Gradient auto tuned takagi–sugeno fuzzy forward control of a hvac system using predicted mean vote index

Energy and Buildings 49 (2012) 254–267
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Energy and Buildings
journal homepage: www.elsevier.com/locate/enbuild
Gradient auto-tuned Takagi–Sugeno Fuzzy Forward control of a HVAC system
using predicted mean vote index
Raad Z. Homoda,1
, Khairul Salleh Mohamed Saharia,1
, Haider A.F. Almuribb,∗
, Farrukh Hafiz Nagia,1
a
Department of Mechanical Engineering, Universiti of Tenaga Nasional, Jalan IKRAM-UNITEN, 43000 Kajang, Malaysia
b
Department of Electrical & Electronic Engineering, The University of Nottingham Malaysia Campus, Jalan Broga, 43500 Semenyih, Selangor Darul Ehsan, Malaysia
a r t i c l e i n f o
Article history:
Received 10 November 2011
Received in revised form 17 January 2012
Accepted 10 February 2012
Keywords:
HVAC system
Building energy control
TSFF control
PMV signal reference
TS Fuzzy identification
a b s t r a c t
Controllers of HVAC systems are expected to be able to manipulate the inherent nonlinear characteristics
of these large scale systems that also have pure lag times, big thermal inertia, uncertain disturbance factors
and constraints. In addition, indoor thermal comfort is affected by both temperature and humidity, which
are coupled properties. To control these coupled characteristics and tackle nonlinearities effectively, this
paper proposes an online tuned Takagi–Sugeno Fuzzy Forward (TSFF) control strategy. The TS model is
first trained offline using Gauss–Newton Method for Nonlinear Regression (GNMNR) algorithm with data
collected from both building and HVAC system equipments. The model is then tuned online using the
gradient algorithm to enhance the stability of the overall system and reject disturbances and uncertainty
effects. As control objective, predicted mean vote (PMV) is adopted to avoid temperature–humidity cou-
pling, thermal sensitivity and to save energy at the same time. The proposed TSFF control method is tested
in simulation taking into account practical variations such as thermal parameters of buildings, weather
conditions and other indoor residential loads. For comparison purposes, normal Takagi–Sugeno fuzzy and
hybrid PID Cascade control schemes were also tested. The results demonstrated superior performance,
adaptation and robustness of the proposed TSFF control strategy.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
The increase in energy consumption and demand in the last few
decades encourages the investigation of new methods to reduce
energy losses. The HVAC systems contribute a significant share of
energy consumed in buildings. So it is advisable to find methods to
reduce the rise of energy consumption in HVAC systems. But energy
and indoor thermal comfort in buildings hold a contradiction; con-
trol devices are expected to balance between energy saving and
achieving occupant satisfaction at the same time [1,2].
For controlling these devices, PID controllers are widely used
because of their simple structure and their relative effectiveness,
which can be easily understood and executed by practical imple-
mentations [3]. However, PID controllers are reliable only if the
parameters of the system under consideration do not vary that
much. On the other hand, variations in the operating condition of
the HVAC system will cause change in the parameters of the sys-
tem. These variations can be due to many factors such as water’s
chilled temperature, weather and occupancy level, which change
∗ Corresponding author. Tel.: +60 3 8924 8613; fax: +60 3 8924 8001.
E-mail addresses: khairuls@uniten.edu.my (K.S.M. Sahari),
haider.abbas@nottingham.edu.my (H.A.F. Almurib).
1
Tel.: +60 3 8921 2020; fax: +60 3 8921 2116.
from day to night. In short, the system is time variable and highly
nonlinear. For these reasons, even for a single HVAC system, the use
of a constant set of PID parameters will not give best results [4,5].
To obtain good PID control performance, the PID parameters should
be tuned continuously, which is time-consuming and dependent on
the experience of the one who adjusts them.
Furthermore, despite the non-stopping continuous research on
improving PID algorithms, requirements for high product quality,
subsystem unification and energy integration have resulted in non-
linearity and pure lag time for most of modern HVAC systems.
These main characteristics have rendered many PID tuning tech-
niques insufficient for dealing with these modern HVAC systems,
which are categorized as Multi-Input Multi-Output (MIMO) pro-
cesses [6,7]. Furthermore, the tuning of PID parameters in MIMO
plants is difficult to obtain because tuning the parameters of one
loop affects the performance of other loops, occasionally destabi-
lizing the entire system. Therefore, most studies in the field of the
HVAC system control tends to belong to artificial intelligence; neu-
ral network (NN) [8,9], fuzzy control [10,11], adaptive fuzzy neural
network [12–14], etc.
Fuzzy logic control is used in HVAC systems for its capability in
dealing with non-linearity as well as its capability to handle MIMO
plants. Moreover, in most cases, fuzzy logic controllers are used
because they are characterized by their flexibility and intuitive use
[15]. Two types of fuzzy inference system (FIS) models, Mamdani
0378-7788/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
doi:10.1016/j.enbuild.2012.02.013

R.Z. Homod et al. / Energy and Buildings 49 (2012) 254–267 255
FIS and Sugeno FIS, are widely used in various applications [16].
The differences between these two FIS models befall in the conse-
quents of their fuzzy rules; and differing in their aggregation and
defuzzification procedures. Researchers found that Sugeno FIS runs
faster, is more dynamic to input changes and is more economical in
the number of input fuzzy sets compared to Mamdnai FIS. It is also
observed that Sugeno FIS is more accurate since the results that
were generated were closer to what was expected [17–19]. Jassbi
et al. [20] concluded that Sugeno FIS performs better than Mam-
dani FIS with respect to noisy input data. Furthermore, Sugeno FIS
is more responsive and that is due to the fact that when the noise
becomes too high (i.e. when the input data has drastically changed),
Sugeno FIS reacts more strongly and its response is more realistic.
In recent years, the learning methods based on using fuzzy control
emerged as a vital tool in application to control nonlinear systems,
including HVAC systems. For large scale HVAC systems, iteration
tuning makes better a control system and gets minimum cost on a
system level [21].
A Takagi–Sugeno Fuzzy Forward (TSFF) controller is a forward
type controller. The main benefits of implementing such controllers
are to speed up system response and reduce any overshot [22].
These controllers can be made more robust by auto-tuning them
online to deal with any change of plant parameters, disturbances
and heating/cooling loads. The speed of TSFF tuning is higher than
the conventional backpropagation type neural network [23].
Temperature and relative humidity are correlated variables, so
to control them at specific values is a complex task. One of the
proposed solutions is the addition of a reheating coil to overcome
this coupling relation. However, this increases the power con-
sumed to control the conditioning space. A better solution would
be the use of predicted mean vote (PMV) as a reference for the
HVAC system, which will result in several features and advantages;
first of all, it means that the thermal sensation of the conditioned
space is controlled directly compared to previous methods, such
as the widely used indoor temperature variable as reference signal
method, where the thermal sensation is controlled inefficiently.
Another direct advantage is the flexibility to control coupled vari-
ables like temperature and relative humidity without the need
to decouple them. In this way, the controller can easily track the
desired thermal sensation for the conditioned space by controlling
more controllable variables like the indoor air velocity and the flow
rate of the refresh air [24]. Moreover, these controlled variables can
be fitted (optimized) by the controller according to the amount of
impact on the reference output. Therefore, using the PMV index
as the target set value for the indoor conditioned space is a better
and more suitable choice than using temperature because the PMV
changes dynamically so as to suit the constantly changing indoor
environment, and this will be useful to HVAC control systems aimed
at both controlling thermal comfort and saving energy [25].
There are numerous mathematical relationships to represent
thermal comfort. Fanger [26] representation was accepted to be
the closest one to the real behavior of the indoor actual model, and
that is the reason why it is adopted in ASHRAE Standard 55-92 [27]
and ISO-7730 [28]. In this study, the HVAc system and PMV mod-
els are integrated to evaluate indoor thermal comfort situations.
The first model is an extensive and elaborate model of the building
and air handling unite (AHU). It is designed to represent the real
system by the consolidation of five subsystems (pre-cooling coil,
mixing air chamber, main cooling coil, building structure and con-
ditioned space) interacting with each other. The second model is a
fuzzy PMV model which is regarded as a white-box model. Then
a TSFF control system is designed. The construction of the TSFF
is based on two types of learning; offline and online. The offline
learning method is performed using the Gauss–Newton Method for
Nonlinear Regression (GNMNR) algorithm which has the capability
to express the knowledge acquired from input–output data in the
form of layers of parameters. The online tuning of the TSFF is accom-
plished using the gradient method to modulate the parameters of
layers obtained from the offline learning. Most of control research
in HVAC systems use PID control as a benchmark to compare with
their new controllers [29–38]. In our case, a conventional PID con-
trol method would exhibit disability to control the plant system
due to the fact that the MIMO model used in this study is a large
scale system model that has nonlinearity, pure lag time, big ther-
mal inertia, uncertain disturbance factors, and constraints in valves
and dampers. Therefore, this work considers a Takagi–Sugeno fuzzy
controller with fixed parameters and a hybrid PID-cascade con-
troller [29–33] as benchmarks.
In general, when comparing the performance of fuzzy logic
and PID with anti wind-up control systems, the fuzzy algorithm
achieves the imposed overshot and a settling time required speci-
fications [39]. Furthermore, the performance of the traditional PID
controller scheme is limited when applied to AHU processes due
to the coupled temperature and humidity properties of the system,
especially when significant load variations and disturbances occur
[40]. Therefore, it is the aim of this study to investigate techniques
to enhance the control performance of indoor thermal comfort in
comparison to the conventional hybrid PID-cascade and the nor-
mal Takagi–Sugeno fuzzy controllers. Tests were conducted taking
into account practical variations such as thermal parameters of
buildings, weather conditions and other indoor residential loads;
between the three different schemes (the proposed method and
the two conventional methods) to demonstrate the efficiency of
the proposed method.
2. Model
The frame work of this section is to describe the HVAC sys-
tem and PMV sensor models, which are the plant on which indoor
thermal sensation is to be controlled.
2.1. HVAC system model
The HVAC mode consists of enormous number of variables and
parameters that make modeling a difficult task. To reduce the com-
plexity of building the model, Homod et al. [41,42] divided the
HVAC system into five parts; pre-cooling coil, mixing air chamber,
main cooling coil, building structure (opaque surfaces structure,
slab floor structure and transparent fenestration surface at the
structure) and conditioned space. The consolidation of the five parts
together is discussed in the previous paper [41] to provide the
overall equation model as follows:
Tr (s)
ωr (s)
=
T1,1(s) T1,2(s) T1,3(s) T1,4(s) T1,5(s) · · · T1,10(s) T1,11(s) T1,12(s)
T2,1(s) T2,2(s) T2,3(s) T2,4(s) T2,5(s) · · · T2,10(s) T2,11(s) T2,12(s)
×
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
˙mw(s)
˙mmw(s)
˙mos(s)
˙mr (s)
To(s)
ωo(s)
f4
˙Qig,l
Aslab
fDR
k2
Tr (s)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(1)
where
T1,1(s) T1,2(s) · · · T1,12(s) and T2,1(s) T2,2(s) · · · T2,12(s)
represent the input factors. For a detailed description of all

256 R.Z. Homod et al. / Energy and Buildings 49 (2012) 254–267
Fig. 1. HVAC system model block diagram.
parameters and factors of Eq. (1) and the steps to obtain it,
the reader is encouraged to refer to the previous work in
Homod et al. [41].
To obtain a good overall picture of the relationships among
inputs and outputs of the developed model, the block diagram of
Fig. 1 represent the transfer functions variables of Eq. (1).
Eq. (1) and Fig. 1 imply that the system has twelve input vari-
ables and two outputs;
The input variables are:
1. ˙mw(s) = flow rate of chilled water supply to pre-cooling coil,
2. ˙mmw(s) = flow flow rate of chilled water supply to main cooling
coil,
3. ˙mos(s) = flow rate of return air to conditioned space,
4. ˙mr(s) = flow rate of outside air to conditioned space,
5. To(s) = perturbations in outside temperature,
6. k2 = perturbations due to thermal resistance of building enve-
lope,
7. f4 = perturbations of internal sensible heat gain,
8. Aslab = area of slab floors,
9. fDR = location factor,
10. ωo(s) = perturbations in outside air humidity ratio,
11. ˙Qig,l = perturbations of internal latent heat gain, and
12. Tr(s) = conditioned space temperature.
The output variables are:
1. Tr(s) = room temperature or conditioned space temperature, and
2. ωr(s) = room humidity ratio or conditioned space humidity ratio.
Eq. (1) and Fig. 1 also indicate that the model is built on the basis
of variable air volume (VAV) and variable water volume (VWV). So
the HVAC system model is wholly analyzed with large scale system
theory based on “decomposition and coordination” strategy [38].
2.2. PMV sensor model
In the last two decades, temperature and relative humidity are
preferred to be a reference instead of temperature only sensing
commonly used in the earlier HVAC systems. However, although
it is one of the factors involved in affecting human’s comfort,
temperature does not solely represent human’s thermal comfort.
Furthermore, temperature and relative humidity are coupled;
it is difficult to control them separately for a certain desired

Table 1
Input parameters range and increments.
Parameters Symbols Parameter range Steps Units
Air temperature (ta) x1 2–45 0.25 ◦
C
Relative humidity (RH) x2 10–90 0.5 %
Radiant temperature (tr) x3 10–53 0.25 ◦
C
Relative air velocity (var) x4 0–1.0 0.0055 m/s
Clothing insulation (Icl) x5 0–0.31 0.0017 m2 ◦
C/W (1 clo = 0.155 m2 ◦
C/W)
Metabolic rate (M) x6 46–235 1.1 W/m2
(1 met = 58.2 W/m2
)
thermal sensation. On the contrary, when the PMV is used as a
reference, human’s thermal comfort in the conditioned space can
be controlled accurately and efficiently by optimizing between
temperature and relative humidity simultaneously, i.e. there is no
specific temperature or humidity ratio that act as a control refer-
ence. Furthermore, the PMV or TS model exploits the fresh air flow
rate and the indoor air velocity and its effect on thermal comfort
levels. In addition, using temperature and relative humidity as
separate reference signals is unsuitable for energy saving due to
their static values for a daily period. Because in a humid climate or
a high indoor latent gain, latent cooling loads are unavoidable and
therefore the use of reheating coil to overcome the coupling which
exists between the temperature and relative humidity. Therefore,
the PMV model is used for energy savings also by varying air
velocity that can result in an energy savings of around 30% while
maintaining the same comfort level [43].
The PMV sensor model composed based on the Fanger’s empir-
ical model. The Fanger’s model has been used widely to predict
indoor thermal comfort. Furthermore the Fanger’s PMV model has
been a general standard since the 1980s [27,28]. The range value of
PMV is from −3 to +3, where a cold sensation is a negative value,
the comfort situation is close to zero and hot sensation is a pos-
itive value. The PMV can be estimated by empirical equation as
presented in [26] by:
PMV = (0.303e−0.036M
+ 0.028)[(M − W) − 3.05 × 10−3
{5733
− 6.99(M − W) − pa} − 0.42{(M − W) − 58.15}
− 1.7 × 10−5
M(5867 − pa) − 0.0014M{34 − tr} − 3.96 × 10−8
+ fcl{(tcl + 273)4
− (trr + 273)4
} − fclhc(tcl − tr)] (2)
tcl, pa, hc and fcl are given by equations:
tcl = 35.7 − 0.028(M − W) − 0.155Icl[3.96 × 10−8
fcl{(tcl + 273)4
− (trr + 273)4
} + fclhc(tcl − tr)]
pa =
psRH
100
and
ps =
C1
T
+ C2 + C3T + C4T2
+ C5T3
+ C6T4
+ C7 ln(T)
hc =
2.38(tcl − tr)0.25
for 2.38(tcl − tr)0.25
> 12.1
√
va
12.1
√
va for 2.38(tcl − tr)0.25
< 12.1
√
va
hc =
1.00 + 0.2Icl for Icl < 0.5 clo
1.05 + 0.1Icl for Icl > 0.5 clo
where PMV is the predict mean vote, M is the metabolism
(W/m2), W is the external work (W/m2), Icl is the thermal resis-
tance of clothing (m2 K/W), fcl is the ratio of the surface area of the
clothed body to the surface area of the nude body, tr is the room
temperature (◦C), trr is the room mean radiant temperature (◦C),
va is the relative air velocity (m/s), Pa is the water vapor pressure
(Pa), Ps is saturated vapor pressure at specific temperature (Pa), RH
is the relative humidity in percentage, C1, C2, . . ., C7 are empirical
constant that can be found from [44], T is the absolute dry bulb tem-
perature in kelvins (K), hc is the convective heat transfer coefficient
(W/(m2 K)) and tcl is the surface temperature of clothing (◦C).
The solution of Fanger’s model Eq. (2) requires a lot of compu-
tational effort and time. For these reasons, the Fanger’s model is
difficult to use in real time application or to represent it on mod-
ern computers. One of the ways to apply such nonlinear models in
real-time is to use a nonlinear system identification method such as
Fuzzy Logic identification. To clarify the model identification pro-
cess, we follow the procedure presented by Homod et al. [24], which
can be summarized by the following steps:
1. Prepare the training data set for the input–output PMV sen-
sor model from Fanger’s model with a feasible range for input
parameters as shown in Table 1.
2. Break up the output into clusters, and then represent each cluster
by Takagi–Sugeno fuzzy rules. The weights and clusters param-
eters are obtained from these rules.
3. The parameters and weight layers, obtained from the training
data set and optimized by GNMNR can be structured as a lay-
ered framework. Fig. 2 shows the architecture of the PMV sensor
model including input space, parameters memory space, weight
memory space and output space.
Fig. 2 can help to show the identification of any package of
parameter layers by knowing the set of inputs x6, x5 and x4. Then,
x3 will specify the parameters layer, after which the parameters
can be obtaining by inputs x2 and x1. Then from these parameters
and the weights of clusters, one can attain the output. A detailed
explanation will follow in Section 3.2.
3. TSFF control design
3.1. Control structure
The aim and main task of the controller is to react to the dynamic
loads on the system and regain stability without the necessity to
maintain the same temperature but rather achieving it through the
introduction of new operation conditions (a combination of new
temperature, humidity, air velocity, etc.) as long as thermal com-
fort is achieved. The steady-state values can be measured and fed
back to the iterative tuning algorithm, enabling the tuning func-
tion to run after the transient response lapses away [38,45]. The
purpose of the TS model is to shorten the tuning time, while the
online tuning is to alleviate disturbance influences due to contin-
ually changing stable states. When there are missing parameters
from the TS model, the settling time will take longer because the
tuning process takes longer time. Therefore, the control structure
consists of two parts; TS model and tuning algorithm. Time variable
is the input of the TS model since the cooling/heating load is a peri-
odical process with time, while the inputs of the tuning algorithm
are the PMV index and the error as shown in Fig. 3.

Fig. 2. The PMV sensor model structure.
3.2. TS model
This section describes the TS model and its structure, identifica-
tion, and tuning. The numerical example model will be described
later in Section 4.
3.2.1. General idea
From the data set it is obvious that the model has one input
and five outputs; supplied chilled water to pre-cooling coil and
main cooling coil, return and fresh supplied air to conditioned space
and indoor air velocity. These outputs can be clustered into seven
groups within a time frame of 24 h. Each of these clusters for each
output can be represented by Takagi–Sugeno fuzzy rules, where
each rule of the cluster can be formulated as follows:
i : if x1 is Ak(x1)
i
and x2 is Ak(x2)
i
· · · and xm is Ak(xm)
i
then Yj(X) = ωiyi, yi = f (x; ai, bi) (3)
where Ai is the set of linguistic terms defined for an antecedent
variable x, m is the number of input variables, i is a rule number
subscript, ai and bi are the Takagi–Sugeno parameters function, ωi is
the basis functions, X = [x1, x2, . . ., xm]T is the input variables vector,
j is the cluster number subscript, f (x; ai bi) is a nonlinear function of
the TS parameters and the independent variable x and a nonlinear
function of the parameters, and k(xi), . . ., k(xm) are linguistic values
and are generally descriptive terms such as negative big or positive
large and so on.
The basis functions ωi can be described by the degrees of
antecedents rule fulfillment and the outputs of the model Yj(X) are
the consequents. The basis and premise membership functions can
be represented with relation to clustering data set as shown in Fig. 4.
The outputs Yj(X) must fit those of the data set. This can be
achieved by modulating the nonlinear equation yi. The modulation
can be attained by tuning the parameters ai and bi. Manual tun-
ing is time consuming and needs patience to balance between the
parameters which are nonlinearly related. Thus, we prefer using an
Fig. 3. Control structure of TSFF.

Fig. 4. Basis and premise membership functions with relation to main cooling coil clustering data.
algorithm to optimize the factors of the model outputs. This will
be done based on the residual error between the model and the
calculated data set. The GNMNR tuning method will be discussed
shortly in Section 3.2.3.
3.2.2. Identiﬁcation of TS model
As described in Section 3.2.1, the number of rules or membership
functions is related to each cluster. The overall model output can
be represented by aggregating clusters’ outputs as follows:
i : if x1 is Ak(x1)
i
and x2 is Ak(x2)
i
· · · and
xm is Ak(xm)
i
then Y(X) =
j
Yi(X) (4)
The defuzziﬁcation for the singleton model can be used as center
of gravity (COG) in the fuzzy-mean method:
Y(X) =
N
i=1
ˇiyi
N
i=1
ˇi
(5)
where N is a set of linguistic terms, ˇi is the consequent upon all
the rules and can be expressed as follows:
ˇi = Ak(x1)
i
(x1) ∧ A
k(x2)
i
(x2) ∧ · · · ∧ Ak(xm)
i
(xm), 1 ≤ i ≤ N (6)
Based on the basis function’s expansion [46], the singleton fuzzy
model belonging to a general class of universal model output can
be obtained as follows:
Y(X) =
N
i=1
ˇiyi
N
i=1
ˇi
=
N
i=1
ωiyi (7)
where ωi =
ˇi
N
i=1
ˇi
.when yi is imposed as a nonlinear equation the
above output model can be presented as follows:
Y(X) =
N
i=1
ωiai(1 − e−bix
) (8)
From Eq. (8), the consequent parameters can be obtained by map-
ping from the antecedent space to consequent space. The obtained
parameters of consequent space are organized as layers in memory
space. The parameters in these layers are functions to the input of
the model (time).
The TS model can be structured as a layers and weights frame-
work as shows Fig. 5. The layers’ parameters are obtained from
training data set and optimized by GNMNR method, whereas
the weights are represented by basis function. The architecture
includes input space, parameters memory space, weight memory
Fig. 5. The TS model structure.

space and output space. It is obvious that the model is a single-input
(time) multi-output signal (pre-cooling coil valve, main cooling
coil valve, returned air damper, fresh air damper and fan air seed)
model.
3.2.3. Offline learning of TS model
The processed data sets are clustered into seven hyper-
ellipsoidal clusters as shown in Fig. 4. The singleton TS model output
can be expressed as:
i : if x1 is Ak(x1)
i
and x2 is Ak(x2)
i
· · · and
xm is Ak(xm)
i
then Y(X) =
N
i=1
ωiai(1 − e−bix
) (9)
Consequents of i are piece-wise outputs to the parabola defined
by Y(X) in the respective cluster centers. The output model Y(X) is
tuned by optimizing ai and bi in Eq. (9) using the GNMNR algorithm,
which is an iteration based method.
This nonlinear regression algorithm is based on determining the
values of the parameters that minimize the sum of squares of the
residuals in an iteration fashion. The nonlinear output model must
fit the calculated output of the data set. To illustrate how this is
done, first the relation between the nonlinear equation and the
data can be expressed as
yi = f (xi; a, b) + ei (10)
where yi is a measured value of the dependent variable, f(xi ; a, b)
is the equation that is a function of the independent variable xi
and a nonlinear function of the parameters a and b, and ei is a ran-
dom error. The nonlinear model can be expanded in a Taylor series
around the parameter values and curtailed after the first derivative
as follows:
f (xi)j+1 = f (xi)j +
∂f (xi)j
∂a
a +
∂f (xi)j
∂b
b (11)
where j is the initial guess, j + 1 is the prediction, a = aj+1 − aj and
b = bj+1 − bj.
Eq. (11) can be substracted from Eq. (10) to yield
yi − f (xi)j =
∂f (xi)j
∂a
a +
∂f (xi)j
∂b
b + ei (12)
or can be expressed in matrix notation as
{D} = [Zj]{ A} + {E} (13)
where [Zj] is the matrix of partial derivatives of the function eval-
uated at the initial guess j, vector {D} contains the differences
between the measurements and the function values, vector { A}
contains parameters a and b and {E} is a random error vector.
Applying linear least-squares theory to Eq. (13) results in the
following normal equation:
A =
1
[Zj]T
[Zj]
{[Zj]T
D} (14)
Thus, the approach consists of solving Eq. (14) for { A}, which
can be employed using the GNMNR method to compute { A} by
iteration to correctly approximate the parameters [47].
3.3. Online tuning parameters
The optimization algorithm is to seek for the vector V which
minimizes an objective function F(V), where V is a p dimensional
vector of the tuning parameters. Gradient method has been devel-
oped to optimize the continuous output functions, which depend
on the slope information of the output error. The slope or gradient
of a function is used to guide the direction of maximum change
in vector parameters, known as hill climbing search. The vector
parameters’ changes are proportional to the derivatives of the plant
output error with respect to the set point. This is done to minimize
the output error.
In other words, first we need to find the direction of the steepest
descent, where the direction is the gradient of the function. Once
this direction has been established, we need to find out the step
length.
In general the vector tuning rule is expressed using the following
formula:
Vi(k + 1) = Vi(k) + Si(k) ·
∂Y(t(k); ai, bi)
∂Vi(k)
(i = 1, . . . , p) (15)
where k is the number of tuning iterations, V is the parameters vec-
tor, S is the step length along the steepest ascent axis, ∂Y(t(k) ; ai,
bi) is the output of the plant which, can be related to the error.
Here, the error is related to the [ai bi]T vector which represents
the TSFF parameters. To minimize this error, one should calcu-
late the gradient of the error itself. To do so, we use the following
optimization of the parameters of the vector;
First, the gradient is calculated,
∇ek+1 =
∂
∂ai
Y(tk+1; ai, bi)
∂
∂bi
Y(tk+1; ai, bi)
T
(16)
Then, to obtain the parameters of the [ai bi]T vector, Eq. (16) is
iterated for minimum errors;
ai
bi
k+1
=
ai
bi
k
+
ai
bi
k
(17)
where
ai
bi
k
can be obtained using the gradient as in Eq. (18);
ai
bi
k
= Sk∇ek+1 (18)
while the error and the step length S can be calculated from the
plant diagram relationship as follows:
ek+1 = ek + e = mpv(tk+1) − S · P·k
=
N
i=1
ωif (tk+1; ai, bi) − S · P·k = Y(tk+1; ai, bi) − S · P·k (19)
where pmv(tk+1) is the plant model output at iteration k + 1, S · P · k
is the set point at iteration k, N is a set of linguistic terms, which is
equal to the number of clusters.
Eq. (19) is nonlinear but can be expanded in a Taylor series
around the parameter values and curtailed after the first derivative
as follows:
ek+1 = ek + e = Y(tk; ai, bi) +
∂
∂ai
Y(tk; ai, bi) ai
+
∂
∂bi
Y(tk; ai, bi) bi − S · P·k (20)
ek+1 = ek +
∂
∂ai
Y(tk; ai, bi) ai +
∂
∂bi
Y(tk; ai, bi) bi (21)
Now, to achieve best value of the step length, we substitute the [ ai
bi]T vector values of Eq. (18) into Eq. (21) and make the error equal
to zero as in the following steps:
ek+1 = ek + ∇
∂
∂ai
Y(tk; ai, bi) Sk +
∂
∂bi
Y(tk; ai, bi) Sk (22)
ek+1 = ek +
∂2
∂a2
Y(tk; ai, bi) Sk +
∂2
∂b2
Y(tk; ai, bi) Sk (23)

Fig. 6. Comparison of chilled water flow rate between TS model and calculated result with absolute error.
ek +
∂2
∂a2
Y(tk; ai, bi) Sk +
∂2
∂b2
Y(tk; ai, bi) Sk = 0 (24)
By substituting for the TS model outputs’ values and solving Eq.
(24) will yield;
Sk =
−ek
(∂2/∂a2)Y(tk; ai, bi) + (∂2/∂b2)Y(tk; ai, bi)
=
S · P·k − ωi(k)ai(k)(1 − e−bi(k)tk )
ωi(k)ai(k)e−bi(k)tk
=
ωi(k)ai(k) − S · P·k
− 1 (25)
Eq. (25) can also provide the initial value of Sk that can be
obtained from the parameters and weight layers of the model in
order to start online tuning to improve the values of the parameters.
The tuning process is done by optimizing the parameters of Eq. (17)
by iterative mode [46]. So Eq. (17) becomes as follows:
ai
bi
k+1
=
ai
bi
k
+
ωi(k)ai(k) − S · P·k
− 1 ∇ek+1 (26)
From Eq. (26), it is obvious that all obtained parameters are the
result of previous iteration except the gradient error, which is the
current value of gradient error. So the gradient error is used as a
feedback for the online tuning.
4. Simulation results and discussion
The data set measurement is related to both supply air and
chilled water flow rate for comfortable indoor conditions. This is
done by analyzing and evaluating the cooling load for the model.
The model is a typical one-story house that has a simple struc-
ture. The overall area is 248.6 m2 while the net area excluding the
garage is 195.3 m2. The gross windows and walls exposed area is
126.2 m2 while walls exterior area is 108.5 m2. The house volume
excluding the garage is 468.7 m3. The HVAC equipments include
pre-cooling coils, main cooling coils and mixing air chambers. The
mixing air chamber controls the air supply to conditioned space by
two dampers, one for returned air and the other for fresh air. Adding
the pre-cooling coils is to cater for humid climate environment and
to economically control the indoor relative humidity.
The cooling loads are calculated every 20 min for 24 h; the cal-
culation is based on the outdoor conditions variation. The dry bulb
temperature varies from 17 to 32 ◦C and humidity ratio varies from
0.01 to 0.01909 kg moisture per kg dry air. After obtaining cooling
loads the output model (supplied chilled water to pre-cooling coil
and main cooling coil, return and fresh supplied air to conditioned
space and indoor air velocity) can be calculated depending on the
desired indoor condition [39].
4.1. TS model validation
The TS model has one input (time) and five outputs, which are
chilled water flow rate for pre-cooling coil valve, chilled water flow
rate for main cooling coil valve, air flow rate for fresh air damper, air
flow rate for return air damper and fan relative air velocity. The per-
formance of the TS model for a flow rate of main cooling coil valve
is tested by comparing it with the calculated results. The results of
this comparison and absolute errors are shown in Fig. 6 where it can
be observed that the implementation of the GNMNR algorithm to
tune model parameters illustrate considerable performance. Here,
the maximum absolute error, mean square error and mean abso-
lute error between the calculated values and the values obtained
from the TS model were 0.0193, 0.0081 and 0.0065, respectively.
4.2. Nominal operation conditions
To validate the designed TSFF control system, we choose the
summer work conditions and compare the designed TSFF to the
normal Takagi–Sugeno fuzzy (fixed parameters) and hybrid PID-
cascade control. The reason for not comparing with classical PID
control schemes is that these controllers are limited by the use of
first order or second order plus time delay models to represent the
dynamics of the process [40]. However, as explained earlier, the
HVAC model is a sophisticated MIMO model that is from the 13th
order and therefore classical control strategies like a PID controller
would fail to control it effectively.
The normal Takagi–Sugeno fuzzy (fixed parameters) uses two
fuzzy input variables, error and change of error as inputs, and five
fuzzy output variables. Five antecedent linguistic variables (from
Big Negative to Big Positive) are used. The consequent member-
ship’s functions are a first-order polynomial type, i.e. a first-order
Sugeno fuzzy model. The resulting fuzzy interference system is
a singleton with vector [a b c]. For a fair comparison, the same
Gauss–Newton method is used to tune the fixed parameters of the
[a b c] vector where c is the shifted parameter and a and b are the
inputs parameters.
As for the hybrid PID-cascade control, the PID parameters are
tuned using the robust PID tuning method in [48]. The parameters
of the outer-loop PID controller are tuned to Kp = 50, Ki = 30 and
Kd = 3.5. Following the procedure presented by Homod et al. [30]
and Wang et al. [32], the internal loop controller (F) can be obtained
as follows:
Transfer function from input 1 to output:
8.843 × 10−0.007
S + 0.0002261
e−0.2×S

Fig. 7. Comparison of the control performances of the HVAC system process when TSFF, normal Sugeno and hybrid PID-cascade were used.
0.0943
S + 0.012261
e−3.7×S
−0.2675
where the inputs internal loop controller F are ts, ωs and var, respec-
tively, which are the outputs of air-handling unit in Fig. 3 and the
output is the rectification of the PMV.
The control performances of the TSFF algorithm, fixed parame-
ters fuzzy and the hybrid PID-cascade controller were applied to the
HVAC system and compared. The HVAC system was manipulated
by the controllers to track a PMV set point which changes from 0 to
0.5 during 24 h of the day as shown in Fig. 7. It can be seen that both
the fuzzy fixed parameters and the hybrid PID-cascade controllers
did not produce satisfactory results due to the indoor and outdoor
cooling load variations. This is clear from the fluctuations around
the set point and the sluggish response within the 24 h time frame.
These performances are contrary to the TSFF performance, where
the PMV is tightly controlled to its set-point as shown in Fig. 7.
To have a clearer assessment of the performance of the proposed
controller against the conventional controllers, the study chose the
following statistical indices:
• Maximum absolute error;
Max. AE = max |yi − yi| (27)
• Mean absolute error;
Mean AE =
1
N
N
i=1
|yi − yi| (28)
• Mean squared error;
Mean SE =
1
N − 1
N
i=1
|yi − yi|
2
(29)
• Coefficient of determination;
r2
=
N yi ¯yi − yi ¯yi
2
N yi
2 − yi
2
N ¯y2
i
− ¯yi
2
(30)
where yi is the objective of the controller, ¯yi is the controlled plant
output, and N is the number of testing samples.
The controlled plant outputs accuracy are compared by com-
puting mean square error, mean absolute error, maximum absolute
error and coefficient determination for three controllers as shown
in Table 2.
The indoor PMV fluctuations under the fixed parameters fuzzy
and the hybrid PID-cascade controllers are reflected on their indoor
temperature and relative humidity as shown in Fig. 8 and Table 2.
It can be seen that the TSFF is trying to keep the temperature up at
23 ◦C when the PMV is zero while the other controllers fluctuates
around 23 ◦C. The TSFF exhibits perfect manipulating of indoor rel-
ative humidity in spite of high outdoor relative humidity because
of the utilization of the pre-cooling coil for better manipulation.
The TSFF was trying to keep the indoor relative humidity at 50%
when the PMV is zero while the fixed parameters fuzzy controller
is affected by the indoor and outdoor humidity. The hybrid PID-
cascade controller exhibits fluctuations with a wide range around
60%. The chilled water valves, the air duct dampers and the air
velocity fans were the control actuators of the system. As indicated
by the last control parameter of Table 2 and as can be concluded
from Fig. 8, the fixed parameters fuzzy and the hybrid PID-cascade
control signals acted like a Bang–Bang controller. In other words,
the modulating valve continuously fluctuated (ON–OFF). This will
wear out the valve and shorten its life. On the other hand, the TSFF
output control signal worked smoothly to provide good control
performance.
4.3. Validating robustness and disturbance rejection.
To validate the robustness of the proposed TSFF controller,
the controlled process parameters are changed, and simulations
are then conducted. Here, the gain of the controlled process and
its time constant are both increased by 20%. As for disturbance

Table 2
Performance indices comparison results of TSFF, hybrid PID and fuzzy fixed for controlling indoor PMV in nominal state of operation.
Control parameter Performance index Control objective
Max. AE Mean SE Mean AE r2
Indoor PMV
Hybrid PID 0.8409 0.1979 0.2340 0.2772
PMV = Zero [27,28]Fuzzy fixed 0.6025 0.1220 0.1424 0.4548
Proposed TSFF 0.4957 0.0254 0.0495 0.8773
Indoor temperature
Hybrid PID 4.4393 1.0177 1.2992 0.8005 Indoor
temperature = 23 ◦
C
[27,28]
Fuzzy fixed 2.8785 0.5508 0.7100 0.9356
Proposed TSFF 2.6902 0.2971 0.6232 0.9892
Indoor relative
humidity
Hybrid PID 0.2048 0.1059 0.1369 0.5973
Indoor relative
humidity = 50% [27,28]
Fuzzy fixed 0.4174 0.0880 0.1212 0.0010
Proposed TSFF 0.1495 0.0614 0.0640 0.8679
Signal valve opening
position
Hybrid PID 0.5000 0.2542 0.285 0.5896 Signal valve position = 50% for
energy saving, highest coefficient
of performance (COP)
Fuzzy fixed 0.5000 0.3931 0.4601 0.1135
Proposed TSFF 0.4800 0.1992 0.2634 0.8834
Fig. 8. Comparison of the indoor temperature behavior when TSFF, normal Sugeno and hybrid PID-cascade were used.
rejection, pulses of 6 min duration each are injected every
175 min to the input of the process and the results are recorded in
Figs. 9 and 10 and analyzed in Table 3. As can be seen from Fig. 9, the
effect of the disturbance disappears quickly under TSFF control in
spite of the large applied signal. The other controllers demonstrate
bad response compared to the proposed TSFF scheme. For the
indoor PMV set point response for example, both show an increase
in fluctuation reflected on the indoor temperature as illustrated in
Table 3
Performance indices comparison results of TSFF, hybrid PID and fuzzy fixed for controlling indoor PMV under disturbance.
Indoor PMV
Hybrid PID 1.9381 0.2674 0.3739 0.2035
Proposed TSFF 1.0831 0.0492 0.1551 0.7558
Indoor temperature
C
[27,28]
Fuzzy fixed 7.1020 1.0061 1.4390 0.6839
Proposed TSFF 5.3884 0.4895 0.9973 0.9824
Indoor relative
humidity
Hybrid PID 0.2340 0.1061 0.1382 0.4146
Indoor relative
humidity = 50% [27,28]
Fuzzy fixed 0.5000 0.1396 0.1933 0.1026
Proposed TSFF 0.1740 0.0932 0.0998 0.5817
position
Fuzzy fixed 0.4371 0.3184 0.3396 0.3616
Proposed TSFF 0.3621 0.1846 0.2047 0.4668

Fig. 9. Comparison of the control performances of the HVAC system process for the robustness and disturbance rejection.
Fig. 10. The benefit of the proposed method is that the parameters
of the TSFF controller are dynamically modified and tuned to cope
with disturbances. The proposed controller reacts to these distur-
bances by controlling the valve opening position that reaches its
maximum if necessary to overcome the applied disturbance. The
controller then brings it back to normal conditions and that is the
result of the tuning process. In contrast, the fixed parameters fuzzy
controller does not respond to changes in the opening position,
except only when there is a significant change in the reference.
On the other hand, the hybrid PID-cascade controller increases the
amplitude of the control signal in order to become stable.
4.4. Sensitivity to noise and sensor deterioration
Control performance is directly related to the amount of sensor
deteriorations and noise in the control loop. To test the sensitivity
of the proposed control system algorithm to noise, the system is
subjected to a noisy environment by adding a continuous noise
signal to the controlled signal. The standard of noise measurement
is the noise-to-signal ratio (NSR), defined as NSR = mean (abs
(noise))/mean (abs (signal)) [49]. In this simulation test, a 10% NSR
is applied to the plant and the controllers are compared. The PMV
value is calculated by collecting all variables by the sensors. In
Fig. 10. Comparison of the indoor temperature behavior of the HVAC system process for the robustness and disturbance rejection.

Fig. 11. Comparison of the control performances of the HVAC system process due to applied noise and sensor deterioration.
addition, to verify the ability of the proposed controller to rectify
sensor performance deterioration, we changed the sensor gain to
0.8, i.e. an introduction of 20% sensor fault (sensor gain = 1 when
the sensor is in perfect condition). The implementation results are
recorded in Figs. 11 and 12 and analyzed in Table 4. Fig. 11 shows
the performance of the three controllers with applied noise and
sensor deterioration. It can be seen from the figure that the HVAC
system with TSFF control is much less sensitive to noise and sensor
deterioration than the other schemes. The performance of the TSFF
is affected in the first 2 h of the simulation. This is due to the large
difference between the TS model parameters and the final model
parameters that leads to inaccurate results in Eq. (26). However,
later, the TSFF controller produces a fine adjustment to fit the grad-
ual change of the space thermal conditions and maintain excellent
thermal comfort level, as evident in Fig. 11. The other controllers
exhibited deterioration in their performances and consequently
violated ASHRAE 55-92 [27] and ISO-7730 [28] standards for
indoor thermal comfort. These standards recommend that the
acceptable range of the comfort of thermal sensation is limited
between −0.5 ≤ PMV ≤ 0.5. This violation is evident by observing
the behavior of the three temperature curves of the simulation
results in Fig. 12. Table 4 illustrates the results of the three control
Fig. 12. Comparison between three temperature curves of the HVAC system process due to applied noise and sensor deterioration.

Table 4
Performance indices comparison results of TSFF, hybrid PID and fuzzy fixed for controlling indoor PMV under noise and sensor deterioration conditions.
Indoor PMV
Hybrid PID 1.7836 0.3704 0.4234 0.3654
Proposed TSFF 0.5256 0.0212 0.0452 0.9841
Indoor temperature
C
[27,28]
Fuzzy fixed 5.2146 1.2868 1.6144 0.1569
Proposed TSFF 2.2631 1.0008 1.3384 0.5879
Indoor relative
humidity
Hybrid PID 0.2492 0.1058 0.1397 0.4308
Indoor relative
humidity = 50% [27,28]
Fuzzy fixed 0.5000 0.2176 0.2717 0.0873
Proposed TSFF 0.1892 0.0945 0.1021 0.6270
position
Fuzzy fixed 0.4371 0.3147 0.3407 0.3201
Proposed TSFF 0.3621 0.2595 0.2763 0.7369
signals of the valve opening position for the main cooling coil. The
results indicate excellent and reliable online tuning of the TSFF
controller to respond rapidly to tune the TS model parameters
and resist the abrupt noise and sensor gain changes, as evident in
Fig. 10 and Table 4.
5. Conclusion
This paper introduces a controller that is built by converting a
Takagi–Sugeno fuzzy inference system (TSFIS) model into memory
layers parameters (TS model). The outputs routine of the classical
TSFIS model requires numerical and logical operation tasks and this
consumes time. Contrarily, the proposed TS model uses the gradient
algorithm, a faster online tuning method that requires less math-
ematical manipulations compared to other methods such as the
backpropagation method for neural networks. What is most impor-
tant is that this online tuning can tune a multivariable controller
with multi outputs. With this tuner, a good control performance
can be expected even though the process is a large scale system and
heavily coupled, which are properties common in HVAC systems.
The simulation results show that the proposed TSFF controller is
more suitable for HVAC systems when compared to a hybrid PID-
cascade and fixed parameters fuzzy controllers. On the other hand,
the TSFF demonstrated the ability of dealing with MIMO models
that possess nonlinearity, pure lag time, big thermal inertia, uncer-
tain disturbance factors, and constraints. The procedure of using
TSFF is straightforward and easy to carry out; offline learning of
the TS model layers and online tuning of the parameters layers.
This study considered the use of TSFF to control PMV as an objec-
tive instead of temperature and relative humidity. This solves the
temperature–humidity coupling; evident behavior of temperature
and relative humidity shown in Fig. 8 and Table 2. The most signif-
icant aspect of using PMV as a reference is because the controller
can be controlled accurately and efficiently by optimizing between
the temperature and relative humidity, i.e. there is no specific tem-
perature or humidity ratio that acts as a control reference that will
be decided by the PMV. Furthermore, using PMV makes it possible
to exploit the effects of controlling air velocity and manipulation
of flow rate of fresh air on thermal comfort levels, and that will
help to save energy. In addition, another advantage of the pro-
posed method is that the proposed TSFF offers computational cost
reduction in real time implementation. This is possible because the
proposed method requires less number of iterations to perform the
online tuning procedure, which is carried out using the gradient
algorithm. Furthermore, the controller is based on forward type
conception, which is well known to be faster than feedback type
and the TSFF algorithm can be oriented to extend to more general
nonlinear plants.
As for future work on the proposed model, the authors are
considering improving this type of controller further by integrat-
ing other building variable controlled devices, e.g., lights, shading
blinds and natural ventilation, which has a significant effect on
energy building cost. The optimization control of these devices is
becoming more and more important because the effects of different
devices on energy costs and human comfort are coupled. Therefore,
one integrated controller for these devices should work better than
the traditionally used non-integrated controllers.
Appendix A. Supplementary data
Supplementary data associated with this article can be found, in
the online version, at doi:10.1016/j.enbuild.2012.02.013.
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