Active Vibration Control and Stability Analysis of Flexible Beam Systems

202 Pages·2019·8.092 MB·English

by  Wei He, Jinkun Liu

Wei He · Jinkun Liu
Active Vibration
Control and
Stability Analysis
of Flexible Beam
Systems
Active Vibration Control and Stability Analysis
of Flexible Beam Systems
Wei He Jinkun Liu
(cid:129)
Active Vibration Control
and Stability Analysis
of Flexible Beam Systems
123
Wei He Jinkun Liu
Schoolof Automation andElectrical Schoolof Automation Science
Engineering andElectrical Engineering
University of Science andTechnology Beihang University
Beijing Beijing,China
Beijing,China
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Preface
Control of flexible beam systems constitutes an important benchmark problem in
manyapplicationareas,suchasflexiblemanipulatorsforgrasping,marinemooring
lines for station keeping, marine risers for oil transportation, and crane cables for
positioning the payload. Flexible systems have a number of advantages such as
lightweight,efficiency,higheroperationspeed,andlowercost.Thedynamicsofthe
flexiblebeamsystemisessentiallyadistributedparametersystem(DPS).Different
from lumped parameter systems, the DPS has an infinite-dimensional state space.
The dynamics of DPS modeled by the coupled partial differential equations
(PDEs)–ordinary differential equations (ODEs) is difficult to control due to the
infinite dimensionality of the beam system.
Since the flexible beam system becomes lighter and more flexible, the external
disturbances will lead to the mechanical vibrations of flexible beam systems.
Therefore, the control strategy should be designed to suppress the vibrations of
flexiblebeamsystems.Drivenbypracticalneedsandtheoreticalchallenges,flexible
beam systems and their vibration suppression have received great attention.
Boundary control is an effective control strategy for vibration suppression of the
flexiblebeamsystemwhichisdescribed byhybridPDEs–ODEs.Byusingsensors
andactuatorsattheboundary,thedynamicmodelofthesystemisnotaffected,and
boundarycontrolcanbederivedfromaLyapunovfunctionwhichisrelevanttothe
mechanical energy based on the dynamics of the system.
The purpose of this book is to investigate the fundamental issues including
dynamic analysis and control design for flexible beam systems by theoretical
analysisandnumericalsimulations.Acomprehensivestudyisprovidedtodevelop
boundary control methods for the vibration suppression of flexible beam systems
with input nonlinearities and output constraint. In addition, the book presents
theoretical explorations for advanced control methods of flexible beam systems,
includingdistributedcontrol,iterativelearningcontrol,andneuralnetworkcontrol.
The control designs are coupled with numerical simulations to illustrate the
effectiveness.
The book starts with a brief introduction of modeling methods and control
techniques for a class offlexible beam systems in Chap. 1.
v
vi Preface
Chapter 2 presents the preliminaries and several lemmas for the subsequent
development to simplify the dynamical modeling and further stability analysis for
the beam structures, and the dynamic model is also introduced.
In Chap. 3, we investigate the boundary control design problem for an Euler–
Bernoulli beam model under both the unknown spatiotemporally varying dis-
tributed disturbance and time-varying boundary disturbance. The adaptive control
laws and disturbance observers are proposed to deal with system parametric
uncertainties and external disturbances, respectively.
In Chap. 4, the control problem of an Euler–Bernoulli beam with boundary
output constraint is addressed. We design a boundary barrier control scheme and
apply the proposed Lyapunov function to the original partial differential equations
inordertoavoidthespilloverproblem.TheformofLyapunovfunctioncombining
boththeintegralLyapunovfunctionandthebarrierLyapunovfunctionisemployed
for the control design and stability analysis of the system.
In Chap. 5, the largest challenge is how to eliminate the nonlinear input satu-
ration characteristic and to design an effective active control law for the flexible
beam system. We propose the hyperbolic tangent function in the control design
of the flexible system to reduce the influence induced by the input saturation.
In Chap. 6, the boundary control architecture is proposed for continuous-time
PDEsystemsprecededbyinputdead-zonenonlinearity.Byusinganewdescription
of the dead-zone, boundary control schemes are developed to regulate the defor-
mation of Euler–Bernoulli beam system even in the presence of the external dis-
turbance.Itisprovedthattheproposedcontrollawcanensureuniformlyultimately
boundness of the entire system and achieve stabilization.
InChap.7,wediscussaproblemofaflexiblebeamsystemwithinputbacklash
characteristic. By transforming the nonlinear input backlash to linear input, the
boundary control law and disturbance observer are designed at the top offlexible
beamviaconstructingaLyapunovcandidatefunctiontoreducethevibrationofthe
system. The transverse displacement of the closed-loop system is proved to con-
verge to a small neighborhood of zero.
In Chap. 8, we present the control design for the flexible Euler–Bernoulli beam
system with global constraint and uncertain tip payload. To prevent the constraint
violation,theintegralbarrierLyapunovfunctionisemployedforthecontroldesign
andstabilityanalysis.Weproposeadistributedcontrolsothatthedeflectionofthe
mechanical system can track the desired signal. Exponential stability is well
achieved without violation of the constraint.
InChap.9,weproposeanadaptiveboundaryiterativelearningcontrol(ABILC)
scheme for an Euler–Bernoulli beam system with an aperiodic distributed distur-
bance, an aperiodic boundary disturbance, and an unknown system parameter. In
order to tackle the input saturation, a hyperbolic tangent function and a saturation
function are utilized. Three adaptive laws are proposed and learned along the
iteration axis. Based on a time-weighted Lyapunov–Krasovskii-like composite
energyfunction,therestrainedABILClawisdesigned.Fortheclosed-loopsystem,
the boundedness of all the signals in each iteration is guaranteed. Along the
Preface vii
iteration axis, the displacements converge to zero in the presence of the external
disturbances.
In Chap. 10, in order to analyze the flexible beam structure, the assumed mode
method (AMM) is employed to develop the dynamic model. Based on the
N-dimensional discrete dynamic model, neural network control is investigated to
track the desired trajectory accurately and to suppress the flexible vibration maxi-
mally.Toensurestabilityrigorouslyasthegoal,thesystemisprovedtobeuniform
ultimate boundedness (UUB) by Lyapunov stability method. Eventually, simula-
tions verify that the proposed control strategy is effective.
In Chap. 11, a model of a coupled three-dimensional flexible beam with a tip
payload is derived by using Hamilton principle and described by a set of partial
differentialequationsandordinarydifferentialequations.Boundarycontrollawsare
proposed to suppress the vibration of the beam in both longitudinal and transverse
directions with the environmental disturbance, and the uniform boundedness and
theuniformlyultimateboundednessoftheclosed-loopsystemareproved.Adaptive
control laws are proposed to overcome the uncertainty of the tip payload, and the
uniform boundedness and the uniformly ultimate boundedness of the system are
proved.
In Chap. 12, the conclusions are summarized and the future works are given.
In summary, this book covers the dynamical analysis and control design for
flexiblebeamsystems.Thebookisprimarilyintendedforresearchersandengineers
in the control system. It can also serve as a complementary reading on modeling
and control offlexible beam systems at the postgraduate level.
Beijing, China Wei He
Jinkun Liu
Acknowledgements
For this book, we have had the great fortune of working with brilliant people who
are generous with their time and friendship, through many discussions filled with
creativityandinspiration.Firstofall,wewouldliketoexpressourgratitudetoour
co-workers who have contributed to the collaborative studies of this book.
We would also like to express our sincere appreciation to our colleagues who
havecontributedtothecollaborativeresearch.Inparticular,wewouldliketothank
Shuzhi Sam Ge, from the National University of Singapore, Singapore; Miroslav
Krstic, from the University of California, San Diego, US; Khac Duc Do, from the
Curtin University, Australia; Keum-Shik Hong, from the Pusan National
University, Korea; Bao-Zhu Guo, from the Academy of Mathematics and Systems
Science, Chinese Academy of Sciences, China; Huai-Ning Wu, from the Beihang
University,China;Jun-MinWang,fromtheBeijingInstituteofTechnology,China,
and their research groups for their excellent research works, and helpful advice on
ourresearch.SpecialthanksgotoDongliangShengforhisassistanceandeffortson
the process of publishing this book.
Appreciations must be made to Xiuyu He, Yuhua Song, Tingting Meng, Kai
Huang, Hejia Gao, Zhe Jing, Weijie Xiang, Hui Qin, Hoang Minh Vu, Yu Liu,
Zhijia Zhao, Zhijie Liu, Yuncheng Ouyang, Linghuan Kong, Xinling Yue, Jiali
Feng, and Qingyu Zhou for the constructive discussions and sharing of ideas.
ThisworkissupportedbytheNationalKeyResearchandDevelopmentProgram
ofChina under Grant 2017YFB1300102, theNationalNaturalScience Foundation
of China under Grant 61522302, 61873298, 61761130080, the Newton Advanced
Fellowship from The Royal Society, UK, under Grant NA160436, the Beijing
Natural Science Foundation under Grant 4172041, and the Fundamental Research
FundsfortheChinaCentral UniversitiesofUSTBunder GrantFRF-BD-17-002A.
Haidian District, Beijing, China Wei He
Jinkun Liu
ix
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Application of Flexible Beam Systems. . . . . . . . . . . . . . . . . . . 3
1.3 Conventional Modelling Methods . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Model Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 A Brief Survey of Control Approaches . . . . . . . . . . . . . . . . . . 11
1.4.1 Implementation Location . . . . . . . . . . . . . . . . . . . . . . 11
1.4.2 Control Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 The Hamilton Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Lyapunov Stability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Useful Lemmas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Modeling of a Flexible Beam System . . . . . . . . . . . . . . . . . . . 28
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Vibration Control of a Flexible Beam. . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Control Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.1 Robust Boundary Control with a Disturbance
Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.2 Adaptive Boundary Control with the System
Parametric Uncertainties . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
xi
xii Contents
3.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 Vibration Control of a Flexible Beam with Output Constraint. . . . 59
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Control Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.1 Model-Based Boundary Control . . . . . . . . . . . . . . . . . 61
4.2.2 Adaptive Boundary Control . . . . . . . . . . . . . . . . . . . . 65
4.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5 Vibration Control of a Flexible Beam with Input Saturation . . . . . 75
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3 Control Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6 Vibration Control of a Flexible Beam with Input Dead-Zone . . . . . 85
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.3 Control Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7 Vibration Control of a Flexible Beam with Input Backlash . . . . . . 97
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.2 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.3 Control Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8 Distributed Control of a Flexible Beam. . . . . . . . . . . . . . . . . . . . . . 113
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.2 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.3 Control Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
8.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124