Nonlinear multivariable control and performance analysis of an air-handling unit

Energy and Buildings 43 (2011) 805–813
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Energy and Buildings
journal homepage: www.elsevier.com/locate/enbuild
Nonlinear multivariable control and performance analysis of an air-handling unit
Hamed Moradi∗
, Majid Saffar-Avval, Firooz Bakhtiari-Nejad
Energy and Control Centre of Excellence, Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran
a r t i c l e i n f o
Article history:
Received 26 September 2010
Received in revised form 4 November 2010
Accepted 25 November 2010
Keywords:
Air-handling unit
Nonlinear model
Multivariable control
Tracking
Gain scheduling
Feedback linearization
a b s t r a c t
To maintain satisfactory comfort conditions in buildings with low energy consumption and operation
cost, control of air-conditioner units is required. In this paper, nonlinear control of an air-handling unit
(AHU) is investigated and compared for two control approaches: gain scheduling and feedback lineariza-
tion. A nonlinear multi input–multi output model (MIMO) of an air-handling unit (AHU) is considered.
Both indoor temperature and relative humidity are controlled via manipulation of valve positions of air
and cold water flow rates. Using an observer to estimate state variables, a hybrid control system includ-
ing regulation system for disturbance rejection and nonlinear control system for tracking objectives is
designed. Achievement of tracking objectives is investigated for various desired commands of indoor
temperature and relative humidity; including a sequence of steps and ramps-steps. According to results,
more quick time responses with a bit more overshoot in tracking set-points/paths are achieved by using
feedback linearization method (especially for temperature). However, valves position as input control
signals are associated with less oscillation (and consequently less energy consumption) when the con-
troller designed based on gain scheduling approach is used. Finally, it is shown through phase portrait of
the system that the controller designed based on feedback linearization shows a robust performance in
the presence of random uncertainty in model parameters.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
With the improvement of standard of living, air-conditioning
systems are extensively used to provide comfort and accept-
able indoor air quality (IAQ). A comprehensive review on
air-conditioning systems and indoor air quality control to preserve
human health has been done [1]. Since energy and operation costs
of buildings are directly influenced by how well an air-conditioning
system perform, effective thermal management is of great impor-
tance. It is estimated that avoidable energy waste in buildings is
about 20–50% and that 15% of the energy waste can be recovered
by effective control of the air-conditioning system [2,3].
Air-handling units (AHU) have an essential role for provid-
ing supply air with specific temperature and humidity in heating,
ventilating and air-conditioning (HVAC) operations. At the design
stage, for studying the system performance, finding an approximate
mathematical model of system components is essential. However,
due to the complex nonlinear nature of an AHU with multivari-
able parameters and time varying characteristics of its components,
finding an exact mathematical model is difficult [4]. For testing,
commissioning and evaluating control strategies implemented in
energy management and control systems (EMCS), dynamic simu-
lation of HVAC systems is a convenient and low cost tool.
∗ Corresponding author. Tel.: +98 21 64543417; fax: +98 21 66419736.
E-mail address: hamedmoradi@mech.sharif.ir (H. Moradi).
Several investigations have been accomplished for dynamic
modelling and simulation of HVAC systems and its components.
Dynamic model and transient response for space heating and cool-
ing zones [5] and dynamic simulation and evaluation of EMCS
on-line [6,7] for variable air volume (VAV) air-conditioning sys-
tems have been presented. Using automatic data acquisition system
for on-line training and artificial neural network [8] and grey-box
identification approach [3], air-handling unit has been modelled.
In addition, in other works of this area, a dynamic model of
cooling coil unit (the basic element in AHU unit) has been devel-
oped by extending the engineering model and combining with
the mass and energy balance equations [9]. Recently, simula-
tion of a VAV air-conditioning system for the cooling mode has
been carried out through an energetic study [10] (including sev-
eral references dealing with innovative work on VAV). Also, an
overview of current approaches used for modelling and simula-
tion of HVAC systems including control aspects has been presented
[11].
Improving system energy efficiency (by reducing energy con-
sumption) and indoor comfort conditions are major concerns in
any HVAC control system. Many control strategies have been imple-
mented to improve dynamic behaviour of air-conditioning systems.
Model based analysis and simulation of airflow control of AHU units
using PI controllers [12], multivariable control of indoor air qual-
ity in a direct expansion air-conditioning system [13] and control
tuning of a simplified VAV system [14] have been studied. Also,
cascade control algorithm and gain scheduling [15], analysis of dif-
0378-7788/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.enbuild.2010.11.022

806 H. Moradi et al. / Energy and Buildings 43 (2011) 805–813
Fig. 1. Schematic view of an air-handling unit having one zone (indoor) in VAV system.
ferent control schedules on EMCS [16] and model predictive control
[17] of air-handling units have been investigated.
In other control strategies, rule development and adjustment of
a fuzzy controller [18], fuzzy control optimized by genetic algo-
rithms (GA) [19], using a combination of artificial neural fuzzy
interface modelling and a PID controller [20] and developing an
adaptive fuzzy controller based on GA [21] have been implemented
on air-handling units. In addition, optimal control [22], propor-
tional optimal control [23] and adaptive self-tuning PI control [24]
are other control approaches used for HVAC systems.
Since tuned control parameters cannot cover all the operat-
ing range of the air-conditioning systems, using traditional control
approaches may result in aggressive or sluggish response at other
operating conditions. Moreover, for a constant provided ventilation
flow rate, they may cause over ventilation or insufficient ventilation
when the occupancy is lower or higher than the expected maximum
occupancy (consequently leads to energy waste and/or unsatisfied
IAQ) [17,25]. On the other hand, due to non-stationary plant oper-
ation associated with the nonlinearity of AHU components and the
coupling of the controlled variables, AHU control is a non-trivial
problem [3,4].
In this paper, unlike the previous mentioned works, a com-
parison between two control approaches (gain scheduling and
feedback linearization) applying on a nonlinear MIMO dynamic
model of an AHU is investigated. Advantages and disadvantages
of these methods are compared from various points of view such
as achieving control objectives and energy consumption. After the
state space formulation of the problem, an observer is designed
to estimate state variables of the system and a regulator system is
designed for disturbance rejection. Nonlinear control strategy of
the system for tracking objective is developed through feedback
linearization and gain scheduling approaches. Various desired
commands of indoor temperature and humidity ratio (including
a sequence of steps and ramps-steps) are tracked by manipulation
of air and cold water flow rates.
According to results, controlled system based on feedback lin-
earization shows more quick time responses in tracking desired
set-points/paths (especially for temperature). However, using the
controller designed based on gain scheduling leads to less oscilla-
tion of valves position of air and cold water (and consequently less
energy consumption). It is also shown that in the presence of an
arbitrary random uncertainty in model parameters, the controller
designed based on feedback linearization is robust.
2. System description of the air-handling unit
A schematic view of an air-handling unit having one zone
(indoor) in VAV system is shown in Fig. 1 [9]. This unit consists
of supply/return fans, cooling coil, filter, ductwork, humidifier and
dehumidifying coil (not shown). Since in this paper, AHU is essen-
tially designed for operation in summer, chilled water and air loops
exist. After the entrance and passing of the hot and humid air
through the cooling and dehumidification coil, its temperature and
humidity ratio decrease. For proper performance of the AHU, 25%
of fresh air with 75% of returned air are mixed and passed through
cooling unit. Finally satisfying supply air is provided and delivered
to the ventilated space through output channel.
3. Dynamic modelling of the nonlinear air-handling unit
For the formulation of the problem, it is assumed that gases are
ideal and mixed completely; air flow is homogeneous; the effect of
air speed variations on the zone pressure is negligible and there is
no air leakage except in the exhaust valves of the zone [26]. Using
thermodynamics, heat and mass transfer laws, differential equa-
tions describing dynamic behaviour of the air-handling unit are
determined as follows [26–28]:
˙Ts =
˙fa
Vc
(Tt − Ts) +
0.25 ˙fa
Vc
(To − Tt) −
˙fahw
CpaVc
(0.25wo + 0.75wt − ws)
− ˙fw
wCpw Tc
aCpaVc
˙Tt =
1
tCpaVt
( ˙Qo − hfg
˙Mo) +
˙fahfg
CpaVt
(wt − ws) −
˙fa
Vt
(Tt − Ts)
˙wt =
˙Mo
a Vt
−
˙fa
Vt
(wt − ws)
(1)
where Ts/ws, Tt/wt and To/wo are the temperature/humidity
ratio of the supply air, indoor air (zone) and environment, respec-
tively; Tc is the temperature gradient in cooling unit; ˙fa and ˙fw
are the air and cold water flow rates; Vc and Vt are the volume of
the cold unit and indoor space (zone); ˙Qo and ˙Mo are the strength
of heat load and humidity load; a/Cpa, w/Cpw are the mass den-
sity/specific heat of the air and cold water. hw and hfg are the
enthalpy of saturated water and vaporization (the list of thermo-
fluid parameters are given in Table 1). To simplify Eq. (1), following
terms are defined:
˛1 =
1
Vt
, ˛2 =
1
aVt
, ˛3 =
1
Vc
ˇ1 =
hfg
CpaVt
, ˇ2 =
wCpw Tc
aCpaVc
1 =
1
tCpaVt
, 2 =
hw
CpaVc
(2)

H. Moradi et al. / Energy and Buildings 43 (2011) 805–813 807
Fig. 2. Control system structure for the air-handling unit.
To represent state space formulation of the dynamic system, input,
output and state variables are considered as:
u1 = ˙fa, u2 = ˙fw
y1 = wt, y2 = Tt
x1 = Tt, x2 = wt, x3 = Ts
(3)
Using Eqs. (1)–(3), state space equations of the system are described
as [29]:
˙x1 = 1( ˙Qo − hfg
˙Mo) + ˇ1u1(x2 − ws) − ˛1u1(x1 − x3)
˙x2 = ˛2
˙Mo − ˛1u1(x2 − ws)
˙x3 = ˛3u1(x1 − x3) + 0.25˛3u1(To − x1)
− 2u1[0.25wo + 0.75x2 − ws] − ˇ2u2
(4)
4. Nonlinear control of the system
4.1. Regulator and observer systems design
Fig. 2 shows a schematic of the feedback control system
designed for the AHU system. For disturbance rejection of the prob-
lem, a regulator is designed, with the procedure for MIMO systems,
as explained with details in [32]. Fig. 3 shows the variation of
state variables (disturbance rejection) around an operating point
as given in Table 2 where the state vector is ¯x = [20, 0.00804, 17]
and for an arbitrary initial disturbance as ¯x = [−1, 0.001, −0.5]
(although the variation in temperature values is low, humidity ratio
disturbance is considerable). The required manipulation of valves
position of air and cold water ﬂow rates, for the mentioned distur-
bance rejection, is shown in Fig. 4.
State feedback control laws are constructed based on the
assumption that all state variables are available for measurement.
Fig. 3. Variation of state variables around the operating point ¯x = [20, 0.00804, 17] (given in Table 2), for the disturbance of ¯x(0) = [−1, 0.001, −0.5].

Table 1
Thermo-fluid parameters of the air-handling unit.
wo Environment humidity ratio
ws Supply air humidity ratio
wt Indoor humidity ratio (zone)
To Environment temperature
Ts Supply air temperature
Tt Indoor temperature (zone)
Tc Temperature gradient in cooling unit
˙Ma Strength of the humidity source
˙Qo Heat load
˙fa Air flow rate
˙fw Cooling water flow rate
Cpa Specific heat of the air
Cpw Specific heat of the water
hw Enthalpy of the saturated water
hfg Enthalpy of the vaporization
a Air mass density
w Water mass density
Vc Volume of the cooling unit
Vt Volume of the indoor space (zone)
Table 2
Values of the AHU thermo-fluid parameters around an operating point.
wO = 0.0082kg H2O/kg dry air ˙fa = 2.6 m3
/s
ws = 0.0080 kg H2O/kg dry air ˙fw = 0.9 × 10−3
m3
/s
wt = 0.00804 kg H2O/kg dry air To = 32 ◦
C
Cpa = 1000 J/kg ◦
C Ts = 17 ◦
C
Cpw = 4180 J/kg ◦
C Tt = 20 ◦
C
hw = 80 kJ/kg Tc = 6 ◦
C
hfg = 2500 kJ/kg ˙Mo = 0.000115 kg/s
a = 1.18 kg/m3 ˙Qo = 20 kW
w = 1000 kg/m
3
Vc = 1 m3
Vt = 400 m3
In practice, it may be impossible or too expensive to measure
all state variables. Under such condition, a state observer can
be used to estimate the process states. For such a MIMO sys-
tem, using the duality principle, state observer design is straight
forward as explained in the [32]. Fig. 5 shows the variation
Fig. 4. Variation of the required air and cold water flow rates for disturbance rejec-
tion around the operating point ¯x = [20, 0.00804, 17] (given in Table 2).
of real and estimated state variables for the same disturbance
¯x = [−1, 0.001, −0.5] and an arbitrary error vector ¯e = ¯x − ˜¯x =
[0, 0.001, 0], in which ¯x and ˜¯x are the real and estimated state vec-
tors, respectively (for the variation of real state variables, Fig. 5
shows the first 200 s and 20 s of the Fig. 3 for temperatures and
humidity ratio). As it is shown, the approximated humidity ratio
approaches to its real value in less time, in comparison with the
temperature behaviour.
4.2. Development of the controller based on gain scheduling
Gain scheduling technique can guarantee the validity of lin-
earization approach to a range of operating points. It may even
be possible to parameterize the operating points by one or more
Fig. 5. Variation of the real (solid line) and estimated (dashed line) state variables around the operating point ¯x = [20, 0.00804, 17].

Fig. 6. Desired commands for temperature and humidity ratio for a sequence of steps (a), step-ramp-step (b) and a combination of them (c).
scheduling variables. Under such condition, the system is linearized
at several operating points and a linear feedback controller is
designed at each point. This family of linear controllers can be
implemented as a single controller whose parameters changed by
monitoring the scheduling variables [30].
Consider again the dynamic model given by Eq. (4). To maintain
the system on each operating point as that of given in Table 2 at
desired state vector ¯x0 = [x1
0x2
0x3
0], a constant input vector ¯u0 =
[u1
0u2
0] must be imposed. For simplification, new variables are
defined as:
1 = x1
0, 2 = x2
0, 3 = x3
0
Á1 = u1
0, Á2 = u2
0 (5)
Linearizing Eq. (4) around the operating point given in Table 2,
yields:
˙¯xı = A( i, Áj)¯xı + B( i, Áj)¯uı i = 1, 2, 3; j = 1, 2
¯xı = ¯x − ¯x0, ¯uı = ¯u − ¯u0 (6)
where
A( i, Áj) = Á1
−˛1 ˇ1 ˛1
0 −˛1 0
0.75˛3 −0.75 2 −˛3
B( i, Áj) =
˛1( 3 − 1) − ˇ1(ws − 2) 0
˛1(ws − 2) 0
∗ −ˇ2
,
∗ = ˛3(0.25To + 0.75 1 − 3) − 2(0.25wo + 0.75 2 − ws),
i = 1, 2, 3; j = 1, 2
(7)
In the state feedback control scheme, to achieve desired locations of
closed-loop control system and consequently desired performance
of the system, the control vector ¯uı is constructed as:
¯uı = −K( i, Áj) ¯ ,
¯ = ¯xı − ¯rı, ¯rı = ¯yR − ¯y0 (8)
where K( i, Áj) is the variable gain matrix adjusted according to
the monitored scheduling variables, ¯ is the error vector, ¯yR is the
command vector signal that must be tracked and ¯y0 = [y1
0y2
0] is
the output vector defined in terms of state variables given by Eq.
(3) at each operating point. Substituting Eqs. (7) and (8) in the first

Fig. 7. Time response of the output temperature (a) and humidity ratio (b) in track-
ing a sequence of desired step set-points (case ‘a’ of Fig. 6).
derivative of Eq. (6) yields:
˙¯xı = [A( i, Áj) − B( i, Áj)K( i, Áj)]¯xı + B( i, Áj)K( i, Áj)¯rı (9)
It is assumed that a maximum overshoot of Mp = 10 % and rise time
tr ≤ 100 s in tracking behaviour of room temperature and humidity
ratio are desired. To achieve this, closed-loop poles of the system
(including a far non-dominant pole as ˆ 3 = −0.5) are assigned as:
ˆ 1,2 = −0.018 ± 0.024j, ˆ 3 = −0.5
Using the procedure given in Appendix A, instantaneous adjustable
feedback gain matrix K ( i, Áj) is found.
4.3. Development of the controller based on feedback
linearization
In feedback linearization approach, the nonlinear terms of the
dynamic system are eliminated by means of state variables feed-
back. Then a suitable controller is designed to stabilize the desired
trajectories of the system. Consider a square MIMO system in the
Fig. 8. Variation of required air (a) and cold water (b) flow rates for tracking a
sequence of desired step set-points (case ‘a’ of Fig. 6).
neighbourhood of a the operating point ¯x0 as [31]:
˙¯x = f (¯x) + G(¯x)¯u
¯y = h(¯x)
(10)
where ¯x is n × 1 the state vector, ¯u is r × 1 control input vector, ¯y
is m × 1 system outputs vector, f and h are smooth vector fields
and G is a n × r matrix whose columns are smooth vector fields gi
(in this paper, m = r = 2). Input–output linearization is obtained by
differentiating the outputs yi until the inputs appear. Assume that
i is the smallest integer that at least one of the inputs appears in
yi
( i), then (in this paper, yi
j represents output yi at operating point
j while y
(j)
i
represents the differentiation of yi of order j):
yi
( i)
= Lf i hi +
r
j=1
Lgj
Lf i−1 hiuj (11)
with Lgj
Lf i−1 hi(x) /= 0 for at least one j in a neighbourhood ˝i of
the operating point ¯x0 (operations Lfh, Lf i h and LgLf i h are defined
in Appendix A). Applying the same procedure for each output yi,
yields:
[y1
( 1)
· · ·ym
( m)
]T
= Lf 1 h1(¯x) Lf 2 h2(¯x) · · · Lf m hm(¯x)
T
+E(¯x)¯u (12)
where the r × r matrix E(¯x) is defined. Let ˝ represents the intersec-
tion of the ˝i. If all the partial relative degrees i are well defined,
then ˝ is a finite neighbourhood of ¯x0. In addition, if E(¯x) is invert-

Fig. 9. Time response of the output temperature (a) and humidity ratio (b) in track-
ing a desired step-ramp-step set-path (case ‘b’ of Fig. 6).
ible over the region ˝, the input transformation:
¯u = E−1
[v1 − Lf 1 h1 v2 − Lf 2 h2 · · · vm − Lf m hm]
T
(13)
yields a simpler form of m equations as:
y( i)
i
= vi (14)
Eq. (13) is called a decoupling control law, because the output yi is
only affected by the input vi, after applying the invertible decou-
pling matrix E(¯x). By differentiating from yi, inputs will appear after
first differentiation (as given by Eq. (4)) as:
y1
(1)
y2
(1) =
˛2Mo
0
+
˛1(ws − x2) 0
∗(x1, x2, x3) − ˇ2
u1
u2
(15)
∗
(x1, x2, x3) = ˛3(0.25To + 0.75x1 − x3) − 2(0.25wo + 0.75x2 − ws)
According to Eq. (13), control signal ¯u is constructed as:
[u1 u2]T
= E−1
[v1 − ˛2Mo v2]T
(16)
where
E =
˛1(ws − x2) 0
∗(x1, x2, x3) − ˇ2
(17)
Fig. 10. Variation of required air (a) and cold water (b) flow rates for tracking a
desired step-ramp-step set-path (case ‘b’ of Fig. 6).
using this control law results in two separate dynamics for outputs
as:
yi
(1)
= vi i = 1, 2 (18)
after decoupling the outputs dynamics, a PI controller is designed
as:
vi = −K1i i − K2i i, ˙ i = i = yi − ri (19)
where ri is the command input signal that is desired to be tracked.
Differentiating from Eq. (18) yields:
ÿi + K1i ˙yi + K2iyi = K1i ˙ri + K2iri (20)
Transforming this equation into the Laplace domain yields:
Yi(s)
Ri(s)
=
K1is + K2i
s2 + K1is + K2i
(21)
If the closed loop system is expected to have a behaviour similar to
the system with the following characteristic equation
s2
+ 2 ωns + ωn
2
= 0, ωn > 0, 0 < < 1 (22)
Control signal gains must be adjusted as:
K1i = 2 iωi, K2i = ωi
2
(23)
Again, to have a maximum overshoot of Mp = 10 % and rise time
tr ≤ 100 s in tracking behaviour of all output variables, param-

Fig. 11. Time response of the output temperature (a) and humidity ratio (b) in
tracking a desired set-path constituted of steps and ramps (case ‘c’ of Fig. 6).
eters of Eq. (23) must be selected as ωi = 0.03, i = 0.6, i = 1,
2.
5. Results and discussion
For investigation of tracking objectives, three practical cases of
desired commands for temperature and humidity ratio including a
sequence of steps, ramp-step and a combination of them are consid-
ered. Fig. 6 shows three arbitrary cases of desired set-points/paths
which results in comfort condition (since in summer season, the
majority of metropolitan cities of Iran associate with high temper-
ature and low relative humidity weather). However, it should be
mentioned that the designed controllers based on gain scheduling
and feedback linearization can be effectively used in other weather
conditions.
Fig. 7 shows the time response of the output temperature and
humidity ratio in tracking a sequence of desired step set-points
(case ‘a’ of Fig. 6), after applying two control approaches. As it
is shown, in comparison with feedback linearization, using gain
scheduling approach leads to slow time responses with less over-
shoot. Variations of the required air and cold water flow rates for
tracking of such sequence of step set-points are shown in Fig. 8. As
it is shown, although the variation of cold water is rather similar
for two control approaches, more air flow rate variation is required
when feedback linearization is used.
Fig. 12. Phase portrait of the air-handling unit using gain scheduling (dashed-dot
blue line) and feedback linearization (dot black line) for tracking a desired set-path
constituted of steps and ramps (case ‘c’ of Fig. 6).
Time responses of the output temperature and humidity ratio
in tracking a desired step-ramp-step set-path (case ‘b’ of Fig. 6) are
shown in Fig. 9. As it is shown, time responses speed in tracking is
similar for both control approaches. However, controller designed
based on feedback linearization shows more overshoots, especially
in tracking the ramp section of desired set-path for humidity ratio.
Fig. 10 shows the variations of the required air and cold water flow
rates for tracking of such step-ramp-step set-path. As it is shown,
both required air and cold water flow rates associate with less
oscillation (and consequently less energy consumption) when the
controller designed based on gain scheduling is used.
Fig. 11 shows the time response of the output temperature and
humidity ratio in tracking a desired set-path constituted of steps
and ramps (case ‘c’ of Fig. 6). Phase portrait of the air-handling unit
using gain scheduling and feedback linearization for tracking such a
desired set-path is shown in Fig. 12. As it is shown in Figs. 11 and 12,
generally controller designed based on feedback linearization leads
to more quick time responses with more overshoot (due to similar
general behaviour of required air and cold water flow rates to that
of previous case, their variation is not shown).
To investigate the robustness of the designed control systems,
two arbitrary 10% and 20% uncertainty in model parameters of
the air-handling unit (parameters defined by Eq. (2)), environment
temperature (To) and humidity ratio (wo) are considered. Fig. 13
shows 2D phase portrait of the air-handling unit in tracking the set-
Fig. 13. 2D phase portrait of the air handling unit for the model without uncertainty
(dashed black line), 10% (dot blue line) and 20% (dashed-dot red line) uncertainty
in model parameters (in tracking case ‘c’ of Fig. 6).

path of case ‘c’ of Fig. 6 for the model with/without uncertainties.
As it is shown, controller designed based on feedback linearization
approach shows a robust performance in the presence of para-
metric uncertainties (while the controller designed based on gain
scheduling does not work properly for the uncertain system).
6. Conclusions
In this paper, unlike the previous mentioned works, a compari-
son between two control approaches (gain scheduling and feedback
linearization) applying on a nonlinear MIMO dynamic model of
an AHU is investigated. Advantages and disadvantages of these
methods are compared from various points of view such as control
objectives and energy consumption. In this MIMO model, air and
cold water flow rates are manipulated to achieve desired tracking
objectives of indoor temperature and humidity ratio.
Three practical cases of desired commands of temperature and
humidity ratio including a sequence of steps, ramp-step and a
combination of them are considered for investigation of tracking
objectives. Comparing the results of various cases of the tracking
problem for gain scheduling and feedback linearization approaches,
the following conclusions are extracted:
1 In general, using the controller based on feedback linearization
approach leads to more quick time responses of temperature and
humidity ratio in tracking objectives, but with more overshoots.
Especially in tracking the ramp sections of desired set-paths for
humidity ratio, more overshoots can be seen (Fig. 9b).
2 Using the controller designed based on gain scheduling leads
to less consumption of energy in AHU. This is because, accord-
ing to Figs. 8 and 10, less variation of air and cold water flow
rates are required for achieving tracking objectives when the gain
scheduled-based controller is used.
3 In the presence of model uncertainties, the controller designed
based on feedback linearization guarantee the robust per-
formance of the air-handling unit in tracking the desired
set-points/paths. But as expected, due to inaccurate parameter
adjustment, the gain scheduled controller cannot show a robust
performance in the presence of parametric uncertainties.
Appendix A. Lie derivative definition
Let h : Rn → R be a smooth scalar function and f : Rn → R be a
smooth vector field on Rn, then the Lie derivative of h with respect
to f is a scalar function defined by [31]
Lf h = ∇h.f
Repeated Lie derivatives can be defined recursively:
Lf 0 h = h
Lf i h = Lf (Lf i−1 h) = ∇(Lf i−1 h).f
Similarly, if g is another vector field, then the scalar function LgLfh(x)
is
LgLf h = ∇(Lf h).g
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