Robust control of a Laboratory Helicopter

1. Available online at www.sciencedirect.com
Journal of the Franklin Institute 351 (2014) 259–276
Robust hierarchical control of a laboratory helicopter
Hao Liua,b
, Jianxiang Xic,n
, Yisheng Zhongb
a
School of Astronautics, Beihang University, Beijing 100191, PR China
b
Department of Automation, TNList, Tsinghua University, Beijing 100084, PR China
c
High-Tech Institute of Xi'an, Xi'an 710025, PR China
Received 12 December 2012; received in revised form 7 August 2013; accepted 23 August 2013
Available online 5 September 2013
Abstract
This paper deals with the robust position control problem for a three degree-of-freedom (3DOF)
laboratory helicopter. The 3DOF helicopter system is a nonlinear multiple-input multiple-output (MIMO)
uncertain system, and has the elevation, pitch, and travel angles. The proposed robust controller is a
hierarchical controller including an attitude controller and a position controller. The position controller
generates the desired reference of the pitch angle based on the tracking error of the travel angle, while the
attitude controller achieves the reference tracking of the pitch and elevation angles. It is proven that the
tracking errors of the three angles can converge into the given neighborhoods ultimately. Experimental
results on the laboratory helicopter demonstrate the effectiveness of the proposed hierarchical control
strategy.
& 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Unmanned aerial vehicles (UAVs) are suitable for various civil and military tasks and have
received much attention in the academic domain (see, e.g., [1–11]). Unmanned helicopters have
advantages over the fixed-wing UAVs because of their abilities to hover and vertically take-off
and land. However, the unmanned helicopter is an underactuated multiple-input multiple-output
(MIMO) system and its dynamics involves various uncertainties such as nonlinearity, coupling,
parametric uncertainties, and external disturbances. Its flight controller design is a challenge in
the control community.
www.elsevier.com/locate/jfranklin
0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jfranklin.2013.08.020
n
Corresponding author. Tel./fax: þ86 029 84744111.
E-mail addresses: liuhao13@buaa.edu.cn, liuhao9141@163.com (H. Liu), xijx07@mails.tsinghua.edu.cn (J. Xi),
zys-dau@mail.tsinghua.edu.cn (Y. Zhong).

2. The hierarchical control method has gained a growing interest for unmanned helicopters to
achieve trajectory tracking in recent years. By separating the helicopter system into two
subsystems with slow and fast dynamics respectively, hierarchical control can be achieved.
Based on the assumption that the closed-loop subsystem with fast dynamics can converge much
faster than the closed-loop subsystem with slow dynamics, the controllers for the two subsystems
can be designed separately as shown in [11]. In [8], a nonlinear hierarchical controller was
designed for a reduced-order model helicopter under the assumption that the motor dynamics
was much slower than that of the main rotor. The attitude and position subsystems, in general,
are selected as the subsystems with fast and slow dynamics respectively. In [9], the position and
attitude controllers were designed for the two subsystems of the HeLion helicopter respectively.
A hierarchical control approach was proposed in [10] with a model predictive controller to track
the desired trajectory references and a nonlinear robust controller to stabilize the attitude. In [12],
based on the trajectory linearization control method, translational motion and rotational motion
of a tripropeller vertical-takeoff-and-landing UAV were controlled respectively. The hierarchical
control schemes were also discussed for four-rotor helicopters as shown in [13–15]. In these
previous studies on the hierarchical control method as shown in [8–15], the influence of the inner
dynamics on the tracking performance has not been fully discussed. Besides, many works mainly
focus on restraining uncertainties involved in the rotational subsystem, whereas the tracking
performance of the closed-loop position subsystem cannot be guaranteed.
In this paper, a laboratory reduced-order helicopter with three degree-of-freedom (3DOF) is
used (see, Fig. 1). This helicopter is produced by the Quanser Company and has attracted an
increasing interest in the control community (see [16–25] to mention a few). As depicted in
Fig. 2, the 3DOF helicopter has three outputs: the elevation angle, the pitch angle, and the travel
angle, while possesses two control inputs: the voltages applied on the front motor and the back
motor as shown in [26]. Thus, this helicopter system is an underactuated MIMO system. If the
elevation and pitch angles are chosen as the interested outputs, it is an attitude control problem as
depicted in [23]. Besides, the position controller focuses on tracking the desired references for
the elevation and travel angles and pitch angle dynamics is considered as the inner dynamics.
Furthermore, the 3DOF helicopter system is an uncertain system containing nonlinear and
Fig. 1. The 3DOF helicopter.
H. Liu et al. / Journal of the Franklin Institute 351 (2014) 259–276260

3. unmodeled dynamics, coupling, parametric perturbations, and external disturbances. Therefore, it
serves as an ideal experimental platform to examine the effectiveness of the hierarchical control
scheme for unmanned helicopters.
Among previous research studies on this helicopter, a single-input single-output (SISO)
adaptive control method was applied to address the pitch angle control problem in [16]. A linear
quadratic regulation (LQR) controller was designed in [19] under input and state constraints
without discussing the influence of uncertainties. Adaptive motion control and nonlinear adaptive
model following control was achieved in [17,18] for the 3DOF helicopter respectively. In [24], a
robust controller is designed based on the H1 control theory for the 3DOF helicopter with
limitations on restraining the effects of nonlinear dynamics, coupling, and external disturbances.
In [25], a sliding mode observer was designed to achieve robust regulation of the 3DOF
helicopter. Although promising, the dynamical tracking performance cannot be specified by their
adaptive or sliding mode control methods. As shown in [20–23], our early research focused on
the robust attitude controller design for this lab helicopter. If the elevation and travel angles are
chosen as the outputs, the position of the pitch angle is required to change in order to achieve the
desired response of the travel angle. Therefore, the position control problem considered here is
more challenging, because the travel angle dynamics is coupled with the inner dynamics, i.e., the
pitch angle dynamics.
In this paper, based on the robust compensation technique, a robust hierarchical controller is
proposed. It consists of two controllers: an attitude controller and a position controller. The
attitude controller tracks the desired references of the elevation and pitch angles and the position
controller generates the desired pitch reference based on the tracking error of the travel angle.
The attitude or position controller is designed with a nominal controller and a robust
compensator. The nominal controllers are designed based on the nominal system for the desired
tracking, whereas the robust compensators are introduced to restrain the influence of the
uncertainties. The tracking errors of the elevation and travel angles are proven to be bounded
with specified boundaries under the effects of multiple uncertainties. The inner dynamics can
also be stabilized by selecting the robust compensator parameters.
The remaining parts of this paper are structured as follows. In Section 2, the 3DOF helicopter
mathematical model is described. The attitude and position controllers are designed in Section 3.
In Section 4, the robust tracking properties of the closed-loop system are analyzed. Experimental
results on the 3DOF helicopter are shown in Section 5. Conclusions are drawn in Section 6.
Counterweight
Arm
Base
Front
Motor
Back
Motor
Elevation
Axis
Travel
Axis
Pitch
Axis
Helicopter
Frame
Fig. 2. Schematic of the 3DOF helicopter.
H. Liu et al. / Journal of the Franklin Institute 351 (2014) 259–276 261

4. 2. Model description
As depicted in Fig. 2, the 3DOF helicopter includes two motors, a helicopter frame, an arm, a
counterweight, and a base. The arm is free to rotate around the elevation and travel axes. At one
end of the arm, a counterweight is attached. The helicopter frame is attached at the other end of
the arm and is free to rotate about its center (the pitch axis). The front and back motors are
installed at the two ends of the helicopter frame, applied with voltages vf ðtÞ and vbðtÞ
respectively. The twisting moment about the elevation axis can be generated by the sum of the
forces produced by the front and back motors; the pitch motion is resulted from the difference
of the forces produced by the two motors; the position of the pitch angle can change the
torque about the travel axis as shown in [21]. The dynamical model of the elevation angle ξðtÞ,
the pitch angle p(t), and the travel angle ψðtÞ can be described by the following equations (see
also in [17]):
€ξðtÞ ¼ a1ξðvf ðtÞ þ vbðtÞÞ cos pðtÞ þ a2ξ
_ξðtÞ þ a3ξ sin ξðtÞ þ dξðtÞ;
€pðtÞ ¼ a1pðvf ðtÞÀvbðtÞÞ þ a2p _pðtÞ þ a3p sin pðtÞ þ dpðtÞ;
€ψðtÞ ¼ a1ψ ðvf ðtÞ þ vbðtÞÞ sin pðtÞ þ a1ψ Vop sin pðtÞ þ a2ψ _ψðtÞ þ dψ ðtÞ;
8
><
>:
ð1Þ
where diði ¼ ξ; p; ψÞ are external disturbances, a1i; a2iði ¼ ξ; p; ψÞ; a3ξ, and a3p are the helicopter
parameters, and Vop is the sum of quiescent voltages of the front and back motors, which is a
positive constant. Let bξ ¼ a1ξ; bp ¼ a1p, and bψ ¼ a1ψ Vop. The parameters bi ði ¼ ξ; p; ψÞ can be
split up into the given nominal parts and the uncertain parts, which are denoted by N and Δ
respectively
bi ¼ bN
i þ Δbi; i ¼ ξ; p; ψ:
Define uξðtÞ ¼ vf ðtÞ þ vbðtÞ and upðtÞ ¼ vf ðtÞÀvbðtÞ. Then, the dynamical model (1) can be
rewritten as follows:
€ξðtÞ ¼ bN
ξ uξðtÞ þ qξðtÞ;
€pðtÞ ¼ bN
p upðtÞ þ qpðtÞ;
€ψðtÞ ¼ bN
ψ pðtÞ þ qψ ðtÞ;
8
>><
>>:
ð2Þ
where qiðtÞ ði ¼ ξ; p; ψÞ are the named equivalent disturbances and take the following forms:
qξðtÞ ¼ ðbξ cos pðtÞÀbN
ξ ÞuξðtÞ þ a2ξ
_ξðtÞ þ a3ξ sin ξðtÞ þ dξðtÞ;
qpðtÞ ¼ ðbpÀbN
p ÞupðtÞ þ a2p _pðtÞ þ a3p sin pðtÞ þ dpðtÞ;
qψ ðtÞ ¼ a1ψ uξðtÞ sin pðtÞ þ bψ sin pðtÞÀbN
ψ pðtÞ þ a2ψ _ψðtÞ þ dψ ðtÞ:
8
>><
>>:
ð3Þ
Remark 1. From Eq. (2), one can see that the desired response of the travel angle can be
achieved by changing the position of the pitch angle. Thus, if the elevation and travel angles are
selected as outputs, the pitch angle dynamics can be considered as the inner dynamics. Besides, it
should be noted that the nominal model (2) only contains primary linear terms, whereas other
known terms are considered in equivalent disturbances in order to simplify the nominal model.
Assumption 1. The uncertain parameters a2i; bi ði ¼ ξ; p; ψÞ; a3ξ, and a3p are bounded and
bN
i ði ¼ ξ; p; ψÞ are positive and satisfy that jΔbijobN
i .
H. Liu et al. / Journal of the Franklin Institute 351 (2014) 259–276262

5. Assumption 2. The elevation and pitch angles satisfy that ξðtÞA½Àπ=2 þ δξ; π=2ÀδξŠ and
pðtÞA½Àπ=2 þ δp; π=2ÀδpŠ with δξ and δp positive constants.
Remark 2. There exist mechanical limits for the elevation and pitch angles and the helicopter is
required to avoid overturning during the flight. The constants δξ and δp can be measured easily
and δξ ¼ π=4 and δp ¼ π=36.
Define the positive constants ρi ði ¼ ξ; p; ψÞ as ρξ ¼ maxjbξ cos pðtÞÀbN
ξ j=bN
ξ , ρp ¼ jbpÀbN
p j=
bN
p , and ρψ ¼ maxjbψ sin pðtÞ=pðtÞÀbN
ψ j=bN
ψ .
Remark 3. If Assumptions 1 and 2 hold, one can obtain that 0rρio1 ði ¼ ξ; p; ψÞ.
Assumption 3. The external disturbances di ði ¼ ξ; p; ψÞ are additional torques acting on the
helicopter from the wind gusts, which are continuously differentiable. The external disturbances
and their first and second order derivatives dðkÞ
i ði ¼ ξ; p; ψ; k ¼ 0; 1; 2Þ are bounded almost
everywhere.
In this paper, the elevation and travel angles are required to track the desired references,
denoted by riði ¼ ξ; ψÞ respectively.
Assumption 4. The desired references and their derivatives rðjÞ
ξ ðj ¼ 0; 1; 2Þ and rðkÞ
ψ ðk ¼ 0; 1;
2; 3; 4Þ are piecewise uniformly bounded.
3. Robust hierarchical controller design
In this section, the attitude and position controllers are designed respectively in order to
achieve the position control for the elevation and travel angels. From the 3DOF helicopter model
(2), one can see that the elevation angle ξðtÞ can be controlled by the input uξðtÞ directly. Besides,
the position of the pitch angle p(t) results in the motion of the travel angle ψðtÞ, whereas p(t) can
be controlled by the input upðtÞ directly. Therefore, the position controller generates the desired
reference of the pitch angle based on the tracking error of the travel angle and the attitude
controller aims to achieve the desired reference tracking for the elevation and pitch angles.
3.1. Position controller design
From Eq. (2), the mathematical model of the travel channel can be rewritten as
€ψðtÞ ¼ bN
ψ bpðtÞ þ bN
ψ ðpðtÞÀbpðtÞÞ þ qψ ðtÞ; ð4Þ
where bpðtÞ is the virtual control input to control the travel angle. bpðtÞ consists of the nominal
control input uN
ψ ðtÞ and the robust compensating input uRC
ψ ðtÞ as
bpðtÞ ¼ uψ ðtÞ ¼ uN
ψ ðtÞ þ uRC
ψ ðtÞ: ð5Þ
Define eψ1ðtÞ ¼ ψðtÞÀrψ ðtÞ and eψ2ðtÞ ¼ _eψ1ðtÞ. Design the nominal control law of the travel
angle as
uN
ψ ðtÞ ¼ Àðkψ1eψ1ðtÞ þ kψ2eψ2ðtÞÞ=bN
ψ ; ð6Þ
H. Liu et al. / Journal of the Franklin Institute 351 (2014) 259–276 263

6. where kψj ðj ¼ 1; 2Þ are positive constants to be determined. The robust filter Fψ ðsÞ is introduced
to restrain the effects of the uncertainties in the travel channel and is given by
Fψ ðsÞ ¼ Ff ψ ðsÞFgψ ðsÞ ¼
f ψ
s þ f ψ
gψ
s þ gψ
;
where s is the Laplace operator and the robust compensator parameters f ψ and gψ are sufficiently
large positive constants.
Design the robust compensating input as follows:
uRC
ψ ðsÞ ¼ ÀFψ ðsÞbqψ ðsÞ=bN
ψ ; ð7Þ
where
bqψ ðtÞ ¼ qψ ðtÞ þ bN
ψ pðtÞÀbN
ψ bpðtÞÀ€rψ ðtÞ: ð8Þ
If f ψ and gψ are sufficiently large, one can expect that Ff ψ ðsÞ and Fgψ ðsÞ have sufficiently wide
frequency bandwidths and the primary components of the interested signals can pass the filters.
Therefore, the gains of these filters would approximate 1 (see [23,27] to mention a few). In this
case, the robust compensating input uRC
ψ ðtÞ would approximate Àbqψ ðtÞ=bN
ψ and the effects of the
uncertainties bqψ ðtÞ could be restrained. However, in practical applications, bqψ ðtÞ cannot be
measured or obtained directly and thereby the robust compensating control input uRC
ψ ðtÞ in Eq. (7)
is not implementable. Therefore, from Eqs. (4) and (8), one can obtain that
bqψ ðtÞ ¼ €eψ1ðtÞÀbN
ψ uψ ðtÞ: ð9Þ
In the above expression of bqψ ðtÞ, €eψ1 still cannot be obtained. Therefore, a new realization of the
robust compensating control input uRC
ψ ðtÞ is required, which is independent of bqψ ðtÞ or €eψ1ðtÞ.
Then, from Eq. (7) and (9), one can have that
uRC
ψ ðsÞ ¼ À
f ψ gψ
bN
ψ ðs þ f ψ Þðs þ gψ Þ
bqψ ðsÞ ¼ À
f ψ gψ
bN
ψ ðs þ f ψ Þðs þ gψ Þ
ðs2
eψ1ðsÞÀbN
ψ uψ ðsÞÞ
¼ À
f ψ gψ
bN
ψ
eψ1ðsÞ þ
f ψ gψ
bN
ψ
zψ2ðsÞ;
where
zψ2ðsÞ ¼
1
s þ f ψ
ðf ψ þ gψ Þseψ1ðsÞ þ f ψ gψ eψ1ðsÞ þ bN
ψ uψ ðsÞ
s þ gψ
¼
1
s þ f ψ
ðf ψ þ gψ Þeψ1ðsÞ þ
Àg2
ψ eψ1ðsÞ þ bN
ψ uψ ðsÞ
s þ gψ
!
:
If define
zψ1ðsÞ ¼
Àg2
ψ eψ1ðsÞ þ bN
ψ uψ ðsÞ
s þ gψ
;
then it follows that
zψ2ðsÞ ¼
1
s þ f ψ
ððf ψ þ gψ Þeψ1ðsÞ þ zψ1ðsÞÞ:
H. Liu et al. / Journal of the Franklin Institute 351 (2014) 259–276264

7. Therefore, from the above expressions, uRC
ψ ðtÞ can be realized with zψ1ðtÞ and zψ2ðtÞ as
_zψ1ðtÞ ¼ Àgψ zψ1ðtÞÀg2
ψ eψ1ðtÞ þ bN
ψ uψ ðtÞ;
_zψ2ðtÞ ¼ Àf ψ zψ2ðtÞ þ ðf ψ þ gψ Þeψ1ðtÞ þ zψ1ðtÞ;
uRC
ψ ðtÞ ¼ f ψ gψ ðzψ2ðtÞÀeψ1ðtÞÞ=bN
ψ :
8
>><
>>:
ð10Þ
From Eq. (10), one can see that uRC
ψ ðtÞ can be realized without bqψ ðtÞ or €eψ1ðtÞ.
3.2. Attitude controller design
In this section, the elevation angle ξðtÞ and the pitch angle p(t) are required to track the desired
references rξðtÞ and rpðtÞ respectively, where rpðtÞ ¼ bpðtÞ is generated by the position controller.
Define eξ1ðtÞ ¼ ξðtÞÀrξðtÞ; ep1ðtÞ ¼ pðtÞÀrpðtÞ, and ei2ðtÞ ¼ _ei1ðtÞ ði ¼ ξ; pÞ. Similarly, the attitude
control inputs uiðtÞ ði ¼ ξ; pÞ also consist of two parts as
uiðtÞ ¼ uN
i ðtÞ þ uRC
i ðtÞ; i ¼ ξ; p; ð11Þ
where
uN
i ðtÞ ¼ Àðki1ei1ðtÞ þ ki2ei2ðtÞÞ=bN
i ;
uRC
i ðsÞ ¼ ÀFiðsÞbqiðsÞ=bN
i ;
bqiðtÞ ¼ qiðtÞÀ€riðtÞ; i ¼ ξ; p;
8
><
>:
ð12Þ
and FiðsÞ ¼ f igi=ðs þ f iÞ=ðs þ giÞ. The parameters kij ði ¼ ξ; p; j ¼ 1; 2Þ are positive constants to
be determined and f i and gi ði ¼ ξ; pÞ have sufficiently large positive values. However,
bqi ði ¼ ξ; pÞ involved in Eq. (12) cannot be measured directly. Therefore, the robust
compensating input uRC
i ðtÞ ði ¼ ξ; pÞ can be realized in a similar way as Eq. (10).
3DOF
Helicopter
, , ,p p
u ,
Nominal
Controller
Robust
Compensator
Nominal
Controller
Robust
Compensator
,p pr r
,r r
N
u
RC
u
N
pu
RC
pu
pu
Robust
Compensator
Nominal
Controller
,r rN
u
RC
uu
Travel
Angle
Pitch
Angle
Elevation
Angle
Attitude
Controller
Position
Controller
Fig. 3. The block diagram of the robust hierarchical control system.
H. Liu et al. / Journal of the Franklin Institute 351 (2014) 259–276 265

8. The block diagram of the robust hierarchical control system is depicted in Fig. 3.
Remark 4. The resulted hierarchical controller is a linear time-invariant one. In practical
applications, it is easy to be implemented.
4. Robust properties analysis
In this section, the robust position tracking properties of the closed-loop system constructed as
in the previous section will be analyzed in two steps. Firstly, it will prove that the tracking error
of the elevation channel will be bounded with the given boundary ultimately. Then, the tracking
errors of the pitch and travel angles will also be proven to converge into the specified
neighborhoods of the origin in a finite time.
The norms used in this section are defined as follows:
‖y‖1 ¼ max
i
sup
t Z0
jyiðtÞj;
‖HðsÞ‖1 ¼ ‖h‖1 ¼ max
i
∑
j
R1
0 jhijðtÞj dt
!
;
8
>>><
>>>:
where yðtÞ ¼ ½yiðtÞŠALn
1, hðtÞ ¼ ½hijðtÞŠ ¼ ℓÀ1
ðHðsÞÞALnÂm
1 , and ℓÀ1
ðÁÞ is the inverse Laplace
transform. If y ¼ HðsÞu, one can obtain that ‖y‖1 r‖HðsÞ‖1‖u‖1.
4.1. The robust properties of the elevation channel
From Eqs. (2), (11) and (12), one can obtain that
_eξðtÞ ¼ AξHeξðtÞ þ BξðbN
ξ uRC
ξ ðtÞ þ bqξðtÞÞ; ð13Þ
where eξðtÞ ¼ ½eξ1ðtÞ eξ2ðtÞŠT
and
AξH ¼
0 1
Àkξ1 Àkξ2
" #
; Bξ ¼
0
1
:
Remark 5. It should be noted that the nominal controller parameters kξ1 and kξ2 are required to
be selected such that the matrix AξH is a Hurwitz matrix.
Lemma 1. If f ξ and gξ are sufficiently large and satisfy f ξbgξ40, one can obtain a positive
constant λδξ such that δξ ¼ JðsI2Â2ÀAξHÞÀ1
Bξð1ÀFξÞJ1 rλδξ=gξ and δξ can be made as small
as desired.
Proof. See Appendix A. □
Theorem 1. If Assumptions 1–4 are met, for given bounded initial state eξð0Þ and a given
positive constant εξ, there exist a positive constant Tn
ξ and sufficiently large positive parameters
f ξ and gξ satisfying f ξbgξ, such that eξðtÞ is bounded and satisfies that jeξðtÞjrεξ; 8tZTn
ξ .
Proof. Substitute the expression of uRC
ξ ðsÞ in Eq. (12) into Eq. (13), one can obtain that
Jeξ J1 rλeξð0Þ þ δξ Jbqξ J1; ð14Þ
where λeξð0Þ ¼ maxjsupt Z0jcT
ξjeAξHt
eξð0Þj, cξj is a 2 Â 1 vector with one on the jth row and zero
elsewhere, and I2Â2 is a 2 Â 2 unit matrix.
H. Liu et al. / Journal of the Franklin Institute 351 (2014) 259–276266

9. From Eqs. (3) and (12), one can obtain positive constants ζqξe and ζqξc such that
‖bqξ‖1 rbN
ξ ρξ‖uξ‖1 þ ζqξe‖eξ‖1 þ ζqξc; ð15Þ
where ζqξe and ζqξc satisfy that ζqξe Zja2ξj and ζqξc Z‖a2ξ _rξ‖1 þ ja3ξj þ ‖dξ‖1 þ ‖€rξ‖1
respectively. From Eqs. (11) and (12), one has that
‖uξ‖1 rζuξe‖eξ‖1=bN
ξ þ ‖bqξ‖1=bN
ξ ; ð16Þ
where ζuξe ¼ jkξ1j þ jkξ2j. Combining Eqs. (15) and (16), one has that
‖bqξ‖1 rλeξ‖eξ‖1 þ λcξ; ð17Þ
where λeξ ¼ ðρξζuξe þ ζqξeÞ=ð1ÀρξÞ and λcξ ¼ ζqξc=ð1ÀρξÞ. Then, from Eqs. (14) and (17), and
Lemma 1, one can obtain that
‖bqξ‖1 rðλeξλeξð0Þ þ λcξÞ=ð1ÀδξλeξÞ;
‖eξ‖1 rðλeξð0Þ þ δξλcξÞ=ð1ÀδξλeξÞ:
(
ð18Þ
From Eq. (18), one can see that bqξðtÞ and eξðtÞ are bounded. Then, from Eq. (16), it follows that
uξðtÞ is also bounded. Therefore, there exist positive constants ηqξ; ηeξ, and ηuξ satisfying that
‖bqξ‖1 rηqξ;
‖eξ‖1 rηeξ;
‖uξ‖1 rηuξ:
8
:
ð19Þ
Moreover, from Eqs. (12), (13), and (19), one can obtain that
max
j
jeξjðtÞjrmax
j
jcT
ξjeAξHt
eξð0Þj þ δξηqξ:
Thus, for the given positive constant εξ and given initial state eξð0Þ, one can find positive
constant Tn
ξ and sufficiently large positive parameters f ξ and gξ satisfying that
f ξbgξ42λδξηqξ=εξ, such that jeξðtÞjrεξ; 8tZTn
ξ . □
Remark 6. It should be noted that the positions of the pitch and travel angles do not influence
the robust tracking properties of the elevation channel. Therefore, the values of f ξ and gξ can be
determined before the robust properties of the pitch and travel angles are analyzed.
4.2. The robust properties of the pitch and travel angles
In this section, the robust tracking properties of the pitch and travel angles are discussed
together. Thus, define eψ ðtÞ ¼ ½eψ1ðtÞ eψ2ðtÞ ep1ðtÞ ep2ðtÞŠT
.
Similarly, from Eqs. (2), (5), (6), (7), (11), and (12), one can have that
_eψ ðtÞ ¼ AψHeψ ðtÞ þ Bψ ½bN
ψ uRC
ψ ðtÞ þ bqψ ðtÞ bN
p uRC
p ðtÞ þ bqpðtÞŠT
; ð20Þ
where AψH ¼ diagðA′
ψH; A′
pHÞ; Bψ ¼ diagðB′
ψ ; B′
pÞ, and
A′
iH ¼
0 1
Àki1 Àki2
#
; B′
i ¼
0
1
; i ¼ ψ; p:
The parameters kψ1; kψ2; kp1, and kp2 are selected such that AψH is a Hurwitz matrix.
H. Liu et al. / Journal of the Franklin Institute 351 (2014) 259–276 267

10. Then, from Eqs. (7), (12), and (20), one can obtain that
‖eψ ‖1 rλeψð0Þ þ δψ ‖Δψ ‖1; ð21Þ
where Δψ ðtÞ ¼ ½bqψ ðtÞ bqpðtÞ=f ψ =gψ ŠT
, δψ ¼ maxfδ′
ψ ; δ′
pg, δ′
ψ ¼ ‖ðsI2Â2ÀA′
ψHÞÀ1
B′
ψ ð1ÀFψ Þ‖1,
δ′
p ¼ f ψ gψ ‖ðsI2Â2ÀA′
pHÞÀ1
B′
pð1ÀFpÞ‖1, λeψð0Þ ¼ maxjsupt Z0jcT
ψjeAψHt
eψ ð0Þj, and cψj is a 4 Â 1
vector with one on the jth row and zeros elsewhere.
Lemma 2. If the robust compensator parameters f p; gp; f ψ , and gψ satisfy that
f ibgi40 ði ¼ p; ψÞ and f p; gpbf ψ ; gψ , there exist positive constants λeψ ; λcψ ; λep, and λcp such
that
‖bqψ ‖1 rλeψ ‖eψ ‖1 þ λcψ ;
‖bqp‖1 rλepgψ f ψ ‖eψ ‖1 þ λcpgψ f ψ :
(
ð22Þ
Proof. See Appendix B. □
Theorem 2. If Assumptions 1–4 hold, for a given positive constant εψ and given bounded
initial states of the pitch and travel angles, there exist a positive constant Tn
ψ and sufficiently
large positive parameters f p; gp; f ψ , and gψ , which satisfy that f ibgi ði ¼ p; ψÞ and f p; gpb
f ψ ; gψ , such that the tracking error eψ is bounded and satisfies that jeψ ðtÞjrεψ ; 8tZTn
ψ .
Proof. If f p and gp are much larger than f ψ and gψ , and f i ði ¼ p; ψÞ are much larger than
gi ði ¼ p; ψÞ, one can obtain that δψ can be made as small as desired in a similar way. In this case,
define λepψ ¼ maxfλeψ ; λepg and λcpψ ¼ maxfλcψ ; λcpg. Similarly with the elevation channel, from
Eqs. (21) and (22), and Lemma 2, one can have that
‖bqψ ‖1 rðλepψ λeψð0Þ þ λcpψ Þ=ð1Àδψ λepψ Þ;
‖eψ ‖1 rðλeψð0Þ þ δψ λcpψ Þ=ð1Àδψ λepψ Þ:
(
ð23Þ
Therefore, one can see that bqψ ðtÞ and eψ ðtÞ are bounded, that is, one can find positive constants
ηqψ and ηeψ satisfying that
‖bqψ ‖1 rηqψ ;
‖eψ ‖1 rηeψ :
(
ð24Þ
Moreover, from Eqs. (7), (12), (20), and (24), one can have that
max
j
jeψjðtÞjrmax
j
jcT
ψjeAψH t
eψ ð0Þj þ δψ ηqψ :
For the given positive constant εψ and the given initial condition, one can find positive constant
Tn
ψ and positive parameters f p; gp; f ψ , and gψ with sufficiently large values and satisfy that
f ibgi ði ¼ p; ψÞ and f p; gpbf ψ ; gψ , such that jeψ ðtÞjrεψ ; 8tZTn
ψ . □
Remark 7. One can see that the tracking errors of the elevation and travel angles are proven to
converge into the given neighborhoods of the origin under the influence of various uncertainties.
Furthermore, it should be noted that the robust tracking performance of the pitch angle can also
be guaranteed by selecting the robust compensator parameters.
H. Liu et al. / Journal of the Franklin Institute 351 (2014) 259–276268

11. 5. Experimental results and discussions
In this section, results of real-time implementation of the proposed hierarchical controller
on the 3DOF helicopter are given to verify the effectiveness of the designed hierarchical
control scheme. The three angles are measured by three encoders respectively. The effective
resolutions of encoders for the elevation, travel, and pitch angles are 0.08791, 0.04391,
and 0.08791 respectively as shown in [26]. The designed control scheme is implemented
on a dSPACE system with a sampling time 10 ms. Then, the control signals are outputted to the
two motors via amplifying. _rpðtÞ is estimated by the change rate of rpðtÞ. In practical applica-
tions, let rpðsÞ ¼ bpðsÞ=ð0:01s þ 1Þ2
to reduce the noise in bpðsÞ. The nominal values of helicopter
parameters are bN
ξ ¼ 0:0858, bN
p ¼ 0:581, and bN
ψ ¼ 0:0825. The nominal controller para-
meters are selected as kξ1 ¼ 2:79; kξ2 ¼ 2:38, kp1 ¼ 34:9; kp2 ¼ 31:9; kψ1 ¼ 0:496, and
kψ2 ¼ 0:330. Choose the robust compensator parameters as f ξ ¼ 25; gξ ¼ 5; f p ¼ 25;
gp ¼ 5; f ψ ¼ 1, and gψ ¼ 0:2 in order to guarantee the robust tracking performance of the
closed-loop system. In practical applications, by the following approach, one can try to make the
3DOF helicopter avoiding overturning during the flight: generate the desired references without
overturning missions, make Assumption 4 hold, and try to make eið0Þ ði ¼ ξ; p; ψÞ as small as
possible.
0 10 20 30 40 50
−0.1
0
0.1
0.2
Time (s)
Elevationangle(deg)
Reference signal
Elevation angle
0 10 20 30 40 50
−0.05
0
0.05
0.1
Time (s)
Travelangle(deg)
Reference signal
Travel angle
0 10 20 30 40 50
0
5
10
15
Time (s)
Pitchangle(deg)
Reference signal
Pitch angle
Fig. 4. Responses of the three angles with RCITA. (a) Response of the elevation angle, (b) response of the travel angle
and (c) response of the pitch angle.
H. Liu et al. / Journal of the Franklin Institute 351 (2014) 259–276 269

12. Case 1. Hovering mission: Firstly, the 3DOF helicopter is needed to carry out a simple task. The
elevation and travel angles are required to be stabilized at 01. Corresponding responses are
depicted in Fig. 4. Besides, the responses without robust compensating input uRC
ψ ðtÞ in the travel
angle (RCITA) are presented in Fig. 5. From Fig. 4, one can see that the steady-state errors are
less than 70.11 and 70.051 for the elevation angle and the travel angle with RCITA
respectively. In contrast, the steady-state errors without RCITA are less than 70.11 in the
elevation and travel channels. Both of the closed-loop systems achieve good steady-state
performance. From the third subfigures of Figs. 4 and 5, one can see that the pitch angle is not 01
in the hovering mission. When the designed control algorithm is started to implement on the
dSPACE processor, the initial position of the pitch angle is 01. Since it is difficult to place
the laboratory helicopter in an absolute horizontal plane at the initial time, the pitch angle may be
nonzero in this mission. On the other hand, position measurements of the travel angle are
discontinuous and the travel angular velocity cannot be obtained directly. Therefore, the
derivative of the travel angle is used by the nominal position controller to generate the pitch
angle reference and the pitch angle reference involves some jumps.
Case 2. Trajectory tracking mission: In this experiment, large-angle references are required to
track for the elevation and travel angles in the coupling condition. The desired signals
0 10 20 30 40 50
−0.1
0
0.1
0.2
Time (s)
Elevationangle(deg)
Reference signal
Elevation angle
0 10 20 30 40 50
−0.1
0
0.1
0.2
Time (s)
Travelangle(deg)
Reference signal
Travel angle
0 10 20 30 40 50
0
5
10
15
Time (s)
Pitchangle(deg)
Reference signal
Pitch angle
Fig. 5. Responses of the three angles without RCITA. (a) Response of the elevation angle, (b) response of the travel angle
and (c) response of the pitch angle.
H. Liu et al. / Journal of the Franklin Institute 351 (2014) 259–276270

13. riðsÞ ði ¼ ξ; ψÞ are generated by riðsÞ ¼ rswr
i ðsÞ=ð3s þ 1Þ2
, where rswr
i ðsÞ is the square waveform
reference with amplitude of 201. The tracking performance of the closed-loop system is seriously
affected by the uncertainties such as parametric uncertainties, nonlinearity, coupling, and
external disturbances in the three angle dynamics. The responses with RCITA and without
RCITA are depicted in Figs. 6 and 7 respectively. In order to compare the tracking performance,
the tracking errors of the elevation angle and the travel angle are presented in Fig. 8. From these
figures, one can see that both methods can achieve good dynamical tracking performance for the
two angles, but better steady-state performance is achieved for the closed-loop system of
the travel angle with RCITA. The third subfigures of Figs. 6 and 7 show the inner dynamics, i.e.,
the pitch angle dynamics. One can see that the pitch angle tracks the reference generated by the
tracking error of the travel angle well.
6. Conclusions
A robust hierarchical controller was proposed to achieve the position control for a laboratory
helicopter. It includes an attitude controller and a position controller. Both the attitude and
position controllers consist of a nominal controller and a robust compensator. It is proven that the
tracking errors of the three angles are ultimately bounded and the boundaries can be specified.
0 50 100 150 200
−10
0
10
20
Time (s)
Elevationangle(deg)
Reference signal
Elevation angle
0 50 100 150 200
0
10
20
30
Time (s)
Travelangle(deg)
Reference signal
Travel angle
0 50 100 150 200
−10
0
10
20
30
Time (s)
Pitchangle(deg)
Reference signal
Pitch angle
Fig. 6. Responses of the three angles with RCITA. (a) Response of the elevation angle, (b) response of the travel angle
and (c) response of the pitch angle.
H. Liu et al. / Journal of the Franklin Institute 351 (2014) 259–276 271

14. 0 50 100 150 200
−10
0
10
20
Time (s)
Elevationangle(deg)
Reference signal
Elevation angle
0 50 100 150 200
0
10
20
30
Time (s)
Travelangle(deg)
Reference signal
Travel angle
0 50 100 150 200
−20
−10
0
10
20
30
Time (s)
Pitchangle(deg)
Reference signal
Pitch angle
Fig. 7. Responses of the three angles without RCITA. (a) Response of the elevation angle, (b) response of the travel angle
and (c) response of the pitch angle.
0 50 100 150 200
−2
0
2
4
6
Time (s)
Trackingerrors(deg)
Tracking errors without RCITA
Tracking errors with RCITA
0 50 100 150 200
−4
−2
0
2
4
6
8
Time (s)
Trackingerrors(deg)
Tracking errors without RCITA
Tracking errors with RCITA
Fig. 8. Tracking error comparison. (a) Tracking errors of the elevation angle and (b) tracking errors of the travel angle.
H. Liu et al. / Journal of the Franklin Institute 351 (2014) 259–276272

15. Experimental results on the 3DOF helicopter demonstrated the effectiveness of the proposed
control method.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grants
61174067, 61203071, and 61374054, as well as Shaanxi Province Natural Science Foundation
Research Projection under Grants 2013JQ8038.
Appendix A. Proof of Lemma 1
Let
dξHðsÞ ¼ detðsI2Â2ÀAξHÞ ¼ ðs þ sξ1Þðs þ sξ2Þ ¼ ðs2
þ kξ2s þ kξ1Þ; ðA:1Þ
and
χξðsÞ ¼ ðs þ 1ÞðsI2Â2ÀAξHÞÀ1
ÀI2Â2: ðA:2Þ
It follows that
δξ rð‖χξðsÞ‖1 þ ‖I2Â2‖1Þ‖Bξ‖1 ð1ÀFξÞ
1
s þ 1
1
: ðA:3Þ
The term ðsI2Â2ÀAξHÞÀ1
can be expressed as
ðsI2Â2ÀAξHÞÀ1
¼
NξHðsÞ
dξHðsÞ
; ðA:4Þ
where
NξHðsÞ ¼
s þ kξ2 1
Àkξ1 s
#
: ðA:5Þ
Let χξðsÞ ¼ ½χξ;jkðsÞŠ2Â2. Then, χξ;jkðsÞ has the following form:
χξ;jkðsÞ ¼
χξ1
ξ;jk
s þ sξ1
þ
χξ2
ξ;jk
s þ sξ2
;
where χξ1
ξ;jk and χξ2
ξ;jk are constants. It follows that
‖χξðsÞ‖1 rmax
j
∑
2
k ¼ 1
‖χξ;jkðsÞ‖1
rmax
j
∑
2
k ¼ 1

18. χξ1
ξ;jk
sξ1

21. þ

24. χξ2
ξ;jk
sξ2

27. !
:
Therefore, there exists a positive constant λ1δξ, such that
ð‖χξðsÞ‖1 þ ‖I2Â2‖1Þrλ1δξ: ðA:6Þ
Furthermore, one can obtain that ‖Bξ‖1 ¼ 1 and
ð1ÀFξÞ
1
s þ 1
1
¼
ηf ξ
s þ f ξ
þ
ηgξ
s þ gξ
þ
1
gξ
η1
s þ 1
1
;
H. Liu et al. / Journal of the Franklin Institute 351 (2014) 259–276 273

28. where
ηf ξ
¼ Àf ξgξ=ðf ξÀgξÞ=ðf ξÀ1Þ;
ηgξ
¼ f ξgξ=ðf ξÀgξÞ=ðgξÀ1Þ;
η1 ¼ ðÀf ξgξÀg2
ξ þ gξÞ=ðf ξÀ1Þ=ðgξÀ1Þ:
8
:
If the positive constants f ξ and gξ are sufficiently large, one can obtain a positive constant λ2δξ
such that
‖Bξ‖1 ð1ÀFξÞ
1
s þ sa
1
r
λ2δξ
gξ
: ðA:7Þ
Let λδξ ¼ λ1δξλ2δξ. From Eqs. (A.(3), A.6), and (A.7), one can obtain that
δξ r
λδξ
gξ
: ðA:8Þ
If the positive constants f ξ and gξ are sufficiently large, the constant λδξ is irrelevant of f ξ and gξ.
In this case, if gξ is sufficiently large, the variable δξ can be made as small as desired. It follows
that Lemma 1 can hold.
Appendix B. Proof of Lemma 2
From Eqs. (5) and (6), one has that
rpðtÞ ¼ uψ ðtÞ ¼ Àðkψ1eψ1ðtÞ þ kψ2eψ2ðtÞÞ=bN
ψ þ uRC
ψ ðtÞ: ðB:1Þ
From Eqs. (3), (8), and (19), one can obtain that
‖bqψ ‖1 rζqψe‖eψ ‖1 þ ρψ bN
ψ ‖rp‖1 þ ζqψc; ðB:2Þ
where ζqψe and ζqψc satisfy that
ζqψe Zja2ψ j þ bN
ψ þ ρψ bN
ψ ;
ζqψc Zja1ψ jηuξ þ ja2ψ j‖_rψ ‖1 þ ‖€rψ ‖1 þ ‖dψ ‖1:
(
Combining Eqs. ((7) and B.2), one can have that
‖uRC
ψ ‖1 rζqψe‖eψ ‖1=bN
ψ þ ρψ ‖rp‖1 þ ζqψc=bN
ψ : ðB:3Þ
Let
λr0ep ¼ ðjkψ1j þ jkψ2j þ ζqψeÞ=bN
ψ =ð1Àρψ Þ;
λr0cp ¼ ζqψc=bN
ψ =ð1Àρψ Þ:
(
Then, from Eqs. (B.(1) and B.3), one can obtain that
‖rp‖1 rλr0ep‖eψ ‖1 þ λr0cp: ðB:4Þ
Differentiating both sides of Eq. (B.1) leads to
_rpðtÞ ¼ Àðkψ1eψ2ðtÞ þ kψ2 _eψ2ðtÞÞ=bN
ψ þ _uRC
ψ ðtÞ: ðB:5Þ
Besides, from Eq. (20), one has that
_eψ2ðtÞ ¼ Àkψ1eψ1ðtÞÀkψ2eψ2ðtÞ þ bN
ψ uRC
ψ ðtÞ þ bqψ ðtÞ: ðB:6Þ
H. Liu et al. / Journal of the Franklin Institute 351 (2014) 259–276274

29. From Eqs. ((7), B.5), and (B.6), one can obtain positive constants ζr1ep; ζr1qp, and ζr1cp such that
‖_rp‖1 rζr1ep‖eψ ‖1 þ ζr1qp‖1ÀFψ ‖1‖bqψ ‖1 þ ‖_uRC
ψ ‖1 þ ζr1cp: ðB:7Þ
From Eq. (7), one has that
‖_uRC
ψ ‖1 rgψ ‖
f ψ
s þ f ψ
s
s þ gψ
‖1‖bqψ ‖1=bN
ψ rgψ ‖1ÀFgψ ‖1‖bqψ ‖1=bN
ψ :
It follows that
‖_rp‖1 rζr1ep‖eψ ‖1 þ ζr1qp‖1ÀFψ ‖1‖bqψ ‖1 þ gψ ‖1ÀFgψ ‖1‖bqψ ‖1=bN
ψ þ ζr1cp: ðB:8Þ
Moreover, if the positive parameters f ψ and gψ are sufficiently large, one can see that
‖1ÀFψ ‖1 and ‖1ÀFgψ ‖1 are bounded. Thus, there exist positive constants λr1ep and λr1cp
satisfying
λr1epgψ 4ζr1ep þ ðζr1qp‖1ÀFψ ‖1 þ gψ ‖1ÀFgψ ‖1=bN
ψ Þðζqψe þ ρψ bN
ψ λr0epÞ;
λr1cpgψ 4ζr1cp þ ðζr1qp‖1ÀFψ ‖1 þ gψ ‖1ÀFgψ ‖1=bN
ψ Þðζqψc þ ρψ bN
ψ λr0cpÞ:
(
Then, from Eqs. (B.(2), B.4), and (B.8), it follows that
‖_rp‖1 rλr1epgψ ‖eψ ‖1 þ λr1cpgψ : ðB:9Þ
Similarly, there exist positive constants λr2ep and λr2cp such that
‖€rp‖1 rλr2epgψ f ψ ‖eψ ‖1 þ λr2cpgψ f ψ : ðB:10Þ
From Eqs. (B.(2) and B.4), one can obtain positive constants λeψ and λcψ such that
‖bqψ ‖1 rλeψ ‖eψ ‖1 þ λcψ : ðB:11Þ
Furthermore, from Eqs. (3), (11), and (12), there exist positive constants ζr1p; ζr2p; ζrep, and
ζrcp such that
‖bqp‖1 rζr1p‖_rp‖1 þ ζr2p‖€rp‖1 þ ζrep‖eψ ‖1 þ ζrcp: ðB:12Þ
If f ψ and gψ are sufficiently large and satisfy that f ψ bgψ 40, from Eqs. (B.(9), B.10), and
(B.12), one can obtain positive constants λep and λcp such that
‖bqp‖1 rλepgψ f ψ ‖eψ ‖1 þ λcpgψ f ψ : ðB:13Þ
From Eqs. (B.(11) and B.13), one can see that Lemma 2 holds.
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