Development of DMC controllers for temperature control of a room deploying the displacement ventilation HVAC system

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I
NTERNATIONAL
J
OURNAL OF

E
NERGY AND
E
NVIRONMENT



Volume 4, Issue 3, 2013 pp.415-426

Journal homepage: www.IJEE.IEEFoundation.org


ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2013 International Energy & Environment Foundation. All rights reserved.
Development of DMC controllers for temperature control of
a room deploying the displacement ventilation HVAC
system


Zhicheng Li
1
, Ramesh K. Agarwal
1
, Huijun Gao
2


1
Department of Mechanical Engineering and Materials Science, Washington University in Saint Louis,
MO 63130, USA.
2
Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001,
China.


Abstract
In this paper, by developing a new Dynamic Matrix Control (DMC) method, we develop a controller for
temperature control of a room cooled by a displacement ventilation HVAC system. The fluid flow and
heat transfer inside the room are calculated by solving the Reynolds-Averaged Navier-Stokes (RANS)
equations including the effects of buoyancy in conjunction with a two-equation realizable k - epsilon
turbulence model. Thus the physical environment is represented by a nonlinear system of partial
differential equations. The system also has a large time delay because of the slowness of the heat
exchange. The goal of the paper is to develop a controller that will maintain the temperature at three
points near three different walls in a room within the specified upper and lower bounds. In order to solve
this temperature control problem at three different points in the room, we develop a special DMC
method. The results show that the newly developed DMC controller is an effective controller to maintain
temperature within desired bounds at multiple points in the room and also saves energy when compared
to other controllers. This DMC method can also be employed to develop controllers for other HVAC
systems such as the overhead VAV (Variable Air Volume) system and the radiant cooling hydronic
system.
Copyright © 2013 International Energy and Environment Foundation - All rights reserved.

Keywords: Computational fluid dynamics; Dynamic matrix control method; Energy efficiency of
buildings; Temperature control in enclosures.



1. Introduction
Effective energy management for facilities such as hospitals, factories, malls, or schools is becoming
increasingly important due to rising energy costs and increase in the associated greenhouse gas (GHG)
emissions. One of the major users of energy is buildings. Most modern buildings employ a heating and
cooling system depending upon the climate and time of the year. The focus of this paper is on control of
HVAC units in buildings deployed for cooling during summer months to maintain temperature inside the
building for human comfort and other operational requirements. In many climates around the world, the
air-conditioning requirements for cooling the buildings can be very high during the summer months, and
it turns out that the major portion of energy consumption of a building is from HVAC units. For
example, it has been reported that the energy consumption of HVAC units in general accounts for 40%
International Journal of Energy and Environment (IJEE), Volume 4, Issue 3, 2013, pp.415-426
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2013 International Energy & Environment Foundation. All rights reserved.

416
of total energy use by a building [18] and on an extremely hot day it could be as high as 65% [19].
Improvement in the control of HVAC systems can therefore result in significant savings (e.g. 25% in
energy use, see [20]).
To control HVAC systems, the traditional method is the on/off control at the level of HVAC
components, for example an air-conditioning unit. This kind of control is a very low-level control. In
recent years, some advanced control strategies have been developed that can be implemented in
operating the HVAC systems in an integrated fashion for commercial buildings to improve their energy
efficiency. There have been some results reported in the literature to investigate the energy requirements
of buildings using different HVAC systems, see [14-16] and the references there in. Our goal is to
control the temperature inside the building as well as well as save energy. There are many types of
methods, which can be employed to control the operation of HVAC systems. To mention a few from the
literature, an immune PID adaptive controller has been presented in Reference [9], which is quite
different from the traditional PID controller [8]. References [2, 4-6, 10-12] introduce Model Predictive
Control (MPC) method for building cooling systems. In particular, the DMC method, as one of MPC
methods has been widely employed in the study of HVAC control systems involving large time delays,
see for instance [11-12] and the references therein. In another study [7], the authors have used Artificial
Neural Network (ANN) based models to control the temperature of a building and have obtained
impressive results. The fuzzy control method of Zadeh [17] has also been widely used for control of
many nonlinear systems; a fuzzy control method is given in Reference [3] which shows promise for
temperature control in buildings using different HVAC systems. However, all these studies have
limitations with respect to the nature of the disturbance and the time delay; they are limited to small
disturbance in temperature as well as small time delay in heat exchange. Thus, it remains an important
and challenging problem to design good controllers, which can keep the temperature stable in a smaller
time interval as well as result in more savings in energy.
In this paper, we develop a controller for temperature control inside a room within a desired band of
temperatures for comfort. The details of the geometry of the room and the HVAC system based on
displacement ventilation for cooling the room are taken from Reference [1]. The control of this system is
difficult since the HVAC system has no heater, which means that we can only cool the room, but not heat
it. In addition, the time delay in heat exchange also exists in the system. All of these factors make it
difficult in achieving the temperature control objective using the methods described in the references
listed above. After many computational experiments, we have determined and developed an effective
method to solve this control problem. The DMC controller is developed based on the traditional one for
controlling a one-input three-output system. We employ two groups of model systems to illustrate the
effectiveness and disadvantages of this method, and finally show the effectiveness of the controller for
not only temperature control but also in energy savings for the HVAC system under consideration.

2. Fluid flow simulation in the room
The flow field inside the room with and without displacement ventilation was simulated by the CFD
software FLUENT, which solves the Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations
employing the finite-volume method on a collocated grid. In Fluent, URANS equations are solved using
the second-order upwind scheme and the pressure is calculated using the PRESTO scheme. The SIMPLE
algorithm is employed for the coupling of the velocity and pressure. In our calculations, both the one-
equation Spalart Allmaras (S-A) turbulence model and the two-equation k-ε realizable turbulence model
were employed. The S-A model is a simpler turbulence model, which only uses one equation to describe
the turbulent eddy viscosity, compared to the k-ε realizable model, which uses two equations to calculate
the eddy viscosity. We computed the flow field using both the S-A and k-ε realizable models on the same
grid and found little difference in the results. The geometry of the room and other details of displacement
ventilation are taken from Reference [1]. Figure 1 shows the schematic of the room with the two outlet
vents in the ceiling and six inlet vents on the floor. The dimensions of the room are 12 ft x 12 ft x 9.5 ft
with a surface are of 804 ft² and volume of 1368 ft
3
.The inlet vents on floor of the room are 6"×9" in
cross-section, which gives an area of 2.25ft² for the six vents. The air flow in the room meets the
ASHRAE guidelines of air movement. The six inlet vents are placed on the floor near the adiabatic walls.
This is done in order to keep the installation of the vents on the floor practical, so that the vents may not
be blocked by the furniture in the room. The two outlet vents in the ceiling are 1′-6"×1′-6" in size, giving
an area of 4.5ft² (0.418m²)) for the 2 outlet vents. We set three sensors in the room to monitor the
temperature at three points close to three walls, whose locations are shown in Figure 1. The temperature
International Journal of Energy and Environment (IJEE), Volume 4, Issue 3, 2013, pp.415-426
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2013 International Energy & Environment Foundation. All rights reserved.

417
of the exterior wall of the room was kept at a constant temperature while the other five walls were
considered adiabatic. Figure 2 show the 3-D Cartesian mesh inside the room.
A Fluent UDF (User Defined Function) was created to simulate the temperature of the exterior wall of
the room. This temperature curve simulated the exterior surface and was assumed to be at a constant
value of 320K. A 3-D Cartesian mesh inside room was generated by GAMBIT with a uniform grid
spacing of 3".
In the following sections, we develop the DMC method to control the temperature of this room with
three temperature sensors.
Remark 1: The temperature sensors are not real. We assume that there are three sensors, which can give
us temperature data, which is obtained from CFD simulations using FLUENT. We only use these three
points’ temperature as reference temperature for the present control method.

Figure 1. 3D view of the room with three sensors, two outlet vents and six inlet vents employed in
displacement ventilation



Figure 2. The 3-D Cartesian mesh inside the room

3. Dynamic matrix control (DMC) method
Dynamic Matrix Control (DMC) has been shown to be an effective advanced control technique in many
industrial process control applications and has recently been extended to the procedure control systems
which often have large time delay and uncertainty. Our HVAC system has these characteristics. We
consider designing a DMC controller, which is a model-based control method [11-12]. Traditional DMC
method can be used in single-input-single-output (SISO) systems, and there are some theories about the
DMC controllers for multi-input-multi-output (MIMO) systems, for example in Reference [21]. However
the DMC controllers for MIMO systems have only been discussed from a theoretical point of view. In
this paper it is developed for a single-input-multiple-output (SIMO) system and is applied to an
application governed by a set of highly nonlinear partial differential equations governing fluid flow.
International Journal of Energy and Environment (IJEE), Volume 4, Issue 3, 2013, pp.415-426
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2013 International Energy & Environment Foundation. All rights reserved.

418
3.1 Model foundation
In DMC based controller, we first need to determine a system model. The model in DMC is determined
by the step response, which is similar to the traditional model composed of the difference equation. From
the change of exterior wall temperature in Figure 3, we know that the average exterior wall temperature
is 320K. Thus, we set the exterior wall temperature equal to 320K without control, and when the room
temperature is near 320K, we give a step signal to mass-flux (0.1) to make the HVAC system cool the
room. At this time we can obtain three temperatures from three sensors that can be used in the model as
the step response data, which is shown in Figure 3.

0 100 200 300 400 500 600 700 800 900 1000 1100
-0.01
0
0.01
0.02
0.03
Time [60s/step]
Mass flow Rate [Kg/s]


Step Input
0 200 400 600 800 1000 1200
290
300
310
320
Time [60s/step]
Temperature [K.]


Point 1
Point 2
Point 3


Figure 3. The input signal and the step responses

According to the superimposition principle of the linear system, suppose the original output value of the
system at k is y
0
(k), the control value u(k) (here it is the mass flow rate) has an increment ∆u(k) at k. The
output predictive values Y
n
(k) with n = 1, 2, 3 (here they are temperature) at future time steps are:

1
10 1
2
20 2
3
30 3
() () (),
() () (),
() () (),
Yk Y k uk
Yk Y k uk
Yk Y k uk
⎧
=+Γ∆
⎪
=+Γ∆
⎨
⎪
=+Γ∆
⎩
(1)

where
( ) ( 1) ( 2) ( ) ,
T
TT T
nn n n
Yk yk yk ykN
⎡⎤
=+ + +
⎣⎦

0,0 ,0 ,0
() ( 1) ( 2) ( ) ,
T
nT T T
nn n
Yk yk yk ykN
⎡ ⎤
=+ + +
⎣ ⎦

,1 , 2 ,
, 1, 2, 3.
T
nnn nN
aa a n
⎡⎤
Γ= =
⎣⎦

Γ
n
is the dynamic coefficient vector of the point n’s step response. Y
n
(k) expresses the predictive system
output of the future N moments. The equation (1) has been obtained assuming that ∆u(k) doesn’t change
any more. If the added control quantity changes at M sample intervals: ∆u(k), ∆u(k+1),…, ∆u(k+M+1),
then the model output value would be

1, 1,0 1, 1
1
2, 2,0 2, 1
1
3, 3,0 3, 1
1
() () ( 1) , (1,2, ,)
() () ( 1) , (1,2, ,)
( ) ( ) ( 1) , ( 1, 2, , )
i
Mij
j
i
Mij
j
i
Mij
j
ykiyki a ukj i M
ykiyki a ukj i M
ykiyki a ukj i M
−+
=
−+
=
−+
=
⎧
+= ++ ∆ +− =
⎪
⎪
⎪
⎪
+= ++ ∆ +− =
⎨
⎪
⎪
+= ++ ∆ +− =
⎪
⎪
⎩
∑
∑
∑
(2)

Thus we obtain the three points’ predictive model described above. Note that the subscript M of y
n,M
(n =
1 ,2 ,3) symbolizes change with time of the control value ∆u(k) (M<N), which means that to calculate
∆u(k), we need to know ∆u(k-i),(i=1,2,…M).
International Journal of Energy and Environment (IJEE), Volume 4, Issue 3, 2013, pp.415-426
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2013 International Energy & Environment Foundation. All rights reserved.

419
3.2 Rolling optimization
The DMC controller has the ability to adapt if we use certain optimal criterion to calculate the control
value. Our goal is to make the predictive output value y
n,M
(k+i) (i =1,2,…N, n = 1,2,3) track the expected
orbit y
n,r
(k+i) (i =1,2,…N, n = 1,2,3). To ensure that ∆u(k+i-1) does not change significantly, we employ
the following quadratic optimization objective function:

() () () ()
3
2
2
,, ,
11 1
mi n mi n ,
NM
ni nr nM i
ni i
J k q y ki y ki ruki
== =
⎛⎞
⎛⎞
⎡⎤
=+−++∆+
⎜⎟
⎜⎟
⎣⎦
⎝⎠
⎝⎠
∑∑ ∑
(3)

where y
n,r
(k) is the expected output, y
n,M
(k) is the predictive output, and ∆u(k) is the increment of input.
We can rewrite the function in a vector form as follows:

()
(
)
123
222
2
1, 1, 2, 2, 3, 3,
min min () () () () () () () ,
rM r M rM
R
QQQ
Jk Y k Y k Y k Y k Y k Y k Uk=−+−+−+∆
(4)

where
[]
[]
,, , ,
,, , ,
,1 ,2 , 1 2
( ) ( 1) ( 2) ( ) ,
( ) ( 1) ( 2) ( ) ,
() () ( 1) ( 1) ,
, 1,2,3, .
T
nr nr nr nr
T
nM nM nM nM
T
nnn nN M
Yk yk yk ykN
Yk yk yk ykN
Uk uk uk uk M
Qqq q n Rrr r
⎡⎤
=+ + +
⎣⎦
⎡⎤
=+ + +
⎣⎦
∆=∆ ∆+ ∆+−
⎡⎤
===
⎣⎦

Q
n
is the error weight matrix and R is the control weight matrix.
From the formula in equation (2), we obtain:

,,0
,0 ,0 ,0 ,0
,1
,2 ,1
,,1
,,1
() () ()
() ( 1) ( 2) ( )
0000
000
, 1,2,3.
0
nM n n
T
nn n n
n
nn
n
nM n
nN nN M
YkYkAUk
Yk yk yk ykN
a
aa
An
aa
aa
−+
=+∆
⎡⎤
=+ + +
⎣⎦
⎡⎤
⎢⎥
⎢⎥
⎢⎥
==
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎣⎦
"
"
# #### #
"" "
# #### #
""""
(5)

Then we obtain another form of J(k):

()
3
,, ,,
1
3
,,0 ,,0
1
() () () () () ()
() () () () () () () ().
T
T
jr jM j jr jM
j
T
T
jr j j j jr j j
j
Jk Y k Y k Q Y k Y k Uk RUk
YkYk AUk QYkYk AUk UkRUk
=
=
⎡⎤⎡⎤
=− −+∆∆
⎣⎦⎣⎦
⎡⎤⎡⎤
=−−∆ −−∆+∆∆
⎣⎦⎣⎦
∑
∑
(6)

We employ the formula in equation (6) to get the optimal increment ∆U*(k) by the following operation:

( )
(
)
3
**
,,0 ,,0
1
() () () () 2 () 2 () 0.
()
T
TT
jr j j j j j jr j j j j
j
Jk
YkYk QAAQYkYk AQAUk RUk
Uk
=
∂
⎡⎤⎡⎤
=− − + − + ∆ + ∆ =
⎣⎦⎣⎦
∂∆
∑
(7)

We obtain:

()
1
33
*
,,0
11
() () ().
TT
jjj jj jr j
jj
U k AQA R AQ Y k Y k
−
==
⎡⎤
⎡⎤
∆= + −
⎢⎥
⎣⎦
⎣⎦
∑∑
(8)

International Journal of Energy and Environment (IJEE), Volume 4, Issue 3, 2013, pp.415-426
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2013 International Energy & Environment Foundation. All rights reserved.

420
The formula in equation (8) can be used for calculating the input increments ∆U
*
(k) for all M steps.
However, we only need to use the first ∆u(k) to get the next step’s inputs and thus we get:

()
1
33
,,0
11
() () (),
TT T
jjj jj jr j
jj
uk C AQA R AQ Y k Y k
−
==
⎡⎤
⎡⎤
∆= + −
⎢⎥
⎣⎦
⎣⎦
∑∑
(9)

where C=[1, 0, …, 0]. In equation (9), C, A, Q
j
, and R can be determined a-priori by off-line calculations.
Thus if we can keep ∆u(k) updated at all instances, then the system can be very well controlled.
There are many sources, which influence the output of the system. Thus, if the output y
n
(k+1) (n = 1,2,3)
is not corrected, the error will be larger, and it will not assure that actual output gets close or tracks the
expected value. The dynamic correction is used to correct the error. Then we can get the error:

,
(1) (1) (1), 1,2,3,
nnnM
ek yk y k n+= +− + =
(10)

where y
n
(k+1) is the output and y
n,M
(k+1) is the predictive output. We have the predictive value:

()
,,
y1 (1)(1), 1,2,3.
nc nM n n
kykhekn+= ++ + =
(11)

In equation (11), h
n
is correction parameter. Thus, the predictive value after correction is as follows:
()
,0 ,
y ( 1), 1, , 1, 1, 2,3.
nnc
ki y ki i N n+= ++ = − =

Then, we get:

()
1
33
,0
11
() () ( 1) (),
TT T
jjj jj jr jj
jj
uk C AQA R AQ Y k AUk he k
−
==
⎡⎤
⎡⎤
∆= + − −−
⎢⎥
⎣⎦
⎣⎦
∑∑
(12)

where
()
[ ]
1 ( 1) ( 2) ( 1) ,U k uk N uk N uk−= −+ −+ −

and Y
n,r
(k) and e
n
(k) are defined by eqns. (4) and (10) respectively.

Remark 2: The DMC method introduced in this article is different from traditional one. First, traditional
DMC method can only be used in SISO systems; however our method can be used in SIMO systems.
Second, there are still errors when the systems are stable in our method, while in the traditional method
one can get a zero-error result. That is because we only use one input to control three outputs. Third, for
SIMO system, the three outputs’ performances must be similar otherwise the errors will be too large.

4. Results
In this section, we first employ a model example to discuss the effectiveness of the new DMC controller.
This simple example is used to illustrate the method’s limitations. Next we show the effectiveness of the
DMC controller in controlling the temperature in the room using the displacement ventilation HVAC
system.

4.1 Model example
We employ two groups of systems:

( )
()
()
35 1
1
12
35 1
2
12
35 1
3
12
0.035 0.0307
()
1 1.638 0.6703
0.135 0.0307
1: ( )
1 1.7 0.8
0.1 0.05
()
1 1.5 0.6
zz
Gz
zz
zz
Group G z
zz
zz
Gz
zz
−−
−−
−−
−−
−−
−−
⎧
−−
⎪
=
−+
⎪
⎪
−−
⎪
=
⎨
−+
⎪
⎪
−−
⎪
=
−+
⎪
⎩
(13)

International Journal of Energy and Environment (IJEE), Volume 4, Issue 3, 2013, pp.415-426
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2013 International Energy & Environment Foundation. All rights reserved.

421
( )
()
()
35 1
1
12
35 1
2
12
35 1
3
12
0.035 0.0307
()
1 1.638 0.6703
0.135 0.0307
2: ( )
11.7 0.8
0.1 0.25
()
11.6 0.7
zz
Gz
zz
zz
Group G z
zz
zz
Gz
zz
−−
−−
−−
−−
−−
−−
⎧
−−
⎪
=
−+
⎪
⎪
−−
⎪
=
⎨
−+
⎪
⎪
−−
⎪
=
−+
⎪
⎩
(14)

Now, if we give the three systems in group 1 a step input, then we can get the open-loop systems’
performance shown in Figure 4. We obtain

()
1
3
1
0.0714 -0.1380 0.0669
-0.1380 0.2789 -0.1418 .
0.0669 -0.1418 0.0756
T
jjj
j
AQA R
−
=
⎡ ⎤
⎡⎤
⎢ ⎥
+=
⎢⎥
⎢ ⎥
⎣⎦
⎢ ⎥
⎣ ⎦
∑
(15)

After using the DMC method, we can obtain the closed-loop systems’ performance shown in Figure 5.
When all of the systems are stable, there are still errors compared to the input shown in Figure 5, since it
is one-input-three-output system. If we only consider one of the system’s performances, for example G
3
(z) and employ the traditional DMC method, we can get the systems’ performance in group 2. From
Figure 6, we know that the error of G
1
(z) is very large. Thus, if we only consider one output to employ
DMC method, other outputs’ errors may unacceptable. At the same time, if the three systems are quite
similar, the controller design method is very effective. To show this, we use the DMC method for the
systems in group 2, and we get an open-loop systems’ step input performance shown in Figure 6. Using
our method, we obtain

()
1
3
1
0.0155 -0.0303 0.0149
-0.0303 0.0611 -0.0310 .
0.0149 -0.0310 0.0162
T
jjj
j
AQA R
−
=
⎡ ⎤
⎡⎤
⎢ ⎥
+=
⎢⎥
⎢ ⎥
⎣⎦
⎢ ⎥
⎣ ⎦
∑
(16)

From Figure 7, we know that G
3
(z)’s performance is different from the other two systems. As Figure 8
shows, the closed-loop systems’ errors are still very large with our method. One way to solve this
problem is to introduce some other inputs. Only one input cannot satisfy all the requirement of these
three outputs.

0 20 40 60 80 100 120 140 160 180 200
-2.5
-2
-1.5
-1
-0.5
0
Simulation Unit
Step


r
system1
system2
system3


Figure 4. The three systems’ open-loop step responses in group 1
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ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2013 International Energy & Environment Foundation. All rights reserved.

422
0 20 40 60 80 100 120 140 160 180 200
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
Output of the process
step
r(t),y
1
(t),y
2
(t),y
3
(t)


Step input r(t)
Performance of G
1
(z)
Performance of G
2
(z)
Performance of G
3
(z)


Figure 5. The three systems’ closed-loop step responses considering three points’ performance in group 1


0 50 100 150 200
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
Output of the process
step
r(t),y
1
(t),y
2
(t),y
3
(t)


Step input r(t)
Performance of G
1
(z)
Performance of G
2
(z)
Performance of G
3
(z)


Figure 6. The three systems’ closed-loop step responses only considering G
3
(z)’s performance in group 1


0 20 40 60 80 100 120 140 160 180 200
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
Simulation Unit
Step


r
Responce of G
1
(z)
Responce of G
2
(z)
Responce of G
3
(z)


Figure 7. The three systems’ open-loop step responses in group 2

International Journal of Energy and Environment (IJEE), Volume 4, Issue 3, 2013, pp.415-426
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2013 International Energy & Environment Foundation. All rights reserved.

423
0 20 40 60 80 100 120 140 160 180 200
-1.5
-1
-0.5
0
Output of the process
step
r(t),y
1
(t),y
2
(t),y
3
(t)


Step input r(t)
Performance of G
1
(z)
Performance of G
2
(z)
Performance of G
3
(z)


Figure 8. The three systems’ closed-loop step responses considering three points’ performance in group 2


4.2 HVAC application
We employ the DMC controller to control the temperature in the room deploying displacement
ventilation HVAC system described before in section 2 titled “Flow Simulation in a Room.” Using the
DMC controller by UDF in FLUENT, we control the temperature in the room between 295.8K and
297.14K as shown in Figure 9; the error due to DMC controller is smaller than from the controller
employed in Reference [1]. From Figure 10, it can be noted that the DMC controller saves more energy
than the controller employed in Reference [1], since the DMC controller requires less input to get better
performance. From Figure 11, it can be seen further that the DMC controller can save more energy.
The advantages of the DMC controller are as follows. First, it is an optimal controller using the minimal
input to get better performance since the quadratic optimization objective function considers the input
information. Second, it is a self-adapting controller which changes as the input changes. There are two
disadvantages of the DMC controller. First, the DMC controller is a local controller which can only
guarantee the stability of the system in a local area. Second, the DMC controller is a model-based
controller whose model is linear. But our system is a highly nonlinear system and therefore there are
errors if a linear model is employed to describe the nonlinear system. Furthermore, as the disturbance
becomes bigger and bigger, we need to find another model and design a new controller. In the DMC
controller, we set N = 80, M = 5, and Q
j
and R are defined as the identity matrices with proper dimension.
Then we obtain Γ
n
defined in equation (1). Because Γ
n
is a long vector, we don’t give all the values here.
Now, it is straightforward to get the following matrix:

()
1
3
1
0.461 0.519 0.201 0.068 0.192
0.413 0.848 0.324 0.176 0.064
.
0.206 0.206 0.943 0.325 0.207
0.034 0.083 0.209 0.846 0.520
0.191 0.039 0.210 0.412 0.471
T
jjj
j
AQA R
−
=
−−
⎡ ⎤
⎢ ⎥
−−−
⎢ ⎥
⎡⎤
⎢ ⎥
+=
−− −−
⎢⎥
⎢ ⎥
⎣⎦
−−− −
⎢ ⎥
⎢ ⎥
−−−
⎣ ⎦
∑
(17)

We want the temperature to stay at 296.6K, and therefore we set Y
r
(k) = [296.6, 296.6, … 296.6]. Then
we can obtain every control value in real time. It should be noted that the three temperature systems in
room are similar, since temperatures of all three points are obtained from the same flow model for the
room. Figure 12 shows the temperature distribution after control. In the systems, we assume that all six
inlets’ mass flow rates are controlled by the same u(k) in the model. If we want to further promote the
control performance, the six inlets’ mass flow rates need to be controlled separately.

International Journal of Energy and Environment (IJEE), Volume 4, Issue 3, 2013, pp.415-426
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2013 International Energy & Environment Foundation. All rights reserved.

424
0 200 400 600 800 1000 1200
292.5
293
293.5
294
294.5
295
295.5
296
296.5
297
297.5
298
Time [60s/step]
Temperature [K.]


Point 1 under DMC control
Point 2 under DMC control
Point 3 under DMC control
Point 1 under Ref.[1]
Point 2 under Ref.[1]
Point 3 under Ref.[1]


Figure 9. The comparison of temperature control results from Ref. [1] and the present fuzzy controller


0 100 200 300 400 500 600 700 800 900 1000
0
200
400
600
800
1000
1200
1400
1600
Time step [60s/step]
Total air in [kg]


DMC controller
Controller in Ref.(Lee et al., 2010)


Figure 10. The cost of the cold air or energy saving using two different controllers


0 100 200 300 400 500 600 700 800 900 1000
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07


DMC Controller input
Controller input in Ref.[1](Lee et al., 2010)

Figure 11. The control inputs of two different controllers

International Journal of Energy and Environment (IJEE), Volume 4, Issue 3, 2013, pp.415-426
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2013 International Energy & Environment Foundation. All rights reserved.

425


Figure 12. The temperature of room after control

5. Conclusion
This paper has developed a DMC controller for controlling the temperature in a room deploying a
displacement ventilation HVAC system without heater. It is a nonlinear system with large disturbance,
which has delay in the control variable and in the environment disturbance. By analyzing the temperature
at three points in the room and a proper estimation of the environmental disturbance and the heat
exchange delay, we have designed a DMC controller to control the displacement ventilation HVAC
system. The control results using the DMC controller are analyzed and compared with those obtained
from another controller. The DMC controller performs quite well and results in greater energy savings
compared to those reported in Reference [1].

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