Research Article Active Control of Contact Force for a Pantograph-Catenary...

Research ArticleActive Control of Contact Force fora Pantograph-Catenary System

Jiqiang Wang

Jiangsu Province Key Laboratory of Aerospace Power Systems College of Energy and Power EngineeringNanjing University of Aeronautics and Astronautics Nanjing 210016 China

Correspondence should be addressed to Jiqiang Wang jiqiang wanghotmailcom

Received 18 November 2015 Accepted 20 December 2015

Academic Editor Mickael Lallart

Copyright copy 2016 Jiqiang WangThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The performance of the high speed trains depends critically on the quality of the contact in the pantograph-catenary interactionMaintaining a constant contact force needs taking specialmeasures and one of themethods is to utilize active control to optimize thecontact force A number of active control methods have been proposed in the past decade However the primary objective of thesemethods has been to reduce the variation of the contact force in the pantograph-catenary system ignoring the effects of locomotivevibrations on pantograph-catenary dynamics Motivated by the problems in active control of vibration in large scale structuresthe author has developed a geometric framework specifically targeting the remote vibration suppression problem based only onlocal control action It is the intention of the paper to demonstrate its potential in the active control of the pantograph-catenaryinteraction aiming to minimize the variation of the contact force while simultaneously suppressing the vibration disturbance fromthe train A numerical study is provided through the application to a simplified pantograph-catenary model

1 Introduction

The quality of the contact between pantograph and catenaryhas played an essential role in delivering electrical energyto high speed trains However the contact force variesstrongly with speed and over spans causing excessive wearsor even lost contact Measures have to be taken to main-tain a relatively constant contact force in the pantograph-catenary interaction A traditional pantograph system ispassive consisting of viscous-elastic components with fixedparameters designed for fixed type of catenaries and loco-motive velocities This obviously prevents its utilization inmodern high speed trains such as ICE TGV or CRH Onemethod to achieve the uniformity of the contact force isto make use of advanced materials for pantograph slideplate (see [1] for a review on recent development) whileanother is to take advantage of the development in controltheorymdashthis is the concept of active control of pantograph-catenary system in order to maintain an optimal contactforce Indeed this has attracted much attention in the studyof pantograph-catenary interaction For example simple statefeedback control strategy is considered in [2 3] in [4]

an LQR is designed using a linear pantograph-catenarymodel while in [5] a high order sliding mode variable struc-ture controller is constructed for the active control of panto-graph an evolutionarymultiobjective optimization approachis utilized taking into account the perturbations caused by thetime-varying stiffness of catenary [6] differential-geometrictheory is used for output-perturbation decoupling for anonlinear pantograph-catenary model in [7] very recentlyfuzzy logic has also found application in the active controlstrategies such as in [8 9]

In the above papers the primary objective of activecontrol has been to reduce the variation of the contactforce in the pantograph-catenary system while it is alsowell-known that the locomotive vibrations on pantograph-catenary dynamics are nonnegligible and the effect becomesparticularly violent with the increasing speed of the train [1011] As a consequence it is essential to attenuate the vibrationdisturbance while minimizing the variation of the contactforce In principle this can be achieved through decouplingcontrol with each controller dealing with contact force andvibration disturbance separately but implementation costincreases with system complexity In this paper a geometric

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 2735297 7 pageshttpdxdoiorg10115520162735297

2 Shock and Vibration

design framework is introduced aiming tominimize the vari-ation of the contact force while simultaneously suppressingthe vibration disturbance from the train The paper is struc-tured as follows Section 2 describes the pantograph-catenarymodel Section 3 briefly reviews the design procedures andpresents the simulation results finally Section 4 concludes thepaper and comments the further development

2 Simplified Pantograph-Catenary Model

21 Pantograph Model A DSA250 pantograph is shown inFigure 1(a) which is used in CRH1 CRH2 and CRH5EMUs with a designed speed of 250 kmh Some complexpantograph models exist but to be elementary a pantographcan be modelled as a dual-mass system with known massesrepresenting the head and frame along with appropriatesprings and dampers A configuration is illustrated in Fig-ure 1(b)

In Figure 1 1198981and 119898

2are the equivalent mass of

the collector head and pantograph frame with 1198851and 119885

2

denoting their displacement relative to the equilibrium 1198881

and 119896119896are the parameters of the spring-damper structure

connecting the collector head and pantograph frame 1198882is

the viscosity coefficient of the pantograph frame 1198650 119891119896

and 119891119888are static lifting force elastic force and damping

force respectively 119885119903represents the vibration disturbance

transmitting from the locomotive and finally 119906 is the requiredforce for active control

22 Catenary Model The suspension catenary in a singlespan is shown in Figure 2

Usually the rigidity of the suspension catenary 119896(119905) is atime-varying function affected by the speed of the locomo-tive type of the catenaries and its corresponding parametersand so forth It can be obtained One of the methods toobtain an expression of 119896(119905) is through the application of finiteelement method to practical catenaries leading to

119896 (119905) = 1198960(1 + 119886

1cos 2120587

119871

119905 + 1198862cos 2120587

1198711

119905

+ 1198863(cos2120587

119871

119905)

2

+ 1198864(cos 120587

119871

119905)

2

+ 1198865(cos 120587

1198711

119905)

2

+ sdot sdot sdot)

(1)

and the parameters are then estimated by a nonlinear leastsquares approximation

23 Pantograph-Catenary Interaction Model For panto-graph-catenary interaction the catenary can be modeled asa spring with time varying rigidity 119896(119905) With the expressionfor 119896(119905) the contact force 119865 shown in Figure 1 is

119865 = 119896 (119905) 1198851 (2)

Hence combining the pantograph model (Figure 1(b)) withthe contact force from suspension catenary a simplified

pantograph-catenary interaction model can be obtained asshown in Figure 3

Then it is easy to show the above model can be describedby the following equations

11989811198851+ 119896 (119905) 119885

1+ 119891119896+ 119891119888= 119906

11989811198852+ 1198882(1198852minus

119885119903) minus 119891119896minus 119891119888= 1198650minus 119906

119891119896= 119896119896(1198851minus 1198852)

119891119888= 1198881(1198851minus

1198852)

(3)

The objective of active control can then be stated as followsfinding a control action 119906 based only on the displacement119885

1

such that the variation of both the contact force 119865 includingthe effects of the disturbance transmission force

119903is reduced

simultaneouslyRewriting (3) in state space representation is as follows

(119905) = 119860119909 (119905) + 119861120577 (119905)

119910 (119905) = 119862119909 (119905)

(4)

where 119909119879 = [1198851

11988511198852

1198852]119879 120577119879 = [119906 119889]

119879 and 119910119879 =

[11988511198852]119879 with 119889 equiv 119865

01198882+

119885119903 the corresponding matrices

are

119860 =

[

[

[

[

[

[

[

[

[

0 1 0 0

minus

119896119896+ 1198960

1198981

minus

1198881

1198981

119896119896

1198981

1198881

1198981

0 0 0 1

119896119896

1198982

1198881

1198982

minus

119896119896

1198982

minus

1198881+ 1198882

1198982

]

]

]

]

]

]

]

]

]

119861 =

[

[

[

[

[

[

[

[

[

0 0

1

1198981

0

0 0

minus

1

1198982

1198882

1198982

]

]

]

]

]

]

]

]

]

119862 = [

1

0

0

0

0

1

0

0

]

(5)

The objective of active control presented above can thenbe restated as finding a local control action using only theinformation from 119885

1or

119906 = minus1198701198851 (6)

such that both 1198851and 119885

2are attenuated simultaneously

Shock and Vibration 3

(a)

F

m1

fk u fc

m2

c2F0

Z1

Z2

Zr

(b)

Figure 1 (a) DSA250 produced by CED Railway Electric Tech Co (b) A simplified dual-mass pantograph model

Carrier

Support

Contact wire

Figure 2 The suspension catenary in a single span

m1

fk u fc

m2

c2F0

Z1

Z2

Zr

k(t)

Figure 3 A simplified pantograph-catenary interaction model

3 Geometric Design for Active Control

31 Geometric Design Procedures To proceed first considerthe dynamics at a certain frequency 120596 and then (4) can bewritten in the following general form

[

1199111(119895120596)

1199112(119895120596)

] = [

11989211(119895120596) 119892

12(119895120596)

11989221(119895120596) 119892

22(119895120596)

] [

119906 (119895120596)

119889 (119895120596)

] (7)

and the required control action has the form

119906 (119895120596) = minus119870 (119895120596) 1199111(119895120596) (8)

Now denote the sensitivity at a discrete frequency 120596 = 1205960

by 119878(1198951205960) and further define

120572 (1198951205960) = 119878 (119895120596

0) minus 1 (9)

120572 (1198951205960) = 120573 (119895120596

0) 119892 (119895120596

0) (10)

with 119892 (1198951205960) =

11989211(1198951205960) 11989222(1198951205960)

11989212(1198951205960) 11989221(1198951205960)

(11)

Hence the reduction in 1199111(119895120596) and 119911

2(119895120596) for the discrete

frequency 120596 = 1205960is equivalent to satisfying the following

conditions respectively1003816100381610038161003816120572 (119895120596) + 1

1003816100381610038161003816lt 1

1003816100381610038161003816120573 (119895120596) + 1

1003816100381610038161003816lt 1

(12)

As (11) relating 120572(119895120596) to 120573(119895120596) defines a Mobius trans-formation then the mapping of (10) on complex 120572-119901119897119886119899119890is a circle (and its interior) with centre at minus119892(119895120596

0) and

radius |119892(1198951205960)| Hence it can be concluded that simultaneous

reduction of 1199111(119895120596) and 119911

2(119895120596) is achievable for this discrete

frequency 120596 = 1205960if and only if the mapping of the unit circle


(and its interior) |120573(119895120596) + 1| lt 1 on the complex 120572-119901119897119886119899119890intersects the unit 120572 circle (and its interior) |120572(119895120596) + 1| lt1 Finally it is noted that an optimal line jointing (minus1 0)with (minus119892(119895120596

0)) on the complex 120572-119901119897119886119899119890 can be defined and

desired levels of attenuation can be obtained by choosingan appropriate point on the line for example a choice atpoint minus119892(119895120596

0) provides infinite attenuation in 119911

2(1198951205960) This

situation is illustrated in Figure 4Based on the development for the discrete frequency 120596 =

1205960 the aim of reducing 119911

1(119895120596) and 119911

2(119895120596) over an arbitrary

frequency band [1205961 120596119873] can be approached on a frequency-

by-frequency basis and this results in an optimal trajectory onthe complex 120572-119901119897119886119899119890 see Figure 4 However given the optimaltrajectory a problem arises whether there exists a transferfunction 120572(119904) isin 119867

infinsuch that 120572(119895120596

119894) = 120572

119894forall119894 isin [1119873] and

sup119904|120572(119904)| le 119872 where 119867

infinis the Hardy space of bounded

analytic functions in the right half planeR(119904) ge 0 and119872 gt 0

is a real numberThis is in fact a well-known Nevanlinna-Pick interpola-

tion problem The problem of the existence of an interpo-lating transfer function can be answered by the followingmodified Pick condition

Modified Pick condition for the existence of a stableinterpolating transfer function there exists a stable transferfunction 120572(119904) isin 119867

infinthat interpolates the prescribed trajec-

tory on the complex 120572-119901119897119886119899119890 if and only if the Pick matrix

119875 = [

1 minus 1205721198961205721198971198722

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

(13)

is positive definite

Remark 1 The minimal degree of stability 119886 and 119872 canalso be considered as design freedom since they can effec-tively influence the performance outside the frequency band[1205961 120596119873]

The positive definiteness of the Pick matrix will ensurethe existence of an interpolating transfer function But thequestion of how to modify the optimal trajectory in thecase of nonpositive definiteness of the Pick matrix remainsto be answered if an optimal solution is not feasible asuboptimal one will have to be sought to provide the bestapproximation to the initial optimal choice The solutioncomes from using an LMI solver However the Pick conditionmust be transformed to an LMI form to use the LMI solver

Rewrite the Pick matrix as follows

119875 = [

1 minus 1205721198961205721198971198722

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

= [

1

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

minus [

(120572119896119872) times (120572

119897119872)

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

(14)

Define

1198790= [

1

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

119876 = blockdiag (1205721119872

120572119873

119872

)

(15)

We then have

119875 = 1198790minus 1198761198790119876lowast

(lowast is the Hermitian operator) (16)

It is known that 1198790gt 0 hence by applying the Schur

complement to (16) We have the following 119875 is positivedefinite if and only if

[

1198790

119876

119876lowast1198790

minus1] gt 0 (17)

The solution to the above LMIs will result in an optimalsolution that guarantees a stable interpolating transfer func-tion 120572(119904) The controller119870(119904) is then obtained

119870 (119904) = minus

120572 (119904)

[120572 (119904) + 1] 11989211 (119904)

(18)

The above design procedures are summarized as follows

Step 1 Choose control architecture so that 1199111minus119906 is collocated

(a stable 11989211

can facilitate the design but is not an absoluterequirement)

Step 2 From the system plant determine the geometry of 120572and 120573

Step 3 Depending on the design objective choose appropri-ate 120572(119895120596) trajectory

Step 4 Find a stable function120572(119904) such that it interpolates theoptimal trajectory 120572 = 119878 minus 1 designed in Step 3

Step 5 Implement the feedback controller119870 as in (13)

32 Simulation Results To verify the effectiveness of thedesignmethodology consider the following numerical valuesin system dynamics

1198981= 1198982= 1 kg

1198650= 80N

1198882= 30N sdot sm

119896119896= 10

1198881= 10

(19)

The rigidity of the suspension catenary is assumed to be

119896 (119905) = 1198960(1 + 119886

1cos 2120587

119871

119905 + 1198862cos 2120587

1198711

119905

+ 1198863(cos 2120587

119871

119905)

2

+ 1198864(cos 120587

119871

119905)

2

)

(20)


Im

Re

120572-plane

120573-circle

Unit 120572-circle

minus1

120572opt

minusg(j1205960) = minusg11(j1205960)g22(j1205960)

g12(j1205960)g21(j1205960)

(a)

Im

Re

Complex 120572-plane

120573-circles

Unit 120572-circle

minus1

120572opt(j120596)

(b)

Figure 4 (a) Mapping of |120573(119895120596) + 1| lt 1 (b) optimal solution 120572opt(119895120596) on the complex 120572-119901119897119886119899119890 for broad band control

where 1198960= 30 119871 = 65m 119871

1= 6m 119886

1= 05 119886

2= 01

1198863= 02 and 119886

4= minus02

Further assume the scenario of optimal control is toprovide optimal reduction in 119911

1(hence the variation of the

contact force 119865 is minimized) but without causing vibrationenhancement in 119911

2(hence the transmission force from the

locomotive to pantograph is attenuated) Then a series ofLMIs constraining the values of 120572(119895120596) within the desiredlimit of optimal choices is used to obtain the optimal values120572(119895120596) over the desired frequency band an application of theNP algorithm results in a transfer function 120572(119904) with 5thorder that interpolates the optimal 120572(119895120596) valuesThe optimalcontroller119870(119904) can then be obtained from (18)

With this optimal controller 119870(119904) consider the case thatthe velocity of the locomotive is V = 200ms then theperformance of the active controller is shown in Figure 5From Figure 5 it is seen clearly that simultaneous vibrationreduction in both 119911

1(119895120596) and 119911

2(119895120596) is achieved using only

the feedback of the displacement 1198851 which is remarkable

(almost 6 dB attenuation is achieved) The vibration reduc-tion in 119885

1leads to the reduction of the variation in contact

force 119865

119865 = 119896 (119905) 1198851= [1198960(1 + 119886

1cos 2120587

119871

119905 + 1198862cos 2120587

1198711

119905

+ 1198863(cos 2120587

119871

119905)

2

+ 1198864(cos 120587

119871

119905)

2

)] times 1198851

(21)

This is validated in Figure 6Now increase the speed of the locomotive to V = 400ms

and the variation of the contact force 119865 is shown in Figure 7where again the active controller is turned on around 500thsecond

Z1 control starts around 500 s

100 200 300 400 500 600 700 800 900 10000Time (s)

minus15

minus1

minus05

0

05

1

15

Mag

nitu

de

(a)


minus5minus4minus3minus2minus1

012345

Mag

nitu

de

100 200 300 400 500 600 700 800 900 10000Time (s)

(b)

Figure 5 The performance of the active controller (a) showsthe reduction of variation in 119911

1and (b) shows the attenuation

of vibration transmission force in 1199112 The controller is turned on

around 500th s


minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

Forc

e (N

)

100 200 300 400 500 600 700 800 900 10000Time (s)

Variation of contact force F control starts around 500 s

Figure 6 The reduction of variation in contact force 119865 for V =

200ms


012345

Forc

e (N

)

100 200 300 400 500 600 700 800 900 10000Time (s)

Variation of contact force F = 400ms


400ms

It is seen that the same controller is capable of reducingthe variation of the contact force over a wide range of thelocomotive speed demonstrating the potential practicality ofthe above design methodology

Remark 2 The disturbance in the above simulation isassumed to be a random forcewith a spectrumover [2 10]Hzfrequency band

Remark 3 In fact simulation results also conform with theabove declaration when the locomotive speed varies with anyvelocityThis is a strong indication for the effectiveness of theactive control strategy in practical applications

4 Conclusion

A simplified model for the pantograph-catenary system hasbeen presented and a geometric design methodology hasbeen introduced for simultaneous variation reduction in boththe contact force and the transmission force from the locomo-tiveThe proposed method has been formulated in frequencydomain permitting both harmonic and broad band vibration

force attenuationThis feature is very desirable since it allowsperformance shaping at any discrete frequency or over anydesignated frequency band Conventional design approachescan only handle this issue through the use of weightingfunctions and this is often of rule-of-thumb nature Indeedas in the high speed train pantograph-catenary structureconsidered in the paper the frequency band can often beestimated for the problematic disturbance forces in manypractical industrial systems And the proposed method thuspresents a novel solution to the active control problems Thisforms one contribution Another contribution of the paper isthat the proposed strategy is different from the conventionalapproaches where the objective has been solely to reducethe variation of the contact force in the pantograph-catenarysystem without consideration of vibration transmission fromthe locomotive Such an active control strategy is novel anddeserves further exploration The simulation results clearlyvalidate the claim that the proposed methodology is veryeffective in active control of pantograph-catenary interaction

Further development must utilize a more accurate andrealistic model than the one in this note Indeed modelingof pantograph-catenary interaction has found much interestand many aspects have been considered for example [12]considers a partial differential algebraic equation (a partialdifferential equation for the catenary and a differentialalgebraic equation for the pantograph) for the pantograph-catenary system while both static and dynamic analyses ofa catenaries system have been conducted through a highperformance computing algorithm in [13] a finite elementmodel of the catenary system is described with the pan-tograph system detailed by a multibody model [14] eventhe effect of locomotive vibrations on pantograph-catenarysystem dynamics has been studied in [15] demonstratingthe nonnegligibility of locomotive vibrations in high speedoperation (as considered in this note) as yet another example[16] studies the pantograph-catenary interaction consideringtwo independent series of the catenary spanswhere transitionspans are overlapped Utilization of these models will pavethe way of applying the proposed design methodology to thepractical pantograph-catenary interaction system

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationof Jiangsu Province (No BK20140829) Jiangsu Postdoc-toral Science Foundation (No 1401017B) and the Funda-mental Research Funds for the Central Universities (NoNS2016024)

References

[1] J Xiao P Zhang Y Du and H Liu ldquoDevelopment of materialsfor electric locomotive pantograph slide platerdquo Railway Loco-motive amp Car vol 25 no 6 pp 65ndash68 2005 (Chinese)


[2] TWuX andM J Brennan ldquoActive vibration control of a railwaypantographrdquo Proceedings of the IME Part F Journal of Rail ampRapid Transit vol 211 no 2 pp 117ndash130 1997

[3] G Poetsch J Evans R Meisinger et al ldquoPantographcatenarydynamics and controlrdquo Vehicle System Dynamics vol 28 no 2-3 pp 159ndash195 1997

[4] D N OrsquoConnor S D Eppinger W P Seering and D NWormley ldquoActive control of a high-speed pantographrdquo Journalof Dynamic Systems Measurement and Control vol 119 no 1pp 1ndash4 1997

[5] A Levant A Pisano and E Usai ldquoOutput-feedback control ofthe contact-force in high-speed-train pantographsrdquo in Proceed-ings of the 40th IEEE Conference on Decision and Control (CDCrsquo01) pp 1831ndash1836 Orlando Fla USA December 2001

[6] Y Lin C Lin and N Shieh ldquoRobust optimal design for activepantograph suspension of light rail vehiclesrdquo in Proceedings ofthe 25th IASTED International Conference on Modeling Identi-fication and Control pp 396ndash401 Lanzarote Spain February2006

[7] J Guo S Yang and Q Zhu ldquoResearch on high speed trainpantographcatenary contact under control with differentialgeometry theoryrdquo Journal of Mechanical Strength vol 27 no 3pp 320ndash323 2005

[8] S Rusu-Anghel C Miklos J Averseng and G O TirianldquoControl system for catenary-pantograph dynamic interactionforcerdquo in Proceedings of the IEEE International Joint Conferenceson Computational Cybernetics and Technical Informatics (ICCC-CONTI rsquo10) pp 181ndash186 Timisoara Romania May 2010

[9] S Walters ldquoSimulation of fuzzy control applied to a railwaypantograph-catenary systemrdquo in Knowledge-Based and Intelli-gent Information and Engineering Systems vol 6277 of LectureNotes in Computer Science pp 322ndash330 Springer BerlinGermany 2010

[10] W Zhang J Zeng and Y Li ldquoA review of vehicle systemdynamics in the development of high-speed trains in ChinardquoInternational Journal of Dynamics and Control vol 1 no 1 pp81ndash97 2013

[11] W Zhang and S Zhang ldquoDynamics and service simulationfor general coupling system of high-speed trainsrdquo Journal ofSouthwest Jiaotong University vol 43 no 2 pp 147ndash152 2008(Chinese)

[12] M Arnold and B Simeon ldquoThe simulation of pantographand catenary a PDAE approachrdquo Tech Rep 1990 TechnischeUniversitat Darmstadt Fachbereich Mathematik 1998

[13] A Alberto J Benet E Arias D Cebrian T Rojo and FCuartero ldquoA high performance tool for the simulation of thedynamic pantograph-catenary interactionrdquo Mathematics andComputers in Simulation vol 79 no 3 pp 652ndash667 2008

[14] F G Rauter J Pombo J Ambrosio andM Pereira ldquoMultibodymodeling of pantographs for pantograph-catenary interactionrdquoin IUTAM Symposium on Multiscale Problems in MultibodySystem Contacts vol 1 of IUTAM Bookseries pp 205ndash226Springer Dordrecht The Netherlands 2007

[15] W M Zhai and C B Cai ldquoEffect of locomotive vibrations onpantograph-catenary systemdynamicsrdquoVehicle SystemDynam-ics vol 29 no 1 pp 47ndash58 1998

[16] J Benet A Alberto E Arias and T Rojo ldquoA mathematicalmodel of the pantograph-catenary dynamic interaction withseveral contact wiresrdquo IAENG International Journal of AppliedMathematics vol 37 no 2 2007

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design framework is introduced aiming tominimize the vari-ation of the contact force while simultaneously suppressingthe vibration disturbance from the train The paper is struc-tured as follows Section 2 describes the pantograph-catenarymodel Section 3 briefly reviews the design procedures andpresents the simulation results finally Section 4 concludes thepaper and comments the further development

2 Simplified Pantograph-Catenary Model

21 Pantograph Model A DSA250 pantograph is shown inFigure 1(a) which is used in CRH1 CRH2 and CRH5EMUs with a designed speed of 250 kmh Some complexpantograph models exist but to be elementary a pantographcan be modelled as a dual-mass system with known massesrepresenting the head and frame along with appropriatesprings and dampers A configuration is illustrated in Fig-ure 1(b)

In Figure 1 1198981and 119898

2are the equivalent mass of

the collector head and pantograph frame with 1198851and 119885

2

denoting their displacement relative to the equilibrium 1198881

and 119896119896are the parameters of the spring-damper structure

connecting the collector head and pantograph frame 1198882is

the viscosity coefficient of the pantograph frame 1198650 119891119896

and 119891119888are static lifting force elastic force and damping

force respectively 119885119903represents the vibration disturbance

transmitting from the locomotive and finally 119906 is the requiredforce for active control

22 Catenary Model The suspension catenary in a singlespan is shown in Figure 2

Usually the rigidity of the suspension catenary 119896(119905) is atime-varying function affected by the speed of the locomo-tive type of the catenaries and its corresponding parametersand so forth It can be obtained One of the methods toobtain an expression of 119896(119905) is through the application of finiteelement method to practical catenaries leading to

119896 (119905) = 1198960(1 + 119886

1cos 2120587

119871

119905 + 1198862cos 2120587

1198711

119905

+ 1198863(cos2120587

119871

119905)

2

+ 1198864(cos 120587

119871

119905)

2

+ 1198865(cos 120587

1198711

119905)

2

+ sdot sdot sdot)

(1)

and the parameters are then estimated by a nonlinear leastsquares approximation

23 Pantograph-Catenary Interaction Model For panto-graph-catenary interaction the catenary can be modeled asa spring with time varying rigidity 119896(119905) With the expressionfor 119896(119905) the contact force 119865 shown in Figure 1 is

119865 = 119896 (119905) 1198851 (2)

Hence combining the pantograph model (Figure 1(b)) withthe contact force from suspension catenary a simplified

pantograph-catenary interaction model can be obtained asshown in Figure 3

Then it is easy to show the above model can be describedby the following equations

11989811198851+ 119896 (119905) 119885

1+ 119891119896+ 119891119888= 119906

11989811198852+ 1198882(1198852minus

119885119903) minus 119891119896minus 119891119888= 1198650minus 119906

119891119896= 119896119896(1198851minus 1198852)

119891119888= 1198881(1198851minus

1198852)

(3)

The objective of active control can then be stated as followsfinding a control action 119906 based only on the displacement119885

1

such that the variation of both the contact force 119865 includingthe effects of the disturbance transmission force

119903is reduced

simultaneouslyRewriting (3) in state space representation is as follows

(119905) = 119860119909 (119905) + 119861120577 (119905)

119910 (119905) = 119862119909 (119905)

(4)

where 119909119879 = [1198851

11988511198852

1198852]119879 120577119879 = [119906 119889]

119879 and 119910119879 =

[11988511198852]119879 with 119889 equiv 119865

01198882+

119885119903 the corresponding matrices

are

119860 =

[

[

[

[

[

[

[

[

[

0 1 0 0

minus

119896119896+ 1198960

1198981

minus

1198881

1198981

119896119896

1198981

1198881

1198981

0 0 0 1

119896119896

1198982

1198881

1198982

minus

119896119896

1198982

minus

1198881+ 1198882

1198982

]

]

]

]

]

]

]

]

]

119861 =

[

[

[

[

[

[

[

[

[

0 0

1

1198981

0

0 0

minus

1

1198982

1198882

1198982

]

]

]

]

]

]

]

]

]

119862 = [

1

0

0

0

0

1

0

0

]

(5)

The objective of active control presented above can thenbe restated as finding a local control action using only theinformation from 119885

1or

119906 = minus1198701198851 (6)

such that both 1198851and 119885

2are attenuated simultaneously


(a)

F

m1

fk u fc

m2

c2F0

Z1

Z2

Zr

(b)


Carrier

Support

Contact wire


m1

fk u fc

m2

c2F0

Z1

Z2

Zr

k(t)




[

1199111(119895120596)

1199112(119895120596)

] = [

11989211(119895120596) 119892

12(119895120596)

11989221(119895120596) 119892

22(119895120596)

] [

119906 (119895120596)

119889 (119895120596)

] (7)


119906 (119895120596) = minus119870 (119895120596) 1199111(119895120596) (8)



120572 (1198951205960) = 119878 (119895120596

0) minus 1 (9)

120572 (1198951205960) = 120573 (119895120596

0) 119892 (119895120596

0) (10)

with 119892 (1198951205960) =

11989211(1198951205960) 11989222(1198951205960)

11989212(1198951205960) 11989221(1198951205960)

(11)





1003816100381610038161003816lt 1

1003816100381610038161003816120573 (119895120596) + 1

1003816100381610038161003816lt 1

(12)


0) and


reduction of 1199111(119895120596) and 119911








2(1198951205960) This



1(119895120596) and 119911




infinsuch that 120572(119895120596

119894) = 120572

119894forall119894 isin [1119873] and

sup119904|120572(119904)| le 119872 where 119867








119875 = [

1 minus 1205721198961205721198971198722

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

(13)





119875 = [

1 minus 1205721198961205721198971198722

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

= [

1

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

minus [

(120572119896119872) times (120572

119897119872)

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

(14)

Define

1198790= [

1

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

119876 = blockdiag (1205721119872

120572119873

119872

)

(15)

We then have

119875 = 1198790minus 1198761198790119876lowast




[

1198790

119876

119876lowast1198790

minus1] gt 0 (17)


119870 (119904) = minus

120572 (119904)

[120572 (119904) + 1] 11989211 (119904)

(18)



(a stable 11989211







1198981= 1198982= 1 kg

1198650= 80N

1198882= 30N sdot sm

119896119896= 10

1198881= 10

(19)


119896 (119905) = 1198960(1 + 119886

1cos 2120587

119871

119905 + 1198862cos 2120587

1198711

119905

+ 1198863(cos 2120587

119871

119905)

2

+ 1198864(cos 120587

119871

119905)

2

)

(20)


Im

Re

120572-plane

120573-circle

Unit 120572-circle

minus1

120572opt


g12(j1205960)g21(j1205960)

(a)

Im

Re


120573-circles

Unit 120572-circle

minus1

120572opt(j120596)

(b)


where 1198960= 30 119871 = 65m 119871

1= 6m 119886

1= 05 119886

2= 01

1198863= 02 and 119886

4= minus02







1(119895120596) and 119911





force 119865

119865 = 119896 (119905) 1198851= [1198960(1 + 119886

1cos 2120587

119871

119905 + 1198862cos 2120587

1198711

119905

+ 1198863(cos 2120587

119871

119905)

2

+ 1198864(cos 120587

119871

119905)

2

)] times 1198851

(21)




100 200 300 400 500 600 700 800 900 10000Time (s)

minus15

minus1

minus05

0

05

1

15

Mag

nitu

de

(a)



012345

Mag

nitu

de

100 200 300 400 500 600 700 800 900 10000Time (s)

(b)




around 500th s


minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

Forc

e (N

)

100 200 300 400 500 600 700 800 900 10000Time (s)



200ms


012345

Forc

e (N

)

100 200 300 400 500 600 700 800 900 10000Time (s)



400ms




4 Conclusion






Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a)

F

m1

fk u fc

m2

c2F0

Z1

Z2

Zr

(b)


Carrier

Support

Contact wire


m1

fk u fc

m2

c2F0

Z1

Z2

Zr

k(t)




[

1199111(119895120596)

1199112(119895120596)

] = [

11989211(119895120596) 119892

12(119895120596)

11989221(119895120596) 119892

22(119895120596)

] [

119906 (119895120596)

119889 (119895120596)

] (7)


119906 (119895120596) = minus119870 (119895120596) 1199111(119895120596) (8)



120572 (1198951205960) = 119878 (119895120596

0) minus 1 (9)

120572 (1198951205960) = 120573 (119895120596

0) 119892 (119895120596

0) (10)

with 119892 (1198951205960) =

11989211(1198951205960) 11989222(1198951205960)

11989212(1198951205960) 11989221(1198951205960)

(11)





1003816100381610038161003816lt 1

1003816100381610038161003816120573 (119895120596) + 1

1003816100381610038161003816lt 1

(12)


0) and


reduction of 1199111(119895120596) and 119911








2(1198951205960) This



1(119895120596) and 119911




infinsuch that 120572(119895120596

119894) = 120572

119894forall119894 isin [1119873] and

sup119904|120572(119904)| le 119872 where 119867








119875 = [

1 minus 1205721198961205721198971198722

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

(13)





119875 = [

1 minus 1205721198961205721198971198722

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

= [

1

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

minus [

(120572119896119872) times (120572

119897119872)

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

(14)

Define

1198790= [

1

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

119876 = blockdiag (1205721119872

120572119873

119872

)

(15)

We then have

119875 = 1198790minus 1198761198790119876lowast




[

1198790

119876

119876lowast1198790

minus1] gt 0 (17)


119870 (119904) = minus

120572 (119904)

[120572 (119904) + 1] 11989211 (119904)

(18)



(a stable 11989211







1198981= 1198982= 1 kg

1198650= 80N

1198882= 30N sdot sm

119896119896= 10

1198881= 10

(19)


119896 (119905) = 1198960(1 + 119886

1cos 2120587

119871

119905 + 1198862cos 2120587

1198711

119905

+ 1198863(cos 2120587

119871

119905)

2

+ 1198864(cos 120587

119871

119905)

2

)

(20)


Im

Re

120572-plane

120573-circle

Unit 120572-circle

minus1

120572opt


g12(j1205960)g21(j1205960)

(a)

Im

Re


120573-circles

Unit 120572-circle

minus1

120572opt(j120596)

(b)


where 1198960= 30 119871 = 65m 119871

1= 6m 119886

1= 05 119886

2= 01

1198863= 02 and 119886

4= minus02







1(119895120596) and 119911





force 119865

119865 = 119896 (119905) 1198851= [1198960(1 + 119886

1cos 2120587

119871

119905 + 1198862cos 2120587

1198711

119905

+ 1198863(cos 2120587

119871

119905)

2

+ 1198864(cos 120587

119871

119905)

2

)] times 1198851

(21)




100 200 300 400 500 600 700 800 900 10000Time (s)

minus15

minus1

minus05

0

05

1

15

Mag

nitu

de

(a)



012345

Mag

nitu

de

100 200 300 400 500 600 700 800 900 10000Time (s)

(b)




around 500th s


minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

Forc

e (N

)

100 200 300 400 500 600 700 800 900 10000Time (s)



200ms


012345

Forc

e (N

)

100 200 300 400 500 600 700 800 900 10000Time (s)



400ms




4 Conclusion






Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














2(1198951205960) This



1(119895120596) and 119911




infinsuch that 120572(119895120596

119894) = 120572

119894forall119894 isin [1119873] and

sup119904|120572(119904)| le 119872 where 119867








119875 = [

1 minus 1205721198961205721198971198722

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

(13)





119875 = [

1 minus 1205721198961205721198971198722

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

= [

1

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

minus [

(120572119896119872) times (120572

119897119872)

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

(14)

Define

1198790= [

1

119895 (120596119896minus 120596119897) + 2119886

]

1le119896119897le119873

119876 = blockdiag (1205721119872

120572119873

119872

)

(15)

We then have

119875 = 1198790minus 1198761198790119876lowast




[

1198790

119876

119876lowast1198790

minus1] gt 0 (17)


119870 (119904) = minus

120572 (119904)

[120572 (119904) + 1] 11989211 (119904)

(18)



(a stable 11989211







1198981= 1198982= 1 kg

1198650= 80N

1198882= 30N sdot sm

119896119896= 10

1198881= 10

(19)


119896 (119905) = 1198960(1 + 119886

1cos 2120587

119871

119905 + 1198862cos 2120587

1198711

119905

+ 1198863(cos 2120587

119871

119905)

2

+ 1198864(cos 120587

119871

119905)

2

)

(20)


Im

Re

120572-plane

120573-circle

Unit 120572-circle

minus1

120572opt


g12(j1205960)g21(j1205960)

(a)

Im

Re


120573-circles

Unit 120572-circle

minus1

120572opt(j120596)

(b)


where 1198960= 30 119871 = 65m 119871

1= 6m 119886

1= 05 119886

2= 01

1198863= 02 and 119886

4= minus02







1(119895120596) and 119911





force 119865

119865 = 119896 (119905) 1198851= [1198960(1 + 119886

1cos 2120587

119871

119905 + 1198862cos 2120587

1198711

119905

+ 1198863(cos 2120587

119871

119905)

2

+ 1198864(cos 120587

119871

119905)

2

)] times 1198851

(21)




100 200 300 400 500 600 700 800 900 10000Time (s)

minus15

minus1

minus05

0

05

1

15

Mag

nitu

de

(a)



012345

Mag

nitu

de

100 200 300 400 500 600 700 800 900 10000Time (s)

(b)




around 500th s


minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

Forc

e (N

)

100 200 300 400 500 600 700 800 900 10000Time (s)



200ms


012345

Forc

e (N

)

100 200 300 400 500 600 700 800 900 10000Time (s)



400ms




4 Conclusion






Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Im

Re

120572-plane

120573-circle

Unit 120572-circle

minus1

120572opt


g12(j1205960)g21(j1205960)

(a)

Im

Re


120573-circles

Unit 120572-circle

minus1

120572opt(j120596)

(b)


where 1198960= 30 119871 = 65m 119871

1= 6m 119886

1= 05 119886

2= 01

1198863= 02 and 119886

4= minus02







1(119895120596) and 119911





force 119865

119865 = 119896 (119905) 1198851= [1198960(1 + 119886

1cos 2120587

119871

119905 + 1198862cos 2120587

1198711

119905

+ 1198863(cos 2120587

119871

119905)

2

+ 1198864(cos 120587

119871

119905)

2

)] times 1198851

(21)




100 200 300 400 500 600 700 800 900 10000Time (s)

minus15

minus1

minus05

0

05

1

15

Mag

nitu

de

(a)



012345

Mag

nitu

de

100 200 300 400 500 600 700 800 900 10000Time (s)

(b)




around 500th s


minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

Forc

e (N

)

100 200 300 400 500 600 700 800 900 10000Time (s)



200ms


012345

Forc

e (N

)

100 200 300 400 500 600 700 800 900 10000Time (s)



400ms




4 Conclusion






Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus5

minus4

minus3

minus2

minus1

0

1

2

3

4

5

Forc

e (N

)

100 200 300 400 500 600 700 800 900 10000Time (s)



200ms


012345

Forc

e (N

)

100 200 300 400 500 600 700 800 900 10000Time (s)



400ms




4 Conclusion






Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Active Control of Contact Force for a Pantograph-Catenary...

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Transcript of Research Article Active Control of Contact Force for a Pantograph-Catenary...