Modeling and control approach to a distinctive quadrotor helicopter

Practice Article
Modeling and control approach to a distinctive quadrotor helicopter$
Jun Wu a,b
, Hui Peng a,c,n
, Qing Chen d
, Xiaoyan Peng e
a
School of Information Science & Engineering, Central South University, Changsha, Hunan 410083, China
b
School of Electrical & Information Engineering, Changsha University of Science & Technology, Changsha, Hunan 410004, China
c
Hunan Engineering Laboratory for Advanced Control and Intelligent Automation, Changsha, Hunan 410083, China
d
China Machinery International Engineering Design & Research Institute, Changsha, Hunan 410007, China
e
College of Mechanical and Automobile Engineering, Hunan University, Changsha, Hunan 410082, China
a r t i c l e i n f o
Article history:
Received 19 February 2012
Received in revised form
15 August 2013
Accepted 15 August 2013
Available online 7 September 2013
This paper was recommended for
publication by Prof. A.B. Rad.
Keywords:
ARX model
Nonlinear system
Physical model
Quadrotor helicopter
RBF-ARX model
a b s t r a c t
The referenced quadrotor helicopter in this paper has a unique configuration. It is more complex than
commonly used quadrotors because of its inaccurate parameters, unideal symmetrical structure
and unknown nonlinear dynamics. A novel method was presented to handle its modeling and control
problems in this paper, which adopts a MIMO RBF neural nets-based state-dependent ARX (RBF-ARX)
model to represent its nonlinear dynamics, and then a MIMO RBF-ARX model-based global LQR
controller is proposed to stabilize the quadrotor's attitude. By comparing with a physical model-based
LQR controller and an ARX model-set-based gain scheduling LQR controller, superiority of the MIMO
RBF-ARX model-based control approach was confirmed. This successful application verified the validity
of the MIMO RBF-ARX modeling method to the quadrotor helicopter with complex nonlinearity.
& 2013 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Quadrotor helicopter is a kind of multicopter that is lifted and
propelled by four rotors. Compared with the classical helicopter
which only has a main rotor and a tail rotor, it is much easier to be
constructed because it does not require mechanical linkages to
vary rotor angle of attack as they spin. The vehicle motion control
is easier too, which can be achieved by varying the relative speed
of each rotor.
Quadrotor helicopters are commonly designed to be micro-
unmanned aerial vehicles (UAVs). With their small size and agile
maneuverability, quadrotor helicopters are capable of small-area
monitoring and exploration. Recent years, researchers are trying to
expand its applications both in commercial fields and in industrial
fields. A lot of smart quadrotor helicopters appended with all kinds
of special mechanical equipments were designed to accomplish
many complicated tasks such as gripping and perching. In near
future, quadrotor helicopters may even be used as human carrying
transportation devices.
Undoubtedly, quadrotor helicopter is a useful class of flying
robots, and it is also a typical multivariable nonlinear plant.
Generally speaking, we have two ways to improve their control
performances, building more complete models and designing con-
trollers that do not need an accurate model. Recent researches were
mainly focused on nonlinear controllers design. The application of
two different control techniques (PID and LQ) to OS4 (Omnidirec-
tional Stationary Flying Outstretched Robot) were presented in [1].
The results of two model-based control techniques were shown.
Tayebi and McGilvray [2] provided a PD2
feedback structure, which
had the exponential convergence property due to the compensation
of the Coriolis and gyroscopic torques. Bouchoucha et al. [3]
presented an approach which is based on the combination of a
backstepping technique and a robust nonlinear PI controller to
stabilize the quadrotor attitude. A switching function was con-
structed to achieve a robust behavior for the overall control law, but
the choice of the PI gains would be a restriction of this method. In
[4], dynamic inversion was applied, which is effective in the control
of both linear and nonlinear systems, to tackle the coupling in
quadrotor dynamics. Unlike standard dynamic inversion, the linear
controller gains are chosen uniquely to satisfy the tracking perfor-
mance. Guenard et al. [5] presented a visual servo control using
backstepping techniques for stabilization of a quadrotor. Alexis et al.
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2013.08.010
☆
This work was supported by the International Science & Technology
Cooperation Program (2011DFA10440), and the National Natural Science
Foundation of China (71271215, 70921001).
n
Corresponding author. Tel./Fax: +86 731 88830642.
E-mail addresses: jun.wu@csu.edu.cn (J. Wu),
huipeng@mail.csu.edu.cn (H. Peng), coolchen302@qq.com (Q. Chen),
xiaoyan_p@126.com (X. Peng).
ISA Transactions 53 (2014) 173–185

[6] presented a switching model predictive attitude controller for
an unmanned quadrotor helicopter subject to atmospheric distur-
bances. The switchings among the piecewise affine models are
ruled by the rate of the rotation angles. To attenuate time-varying
and non-vanished disturbance torques, Zhang et al. [7] designed a
feedback controller with a sliding mode term to stabilize the
attitude of the quadrotor. Some literatures discussed neural net-
work (NN) based controller design to stabilize an aircraft against
modeling error and considerable wind disturbance [8–11].
However, dynamic models built or adopted in recent researches
are very similar and may be classified into three types according to
different simplifications from quadrotor dynamics. The first type is
shown in [12], which neglected the air friction and the gyroscopic
effects resulting from both the rigid body rotation and the
propellers rotation. The second type neglected the gyroscopic
effect resulting from the four propellers rotation [5,9,13–15]. The
third type ideally included the gyroscopic effects resulting from
both the rigid body rotation in space and the four propellers'
rotation [3,4,8,16,17]. But the relation between the rotor thrust and
the rotor input voltage was simplified.
If accurate parameters of quadrotor can be obtained and its
nonlinearities are clearly known, some physical model-based
methods can achieve very good control performances. However,
in some cases the quadrotor structure could be very complex so
that it is difficult to obtain its accurate physical model. First, the
physical model needs some accurate physical quantities, such as
the moment of inertia and the motor force constant, which are not
easy or even unable to obtain in real application. Some groups had
to try to construct their own prototype to get the accurate
quantities and the symmetric structure [17], but in most applica-
tions it is unfeasible to reconstruct the controlled objects. Second,
in order to establish the moment equilibrium equations, we need
to know the thrust forces generated by propellers. It is obvious
that the relation between the thrust force and the control voltage
is complicated and related to many factors, such as blade area,
density of air and radius of blade. Hoffmann et al. [18] introduced a
method to measure the thrusts and the torques using a load cell,
but developing a thrust test stand is a big challenge itself. In
addition, some quadrotors may have different configurations and
some of them may be difficult to be taken apart (see Fig. 1). Third,
in different flying postures, especially in the condition of large
operating angle, the coupling dynamics among the outputs are
also varying, and uncertain nonlinear terms like aerodynamic
friction and blade flapping can hardly be taken into account.
Besides, in different applications, there might be some complex
mechanical equipments appended to the quadrotor helicopters for
their special purposes, these equipments could also bring some
uncertain nonlinearities. Sometimes, the simplified physical mod-
els may be rough and inaccurate so that the control performance
would be degraded accordingly.
To overcome the restrictions of the physical models, system
identification in control engineering was proposed for under-
standing and controlling those unknown nonlinear dynamical
systems. In [19], neural networks with linear filter also known as
Narendra's model and recurrent neural networks with internal
memory (Memory Neuron Networks) are used for the purpose.
From the simulation studies it is shown that MNN approach is
better than Narendra's approach. In [20], a comparative study and
analysis of different Recurrent Neural Networks (RNN) for the
identification of helicopter dynamics using flight data is presented.
Three different RNN architectures, namely Nonlinear AutoRegres-
sive eXogenous input (NARX) model, neural network with internal
memory known as Memory Neuron Networks (MNN), and Recur-
rent MultiLayer Perceptron (RMLP) networks, are used to identify
dynamics of the helicopter at various flight conditions. Based on
the results, the practical utility, advantages and limitations of
the three models are critically appraised and it is found that the
NARX model is most suitable for the identification of helicopter
dynamics. In this paper, a novel NARX model is proposed to handle
the modeling and control problems to an unknown nonlinear
dynamical quadrotor helicopter.
The referenced quadrotor helicopter in this paper is a testbed
shown in Fig. 1. It is a very useful experimental equipment to test
and verify all kinds of modeling and control methods to the
quadrotor helicopter. Three degrees of freedom were locked in
order to reduce control complexity and avoid system damage. The
issue we are concerned with is obviously attitude stabilization,
which is very important for control of a quadrotor helicopter since
it allows the vehicle to maintain a desired orientation and results
in lateral or sideways motion [2]. It has 4 propellers, 3 of which are
horizontally mounted to control its pitch and roll rotation while
the last one is vertically mounted to control its yaw rotation.
Therefore, it has the advantage of classical helicopters in yaw
motion control and also has the advantage of quadrotors in pitch
and roll motion control [18].
Therefore, this quadrotor helicopter has 3 outputs and 4 inputs.
The outputs are the pitch angle, roll angle and yaw angle, and
the inputs are the control voltages of the 4 propellers' motors
equipped at the 4 ends of the quadrotor helicopter. It is easier to be
controlled compared with the traditional underactuated quadrotor
which only has two inputs [21]. The quadrotor helicopter's motion
is captured by a 3D universal joint and decoded to extract absolute
orientation information. An intelligent data acquisition card on a
PCI slot of the upper computer was used to collect real-time data
and send orders to the testbed. Thanks to the Real-Time Workshop
(RTW) of Matlab, we can construct real-time control system based
on Matlab/Simulink environment to implement many control
strategies conveniently.
In this paper, we handle its modeling and control problems
step by step. In Section 2, we shall first briefly introduce its
physical model-based LQR controller design, from which we can
clearly see that the quadrotor helicopter is a complex system
whose accurate physical model can hardly be obtained. Therefore,
in Section 3 the second method using system identification
technique was presented for the first time. According to the flyingFig. 1. The testbed for modeling research, 3DOF are locked.
J. Wu et al. / ISA Transactions 53 (2014) 173–185174

posture of the quadrotor helicopter, we divide its working states
into 16 regions averagely and a set of ARX models is identified in
each region to approximate its global nonlinear behaviors, which
may outperform the physical model. Then the ARX model-set-
based state-feedback control law is calculated using LQR. After
debugging in each region to get the optimal parameters, a gain
scheduling controller integrating all the regional models and state-
feedback control laws is designed, which shows good control
performance in the full range of the quadrotor's attitude control.
The essence of the second method is to represent the quadrotor
helicopter's nonlinear characteristics with several linear ARX
models which were obtained in some different working regions.
For this kind of nonlinear process whose dynamic behavior can be
represented by several local linear models at each time-varying
operating point, one can use the Gaussian radial basis function
(RBF) neural networks-based local linearization autoregressive
with eXogenous (ARX) model (RBF-ARX model) [22] to effectively
characterize such nonlinear system. In other words, the RBF-ARX
model is a type of hybrid pseudo-linear time-varying model. The
SISO RBF-ARX modeling method and its nonlinear MPC design
method had been investigated in both simulation studies and real
applications [23,24]. Furthermore, some stability conditions on the
offline identified RBF-ARX model-based NMPC were investigated in
[25]. On the basis of that the MIMO RBF-ARX model-based MPC
controller had been also proposed and its effectiveness were demon-
strated by a simulation study on thermal power plant [26], but it
does not mean that the MIMO RBF-ARX model works for the
quadrotor helicopter as well. However, it is a good motivation to
try it. In Section 4 we proposed a MIMO RBF-ARX model-based
global LQR control strategy (a kind of infinite horizon predictive
controller) for the first time in order to improve the control
performance of the quadrotor helicopter further.
The essence of the last method is to describe the quadrotor
helicopter's nonlinear characteristics via the global MIMO RBF-
ARX model. By using a set of RBF networks to approximate the
coefficients of a state-dependent ARX model, the RBF-ARX model
is yielded, which has the advantage of the state-dependent ARX
model in the description of nonlinear dynamics. It also has the
advantage of RBF networks in function approximation [27,28]. In
general, a RBF-ARX model uses far fewer RBF centers compared
with a single RBF network model, because the complexity of the
model is dispersed into the lags of the autoregressive parts of
the model. The RBF-ARX model is identified offline, which avoids
potential problems led by online identification and high costs
for real-time computation. Moreover, it provides enough time for
analysis and optimizations. Based on the MIMO RBF-ARX model,
an infinite horizon predictive controller was designed. The com-
parison of the real-time control results of the three methods given
in this paper showed the advantages of the MIMO RBF-ARX
model-based method and confirmed the validity of the MIMO
RBF-ARX modeling method to this class of fast nonlinear systems.
2. The physical model-based LQR controller
The coordinate of the quadrotor helicopter is shown in Fig. 2,
where Fx ðx ¼ f ; l; r; bÞ denotes the thrust forces generated by
4 propellers, and its suffixes mean its locations which are front,
left, right, and back. From Fig. 2, we can see that the pitch is
defined to be the angle circled around the Y-axis, and the anti-
clockwise rotation round Y-axis is defined to be positive. The roll is
defined to be the angle circled around the X-axis and the anti-
clockwise rotation round X-axis is defined to be positive. The yaw
is defined to be the angle circled around the Z-axis and the anti-
clockwise rotation round Z-axis is defined to be positive as well.
In order to reduce control complexity and avoid system damage,
three degrees of freedom of the translations subsystem were locked.
The structure is assumed to be symmetrical, the origin is assumed
to coincide with the quadrotor's centroid. According to the torque
equilibrium equation of each axis, three differential equations can
be formulated as follows [29]:
Jpp″ ¼ ðFl þFrÞLcÀFf Lf
Jrr″ ¼ FrLaÀFlLa
Jyy″ ¼ FbLb:
8
><
>:
ð1Þ
And other known conditions of the quadrotor helicopter are as
follows
Lc ¼ 1
2 Lf ¼
ffiffi
3
p
3 La ð2Þ
Lb ¼ Lf ð3Þ
Fx ¼ f ðVxÞ ¼ KfcVx ðx ¼ f ; l; r; bÞ: ð4Þ
Definitions of symbols are detailed in Table 1.
Notice that the relation between Fx and Vx in Eq. (4) is assumed
to be linear and time-invariant. Substituting Eqs. (2)–(4) to Eq. (1),
we obtain
p″ ¼
1
2
ðVl þVrÞÀVf

Lf
Kfc
Jp
r″ ¼
ffiffiffi
3
p
KfcLf
2Jr
ðVrÀVlÞ
y″ ¼
KfcLb
Jy
Vb:
8

:
ð5Þ
In addition, if the quadrotor helicopter is stabilized at a steady
state where the output is Ys ¼ ½ps rs ysŠT
and the input is Us ¼
½Vfs Vrs Vls VbsŠT
, then according to Eq. (5) we can also obtain the
Fig. 2. Coordinate of the quadrotor helicopter.
Table 1
Symbols and definitions.
Symbol Definition
Jp;r;y Body inertia
p Pitch angle
r Roll angle
y Yaw angle
Fx Thrust force
Kfc Thrust factor
Vx Motor input
Lc;f ;a;b Lever
J. Wu et al. / ISA Transactions 53 (2014) 173–185 175

equations below:
ðpÀpsÞ″ ¼
1
2
ðVlÀVls þVrÀVrsÞÀVf þVfs

lf
Kfc
JP
ðrÀrsÞ″ ¼
ffiffiffi
3
p
Kfclf
2Jr
ðVrÀVrsÀVl þVlsÞ
ðyÀysÞ″ ¼
Kfclb
Jy
ðVbÀVbsÞ:
8

:
ð6Þ
A state-space equation model can be built by defining the state
vector as
xðtÞ ¼ pÀps ðpÀpsÞ′
Z
ðpÀpsÞ rÀrs ðrÀrsÞ′

Â
Z
ðrÀrsÞ yÀys ðyÀysÞ′
Z
ðyÀysÞ
T
: ð7Þ
And the inputs and outputs are
ΔUðtÞ ¼ UÀUs
¼ ½Vf ÀVfs VrÀVrs VlÀVlsVbÀVbsŠT
ΔYðtÞ ¼ YÀYs ¼ ½pÀps rÀrsyÀysŠT
:
8

:
ð8Þ
Therefore, the state-space model of this linear time-invariant
(LTI) system is
_xðtÞ ¼ AxðtÞþBΔUðtÞ
ΔYðtÞ ¼ CxðtÞ
(
ð9Þ
where
A ¼
0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
B ¼
0 0 0 0
À
Kfclf
JP
Kfclf
2JP
Kfclf
2JP
0
0 0 0 0
0 0 0 0
0
ffiffi
3
p
Kfclf
2Jr
ffiffi
3
p
Kfclf
2Jr
0
0 0 0 0
0 0 0 0
0 0 0
Kfclb
Jy
0 0 0 0
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
C ¼
1 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0 0
2
6
4
3
7
5
:
8

:
ð10Þ
On the basis of the above state-space model, we can design
an infinite-time state regulator for this LTI system. The objective
function incorporating the states and the control efforts is
min
ΔUðtÞ
J ¼
1
2
Z 1
0
½xT
ðtÞQxðtÞþΔUT
ðtÞRΔUðtÞŠ dt ð11Þ
where the Q 40 and R40 are the diagonal weighing matrix for
xðtÞ and ΔUðtÞ respectively.
It was verified that ðA; BÞ is stabilizable and ðA; Q1=2
Þ is detect-
able, thus ΔUðtÞn
exists and be unique
ΔUðtÞn
¼ ÀRÀ1
BT
PxðtÞ; ð12Þ
where the constant matrix PZ0 is the solution of the following
algebraic Riccati equation:
PAþAT
PÀPBRÀ1
BT
PþQ ¼ 0: ð13Þ
The state feedback control law is
ΔUðtÞn
¼ ÀKxðtÞ ¼ ÀRÀ1
BT
PxðtÞ
UðtÞ ¼ Us þΔUðtÞn
:
(
ð14Þ
When the system goes to a steady state, then x-0 so that ΔY-0,
and the tracking goal is achieved. Notice that Ys and Us are desired
output and input. In the control strategy presented above, Us is
always equal to 0, because 3DOF are locked and the gravity factor is
not taken into account in the physical model. The LQR control results
based on the model will be presented later (see Figs. 10 and 11).
3. ARX model-set-based gain scheduling LQR controller
3.1. ARX model-set of the quadrotor
Though the physical model had been given, it was obviously
an inaccurate model. For one thing, the quadrotor helicopter
(see Fig. 2) has a complex structure, so that some components can
hardly be taken into account when calculating the inertia.
And the fixed rotational pivot may deviate from its actual centroid
because of its unideal symmetric structure. This would result in the
inaccuracy of the physical quantities. Actually, in many real applica-
tions there might be all kinds of mechanisms, such as cameras and
claws, appended to the quadrotors for accomplishing different tasks.
This would make their structures very complex so that the accurate
physical models can hardly be obtained. For another, many nonlinear
factors or unmodeled dynamics had to be simplified or totally
ignored. For example, in Eq. (4) the relation between the voltage
and the thrust force was simplified as a linear relation. Additional
dynamics introduced by the fixed rotational pivot, such as the
damping rotations due to friction, had been totally ignored. And
the gyroscopic effects resulting from both the rigid body rotation and
the propellers rotation were totally ignored. So we can see that
sometimes it is difficult or even unable to obtain the accurate physical
model, and the control performance would be degraded accordingly.
Therefore, a set of identified ARX models is proposed in this
section to approximate the global nonlinear dynamics of the
quadrotor helicopter, which may outperform the physical model.
Deng et al. [30] presented a discrete-time linear time-invariant (LTI)
model to approximate an actual continuous-time nonlinear system
and based on the identified model an output feedback LQR regulator
was designed. In this paper, a set of ARX models is identified, each of
them describes a local dynamics, and all the models may represent
the global nonlinear behaviors well. The nonlinear characteristics is
chiefly determined by the flying posture, which is mainly related
with the pitch and roll angles. Thus, according to the range of pitch
and roll angle, we divide the working state into 16 regions as
follows, and ξ denotes the region number
ξ ¼ fix
pÀ2
10

Â 4þfix
rÀ5
7:5

pAð2; 42Þ; rAð5; 35Þ
8

:
ð15Þ
where fixðαÞ is an operator that rounds α to the nearest integer
towards zero. When p¼22 and r¼20, the quadrotor helicopter is
horizontally postured.
In each region one ARX model is identified offline. Because the
quadrotor helicopter is a multiple-output multiple-input (MIMO)
system, its locally linear ARX structure is designed as follows:
YðtÞþA1YðtÀ1Þþ⋯þAny
YðtÀnyÞ ¼ Y0
þB1UðtÀnkÞþ⋯þBnu
UðtÀnkÀnu þ1ÞþΞðtÞ ð16Þ

where
Ak ¼
a11;k a12;k a13;k
a21;k a22;k a23;k
a31;k a32;k a33;k
2
6
4
3
7
5
Bk ¼
b11;k b12;k b13;k b14;k
b21;k b22;k b23;k b24;k
b31;k b32;k b33;k b34;k
2
6
4
3
7
5
8

:
ð17Þ
and YðtÞ ¼ ½pðtÞ rðtÞ yðtÞŠT
are the outputs which include the
pitch angle, the roll angle and the yaw angle; UðtÞ ¼
½Vf ðtÞ VrðtÞ VlðtÞ VbðtÞŠT
are the inputs that denote the voltage
of 4 propellers; nu, ny and nk 40 are the system orders;
Ak ðk ¼ 1; 2; …; nyÞ and Bk ðk ¼ 1; 2; …; nuÞ are the coefficient
matrixes; ΞðtÞ is the modeling residual.
Assuming that at a steady state, the input and output are Us
and Ys, respectively. From model (16) one can has the following
ARX model:
ΔYðtÞþA1ΔYðtÀ1Þþ⋯þAny ΔYðtÀnyÞ
¼ B1ΔUðtÀnkÞþ⋯þBnu ΔUðtÀnkÀnu þ1Þþ ^ΞðtÞ ð18Þ
where ΔUðtÞ ¼ UÀUs, ΔYðtÞ ¼ YÀYs. ARX model (18) can be
transformed into a state-space model by defining the state vector
as follows:
XðtÞ ¼ x1
1;t x1
2;t x1
3;t ∑
t
i ¼ 0
x1
1;i;

x2
1;t x2
2;t x2
3;t ∑
t
i ¼ 0
x2
1;i;
x3
1;t x3
2;t x3
3;t ∑
t
i ¼ 0
x3
1;i
#T
x1
1;t ¼ pðtÞÀps; x2
1;t ¼ rðtÞÀrs; x3
1;t ¼ yðtÞÀys
xl
k;t ¼ ∑
nþ 1Àk
i ¼ 1
∑
3
j ¼ 1
a
_
lj;k þiÀ1 xj
1;tÀ1⋯
þ ∑
nþ 1Àk
i ¼ 1
∑
4
j ¼ 1
b
_
lj;k þ iÀ1 uj;tÀ1
n ¼ maxðny; nu þnkÀ1Þ
a
_
ij;k ¼
Àaij;k; krny
0; k4ny
(
b
_
ij;k ¼
bij;k; nk rkrnu þnkÀ1
0 else

k ¼ 2; 3; …; n; l ¼ 1; 2; 3
8

:
ð19Þ
aij;k and bij;k are the elements in Ak and Bk of Eq. (17).
The state-space equation corresponding to model (18) may
then be given by
Xðtþ1Þ ¼ AXðtÞþBΔUðtÞ
ΔYðtÞ ¼ CXðtÞþ ^ΞðtÞ
(
ð20Þ
where
A ¼
α11 ~α12 ~α13
~α21 α22 ~α23
~α31 ~α32 α33
2
6
4
3
7
5
B ¼
β11 β12 β13 β14
β21 β22 β23 β24
β31 β32 β33 β34
2
6
4
3
7
5
C ¼
χ 0 0
0 χ 0
0 0 χ
2
6
4
3
7
5
8

:
αii ¼
Àa
_
ii;1 1 0 ⋯ 0 0
Àa
_
ii;2 0 1 ⋮ ⋮
⋮ ⋮ ⋱ 1 ⋮
Àa
_
ii;n 0 0 ⋯ 0 0
1 0 ⋯ ⋯ 0 1
2
6
6
6
4
3
7
7
7
5
ðn þ1ÞÂðnþ 1Þ
~αij ¼
Àa
_
ij;1 0 0 ⋯ 0 0
Àa
_
ij;2 0 0 ⋮ ⋮
⋮ ⋮ ⋱ 0 ⋮
Àa
_
ij;n 0 0 ⋯ 0 ⋮
0 ⋯ ⋯ ⋯ ⋯ 0
2
6
6
6
4
3
7
7
7
5
ðnþ1ÞÂðnþ 1Þ
βij ¼ ½b
_
ij;1 b
_
ij;2 ⋯ b
_
ij;n 0ŠT
ðnþ 1ÞÂ1
χ ¼ ½1 0 ⋯ 0Š1Âðn þ1Þ
There are many ways to determine the orders [31], such as Loss
Function, Akaike Final Prediction Error and Akaike Information
Criterion (AIC). We adopt the AIC to select the orders of the ARX
models.
To start identification, the valid data is needed. Under the
physical model, a controller had been designed. Although the
control performance is not good, it does not affect the data
acquisition. First, one made the quadrotor helicopter stabilized in
one divided region, and then a noise signal was added to make it
swing a little but not to exceed the divided region. By sampling
a length of inputs and outputs, the data for identification was
obtained. Using the same method, we sampled 16 groups of local
data for identifying all 16 local ARX models.
Under the different orders, the identified models' AICs were
calculated. By taking the trade-off between the smallest AIC
and the complexity of the model, also considering the real-time
control performance, the system orders are selected as ny ¼3,
nu ¼1, nk ¼2 and n ¼ maxðny; nu þnkÀ1Þ ¼ 3.
We use the least square method to estimate coefficients of the
ARX models. Figs. 3–5 showed a comparison between the actual
outputs and the one-step-ahead prediction of the local ARX model
in three degrees of freedom when ξ ¼ 15.
Figs. 3–5 revealed that the ARX model can represent the local
dynamics of the quadrotor helicopter very well. By using the same
method, we obtained and tested 16 local ARX models one by one,
and Table 2 shows the standard deviations of the one-step-ahead
predictive errors in 16 local regions. The global dynamics of the
quadrotor helicopter could be approximated by these local ARX
models.
The 16 groups of actual data sampled in 16 local regions can
also be used to test the physical model presented in Section 2.
Fig. 3. Comparison of actual outputs and the ARX model outputs.

Table 3 shows the standard deviations of the one-step-ahead
predictive errors of the physical model in 16 local regions.
By comparing the predictive errors shown in Tables 2 and 3,
one can see that the ARX model-set modeling method improved
modeling accuracy considerably. Therefore, it is meaningful to use
system identification technologies to build a better model for the
quadrotor.
3.2. Gain scheduling LQR controller
Using the same method shown in Section 2, we can design an
infinite-time quadratic regulator based-on the time-invariant state-
space model (20) in each divided working region, the objective
function in discrete form is designed as
min
ΔUðkÞn
J ¼
1
2
∑
1
k ¼ 0
½XT
ðkÞQXðkÞþΔUT
ðkÞRΔUðkÞŠ: ð21Þ
By solving the discrete Riccati equation
P ¼ Q þAT
PðIþBRÀ1
BT
PÞÀ1
A; ð22Þ
the state-feedback gain matrix K based on the ARX model at a
flying posture can be obtained, and the state-feedback optimal
control law at this posture is given by
ΔUðkÞn
¼ ÀKXðkÞ
¼ ÀRÀ1
BT
AÀT
ðPÀQÞXðkÞ
UðkÞ ¼ Us þΔUðkÞn
:
8

:
ð23Þ
When the system goes to a steady state, XðtÞ-0, and the states
x1
1;t, x2
1;t and x3
1;t also go to zero. This means the achievement of the
tracking goal.
We model the quadrotor helicopter at 16 operating regions, and
the corresponding 16 linear ARX models are then built to describe
the nonlinear characteristics of the quadrotor helicopter. In each
divided region, an ARX model is identified and a state-feedback
control law is obtained, which is debugged to work well in its
region. A desired control performance may be easily achieved
by adjusting the weighing matrix Q 40 and R40 in Eq. (21)
appropriately.
Q and R in Eq. (21) represent the relative importance to be
assigned to the structural response and the control effort respec-
tively. A relative larger weight would impose a higher penalty
on the corresponding term for optimization of the cost function.
Hence, if the reduction of the structural response is the prime
concern irrespective of the cost of control or even at the expense
of higher cost of control, a lower weighting force should be
assigned to the term associated with the calculation of the control
effort and vice versa.
Because the ARX model is a local model, when the quadrotor
helicopter's working region is exceeded its modeling range, a good
control performance cannot be guaranteed. There is a necessity
to design a global controller which integrates all the models and
Table 2
Standard deviations of modeling residuals of the 16 ARX models.
Region num. Pitch (deg) Roll (deg) Yaw (deg)
1 0.08409 0.15942 0.05988
2 0.08137 0.14798 0.06425
3 0.07995 0.14044 0.06481
4 0.07387 0.13977 0.05696
5 0.08155 0.15697 0.05336
6 0.07119 0.11337 0.05543
7 0.06803 0.12433 0.06166
8 0.07525 0.13354 0.04781
9 0.07910 0.13902 0.05717
10 0.09002 0.09742 0.05270
11 0.07823 0.11515 0.05529
12 0.08965 0.14741 0.05134
13 0.09584 0.12968 0.06395
14 0.09611 0.09736 0.05857
15 0.10190 0.10607 0.05354
16 0.10145 0.13660 0.06416
Table 3
Standard deviations of modeling residuals of the physical model.
1 0.27540 0.51759 0.15524
2 0.24629 0.30584 0.12272
3 0.24464 0.28601 0.12955
4 0.25461 0.43836 0.14071
5 0.21382 0.54285 0.11951
6 0.20383 0.36895 0.12767
7 0.19917 0.30248 0.11814
8 0.18682 0.50068 0.12215
9 0.17178 0.52032 0.11861
10 0.15259 0.41090 0.13090
11 0.16282 0.35907 0.12595
12 0.17483 0.49034 0.12552
13 0.19508 0.54313 0.14181
14 0.18419 0.54569 0.15156
15 0.19212 0.51051 0.15662
16 0.19985 0.43155 0.14126
Fig. 4. Outputs error of the ARX model.
Fig. 5. Histograms of the residuals of the ARX model.

operates well in the full range. Therefore, a gain scheduling
controller is proposed, which can change its state feedback gain
properly according to the posture of the quadrotor helicopter. To
avoid the possible instability caused by repeatedly switching from
one regional model to another at the fringes, the controller uses a
gain switch law like a pivoted loop of relay. With appropriate
thresholds, the problems mentioned above can be eliminated. The
structure of the system is given in Fig. 6.
After accomplishing all the tasks mentioned above, the real-
time control is carried out. The sample period is 0.1 s. In practice,
firstly the debugging was undertaken in each region to choose the
optimal parameters, and then the global controller integrated all
the state feedback gain matrices K. The real-time control results
of the ARX model-set-based gain scheduling LQR controller are
shown in Figs. 7–9.
From Figs. 7 and 8 we can see that the ARX model-set-based
LQR controller can stabilize the quadrotor helicopter in any given
postures when the 3 outputs were changing one by one. Fig. 9
showed that the region number (or the model index) was chan-
ging with the varying of the quadrotor's postures.
The comparison between the new method and the former one
has been illustrated in Figs. 10–13. All the parameters have been
adjusted to be optimal already. The controllers accomplished the
same task, which is: the quadrotor helicopter goes to a given
posture, and stabilizes for a while, then returns to the original
posture and stabilizes again. Notice that the 3 outputs were
changing simultaneously, which is different from the control
results shown in Fig. 7.
From the real-time control results, one can see that the new
method is feasible, and the control performance is pretty good.
From the structure of the quadrotor helicopter, one can easily
see that the yaw is comparably easier to be controlled than the
pitch and roll, because the yaw is little coupled with pitch and roll,
and is mainly affected by the back propeller. Meanwhile, the pitch
and roll, which mainly represent the flying posture, are mostly
related with the front, left and right propellers, so the complex
couplings exist, which make them more difficult to be controlled
relatively.
In Fig. 10, we can see that there are big oscillations when
the quadrotor helicopter was stabilized in the horizontal posture,
Fig. 7. Outputs of real-time control based on identified ARX models. Values in
square bracket denote the desired values of pitch, roll, and yaw angle; the real line
represents the pitch angle, the dashed line represents the roll angle, and the dotted
line represents the yaw angle.
Fig. 8. Inputs of real-time control based on identified ARX models.
Fig. 9. The index in (15) in real-time control based on identified ARX models,
which reflects the region of the quadrotor helicopter posture.
Fig. 10. The control result based on the physical model.
Fig. 6. Structure of the control system.

because the physical model is just a simplified model and it could
not perform well at all postures. From Figs. 7 and 12 one can see
that the second method shows a comparably better control perfor-
mances, and it can stabilize the quadrotor helicopter at any given
working point quickly without large oscillations. In Fig. 12, from the
20th second to the 30th second, one can see that the transient
procedure is not smooth, because the simultaneously changing of
the pitch and roll resulted in the rapid changing of the quadrotor's
working regions. In order to avoid the jumping of the state feedback
gain matrix K and make the transient procedure smooth, it is
needed to identify the local ARX models as many as possible.
4. MIMO RBF-ARX model-based global LQR controller
The essence of the second method in Section 3 is to approx-
imate the quadrotor helicopter's global nonlinear dynamics with
several LTI models identified in different operating points. How-
ever, identifying so many local models is really a tough work and
sometimes even unable to realize in real applications.
The MIMO RBF-ARX model is constructed as a global model.
It treats a nonlinear process by splitting the state space up into
a large number of small segments, and regards the process as
locally linear within each segment. Therefore the quadrotor
helicopter's global nonlinear dynamics can be represented by
using only one MIMO RBF-ARX model. The RBF-ARX model may
be estimated by the structured nonlinear parameter optimization
method (SNPOM) [26].
4.1. RBF-ARX model of the quadrotor
The quadrotor helicopter is a multiple-outputs multiple-inputs
system, and its MIMO RBF-ARX structure is given as follows:
YðtÞ ¼ CðwðtÀ1ÞÞþ ∑
ny
k ¼ 1
AkðwðtÀ1ÞÞYðtÀkÞ
þ ∑
nu þnkÀ1
k ¼ nk
BkðwðtÀ1ÞÞUðtÀkÞþΞðtÞ
CðwðtÀ1ÞÞ ¼ ½ϕ1
0ðwðtÀ1ÞÞ ϕ2
0ðwðtÀ1ÞÞ ϕ3
0ððwðtÀ1ÞÞÞŠT
ϕi
0ðwðtÀ1ÞÞ ¼ ci
0 þ ∑
h
m ¼ 1
ci
m expfÀJwðtÀ1ÞÀZY;m J2
λY;m
g
AkðwðtÀ1ÞÞ ¼
a11;kðwðtÀ1ÞÞ ⋯ a13;kðwðtÀ1ÞÞ
⋮ ⋱ ⋮
a31;kðwðtÀ1ÞÞ ⋯ a33;kðwðtÀ1ÞÞ
2
6
4
3
7
5
BkðwðtÀ1ÞÞ ¼
b11;kðwðtÀ1ÞÞ ⋯ b14;kðwðtÀ1ÞÞ
⋮ ⋯ ⋮
b31;kðwðtÀ1ÞÞ ⋯ b34;kðwðtÀ1ÞÞ
2
6
4
3
7
5
aij;kðwðtÀ1ÞÞ ¼ cij
k;0
þ ∑
h
m ¼ 1
cij
k;m
expfÀJwðtÀ1ÞÀZY;m J2
λY;m
g
bij;kðwðtÀ1ÞÞ ¼ d
ij
k;0 þ ∑
h
m ¼ 1
d
ij
k;m expfÀJwðtÀ1ÞÀZU;m J2
λU;m
g
Zj;m ¼ ½zj;m;1 … zj;m;dimðwðtÀ1ÞÞŠ; j ¼ Y; U
8

:
ð24Þ
where UðtÞ ¼ ½Vf ðtÞ VrðtÞ VlðtÞ VbðtÞŠT
are the inputs, which denote
the voltage of 4 propellers; YðtÞ ¼ ½pðtÞ rðtÞ yðtÞŠT
are the outputs,
which are the pitch angle, the roll angle and the yaw angle
respectively; ny, nu, nk and h are the orders; zj;m's are the centers
of Gaussian RBF networks; c1
m, c2
m, c3
m, cij
k;m
's, and d
ij
k;m's are the
weighting coefficient matrices of suitable dimensions; JxJ2
^λ
9
xT ^λx, ^λ ¼ diagð^λ1
^λ2 … ^λdimðxÞÞ, and f^λ1
^λ2 ⋯ ^λdimðxÞg are the scal-
ing factors; ΞðtÞAR3
denotes noise usually regarded as Gaussian
white noise. The signal wðtÀ1Þ in Eq. (24) is the index of the
model, which is the variable causing nonlinearity. wðtÀ1Þ could be
a system variable causing the operation-point of the system
to change with time. wðtÀ1Þ has direct or indirect relation with
inputs or outputs of the system, in some cases probably being just
Fig. 12. The control result based on the ARX model.
Fig. 13. Inputs of the control result based on the ARX model.Fig. 11. Inputs of the control result based on the physical model.

the input or/and output itself. For this quadrotor helicopter, we
choose pitch angle and roll angle as the model index, because the
nonlinear characteristics may change with the flying postures,
which are mainly related with the pitch and roll.
It is easy to see that the local linearization of the model (24) is a
linear MIMO ARX model by fixing wðtÀ1Þ at time tÀ1. It is natural
and appealing to interpret model (24) as a locally linear MIMO
ARX model in which the evolution of the process at time t is
governed by a set of AR coefficient matrices fAk; Bkg at a local mean
ϕ0, all of which depend on the ‘working-point’ of the process at
time tÀ1. Thus the structure of MIMO RBF-ARX resembles the ARX
in application.
Assuming that, at a steady state, Us, Ys, and ws are the values of
the related variables, then from the MIMO RBF-ARX model (24)
yields
Ys ¼ CðwsÞþ ∑
ny
k ¼ 1
AkðwsÞYs þ ∑
nu þ nkÀ1
k ¼ nk
BkðwsÞUs ð25Þ
from which we can easily get the Us with a desired Ys. And an
increment equation around the steady state may be derived from
Eq. (24) and (25) as follows:
ΔYðtÞÀΔCðwðtÀ1ÞÞ ¼ ∑
ny
k ¼ 1
AkðwsÞΔYðtÀkÞ
þ ∑
nu þ nkÀ1
k ¼ nk
BkðwsÞΔUðtÀkÞþ ^ΞðtÞ ð26Þ
where
ΔYðtÞ ¼ YðtÞÀYs
ΔUðtÞ ¼ UðtÞÀUs
ΔCðwðtÀ1ÞÞ ¼ CðwðtÀ1ÞÞÀCðwsÞ:
8

:
The increment MIMO RBF-ARX model (26) can be transformed
into a state-space equation model by defining state vector as
follows:
XðtÞ ¼ x1
1;t x1
2;t ⋯ x1
n;t ∑
t
i ¼ 0
x1
1;i;

x2
1;t x2
2;t ⋯ x2
n;t ∑
t
i ¼ 0
x2
1;i;
x3
1;t x3
2;t ⋯ x3
n;t ∑
t
i ¼ 0
x3
1;i
#T
x1
1;t ¼ pðtÞÀpsÀϕ1
0ðwðtÞÞþϕ1
0ðwsÞ
x2
1;t ¼ rðtÞÀrsÀϕ2
0ðwðtÞÞþϕ2
0ðwsÞ
x3
1;t ¼ yðtÞÀysÀϕ3
0ðwðtÞÞþϕ3
0ðwsÞ
xl
k;t ¼ ∑
nþ 1Àk
i ¼ 1
∑
3
j ¼ 1
a
_
lj;k þiÀ1 xj
1;tÀ1⋯
þ ∑
nþ 1Àk
i ¼ 1
∑
4
j ¼ 1
b
_
lj;k þ iÀ1uj;tÀ1
n ¼ maxðny; nu þnkÀ1Þ
a
_
ij;k ¼
Àaij;k; krny
0; k4ny
(
b
_
ij;k ¼
bij;k; drkrnu þdÀ1
0 else

k ¼ 2; 3; …; n; l ¼ 1; 2; 3
8

:
where aij;k and bij;k are the elements in Ak and Bk of Eq. (24).
Notice that ΔCðwðtÀ1ÞÞ in Eq. (26) is included in the state
variables, so state-space equation corresponding to model (26)
may then be given by
Xðtþ1Þ ¼ AXðtÞþBΔUðtÞ
ΔYðtÞ ¼ CXðtÞþ ^ΞðtÞ
(
ð27Þ
where
A ¼
α11 ~α12 ~α13
~α21 α22 ~α23
~α31 ~α32 α33
2
6
4
3
7
5
B ¼
β11 β12 β13 β14
β21 β22 β23 β24
β31 β32 β33 β34
2
6
4
3
7
5
C ¼
χ 0 0
0 χ 0
0 0 χ
2
6
4
3
7
5
8

:
αii ¼
Àa
_
ii;1 1 0 ⋯ 0 0
Àa
_
ii;2 0 1 ⋮ ⋮
⋮ ⋮ ⋱ 1 ⋮
Àa
_
ii;n 0 0 ⋯ 0 0
1 0 ⋯ ⋯ 0 1
2
6
6
6
4
3
7
7
7
5
ðn þ1ÞÂðnþ 1Þ
~αij ¼
Àa
_
ij;1 0 0 ⋯ 0 0
Àa
_
ij;2 0 0 ⋮ ⋮
⋮ ⋮ ⋱ 0 ⋮
Àa
_
ij;n 0 0 ⋯ 0 ⋮
0 ⋯ ⋯ ⋯ ⋯ 0
2
6
6
6
4
3
7
7
7
5
ðnþ1ÞÂðnþ 1Þ
βij ¼ ½b
_
ij;1 b
_
ij;2 ⋯ b
_
ij;n 0ŠT
ðnþ 1ÞÂ1
χ ¼ ½1 0 ⋯ 0Š1Âðn þ1Þ
We also adopt the AIC to select the order of the MIMO RBF-ARX
models, whose expression on the quadrotor helicopter is defined by
AIC ¼ N log jΣjþ2ðð1þhÞð3þ9ny þ12nuÞþ2m dimðwðtÀ1ÞÞþnyÞ
ð28Þ
where N is the data length; jΣj is the determinant of variance–
covariance matrix of modeling residuals.
In order to get a global data for identification, a set of sine
signals is set to make the quadrotor helicopter swing in the full
range. And so as to meet the persistent excitation condition, a set
of Gauss White Noises with small power is added. By sampling
a length of inputs and outputs, the data for identification is
obtained, which is shown in Fig. 14 where the front 800 data
points are used to identify the RBF-ARX model, and the back 600
data points are used to test the model.
Under different model orders the identified model's AICs are
calculated. By taking the trade-off between the smallest AIC and
the real-time control performance, the system orders are selected
as ny ¼3, nu ¼1, nk ¼2, h¼1, and n ¼ max ðny; nu þnkÀ1Þ ¼ 3. The
corresponding AIC value is À10 904.
The modeling results of the MIMO RBF-ARX model are shown
in Figs. 15 and 16. From the figures, one can see that the MIMO
RBF-ARX model has an excellent modeling accuracy. We use the 16
groups of actual data sampled in 16 different working regions to
test the MIMO RBF-ARX model too, just the same work as we did
in Tables 2 and 3. Table 4 shows the standard deviations of the
one-step-ahead predictive errors of the MIMO RBF-ARX model in
the 16 working regions.
By comparing Tables 2 and 4, we can see that the MIMO RBF-
ARX model and the ARX model-set have close modeling accuracy.
Both of them show much better modeling accuracy than the
physical model does in all 16 local working regions.

4.2. Global LQR controller
Based on the offline identified MIMO RBF-ARX model, a global
LQR controller is designed, which can self-adjust the LQR gain
according to the flying posture of the quadrotor helicopter. The
system structure is given in Fig. 17.
At any working point, by using the same method introduced in
Section 3 we can design an infinite-time quadratic regulator based
on the locally linear time-invariant state-space model (27). The
objective function of LQR in discrete-time form is given by
min
ΔUðkÞn
J ¼
1
2
∑
1
k ¼ 0
½XT
ðkÞQXðkÞþΔUT
ðkÞRΔUðkÞŠ ð29Þ
By solving the discrete Riccati equation (22) at a working-point,
the state-feedback matrix K at the working-point can be obtained
as follows:
K ¼ RÀ1
BT
AÀT
ðPÀQÞ ð30Þ
and the state-feedback optimal control law at this working-point is
ΔUðkÞn
¼ ÀKXðkÞ
UðkÞ ¼ Us þΔUðkÞn
:
(
ð31Þ
When the system goes to a steady state, i.e. XðtÞ-0, and the
states x1
1;t, x2
1;t, and x3
1;t also go to zero, this means the achievement
of the tracking goal.
The RBF-ARX model-based global LQR controller is a special
case of the RBF-ARX model-based predictive controller (RBF-ARX-
MPC). Peng et al. had discussed its stability in [25], and this paper
would focus on its practical application.
After all the tasks mentioned above have been accomplished,
the real-time control can be carried out. The control sample period
is 0.1 s in which dynamics of quadrotor varies slightly and enough
time for control law calculation is guaranteed. The results of real-
time control based on the identified MIMO RBF-ARX model are
shown in Figs. 18–21.
From Fig. 18 we can see that the quadrotor helicopter can be
stabilized at any given point very quickly and smoothly by using the
MIMO RBF-ARX model-based global LQR control strategy when the
3 outputs were changing one by one. Compared with the ARX
model-set-based gain-scheduling LQR control strategy, it used only
one model but obtained better control results. From Fig. 20 one can
see that when the pitch and roll changed simultaneously, from the
Table 4
Standard deviations of modeling residuals of the RBF-ARX model.
1 0.12069 0.14334 0.07597
2 0.12040 0.15015 0.06463
3 0.10845 0.14612 0.07327
4 0.11264 0.16021 0.07580
5 0.11078 0.16360 0.06646
6 0.09537 0.14166 0.06749
7 0.09811 0.13789 0.06082
8 0.09006 0.15414 0.06874
9 0.12415 0.14965 0.06072
10 0.09894 0.13478 0.07016
11 0.08680 0.14197 0.06472
12 0.09036 0.16699 0.06667
13 0.10610 0.15922 0.07072
14 0.09913 0.16542 0.07845
15 0.09094 0.16232 0.07627
16 0.08848 0.17551 0.06616
Fig. 17. Structure of the control system based on RBF-ARX model.
Fig. 15. Residuals of MIMO RBF-ARX model for test data.
Fig. 16. Histograms of the residuals of MIMO RBF-ARX model for test data.
Fig. 14. Identification data.

20th second to the 25th second, the transition process is very
smooth and fast, which is much better than the control results of
the ARX model-set-based gain scheduling LQR controller (see
Fig. 12). The detailed comparisons of the control results shown in
Figs. 10, 12 and 20 are given in Tables 5–7.
From Tables 5 and 6, one can see that the ARX model-set-based
method and the MIMO RBF-ARX model-based method show a
much better control performance compared with the physical
model-based one. And the MIMO RBF-ARX model-based method is
close to the ARX model-set-based one in general. From Table 7,
one can also see that the dynamic process of the MIMO RBF-ARX
model-based method is smoother and faster than that of the ARX
model-set-based method.
We did not compare the control results with those presented in
other literatures, because the referenced quadrotor helicopter in
this paper has not only a unique configuration but also larger
inertia, and it is quite different from those classic quadrotors.
Anti-disturbance tests are very important for this kind of
vehicles, because it is easily affected by the wind disturbance or
encounter sudden collisions in obstacle-dense environments. For
testing anti-disturbance performance, the pulse-type disturbance
Fig. 19. Inputs of real-time control based on identified RBF-ARX model.
Fig. 20. Control result based on RBF-ARX model.
Fig. 21. Inputs of control result based on RBF-ARX model.
Table 5
Standard deviation of steady-state errors.
Models Pitch (deg) Roll (deg) Yaw (deg)
Physical 0.3818 1.0347 0.2397
ARX model-set 0.1957 0.2295 0.0372
MIMO RBF-ARX 0.1465 0.1824 0.1825
Table 6
Overshoot of dynamic transition processes.
Models Pitch (deg) Roll (deg) Yaw (deg)
Physical 2.02 3.39 1.07
ARX model-set 0.85 2.04 1.29
MIMO RBF-ARX 1.12 0.51 1.43
Table 7
Transient time of dynamic transition processes (75% error band).
Models Pitch (s) Roll (s) Yaw (s)
Physical 4.3 6.7 5.7
ARX model-set 4.3 4.1 4.5
MIMO RBF-ARX 2.3 1.9 2.7
Fig. 18. Outputs of real-time control based on identified MIMO RBF-ARX model.
Values in square bracket denote the desired values of pitch, roll, and yaw angles;
the real line represents the pitch angle, the dashed line represents the roll angle,
and the dotted line represents the yaw angle.

with 720 V voltage and 1 s duration time are added on each
motor's control input. Fig. 22 shows the disturbance signals
(dotted lines) together with the input signals of the physical
model-based control approach (dash-dotted lines) and the
MIMO RBF-ARX model-based control approach (solid lines). Fig. 23
compares the anti-disturbance results of the physical model-based
control approach (dash-dotted lines) and the MIMO RBF-ARX
model-based control approach (solid lines). It is clear that the
MIMO RBF-ARX model-based LQR controller can stabilize the
quadrotor helicopter much faster than the physical model-based
one against the sudden disturbance. It is attributed to the
remarkable capability of the MIMO RBF-ARX model in the descrip-
tion of nonlinear dynamics of the quadrotor helicopter, not only at
some working-points but also in the full range of the quadrotor's
working area. The MIMO RBF-ARX model has the time-varying AR
coefficient matrices and can be locally linearized in any sampling
period easily. In other words, one MIMO RBF-ARX model can
be regarded as a composition of infinite amount of ARX models.
Therefore, the method of using one MIMO RBF-ARX model to
represent the global nonlinear dynamics of the quadrotor heli-
copter is far better than that of using only 16 ARX models or using
one linear physical model.
5. Conclusions
The referenced quadrotor helicopter in this paper has a unique
configuration. It has 4 propellers, 3 of which are horizontally
mounted to control its pitch and roll rotation while the last one is
vertically mounted to control its yaw rotation. It is also an unknown
nonlinear dynamical system whose physical model is not accurate.
By comparing the modeling accuracy of three modeling methods in
16 working regions, it is demonstrated that the ARX model-set and
the MIMO RBF-ARX model are much better than the physical
model. The MIMO RBF-ARX model has a close modeling accuracy
with the ARX model-set, besides, it avoids the tough work of the
ARX model-set identifications. By comparing the real-time control
results of the three model-based LQR controller, it is concluded that
the MIMO RBF-ARX model-based LQR control strategy presented in
this paper is better. The anti-disturbance tests also demonstrated
the superiority of the MIMO RBF-ARX model-based method. The
validity of the MIMO RBF-ARX modeling method for such kind of
plants was confirmed by this successful application.
Acknowledgments
The authors would like to thank the editors and reviewers for
his valuable comments.
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