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Studies in Computational Intelligence 576
AhmadTaher Azar
Quanmin Zhu Editors
Advances and
Applications in
Sliding Mode
Control systems

Studies in Computational Intelligence
Volume 576
Series editor
Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland
e-mail: kacprzyk@ibspan.waw.pl

About this Series
The series “Studies in Computational Intelligence” (SCI) publishes new developments
and advances in the various areas of computational intelligence—quickly and with a
high quality. The intent is to cover the theory, applications, and design methods of
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contains monographs, lecture notes and edited volumes in computational intelligence
spanning the areas of neural networks, connectionist systems, genetic algorithms,
evolutionary computation, artificial intelligence, cellular automata, self-organizing
systems, soft computing, fuzzy systems, and hybrid intelligent systems. Of particular
value to both the contributors and the readership are the short publication timeframe
and the world-wide distribution, which enable both wide and rapid dissemination of
research output.
More information about this series at http://www.springer.com/series/7092

Ahmad Taher Azar • Quanmin Zhu
Editors
Advances and Applications
in Sliding Mode Control
systems
123

Editors
Ahmad Taher Azar
Faculty of Computers and Information
Benha University
Benha
Egypt
Quanmin Zhu
Department of Engineering Design
and Mathematics
University of the West of England
Bristol
UK
ISSN 1860-949X ISSN 1860-9503 (electronic)
ISBN 978-3-319-11172-8 ISBN 978-3-319-11173-5 (eBook)
DOI 10.1007/978-3-319-11173-5
Library of Congress Control Number: 2014953225
Springer Cham Heidelberg New York Dordrecht London
© Springer International Publishing Switzerland 2015
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Springer is part of Springer Science+Business Media (www.springer.com)

Foreword
In control theory of linear and nonlinear dynamical systems, sliding mode control
(SMC) is a nonlinear control method. The sliding mode control method alters the
dynamics of a given dynamical system (linear or nonlinear) by applying a dis-
continuous control signal that forces the system to “slide” along a cross-section
(manifold) of the system’s normal behaviour.
Sliding mode control (SMC) is a special class of variable-structure systems
(VSS). In sliding mode control method, the state feedback control law is not a
continuous function of time. Instead, the state feedback control law can switch from
one continuous structure to another based on the current position in the state space.
Variable-structure systems (VSS) and the associated sliding mode behavior was
first investigated in the early 1950s in the USSR and seminal papers on SMC were
first published by Profs. S.V. Emelyanov (1967) and V.I. Utkin (1968). The early
research on VSS dealt with single-input and single-output (SISO) systems. In recent
years, the majority of research in SMC deals with multi-input and multi-output
(MIMO) systems.
For over 50 years, the sliding mode control (SMC) has been extensively studied
and widely used in many scientific and industrial applications due to its simplicity
and robustness against parameter variations and disturbances.
The design procedure of the sliding mode control (SMC) consists of two major
steps, (A) Reaching phase and (B) Sliding-mode phase. In the reaching phase, the
control system state is driven from any initial state to reach the sliding manifold in
finite time. In the sliding-mode phase, the system is confined into the sliding motion
on the sliding manifold. The stability results associated with the sliding mode
control are established using the direct method of the Lyapunov stability theory.
Hence, the sliding mode control scheme involves (1) the selection of a hyper-
surface or a manifold (i.e. the sliding manifold) such that the system trajectory
exhibits desirable behavior when confined to this sliding manifold and (2) finding
feedback gains so that the system trajectory intersects and stays on the sliding
manifold.
v

The merits of sliding mode control (SMC) are robustness against disturbances
and parameter variations, reduced-order system design, and simple control struc-
ture. Some of the key technical problems associated with sliding mode control
(SMC) are chattering, matched and unmatched uncertainties, unmodeled dynamics,
etc. Many new approaches have been developed in the last decade to address these
problems.
Important types of sliding mode control (SMC) are classical sliding mode
control, integral sliding mode control, second-order sliding mode control, and
higher order sliding mode control. The new SMC approaches show promising
dynamical properties such as ﬁnite time convergence and chattering alleviation.
Sliding mode control has applications in several branches of Engineering like
Mechanical Engineering, Robotics, Electrical Engineering, Control Systems, Chaos
Theory, Network Engineering, etc.
One of the key objectives in the recent research on sliding mode control (SMC)
is to make it more intelligent. Soft computing (SC) techniques include neural
networks (NN), fuzzy logic (FL), and evolutionary algorithms like genetic algo-
rithms (GA), etc. The integration of sliding mode control and soft computing
alleviates the shortcomings associated with the classical SMC techniques.
It has been a long road for the sliding mode control (SMC) from early VSS
investigations in the 1950s to the present-day investigations and applications. In
this book, Dr. Ahmad Taher Azar and Dr. Quanmin Zhu have collected and edited
contributions of well-known researchers and experts in the ﬁeld of sliding mode
control theory in order to provide a comprehensive view of the recent research
trends in sliding mode control theory. Their efforts have been very successful.
Therefore, it has been a great pleasure for me to write the Foreword for this book.
Professor and Dean, R & D Centre
Vel Tech University
Chennai
Tamil Nadu, India
vi Foreword

Preface
Sliding mode control, also known as variable structure control, is an important
robust control approach and has attractive features to keep systems insensitive to
uncertainties on the sliding surface. For the class of systems to which it applies,
sliding mode controller design provides a systematic approach to the problem of
maintaining stability and consistent performance in the face of modeling impreci-
sion. On the other hand, by allowing tradeoffs between modeling and performance
to be quantified in a simple fashion, it can illuminate the whole design process.
Sliding mode schemes have become one of the most exciting research topics in
several fields such as electric drives and actuators, power systems, aerospace
vehicles, robotic manipulators, biomedical systems, etc. In its earlier approach, an
infinite frequency control switching was required to maintain the trajectories on a
prescribed sliding surface and then eventually to enforce the orbit tending to the
equilibrium point along the sliding surface. However, in practice the system states
do not really locate on the designed sliding surface after reaching it due to
numerically discretizing errors, signal noise, as well as structural uncertainties in the
dynamical equations. Since the controller was fast switched during operation, the
system underwent oscillation crossing the sliding plane. Around the sliding surface
is often irritated by high frequency and small amplitude oscillations known as
chattering. The phenomenon of chattering is a major drawback of SMC, which
makes the control power unnecessarily large. To eliminate chattering, some
methods are being developed.
This book consists of 21 contributed chapters by subject experts specialized in
the various topics addressed in this book. The special chapters have been brought
out in this book after a rigorous review process. Special importance was given to
chapters offering practical solutions and novel methods for recent research prob-
lems in the main areas of this book. The objective of this book is to present recent
theoretical developments in sliding mode control and estimation techniques as well
as practical solutions to real-world control engineering problems using sliding mode
methods. The contributed chapters provide new ideas and approaches, clearly
indicating the advances made in problem statements, methodologies, or applica-
tions with respect to the existing results. The book is not only a valuable title on the
vii

publishing market, but is also a successful synthesis of sliding mode control in the
world literature.
As the editors, we hope that the chapters in this book will stimulate further
research in sliding mode control methods for use in real-world applications. We
hope that this book, covering so many different aspects, will be of value to all
readers.
We would like to thank also the reviewers for their diligence in reviewing the
chapters.
Special thanks go to Springer publisher, especially for the tireless work of the
series editor “Studies in Computational Intelligence,” Dr. Thomas Ditzinger.
Benha, Egypt Ahmad Taher Azar
Bristol, UK Quanmin Zhu
viii Preface

Contents
Adaptive Sliding Mode Control of the Furuta Pendulum. . . . . . . . . . . 1
Ahmad Taher Azar and Fernando E. Serrano
Optimal Sliding and Decoupled Sliding Mode Tracking
Control by Multi-objective Particle Swarm Optimization
and Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
M. Taherkhorsandi, K.K. Castillo-Villar, M.J. Mahmoodabadi,
F. Janaghaei and S.M. Mortazavi Yazdi
Robust Control of Robot Arms via Quasi Sliding Modes
and Neural Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Maria Letizia Corradini, Andrea Giantomassi, Gianluca Ippoliti,
Sauro Longhi and Giuseppe Orlando
A Robust Adaptive Self-tuning Sliding Mode Control for a Hybrid
Actuator in Camless Internal Combustion Engines . . . . . . . . . . . . . . . 107
Benedikt Haus, Paolo Mercorelli and Nils Werner
Sliding Mode Control of Class of Linear Uncertain Saturated
Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Bourhen Torchani, Anis Sellami and Germain Garcia
Sliding Mode Control Scheme of Variable Speed Wind Energy
Conversion System Based on the PMSG for Utility Network
Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Youssef Errami, Mohammed Ouassaid, Mohamed Cherkaoui
and Mohamed Maaroufi
Super-Twisting Air/Fuel Ratio Control for Spark Ignition Engines . . . 201
Jorge Rivera, Javier Espinoza-Jurado and Alexander Loukianov
ix

Robust Output Feedback Stabilization of a Magnetic Levitation
System Using Higher Order Sliding Mode Control Strategy . . . . . . . . 227
Muhammad Ahsan and Attaullah Y. Memon
Design and Application of Discrete Sliding Mode Controller
for TITO Process Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
A.A. Khandekar and B.M. Patre
Dynamic Fuzzy Sliding Mode Control of Underwater Vehicles . . . . . . 279
G.V. Lakhekar and L.M. Waghmare
An Indirect Adaptive Fuzzy Sliding Mode Power System
Stabilizer for Single and Multi-machine Power Systems. . . . . . . . . . . . 305
Saoudi Kamel, Bouchama Ziyad and Harmas Mohamed Naguib
Higher Order Sliding Mode Control of Uncertain Robot
Manipulators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
Neila Mezghani Ben Romdhane and Tarak Damak
Generalized H2 Sliding Mode Control for a Class of (TS) Fuzzy
Descriptor Systems with Time-Varying Delay and Nonlinear
Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
Mourad Kchaou and Ahmed Toumi
Rigid Spacecraft Fault-Tolerant Control Using Adaptive Fast
Terminal Sliding Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
Pyare Mohan Tiwari, S. Janardhanan and Mashuq un-Nabi
Sliding Modes for Fault Tolerant Control. . . . . . . . . . . . . . . . . . . . . . 407
Hemza Mekki, Djamel Boukhetala and Ahmad Taher Azar
Transient Stability Enhancement of Power Systems Using
Observer-Based Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . 435
M. Ouassaid, M. Maaroufi and M. Cherkaoui
Switching Function Optimization of Sliding Mode Control
to a Photovoltaic Pumping System . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
Asma Chihi, Adel Chbeb and Anis Sellami
Contribution to Study Performance of the Induction Motor
by Sliding Mode Control and Field Oriented Control . . . . . . . . . . . . . 495
Oukaci Assia, Toufouti Riad and Dib Djalel
x Contents

Anti-synchronization of Identical Chaotic Systems Using Sliding
Mode Control and an Application to Vaidyanathan–Madhavan
Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
Sundarapandian Vaidyanathan and Ahmad Taher Azar
Hybrid Synchronization of Identical Chaotic Systems Using
Sliding Mode Control and an Application to Vaidyanathan
Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
Sundarapandian Vaidyanathan and Ahmad Taher Azar
Global Chaos Control of a Novel Nine-Term Chaotic System
via Sliding Mode Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
Sundarapandian Vaidyanathan, Christos K. Volos and Viet-Thanh Pham
Contents xi

Adaptive Sliding Mode Control of the Furuta
Pendulum
Ahmad Taher Azar and Fernando E. Serrano
Abstract InthischapteranadaptiveslidingmodecontrollerfortheFurutapendulum
is proposed. The Furuta pendulum is a class of underactuated mechanical systems
commonly used in many control systems laboratories due to its complex stabiliza-
tion which allows the analysis and design of different nonlinear and multivariable
controllers that are useful in some fields such as aerospace and robotics. Sliding
mode control has been extensively used in the control of mechanical systems as an
alternative to other nonlinear control strategies such as backstepping, passivity based
control etc. The design and implementation of an adaptive sliding mode controller
for this kind of system is explained in this chapter, along with other sliding mode
controller variations such as second order sliding mode (SOSMC) and PD plus slid-
ing mode control (PD + SMC) in order to compare their performance under different
system conditions. These control techniques are developed using the Lyapunov sta-
bility theorem and the variable structure design procedure to obtain asymptotically
stable system trajectories. In this chapter the adaptive sliding mode consist of a slid-
ing mode control law with an adaptive gain that makes the controller more flexible
and reliable than other sliding mode control (SMC) algorithms and nonlinear con-
trol strategies. The adaptive sliding mode control (ASMC) of the Furuta pendulum,
and the other SMC strategies shown in this chapter, are derived according to the
Furuta’s pendulum dynamic equations making the sliding variables, position errors
and velocity errors reach the zero value in a specified reaching time. The main reason
of deriving two well known sliding mode control strategy apart from the proposed
control strategy of this chapter (adaptive sliding mode control) is for comparison
purposes and to evince the advantages and disadvantages of adaptive sliding mode
control over other sliding mode control strategies for the stabilization of the Furuta
A.T. Azar (B)
Faculty of Computers and Information, Benha University, Benha, Egypt
e-mail: ahmad_t_azar@ieee.org
F.E. Serrano
Department of Electrical Engineering, Florida International University,
10555 West Flagler St, Miami, FL 33174, USA
e-mail: fserr002@fiu.edu
© Springer International Publishing Switzerland 2015
A.T. Azar and Q. Zhu (eds.), Advances and Applications in Sliding Mode Control systems,
Studies in Computational Intelligence 576, DOI 10.1007/978-3-319-11173-5_1
1

2 A.T. Azar and F.E. Serrano
pendulum. A chattering analysis of the three SMC variations is done, to examine
the response of the system, and to test the performance of the ASMC in comparison
with the other control strategies explained in this chapter.
1 Introduction
In this chapter an adaptive sliding mode control of the Furuta pendulum is proposed.
The Furuta pendulum is a class of underactuated mechanical system used in labora-
tories to test different kinds of control strategies that are implemented in aerospace,
mechanical and robotics applications. A mechanical system is underactuated when
the number of actuators is less than the degrees of freedom of the system, for this
reason, the research on the control of this kind of systems is extensively studied.
There are different kinds of control strategies for the Furuta pendulum found
in literature, these approaches take in count the complexity of the dynamic model
considering that is coupled and nonlinear. In Ramirez-Neria et al. 2013 an active
disturbance rejection control (ADRC) is proposed for the tracking of a Furuta pen-
dulum, specially, when there are disturbances on the system; the ADRC cancels the
effects of the disturbance on the system by an on line estimation of the controller
parameters. In Hera et al. (2009), the stabilization of a Furuta pendulum applying an
efficient control law to obtain the desired trajectory tracking is corroborated by the
respective phase portraits. Some authors propose the parameter identification of the
model (Garcia-Alarcon et al. 2012) implementing a least square algorithm; becom-
ing an important technique that can be used in adaptive control strategies. Another
significant control approach is implemented by Fu and Lin (2005) where a back-
stepping controller is applied for the stabilization of the Furuta pendulum where a
linearized model of the pendulum is used to stabilizes this mechanical system around
the equilibrium point.
Sliding mode control SMC has been extensively implemented in different kinds of
systems, including mechanical, power systems, etc. this is a kind of variable structure
controllers VSC that is becoming very popular in the control systems community due
to its disturbance rejection properties yielded by external disturbance or unmodeled
dynamics (Shtessel et al. 2014). It consist on stabilizing the system by selecting an
appropriate sliding manifold until these variables reach the origin in a determined
convergence time; during the last decades the SMC control strategy has evolved,
from first order SMC to higher order sliding mode control HOSMC (Kunusch et al.
2012), which has been implemented in recent years due to its chattering avoidance
properties (Bartolini and Ferrara 1996). Due to the discontinous control action of the
SMC, sometimes the chattering effect is found in the system producing unwanted
system responses. Chattering basically is a high frequency oscillations in the control
input that can yield instability and unwanted system response, due to the chattering
avoidance properties of the HOSMC and their disturbance rejection (Utkin 2008),
this control technique has replaced the classical sliding mode approach.

Adaptive Sliding Mode Control of the Furuta Pendulum 3
In order to solve the chattering problem, some SMC techniques have been pro-
posed to deal with this effect, like the twisting and super twisting algorithms (Fridman
2012), even when they are first generation algorithms, they have some advantages and
disadvantages when they are implemented in the control of underactuated mechanical
systems. Even when the previous algorithms are commonly implemented, a second
order sliding mode control SOSMC for the stabilization of the Furuta pendulum is
proposed in Sect. 2 where an specially design control algorithm is implemented in
the control and tracking of this mechanism (Moreno 2012). This approach is devel-
oped in this section to provide a different point of view on how to deal with this
kind of problem and because this is the theoretical background for the development
of other control algorithms, including adaptive sliding mode control (Ferrara and
Capisani 2012). In Sect. 3 the derivation and application of a proportional derivative
plus sliding mode control (PD + SMC) for the stabilization of a Furuta pendulum is
explained to show the advantages and disadvantages of this hybrid control strategy
over SOSMC and compare it with the adaptive sliding mode control strategy pro-
posed in this article. The reason because these two sliding mode control approaches
are explained in this chapter, is because it is necessary to compare these two sliding
mode control strategy with the main contribution of this chapter, in which the sta-
bilization of the Furuta pendulum by an adaptive sliding mode control strategy for
the Furuta pendulum is proposed to be compared and analyzed with other sliding
mode control approaches and to understand the theoretical background of adaptive
sliding mode controllers for mechanical underactuated systems. The derivation of
the adaptive sliding mode control ASMC strategy for the Furuta pendulum is shown
in Sect. 4, where an adaptive gain control strategy is obtained (Fei and Wu 2013; Liu
et al. 2013; Chen et al. 2014) exploiting the advantage of a classical sliding mode
controller with the on line tuning of a variable parameter controller. ASMC has been
demonstrated to be an effective control strategy for similar mechanical systems (Yao
and Tomizuka 1994) and other mechanical devices (Jing 2009; Li et al. 2011) where
the improved parameter adjustment make this strategy ideal for the control and sta-
bilization of this underactuated mechanical systems. In Sect. 5 a chattering analysis
of the three control approaches shown in this chapter is done to find the oscillation
period yielded by the discontinuous control law, then some conclusions are obtained
from this controller’s comparison. In Sect. 6 a discussion about the performance of
the three approaches explained in this chapter are analyzed to explain the advantages,
disadvantages and characteristic of the proposed control technique; finally, in Sect. 7
the conclusions of this chapter are shown to summarize the results obtained in this
chapter.
2 Second Order Sliding Mode Control of the Furuta Pendulum
In this section the derivations of a second order sliding mode control (SOSMC) is
shown to stabilizes the Furuta pendulum. The main idea of this control approach
is to find a suitable control law which stabilizes the system reducing the chattering

effects and making the sliding manifold to reach the origin in finite time (Bartolini
et al. 1998). SOSMC has been extensively implemented in the control of different
kind of mechanisms, where the dynamic model of the system is considered to develop
anappropriateswitchingcontrol law(SuandLeung 1992; Zhihonget al. 1994; Gracia
et al. 2014) therefore it has became in an attractive strategy for the control of the
Furuta pendulum.
Apart from the chattering avoidance nature of the SOSMC, another advantage
of the SOSMC is the disturbance rejection properties of this approach, making it
a suitable choice for the control of mechanical systems, (Punta 2006; Chang 2013;
Estrada and Plestan 2013), considering that the Furuta pendulum is an underactuated
system (Nersesov et al. 2010), generating exponentially stable sliding manifolds to
reach the origin in a prescribed time. The SOSMC strategy allows the design of
appropriate sliding manifolds which converge to zero in a defined time, for MIMO
and coupled dynamic systems (Bartolini and Ferrara 1996) making this approach
ideal for the control of the Furuta pendulum.
Higher order sliding mode control HOSMC (Levant 2005) has demonstrated
its effectiveness in the control of different kinds of systems (Rundell et al. 1996;
Shkolnikov et al. 2001; Fossas and Ras 2002), for this reason a SOSMC is designed
to stabilizes the Furuta pendulum with specific initial conditions implementing a
Lyapunov approach to obtain a suitable control law. This control strategy is designed
considering the dynamics of the Furuta pendulum (Fridman 2012) instead of imple-
menting well known SOSMC control algorithms such as the twisting and the super
twisting algorithms (Moreno 2012). The SOSMC strategy is done by designing a
control algorithm for arbitrary order SMC (Levant 2005; Fridman 2012) and test the
stability of the SOSMC by the Lyapunov theorem. The stabilization of the Furuta
pendulum is not a trivial task, even when different nonlinear control techniques are
proposed by some authors (Fu and Lin 2005; Ramirez-Neria et al. 2013) an ideal
control law that improves the system performance and reduces the tracking error with
smaller oscillations in the system that can be harmful for the mechanical system. It
is important to avoid these unwanted effects on the system considering an appropri-
ate second order sliding mode control law which decreases the deterioration of the
system performance, then a stabilizing control law that makes the sliding variables
and its first derivative to reach the origin in finite time is chosen for the stabilization
of this underactuated mechanical system.
The first subsection of this chapter is intended to explain the dynamic equations
of the Furuta pendulum that are determined by the respective kinematics equations
and the Euler–Lagrange formulation of this mechanical system. This model is imple-
mented in the rest of this chapter to derive the sliding mode controllers explained in
the following sections. In Sect. 2.2 the design of a second order sliding mode con-
troller for the Furuta pendulum is explained where this control strategy is designed
according to the system dynamics of the model while keeping the tracking error
as small as possible and driving the sliding variable to zero in finite time. Finally
in Sect. 2.3 an illustrative example of this control approach is done visualizing the
system performance and analyzing the controlled variables behavior; such as the
angular position, velocity and tracking error of this mechanical system.

TheintentioninthissectionistocomparetheSOSMCalgorithmwiththeproposed
strategy of this chapter, then the discussion and analysis of this control approach are
explained in Sects. 5 and 6.
2.1 Dynamic Model of the Furuta Pendulum
The Furuta pendulum is an underactuated mechanism which consists of a rotary base
with a pendulum connected to a arm. The angle of the rotary base is denoted as φ and
the angle of the pendulum is denoted as θ. This mechanism is a perfect example of
underactuated nonlinear mechanical system that is implemented in the development
anddesignofdifferentkindofnonlineararchitecturesforseveralkindsofapplications
such as aeronautics, aerospace, robotics and other areas in the control systems field.
This mechanism works by rotating the base of the pendulum and then the arm rotates
according to the interaction of the pendulum and base of the arm. As it is explained
in the introduction of this section the stabilization of this mechanical system is a
difficult task, so in this section it is proved that a suitable second order sliding mode
control for the stabilization of this system is possible, while considering the system
dynamics of the model. The dynamical model shown in this section has two angles,
that must be controlled efficiently in order to keep the base and pendulum positions
in the desired values. This mathematical model is necessary for the design of efficient
sliding mode techniques where in order to design the proposed control strategy the
linearization of the model is essential to develop the adaptive gain SMC technique.
The design of a second order sliding mode controller for the Furuta pendulum leads
the path to the development of other sliding mode controller variations, so an efficient
control system design is important in this section to improve the performance of the
controlled system, therefore well defined dynamical systems equations lead to an
efficient design of the sliding mode control strategies that are developed in this and
the following sections.
In Fig.1 the Furuta pendulum configuration is depicted showing the respective
rotational angles; meanwhile, in Fig.2 a CAD model of the Furuta pendulum is
depicted for a clear understanding of the model.
The dynamic equations of the Furuta pendulum are given by (Fu and Lin 2005;
Hera et al. 2009; Garcia-Alarcon et al. 2012; Ramirez-Neria et al. 2013):
(p1 + p2sin2
(θ))φ̈ + p3cos(θ)θ̈ + 2p2sin(θ)cos(θ)θ̇φ̇ − p3sin(θ)θ̇2
= τφ (1)
p3cos(θ)φ̈ + (p2 + p5)θ̈ − p2sin(θ)cos(θ)φ̇2
− p4sin(θ) = 0 (2)
where:
p1 = (M + mp)2
a (3)
p2 = (M + (1/4)mp)2
p (4)
p3 = (M + (1/2)mp)pa (5)

Fig. 1 Furuta pendulum
system
x
y
z
h
1
2
Fig. 2 CAD drawing of the
furuta pendulum
p4 = (M + (1/2)mp)pg (6)
p5 = (1/12)mp2
p (7)
where a is the length of the arm, p is the length of the pendulum, mp is the
pendulum mass, M is the mass of the bob at the end of the pendulum and g is the
gravity constant.
Now, to establish the dynamic equations in the standard form it is necessary to
define the next vector q = [φ, θ]T = [q1, q2]T, then the dynamic equations are
represented by:
D(q)q̈ + C(q, q̇)q̇ + g(q) =

τφ
0

(8)
defining the following state variables:

x1 = q
x2 = q̇
The following state space representation is obtained:
ẋ1 = x2
ẋ2 = −D−1
(x1)C(x1, x2)x2 − D−1
(x1)g(x1) + D−1
(x1)

1 0
0 0

τ
where τ, the inertia matrix, coriolis matrix and gravity vector are defined as:
D(q) =

(p1 + p2sin2(q2)) p3cos(q2)
p3cos(q2) p2 + p5

(9)
C(q, q̇) =

2p2sin(q2)cos(q2)q̇2 −p3sin(q2)q̇2
p2sin(q2)cos(q2)q̇1 0

(10)
g(q) =

0
−p4sin(q2)

(11)
τ =

τφ
τθ

(12)
where D(q) is the inertia matrix, C(q, q̇) is the coriolis matrix and g(q) is the gravity
vector. With these equations, the SMC can be derived in this and the following
sections, stablishing a theoretical background for the development of the sliding
mode controllers because they are settled on the dynamic equations of the Furuta
pendulum. In the next subsection a SOSMC is derived for the stabilization of the
Furuta pendulum, where its performance is analyzed and compared in the following
sections.
2.2 Second Order Sliding Mode Control of the Furuta Pendulum
In this section a SOSMC is designed for the stabilization of the Furuta pendulum.
Second order sliding mode control has been proved to be an effective control strategy
for different kind of mechanical systems (Punta 2006), therefore an appropriate con-
trol algorithm is developed considering the dynamics of the model (Fridman 2012).
The second order sliding mode controller for this mechanism is designed to
ensure that the sliding variables and their derivatives reach the origin in finite time
σ = σ̇ = 0, in order to calculate this convergence time the reader should check
Sect. 6. The convergence of the sliding variables of the system ensures that the con-
trolled variables of the model, angular positions and velocities, reach and keep the
desired values in steady state. Second order sliding mode control (SOSMC) and
higher order sliding mode control (HOSMC) are appropriate control strategies for

this kind of mechanical systems, due to the chattering avoidance properties, distur-
bance rejection and robustness to unmodelled dynamics, therefore an appropriate
SOSMC strategy is implemented in the stabilization of this underactuated mech-
anism to keep the controlled variables in the desired values by moving the joint
positions from their initial conditions to the final position of the pendulum and base.
Despite of the control of the Furuta pendulum with other nonlinear control tech-
niques such as backstepping or robust control, second order sliding mode control
remains acceptable for the control and stabilization of different kind of mechanism
due to the performance enhancement properties such as robustness and disturbance
rejection properties, for these reasons, a higher order sliding mode controller is pro-
posed in this section instead of well known sliding mode control algorithms such
as the twisting and super twisting. The implementation of well defined dynamical
equations of the Furuta pendulum by the Euler Lagrange formulation is an important
fact that must be considered in the design of an efficient second order sliding mode
controller that yields an efficient trajectory tracking by minimizing the system errors.
Another important fact shown in this subsection is the design of an appropriate slid-
ing mode control strategy that reduces chattering and avoids the saturation of the
system actuator, so this SOSMC strategy suppress these effects on the system.
In order to design the desired SOSMC, the first step is the design of the sliding
manifold that in this case is given by:
σ = ė + Φe (13)
where σ is the sliding manifold, q is the position vector, qd is the desired position
vector, Φ is a 2×2 positive definite matrix and:
e = qd(t) − q(t) (14)
Then in order to design the required controller the variable φ must be defined
before deriving the control law (Bartolini et al. 1998; Levant 2005; Fridman 2012):
φ = σ̇ + βi |σ|
1
2 sign(σ) (15)
Then the established control law is given by (Fridman 2012):
u = −σ + αsign(φ) = τ (16)
where α 0 is a positive constant. Before proving the stability of the system the
following property must be explained:
Definition 1 An n-degrees of freedom mechanical system has the following prop-
erty:
σT

1
2
Ḋ(q) − C(q, q̇)

σ = 0 (17)
where σ ∈ n.

In order to test the stability of the system with the proposed control law, the
following theorem is necessary to assure the convergence of the states and sliding
manifold of the system
Theorem 1 The second order sliding mode controller assures the stability of the
system if the defined Lyapunov function indicates that the system is asymptotically
stable.
Proof Define the following Lyapunov function with the established sliding manifold:
V (σ) =
1
2
σT
D(q)σ (18)
where D(q) is the inertia matrix of the system. Then the derivative of the Lyapunov
function yields:
V̇ (σ) = σT
D(q)σ̇ +
1
2
σT
Ḋ(q)σ (19)
The term D(q)σ̇ can be described as:
D(q)σ̇ = −τ + ξ − C(q, q̇)σ (20)
where
ξ = D(q)(q̈d + Φė) + C(q, q̇)(q̇d + Φe) + g(q) (21)
Definition 2 The term ξ has the following property (Liu 1999; Xiang and Siow
2004)
ξ ≤ α1 + α2 e + α3 ė + α4 e ė (22)
where α1,α2,α3, and α4 are positive constants. Then applying (20) and Definition 1
the Lyapunov function derivative becomes in:
V̇ (σ) = −σT
τ + σT
ξ (23)
Then applying the norm and Definition 2, the Lyapunov function derivative becomes:
V̇ (σ) ≤ −

σT

τ +

σT

ξ (24)
Therefore asymptotically stability is assured due to the upper bound of ξ, explained
in Definition 2, implementing the SOSMC.
With these conditions the stability of the SOSMC is assured in the stabilization of
theFurutapendulum.Inthenextsubsectionanillustrativeexampleofthestabilization
of the Furuta pendulum by a SOSMC is shown, the system is tested under certain
initial conditions to analyze the performance of the measured variables and the sliding
manifold.

2.3 Example 1
In this section an example of the stabilization of a Furuta pendulum by a SOSMC
is shown to test the performance of the system under specified initial conditions
(π/2, 0). The purposes of the second order sliding mode control is to make the
sliding variables and their derivatives to reach the origin in a finite time σ = σ̇ = 0
in order to make the controlled variables such as angular position and velocity reach
the desired final value in steady state when a disturbance is applied on the model. In
this example the idea is to illustrate the theoretical background of the second order
sliding mode controller when it is implemented in the control and stabilization of
the Furuta pendulum by a mathematical model of the system. In this example the
simulation results of the Furuta pendulum controlled by a second order sliding mode
controller is shown, depicting the angular position trajectories, the angular velocities,
the phase portraits, the tracking errors and the control input. With these simulation
results the performance of the Furuta pendulum, represented by a mathematical
model, show the system variables performance and evinces important conclusions
on the stabilization of this underactuated system with specified initial conditions.
The Furuta pendulum parameters are given in Table1 The gains of the SOSMC
are given by:
α = 0.01 (25)
Φ =

1000 0
0 1000

(26)
β =

0.7 0
0 0.7

(27)
The simulations where done in M AT L AB® and SimMechanics® and the results
are depicted in Figs.3 and 4
In Figs.3 and 4 the angle position for φ and its angular velocity respectively, show
how these variables reach the final positions in a considerable time. As it is noticed,
evenwhentherearesomeoscillations,thesevariablesreachesthezeroposition.These
results confirm that is possible to stabilize an underactuated mechanical system, in
this case the Furuta pendulum, by a second order sliding mode controller. As it is
Table 1 Furuta pendulum
parameters
Parameter Values
a 0.15 m
ma 0.298 Kg
p 0.26 m
mp 0.032 m
J 0.0007688 Kg.m2
g 9.81 m/s2

Fig. 3 Angular position for φ
Fig. 4 Angular velocity for φ
explained later in this section, these results are obtained due to the performance of
the sliding variables and their derivatives. In the following sections an analysis of the
oscillation or chattering is done to find the oscillation characteristics and compare it
with other control strategies.
In Figs.5 and 6 the variables for the position and angular velocity of θ are shown.
Even when stabilizes the Furuta pendulum variable θ is not an easy task, in this
example the angular rotation and velocity of the pendulum are stabilized satisfactorily
due to the performance of the sliding mode variables and their derivatives. As can
be noticed, these variable reach zero in a specified time, even when there are some

Fig. 5 Angular position for θ
Fig. 6 Angular velocity for θ
oscillations the system reach the origin in steady state, proving that the SOSMC is
effective.
In Figs.7 and 8 the phase portrait of φ and θ are shown depicting the phase trajec-
tories of the measured variables. It can be noticed how the trajectories of the system
reach the equilibrium points, proving that the system is stable under these conditions.
The limit cycles generated by the periodic orbits of the system are stabilized by the
second order sliding mode control that avoids instabilities and the control system
drives the state trajectories of the system until they reach the desired final values
ensuring the asymptotical stability of the system as proved theoretically in the previ-
ous section. This fact is very important for the chattering analysis, because the limit

Fig. 7 Phase portrait of φ
Fig. 8 Phase portrait for θ
cycles yielded by the periodic oscillations provides crucial information that can be
analyzed by Poincare maps as explained in Sect. 5 to calculate the oscillation period.
In Figs.9 and 10 the error signals for φ and θ are shown respectively. The results
depicted in these figures, shown that a very small tracking error for both variables
is obtained and they reach very small values in steady state. While keeping the
tracking errors as small as possible, the trajectory tracking of the two controlled
variables of the system is done effectively by the second order sliding mode control.
As it is explained before, the tracking error is reduced to zero in steady state by the
convergence of the sliding variables in finite time, proving that this control strategy
is suitable for the trajectory tracking of this underactuated mechanical system.

Fig. 9 Error of φ
Fig. 10 Error of θ
The input torque that controls joint 1 φ is shown in Fig.11 where a considerable
control effort is necessary to stabilizes the measured variables in a considerable time.
It is important to notice the oscillations yielded by the control switching function
and the necessary control effort applied by actuator 1 φ in order to keep the joint
in the desired position. As it is expected, even when the sliding mode control law
reduces chattering, it is still present, therefore it is necessary to analyze this effect
for comparison with the other strategies explained in this chapter.
In Figs.12 and 13 the sliding variables for σ1 and σ2 are shown. The sliding
variables reach the origin in a specified time assuring that the state variables achieve

Fig. 11 Torque for input 1
Fig. 12 Sliding variable 1
the zero value in steady state. This fact is very important since the stabilization of the
control variables such as the position and velocity depends on the convergence of the
sliding mode variable, so as it is shown theoretically the selection of an appropriate
control law algorithm is crucial for the efficiency of the SOSMC to stabilize the
Furuta pendulum.
In this section the design of a second order sliding mode controller for the Furuta
pendulum is shown, a higher order sliding mode control law is implemented to make
the state variables to reach the desired steady state value. A convenient control law
is proposed instead of applying classical second order sliding mode approaches such
as the twisting or super twisting algorithms. The objective of the SOSMC design is

to elucidate different SMC strategies for the control and stabilization of the Furuta
pendulum before deriving the adaptive sliding mode controller for this mechanism,
then the performance of the three control strategies are proven in order to evaluate
the chattering effects on the system. In the next section a variation of SMC control
is explained, in order to continue with the evaluation of different approaches before
deriving the proposed strategy of this chapter.
3 Proportional Derivative Plus Sliding Mode Control
of the Furuta Pendulum
In this section the derivation of a PD + sliding mode controller is shown to prove that
is an efficient alternative for the control of underactuated systems such as the Furuta
pendulum. Proportional derivative control (PD) has been proved to be an effective
and simple control architecture for mechanical systems, for this reason, a combined
control strategy along with a sliding mode controller is shown in this section. The
main idea in this section is to show that the Furuta pendulum can be stabilized by
this control law, even when this controller is simple. A combined linear control law,
given by the PD part of the controller, and a nonlinear part, given by the sliding mode
controller (Ouyang et al. 2014) make the system variables to reach the desired values
while the sliding surface reach the origin in a defined time interval.
Descentralized PD controllers are very popular in the control of different kind of
mechanical system; including robotic arms, parallel robots, “etc”, due to the sim-
plicity of their tuning parameters this kind of controllers at least ensure the local
stability of the controlled system. Even when this kind of controllers are very popu-
lar and simple they have some disadvantages such a poor disturbance rejection and

robustness; for this reason in order to improve the properties of the PD controller
sometimes it is necessary to combine this control strategy with a nonlinear control
law. There are some control strategies found in literature in which the PD controller
is combined with nonlinear control to improve the system performance, for example
in Xiang and Siow (2004) a combined PD + nonlinear + neural network control is
implemented for the stabilization of a robotic arm, the hybrid control law improves
the system performance in which the trajectory tracking of a two links robotic arm is
done by following a desired trajectory. In Liu (1999) another PD controller variation
is implemented in the control of a two links robotic arm, where a nonlinear part
is added to the proportional derivative controller for the trajectory tracking of this
mechanism, this descentralized control strategy make the system variables to follow
the desired trajectories when disturbance are applied to the system. Then finally,
a PD + sliding mode controller for the trajectory tracking of a robotic system is
explained in Ouyang et al. (2014), where the controller properties are improved by
adding a nonlinear discontinuous function to the combined control law. Therefore
based on the previous cases a suitable PD + sliding mode controller is suggested in
this section for the stabilization of the Furuta pendulum, considering the similarities
of the properties of some mechanical systems with the Furuta pendulum, the control
approach presented in this section is not only suitable for the control of this underac-
tuated system, it allows the tracking of the mechanical system properties efficiently
while keeping the tracking error as small as possible, with small chattering effect
and control effort.
In the following sections the design of a PD + sliding mode controller is explained,
where a proposed sliding surface is defined to ensure that the system is stable, proved
by an appropriate selection of a Lyapunov function (Liu 1999; Xiang and Siow 2004).
Then, an example of the stabilization of a Furuta pendulum is shown to illustrate
the implementation of this control law in this underactuated system, to analyze its
performance under a specified initial condition. The idea of this section is to provide
an alternative to the adaptive sliding mode control of the Furuta pendulum, that is
analyzed and compared in Sect. 5, then some conclusions are obtained according to
the chattering analysis of these controllers.
3.1 Derivation of the PD + Sliding Mode Controller
In this section the derivation of a proportional derivative plus sliding mode controller
for the Furuta pendulum is developed. A stabilizing PD + sliding mode controller
has been proved to be effective in the control of different kind of mechanical systems,
considering that this is an underactuated mechanical model, the control of this sys-
tem by PD + SMC is appropriate due to the combined advantages and properties of
this control strategy. The development of this control technique consists in designing
an appropriate sliding manifold considering the dynamical system properties of the
model that are common in many mechanical systems. Chattering avoidance is one
of the properties of the model that is required in order to avoid the instability and

system variables deterioration; this controller is very effective in order to cancel this
unwanted effect. Even when this technique is efficient in cancelling the chattering
effects in the system, this phenomenon is still present but with smaller negative results
than classical sliding mode controllers. Therefore, the analysis of this phenomenon
on the system is shown later in this chapter for comparison purposes with the other
sliding mode controllers explained in this chapter. The intention of this section is
to evince a combined sliding mode controller technique to understand and compare
with the main controller derived in this chapter, then some interesting conclusions
are obtained from all of these sliding mode control approaches, so all of these control
strategies are developed to show different alternatives and as a preview and compari-
son with the proposed adaptive sliding mode controller explained in the next section.
A complete analysis of this controller with the respective simulation is shown in
this section in order to clarify the theoretical background of this control approach by
deriving the PD + SMC strategy and show an illustrative example in order to verify
the performance of this controller.
The first step in the derivation of the PD + SMC for the Furuta pendulum, is to
define the following error signal (Liu 1999; Ouyang et al. 2014):
e = qd − q (28)
where based on the error signal the sliding surface r is given by:
r = ė + Φe (29)
where Φ is a positive definite matrix. Then the PD + sliding mode control for the
Furuta pendulum is given by (Liu 1999; Ouyang et al. 2014):
τ = kcr + k1sign(r) (30)
where kc and k1 are positive definite matrices for the PD and the sliding mode parts
of the control law respectively. The stability properties of this control law will be
examined later according to the Lyapunov stability theorem.
Substituting r in the dynamic system of the Furuta pendulum yields:
D(q)ṙ + C(q, q̇)r = −τ + ξ (31)
where
ξ = D(q)(q̈d + Φė) + C(q, q̇)(q̇d + Φe) + g(q) (32)
As explained in the previous section, ξ has a property that is very important for
the analysis of the stability of the closed loop system as described in Definition 2.
Definition 3 An n-degrees of freedom mechanical system has the following prop-
erties according to the dynamical systems characteristics:

μmin I D(q) μmax I (33)
and
C(q) ≤ CH (34)
g(q) ≤ Cg (35)
where μmax μmin 0 and CH , Cg 0
Definitions 2 and 3 are very important in order to prove the stability of the sys-
tems, according to the dynamical systems of the Furuta pendulum. Now, with these
properties and the dynamical system characteristics, the stability of the systems with
the specified control law is done as explained in the following theorem.
Theorem 2 The PD + sliding mode controller ensure the stability of the system if
the defined Lyapunov function indicates that the system is asymptotically stable.
Proof Consider the following Lyapunov function
V (r) =
1
2
rT
D(q)r (36)
The derivative of the Lyapunov function is given by:
V̇ (r) = rT
D(q)ṙ +
1
2
rT
Ḋ(q)r (37)
where
D(q)ṙ = −τ + ξ − C(q, q̇)r (38)
Then by applying Definition 1 and (38) the derivative of the Lyapunov function
becomes in:
V̇ (r) = −rT
τ + rT
ξ (39)
Then applying the norm on both sides of (39) and substituting the control law τφ
yields:
V̇ (r) ≤ −

rT

kcr + k1sign(r) +

rT

ξ (40)
Converting this inequality in:
V̇ (r) ≤ −kcmin

rT

r − k1min

rT

sign(r) +

rT

ξ (41)
where kcmin = mini∈nkc with kcmin 0 and k1min = mini∈nk1 with k1min 0 (Liu
1999; Xiang and Siow 2004).
Using the properties explained in Definition 2 and 3 the Lyapunov function indi-
cates that the system, representing the Furuta pendulum, is asymptotically stable
with the specified PD + sliding mode control law.

In the next section an illustrative example of the control of the Furuta pendulum
with a PD + sliding mode control is done to prove the validity of the theoretical
background demonstrated in this subsection.
3.2 Example 2
In this section an illustrative example of the control of the Furuta pendulum by a
proportional derivative plus sliding mode control is shown to clarify the application
of this controller to this underactuated mechanical system. Even when the control
of underactuated systems is difficult, it is shown theoreticaly and by an illustrative
example that is possible to stabilize this mechanism by selecting an appropriate con-
trol law algorithm. In this example the angle trajectories and velocities are depicted
to prove that these variables are stable and reach the desired values in steady state.
The phase portraits shown in this section, verify the asymptotical stability of the
system while minimizing the tracking error of the model.
In this example the stabilization of a Furuta pendulum with a PD + sliding mode
control is shown with appropriate parameter selection. The parameters of the Furuta
pendulum are specified in Table1 with (π/2, 0) as the initial conditions of the system.
The gains of the PD + SMC are given as follow:
kc =

0.7 0
0 0.7

(42)
k1 =

0.01 0
0 0.01

(43)
Φ =

90 0
0 90

(44)
The simulations were done in M AT L AB® and SimMechanics® where the
specified parameters are used in all the simulation process. In Fig.14 the angle
trajectory φ is depicted, where as it is noticed the trajectory of this variable reaches the
specified value in steady state, proving that PD + sliding mode controller stabilizes
the system with the desired performance. In Fig.15, the angular velocity for the
variable φ is shown, where this variable reaches the zero value in a specified value as
the corresponding variable is stabilized. As it is noticed these variables reaches the
expected values in finite time, this result is achieved due to the appropriate sliding
mode manifold is selected in order to stabilize the controlled variables.
In Fig.16 the angular trajectory for the pendulum angle θ is shown, where this
angle reach the value of zero in steady state as defined by the controller and sys-
tem specifications, so the PD + sliding mode controller of the Furuta stabilizes this
variable in the required time keeping the two controlled variables in the desired

Fig. 14 Angle position for φ
mechanism positions. In Fig.17 the angular velocity of the controlled variable θ is
shown, where this variable reaches the zero value in steady state as defined by the
controller and system specifications. With these results the stability of all the con-
trolled variables is ensured by the implementation of a PD + sliding mode controller,
keeping the Furuta pendulum stable when external disturbances are applied in the
system. As it is proven theoretically, the appropriated sliding manifold selection is
very important in order to stabilize these variables, reaching and keeping the desired
values in finite time.
The respective phase portraits for φ and θ are shown in Figs.18 and 19. As it
is noticed, the two phase portraits show that these variables are stable, according

Fig. 16 Angle position for θ
to their respective phase trajectories. The two limit cycles depicted in these figures
show that the oscillations follow a prescribed trajectory until the variables reach the
desired values when a disturbance is applied on the system. The phase portraits show
that the limit cycles yielded by the periodic oscillations are stable, proving that the
PD + sliding mode control law meets the required specifications according to the
stabilization of the state variables of the system.
In Fig.20 the respective input torque for joint 1 (base) is shown. As it is noticed the
control effort for the joint is reasonable so it is not necessary to saturate the actuator.
The torque input applied to the base joint behaves in an oscillatory manner as it is

Fig. 18 Phase portrait of φ
Fig. 19 Phase portrait of θ
expected generating some oscillations until the system variables reach the desired
values in steady state.
The corresponding oscillation analysis of the variables and the torque inputs is
done in Sect. 5 where the chattering effect is evaluated according to the oscillation
frequencies of this and the other SMC strategies explained in this chapter.
In Figs.21 and 22 the respective sliding variables of the PD + sliding mode
controller are shown, where the two variables converge to zero in a determined time.
Ensuring that the sliding variables reach the origin in an expected time allowing the
system to reach the specified steady state values with a considerable small control
effort generated by the switching control law.

Fig. 20 Input torque 1
The error signals for φ and θ are shown in Figs.23 and 24 respectively, where
the PD + sliding mode controller makes the error signal to reach the zero value in
an expected time, proving the efficiency of this controller to stabilizes mechanical
systems of different kind.
In this section a PD + sliding mode controller for the control and stabilization of
the Furuta pendulum is explained, to prove their suitability in the control of this kind
of underactuated mechanical system. This control approach is advantageous because
it combines the simplicity of a proportional derivative controller and the efficiency of
a nonlinear sliding mode control making this strategy ideal for the stabilization of this
kind of mechanism. The stability of the PD + SMC is corroborated by the selection of

Fig. 23 Error signal for φ
an appropriate Lyapunov function and this fact is confirmed by a numerical example
and simulation of the Furuta pendulum with this control strategy.
As it is confirmed in this chapter, all the variables are stabilized according with
the system design specifications and initial condition of the model; reaching the
expected value in steady state. This system behavior is illustrated in the phase plot of
each variable, where the state trajectories reach the specified point in these diagrams,
so this control strategy yields stable limit cycle oscillations when a disturbance is
applied to the system.
As it is noticed in Example 2, the control effort generated by the controller output
is significantly small to keep the controlled variables in the equilibrium point of the

Fig. 24 Error signal for θ
system. The tracking error of this mechanism reach the desired final value, keeping
the controlled variables in the desired trajectory even when disturbances are present
in the model, proving that this control strategy is efficient in the trajectory tracking
of the Furuta pendulum.
In the next section, the proposed control strategy of this chapter is developed,
an adaptive sliding mode controller ASMC for the stabilization and control of the
Furuta pendulum is designed, where the performance of this model is analyzed when
an ASMC is implemented in the control of this kind of underactuated system. All the
SMC strategies are compared and analyzed in Sects. 5 and 6 to obtain the respective
conclusions of this work.
4 Adaptive Sliding Mode Control of the Furuta Pendulum
In this section the main control technique of this chapter is explained, an adaptive
sliding mode control for the stabilization of the Furuta pendulum (ASMC). Adaptive
sliding mode control is a control approach that has been implemented extensively in
different kinds of applications due to the flexibility of the sliding mode parameters
(Chang 2013; Cheng and Guo 2010); this is a very useful control approach used
in different kinds of systems such as electrical (Liu et al. 2013; Chen et al. 2014)
and mechanical systems (Fei and Wu 2013) yielding the desired performance when
disturbances are applied to the system.
An adaptive gain SMC is implemented for the stabilization of the Furuta pendulum
considering the system dynamics of the model and its stability properties to keep
the mechanism trajectory in the desired position. Another property of this control
approach is that the chattering effect is minimized and then, the oscillations of the
system are cancelled by the controller characteristics.

Adaptive sliding mode control is a control technique that consists in the imple-
mentation of an adaptive gain parameter obtained according to the Lyapunov stability
theorem along with a sliding mode controller to ensure the asymptotical stability of
the system. The design procedure for this kind of controller consists in designing a
feasible controller that reaches the sliding manifold in finite time while stabilizing
the state variables for the trajectory tracking. The advantage of this control strate-
gies is that the effects yielded by perturbation and disturbances are suppressed by
the adaptive control along with the sliding controller action. For this reason, this
control technique is appropriate for the control and stabilization of different kind
of mechanical system, obtaining a very robust controller that adjusts its parame-
ters in real time moving and keeping the state variables in the desired trajectory.
The control law of this controller is designed by combining an adaptive part with a
sliding mode controller that makes the Furuta pendulum variables reach the desired
values in steady state while keeping the system in the sliding manifold to obtain
an asymptotically stable system. Another advantage of this controller is the chatter-
ing suppresion effects yielded by the adaptive sliding mode controller, so the action
produced by the discontinuos sliding mode controller algorithm is cancelled by the
effect of the adaptive gain of the system. Taking in count that the parameters of the
mechanical system such as the gravitational, coriolis and inertia matrices change
in time an adaptive sliding mode controller is suitable for this kind of mechanism
(Yao and Tomizuka 1994) making a flexible control strategy that vary the controller
parameters in real time while ensuring the asymptotical stability of the system. This
control strategy has another advantage that is related to the smaller control effort
that is necessary in order to stabilizes the system variables while keeping the sliding
variables in the origin, this is a desirable property that means that it is not necessary
to saturate the actuator due to higher values of the control action. The adaptive sliding
mode controller is proved to be a strong control strategy that is applied in the control
and stabilization of the Furuta pendulum as demonstrated in this section, the sliding
mode control strategies developed in this chapter proved that are effective and they
are the fundamental control approaches for the stabilization of the Furuta pendulum.
The SMC approaches shown in this chapter are developed to shown the fundamentals
of adaptive sliding mode control for the stabilization of the Furuta pendulum and
for comparison purposes with the proposed control approach of this chapter, that
even when these control strategies are effective in the control of this underactuated
system, they lack of important properties that adaptive sliding mode control has for
the stabilization of the Furuta pendulum; for this reason the comparison and analysis
of these control strategies are shown in Sects. 5 and 6.
In the following sections the development of an ASMC for the Furuta pendulum
is derived and explained as the proposed control strategy of this chapter and then the
system performance is corroborated with an illustrative example of the ASMC for
the Furuta pendulum with specified initial conditions. The proposed control strategy
is compared later in the following sections according to the chattering effects on the
system and other characteristics of the control system.

4.1 Derivation of the Adaptive Sliding Mode Controller of the
Furuta Pendulum
Adaptive sliding mode has successfully proved that is an efficient control technique
for different kinds of systems (Yu and Ozguner 2006) therefore this control approach
has better disturbance rejection properties than classical SMC, for this reason this
control strategy is convenient for the control of underactuated mechanical systems
as explained in this section. Adaptive sliding mode control is suitable for the control
of underactuated mechanical systems due to its robustness and disturbance rejection
properties when unmodelled dynamics and disturbance are present in the system,
this controller updates its adaptive gain online improving the performance of the
controller and therefore the asymptotical stability of the system is ensured by this
control strategy. The derivation of this adaptive gain is done by the Lyapunov stability
theorem in order to ensure the stability of the system and the sliding variables con-
vergence for a better trajectory tracking of the system. The objective of this chapter is
to proved that a feasible adaptive sliding mode controller can be designed in order to
improve the disturbance rejection and chattering avoidance properties of the Furuta
pendulum by ensuring the stability of the system with a small control effort and
reducing the chattering effects on the system.
Before deriving the ASMC for the Furuta pendulum, an important property for
mechanical systems is described as follow: Consider the dynamics equation of the
Furuta pendulum as described in (8), then this system is linearly parametrizable as
described in the next equation (Liu 1999; Xiang and Siow 2004).
D̂(q)q̈r + Ĉ(q, q̇)q̇r + ĝ(q) = Y(q, q̇, q̇r , q̈r )ψ (45)
where D̂(q), Ĉ(q, q̇), ĝ(q) are the estimated dynamical systems matrices and vector
respectively and ψ is the parameter of the dynamical system model that is adjusted
by the adaptive control law. Then q̇r is defined as:
q̇r = q̇d + Φq̃ (46)
where q̃ = qd(t) − q(t), q(t) is the position vector of the Furuta pendulum, qd(t) is
the desired position vector and Φ is a positive definite matrix.
Based on the previous variables, the sliding surface is defined as:
S = q̇r − q̇ = ˙
q̃ + Φq̃ (47)
where the derivative of S is given by:
Ṡ = q̈r − q̈ = ¨
q̃ + Φ ˙
q̃ (48)
The proposed control law for the stabilization of the Furuta pendulum is (Xiang
and Siow 2004; Fei and Wu 2013; Liu et al. 2013; Chen et al. 2014):

τ = D̂(q)q̈r + Ĉ(q, q̇)q̇r + ĝ(q) − kd S − k1sign(S) (49)
Then using the linear parametrization property of the Furuta pendulum dynamics,
(49) is converted to:
τ = Y(q, q̇, q̇r , q̈r )ψ − kd S − k1sign(S) (50)
where kd and k1 are constant positive definite matrices. The following theorem
ensures asymptotical stability of the system with the proposed adaptive sliding mode
control law and it is necessary in order to find the adaptive parameter of the system.
Theorem 3 The adaptive sliding mode controller ensures the stability of the system
if the defined Lyapunov function indicates that the system is asymptotically stable.
Proof Consider the following Lyapunov function:
V (S, ψ) =
1
2
ST
D(q)S +
1
2
ψT
Γ −1
ψ (51)
where Γ is a positive definite adaptive gain matrix. Then the derivative of the
Lyapunov function is given by:
V̇ (S, ψ) = ST
D(q)Ṡ +
1
2
ST
Ḋ(q)S + ψ̇T
Γ −1
ψ (52)
Then with
D(q)Ṡ = −τ + ξ − C(q, q̇)S (53)
where
ξ = D(q)(q̈d + Φ ˙
q̃) + C(q, q̇)(q̇d + Φq̃) + g(q)
ξ = D(q)q̈r + C(q, q̇)q̇r + g(q) (54)
Then V̇ (S, ψ) becomes in
V̇ (S, ψ) = −ST
τ − ST
C(q, q̇)S + ST
ξ +
1
2
ST
Ḋ(q)S + ψ̇T
Γ −1
ψ (55)
Then considering the estimation of the parameter ξ
ξ = Y(q, q̇, q̇r , q̈r )ψ (56)
and rearrenge to apply Definition 1, V̇ (S, ψ) becomes in:
V̇ (S, ψ) = −Sτ + ψ̇T
Γ −1
ψ + ST
Y(q, q̇, q̇r , q̈r )ψ (57)

Then in order to stabilize the system the updating law of the variable parameter
must be:
ψ̇T
= −ST
Y(q, q̇, q̇r , q̈r )Γ (58)
Therefore the derivative of the Lyapunov function becomes in:
V̇ (S, ψ) ≤ − |S| τ (59)
So the system is asymptotically stable with the updating law of the adaptive
parameter ψ. This completes the proof of the theorem.
In the next section an example of the stabilization of the Furuta pendulum by an
ASMC is shown to illustrate the system performance.
4.2 Example 3
In this section an illustrative example of the control and stabilization of the Furuta
pendulum by an ASMC is shown. The main idea of this example is to illustrate
the system performance by a numerical simulation, where the system is tested with
specified initial conditions and trajectories. Then the results obtained for the angular
positions, velocities, phase portraits, tracking errors and input torques are analyzed
to obtain the respective conclusions of the system performance. In this section is
proved that the sliding mode controller with adaptive gain meets the requirement
of the stabilization and tracking error reduction by an appropriate adaptive control
law with a well defined updating gain algorithm. The theoretical background of
the adaptive sliding mode controller for the stabilization of the Furuta pendulum is
corroborated in this example by selecting the appropriate controller parameters for
the adaptive gain, in order to stabilize the system by the on line adaptation of the
adaptive gain.
The controller parameters are:
k1 =

0.0000001 0
0 0.0000001

(60)
kd =

7 0
0 7

(61)
The adaptive gain evolution is shown later in this section and the initial con-
dition of the model is (π, 0). The simulations were done in M AT L AB® and
SimMechanics® where the specified parameters are used in all the simulation
process. In Figs.25 and 26 the angular position and velocity of the base φ are shown
respectively, where as it is noticed these variables reach the desired final values in
steady state, with no oscillations in comparison with the previous SMC alternatives.

Fig. 25 Angle position for φ
The positions and angular velocities of the base are stabilized as it is defined by the
adaptive control law, this requirement is met due to the convergence of the sliding
variables in finite time. In comparison with the previous sliding mode control strate-
gies, the trajectory tracking of these variables evince less oscillations, and a better
system performance.
In Figs.27 and 28 the angular position and velocity of the pendulum θ are shown
respectively where the desired final values in steady state of the system are reached in
a determined time when a disturbance is applied on the system. Practically there are
no oscillations on the pendulum parameters, and then this proves that the proposed
ASMC is an effective technique for the stabilization of this underactuated mechanical

Fig. 27 Angle position for θ
system. These variables reach the desired position due to the convergence of the
sliding variables, and in comparison with the other control strategies, the adaptive
sliding mode controller for the stabilization of the Furuta pendulum shows a better
performance in the stabilization of these variables due to a small chattering, that can
be considered as oscillations of the system, and less control effort.
The input torque for the joint actuator is shown in Fig.29, where the necessary
control effort is necessary to be applied to stabilizes the controlled variables of
the system. As explained before, it can be noticed that practically there are not
oscillations on this control input, so this undesirable effect is eliminated by the
adaptive characteristic of this adaptive gain system.

Fig. 29 Input torque 1
In Figs.30 and 31 the respective sliding variables are shown, where as it is noticed
these variables reach the origin in a considerable time, yielding the convergence of
the controlled variable in finite time. These facts corroborates the theory behind this
control strategy implemented in the control of the Furuta pendulum, where the sliding
manifold must be reached in finite time to ensure the stability of the system.
In Fig.32 the norm of the adaptive gain ψ is depicted in this figure. As can
be noticed, the evolution of the adaptive gain goes from the initial value to the final
value of this parameter until the system variables and the adaptive gain reach the
desired value in finite time.

Fig. 32 Norm of the adaptive gain ψ
Finally in Figs.33 and 34, the error signals of the model reach the zero value in
finite time as specified in the ASMC design, so the controlled variables φ and θ are
stabilized in finite time while keeping the tracking error about the zero.
In this section the proposed control strategy of this chapter is shown, for the
stabilization and control of the Furuta pendulum, keeping the tracking error of the
controlled variable about zero. This objective was proved theoretically and corro-
borated later by a simulation example. The adaptive gain of ASMC improves the
performance of the system considerably in comparison with the control strategies
developed in the previous sections, and as it is confirmed in the following section the

Fig. 33 Error of φ
Fig. 34 Error of θ
performance and chattering avoidance properties of the proposed controller is better
than the other two control alternatives explained in this chapter.
The main objective of this chapter is to analyze and develop different sliding
mode control strategies, from the classical to new kinds of SMC variations to find the
theoretical basis of well known sliding mode control algorithms and compare them
with novel SMC strategies for the control of underactuated mechanical systems.
In this section the adaptive sliding mode controller is developed exploiting the
linear parametrization of the system, that is an important properties of different
kinds of mechanical systems, and it is a contrasting characteristic of the ASMC in
comparison with the other algorithms explained in this chapter. The adaptive gain

increases the disturbance rejection properties of the system so the system is more
robust in comparison with the other algorithms explained in this chapter.
In the next section a chattering analysis of the three SMC approaches is clearly
explained to test the system performance when one of these techniques is imple-
mented for the control and stabilization of the Furuta pendulum, then some impor-
tant conclusions are obtained in the design of an appropriate control strategy for this
mechanical system.
5 Chattering Analysis
In this section a chattering analysis of the SMC developed in this chapter is done.
Chattering is basically a high frequency oscillation effect yielded by the switching
inputs of the sliding mode control law. This unwanted effect deteriorates the system
performance and could lead to the instability of the system. One way to avoid this
effect is by designing appropriated control strategies such as the second order sliding
mode control (Bartolini et al. 1998) or high order sliding mode control instead of
implementing classical sliding mode controller strategies. Another way to solve
this problem is by selecting appropriate sliding mode control laws that reduce this
unwanted effect such as implementing second order sliding mode algorithms like the
twisting and super twisting algorithms (Fridman 2012).
The chattering oscillations yielded by the chattering effect have been studied
by different authors and basically there are two methods that can be implemented
to analyze this effect; the describing function analysis (Boiko and Fridman 2005;
Boiko et al. 2007, 2008) and the Poincare map analysis (Boiko et al. 2008; Haddad
and Chellaboina 2014) where the frequency, period and stability of the limit cycle
oscillations yielded by chattering can be analyzed.
The purpose of this section is to find the period and frequency of the limit cycle
oscillations generated by chattering in each of the SMC explained in this section.
In the previous sections the performance of the system was proved analytically and
corroborated by examples; therefore there is a clear idea of the limit cycle properties
yielded by chattering in the input of the mechanical system, so the intention here is
to show an analytical procedure to find the oscillation periods by Poincare maps.
In order to define the Poincare maps, the first step is to transform the dynamical
system (8) of the Furuta pendulum to state space by linearizing the model
ẋ = Ax + Bτ (62)
Y = Cx (63)

where
A =
⎡
⎢
⎢
⎢
⎣
0 0 1 0
0 0 0 1
0 −p3 p4
p1 p2+p1 p5−p2
3
0 0
0 p1 p4
p1 p2+p1 p5−p2
3
0 0
⎤
⎥
⎥
⎥
⎦
(64)
B =
⎡
⎢
⎢
⎢
⎣
0 0
0 0
p2+p5
p1 p2+p1 p5−p2
3
−p3
p1 p2+p1 p5−p2
3
p3
p1 p2+p1 p5−p2
3
p1
p1 p2+p1 p5−p2
3
⎤
⎥
⎥
⎥
⎦
(65)
C = I4×4 (66)
where I4×4 is an identity matrix. Now defining the solution of the linear system in a
specified point by (Haddad and Chellaboina 2014):
s(t, p) = x(o)et
+
t
0
e(t−λ)A
B(λ)τ(λ)dλ (67)
Define the function:
ζ(x)

=

ζ̂ 0 : S(ζ̂, x) ∈ Sand S(t, x) /
∈ S, 0 t ζ̂

(68)
Then the Poincare map is given by:
P(x)

= s(ζ(x), x) (69)
Finally, from the Poincare map shown in (70) a dynamic system in discrete time
is obtained as shown in (71)
z(k + 1) = P(z(k)) (70)
Therefore proving the stability of the discrete function shown in (71) the stability
of the periodic orbit can be determined in a fixed point x = p since the period is
T = ζ(p) and consequently p = P(p) (Haddad and Chellaboina 2014).
Then making the Poincare map
P(z(k)) = 0 (71)
With t = T , then the smaller positive period T obtained from this equation is
the resulting period of the chattering oscillations. Making the Poincare map for each

Table 2 Period and
frequency of the chattering
oscillations
Example Chattering Chattering
period (s) frequency (rad/s)
1 3.10 2.02
2 5.10 1.23
3 8.10 0.77
example of this chapter the following periods and chattering oscillation frequencies
are obtained:
In Table2 the chattering period and frequency is shown, as can be noticed, the
frequency of the oscillations of Example 3 is the smallest of the three examples
corroborating that the adaptive sliding mode controller avoids the chattering effects
better than the other two approaches as it is seen in the simulation results. It is impor-
tant to notice, that the PD + sliding mode controller implemented in Example 2
yield a small chattering frequency in comparison with the second order sliding mode
controller of Example 1, so this combined control strategy yields better results than
the approach shown in Sect. 2. With this analysis the results obtained by simulation
were proved by an analytical method, therefore the proposed adaptive sliding mode
controller for the Furuta pendulum yields better results than the other control strate-
gies as it is corroborated in Sect. 4 due to the chattering avoidance properties of this
controller.
6 Discussion
In this chapter, three control strategies are shown for the control and stabilization of
the Furuta pendulum. A SOSMC, a PD + SMC; and the proposed control strategy
of this paper, the ASMC for the Furuta pendulum is shown. In order to verify the
performance and properties of each controller there are two important properties
that must be considered to evaluate the system performance; these parameters are
the convergence time, chattering period and frequency.
The converge time can be computed considering the Lyapunov function V as
follow (Shtessel et al. 2014):
tr ≤
2V
1
2 (0)
α
(72)
where α is a positive constant and tr is the upper bound of the convergence time.
All the sliding mode controllers for the stabilization of the Furuta pendulum reach
the sliding manifold with this convergence time, so all the alternatives shown in
this chapter meet the requirement to control this underactuated mechanical system.
This is an important property that must be considered in the design of sliding mode
controller for coupled nonlinear systems because it ensures the stability and converge
of the controlled variables in finite time and keeping the desired final value in steady

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Les formalités du mariage
catholique.
ÉGLISE ne bénit pas de mariages pendant le carême jusqu'à
l'octave de Pâques; non plus, entre le premier dimanche de
l'Avent et la fête de l'Épiphanie.
Cependant, avec une dispense, on peut se marier pendant ces
temps.
Il faut également une dispense pour les mariages entre parents.
Cette dispense s'obtient par l'entremise du curé de la paroisse
moyennant une somme qui varie, selon le degré de parenté.
Les bans doivent être annoncés au prône, pendant trois dimanches
consécutifs, aux paroisses des deux époux; on peut racheter deux
bans et même trois bans; mais ce dernier rachat n'est admis que
dans certains cas très graves.
Le prix de rachat des bans est fixé, selon les usages, par chaque
paroisse.
Si on laisse écouler trois mois après la publication des bans, avant
de célébrer le mariage, il faut les renouveler; dans certaines

paroisses, on a six mois.
Pour pouvoir publier les bans à l'église, il faut un certificat de
publication à la mairie.
Aucun ministre d'une religion n'a, en France, le droit de bénir une
union, si elle n'a pas été célébrée devant l'officier de l'état civil.
En passant outre, il aurait d'abord une amende de 16 francs; et, en
cas de récidive, il serait condamné la première fois à un
emprisonnement, variant de un à cinq ans, la seconde fois, à la
déportation.
Il faut donc faire déposer à la sacristie un certificat du mariage civil,
avec le certificat de publication des bans dans les deux paroisses, les
dispenses de l'évêque, en cas de mariage entre parents ou en temps
prohibé; les extraits de baptême des deux époux, le certificat de
première communion qui peut, à la rigueur, remplacer l'extrait de
baptême; le billet de confession.
Si la cérémonie a lieu dans une église autre que la paroisse de l'un
des mariés, le consentement des curés de ces églises est nécessaire,
ou, en cas de refus de leur part, l'autorisation de l'évêque.
L'acte du précédent mariage et l'acte de décès de l'époux pour les
veufs ou veuves qui se remarient, remplacent les certificats de
baptême et de première communion.
Il faut aussi une dispense papale pour épouser une personne
appartenant à une autre religion.
Il y a plusieurs classes pour le mariage religieux. Les premières
classes voient le mariage célébré au maître-autel avec une plus ou
moins grande profusion de fleurs, de lumière, de chants, de
musique. Les autres classes sont pour la chapelle de la Sainte-
Vierge, de Saint-Joseph et autres chapelles.
On s'entend à l'avance avec le prêtre qui représente le curé et la
fabrique; on paye tous les frais: les gratifications aux employés de

l'église et le prix des chaises sont compris dans ces frais.

A l'Autel.
I l'on désire que la bénédiction nuptiale soit donnée par un
prêtre étranger à la paroisse, il y a un arrangement à
prendre: c'est alors le prêtre qui eût béni l'union qui dit la
messe.
Le mariage religieux se célèbre toujours dans la matinée; dans
quelques châteaux, chez les partisans des innovations, on voit des
mariages qui sont célébrés à minuit. Je n'aime pas cela; la lumière
du jour, le soleil me semblent indispensables.
Le marié et sa famille se rendent les premiers au domicile de la
mariée.
Le marié apporte à sa femme le bouquet nuptial qui doit être
entièrement blanc, mais non composé exclusivement de fleurs
d'oranger; ce n'est plus le volumineux bouquet rond, entouré de
papier découpé; c'est une gracieuse gerbe de moyenne grosseur,
aux tiges flexibles; les tiges sont entourées d'un mouchoir de batiste
garni de haute dentelle qui retombe en collerette autour des fleurs;
un nœud de ruban de satin ou de moire blanc, à longs pans, attache
le mouchoir, charmant cadeau pour la jeune femme.
Le marié doit envoyer prendre chez eux, en voiture, les témoins, les
parents, les amis, les garçons d'honneur qui doivent former le
cortège de la mariée.

Dans les mariages modestes, ces personnes se rendent à pied au
domicile de la mariée.
Seuls, les garçons d'honneur vont toujours en voiture chercher leurs
demoiselles d'honneur. Bien entendu, les jeunes personnes ne s'en
vont jamais seules, avec leurs garçons d'honneur, fussent-elles deux;
il leur faut toujours un chaperon.
Le matin de la cérémonie, le garçon d'honneur envoie à sa
demoiselle d'honneur un bouquet ROSÉ et non pas blanc, entouré de
dentelle et lié de ruban blanc; ce bouquet doit être plus petit que
celui de la mariée.
A moins de relations étroites entre la famille du garçon d'honneur et
celle de la demoiselle d'honneur, celui-ci ne doit pas envoyer le
moindre présent. En cas d'intimité ou de parenté, il peut envoyer des
gants, une jardinière, mais jamais de bijoux.
Le lendemain du mariage, le garçon d'honneur rend une visite à la
famille de la demoiselle d'honneur.
On doit ne jamais prendre frère et sœur pour être garçon et
demoiselle d'honneur; il faut intervertir l'ordre des familles.
Il n'est pas nécessaire d'être de la famille pour remplir ces fonctions.
Le nombre des demoiselles d'honneur est illimité, deux, quatre, six;
en Angleterre, elles sont quelquefois douze, habillées de mêmes
nuances claires formant un charmant escadron volant à la jeune
épousée.

Toilette de la mariée.
ORSQUEtout le monde est arrivé, le père de la mariée va
chercher sa fille et la conduit au salon où la mère de la
mariée reçoit ses invités.
La mariée doit porter une robe de satin, de brocart, de moire, de
faille, de cachemire, de mousseline blanche; quelle que soit la valeur
de l'étoffe, la façon doit être simple.
Les dessous de la mariée doivent être entièrement blancs; chemise
de batiste ou de surah blanc, avec nœuds bébé en satin blanc, aux
épaulettes, pantalon assorti; corset de faille ou de satin blanc, petit
jupon assorti et grand jupon de satin, de faille, de nansouk, tout
floconneux de multiples volants qui doivent soutenir la traîne; la
robe à traîne ronde ou carrée peut être garnie dans le bas d'une
draperie retenue par de minuscules piquets de fleurs d'oranger, ou
tout unie. Le corsage et les manches, selon la mode de l'année où
on se marie; un tout petit bouquet d'oranger au côté gauche du
corsage; pas de brillants, pas de bijoux, à peine des perles aux
oreilles.
Dans quelques mariages fastueux, on garnit la robe des dentelles de
famille et la mariée a un voile de vieux point d'Angleterre ou de
toute autre dentelle précieuse.

Les grosses couronnes qui casquaient si lourdement les fronts des
mariées ne se portent plus à Paris; on préfère la petite couronne
comtesse, posée très en arrière. Le piquet de côté en aigrette ou un
tout petit semis de fleurs d'oranger est joli et à la mode.
Le voile se pose à la juive, à la mauresque, à l'espagnole, mais je
préfère à la juive.
Bas de soie, souliers de satin blanc; gants longs en chevreau blanc.
Tout au plus, en fait de joyaux, un rang de perles.
Le marié porte le classique costume: souliers vernis, chaussettes de
soie noire, pantalon noir, gilet noir, cravate de soie blanche, habit,
gants blancs, claque.
Si le futur appartient à l'armée, il se marie en grand uniforme.
Les modes plus ou moins exotiques qui ont essayé de prévaloir
contre le costume traditionnel n'ont pas réussi.
L'habit rouge que quelques sportsmen anglais ont adopté ferait en
France assez mauvais effet.
De même quelques élégants chez nous ont vainement essayé de
mettre à la mode la redingote longue et le pantalon gris, ce costume
matrimonial n'a point été adopté.

Cortège.
OMME pour aller à la mairie, la mariée occupe la droite dans
la première voiture à côté de sa mère; son père et sa sœur
ou une jeune parente prennent place sur le devant.
Il va sans dire que, si la jeune fille est orpheline, la dame qui lui sert
de mère a tous les honneurs.
Dans la seconde voiture monte le marié, à côté de sa mère; en face,
son père et sa sœur s'il en a une, ou tout autre proche parent.
Si les demoiselles d'honneur ne sont pas dans les voitures des
mariés, elles sont avec leur famille et leurs garçons d'honneur dans
les voitures qui suivent immédiatement celle de la mariée; après
viennent les témoins, puis, un peu à leur guise, les invités.
Les cochers, les serviteurs ont à leur boutonnière un très petit
bouquet de fleurs d'oranger.
Lorsque la mariée descend de voiture, il doit y avoir, sous le porche
de l'église, une femme de chambre avec des aiguilles enfilées, des
épingles, de manière à pouvoir réparer toute avarie à la virginale
toilette ou tout au moins arranger le voile, la traîne. C'est à ce
moment que se forme le cortège.

Toujours la mariée doit laisser se former le cortège avant de
descendre de voiture.
Au son d'une marche triomphale la mariée effectue son entrée au
bras gauche de son père; dans le cas où il porterait l'épée, au bras
droit.
Elle ne doit pas distribuer des signes de tête et des sourires de
droite et de gauche; elle doit s'avancer d'un pas cadencé, les yeux
baissés sans ostentation.
Le marié vient ensuite avec sa mère, puis la mère de la jeune femme
avec le père du marié, les deux couples de garçons et de
demoiselles d'honneur, les plus proches parents des deux familles,
assortis d'âge et de goût autant que possible, le flot des amis et en
serre-file les hommes qui n'ont pas de cavalières, chose qu'il faut
éviter autant que possible.
Lorsque la jeune femme arrive à sa place, le suisse ou mieux le
garçon d'honneur doit arranger son voile, sa traîne.
Au reste, pour être digne de cette fonction, enviée et pourtant
difficile de garçon d'honneur, il faut payer de sa personne; non
seulement le matin on doit aller chercher sa demoiselle d'honneur,
mais encore les autres dames.
Au signal donné par le suisse d'un coup de hallebarde, tous les
assistants se sont levés, ils se tournent à demi pour regarder le
défilé.
Le père de la mariée la conduit à sa place; le prie-Dieu est à gauche,
un cierge à poignée blanche brûle auprès; le marié est à droite avec
ses témoins.
Il est à remarquer que les amis et invités du marié sont du côté
droit, ceux de la mariée du côté gauche.
Les parents sont dans le chœur le plus près possible de leur enfant;
les garçons d'honneur doivent placer les invités selon les rangs de

Le Cérémonial.
E suisse et le bedeau indiquent aux assistants le moment où
il faut se lever, s'agenouiller, s'asseoir.
A l'église, des parents peuvent remplir le rôle de témoins; il suffit
donc d'en avoir deux, au lieu de quatre comme à la mairie.
La jeune mariée doit éviter de tourner la tête pour voir ce qui se
passe derrière elle; le soin de son voile, de sa robe ne doit pas
l'occuper.
Si, dans l'église où a lieu le mariage, on tend le poêle (bande
d'étoffe) au-dessus de la tête des mariés, je recommande vivement
au garçon d'honneur de faire attention à la coiffure de la mariée.
Les mariés sont assis pour écouter l'allocution du prêtre au sujet de
leurs devoirs réciproques et des obligations qu'ils auront envers les
enfants qui leur naîtront.
Pour la consécration du mariage le prêtre vient aux jeunes époux,
qui se tiennent par la main droite (dégantée), et c'est ainsi qu'ils
doivent répondre aux questions sacramentelles.
De même, lorsqu'ils s'agenouillent sur leurs prie-Dieu pour recevoir
la bénédiction.
Le oui doit être articulé à mi-voix mais distinctement.

Lorsque les anneaux sont bénis, le prêtre les remet à l'époux; celui-
ci passe l'alliance symbolique au quatrième doigt de la main
dégantée de sa femme. Il serait logique que celle-ci passât de même
la bague au doigt de son mari, mais c'est lui-même qui s'en charge.
Les mariés peuvent ensuite se reganter.
Tantôt on applique la pièce d'or ou d'argent à la cire du cierge que
tiennent les époux pour aller baiser la patène, tantôt on la dépose
dans le plat de vermeil que tient l'enfant de chœur.
Pour les quêtes dans l'église, faites par les garçons et les demoiselles
d'honneur, il y a certaines nuances à observer.
Disons, à ce propos, que, si les garçons d'honneur sont de tout
petits garçons et de toutes petites filles, et rien de plus charmant, on
peut se livrer à la fantaisie pour les habiller.
Lorsqu'il s'agit de demoiselles pour de bon, elles devront éviter
d'être en blanc, sauf les gants qui, ainsi que ceux des garçons
d'honneur, doivent toujours être de cette couleur; la nuance paille ou
crème n'est même pas admise. La bourse de quêteuse est faite en
étoffe semblable à la robe avec petit bouquet d'oranger et nœud de
ruban.
Passé trente ans pour les demoiselles et quarante ans pour les
garçons, il n'est guère possible d'accepter ces fonctions.
Lorsque le suisse (pour le couple qui appartient au côté de la
mariée) et le bedeau (pour celui qui appartient au côté du marié)
viennent chercher les garçons et les demoiselles d'honneur pour la
quête, ceux-ci doivent tout d'abord déposer leur offrande
personnelle au fond de la bourse, puis la présenter au jeune couple,
aux parents qui sont dans le chœur, enfin descendre dans la nef et
s'arrêter devant chaque rang d'invités, qui à droite, qui à gauche.
Le garçon d'honneur tient de la main gauche le bouquet de sa
compagne et son claque et il lui offre le poing droit fermé; la jeune

fille y pose sa main gauche: cette main doit être maintenue à une
certaine hauteur.
Cette position, très gracieuse, vous a un petit air moyen âge plus joli
en vérité que l'attitude de jeunes gens marchant la main dans la
main comme des enfants qui vont à l'école.
La jeune fille tend la bourse avec une grande discrétion; elle ne doit
pas l'agiter violemment en façon d'appel aux pièces, surtout ne
jamais jeter un coup d'œil dans l'intérieur, et son remerciement doit
être également gracieux si elle a entrevu l'éclair d'un louis ou si elle
a perçu le son d'une pièce de dix centimes.
Si l'une des demoiselles d'honneur a une récolte d'argent plus
abondante que celle de sa compagne, il serait d'une grande
inconvenance de faire sonner (c'est le mot) ce petit triomphe
d'amour-propre devant celle qui a été moins favorisée.
Le rôle de garçon d'honneur est d'avoir l'œil à tout, de prévenir les
désirs des dames, de faire danser toutes les invitées, s'il y a un bal.
Lorsque la cérémonie religieuse est terminée, la mariée, au bras de
son beau-père et non à celui de son mari, passe à la sacristie; le
jeune marié offre le bras à sa belle-mère, le père de la jeune fille à
la mère du jeune homme.
Arrivé à la sacristie, après avoir signé sur le registre, le jeune couple
ayant ses parents réciproques de chaque côté, attend le défilé, les
félicitations et les baisers.
Le registre reste ouvert pour tous, mais on ne doit signer que si l'on
vous en prie, à moins que vous soyez un très grand personnage et
que votre signature ne soit un grand honneur.
Lorsque les derniers invités sont partis de la sacristie pour aller
reprendre leur place à l'église, la mariée, au bras de son mari cette
fois, et précédée du suisse, traverse l'église de nouveau aux sons de
l'orgue.

Le marié monte avec sa femme, sa mère et son père dans une
voiture, les deux femmes au fond, bien entendu.
Si le marié a une voiture, il part seul avec sa femme dans son coupé.
Mariage protestant.
On commence par aller à l'église, si l'un des deux conjoints est
catholique, on peut n'aller qu'au temple ou à l'église, mais le savoir-
vivre veut qu'on aille aux deux.
Les cérémonies sont les mêmes.
On n'exige en fait de pièces que le certificat du mariage civil.
Le prêtre catholique n'est jamais invité aux fêtes de mariage; le
pasteur peut l'être.
Mariage israélite.
Lorsque la mariée juive sort de sa maison, on a la très jolie coutume
de jeter des fleurs sur son passage.
Les hommes qui assistent à un mariage israélite gardent leur
chapeau sur la tête à la synagogue.
La mariée fait son entrée à la synagogue, soutenue et comme
traînée par ses deux témoins, qui lui tiennent les mains très élevées.
Elle monte les degrés du tabernacle et s'assied sous un vaste dais
avec son mari, les parents, les témoins, les garçons et les
demoiselles d'honneur.
Le rabbin, comme le prêtre, prononce un discours, reçoit le
consentement des époux et celui des parents, puis le marié passe
l'anneau au doigt de sa femme en disant qu'il la reconnaît pour sa
légitime épouse devant l'Éternel, devant la loi de Moïse et de l'État.

Le rabbin bénit l'union, fait boire aux époux le vin consacré dans une
même coupe qu'on jette ensuite par terre; lorsqu'elle se brise en
beaucoup de morceaux, c'est signe de prospérité pour le jeune
couple.
L'acte de mariage est lu à haute voix avant la signature.
Lorsque les Israélites appartiennent au rite portugais, la fiancée a
brodé une écharpe qu'on place sur les épaules du marié; la mariée
donne également au jeune marié le linceul dans lequel on
l'ensevelira.
Le mariage russe est très poétique, le marié est couronné de fleurs,
on lâche des colombes.

Autour d'un berceau.
N petit personnage est né, fille ou garçon, lequel, après les
soins d'usage, repose dans son berceau, tendu de rose pour
la future mère de famille et de bleu pour le général ou
l'avocat célèbre à peine éclos.
La nouvelle maman, gardée par sa mère ou par une parente, par sa
domestique ou par une garde, selon les positions de fortune ou de
convenances, ne doit recevoir, les neuf premiers jours, que des
visites de quelques minutes, où à peine entré, après avoir embrassé
l'accouchée, et s'être, suivant l'usage, extasié sur le bébé, il est de
bon goût de se retirer.
Lorsqu'on n'est pas de la famille ou de la stricte intimité de la jeune
femme, il est préférable d'aller demander des nouvelles et de
remettre sa carte sur laquelle on a tracé quelques lignes
affectueuses.
La déclaration de naissance doit être faite sous trois jours à la mairie
de l'arrondissement par le père de l'enfant, et deux témoins français
pouvant signer et étant domiciliés dans l'arrondissement où a eu lieu
la naissance.
Lorsque le père est empêché de se rendre à la mairie, il doit donner
une procuration; s'il était absent, la déclaration doit être faite par le
médecin ou toute autre personne ayant assisté à la naissance.

Faute de faire sa déclaration dans les délais voulus, on peut avoir
une peine correctionnelle variant de trois jours à six mois de prison
et une amende variant de six à trois cents francs.
Le nouveau-né peut être porté à la mairie où l'officier de l'état civil
constate son sexe, mais il est préférable d'attendre le médecin des
naissances, qui vient à domicile, dans les vingt-quatre heures qui
suivent la déclaration.
Une déclaration erronée rend passible des peines les plus graves.
Les prénoms doivent être indiqués dans l'ordre où l'on désire qu'ils
restent.
Autant que possible, on donne à l'enfant trois prénoms au plus, à
moins que, pour des raisons de famille, on ne lui en accorde
quelques-uns en surcroît; mais cette longue énumération n'est plus
guère usitée en France et semble réservée aux grands d'Espagne
qui, dans les siècles passés, entassaient leurs appellations sur des
monceaux de parchemin.
Les noms de fruits, de fleurs, les appellations grotesques sont
interdits.
On donne généralement à l'enfant le prénom de son parrain si c'est
un garçon, ou le prénom de sa marraine si c'est une fille; puis les
prénoms de ses père et mère, ou ceux choisis par ces derniers.
Souvent aussi le goût de la maman domine et le prénom sous lequel
le baby sera dénommé n'appartient à aucun membre de la famille;
en ce cas, les prénoms des parrains et des marraines viennent en
seconde ligne. Du reste, il est de bon goût, pour une marraine, de se
récuser avec grâce de donner son prénom, s'il ne doit pas plaire à la
maman.
Les prénoms bizarres, extraordinaires, sont généralement bannis par
les familles.

L'élégance, pour les jeunes mamans, consiste à avoir une toilette de
nuit très mousseuse, ornée de rubans bleus ou roses, selon, comme
je l'ai dit, que le chérubin est un monsieur ou une demoiselle.
L'oreiller sur lequel elle repose doit être orné de même; la robe de
chambre des relevailles, les rubans de la layette également; mais,
ceci n'est nullement obligatoire et rentre dans le domaine de dame
Fantaisie. Il est bon de dire que presque toutes les femmes aiment
assez ces menus usages qui ne sont pas bien coûteux et qui ornent
la vie.
Pour passer de la chambre à coucher au salon et y faire séjourner
l'enfant, on a d'exquis petits berceaux sans pieds, dénommés
«Moïse».
On doit envoyer des billets de faire part à toutes les personnes avec
lesquelles on est en relation.
La fantaisie est admise pour ces billets qui s'envoient quinze jours
après la naissance.
Pour les amis intimes, la parenté, on prévient, dès le lendemain, par
un mot écrit à la main.
Les billets se font sur de petites feuilles doubles ou sur des cartes
unies ou dorées en genre parchemin.
On peut les envoyer sous enveloppe non cachetée; les initiales du
baby au coin gauche.
Le papier peut être uni ou liseré de rose ou de bleu selon les cas;
lorsqu'on a des armoiries, on les met; quelquefois aussi le
monographe des parents.
On doit retourner une carte dans les deux jours qui suivent la
réception du faire part ou, si l'on veut, une lettre de félicitations;
cela dépend du degré d'intimité.
Une mode, nouvelle et bien gentille, est celle qui consiste à joindre à
la lettre de faire part une carte minuscule cornée, avec le prénom du

baby, c'est une politesse que le nouveau-né fait, d'ores et déjà, à
toutes les personnes qui peuvent s'intéresser à son arrivée en ce
monde.
Voici quelques modèles de billets de faire part.
Monsieur et Madame de B.... ont l'honneur (ou le plaisir)
de vous faire part de la naissance de leur fils Pierre.
Paris, le 25 novembre 18 . 22, rue de l'Arbre-Sec.
Monsieur et Madame R. D.... vous font, avec joie, part de
la naissance de leur fille Marguerite, qui est déjà sage et
jolie.
J'ai le plaisir de vous annoncer que j'ai fait mon entrée en
ce monde le 29 de ce mois de décembre et que ma petite
maman et moi nous nous portons bien.
Marie D....
Nous avons le plaisir de vous annoncer la naissance d'un
gros garçon, qui a reçu les noms de Lucien-Léon-Alfred et
qui se porte à merveille.
Monsieur et
Madame D....
Monsieur et Madame Louis D.... ont le plaisir de vous faire
part de la naissance de leur fils, qui a reçu les prénoms de
Raymond-Gontran.
Parrain: Monsieur Raymond D.....
Marraine: Madame D.....
J'ai la joie de vous annoncer mon heureuse arrivée en ce
monde; j'espère y être heureuse et gâtée. Ma petite mère
et moi nous nous portons à merveille et petit père est très
content.

Laure-Cécile D....
Autant que possible, quand on va faire ses compliments, ne pas
trouver de ressemblance entre le nouveau-né et tel ou tel
ascendant. Savons-nous si le père ou la mère ne trouvent pas ces
personnes-là affreuses?
A l'occasion d'une naissance, les parentes et les femmes de
l'entourage font un cadeau au baby; ce sont souvent des objets
confectionnés par elles-mêmes: bavoir élégant, une initiale
discrètement brodée dans un coin; brassières, petits chaussons de
laine rose ou bleue, bonnet mignon, voire même bracelet d'or avec
une médaille où sont gravés les prénoms de l'enfant.
Cette mode de bracelets est assez abandonnée depuis quelques
années.
Le père offre généralement un présent à la nouvelle maman; c'est
presque toujours un bijou, objet durable, qui perpétue le souvenir de
l'heureux événement.

Sur les fonts baptismaux.
ENGAGERAI toujours à faire célébrer le baptême six
semaines ou deux mois après la naissance et non tout de
suite, ainsi qu'on le faisait il y a quelque vingt ans.
La cérémonie ainsi reculée permet à la jeune maman d'y assister,
d'être avec le chérubin l'héroïne de la fête et, de cette manière,
l'inquiétude étant bannie, on peut être tout à la joie.
Les personnes pieuses qui craignent pour la vie future du baby
doivent le faire ondoyer; mais, je le répète, le baptême étant la fête
de famille par excellence, tout le monde doit y prendre part.
Les parrains et marraines doivent être désignés plusieurs mois à
l'avance. On choisit généralement, pour le premier-né, la grand'mère
maternelle comme marraine et le grand-père paternel comme
parrain; pour le second bébé, c'est l'inverse, grand'mère paternelle
et grand-père maternel.
Faute de ces très proches parents, on prend les frères et sœurs des
époux.
Un frère ou une sœur peuvent très bien être parrain ou marraine de
leur frère ou sœur.
É

L'Église exige l'âge de sept ans pour pouvoir être parrain ou
marraine, mais, par faveur spéciale, elle admet quelquefois des
enfants plus jeunes, à condition qu'ils aient des répondants.
On nomme, d'ancien temps, «compère et commère» le parrain et la
marraine.
Il importe, pour s'assurer un parrainage, de s'y prendre longtemps à
l'avance.
Lorsque le compère et la commère ne se connaissent pas, il est bon
de les présenter l'un à l'autre avant la cérémonie.
Pour le baptême, les prénoms donnés à l'enfant doivent être inscrits
dans le même ordre qu'à la mairie.
Si la marraine ne connaît ni le parrain ni sa famille, un tiers est
nécessaire, lorsque, le jour du baptême, celui-ci va chercher sa
commère en voiture ou à pied, pour l'accompagner au domicile de
l'enfant.
Quelques jours avant la cérémonie, la marraine envoie la robe, le
bonnet, la pelisse, le chapeau que portera l'enfant. Elle peut
n'envoyer que deux de ces objets, chapeau et pelisse, ou robe et
bonnet.
L'élégance diffère suivant les moyens.
Pour mon fils, voilà ce que sa très aimable marraine a donné: robe
de mousseline à tablier d'entre-deux de valenciennes, sur une robe
de soie bleu pâle faisant transparent; petit bonnet tout en entre-
deux de valenciennes et de plumetis sur nansouk avec grosse ruche
de valenciennes et d'étroites coques de ruban comète en satin
crème; pelisse en cachemire crème, brodée au passé d'une
guirlande de fleurs; effilé de soie crème tout autour, doublure en
satin crème, piquée; capote de satin crème avec la même broderie
qu'à la pelisse, garniture de plumes crème.

Les robes de piqué, les pelisses à carreaux, les capelines en laine
peuvent également s'offrir.
Le parrain, suivant ses ressources, offre à son ou à sa filleule tous
les petits ustensiles à son usage: poêlon à bouillie, petite tasse,
coquetier, petite cuillère, petite assiette, hochet, timbale, rond de
serviette en argent, en vermeil, même en or, ou un seul de ces
objets, ou même un simple hochet en ivoire, en os.
Une robe, une pelisse, une capeline confortable, en couleur,
quelques menus objets utiles font le plus grand plaisir aux parents.
Dans la semaine qui précède le baptême, le parrain doit envoyer à la
marraine les boîtes de dragées et un bibelot quelconque.
Il y a vingt ans, le présent était classique: c'était invariablement une
boîte à gants contenant six ou douze paires de gants. Il fallait donc
demander la pointure de la dame, les nuances qu'elle préférait, etc.,
etc. Maintenant la mode a renversé cet usage, et on peut offrir
indifféremment un bronze, une jardinière avec des fleurs, un éventail
et même, si le degré d'intimité est grand, un bijou.
Les père et mère de l'enfant doivent, de leur côté, commander des
boîtes de baptême; l'usage veut que le parrain et la marraine leur en
offrent chacun une.
Les boîtes de baptême sont en papier rose ou bleu; on en fait
aujourd'hui d'adorablement jolies: boîte avec le prénom de l'enfant
et la date de sa naissance estampés en relief or ou argent, ou les
deux mélangés; avec les initiales entrelacées en givré or ou argent
avec le nom en diamanté; avec aquarelle représentant un amour
peignant le nom du nouveau-né sur une boîte de baptême; avec un
cortège XVIe
siècle, violoneux en tête, parrain et marraine, qui sont
un marquis et une marquise falbalatés, jetant les dragées à un
peuple de marmots qui se bousculent; ou bien des anges posant
dans un berceau un petit enfant; une cloche, laissant tomber le
baby, si le baptême se trouve au temps pascal.

On peut aussi offrir en place de boîte un sac de moire ou de satin à
la marraine et à la jeune maman; en tous cas, elles doivent recevoir
toutes les deux un bouquet.
Le parrain a la charge des cadeaux à la marraine, à la garde, aux
domestiques, à l'enfant de chœur, au curé.
La pièce de cinq, dix ou vingt francs qu'on offre au prêtre doit être
placée dans une boîte de dragées, de même pour les autres
personnes, sauf pour l'enfant de chœur auquel on donne un ou deux
francs de la main à la main.
Pour la garde et la nourrice, on peut varier entre cinq et vingt francs.
Pour les domestiques, c'est cinq francs, généralement.
Le jour du baptême, le parrain va prendre la marraine chez elle en
voiture ou à pied et l'amène chez les parents de l'enfant.
Si c'est en voiture, la maman et la nourrice portant l'enfant
monteront dans cette voiture pour aller à l'église; elles occuperont
toutes les deux les places du fond; le parrain et la marraine sur le
devant.
On doit s'entendre à l'avance avec le curé pour le jour et l'heure de
la cérémonie. Pour l'entrée à l'église, c'est la personne qui porte
l'enfant qui ouvre la marche.
Le parrain est placé à droite de la personne qui tient l'enfant, la
marraine à gauche.
Le Pater et le Credo qui sont demandés doivent être récités en
français; le cierge est tenu ensemble, de la main droite, par le
parrain et la marraine.
Il ne faut jamais répondre: «oui, monsieur», mais, oui, tout
simplement.
Le parrain et la marraine mettent leurs mains droites dégantées sur
la tête de l'enfant en même temps que le prêtre.

Après la cérémonie du baptême, on se rend à la sacristie pour signer
l'acte.
A la sortie, mais moins fréquemment qu'autrefois, le parrain et la
marraine jettent des dragées et quelques pièces de monnaie aux
gamins assemblés.
Au dîner de baptême, le parrain et la marraine occupent les places
du maître et de la maîtresse de la maison.
Le dîner doit être servi avec cérémonie; des dragées doivent figurer
au dessert.
Jamais la nourrice ne doit y assister; à la fin du repas, ou plutôt au
commencement, il arrive que l'on fait circuler de main en main le
héros de la fête lequel, généralement, désapprouve fort cette façon
d'aller et le témoigne par des cris perçants.
Un mois après le baptême, si la marraine est mariée, son mari doit
inviter à dîner le parrain et les parents de l'enfant.
En cas de nécessité on peut demander un prêtre pour que le
baptême ait lieu à domicile.
Les usages sont les mêmes pour le baptême protestant, sauf en ce
qui touche la cérémonie du baptême qui est, comme tout le
cérémonial de ce culte, réduite à sa plus simple expression.
Comme chez les chrétiens, l'enfant israélite a un parrain et une
marraine.
Le premier jour de sabbat (c'est-à-dire le samedi), le père qui a eu
un garçon doit porter une offrande à la synagogue.
Dans les deux cas, il y a réunion de parents et d'amis à la maison et
le rabbin ou, à son défaut, le père, appelle solennellement les
bénédictions du dieu d'Abraham et de Jacob, sur le nouveau-né.

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Advances And Applications In Sliding Mode Control Systems 1st Edition Ahmad Taher Azar

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Advances And Applications In Sliding Mode Control Systems 1st Edition Ahmad Taher Azar