper, the load dynamics are assumed to be simple ones, that is, the loads are assumed to be constants. In [3], feedback linearization technique is applied to the same model. In [4], the actuator’s dynamics are considered and the feedback linearization approach is adopted to design the controller. The thermal load is treated as a measurable disturbance and is compensated by the feedback controller. But the humidity ratio of thermal space Ming-Li Chiang is with the Dept. of Electrical Engineering, National
Taiwan University, Taipei, [email protected]
Li-Chen Fu is with the Department of Electrical Engineering, and the Department of Computer Science and Information Engineering, National
Taiwan University, Taipei, [email protected]
is not controlled since the authors chose different output function. In [5], a decentralized nonlinear adaptive controller consists of a fuzzy feedback controller and a frequency-domain adaptive compensator designed in Fourier space is proposed. The control scheme is capable of dealing with the varying thermal loads and is with short setting.
In this paper, we design an adaptive controller for the non-linear MIMO HVAC system which is modelled to have some unknown parameters, to achieve robust and good heating, ventilating, and air conditioning performance. These slowly time varying unknown parameters are the thermal heat load and moisture load. In most of the related literatures, the val-ues of the loads are treated as constants. Therefore, the con-troller proposed here should be more robust, more practical, and still achieving satisfactory performance. In particular, the system with the adaptive controller can operate effectively in the presence of dynamic interactions and disturbances while maintaining the desired thermal comfort level. Adaptive con-trol for nonlinear systems has received a significant research attention and has evolved as a powerful methodology for nonlinear systems with parametric uncertainties. Feedback linearization approach proposed in [6] enhances the robust-ness to the failure of exact linearization. Many applications of nonlinear adaptive control have been presented in the past few years. In [7], the nonlinear adaptive controller based on feedback linearization is applied to the electro-hydraulic servomechanism. The SISO nonlinear system has a strong relative degree of two and trivial zero dynamics. In [8], adaptive control for nonlinear system are combined with the neural network approach and the nonlinear systems are assumed to have the normal form. This approach has the advantage with increasing computation speed and no need to analyze the complex nonlinear behavior of the system. The HVAC system considered in our paper is input, two-output, and has no relative degree. We employ the dynamic extension algorithm to synthesize the feedback controller for the system and then design an adaptive controller to track the desired temperature and humidity ratio.
The remainder of this paper is organized as follows: section II introduces operation process and the hybrid dy-namic model of the HVAC system. In section III, feedback linearization via dynamic extension is applied and a non-adaptive controller is designed to achieve the asymptotical stability for the tracking error. Then, we introduce some parametric uncertainties to replace the ideal model and derive the adaptive controller for the MIMO nonlinear plant.
We also integrate a switching scheme to adjust the CO2
concentration. Section IV shows the simulation results and section V concludes this paper.
Proceedings of the 2006 American Control Conference
II. HYBRIDSYSTEMMODEL FOR THEHVAC SYSTEM
ANDPROBLEMFORMULATION
A. HVAC Model with Temperature and Humidity Ratio
Air in the room is assumed to have an uniform temperature distribution and heat loss between components is neglected. Here we employ the model proposed in [2] which includes both temperature and humidity ratio. The system operates as shown in Fig. 1. Outdoor air enters the system at temperature
T0and with volumetric flow rate fr(t). Air with temperature
T0 and flow rate fr(t) passes through the heat exchanger
where an amount of heat is exchanged with the air. Since we have the assumption of perfect mixing, the air
temper-ature inside and exiting the heat exchanger is T2(t), which
represents the supply air temperature. After being cooled or
heated in the heat exchanger, the air with temperature T2
passes into the thermal space with the help of fan and air
temperature of the thermal space is T3(t). The heat load in
the room is denoted as Qo. To save energy, typically 25% of
the air is drawn out of the thermal space through the help of fan, and 75% of the air is recirculated to be mixed with the fresh air from outdoor.
Filters Heating Coil Cooling Coil Fan Chiller Heat exchanger Thermal space Outdoor air Excluded air Damper Supply air 2 3 4 5 1
Fig. 1. Schematic layout of the HVAC system
Here, we assume the system is operating either with the cooling process and or with the heating process, but both models will be the same except the signs which can be either positive or negative. Control inputs to the system are the pumping rate (gpm) of cold water from chiller to heat exchanger and the air flow rate ( fr) using the variable speed fan. Notations and parameter values used in the dynamic model and simulations here are the same as those in [2] and are given in Table I.
From energy conservation principle, the dynamic equa-tions of the HVAC system are given by
˙ T3 = 60 fr Vs (T2− T3)− 60hf gfr cpVs (Ws−W3) + 1 (1−μ)ρacpVs(Qo− hf gMo) ˙ W3 = 60 fr Vs (Ws−W3) + Mo ρaVs ˙ T2 = 60 fr Vhe (T3− T2) + 60(1−μ) fr Vhe (T0− T3) −60hwfr cpVhe ((1−μ)W0+μW3−Ws)− 6000 gpm ρacpVhe. TABLE I
HVAC SYSTEMVARIABLES ANDPARAMETERS
cp Specific heat of air at constant pressure 0.24 (btu/lb·◦F)
ρa Air mass density 0.074 (lb/ft3)
Vhe Volume of heat exchanger 60.75 (ft3)
Vs Volume of thermal space 58464 (ft3)
W0 Humidity ratio of outdoor air 0.018 (lb/lb)
Ws Humidity ratio of supply air 0.0070 (lb/lb)
W3 Humidity ratio of thermal space (lb/lb)
T0 Temperature of outdoor 85 (◦F)
T2 Temperature of supply air (◦F)
T3 Temperature of thermal space (◦F)
Mo Moisture load 166.06(lb/hr)
Qo Sensible heat load 289897.52(btu/hr)
hw Enthalpy of liquid water (J/lb)
hf g Enthalpy of water vapor (J/lb)
fr Volumetric flow rate of air (cfm=ft3/min)
fr0 Initial of volumetric flow rate of air 4250 (cfm)
gpm Flow rate of chilled water (gal/min)
μ Recirculation rate of air in system(75%)
Cs CO2concentration of thermal space (ppm)
and the state equations can be described as ˙x1 = u1α1(x3− x1)− u1α2(Ws− x2) +α3(Qo− hf gMo) ˙x2 = u1α1(Ws− x2) +α4Mo ˙x3 = u1β1(x1− x3) + (1−μ)u1β1(T0− x1) −u1β3((1−μ)W0+μx2−Ws)− 6000u2β2 y = [h1(x) h2(x)]T = [x1 x2]T (1) where u1= fr, u2= gpm, x1= T3, x2= W3, x3= T2 α1= 60 Vs, α2= 60hf g cpVs , α3= 1 (1−μ)ρacpVs, α4= 60 ρaVs β1= 60 Vhe, β2= 1 ρacpVhe, β3= 60hw cpVhe
This is a bilinear system which is homogeneous in the state.
Heat load Qoand moisture load Mo are usually not
measur-able and hence will be regarded as unknown parameters. In [2], the authors develop an observer for the system to obtain
the estimation of Qo and Mo. Now, consider the actuator
dynamics for the control inputs, namely the valve dynamic as in [4] u1= k1 1+τ1sz1, u2= k2 1+τ2sz2
with u= [u1 u2]T be the control signal applied to the plant and z= [z1 z2]T the input signals applied to the actuator. Hence, we derive an augmented state space model with the new state vector x := [x1x2 x3 u1 u2]T := [x1x2x3 x4 x5]T, and input signal z= [z1z2]T, so that the system model now
becomes ˙x = f(x) + g(x)z = f (x) + g1(x)z1+ g2(x)z2 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ a1(x) a2(x) a3(x) a4(x) a5(x) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦+ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 0 0 k1 τ1 0 0 kτ2 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ z y = h(x) = [x1 x2]T (2) where a1(x) = [α1(x3− x1)−α2(Ws− x2)]u1+α3(Q0− hf gM0) := γ1u1+α3(Q0− hf gM0) a2(x) = α1(Ws− x2)u1+α4M0 := γ2u1+α4M0 a3(x) = [β1(x1− x3) + (1−μ)β1(T0− x1)]u1 +[−β3((1−μ)W0+μx2−Ws)]u1− [6000β2]u2 := γ3u1+γ4u2 a4(x) = − u1 τ1 and a5(x) =−u2 τ2
B. Hybrid System Model for the HVAC System
The quantitative indices of comfort in the room are tem-perature, humidity ratio, and CO2concentration. In fact, the
circulated air which contains too much CO2 will affect the
work efficiency of people. Now, we consider the CO2model
and then propose a hybrid system model for the HVAC
system. From the mass balance equation, the average CO2
concentration Cs in the room can be represented as
˙
Cs=Cq
Vs + (1−μ)(Ci−Co)
where Cq is the amount of CO2 generated in the room
(normally from people entering the room), Ci is the CO2
concentration of inlet air, Co is the CO2 concentration of
air leaving the room, and (1−μ) is the air exchange rate.
Assume that the air in the room is well-mixed and Ci is a
constant, then Co= Cs and the equation can be written as ˙
Cs=Cq
Vs − (1 −μ)C
s (3)
where Cs= Ci−Cs. We can use nonlinear control of(1−μ)
to adjust the CO2 concentration at all operating points. The
value of Cq is dependent on the number and the physical
state of people in the room and there are some reference
data in the ASHRAE standard. Instead of controlling Csto a
desired value, we define three levels of CO2concentration in
the room and adjust the value ofμ according to the level at
which Cs is located. That is, classify the CO2 concentration
into three intervals {CO2Low, CO2Med, CO2High} and
use the supervisory controller S to decide the corresponding value of μ∈ {μ1, μ2, μ3}. Thus, the CO2 concentration is modelled and adjusted by discrete event systems theory. The HVAC system can be modelled as a hybrid system [9]
which contains continuous states x and discrete states μ.
The continuous dynamics in this system is as in (2) and the
discrete dynamics is defined with the supervisory switching logic which will be designed in next section. Thus we have the hybrid system model for the HVAC system as follows:
Continuous dynamics: ˙x= f (x,μ) + gz, μ∈ {μ1, μ2, μ3} y= h(x) Discrete dynamics: μ=μi, i ∈ {1, 2, 3} assigned by S (4)
We will use the continuous adaptive state feedback controller to control the value of y and the discrete event controller S
to decide the value ofμ and thus the CO2 concentration Cs
will be adjusted. Our control object is to track the desired temperature and humidity ratio, and construct a supervisor
to keep the CO2 concentration low.
III. SUPERVISORYADAPTIVECONTROL FOR THE
NONLINEARMIMO HVAC SYSTEM
In this section, we apply the feedback linearization tech-nique [10] to the nonlinear HVAC system. Assume the values of loads Q0and M0are pre-specified constants and we design the non-adaptive dynamic state feedback controller for the HVAC system. Then, the adaptive controller is designed to
cope with the slowly time varying unknown Mo, Qo. We also
integrate a supervisory switching mechanism which aim to
keep the CO2concentration in the HVAC system low.
A. Feedback Linearization via Dynamic Extension for the HVAC System
To reduce the nonlinear system to an aggregate of indepen-dent single-input, single-output channels, which is called the
noninteracting control problem, we differentiate the outputs
yi(t) until the inputs appear. The intuitive concept of relative
degree is the smallest number of times that the output have to differentiate such that at least one of the inputs appears in y(ri i). For the 2-input, 2-output systems of the form (2), the relative degree is defined as r={r1, r2} which satisfies (i) Lgjhi= LgjLfhi= . . . = LgjL(rfi−2)hi= 0, for all j = 1, 2,
i= 1, 2, and (ii) the decoupling matrix
A(x) = Lg1Lrf1−1h1 Lg2Lrf1−1h1 Lg1Lrf2−1h2 Lg2Lrf2−1h2 (5)
is nonsingular near the equilibrium point x= xe. Note that
Lfhi stands for the Lie derivative of hi with respect to f . The definition of relative degree will lead to
y(r11) y(r22) = Lrf1h1 Lrf2h2 + A(X) z1 z2
and thus the feedback control law
z = −A(x)−1 Lrf1h1 Lrf2h2 + A(x)−1v (6) will yield the closed-loop decoupled, linear system
y(r11) y(r22) = v1 v2
Given the desired output yd = [y1d, y2d]T and define the
output error as e= [e1, e2]T := [(y1− y1d), (y2− y2d)]T. Then, we can design the control
vi= y(ridi)− ci1ei(ri−1)− ··· − ciriei (7)
such that the error equation becomes
ei(ri)+ ci1ei(ri−1)+··· + ciriei= 0
The coefficients ci j, j = 1, 2, . . . , ri, are chosen so that
sri+ ci1sri−1+··· + cir
i is a Hurwitz polynomial to meet the
desired performance specification such as transient response or steady state error.
For the HVAC system (2), we have r={2,2}, but the
matrix A(x) = γ1k1/τ1 0 γ2k1/τ1 0
is singular. Thus, the system has no relative degree. To achieve the relative degree and noninteracting control, we resort to the dynamic extension algorithm [10] to incorporate
the dynamic state feedback into this system. Set z1=ψ1,
˙
ψ1=ζ1, and z2=ζ2. Define the new augmented state as
˜x= [x, z1]T ∈ R6and the composed system will be ˙˜x = ˜f( ˜x) + ˜g1( ˜x)ζ1+ ˜g2( ˜x)ζ2
y = h(x) (8)
where ˜f= [ ˜a1, ˜a2, . . . , ˜a6]T is equal to f except the term ˜a4= a4+kτ11z1 and ˜a6= 0. Moreover, the vector field
˜
g = [ ˜g1 g˜2], where ˜g1= [0, 0, 0, 0, 0, 1]T and ˜g2 = [0, 0, 0, 0, k2
τ2, 0]
T. After calculation, we find the relative degree now becomes ˜r={ ˜r1, ˜r2} = {3,3} so that
y(3)1 y(3)2 = L3f˜h1 L3f˜h2 + B( ˜x) ζ1 ζ2 := C( ˜x) + B( ˜x)ζ (9) and the nonsingular decoupling matrix becomes,
B( ˜x) = L˜g1L2f˜h1 L˜g2L2f˜h1 L˜g1L2f˜h2 L˜g2L2f˜h2 = γ 1k1 τ1 (α1γ4x4k2) τ2 γ2k1 τ1 0
Thus, design the control ζ =−B−1C+ B−1v with v =
[v1, v2]T as discussed in (7), the output error will converge to zero and the system will be asymptotically stable.
Remark 1: Since the relative degree ˜r1+ ˜r2= 6 is equal to the number of states, the system has trivial zero dynamics and thus the internal stability and boundedness of states are guaranteed. If the system has relative degree ˜r1+ ˜r2= k < 6, we can obtain the internal dynamics by constructing the local coordinate transform with the functionφisuch that L˜gjφi= 0 for j= 1, 2 and k + 1 ≤ i ≤ 6. The existence of the functions
φi is guaranteed by the Frobenius theorem and the fact that
constant vector fields ˜g1, ˜g2 are always involutive.
B. Adaptive Control for the HVAC system
The drawback of the feedback linearization approach is that the linearizing control law is based on exact cancellation of the nonlinear terms. If there is any uncertainty in the knowledge of the nonlinear functions ˜f, ˜g, the cancellation is not exact and the resulting input-output equation is not
linear in practice. The value of heat thermal load Qo and
that of moisture load Mo are not measurable and are hence
difficult to estimate. In this section, we will use adaptive control techniques [6] to solve this problem.
Define the values of M0and moisture load Q0as unknown
parametersΘ∗:= [θ1∗= Mo, θ2∗= Qo]T. The system is linear with respect to the vector field ˜f and the unknown constant
parameter vectorθ∗, and thus we can rewrite (8) as
˙˜x =
∑
2i=1θ
∗
i fi( ˜x) + f0+ ˜gζ
yi = hi(x), i = 1, 2. (10)
The estimates ofΘ∗is denoted asΘ and thus the estimate of
the vector field ˜f is defined as ˆf=θ1f1+θ2f2+ f0. Thus, the Lie derivatives in our feedback control are replaced by the estimations as the follows:
Lfˆ3hi := 2
∑
j=1 2∑
k=1 2∑
l=1 ∂ ∂˜x ∂ ∂˜x ∂hi ∂˜x fj fk flθjθkθl L˜gL2fˆhi := 2∑
l=1 2∑
j=1 2∑
k=1 ∂ ∂˜x ∂ ∂˜x ∂hi ∂˜x fj fk ˜ glθjθkThe ideal linearizing control law is replaced by ζ = B−1 − L3fˆh1 L3fˆh2 +v (11)
where Bis the estimate of B and the implemented tracking
law v is of the following form
vi= y(3)id + c1(y(2)id − L2fˆhi) + c2(y(1)id − Lfˆhi) + c3(yid− yi), (12)
i= 1, 2. Since in (10) f1and f2are not dependent on ˜x, the terms(θiθj, θiθjθk) will not appear in L2fˆhi, L3fˆhi, Lg˜jL2fˆhi,
and thus the parameter vector we need to estimate is Θ =
[θ1, θ2]T ∈ R2. Substitute the control ζ into the system
and define the parameter error Φ := Θ∗− Θ, then the error
equation with the feedback control will become
e(3)+ c1e(2)+ c2e(1)+ c3e=ΞΦ (13) whereΞ = Ξ11 Ξ12 Ξ21 Ξ22 = Ξ1 Ξ2
∈ R2×2and the terms on
the right hand side are the mismatch between the linearizing law and the actual linearizing law as well as that between the tracking control v and the actual tracking control ˆv. By computation we have ΞiΦ = [L3f˜hi− L3fˆhi] + c1[L2f˜hi− L2fˆhi] + c2[Lf˜hi− Lfˆhi] (14) and Ξ11 = α3hf g(−λ1+α1a˜4+ c1α1u1− c2) +α4(λ2+ α2a˜4+ c1α2u1), Ξ12=α3(λ1−α1a˜4− c1α1u1+ c2), Ξ21=
α4 (α1u1)2− 2α1a˜4− c1α1u1+ c2 , Ξ22 = 0, and λ1 = α1 u21(α1+μβ1)− ˜a4 ,λ2=−2α1α2u21−μβ3α1u21+α2a˜4. For the case of relative degree 3, we define the augmented
error
eiaug= b1e(2)i + b2e(1)i + b3ei (15) such that the transfer function
(b1s2+ b2s+ b3)/(s3+ c1s2+ c2s+ c3) (16) is strictly positive real (SPR). Let
eiaug = ei+ (ΘTL−1ΞTi − L−1ΘTΞTi), i = 1, 2 (17) where the polynomial
L(s) = s(3)+ c1s(2)+ c2s(1)+ c3
is chosen to be Hurwitz and note that ei= L−1(s)ΦTΞTi.
From the fact that Θ∗ is a constant vector, we can obtain
the error equation for adaptation, i.e., eiaug=ΦTL−1ΞTi and define L−1ΞTi =ξiT ∈ R2×1, then eaug= e1aug e2aug = ξ1 ξ2 Φ :=ξΦ (18)
Hence, we can use the normalized gradient adaptive law ˙
Θ = ˙Φ = Γ
(1 +ξξT)(−ξ Te
aug) (19)
where Γ is the adaption gain. Since eaug is chosen so that
(16) is SPR, the estimated matrix B−1= B−1 is away from
singularity and the estimated signals are all bounded, the stability analysis for this adaptive control will yield bounded state ˜x and y→ yd as t→ ∞ [6].
Remark 2: There might be some problem when apply-ing adaptive control with dynamic extension. The problem occurs if the true decoupling matrix B( ˜x) is singular but its estimate B( ˜x) is nonsingular during adaptation. In our case, this will not happen since the unknown parameters only appear in the vector fields ˜a1, ˜a2 and are not coupled with the state variables. This greatly simplify the computation and
the estimated matrix B= B contains no estimate parameters.
C. Supervisory Switching Control for Ventilation and Stabil-ity Analysis
Now we start to design the supervisor to decide the
recirculation rate μ such that the CO2 concentration
in-door will be adjusted. Note that in [2], the
recircula-tion rate μ = 75%. Define three classes of the CO2
concentration as {CO2High, CO2Med, CO2Low} where
CO2High means Cs ≥ 1600 ppm, CO2Med means Cs ∈
(1200 ppm, 1600 ppm) and CO2Low means Cs≤ 1200 ppm.
The ranges can be defined by requirement. We construct a supervisor S with the following switching logic:
S: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ CO2High and y− yd<ε ⇒ μ=μ3= 60% CO2Med and y− yd<ε ⇒ μ=μ2= 65% CO2Low and y− yd<ε ⇒ μ=μ1= 75% Otherwise No switching (20)
whereε is a pre-specified small positive constant. Thus, the
hybrid dynamical system (4) is modelled with the continuous
states ˜x and discrete states μ with the continuous state
feedback controllerζ and the discrete state controller S. The whole hybrid system model and the processing procedure can be clearly represented by the hybrid automaton [11] shown in Fig. 2. Initial 2 CO Med and e H x f g] 3 P P x f g] 1 P P x f g] 2 P P 2 CO Low and e H 2 CO Med and e H 2 CO High and e H 2 CO Low and e H 2 CO High and e H
Fig. 2. Hybrid automaton of the HVAC system
Stability issue of this hybrid system will be discussed in the following. Here the stability means that the continuous states are stable and the discrete state will not switch
infinite times in finite time interval. Since the value of Cs
is continuous and slowly time varying, and our switching will only occur when the tracking error of the current system satisfy the specification, the zeno phenomenon [11] will not occur, that is, the existence of the dwell time guarantees that infinitely switching in finite time will never occur.
As discussed in [12], switching between stable systems may lead to unstable phenomenon. Since the continuous
dynamical system (8) with adaptive controller ζ is stable
with μ=μi, i= 1, 2, 3, in (20), respectively, the possible scenario that leads the system to unstable is that the value of
μswitches sequentially before the continuous states achieves
the temperature and humidity tracking and thus oscillation or divergence might occur. The unstable situation caused from consecutive switching is not allowed in the switching mechanism (20) since the supervisor is designed with the priority rule that the switching can only start when the tracking error converges. Hence, we can conclude that the whole system will be stable with the supervisory switching logic. The simplicity of the stability analysis is attributed
to our switching logic and the CO2 concentration Cs is
not coupled with our continuous state dynamic equation ˙˜x = ˜f+ ˜gζ.
IV. SIMULATIONRESULTS
The equilibrium conditions of the HVAC system are T3e=
71◦F, W3e= 0.0088 lb/lb, T0e= 85◦F, W0e= 0.018 lb/lb,
Wse= 0.0070 lb/lb, ue1= 17000 cfm, ue2= 58 gpm, M0e= 166.06 lb/hr, and Qe0= 289897.52 btu/hr. The initial values
are T2e= 55◦F and ζ1= 15000, ζ2= 40. Figure 3 shows
the first output response of the feedback controller with the design load values Qe0, M0e, and the non-design thermal load
with values Qo= 350000 btu/hr and Mo= 196 lb/lb. It is clear that the controller does not have good tracking performance when the system works upon the environment of non-design load values. The adaptive controller shows its robustness and the transient response is satisfactory compared with [2].
Suppose the values of Qo, Mo are changed from 166 to
176 and from 290000 to 310000, respectively, as shown in the upper side of Fig. 4. In real world, the load changes are in a time scale of hours, here we use these values to show our controller performance. From the bottom of Fig. 4, the proposed adaptive state feedback controller has shown that it can response to the time varying load change. In
Fig. 5, the CO2 concentration changes from CO2Low to
CO2Med at t= 1800 (sec) and thus the supervisor switches
the recirculation rate from μ1 to μ2 at t= 1800 according
to the switching logic. The time response of temperature and humidity ratio shown in Fig. 5 provides the tracking performance with switchings.
0 200 400 600 800 1000 1200 1400 1600 1800 65 70 75 80 85
With design thermal load
nonadaptive controller adaptive controller 0 200 400 600 800 1000 1200 1400 1600 1800 65 70 75 80 85 Time (sec) y1 room temperature ( oF)
With non−design thermal load
nonadaptive controller adaptive controller
Fig. 3. First output response to design and non-design loads
0 1000 2000 3000 160 170 180 190 200 Moisiture load M 0 0 1000 2000 3000 2.8 3 3.2 3.4 x 105 Heat load Q0 0 500 1000 1500 2000 2500 3000 3500 65 70 75 80 85 Room temperature Time (sec) y1 room temperature ( 0F)
Fig. 4. First output response to time varying Qo, Mo
V. CONCLUSIONS
In this paper, we propose a hybrid system model for the HVAC system and apply the feedback linearization technique with dynamic extension to design the continuous adaptive control for the nonlinear MIMO system. Values of thermal
loads Qo and Mo may be time varying or not be known
0 500 1000 1500 2000 2500 3000 3500 65 70 75 80 85 Room temperature y1 room temperature ( 0F) 0 500 1000 1500 2000 2500 3000 3500 0.005 0.01 0.015 0.02 0.025 Moisiture ratio Time (sec) y2 humidity ratio
Fig. 5. Output response of the HVAC system with CO2supervisor.
exactly, and hence adaptive controller is a good choice for HVAC system. Besides, we construct a discrete supervisory
controller to adjust the recirculation rate based on the CO2
concentration and discuss stability of the whole system by hybrid system theory. The adaptive controller tracks the desired temperature and humidity ratio to keep the comfort
of thermal space and the CO2 supervisor improves the
air quality. Computer simulations have proved that such a adaptive controller is superior to the non-adaptive controller due to the ability of adaption to load changes and system perturbation.
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