energy-based controller design of stochastic magnetic

Research ArticleEnergy-Based Controller Design of Stochastic MagneticLevitation System

Weiwei Sun12 Kaili Wang1 Congcong Nie1 and Xuejun Xie12

1 Institute of Automation Qufu Normal University Qufu 273165 China2School of Engineering Qufu Normal University Rizhao 276826 China

Correspondence should be addressed to Weiwei Sun wwsunhotmailcom

Received 9 March 2017 Revised 15 May 2017 Accepted 18 July 2017 Published 20 August 2017

Academic Editor Huanqing Wang

Copyright copy 2017 Weiwei Sun et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper investigates the control problem of magnetic levitation system in which velocity feedback signal is influenced bystochastic disturbance Firstly single-degree-freedom magnetic levitation is regarded as an energy-transform action device Fromthe view of energy-balance relation the magnetic levitation system is transformed into port-controlled Hamiltonian systemmodelNext based on the Hamiltonian structure the control law of magnetic levitation system is designed by applying Lyapunov theoryFinally the simulation verifies the correctness of the proposed results

1 Introduction

Magnetic levitation system is a class of typical nonlinearsystem which is difficult to establish accurate mathematicalmodel for the natural parameter of electromagnetic partdependent times [1 2] Normally a standard magnetic levita-tion system consists of four parts the sensors the controllerthe power amplifier and the electromagnetic drivesThis kindof system is a control system and its control objective is toensure that the soliquoid is in the normal position through aseries of feedback and control activities The general processis as follows firstly after comparing the given signal withthe feedback signal they will be passed along regulatingcircuit to power amplifier circuit then the control currentsflow through the power amplifier finally the electromagnetconverts it to electromagnetic force to control the suspensionposition Shu et al in [3] introduced the working principle ofthe magnetic levitation system Combined with the classicaltheory of dynamics and electromagnet the nonlinear equa-tions of motion of the system are derived by the general formof Lagrange equation and after the linearization processingaround the operating point the state space description of thesystem is obtained

In addition the systems are always affected by stochas-tic disturbance in many practical control problems which

always lead to system instability [4] Therefore more andmore scholars and experts pay attention to the research ofstability and control of stochastic nonlinear systems and alot of research achievements have been made that greatlypromoted the development of the system research [5ndash9] In[9] the nonlinear stochastic Hinfin control of It119900-type differ-ential systems with all the state control input and externaldisturbance is studied and a sufficient condition is given forthe finiteinfinite horizon Hinfin control of such a system bymeans of Hamilton-Jacobi inequality More recently someprogress has been made toward solving the stability analysisand controller design for stochasticHamiltonian systems [10ndash12] Sun and Peng in [12] studied the robust adaptive controlproblem for a class of time-delay stochastic Hamiltoniansystems An uncertainty-independent adaptive control lawis designed to guarantee that the closed-loop Hamiltoniansystem is robustly asymptotically stable in the mean square

Due to the highly nonlinear characteristics of the sys-tem the controller design problem will be solved based onHamiltonian energy theory In fact the energy-based Hamil-tonian system method has been widely used in practicalsystems control [11 13ndash19] Based on the Hamiltonian systemtheory the port-controlled Hamiltonian dissipative modelof magnetic levitation system was established in [19] byusing the Hamiltonian function as the storage function and

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 7838431 6 pageshttpsdoiorg10115520177838431

2 Mathematical Problems in Engineering

the controller was designed which is simple and easy toimplement A key feature of the systems which is usefulfor stability analysis and stabilization is that Hamiltonianfunction in a port-Hamiltonian systems can be used as aLyapunov function which brings great convenience

As is known the controllers and regulators of the systemsare always unavoidably affected by stochastic disturbancesand the study of the controlled systems with stochasticdisturbance is of practical significance Different from whathave been studied this present paper deals with the con-troller design problem of the magnetic levitation systemwith stochastic disturbances Current efforts have been madeto dispose the control problem of the stochastic magneticlevitation system on the basis of Hamiltonian energy theoryWe regard the magnetic suspension as the energy conversiondevice and then derive a mathematical model of the singledegree of freedom stochasticmagnetic levitation system fromthe point of energy balance which is transformed into a port-controlled Hamiltonian system Consequently the controllerof the stochastic magnetic levitation system is designedFinally a simulation example is given to verify the validityof the results

The rest of this paper is organised as follows Section 2provides the problem formulation the Hamiltonian mod-eling process of the stochastic magnetic levitation systemsand some preliminaries Section 3 gives the main results Asimulation example is worked out in Section 4 to illustratethe results Section 5 draws the concluding remarks

Notations sdot stands for either the Euclidean vector normor the induced matrix 2-norm A function 119891(119909) isin C2

means that119891(119909) is a twice differentiable continuous functionThe notation 119883 ge 119884 (resp 119883 gt 119884) where 119883 and119884 are symmetric matrices means that the matrix 119883 minus 119884is positive semidefinite (resp positive definite) 120582max(119875)(120582min(119875)) denotes the maximum (minimum) of eigenvalueof a real symmetric matrix 119875 Throughout the paper thesuperscript ldquoTrdquo stands for matrix transposition In additionfor the sake of simplicity we denote 120597119867120597119909 by nabla119867

2 Problem Description and Transformation

The physical model of the magnetic levitation train systemwhich includes the concentrated mass of train carriages(together with the supporting magnet) suspended on therigid lead rail is shown in Figure 1 where 119898 is the qualityof train carriage (including the supporting magnet) 119892 isgravitational constant 120579119872 is the gap between the supportingmagnet and the guide rail 1205790 is the gap between the guiderail and the reference plane 120579 is the distance between thesupporting magnet and the reference plane 120579119872 = 1205790 + 120579119871(120579) is the self-inductance of magnetic coil which dependson the gap 120579 119894 is the current flowing through magnet spool119877 is the coil resistance 119906 is the voltage at both ends of themagnet spool

By invoking Kirchoff rsquos voltage law and Newtons secondlaw the dynamic equations of the magnetic levitation system

Carriage

Guideway

Magnet

mg

M

0

L () i

u

R

Figure 1 Physical model of magnetic suspension system

can be obtained by taking the vertical upward direction as thepositive direction 119898 120579 = 119865 (119894 120579) minus 119898119892

119906 = 119877119894 + Φ (1)

where Φ = 119871(120579)119894 is the magnetic flux and 119865(119894 120579) is the forcecreated by the electromagnet which is given by

119865 (119894 120579) = 12 120597119871120597120579 (120579) 1198942 (2)

Here we regard the flux Φ as the independent variablethen (1) can be further transformed into the following forms120579 = V

119898V = Φ24119896 minus 119898119892Φ = 119906 minus 119877 Φ119871 (120579)

(3)

where 119896 = 120583011987321198784 1205830 is the permeability of vacuum119873 is coil turns and 119878 is the effective pole area of theelectromagnetic coil

To obtain a port-controlled Hamiltonian model we takea suitable approximation for the inductance is 119871(120579) =2119896(120579119872 minus 120579) As is known the speed of the rigid body willbe affected by stochastic disturbances during the operationof the magnetic levitation system Let 119909 = [Φ 120579119898 120579]T =[1199091 1199092 1199093]T due to the influence of stochastic disturbancethe magnetic levitation system (3) can be modeled as thefollowing algebraic differential equations

d1199091 = minus1198771199091 (120579119872 minus 1199092)2119896 d119905 + 119906d119905 + 1199093d119908 (119905) d1199092 = 1199093119898 d119905d1199093 = 119909214119896d119905 minus 119898119892d119905

(4)

Mathematical Problems in Engineering 3

where 119908(119905) is an independent standard Wiener process andsatisfies 119864d119908(119905) = 0 and 119864d1199082(119905) = d119905 and 119864 is theexpectation operator

The objective of this paper is to find a feedback controllaw as 119906 (119905) = 120572 (119905) (5)

to ensure that the stochastic magnetic levitation system (4)with the controller (5) is asymptotically stable in the meansquare

Obviously system (4) is a nonlinear system In orderto study the control problems of the stochastic magneticlevitation system in view of the energy balance we need toconvert it into a stochastic Hamiltonian system first Takingthe total of electromagnetic energy andmechanical energy asthe Hamiltonian function that is

119867(119909) = 119909214119896 (120579119872 minus 1199092) + 1211989811990923 + 1198981198921199092 (6)

then the magnetic levitation port-controlled Hamiltoniansystem is obtained

d119909 (119905) = (J minus R) nabla119867 (119909) d119905 + g1119906d119905 + g2 (119909) d119908 (119905) (7)

where

J = (0 0 00 0 10 minus1 0)

R = (119877 0 00 0 00 0 0)

g1 = (100)

g2 (119909) = (119909300 )

(8)

According to the equilibrium condition of the system thespeed of the rigid body reduced to zero when the system isstableMeanwhile the electromagnetic force of the rigid bodyis equal to the gravity that acting upon on itThen we can get119909lowast1 = radic4119896119898119892 119909lowast3 = 0 Therefore the equilibrium point of thesystem is 119909lowast = [radic4119896119898119892 119909lowast2 0]T

It is evident that J is a skew symmetric matrix that is J =minusJT and R is a positive semidefinite matrix Consequently if119906 = 0 and 119908(119905) = 0 system (7) is a dissipative Hamiltoniansystem since

L119867(119909) = minusnablaT119867(119909)Rnabla119867 (119909) (9)

In order to design the controller of system (7) weintroduce the following definition

Definition 1 If there exists a controller 119906 such that

lim119905rarrinfin

E 1003817100381710038171003817119909 (119905) minus 119909lowast10038171003817100381710038172 = 0 (10)

the stochastic Hamiltonian system (7) is said to be asymptot-ically stable in the mean square where 119909(119905) is the solution ofsystem (7) at time 119905 under the initial condition 119909(1199050) = 1199090

Next we introduce some auxiliary lemmas which will beused in this paper

Lemma 2 (see [6]) For system

d119909 (119905) = 119891 (119909 (119905)) d119905 + 119892 (119909 (119905)) d119908 (119905) forall119905 ge 0 (11)

assume that 119891(119909) and 119892(119909) are locally Lipschitz in 119909 For aconstant 119870 gt 0 and any 119905 satisfies 119905 ge 0 there exists function119881(119909 119905) isin C21(R119899 times [0infin)R+) such that

L119881 le 119870 (1 + 119881 (119909 (119905) 119905)) lim|119909|rarrinfin

inf119905ge0

119881 (119909 119905) = infin (12)

then from system (11) there exists a unique solution on [0infin)for any initial date 119909(1199050) = 1199090 where

L119881 = 12 tr119892T (119909 (119905)) (1205972119881)1205971199092 119892 (119909 (119905)) + 120597119881120597119905+ 120597119881120597119909 119891 (119909 (119905))

(13)

Lemma 3 (see [4]) Let 119881(119909 119905) isin C21(R119899 times [0infin)R+)1205911 1205912 are the bounded stopping time and satisfy 0 le 1205911 le 1205912If 119881(119909 119905) andL119881(119909 119905) are both bounded on 119905 isin [1205911 1205912] then

119864 119881 (119909 (1205912) 1205912) minus 119881 (119909 (1205911) 1205911)= 119864int12059121205911

L119881 (119909 119905) d119905 (14)

Lemma 4 For any given matrices 119860 isin R119899times119903 and 119866 isin R119899times119899 if119866 ge 0 it follows thattr (119860T119866119860) le 120582max (119866) tr (119860T119860) (15)

3 Controller Design of Stochastic MagneticLevitation System

In this section we will put forward the controller designscheme for stochastic magnetic levitation system (4) Tothis end the stabilization problem of stochastic Hamiltoniansystem (7) is to be discussed first

Consider system (7) Choose Lyapunov function as

119881 (119909) = 119867 (119909) minus 119867 (119909lowast) ge 0 (16)


Suppose that the Hamiltonian function 119867(119909) isin C2 andsatisfies

119867(119909) ge 120572 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (17)

nablaT119867(119909) nabla119867 (119909) ge 120573 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (18)

where 120572 and 120573 are positive scalarsAccording to It119900 differential equations it can be obtained

that

d119881 (119909) = L119881 (119909) d119905 + nabla119881 (119909) g2 (119909) d119908 (119905) (19)

where

L119881 (119909) = minusnablaT119867(119909)Rnabla119867 (119909)+ tr [gT2 (119909)Hess (119867 (119909)) g2 (119909)]+ nablaT119867(119909) 1198921119906

(20)

If we set suitable scalars 120582 and 120583 then we have

tr [gT2 (119909)Hess (119867 (119909)) g2 (119909)]le 12 tr [gT2 (119909)Hess (119867 (119909))HessT (119867 (119909)) g2 (119909)]

+ 12 tr [gT2 (119909) g2 (119909)]le nablaT119867(119909) [12 (120582 + 1) 120583119868] nabla119867 (119909)

(21)

So the stabilization may be achieved by designing a suitablecontroller for system (7) The following theorem provides afeasible scheme

Theorem 5 (consider system (7)) Suppose the Hamiltonianfunction 119867(119909) satisfies (17) and (18) Then the closed-loopstochastic Hamiltonian system of (7) is asymptotic stable in themean square under the feedback control law

119906 = minus [gT1 g1]minus1 gT1 [12 (120582 + 1) 120583119868 + 119868]nabla119867 (119909) (22)

where 120582 and 120583 are scalars which satisfies 120582 =sup119905ge0119867119890119904119904(119867(119909))2 and 120583 ge 120573minus1 tr[gT2 (119909)sdotg2(119909)]sdot119909minus119909lowastminus2Proof Substituting (22) into (7) yields

d119909 (119905) = (J minus R) nabla119867 (119909) d119905minus [12 (120582 + 1) 120583119868 + 119868]nabla119867 (119909) d119905+ g2 (119909) d119908 (119905)

(23)

Combining (18) (20) (21) and (23) we obtain

L119881 (119909) le minusnablaT119867(119909) nabla119867 (119909) le minus120573 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (24)

then 119864 L119881 (119909) le minus120573119864 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (25)

According to Lemma 3 the following formula was estab-lished for all 119905 gt 0

119864 119881 (119905) minus 119864 119881 (0) = int1199050

119864 L119881 (119904) d119904le int1199050

119864 minus120573 1003817100381710038171003817119909 (119904) minus 119909lowast10038171003817100381710038172 d119904 (26)

Therefore the following formula was established

dd119905119864 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 le minus120573120572119864 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (27)

Set 119887 = minus120573120572 and multiplying 119890minus119887119905 to the two sides of theinequality (27) we have

119890minus119887119905 dd119905119864 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 minus 119890minus119887119905119887119864 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 le 0 (28)

that is

dd119905 (119890minus119887119905119864 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172) le 0 (29)

Integrating inequality (29) from 1199050 to 119905 we get119890minus119887119905119864 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 minus 119864 10038171003817100381710038171199090 minus 119909lowast10038171003817100381710038172 le 0 (30)

that is 119864 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 le 119890119887119905119864 10038171003817100381710038171199090 minus 119909lowast10038171003817100381710038172 forall119905 gt 0 (31)

Since 119887 lt 0 which implies that

lim119905rarrinfin

119864 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 = 0 (32)

system (7) is asymptotic stable in the mean square underthe feedback control law (22) This completes the proof

Remark 6 Since 119867(119909) isin C2 g2(119909) are continuous functionsand according to Lemma 2 we can conclude that the solutionof the closed-loop system (23) is unique for any initialcondition in the neighborhood of the equilibrium point 119909lowast

Next we consider the stochastic magnetic levitationsystem (4) Obviously there exist positive scalars 120572 and 120573which make system (4) meet the inequalities (17) and (18) inTheorem 5 thus we can get the following conclusions

Theorem 7 The stochastic magnetic levitation system (4) isasymptotic stable in the mean square under the feedbackcontrol law

119906 = minus [12 (120582 + 1) 120583 + 1] 11990912119896 (120579119872 minus 1199092) (33)

where 120582 = sup119905ge0119867119890119904s((119909214119896)(120579119872 minus 1199092) + (12119898)11990923 +1198981198921199092)2 and 120583 is a scalar and satisfies 120583 ge 11990923(1205731199092)Proof Because of the stochastic magnetic levitation system(4) is equivalent to system (7) substitute g1 into system (7)and the formula (6) into (22) we can get controller (33) Therest of the proof is omitted here


05 1 15 2 250time (s)

x3

minus002

0

002

004

006

008

01ve

loci

ty (c

mlowastＭminus

1)

Figure 2 Velocity response curve

05 1 15 2 250time (s)

minus005

0

005

01

015

02

025

03

disp

lace

men

t (cm

)

x2

Figure 3 Displacement response curve

4 Illustrative Examples

In this section a simulation example is given to verify thecorrectness of the results obtained in this paper The relevantparameters are given as follows 119877 = 4Ω 119898 = 001 g 120579119872 =001m 119896 = 005 119892 = 00098Ng and 1205790 = 01 cm Bycalculating we take 120582 = 100 and 120583 = 10

According to Theorem 7 we can see that system (4) isasymptotically stable in the mean square under the feedbackcontrol law

119906 = minus50601199091 (120579119872 minus 1199092) (34)

Thevelocity curve of the rigid body is shown in Figure 2 Itshows that the designed controller canmake the system reachto the equilibrium point quickly Figure 3 is the displacementcurve of the rigid body the displacement can also quicklyreach to the equilibrium point

5 Conclusion

This paper has investigated the control problem of stochas-tic magnetic levitation system By regarding the magneticlevitation as the energy conversion device we derived themathematical model of single degree of freedom magneticlevitation system with stochastic disturbance from the pointof view of the energy balance and then the model canbe transformed into a port-controlled Hamiltonian systemThen the controller of the stochastic magnetic levitationsystemhas been designed based on the obtainedHamiltoniansystem model Finally the correctness of the conclusion hasbeen verified by simulations The main innovation of thispaper is that we have fully taken into account the effectof random disturbances on the magnetic levitation systemand solve the control problem under Hamiltonian systemsframework by making full use of the dissipative structuralproperties of the Hamiltonian systems

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] L Gentili and L Marconi ldquoRobust nonlinear disturbancesuppression of a magnetic levitation systemrdquo Automatica AJournal of IFAC the International Federation of AutomaticControl vol 39 no 4 pp 735ndash742 2003

[2] F Gomez-Salas Y Wang and Q Zhu ldquoDesign of a discretetracking controller for amagnetic levitation system a nonlinearrational model approachrdquo Mathematical Problems in Engineer-ing vol 2015 Article ID 360783 2015

[3] G Shu W Chen andM Reinhold ldquoThe research on the modelof a magnetic levitation systemrdquo Electric Machines and Controlvol 9 no 3 pp 187ndash195 2005 (Chinese)

[4] X Mao and C Yuan Stochastic Differential Equations withMarkovian Switching Imperial College Press London UK2006

[5] T Li G Li and Q Zhao ldquoAdaptive Fault-Tolerant Stochas-tic Shape Control with Application to Particle DistributionControlrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 45 no 12 pp 1592ndash1604 2015

[6] S Liu S Ge and J Zhang ldquoAdaptive output-feedback controlfor a class of uncertain stochastic non-linear systems with timedelaysrdquo International Journal of Control vol 81 no 8 pp 1210ndash1220 2008

[7] L Liu X Li H Wang and B Niu ldquoGlobal asymptotic stabi-lization of stochastic feedforward nonlinear systems with inputtime-delayrdquo Nonlinear Dynamics An International Journal ofNonlinear Dynamics and Chaos in Engineering Systems vol 83no 3 pp 1503ndash1510 2016

[8] XWei ZWu andHKarimi ldquoDisturbance observer-based dis-turbance attenuation control for a class of stochastic systemsrdquoAutomatica A Journal of IFAC the International Federation ofAutomatic Control vol 63 pp 21ndash25 2016

[9] W Zhang B Chen H Tang L Sheng and M Gao ldquoSomeremarks on general nonlinear stochastic Hinfin control with statecontrol and disturbance-dependent noiserdquo IEEE Transactionson Automatic Control vol 59 no 1 pp 237ndash242 2014


[10] S Satoh and K Fujimoto ldquoPassivity based control of stochasticport-Hamiltonian systemsrdquo IEEE Transactions on AutomaticControl vol 58 no 5 pp 1139ndash1153 2013

[11] W Sun and L Peng ldquoObserver-based robust adaptive controlfor uncertain stochastic Hamiltonian systems with state andinput delaysrdquo Lithuanian Association of Nonlinear Analysts(LANA) Nonlinear Analysis Modelling and Control vol 19 no4 pp 626ndash645 2014

[12] W Sun and L Peng ldquoRobust adaptive control of uncertainstochastic Hamiltonian systems with time varying delayrdquo AsianJournal of Control vol 18 no 2 pp 642ndash651 2016

[13] S Knorn A Donaire J C Aguero and R H MiddletonldquoPassivity-based control for multi-vehicle systems subject tostring constraintsrdquo Automatica vol 50 no 12 pp 3224ndash32302014

[14] R Ortega A J Van der Schaft I Mareels and B MaschkeldquoPutting energy back in controlrdquo IEEE Control Systems Maga-zine vol 21 no 2 pp 18ndash33 2001

[15] H Ramrez Y Le Gorrec B Maschke and F Couenne ldquoOnthe passivity based control of irreversible processes a port-Hamiltonian approachrdquo Automatica vol 64 pp 105ndash111 2016

[16] W Sun L Peng Y Zhang and H Jia ldquoHinfin excitation controldesign for stochastic power systems with input delay based onnonlinear Hamiltonian system theoryrdquoMathematical Problemsin Engineering vol 2015 Article ID 947815 12 pages 2015

[17] W Sun and B Fu ldquoAdapative control of time-varying uncertainnon-linear systems with input delay a Hamiltonian approachrdquoIETControlTheoryampApplications vol 10 no 15 pp 1844ndash18582016

[18] Y Wang and S S Ge ldquoAugmented Hamiltonian formulationand energy-based control design of uncertain mechanicalsystemsrdquo IEEE Transactions on Control Systems Technology vol16 no 2 pp 202ndash213 2008

[19] J Zhang and J Wu ldquoHamiltonian modeling and passivecontrol of magnetic levitation systemrdquo Dianji yu KongzhiXuebaoElectric Machines and Control vol 12 no 4 pp 464ndash472 2008

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Stochastic AnalysisInternational Journal of


the controller was designed which is simple and easy toimplement A key feature of the systems which is usefulfor stability analysis and stabilization is that Hamiltonianfunction in a port-Hamiltonian systems can be used as aLyapunov function which brings great convenience

As is known the controllers and regulators of the systemsare always unavoidably affected by stochastic disturbancesand the study of the controlled systems with stochasticdisturbance is of practical significance Different from whathave been studied this present paper deals with the con-troller design problem of the magnetic levitation systemwith stochastic disturbances Current efforts have been madeto dispose the control problem of the stochastic magneticlevitation system on the basis of Hamiltonian energy theoryWe regard the magnetic suspension as the energy conversiondevice and then derive a mathematical model of the singledegree of freedom stochasticmagnetic levitation system fromthe point of energy balance which is transformed into a port-controlled Hamiltonian system Consequently the controllerof the stochastic magnetic levitation system is designedFinally a simulation example is given to verify the validityof the results

The rest of this paper is organised as follows Section 2provides the problem formulation the Hamiltonian mod-eling process of the stochastic magnetic levitation systemsand some preliminaries Section 3 gives the main results Asimulation example is worked out in Section 4 to illustratethe results Section 5 draws the concluding remarks

Notations sdot stands for either the Euclidean vector normor the induced matrix 2-norm A function 119891(119909) isin C2

means that119891(119909) is a twice differentiable continuous functionThe notation 119883 ge 119884 (resp 119883 gt 119884) where 119883 and119884 are symmetric matrices means that the matrix 119883 minus 119884is positive semidefinite (resp positive definite) 120582max(119875)(120582min(119875)) denotes the maximum (minimum) of eigenvalueof a real symmetric matrix 119875 Throughout the paper thesuperscript ldquoTrdquo stands for matrix transposition In additionfor the sake of simplicity we denote 120597119867120597119909 by nabla119867

2 Problem Description and Transformation

The physical model of the magnetic levitation train systemwhich includes the concentrated mass of train carriages(together with the supporting magnet) suspended on therigid lead rail is shown in Figure 1 where 119898 is the qualityof train carriage (including the supporting magnet) 119892 isgravitational constant 120579119872 is the gap between the supportingmagnet and the guide rail 1205790 is the gap between the guiderail and the reference plane 120579 is the distance between thesupporting magnet and the reference plane 120579119872 = 1205790 + 120579119871(120579) is the self-inductance of magnetic coil which dependson the gap 120579 119894 is the current flowing through magnet spool119877 is the coil resistance 119906 is the voltage at both ends of themagnet spool

By invoking Kirchoff rsquos voltage law and Newtons secondlaw the dynamic equations of the magnetic levitation system

Carriage

Guideway

Magnet

mg

M

0

L () i

u

R

Figure 1 Physical model of magnetic suspension system

can be obtained by taking the vertical upward direction as thepositive direction 119898 120579 = 119865 (119894 120579) minus 119898119892

119906 = 119877119894 + Φ (1)

where Φ = 119871(120579)119894 is the magnetic flux and 119865(119894 120579) is the forcecreated by the electromagnet which is given by

119865 (119894 120579) = 12 120597119871120597120579 (120579) 1198942 (2)

Here we regard the flux Φ as the independent variablethen (1) can be further transformed into the following forms120579 = V

119898V = Φ24119896 minus 119898119892Φ = 119906 minus 119877 Φ119871 (120579)

(3)

where 119896 = 120583011987321198784 1205830 is the permeability of vacuum119873 is coil turns and 119878 is the effective pole area of theelectromagnetic coil

To obtain a port-controlled Hamiltonian model we takea suitable approximation for the inductance is 119871(120579) =2119896(120579119872 minus 120579) As is known the speed of the rigid body willbe affected by stochastic disturbances during the operationof the magnetic levitation system Let 119909 = [Φ 120579119898 120579]T =[1199091 1199092 1199093]T due to the influence of stochastic disturbancethe magnetic levitation system (3) can be modeled as thefollowing algebraic differential equations

d1199091 = minus1198771199091 (120579119872 minus 1199092)2119896 d119905 + 119906d119905 + 1199093d119908 (119905) d1199092 = 1199093119898 d119905d1199093 = 119909214119896d119905 minus 119898119892d119905

(4)






119867(119909) = 119909214119896 (120579119872 minus 1199092) + 1211989811990923 + 1198981198921199092 (6)



where

J = (0 0 00 0 10 minus1 0)

R = (119877 0 00 0 00 0 0)

g1 = (100)

g2 (119909) = (119909300 )

(8)






lim119905rarrinfin

E 1003817100381710038171003817119909 (119905) minus 119909lowast10038171003817100381710038172 = 0 (10)




d119909 (119905) = 119891 (119909 (119905)) d119905 + 119892 (119909 (119905)) d119908 (119905) forall119905 ge 0 (11)



inf119905ge0

119881 (119909 119905) = infin (12)


L119881 = 12 tr119892T (119909 (119905)) (1205972119881)1205971199092 119892 (119909 (119905)) + 120597119881120597119905+ 120597119881120597119909 119891 (119909 (119905))

(13)


119864 119881 (119909 (1205912) 1205912) minus 119881 (119909 (1205911) 1205911)= 119864int12059121205911

L119881 (119909 119905) d119905 (14)





119881 (119909) = 119867 (119909) minus 119867 (119909lowast) ge 0 (16)



119867(119909) ge 120572 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (17)



that

d119881 (119909) = L119881 (119909) d119905 + nabla119881 (119909) g2 (119909) d119908 (119905) (19)

where


(20)




(21)






(23)





119864 119881 (119905) minus 119864 119881 (0) = int1199050

119864 L119881 (119904) d119904le int1199050






that is





lim119905rarrinfin

119864 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 = 0 (32)





119906 = minus [12 (120582 + 1) 120583 + 1] 11990912119896 (120579119872 minus 1199092) (33)



05 1 15 2 250time (s)

x3

minus002

0

002

004

006

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01ve

loci

ty (c

mlowastＭminus

1)


05 1 15 2 250time (s)

minus005

0

005

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015

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t (cm

)

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119906 = minus50601199091 (120579119872 minus 1199092) (34)


5 Conclusion




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









119867(119909) = 119909214119896 (120579119872 minus 1199092) + 1211989811990923 + 1198981198921199092 (6)



where

J = (0 0 00 0 10 minus1 0)

R = (119877 0 00 0 00 0 0)

g1 = (100)

g2 (119909) = (119909300 )

(8)






lim119905rarrinfin

E 1003817100381710038171003817119909 (119905) minus 119909lowast10038171003817100381710038172 = 0 (10)




d119909 (119905) = 119891 (119909 (119905)) d119905 + 119892 (119909 (119905)) d119908 (119905) forall119905 ge 0 (11)



inf119905ge0

119881 (119909 119905) = infin (12)


L119881 = 12 tr119892T (119909 (119905)) (1205972119881)1205971199092 119892 (119909 (119905)) + 120597119881120597119905+ 120597119881120597119909 119891 (119909 (119905))

(13)


119864 119881 (119909 (1205912) 1205912) minus 119881 (119909 (1205911) 1205911)= 119864int12059121205911

L119881 (119909 119905) d119905 (14)





119881 (119909) = 119867 (119909) minus 119867 (119909lowast) ge 0 (16)



119867(119909) ge 120572 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (17)



that

d119881 (119909) = L119881 (119909) d119905 + nabla119881 (119909) g2 (119909) d119908 (119905) (19)

where


(20)




(21)






(23)





119864 119881 (119905) minus 119864 119881 (0) = int1199050

119864 L119881 (119904) d119904le int1199050






that is





lim119905rarrinfin

119864 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 = 0 (32)





119906 = minus [12 (120582 + 1) 120583 + 1] 11990912119896 (120579119872 minus 1199092) (33)



05 1 15 2 250time (s)

x3

minus002

0

002

004

006

008

01ve

loci

ty (c

mlowastＭminus

1)


05 1 15 2 250time (s)

minus005

0

005

01

015

02

025

03

disp

lace

men

t (cm

)

x2





119906 = minus50601199091 (120579119872 minus 1199092) (34)


5 Conclusion




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119867(119909) ge 120572 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (17)



that

d119881 (119909) = L119881 (119909) d119905 + nabla119881 (119909) g2 (119909) d119908 (119905) (19)

where


(20)




(21)






(23)





119864 119881 (119905) minus 119864 119881 (0) = int1199050

119864 L119881 (119904) d119904le int1199050






that is





lim119905rarrinfin

119864 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 = 0 (32)





119906 = minus [12 (120582 + 1) 120583 + 1] 11990912119896 (120579119872 minus 1199092) (33)



05 1 15 2 250time (s)

x3

minus002

0

002

004

006

008

01ve

loci

ty (c

mlowastＭminus

1)


05 1 15 2 250time (s)

minus005

0

005

01

015

02

025

03

disp

lace

men

t (cm

)

x2





119906 = minus50601199091 (120579119872 minus 1199092) (34)


5 Conclusion




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





05 1 15 2 250time (s)

x3

minus002

0

002

004

006

008

01ve

loci

ty (c

mlowastＭminus

1)


05 1 15 2 250time (s)

minus005

0

005

01

015

02

025

03

disp

lace

men

t (cm

)

x2





119906 = minus50601199091 (120579119872 minus 1199092) (34)


5 Conclusion




References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




energy-based controller design of stochastic magnetic

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