Table Of ContentYu Liu · Fengjiao Liu · Yanfang Mei ·
Xiangqian Yao · Wei Zhao
Dynamic
Modeling and
Boundary Control
of Flexible Axially
Moving System
Dynamic Modeling and Boundary Control
of Flexible Axially Moving System
· · ·
Yu Liu Fengjiao Liu Yanfang Mei
·
Xiangqian Yao Wei Zhao
Dynamic Modeling
and Boundary Control
of Flexible Axially Moving
System
YuLiu FengjiaoLiu
SchoolofAutomationScience SchoolofAutomationScience
andEngineering andEngineering
SouthChinaUniversityofTechnology SouthChinaUniversityofTechnology
Guangzhou,Guangdong,China Guangzhou,Guangdong,China
YanfangMei XiangqianYao
SchoolofElectronicsandInformation SchoolofAutomationScience
GuangdongPolytechnicNormalUniversity andEngineering
Guangzhou,Guangdong,China SouthChinaUniversityofTechnology
Guangzhou,Guangdong,China
WeiZhao
SchoolofAutomationScience
andEngineering
SouthChinaUniversityofTechnology
Guangzhou,Guangdong,China
ISBN 978-981-19-6940-9 ISBN 978-981-19-6941-6 (eBook)
https://doi.org/10.1007/978-981-19-6941-6
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Contents
1 Introduction .................................................. 1
1.1 ControloftheAxiallyMovingStructures .................... 1
1.2 OutlineoftheBook ....................................... 5
References .................................................... 6
2 MathematicalPreliminaries .................................... 7
2.1 TheHamiltonPrinciple ................................... 7
2.2 FunctionalandVariation .................................. 8
2.2.1 FunctionalVariatio(cid:2)nRu(cid:3)les .(cid:4)........................ 8
2.2.2 TheExpansionof t2δ 1w2 dt .................... 8
t1 2 t
2.2.3 DefinitionofVariation ............................. 9
2.3 DiscreteSimulationMethod ............................... 9
2.3.1 Discretization of the Vibration Displacement
w(x,t) .......................................... 10
2.3.2 SeveralDiscretizationMethods ..................... 10
2.3.3 DiscretizationofBoundaryConditions ............... 11
2.4 S-CurveAcceleration/Deceleration(Sc-A/D)Method .......... 11
2.5 Preliminaries ............................................ 12
References .................................................... 13
3 PDEModelingfortheAxiallyMovingStructures ................ 15
3.1 PDEModeling ........................................... 15
3.2 SimulationExample ...................................... 21
Appendix1: SimulationProgram ............................... 23
References .................................................... 25
4 BoundaryControlofanAxiallyMovingSystemwithHigh
Acceleration/Deceleration ...................................... 27
4.1 Introduction ............................................. 27
4.2 PDEDynamicModel ..................................... 28
4.3 BoundaryControllerDesign ............................... 28
4.3.1 BoundaryControl ................................. 28
v
vi Contents
4.3.2 StabilityAnalysis ................................. 35
4.4 SimulationExample ...................................... 36
Appendix1: SimulationProgram ............................... 39
References .................................................... 43
5 Robust Boundary Control of an Axially Moving System
with High Acceleration/deceleration and Disturbance
Observer ..................................................... 45
5.1 Introduction ............................................. 45
5.2 PDEDynamicModel ..................................... 45
5.3 BoundaryControllerDesign ............................... 47
5.4 StabilityAnalysis ........................................ 48
5.5 SimulationExample ...................................... 55
Appendix1: SimulationProgram ............................... 60
References .................................................... 65
6 AdaptiveBoundaryControlofanAxiallyMovingSystem
withHighAcceleration/Deceleration ............................ 67
6.1 Introduction ............................................. 67
6.2 PDEDynamicModel ..................................... 67
6.3 BoundaryControllerDesign ............................... 69
6.4 StabilityAnalysis ........................................ 69
6.5 SimulationExample ...................................... 75
Appendix1: SimulationProgram ............................... 79
References .................................................... 84
7 AdaptiveBoundaryControlofanAxiallyMovingSystem
with High Acceleration/Deceleration and Disturbance
Observer ..................................................... 85
7.1 Introduction ............................................. 85
7.2 PDEDynamicModel ..................................... 85
7.3 BoundaryControllerDesign ............................... 87
7.4 StabilityAnalysis ........................................ 88
7.5 SimulationExample ...................................... 95
Appendix1: SimulationProgram ............................... 99
References .................................................... 104
8 Boundary Control of an Axially Moving
Accelerated/DeceleratedBeltSystem ............................ 105
8.1 Introduction ............................................. 105
8.2 PDEDynamicModel ..................................... 106
8.3 BoundaryControllerDesign ............................... 108
8.3.1 StepOne ........................................ 108
8.3.2 StepTwo ........................................ 113
8.4 StabilityAnalysis ........................................ 116
8.5 SimulationExample ...................................... 119
Appendix1: SimulationProgram ............................... 123
References .................................................... 127
Contents vii
9 StabilizationofanAxiallyMovingAccelerated/Decelerated
SystemviaanAdaptiveBoundaryControl ....................... 129
9.1 Introduction ............................................. 129
9.2 PDEDynamicModel ..................................... 130
9.3 BoundaryControllerDesign ............................... 131
9.3.1 StepOne ........................................ 132
9.3.2 StepTwo ........................................ 136
9.4 StabilityAnalysis ........................................ 139
9.5 SimulationExample ...................................... 142
Appendix1: SimulationProgram ............................... 146
References .................................................... 150
10 AdaptiveOutputFeedbackBoundaryControlforaClass
ofAxiallyMovingSystem ...................................... 151
10.1 Introduction ............................................. 151
10.2 PDEDynamicModel ..................................... 152
10.3 BoundaryControllerDesign ............................... 152
10.3.1 StepOne ........................................ 153
10.3.2 StepTwo ........................................ 156
10.4 StabilityAnalysis ........................................ 159
10.5 SimulationExample ...................................... 163
Appendix1: SimulationProgram ............................... 167
References .................................................... 171
11 Vibration Control and Boundary Tension Constraint
ofanAxiallyMovingStringSystem ............................. 173
11.1 Introduction ............................................. 173
11.2 PDEDynamicModel ..................................... 174
11.3 BoundaryControllerDesign ............................... 176
11.4 StabilityAnalysis ........................................ 177
11.5 SimulationExample ...................................... 183
Appendix1: SimulationProgram ............................... 188
References .................................................... 193
12 BoundaryControlforanAxiallyMovingSystemwithInput
RestrictionBasedonDisturbanceObservers ..................... 195
12.1 Introduction ............................................. 195
12.2 PDEDynamicModel ..................................... 196
12.3 BoundaryControllerDesign ............................... 196
12.3.1 DisturbanceObserverDesign ....................... 196
12.3.2 BacksteppingDesign .............................. 198
12.4 StabilityAnalysis ........................................ 202
12.5 SimulationExample ...................................... 205
Appendix1: SimulationProgram ............................... 210
References .................................................... 215
viii Contents
13 Adaptive Neural Network Vibration Control
for an Output-Tension-Constrained Axially Moving
BeltSystemwithInputNonlinearity ............................ 217
13.1 Introduction ............................................. 217
13.2 PDEDynamicModel ..................................... 218
13.3 BoundaryControllerDesign ............................... 218
13.3.1 StepOne ........................................ 219
13.3.2 StepTwo ........................................ 222
13.4 StabilityAnalysis ........................................ 225
13.5 SimulationExample ...................................... 228
Appendix1: SimulationProgram ............................... 234
References .................................................... 243
14 Conclusions ................................................... 245
Chapter 1
Introduction
1.1 ControloftheAxiallyMovingStructures
Precisionelectronicmanufacturingequipmentspansmanydisciplinessuchaselec-
tronics,machinery,automation,optics,computers,andinvolvescoretechnologies,
such as precision visual inspection, precision machining, high-speed and high-
precision control, and computer integrated manufacturing. It plays a vital role in
thesecurityofnationaleconomy,finance,andnationaldefenseinformationsystems,
and is also the core support of mainstream high-tech industries such as solar cells
andoptoelectronicdevices.Asanimportantpartofhigh-speedandhigh-precision
electronicmanufacturingequipment,thevibrationphenomenoncausedbyitselastic
deformation will directly affect the production performance of the equipment and
theproductionqualityofproductsintermsoftimeandspacescale[1,2].Therefore,
howtoquicklyeliminatethevibrationphenomenonoftheaxiallymovingstructure
inprecisionelectronicmanufacturingequipmenthasbecomeabottleneckproblem
thatrestrictsprecisionelectronicmanufacturing,especiallythepackagingprocessof
integratedcircuits(IC).
The axially moving structure, as an essentially typical distributed parameter
system (DPS) with strong coupling, nonlinearity, and infinite dimensions, can be
modeled by a nonlinear partial differential equation (PDE) and a set of ordinary
differential equations (ODEs). This hybrid dynamic model makes it difficult to
developaneffectivecontrolstrategyduetoitsinfinite-dimensionality.Theconven-
tional control methods for PDE system are based on truncated finite-dimensional
modes by using the finite-element method, Galerkin’s method, or assumed modes
method[3–5].However,threemainshortcomingsexistinthecontroldesignofthis
truncatedsystem[6–8].Oneisthatthecontrolorderneedstobeincreasedandthe
numberofflexiblemodesneedstobeconsideredtoachievehigh-precisioncontrol
performance.Theotheristhatthedistributedcontrolforceisrequiredtoovercome
the difficulty of calculating the infinite-dimensional gain matrix. These two draw-
backs make the control difficult to achieve from an engineering perspective. The
third disadvantage is that the control design is restricted to a few key modes and
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https://doi.org/10.1007/978-981-19-6941-6_1
2 1 Introduction
ignoresthehighfrequencymodes,whichleadstothecontrolspilloverinstabilityof
thesystem.
To overcome the aforementioned disadvantages, the boundary control design,
which is to be implemented based on original infinite-dimensional model of
the systems, has been developed for vibration suppression of flexible systems.
Comparing with distributed control, boundary control is considered to be more
economicalandpracticalinvibrationcontrolofflexiblestructuresbecauseitneeds
fewer sensors and actuators. And the kinetic energy, the potential energy, and the
work done by the nonconservative forces used for modeling can be directly used
to construct a Lyapunov function for system stability analysis. In recent years,
the boundary control integrated with other intelligent control methodologies, such
as sliding-mode control, neural network control, robust adaptive control, iterative
learning control, and fuzzy control, has obtained many achievements. But among
theseresults,mostoftheresearchobjectsareaboutflexiblemanipulators.
For axially moving systems, scholars have also made many attempts. In [9], a
robust adaptive boundary control for a class of flexible axially moving string-type
systemsunderunknowntime-varyingdisturbanceisdevelopedtoguaranteeallthe
signals in the closed-loop system are uniformly ultimately bounded. In [10], an
adaptiveisolationschemeforanaxiallymovingsystemisintroducedtocontrolthe
transversevibrationofthecontrolledspantozeroasymptotically.In[11],byusing
Lyapunov’smethodandenergyapproach,anenergy-basedrobustadaptiveboundary
controlisdevelopedforanaxiallymovingbeamtosuppressthevibration.In[12],
robust and adaptive boundary control is developed to stabilize the vibration of a
stretchedstringonamovingtransporter.Simultaneouscontroloftwo-dimensional
vibration, namely longitudinal and transverse, for vibration suppression of axially
movingstringisstudiedin[1].In[13],aniterativelearningboundarycontroltech-
niqueisappliedtoanaxiallymovingsystemtodampoutanystringoscillationduring
transportation.
However, most of the research results above assume that the motion of the
axially moving structure is a uniform motion, which obviously means only the
simplest motion mode is considered. In practice, for high-precision electronics
manufacturing equipments, the axially moving structures work with not only a
variable speed motion, but usually also a high-acceleration/deceleration and high-
speedmovementforimprovingtheefficiencyunderboththedistributeddisturbance
and boundary disturbance in almost all cases. Therefore, the structure has strong
geometricnonlinearcharacteristics,andthecurrentdynamicmodelsandactivevibra-
tioncontrolmethodsforaxiallymovingstructuresathomeandabroadaredifficultto
applytothevibrationcontroloflargeaccelerationanddecelerationaxiallymoving
structures[14].
ThebacksteppingmethodisarecursivealgorithmbasedonLyapunovstability.
ThefeedbackcontrollerisobtainedbyrecursivelyconstructingtheLyapunovfunc-
tion of the closed-loop system, which makes the derivative of the Lyapunov func-
tion along the trajectory of the closed-loop system have some performance, and
