HAL Id: hal-01722001
https://hal.archives-ouvertes.fr/hal-01722001
Submitted on 29 Sep 2020
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
3D trajectory tracking control of quadrotor UAV with
on-line disturbance compensation
Yasser Bouzid, Houria Siguerdidjane, Yasmina Bestaoui
To cite this version:
Yasser Bouzid, Houria Siguerdidjane, Yasmina Bestaoui. 3D trajectory tracking control of quadrotor
UAV with on-line disturbance compensation. 1st IEEE Conference on Control Technology and Ap-
plications (CCTA 2017), Aug 2017, Mauna Lani, HI, United States. �10.1109/CCTA.2017.8062760�.
�hal-01722001�
Abstract:1— In this paper, we propose a revisited form of
the so-called Model-Free Control (MFC). Herein, the
MFC principle is employed to deal with the unknown part
of the plant only (i.e. unmodeled dynamics, disturbances,
etc.) and occurs beside an Interconnection and Damping
Assignment-Passivity Based Control (IDA-PBC) strategy
that is used instead of the PID structure as done in the
classical MFC form. Using the proposed formulation, it is
shown that we can significantly improve the performance
of the control and its robustness level. This problem is
studied in the case of Multi-Inputs Multi-Outputs
(MIMO) system with an application to a small Vertical
Take-Off and Landing (VTOL) vehicle where a stability
analysis is also provided. The numerical simulations have
shown satisfactory results where an in-depth discussion
with respect to the control performance is highlighted by
considering several scenarios and using several metrics.
I. INTRODUCTION
The quadrotors are considered as a good case study to design,
to analyze and to implement flight control strategies.
Moreover, it is necessary to design a controller such that the
quadrotor will be able to efficiently follow a predefined
trajectory, particularly in the presence of disturbances. For this
reason, many studies have led to the development of
sophisticated and robust nonlinear control laws (as for instance
[1-3]). However, most of these proposed strategies require an
accurate model in order to perform a good control, which is
extremely difficult when the system is maneuvering in a harsh
environment.
In this regards, a strategy based on a Model-Free technique is
developed (MFC) (see as for instance [4]). The main
advantage of this control strategy is that it does not require the
knowledge of the system dynamics as it involves a continuous
updating of the input-output of a very local model. Thus, its
use as the basis of control allows the compensation of the
uncertainties as well as other disturbances. It is employed in
many real cases such as mobile robots [4] and quadrotors [5].
In a certain point of view, the control of a system with a model
free has already been used, since many decades, on the basis
of fuzzy logic control or the more popular one for linear
systems through Ziegler-Nichols method [6]. In addition,
assuming no available model is not totally a correct
assumption due to the fact that most of systems, at least, may
*
Y. Bouzid and Y. Bestaoui are with IBISC Laboratory, universitéd’Evry Val
d’Essonne, Université Paris-Saclay, Evry, France (e-mail:
yasseremp@gmil.com,Yasmina.Bestaoui@ufrst.univ-evry.fr).
H. Siguerdidjane iswith L2S, CentraleSupélec, Université Paris-Saclay, Gif
sur yvette, France (e-mails:
Houria.Siguerdidjane@centralesupelec.fr).
be approximated by mathematical models even with poorly
known dynamics.
Using the available information about the system will bring a
notable benefit and significantly improve the performance of
control. Therefore, we propose a Revisited Model-Free
Control (R-MFC) strategy to simultaneously accommodate the
unmodeled and neglected dynamics and external disturbances.
As in general, the tuning of PID parameters allows to meet the
desired specification of control, we use a reference model-
based control technique that achieves the control with the
required specifications, by means of Interconnection and
Damping Assignment-Passivity Based Control (IDA-PBC).
In the last two decades, the use of the so-called Port-Controlled
Hamiltonian (PCH) representation has attracted the attention
of researchers. Many control tools have been developed to deal
with this compact representation. Passivity-Based Control
(PBC) is well known especially in mechanical applications for
controlling nonlinear systems. An improvement was
developed through Interconnection and Damping Assignment
(IDA) where the use of energy shaping was originated in [7].
Recently, the IDA-PBC has become an efficient tool in
nonlinear control applications and has been illustrated in
several real experimentations including electrical motors [8],
magnetic suspension systems [9], etc.
Throughout this paper, a performance assessment is presented
via results of several illustrations, scenarios and numerical
simulations, with complementary comments of the proposed
revisited strategy of control with respect to other techniques.
The remainder of this paper is organized as follows: Section II
concerns the dynamics of the VTOL quadrotor and the control
architecture. Section III and Section IV introduce the design of
our nonlinear control approach. The simulation results are
illustrated in Section V. Finally, the paper is ended with
concluding remarks.
II. QUADROTOR MODELING & CONTROL
ARCHITECTURE
From the fundamental principle of dynamics, we model the
quadrotor as a rigid body for the validation of control
performance, neglecting some aerodynamic effects such as
the gyroscopic and ground effects. The system operates in two
coordinate frames: the Earth-fixed frame
and
the body fixed frame
(see Figure 1). Let
describes the orientation of the aerial vehicle (Roll,
Pitch, Yaw) and χ
denotes its absolute position.
3D Trajectory Tracking Control of Quadrotor UAV with On-Line
Disturbance Compensation
Y. Bouzid*, H. Siguerdidjane, Y. Bestaoui
Figure 1. Frame representation.
In this study, we consider a simplified dynamic model of the
vehicle that is derived in our previous work [1] in order to
make the controller implementation simpler and easier and to
show the efficiency of our control strategy that can deal with
the unmodeled and neglected dynamics. The considered
mathematical model may be written under the general
mechanical equation, as follows
(1)
where
χ
χ
is the generalized
coordinates,
is composed
of the mass
and the inertia matrix,
denotes the Coriolis term,
denotes the gravitational term,
denotes the input matrix, with
(2)
and
(3)
with the mass, the gravity acceleration,
the total thrust
of four rotors,
the control torque vector and
and
abbreviations for and respectively.
The parameters of the system UAV are displayed in Table 1.
Table 1. Quadrotor parameters.
In order to simplify the design of the controller, two virtual
inputs
and
are given as
(4)
The reference angles,
and
are considered
as inputs for the rotation subsystem. From system (4), it
follows
(5)
III. R-MFC FLIGHT CONTROLLER DESIGN
The classic MFC approach proceeds by considering an ultra-
local model, valid in short time that approximates the
nonlinear model via input-output behavior using the
experimental available data without any modeling step.
However, this online numerical differentiator and estimation
may fail with some highly nonlinear and/or time-varying
dynamics that need to be treated carefully. In addition, most
systems have a mathematical model even if it is not accurate.
Note also that the MFC does not distinguish between model
mismatches and perturbations. Therefore, to be more realistic,
our approach employs the model-free principle to only deal
with the unknown part of the controlled system where the use
of the known part brings more benefit and makes the control
more efficient.
This proposed formulation that we can denote by the acronym
R-MFC is explained through a class of systems written under
the general mechanical equation. This class of systems is
widely adopted in the robotics and mechanical fields. For the
sake of clarity, equation (1) is written under a compact form
considering the general case
(6)
where
and
is the output vector,
is
the inertia matrix,
denotes the Coriolis term,
denotes the gravitational term,
denotes
the input matrix and
denotes the input vector.
Usually, model (6) is quite simplified with neglected and
unmodeled dynamics. Moreover, the execution of trajectories
can be easily affected by external disturbances, which
introduces some unknown terms. Therefore, a term could
be added to nominal model (6) gathering the neglected and
unmodeled dynamics and the external disturbances. An
additional effort is requested to deal with this new term.
Thus, the input-output relationship of the anticipated model
may be represented by the following system:
κ (7)
where
is a positive definite diagonal scale matrix
fixed by the user and is an estimated term. In fact, the order
of model (7) is chosen according to the prior knowledge
of the system. It equals to the order of the mathematical model
(6).
Remark 1: For the existing MFC technique, is fixed by the
user and may equal to 1 or 2.
Therefore, the difference between model (6) and (7) lies in the
presence of the unknown modeled part that can be estimated
as
κ
(8)
As the past input vector in the previous time interval and
the actual
derivative of the measured output vector are
known, the value of
is computed. This estimation is valid
for a short period only and it should be continuously
updated at every iteration of the closed loop controller.
Obviously, the estimate of the second-order derivative
yields an estimate of
. Many significant advances on the
numerical differentiation of noisy signals are elaborated in the
literature [10]. This updated term
captures the unknown
dynamics of the system as well as the disturbances during
each period and then brings the required changes in the
control input by compensation. From (7), the control input for
our R-MFC proposed strategy can be split into two parts:
(9)
where the matrix
is non-singular in
.
plays the role of compensator of the unknown part
and
is considered as an auxiliary input that ensures the
asymptotic convergence of the tracking errors of the closed
loop into the origin. Injecting input (9) in (7), leads to a new
and fully known nonlinear model eliminating the unknown
part
(10)
The existing MFC technique employs, in the case of , a
PID controller as an auxiliary input :
(11)
that leads to the intelligent PID where
is the
tracking error vector and
is the reference trajectory
vector.
and
denote the usual tuning gains.
Herein, we proceed with a different way by employing a
sophisticated tool rather than the PID structure where the
main goal is to ensure the asymptotic convergence, towards
the origin, of the tracking errors of closed-loop of model (10).
In this stage, a broad range of strategy can be applied.
The use of PID, in the classic form of MFC, allows to ensure
a given performance of the system time response according to
the given specifications (overshoot, settling time, accuracy)
via the tuning of the control parameters. To achieve the
desired specifications, by using another control procedure, is
more challenging. Therefore, we employ a reference model
based control strategy where the control specifications are a
priori fixed. Then the control input pushes the system to
follow the same behavior as the target model.
IV. IDA-PBC BASED AUXILIARY INPUT
In the following, trajectory tracking control is achieved using
the IDA-PBC approach. In this technique, we modify the total
energy function of (10) to assign the desired equilibrium and
damping injection matrix to meet the asymptotic stability. To
preserve the energy interpretation, the closed-loop system is
presented in a Port-Controlled Hamiltonian (PCH)
representation.
A. System energy and PCH model
Explicitly, system (10) is written as
(12)
The system’s Hamiltonian energy
, the sum of the
kinetic energy,
and the potential one, is written
as:
(13)
So,
(14)
where
is the generalized momentum.
This system has a natural stable equilibrium configuration.
This latter one is related to the minimum of energy. The PCH
model is needed in order to design a controller based on the
IDA-PBC methodology. From (12), the dynamic of the
quadrotor can be written as
(15)
with
(16)
B. Target dynamics
Motivated by equation (13), we propose a desired energy
function as being:
(17)
We modify the total internal energy function of the closed
loop system to assign the desired equilibrium configuration
and we require that
will have a minimum at
,
thus
(18)
where
and
represent the desired closed
loop potential energy function and inertia matrix, respectively
with
(19)
We take the reference trajectories as desired equilibrium
configuration. In another words,
.
To preserve the energy interpretation we also require that the
desired closed loop system be in PCH form
(20)
where
(21)
(22)
is a skew symmetric matrix and
contains the control parameters. Obviously, the matrix
has the same form as the original
system.
C. Energy shaping & damping injection
Commonly, the control input of IDA-PBC is decomposed into
two terms [11].
(23)
where
acts on the damping and
is designed for the
energy shaping.
The controller is obtained by substituting (23) in (15) and
making the resulting equations equal to (20). Thus
(24)
injects a damping into the system via negative feedback of
the passive output
. The damping injection term is
then:
(25)
Thus, the second row leads to the energy shaping
(26)
Finally, the R-MFC is split into three parts as
(27)
where
deals with the unknown parts and allows to maintain
a certain level of robustness,
allows to meet the desired
specification through the target model components and finally
the damping injection term
in order to guarantee a damped
response. Our approach is summarized by Figure 2.
Figure 2. R-MFC scheme.
D. Control design for quadrotor and stability analysis
For the sake of simplification, we keep the interconnection
matrix unchanged, namely
and therefore
becomes
(28)
So,
(29)
is an arbitrary function of and From equation (19), the
necessary condition,
0 and the sufficient
condition,
, will hold at
.
has its minimum
at
.
In our case we choose
to be a quadratic function, which
leads to
(30)
where
and
denote positive definite matrices of the
control parameters.
Now, the auxiliary input for the quadrotor is given by
(31)
where
and
Proposition 1: if
and with
having a
quadratic form (30), the auxiliary input
for the quadrotor,
using IDA-PBC approach, can be given by (31).
The stability of the closed-loop dynamics of the quadrotor
system in form (12) is introduced by the following theorem.
Theorem 1: Closed-loop of system (12) written under PCH
form, using control law (31) is asymptotically stable.
Proof: Following the IDA-PBC approach described above,
closed loop system (12) written under PCH model (15) is
equivalent to the desired PCH model (20) using control law
(31).
is a positive definite function chosen herein as
Lyapunov candidate function where the first time derivative
is
(32)
Using (29), we get
(33)
Then
(34)
So, closed loop of system (12) is asymptotically stable.
V. RESULTS AND DISCUSSION
In this section, we test the effectiveness of the proposed
controller not only in the ideal case but also in the presence of
different disturbances. For the sake of further comparison, we
follow the same protocol and fit the same conditions. The
control parameters are tuned, using Genetic Algorithms (GA),
in the ideal case then kept for the entire proposed scenarios and
for which the objective is to reduce the steady state errors.
Thus, the fitness function is given by
(35)
where
and
denote the initial and the final instants of
optimization respectively. The obtained control parameters
are depicted in Table 2.
Table 2: R-MFC control parameters.
23.33
22.19
21.77
5.05
4.86
4.99
18.77
20.672
19.16
4.51
5.00
4.58
For a significant analysis of the features of the proposed
controller, two additional nonlinear controllers are considered.
The first one is the classic model-free technique that exhibits
System
IDA-PBC
Controller
Eq (23)
R-MFC (9)
Uncertainties
estimator
Eq (8)
Nominal
model
Eq (6)
Target model
Eq (20)