Research Article Decoupling Control Design for the …downloads.hindawi.com/journals/mpe/2015/865650.pdf · Research Article Decoupling Control Design for the ... of the module suspension

Research ArticleDecoupling Control Design for the Module SuspensionControl System in Maglev Train

Guang He Jie Li and Peng Cui

College of Mechatronics Engineering and Automation National University of Defense Technology Changsha 410073 China

Correspondence should be addressed to Jie Li ljmaglevcn

Received 26 August 2014 Revised 11 December 2014 Accepted 26 December 2014

Academic Editor Dan Ye

Copyright copy 2015 Guang He 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

An engineering oriented decoupling control method for the module suspension system is proposed to solve the coupling issues ofthe two levitation units of the module in magnetic levitation (maglev) train According to the format of the system transfer matrixa modified adjoint transfer matrix based decoupler is designedThen a compensated controller is obtained in the light of a desiredclose loop system performance Optimization between the performance index and robustness index is also carried out to determinethe controller parameters However due to the high orders and complexity of the obtained resultant controller model reductionmethod is adopted to get a simplified controller with PID structure Considering the modeling errors of the module suspensionsystem as the uncertainties experiments have been performed to obtain theweighting function of the systemuncertainties By usingthis the robust stability of the decoupled module suspension control system is checked Finally the effectiveness of the proposeddecoupling designmethod is validated by simulations and physical experimentsThe results illustrate that the presented decouplingdesign can result in a satisfactory decoupling and better dynamic performance especially promoting the reliability of the suspensioncontrol system in practical engineering application

1 Introduction

As an urban track transportation vehicle with large appli-cation prospect low speed maglev trains have been devel-oped for almost 30 years [1 2] and there have alreadybeen several commercial operation lines or test lines [3ndash8] Maglev train utilizes suspension controllers to adjustthe electromagnetic forces between the electromagnets andthe track for stable levitation Hence the electromagneticsuspension control system is the most pivotal componentof the maglev train which attracts tremendous attention[9ndash12] As of today the stability problem of the suspensioncontrol system has been basically solved The major workon suspension control is excepted be transferred to theperformance promotion and the practical problem existingin engineering applications On the basis of the mechanicaldecoupling in bogie levitation modules can be consideredas the foundational elements of the low speed maglev trainAt present the main existing suspension control methodsdecompose the module into two single-suspension-controlunits However due to the physical stiffness structure of

levitation module direct coupling between the two single-suspension-control units will attenuate disturbance rejectioncapability of the levitation control system and also becomesa serious obstacle to the performance promotion To someextent the adjustment of one levitation unit may destabilizethewholemodule suspension systemTherefore it is essentialto develop some decoupling control strategy for the modulesuspension control system

By viewing the module as an integrated object thesuspension control system is a two-input-two-output (TITO)control system The engineering oriented research on thedecoupling control of the module suspension system hasrarely been reported Fortunately considerable efforts havealready been devoted to the decoupling control of themultiple-input-multiple-output (MIMO) system for severaldecades Different control strategies have been developed toovercome the complicated couplings between control loopssuch as inverse Nyquist array [13] internal model control[14] inverse based decoupling control [15] and other decou-pler based methods Those methods can allow parameterperturbation and uncertainties in system model with robust

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 865650 13 pageshttpdxdoiorg1011552015865650

2 Mathematical Problems in Engineering

requirement which benefits the decoupling design of themodule suspension system Besides although the differentialgeometry technique is also a feasible approach to deal withthe multivariable decoupling control problems [16 17] theirneed for precise mathematical model is an obstacle to applyit to practical engineering The general decoupling controlapproach is to design the decoupler so that theMIMOcontrolsystem can be treated as multiple single-input-single-output(SISO) loops which allows us to use well developed singleloop controller design methods Many decoupling controldesign methods are developed based on this view as well [18ndash22] The ideal decoupler is to be designed as the inverse ofthe transfer functionmatrix However this kind of decouplerneeds to calculate the inverse of the process transfer functionmatrix resulting in too complicated calculation Shen et alconsidered the adjoint transfer matrix of the original mul-tivariable system as the decoupler [23] where it can avoidcomplicated computation especially for TITO system

In this paper a modified adjoint transfer matrix baseddecoupler was presented First the existing coupling inthe module suspension control system is analyzed and thedynamic model is also given By adopting the modifiedadjoint transfermatrix as the decoupler we divide themodulesuspension control system into two independent SISO con-trol systems Then compensated controllers are designed tomeet the desired loop performance and robustness demandof the module suspension control system The formulationof a resultant decoupling controller is obtained by combingthe decoupler and the compensated controller MultivariablePID structure controller is the most effective technology inengineering applications because of adequate performancewith simple structure [18 19 24 25] which is also adoptedin our practical CMS04 low speed maglev train Hencethe resultant decoupling controller is transformed to PIDtype controller by model reduction Given the parametersuncertainties and nonlinear characteristic in the magneticsuspension system the modeling errors between the lin-earized model and practical physical model have been takeninto accountThemodeling errors aremeasured by frequencysweeping experiments on a real full-scale single bogie ofCMS-04 maglev train based on which the robust stability ofthe decoupledmodule suspension control system is validatedFurthermore simulations and experimental results show thatthe proposed decoupling method can be well applied inthe module suspension system and promote the suspensioncapability in practical engineering application

The rest of the paper is organized as followsThe couplinganalysis and dynamic model of the module suspension sys-tem are given in Section 2 Section 3 describes the decouplingdesign procedure in detail The case study in Section 4 isto determine the parameters of the designed decouplingcontroller Simulations and experiments are presented in thissection Finally Section 5 gives the conclusions of this paper

2 Coupling Analysis and Modeling ofthe Module Suspension System

The low speed electromagnetic suspension (EMS) vehi-cle consists of cabin body levitation bogies secondary

Electromagnets Bogie

TrackCar body

Air spring Buttress

Figure 1 Lateral view of the CMS04 low speed maglev vehicle

Electromagnet

Sensor

xy

z

Box beam

LIM stator

PWM

PWM

Controller 1

Centralized MIMOdecoupling controller

Controller 2

Decentralized SISOcontroller

Air spring supporting point

Figure 2 Schematic of the module suspension control system

suspensions and levitation and guidance magnets A lateralview of the CMS04 low speed maglev vehicle is shownin Figure 1 where it can be founded that the car body issupported by five bogies with each bogie consisting of twolevitation modules Each module contains two pairs of adja-cent electromagnets which are controlled by decentralizedSISO controllers thus there are a total of four levitation unitsin a bogie

As the bogie is the pivotal component of the maglevvehicle analyzing the coupling issue of the bogie is essentialto design decoupling methods The coupling issue of thebogie was investigated through static experiments in theCMS-04 low speed maglev vehicle in [26] from which itcan be founded that the coupling between the two levitationunits in amodule ismuch stronger ([26] Figure 3)Thereforethe paper focuses on the coupling between the two levitationunits in one levitation module The schematic of the modulesuspension control system is shown in Figure 2 where it givestwo kinds of controller construction the decentralized SISOcontroller and the centralized MIMO decoupling controllerThe former is the common control methodology in maglevsystem It utilizes only the sensors message of one levitationunit to realize its stable levitation which leads to the fact thatadjusting the movement of one levitation control unit affectsthe performance of the other one because of themodulersquos stiffstructure To cope with this problem a centralized MIMOdecoupling control scheme described in Figure 2 is presentedand will be discussed in detailThis kind of controller can useboth sensors messages of the levitation units in a module toproduce appropriate control laws

Mathematical Problems in Engineering 3

Generally in the dynamic modeling of the levitationmodule only that the degrees of freedom in pitch andin vertical direction are considered With the purpose ofengineering application some assumptions can be madebelow when developing the dynamic model of the levitationmodule [27]

(1) The mass distribution of the levitation module ishomogenous and the gravity centre of levitationmodule coincides with its geometrical centre Thetrack is considered to be stiff so the flexible distortionof the track can be neglected

(2) The magnetic leakage and edge effect of the electro-magnet are neglectedThat is to say the totalmagneticpotential during the levitation gap is distributedevenly in the 119909 direction

(3) The uniformly distributed electromagnet force can beequated with two concentrated forces acting on thecentre of levitation units in one levitation moduleBesides the force transferred from air spring applieson the measuring point of the gap sensor in the 119910direction

Based on the assumptions above the force diagram ofthe levitation module in lateral is given in Figure 3 119865

119894is

the electromagnet force of the 119894th levitation unit 119911119894is the

gap between the track and the module at the electromagnetforce acting spot of the levitation unit 119894 119873

119894is the force of

the 119894th levitation unit transferred from air spring 120575119894is the

measured gap values of the gap sensors 120579 is the pitch angle ofthe levitation module 119897 is the length of the electromagnet ofone module 119888 is the gap between the centre of the levitationmodule and the track 119898 is the total mass of one levitationmodule

The geometrical relationship of the levitation module isgiven as follows

119888 =

(1205751+ 1205752)

2

120579 =

(1205751minus 1205752)

119897

1199111=

(31205751+ 1205752)

4

1199112=

(1205751+ 3120575

2)

4

(1)

Themotions of the levitationmodule contain the rotationaround themass centre119874 in the119909minus119911 plane and themovementin the vertical direction According to Newtonrsquos law themotion equation of the levitation module is described asfollows

119898119892 minus 1198651minus 1198652+ 119873

1+ 119873

2= 119898 119888

1198652

119897

4

sdot cos 120579 minus 1198651

119897

4

sdot cos 120579 + 1198731

119897

2


119897

2

sdot cos 120579 = 119869 120579(2)

where 119869 is the rotary inertia of the levitation module in thepitch direction 119865

119894= 119896

119890119868119894119911119894 119896119890= 120583

011987321198602 119868

119894is the coil

current of the electromagnets of the 119894th levitation unit 119873is the number of turns of electromagnet winding 119860 is thepole area 120583

0is the permeability of vacuum and 119892 is the

acceleration due to gravity

Track

Levitation moduleN1

1205751 z1 c z2 1205752 N2

O

F1 F2120579

l

Figure 3 The force diagram of one levitation module in lateral

The relationship between the current and the voltage ofthe electromagnets is derived as

119906119894= 119877119868

119894+

2119896119890

119911119894

119868119894minus

2119896119890

1199112

119894

119868119894119894 119894 = 1 2 (3)

where 119906119894and 119877 are the voltage and direct current (DC)

resistance of the electromagnet of the 119894th levitation unitrespectively

Equations (1)sim(3) describe the dynamic behaviour of thelevitation module However the magnetic levitation systemhas been pointed to be inherently unstable without activecontrol In this paper the decoupling control methods aredeveloped on the basis of that the suspension system isalready stable Hence an additional controlling force iscommonly used to stabilize the single levitation unit Herethe PD controller is adopted and the feedback control law isgiven as follows

119868119890119894= 119896119901(120575119894minus 119903119889minus 1205750) + 119896

119889120575119894 119894 = 1 2 (4)

where 119896119901and 119896

119889are the proportional and derivative control

coefficients 119868119890119894

is the desired current 1205750is the desired

levitation gap and 119903119889represents the track disturbances Here

the desired levitation gaps of the two levitation units in onemodule are uniform in steady working condition Let 119868

1198940

denote the steady current and it obtains (when the terms inthe left hand side of (2) are equal to zero)

11986810= radic

(119898119892 + 31198731minus 119873

2) 1199112

0

12058301198732119860

11986820= radic

(119898119892 + 31198732minus 119873

1) 1199112

0

12058301198732119860

(5)

Given the inductance of the electromagnets the currentloop which adopts a proportional control law to adjust thecontrol voltage for achieving the desired current quickly isgiven [28]

119906119894= 119896119888(119868119890119894minus 119868119894) 119894 = 1 2 (6)

where 119896119888is the forward gain of the current loop

The dynamic model of the module suspension controlsystem with decentralized SISO controllers is determinedby (1)sim(6) On the basis of the stable single suspensioncontrol units the decoupling control scheme will be addedto complete the function of decoupling the two levitation


units of one module By this way clear comparisons betweenthe single levitation unit control method and the decouplingmodule suspension control method can be accomplishedThis will be carried out in the following sections

3 Decoupling Control Design

31 Decoupling Methodology In this paper an adjoint matrixbased decoupling control scheme is adopted to provide asimple alternativemethod for practical control engineersThemodule suspension control system is a TITO system and weconsider the TITO process as 119884(119904) = 119866(119904)119880(119904) or

[

1199101 (119904)

1199102 (119904)] = [

11989211 (119904) 11989212 (

119904)

11989221 (119904) 11989222 (

119904)] [

1199061 (119904)

1199062 (119904)] (7)

where 119866(119904) is strictly proper and stable transfer function andcan quantify the proper input-output pairing with diagonaldomination

This section is to design a decoupler119870119889(119904) and a diagonal

compensated controller119862(119904) which guarantees that the resul-tant closed-loop transfer function is stable and decoupledThe ideal decoupler is the inverse of the transfer functionmatrix 119866minus1(119904) However the calculation of 119866minus1(119904) is toocomplicated especially for the system with high dimensionswhichmake it difficult for practical engineering implementa-tion In fact it is clear that the adjoint matrix can be writtenout easily withoutmuch computation burdenThus it ismorepreferable to select the adjoint matrix as the decoupler

119870119889 (119904) = adj119866 (119904) (8)

As for the module suspension control system the lin-earized model around the equilibrium point can be appliedto obtain the nominal transfer function matrix Under thenormal working condition the pitch angle 120579 is smallerthan 05∘ so we can suppose that the cosine of 120579 equals 1approximately in (2) The linearized model is given by

minus2 (1198651+ 1198652) = 119898

1(1205751+1205752)

(

1

4

1198652minus

1

4

1198651) 1198972= 119869 (

1205751minus1205752)

119868119894=

1199110119894

2119896119890

119906119894minus

1198771199110119894

2119896119890

119868119894+

1198680119894

1199110119894

119894

119865119894= 119865119911119911119894minus 119865119868119868119894

119906119894= 119896119901119896119888(120575119894minus 119903119889) + 119896

119889119896119888120575119894

(9)

where 119877 = 119877 + 119896119888 119865119868119894= 2119896

11989011986811989401205752

0 and 119865

119911119894= 2119896

1198901198682

11989401205753

0

During the operation of the maglev train the trackdisturbances have the most significant effect on the suspen-sion performance By omitting the Laplace operator it givesthe transfer function matrix 119866(119904) from 119903

119889to 120575 with SISO

controllers as follows

119866 = 1198660(119868 + 119866

0)minus1 (10)

1198660= 119866

1119866211986651198666+ 119866

111986621198664+ 119866

11198663 (11)

where the meanings of 1198661sim6

can be seen in the AppendixHence the adjoint matrix based decoupler can be rewrittenas

119870119889 (119904) = adj (119866

0 (119904) (119868 + 1198660 (

119904))minus1) (12)

From (11) it is found that the expression of 1198660(119904) is so

huge that the calculation of (119868 + 1198660(119904))minus1 in (12) is still very

complicated Hence we modify the decoupler as follows

119870119889 (119904) = (119868 + 1198660 (

119904)) adj1198660 (119904) (13)

In practice it should be noticed that the format ofdecoupler 119870

119889(119904) in (13) makes its calculation simpler while

the coupler119870119889(119904) contains pure integral terms in the denom-

inators The existence of the pure integral terms in decouplerwill demolish the internal stability of the whole systemTherefore the problem at hand is to improve the decoupler119870119889(119904) such that the pure integral terms can be discarded

Inspired by [29] we make a further modification of thedecoupler 119870

119889(119904) as follows

119870119889 (119904) = (119868 + 1198660 (

119904)) adj1198660 (119904) 119866119886 (119904) (14)

Here119866119886(119904) = diag119892

119886119894(119904) is diagonal stable transfer function

matrix and its elements are designed in the followingexpression

119892119886119894 (119904) =

119904119903119894

(120582119894119904120596119899119894+ 1)

119903119894 119894 = 1 2 (15)

where 119903119894is the maximum order number of the integral terms

included in 119894th column elements of the transfer function(119868 + 119866

0(119904))adj119866

0(119904) The choice of (120582

119894119904 + 1)

119903119894 is to ensurethat the elements of the decoupler 119870

119889(119904) are proper and thus

physically realizable Meanwhile according to the expressionof 119892

119886119894(119904) it can be found that the poles brought by 119892

119886119894(119904)

will affect the practical dynamic performance of the closedloop system And the specification of the parameter 120582

119894will be

discussed in the following procedure according to the desiredclosed loop performance When the decoupler 119870

119889(119904) acts on

the original model the decoupled apparent transfer functionmatrix is diagonal and is obtained as follows

119876 (119904) = 119866 (119904)119870119889 (119904) = det119866

0 (119904) sdot 119866119886 (

119904) (16)

After decoupling the obtained decoupled transfer func-tion may not satisfy the desired dynamic performance Evenworse the decoupling may destabilize the whole modulesuspension control system Thereby after the decoupler isdetermined an extra diagonal compensated controller isexpected to be added to ensure an acceptable dynamicsystem performance The block diagram of the proposeddecoupling feedback control system is given in Figure 4which consists of a decoupler 119870

119889(119904) described in (14) and a

diagonal compensated controller 119862(119904)This scheme is equivalent to a centralized multivariable

controller 119870(119904) Once the decoupler 119870119889(119904) and the diagonal

elements of the compensated controller 119862(119904) are designed


r(s)C(s) Kd(s)

K(s)

Q(s)G(s) = G0(s)(I + G0(s))

minus1y(s)

minus

Figure 4The block diagramof decoupling feedback control system

the resultant decoupling controller 119870(119904) is obtained by thefollowing equations

119870 (119904) = 119870119889 (119904) 119862 (119904)

119896119894119895 (119904) = 119888119894119894 (

119904) 119870119889119894119895(119904) 119894 119895 = 1 2

(17)

According to Figure 4 the closed loop transfer functionmatrix of the decoupled module suspension system is deter-mined as

119867(119904) = 119866 (119904)119870119889 (119904) 119862 (119904) [119868 + 119866 (119904)119870119889 (

119904) 119862 (119904)]minus1

= det1198660 (119904) 119866119886 (

119904) sdot 119862 (119904) [119868 + det1198660 (119904) 119866119886 (

119904) sdot 119862 (119904)]minus1

= diag ℎ11 (119904) ℎ22 (

119904)

(18)

Note that 119867(119904) and 119862(119904) are diagonal By modifying theexpression of (18) the compensated controller 119862(119904) is writtenas follows

119862 (119904) = (119867 (119904)minus1minus 119868)

minus1

sdot det1198660 (119904) 119866119886 (

119904)

= diag ℎ11 (119904)

(1 minus ℎ11 (119904)) sdot 1199021 (

119904)

ℎ22 (119904)

(1 minus ℎ22 (119904)) sdot 1199022 (

119904)

(19)

From (19) once the desired diagonal elements of theclosed loop transfer functionmatrix119867(119904) are determined theelements of the controller 119862(119904) can be ascertained Withoutloss of generality the 119894th decoupled closed loop transferfunction with the undetermined parameters can take theform of [30]

ℎ119894119894 (119904) =

1205962

119899119894

(1199042+ 2120596

119899119894120585119894119904 + 120596

2

119899119894) ((120582

119894120596119899119894) 119904 + 1)

119897119894

prod119898119894

119896=1(119904 + 119911

119896)

prod119902119894

119895=1(119904 + 119901

119895)

119894 = 1 2

(20)

where 120582119894 120585119894 and 120596

119899119894are the adjustable parameters Here the

explanation of the parts of (20) is given Usually the concep-tion of dominant poles is used to specify the parameters ofhigh-order control system The standard 2nd order transferfunction locating a pair of conjugation dominant poles inthe left half of the complex plane represents the dynamicperformance requirement

The term (120582119894119904120596119899119894+ 1)

119897119894 emerges here since it guaranteesthe properness and stability of the decoupler 119870

119889(119904) To make

sure that the elements of resultant centralized controller119870(119904) are strictly proper and physical realizable denoting that119903119889(119866(119904)) is the relative degree of the transfer function 119866(119904)the parameter 119897

119894should satisfy the following relationship

119897119894= max 0 119903119889 (det (119866

0 (119904)))

minusmin 119903119889 (11986611989410(119904)) 119903119889 (119866

1198942

0(119904))

+ 119903119894+ 119898

119894minus 119902119894minus 1 119894 = 1 2

(21)

The parameter 120582119894has an effect on the bandwidth and

the stability margin of the 119894th loop transfer function andalso decides the cut-off frequency in high frequency domainThoughwe can specify the parameter120582

119894to be arbitrarily small

such that the pole brought by (120582119894119904120596119899119894+ 1)

119897119894 can be far awayfrom the dominant poles it will result in a bigger overshot indynamic performance as well

At last the residual part of ℎ119894119894(119904) represents the inherent

characteristic of the original process obtained by stablepole-zero cancellation According to the analysis procedureabove the decoupler 119870

119889(119904) can be obtained by (14) and

the compensated controller 119862(119904) is determined based on(19) when the desired diagonal elements of the closed looptransfer function matrix 119867(119904) are given Then the resultantcontroller 119870(119904) is obtained by (17) Though the proposeddecoupling design is based on TITO system it also providesan alternative solution to unstable MIMO system with highdimensions

32 Practical Considerations on Controllers and Robust Sta-bility Analysis Following the procedure given above thedecoupler 119870

119889(119904) is obtained to decouple the module sus-

pension system to two SISO loops and the compensatedcontroller 119862(119904) is to guarantee the stability and performanceof each loopHowever we adopted the linearizedmodel as thenominal system in the procedure of the decoupling analysisVarious sources of disturbances and unmodeled dynamicsand nonlinearities may cause parametric uncertainties inthe practical control system Hence the specifications ofthe controller parameters should take the robustness toparametric uncertainties and set-point tracking capacity intoconsiderationThe common criterion to measure the robust-ness to process uncertainties is the maximum sensitivity Theoutput sensitivity function and the complementary sensitivityfunction are defined as follows

119878 (119904) = (119868 + 119866 (119904)119870 (119904))minus1

119879 (119904) = 119866 (119904)119870 (119904) (119868 + 119866 (119904)119870 (119904))minus1

(22)

It is obvious that 119878(119904) is diagonal and for the 119894th loop wecan use

119872st119894 = max120596(1003816100381610038161003816119878119894119894(119895120596)

10038161003816100381610038161003816100381610038161003816119879119894119894(119895120596)

1003816100381610038161003816) 119894 = 1 2 forall120596 isin 119877 (23)

as the robustness index [31] The reasonable values of119872119904are

in the range between 12 and 20 A smaller value of 119872119904is

preferred to guarantee the robustness Actually robustness is


usually achieved at the expense of system performance in thenormal working condition which implies that compromisebetween robustness and system performance has to be foundin practical application

For the magnetic suspension system rapid response totrack disturbance is crucial for good dynamic performancewhich means that smaller overshoot and setting time aredemanded In this paper the system performance of thecontrol system is evaluated by calculating the integratedabsolute error (IAE) due to a unit step load disturbance TheIAE index for the 119894th loop is defined as

IAE = intinfin

0

1003816100381610038161003816119903119894 (119905) minus 119910119894 (

119905)1003816100381610038161003816119889119905 119894 = 1 2 (24)

The controller with smaller IAE is considered to have a bettercontrol performance here

From the decoupling design procedure the controller119870(119904) is determined by three parameters 120582

119894 120585119894 and 120596

119899119894 The

determination of those parameters is based on the trade-offbetween robustness and system performance In this workplots are introduced to show the relationship between systemperformance and robustness in regard to the controllerparameters which will be illustrated in Section 41

The decoupling design for the module suspension systemis given in Section 31 and it is obvious that the orders ofthe elements of the resultant controller 119870(119904) are so highHence it is difficult to implement the controller in physicsIn engineering practice it is an advisable way to solve thisproblem by approximating their elements with a reduced-order model However it is difficult to obtain a perfectreduced-order controller whichmatches the original one wellin the whole frequency domain In practice we can requirethat the reduced controller matches the original one in adesired domain of 119904 isin 119863

120596119894≜ 119904 isin 119862 | 119904 = 119895120596 120596

119904119894le 120596 le 120596

119898119894

The value of 120596119898119894

should be well chosen to be large enoughsuch that it is well beyond the system bandwidth [31] In thispaper the frequency domain is chosen as the values between120596119898119894= 120596

11990211989410 and 120596

119898119894= 10120596

119902119894 and 120596

119902119894is the 0 db crossover

frequency of the 119894th loop transfer function 119902119894 The elements

of the resultant decoupled controller119870(119904) have the followingformat

119870119894119895 (119904) = 119860 (119904) =

119887119898119904119898+ sdot sdot sdot + 119887

1119904 + 119860 (0)

119886119899119904119899+ 119886119899minus1119904119899minus1

+ sdot sdot sdot + 1198861119904 + 1

(25)

Assume that the frequency response 119860(119895119908119896) can be

approximated by a reduced-order model 119860119903(119904) with orders

120574 and 119896 in the numerator and denominator respectively Theapproximation rational transfer function is described by (26)with identical stationary gain

119860119903 (119904) =

119873 (119904)

119863 (119904)

=

120573119903119904119903+ sdot sdot sdot + 120573

1119904 + 119860 (0)

120572119896119904119896+ 120572119896minus1119904119896minus1

+ sdot sdot sdot + 1205721119904 + 1

(26)

Denoting that 120579 = (1205721 120572

119896 1205731 120573

119903) is the undeter-

mined parameters vector the proposedmodel approximation

method is to search the optimal parameters to satisfy thefollowing objective function 119869 [32 33]

119869 = min119904isin119863120596119894

1

119873

119873

sum

119896=1

[1003816100381610038161003816119882 (119895119908

119896)1003816100381610038161003816

2sdot1003816100381610038161003816119860 (119895119908

119896) minus 119860

119903(119895119908

119896 120579)1003816100381610038161003816

2]

119882 (119895119908119896) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119863 (119895119908119896 120572119894)

119863 (119895119908119896 120572119894minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(27)

where 119882(119895119908119896) is a weighting function From the above

when the order of the approximation model is chosenthe parameterization for the reduced-order model 119860

119903(119904) is

determined if themean squared error is under preestablishedtolerance

After the model reduction the resultant controller 119870(119904)may have any physically realizable structure PID structureis one of the first developed control strategies with simplestructure and well-known tuning rules which makes itmaintain dominance in practicing engineering applicationsfor several decadesThoughmore advance control algorithmshave been developed the PID controllers are always preferredunless they do not give satisfactory performance [33] Hencein this work the resultant controller119870(119904) is approximated byPID controllers A pure differential term in PID controllerwill cause infinite high frequency gain Also it is undesirableand impossible to realize such a controller In view of thata second order low-pass filter is introduced to replace thecommon procedure of the pure differential terms in thePID controller The second order low-pass filter is shown asfollows

119870119891 (119904) =

1

1 + 119904119879119891+ 1199042(1198792

119891119873)

(28)

Then the PID controller structure used in this paper iswritten as

119870PID (119904) = 119870119891 (119904) (119870119901 +119879119894

119904

+ 119879119889119904)

=

1198791198891199042+ 119870

119901119904 + 119879

119894

119904 (1 + 119904119879119891+ 1199042(1198792

119891119873))

(29)

where 119870119875is the proportional gain 119879

119894is the integration

constant119870119889is the derivative constant and119879

119865is the derivative

filter constant The PID controller is obtained based on themodel reduction method mentioned above Basically theelements of the resultant controller119870(119904) contain integrator toeliminate the stationary error due to setpoint or load changesFor simplicity this paper denotes119870(119904) = 119875(119904)119904 by removingthe integrator Applying the approximation method to theinverse of 119875(119904) the approximation model can be obtained asfollows

119875 (119904) asymp

12057321199042+ 1205731119904 + 119875 (0)

12057221199042+ 1205721119904 + 1

(30)


According to (29) and (30) the PID controller parametersare given as follows

119870119901= 1205731 119879

119894= 119875 (0) 119879

119889= 1205732

119879119891= 1205721 119873 =

1205722

1

1205722

(31)

Since an approximation is adopted to transform the idealdecoupling to the PID structure the robust stability needsto be checked when uncertainties exist in the model ofthe controlled levitation module Assume that the practicalmodule suspension control system has an multiplicativeuncertainty compared with the linearized model 119866(119904) thenit is presented as

119866119901 (119904) = 119866 (119904) (119868 + Δ (119904)) (32)

A rational weighting function which represents theuncertainties of the module suspension control system isgiven as follows

1003816100381610038161003816119882119860(119895120596)

1003816100381610038161003816ge max119866119901isinΛ

120590 (119866minus1(119895120596) [119866

119901(119895120596) minus 119866 (119895120596)]) (33)

where Λ is a set containing all possible real model 119866119901(119904)

Hence the robust stability criterion for the close loop systemis obtained as [34]

120590 (119872(119895120596)) lt 1 forall120596 isin [0infin) (34)

119872(119904) = minus119882119860 (119904) 119870PID (119904) [119868 + 119870PID (119904) 119866 (119904)]

minus1 (35)

In this work the weighting function of the systemuncertainties is obtained by frequency sweeping experimentson the full-scale single bogie of CMS-04 maglev train Theexpression of the weighting function is given in Section 42By this way we can calculate and plot the left hand side of(34) and then compare it with the unity to see whether robuststability is satisfied

4 Case Study

41 Parameters Specification This section gives parametersspecification procedure of the resultant decoupling controller119870(119904)The related calculation is based on the actual parametersof the CMS04 low speed maglev train which are shown inTable 1

According to the design procedure of the decoupler119870119889(119904) it is not necessary to calculate the transfer func-

tion matrix of the module suspension control system 119866(119904)Instead the transfer function matrix 119866

0(119904) is obtained by

using the parameters listed in Table 1

1198660 (119904) =

[

[

[

[

682713 (119904 + 4015)

1199042(119904 + 1135)

55329 (119904 minus 3285)

1199042(119904 + 1135)

55329 (119904 minus 3285)

1199042(119904 + 1135)

682713 (119904 + 4015)

1199042(119904 + 1135)

]

]

]

]

(36)

Table 1 Parameters of the module suspension control system

Property Value119898 1020 kg11987311198732

10 KN119860 00186m2

119897 265m119869 910 kgsdotm2

119873 320119896119888

3001205830

4120587 times 10minus7 Hm119896119901

4000119896119889

331205750

9mm119877 10Ω

From the expression of1198660(119904) it is seen that the nondiago-

nal elements have one nonminimum phase zero So the idealinverted decoupling method may lead to RHP poles in thedecoupler This can be avoided by using the decoupler givenin (14) The decoupler is calculated as follows

119870119889 (119904) = (119868 + 1198660 (

119904)) adj1198660 (119904) 119866119886 (119904)

= [

1198961198891111989611988912

1198961198892111989611988922

]

11989611988911

=

6827131199041199031(119904 + 1074) (119904 + 1524) (119904

2+ 8604119904 + 3768)

1199044(119904 + 1135)

2(12058211199041205961198991+ 1)

1199031

11989611988912=

minus553291199041199031(119904 minus 3285)

1199042(119904 + 1135) (1205822

1199041205961198992+ 1)

1199032

11989611988921=

minus553291199041199031(119904 minus 3285)

1199042(119904 + 1135) (1205821

1199041205961198992+ 1)

1199031

11989611988922

=

6827131199041199032(119904 + 1074) (119904 + 1524) (119904

2+ 8604119904 + 3768)

1199044(119904 + 1135)

2(12058221199041205961198992+ 1)

1199032

(37)

Since the maximum order numbers of the integral termsincluded in the column elements of the transfer function (119868 +1198660(119904))adj119866

0(119904) are both 4 we choose 119903

1= 1199032= 4 After the

decoupler119870119889(119904) is determined the apparent transfer function

matrix of the module suspension control system is decoupledto be two SISO loops and its elements 119902

119894(119904) can be given as

follows

119902119894 (119904) =

466 times 109(1199042+ 8072119904 + 9026)

(119904 + 1135)2(120582119894119904120596119899119894+ 1)

4 119894 = 1 2 (38)

It shows that the zeros and poles of 119902119894(119904) are in the left

side of the complex plane To simplify the structure of


02 03 04 05 06 07 08 09 11

12

14

16

18

2

22

24

26

28

3

120585

Mst

(a) Plots of robustness index (119872st)

02 03 04 05 06 07 08 09 1004

0045

005

0055

006

0065

007

0075

008

0085

009

IAE

120585

120582 = 001

120582 = 005

120582 = 01

(b) Plots of performance index (IEA)

Figure 5 Robustness index (119872st) and performance index as functions of the parameters 120585 and 120582

closed loop transfer function 119867(119904) those dynamics can bediscarded by stable pole-zero cancellation According to (21)the parameter 119897

119894is chosen to be 1 Hence the compensated

controller 119862(119904) is given as follows

119888119894119894 (119904) =

1205962

119899119894

(1199042+ 2120596

119899119894120585119894119904 + 120596

2

119899119894) (120582

119894119904120596119899119894+ 1) minus 120596

2

119899119894

sdot

(119904 + 1135)2(120582119894119904120596119899119894+ 1)

4

466 times 109(1199042+ 8072119904 + 9026)

119894 = 1 2

(39)

Then the resultant controller 119870(119904) is obtained through(17) However the three controller parameters 120582

119894 120585119894 and

120596119899119894

are still not determined So the work at hand is toanalyze the relationship among the controller parameters andthe indexes of robustness and system performance whichhelps us to specify the controller parameters Normally thelevitated units of one module are supposed to have thesame performance so the two diagonal elements of thecompensated controller 119862(119904) are equal Here the subscriptof the parameters is discarded in the following parametersspecification procedure In Section 32 we choose 119872st andIAE as the evaluation indexes of robustness and systemperformance respectively The situation of three controllerparameters is more complicated because it needs threedimensions plots to express the relationships Through themultiple plots we find that the variation of 120596

119899does not affect

the robustness index 119872st described in (23) so we give theplots of robustness index (119872st) and performance index (IAE)as functions of the parameters 120585 and 120582 by setting 120596

119899as a

constant firstlyFigure 5(a) shows the plots of robustness index (119872st)

as function of the parameters 120585 and 120582 and Figure 5(b)shows the plots of performance index (IAE) as function of

the parameters 120585 and 120582 Figure 5(a) shows that the robustnessindex119872st decreases when the parameter 120585 increases When120585 is bigger than almost 03 119872st will be in the reasonablerange between 12 and 20 Besides Figure 5(b) indicates thatthe minimization of IAE will be obtained when 120585 is 066which also gives acceptable robustness For the correlationamong IAE119872st and 120582 IAE and119872st both decrease when theparameter 120582 decreases Once we choose 120585 as 066 a smallervalue of 120582 means a promotion of both the performance andthe robustness

Next we give the plots of robustness index (119872st) andperformance index (IAE) as functions of the parameters 120596

119899

and 120582 in Figures 6(a) and 6(b) when 120585 is 066 The plots inFigure 6(a) show that the level curve of 119872st is horizontalwhich means that robustness index will not be influencedby the variation of the parameter 120596

119899 Figure 6(b) gives the

conclusion that the performance index (IAE) decreases whenthe parameter 120596

119899increases Just like Figure 5 though the

influence caused by the parameter 120582 is much smaller theplots also indicate that robustness index119872st and performanceindex (IAE) will be smaller by setting 120582 on a lower valuewhen the parameters 120585 and 120596

119899are fixed Actually the peak

time is also important for the suspension control systemThe mentioned current loop method above is to shortenthe response time of the electromagnets It is not expectedthat the decoupling design destroys the original intentionof the current loop So the specification of the parametersmust satisfy the demand of the peak time Besides it isobvious that a smaller 120582 and a bigger 120596

119899can both benefit

the robustness and performance while it also give a highercontrol gain For themeasurement noise high gainmeans toolarge control actions So a natural way to solve this problemis to choose acceptable values of 120582 and 120596

119899by considering the

measurement noise


0 5 10 15 20 25 30 35 40 45 501

105

11

115

12

125

13

135

14M

st

120596n


0

005

01

015

02

025

03

IAE

0 5 10 15 20 25 30 35 40 45 50

120582 = 001

120582 = 005

120582 = 01

120596n


Figure 6 Robustness index (119872st) and performance index (IAE) as functions of the parameters 120596119899and 120582

In brief from the plots which present the relationshipamong the robustness index performance index and thethree controller parameters we determine the parameters 120582120585 and 120596

119899as follows

(1) The parameter 120585 = 066 gives an acceptable robust-ness as well as minimization of IAE

(2) The parameter120596119899does not affect the robustness index

119872st It is only determined by the trade-off betweenthe peak time and acceptable control gain for themeasurement noise In practical physical controllerthe filter circuit is already designed to make themeasurement noise in an acceptable degree Denotethat the peak time 119905

119903lt 01 s then we get 120596

119899gt 40

Though a much bigger value of 120596119899can make the peak

time even smaller the induced high control gain willamplify actuator actions caused by the measurementnoise So we choose the parameter 120596

119899as 40

(3) Once the parameters of the standard 2nd ordersystem are specified the parameter 120582

119894can serve as

a monotonously tuning parameter to affect dynamicperformance and robustness In this paper we choosethe desired robustness value119872st = 133 and then 120582will be 005

According to the parameters specified above the robust-ness index is 119872st = 133 and the minimization of IAE is0046 The resultant centralized controller119870PID(119904) is given inTable 2

42 Simulations and Experimental Results To verify thedecoupling effect and the performance the closed loopresponses of the nominal control system subject to a 1mmsquare wave are shown in Figure 7 The initial condition is

Table 2 PID parameters of the controller 119870PID(119904)

Location PID parameters119870119901

119879119894

119879119889

119873 119879119891

1-12-2 391 438 00743 4927 010241-22-1 429 times 10minus6 0 00215 4927 01024

that themodule levitates at a gap of 9mmand the squarewavedisturbance is added at 119905 = 0 s in the first loop and at 119905 = 1 sin the second loop In the ideal case the dynamics of the twolevitated units is decoupled perfectly with the designed highorder centralized controller119870(119904) Compared with the systemresponses generated by traditional decentralized controllersthe approximated centralized controller with PID formulacan provide a better decoupling performance Meanwhilethe peak time is smaller than the desired value of 01 s anda lower overshoot is also obtained Besides a gap disturbancewith a magnitude of 1mm is added in the first loop to showthe disturbance rejection performance at 119905 = 25 s And thesystem responses are also shown in Figure 7 It can be seenthat the disturbance rejection performance is dramaticallyimproved with the approximated centralized PID controller

In practice the carrying mass changes apparently whenpassengers get onoff Hence simulations are alsomadewhenthe mass of the maglev train changes Here it is assumed thatthe steady current of electromagnets can response to the loadchange rapidly which is already realized in CMS04 maglevtrain And the results show that it affects the decoupling of thelevitation units of the module in a very low degree Besidesunder the ideal working condition the mass of the carriageis transferred to each levitation unit through air spring onaverage However more or less difference will exist due tothe installation error in practice which causes directly that


85

9

95

10

105

Gap

1 (m

m)

0 05 1 15 2 25 3Time (s)

(a)

0 05 1 15 2 25 385

9

95

10

105

Time (s)

Gap

2 (m

m)

NondecouplingIdeal decouplingPID decoupling

(b)

Figure 7 Closed loop responses of the nominal linearized controlsystem

the steady currents of the two levitation units in one moduleare different So we give simulations by assuming that thetransformed force on the levitation unit one is 1 KN less thanthat of the others and the simulation results are shown inFigure 8 The results demonstrate that a good decouplingperformance can still be obtained Meanwhile the systemresponses and disturbance rejection performance can also beguaranteed

To evaluate the robust stability of the proposed con-trollers experiments have been performed on the full-scalesingle bogie of CMS-04 maglev train to obtain the weightingfunction of the system uncertainties in this workThe pictureof the full-scale single bogie of CMS-04maglev train is shownin Figure 9

Firstly we adopted the traditional decentralized con-trollers to make the module levitate steadily Then a sineexternal excitation is introduced into one of the levitationpoints in variable frequencies The outputs under variablefrequencies represent the frequency response of the realsuspension control system model 119866

119901(119904) In the experiments

the frequency domain is chosen from 01Hz to 50HzMany repetitive experiments have been finished under theconsiderations such as different steady operation points anddifferent tracks Through those experimental results thepossible set of real model Λ is obtained Because the twolevitation units are symmetrical here the weighting functionfrom multiple experiments is given as follows

119882119860 (119904) =

00318119904 + 04

(0031806) 119904 + 1

sdot 119868 (40)

Gap

1 (m

m)

85

9

95

10

105

0 05 1 15 2 25 3Time (s)

(a)

Gap

2 (m

m)

0 05 1 15 2 25 385

9

95

10

105

Time (s)

NondecouplingPID decoupling

(b)

Figure 8 Closed loop responses of the nominal linearized controlsystem when the transformed forces is different

Figure 9 Full-scale single bogie of CMS-04 maglev train

The weighting function 119882119860(119904) can be loosely inter-

preted as the input of each levitation unit has almost 40uncertainty in the low frequency domain and up to 60uncertainty at the high frequencies The magnitude plot ofthe maximum singular value of119872(119904) is shown in Figure 10It can be seen that the peak value of 120590(119872(119895120596)) is much lessthan the unity which indicates that the proposed controllerdesign indeed provides good robust stability

To validate the practice effect of the proposed decouplingmethod some experiments are also executed on the full-scalesingle bogie of CMS-04 maglev train Firstly we still testthe decoupling capacity when 1mm square wave disturbanceis added In the first stage the module is levitated steadilyaround the operation point 120575

0= 9mm with the designed

decoupling controllerThen a 1mm square wave disturbancelasting 2 s is added to the levitation gap of levitation unitone The curves of the levitation gap and the current areshown in Figure 11 For comparison the same experiments


0

01

02

03

04

05

06

07

Frequency (rads)

Sing

ular

val

ues

10minus1 100 101 102 103

Figure 10 The maximum singular value of119872(119904)

85

9

95

10

105Decoupling

0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 520

22

24

26

28

Time (s)

Curr

ent (

A)

Current 1Current 2

(b)

Figure 11 The response of the levitation gaps and the current whendecoupling design is adopted

with traditional SISO controllers are also carried out and theresponses of the levitation gap and the current are shown inFigure 12

When the module suspension system is controlled withthe traditional SISO controllers it is seen that the gap curve ofthe levitation unit two fluctuates to almost 02mm when thesquare wave disturbance is added into the levitation unit oneFrom the experiment curve with the proposed decouplingdesign the gap of the levitation unit two just varies less than005mmwhich shows that the decoupling design reduces thecoupling between the two levitation units by 75 Because

0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

85

9

95

10

105Nondecoupling

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Current 1Current 2

20

22

24

26

28

Curr

ent (

A)

(b)

Figure 12The response of the levitation gaps and the current whentraditional SISO controllers are adopted

the decoupling controller can receive the sensor message ofboth levitation units the current of levitation unit two can beadjusted accordingly so that the fluctuation of the levitationunit two is suppressed From the experimental results shownin Figure 11 it can be concluded that the proposed decouplingcontrol design can provide a good decoupling performance

Besides experiments assuming that one levitation unitof the module breaks down are also carried out In theexperiments we give a huge disturbance to make the levita-tion unit one collapse artificially and simultaneously recordthe dynamic response of the levitation gap of the levitationunit two The curves of the levitation gap are shown inFigures 13 and 14 In Figure 13 when the levitation unit onebreaks down the large gravity action will also destabilize thelevitation unit two because of the coupling between the twolevitation units Meanwhile the violent collision against thetracks will last a long time until the module starts to levitateagain However when the decoupling design in this paper isused in the suspension control of the module the levitationunit two can be still stable after a short adjustment though thelevitation unit one is destabilized artificially In Figure 14 thelevitation unit one can start to levitate in a short time And thelong time of the levitation procedure turns up because a slowlevitation technology is adopted in the controller to make thelevitation procedure more comfortable

5 Conclusions

In the paper an engineering oriented decoupling controllerdesign has been proposed for the module suspension control


0 05 1 15 2 25 3 35 42

4

6

8

10

12

14

16

18

Gap

(mm

)

Nondecoupling

Gap 1Gap 2

Figure 13 The response of the levitation gaps with traditional SISOcontrollers when levitation unit one breaks down

0 1 2 3 4 5 67

8

9

10

11

12

13

14

15

16

17

18

Gap

(mm

)

Decoupling

Gap 1Gap 2

Figure 14 The response of the levitation gaps with decouplingcontrollers when levitation unit one breaks down

system A modified adjoint transfer matrix based decoupleris used to make the system transfer matrix be diagonal andthen the compensated controller is designed individuallyaccording to the desired close loop system performanceThe resultant controller parameters have been determined bythe trade-offs between the performance index and robust-ness index To avoid high order of the resultant controllermodel reduction method is adopted to simplify it to be amultivariable PID controller Simulations are accomplishedto show that the designed controller has a good decouplingperformance and system response Due to the fact that thelinear model cannot include the uncertainties caused bynonlinearities and other reasons an experimental method

is used to calculate the weighting function of the systemuncertainties based on which the robust stability of theproposed controllers has been checked Finally experimentson the full-scale single bogie of CMS-04 maglev train havebeen executed to verify the effectiveness of the proposedapproach in practical physical model Compared with theexperiment results using the traditional SISO controllersthe presented decoupling design can provide satisfactorydecoupling and set-point tracking performance

Appendix

Themeanings of 119866119894 119894 = 1 sim 6 are shown as follows

1198661=

[

[

[

[

[

minus

1

1198981

minus

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

minus

1198972

8119869

]

]

]

]

]

1198662= [

minus1198651198681

0

0 minus1198651198682

]

1198663=

[

[

[

[

31198651199111

4

1198651199111

4

1198651199112

4

31198651199112

4

]

]

]

]

1198666= [

119896119888(119896119901+ 119896119889119904) 0

0 119896119888(119896119901+ 119896119889119904)

]

1198664=

[

[

[

[

[

311989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

311989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

]

]

]

]

]

1198665=

[

[

[

[

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

0

0

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

]

]

]

]

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Natural Science Foundationof China (nos 11202230 and 11302252) and National KeyTechnology RampDProgram of China (no 2012BAG07B01-13)

References

[1] L G Yan ldquoDevelopment and application of the Maglev trans-portation systemrdquo IEEE Transactions on Applied Superconduc-tivity vol 18 no 2 pp 92ndash99 2008


[2] D M Rote and Y Cai ldquoReview of dynamic stability of repul-sive-force maglev suspension systemsrdquo IEEE Transactions onMagnetics vol 38 no 2 pp 1383ndash1390 2002

[3] D F Zhou CHHansen J Li andW S Chang ldquoReview of cou-pled vibration problems in EMS maglev vehiclesrdquo InternationalJournal of Acoustics and Vibrations vol 15 no 1 pp 10ndash23 2010

[4] L Yan ldquoProgress of the maglev transportation in Chinardquo IEEETransactions on Applied Superconductivity vol 16 no 2 pp1138ndash1141 2006

[5] M Morita M Iwaya and M Fujino ldquoThe characteristics of thelevitation system of linimo (HSST system)rdquo in Proceeding of the18th International Conference on Magnetically Levitated Systemand Linear Drives pp 525ndash532 2004

[6] H-S Han ldquoA study on the dynamic modeling of a mag-netic levitation vehiclerdquo JSME International Journal Series CMechanical Systems Machine Elements and Manufacturing vol46 no 4 pp 1497ndash1501 2003

[7] T E Alberts G Oleszczuk and A M Hanasoge ldquoStablelevitation control of magnetically suspended vehicles withstructural flexibilityrdquo in Proceedings of the American ControlConference (ACC rsquo08) pp 4035ndash4040 Seattle Wash USA June2008

[8] R M Goodall ldquoDynamics and control requirements for EMSmaglev suspensionsrdquo in Proceeding of the 18th InternationalConference on Magnetically Levitated System and Linear Drivespp 925ndash934 Shanghai China 2004

[9] A E Hajjaji and M Ouladsine ldquoModeling and nonlinearcontrol of magnetic levitation systemsrdquo IEEE Transactions onIndustrial Electronics vol 48 no 4 pp 831ndash838 2001

[10] C M Huang J Y Yen and M S Chen ldquoAdaptive nonlinearcontrol of repulsive maglev suspension systemsrdquo Control Engi-neering Practice vol 8 no 12 pp 1357ndash1367 2000

[11] N F Al-Muthairi andM Zribi ldquoSlidingmode control of a mag-netic levitation systemrdquoMathematical Problems in Engineeringvol 2004 no 2 pp 93ndash107 2004

[12] L Gentili and L Marconi ldquoRobust nonlinear disturbancesuppression of a magnetic levitation systemrdquo Automatica vol39 no 4 pp 735ndash742 2003

[13] H H Rosenbrock ldquoDesign of multivariable control systemsusing the inverse nyquist arrayrdquo IEE Proceeding of ControlTheory and Application vol 116 no 11 pp 1929ndash1936 1969

[14] Q-G Wang Y Zhang and M-S Chiu ldquoDecoupling internalmodel control for multivariable systems with multiple timedelaysrdquo Chemical Engineering Science vol 57 no 1 pp 115ndash1242002

[15] H-P Huang and F-Y Lin ldquoDecoupling multivariable controlwith two degrees of freedomrdquo Industrial and EngineeringChemistry Research vol 45 no 9 pp 3161ndash3173 2006

[16] J Descusse and C H Moog ldquoDynamic decoupling for right-invertible nonlinear systemsrdquo Systems amp Control Letters vol 8no 4 pp 345ndash349 1987

[17] G Liu P Liu Y Shen F Wang and M Kang ldquoExperimentalresearch on decoupling control of multi-motor variable fre-quency system based on neural network generalized inverserdquo inProceedings of the IEEE International Conference on NetworkingSensing and Control (ICNSC rsquo08) pp 1476ndash1479 April 2008

[18] B T Jevtovic and M R Matauek ldquoPID controller design ofTITO system based on ideal decouplerrdquo Journal of ProcessControl vol 20 no 7 pp 869ndash876 2010

[19] P Nordfeldt and T Hagglund ldquoDecoupler and PID controllerdesign of TITO systemsrdquo Journal of Process Control vol 16 no9 pp 923ndash936 2006

[20] C Rajapandiyan and M Chidambaram ldquoController designfor MIMO processes based on simple decoupled equivalenttransfer functions and simplified decouplerrdquo Industrial andEngineering Chemistry Research vol 51 no 38 pp 12398ndash124102012

[21] W-J Cai W Ni M-J He and C-Y Ni ldquoNormalizeddecouplingmdasha new approach forMIMOprocess control systemdesignrdquo Industrial and Engineering Chemistry Research vol 47no 19 pp 7347ndash7356 2008

[22] Y Shen W J Cai and S Li ldquoNormalized decoupling controlfor high-dimensional MIMO processes for application in roomtemperature control HVAC systemsrdquo Control Engineering Prac-tice vol 18 no 6 pp 652ndash664 2010

[23] Y L Shen Y X Sun and S Y Li ldquoAdjoint transfer matrix baseddecoupling control for multivariable processesrdquo Industrial ampEngineering Chemistry Research vol 51 pp 16419ndash16426 2012

[24] Q XiongW-J Cai andM-J He ldquoEquivalent transfer functionmethod for PIPID controller design of MIMO processesrdquoJournal of Process Control vol 17 no 8 pp 665ndash673 2007

[25] F Morilla F Vazquez and J Garrido ldquoCentralized PID controlby decoupling for TITO processesrdquo in Proceedings of the 13thIEEE International Conference on Emerging Technologies andFactory Automation (ETFA rsquo08) pp 1318ndash1325 2008

[26] G He J Li Y Li and P Cui ldquoInteractions analysis in themaglev bogie with decentralized controllers using an effectiverelative gain array measurerdquo in Proceedings of the 10th IEEEInternational Conference onControl andAutomation (ICCA rsquo13)pp 1070ndash1075 Hangzhou China June 2013

[27] D F Zhou C H Hansen and J Li ldquoSuppression of maglevvehiclegirder self-excited vibration using a virtual tuned massdamperrdquo Journal of Sound andVibration vol 330 no 5 pp 883ndash901 2011

[28] H Guzman ldquoCurrent loops in a magnetic levitation systemrdquoInternational Journal of Innovative Computing Information andControl vol 5 no 5 pp 1275ndash1283 2009

[29] Y Liu and W Zhang ldquoAnalytical design of two degree-of-freedom decoupling control scheme for two-by-two systemswith integratorsrdquo IET Control Theory and Applications vol 1no 5 pp 1380ndash1389 2007

[30] Q-G Wang Y Zhang and M-S Chiu ldquoNon-interactingcontrol design for multivariable industrial processesrdquo Journal ofProcess Control vol 13 no 3 pp 253ndash265 2003

[31] B Kristiansson and B Lennartson ldquoEvaluation and simpletuning of PID controllers with high-frequency robustnessrdquoJournal of Process Control vol 16 no 2 pp 91ndash102 2006

[32] R Pintelon P Guillaume Y Rolain J Schoukens and H vanHamme ldquoParametric identification of transfer functions in thefrequency domainmdasha surveyrdquo IEEE Transactions on AutomaticControl vol 39 no 11 pp 2245ndash2260 1994

[33] J Garrido F Vazquez and F Morilla ldquoCentralized multi-variable control by simplified decouplingrdquo Journal of ProcessControl vol 22 no 6 pp 1044ndash1062 2012

[34] S Skogestad and I Postlethwaite Multivariable Feedback Con-trol Analysis and Design John Wiley amp Sons New York NYUSA 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


requirement which benefits the decoupling design of themodule suspension system Besides although the differentialgeometry technique is also a feasible approach to deal withthe multivariable decoupling control problems [16 17] theirneed for precise mathematical model is an obstacle to applyit to practical engineering The general decoupling controlapproach is to design the decoupler so that theMIMOcontrolsystem can be treated as multiple single-input-single-output(SISO) loops which allows us to use well developed singleloop controller design methods Many decoupling controldesign methods are developed based on this view as well [18ndash22] The ideal decoupler is to be designed as the inverse ofthe transfer functionmatrix However this kind of decouplerneeds to calculate the inverse of the process transfer functionmatrix resulting in too complicated calculation Shen et alconsidered the adjoint transfer matrix of the original mul-tivariable system as the decoupler [23] where it can avoidcomplicated computation especially for TITO system

In this paper a modified adjoint transfer matrix baseddecoupler was presented First the existing coupling inthe module suspension control system is analyzed and thedynamic model is also given By adopting the modifiedadjoint transfermatrix as the decoupler we divide themodulesuspension control system into two independent SISO con-trol systems Then compensated controllers are designed tomeet the desired loop performance and robustness demandof the module suspension control system The formulationof a resultant decoupling controller is obtained by combingthe decoupler and the compensated controller MultivariablePID structure controller is the most effective technology inengineering applications because of adequate performancewith simple structure [18 19 24 25] which is also adoptedin our practical CMS04 low speed maglev train Hencethe resultant decoupling controller is transformed to PIDtype controller by model reduction Given the parametersuncertainties and nonlinear characteristic in the magneticsuspension system the modeling errors between the lin-earized model and practical physical model have been takeninto accountThemodeling errors aremeasured by frequencysweeping experiments on a real full-scale single bogie ofCMS-04 maglev train based on which the robust stability ofthe decoupledmodule suspension control system is validatedFurthermore simulations and experimental results show thatthe proposed decoupling method can be well applied inthe module suspension system and promote the suspensioncapability in practical engineering application

The rest of the paper is organized as followsThe couplinganalysis and dynamic model of the module suspension sys-tem are given in Section 2 Section 3 describes the decouplingdesign procedure in detail The case study in Section 4 isto determine the parameters of the designed decouplingcontroller Simulations and experiments are presented in thissection Finally Section 5 gives the conclusions of this paper

2 Coupling Analysis and Modeling ofthe Module Suspension System

The low speed electromagnetic suspension (EMS) vehi-cle consists of cabin body levitation bogies secondary

Electromagnets Bogie

TrackCar body

Air spring Buttress

Figure 1 Lateral view of the CMS04 low speed maglev vehicle

Electromagnet

Sensor

xy

z

Box beam

LIM stator

PWM

PWM

Controller 1

Centralized MIMOdecoupling controller

Controller 2

Decentralized SISOcontroller

Air spring supporting point

Figure 2 Schematic of the module suspension control system

suspensions and levitation and guidance magnets A lateralview of the CMS04 low speed maglev vehicle is shownin Figure 1 where it can be founded that the car body issupported by five bogies with each bogie consisting of twolevitation modules Each module contains two pairs of adja-cent electromagnets which are controlled by decentralizedSISO controllers thus there are a total of four levitation unitsin a bogie

As the bogie is the pivotal component of the maglevvehicle analyzing the coupling issue of the bogie is essentialto design decoupling methods The coupling issue of thebogie was investigated through static experiments in theCMS-04 low speed maglev vehicle in [26] from which itcan be founded that the coupling between the two levitationunits in amodule ismuch stronger ([26] Figure 3)Thereforethe paper focuses on the coupling between the two levitationunits in one levitation module The schematic of the modulesuspension control system is shown in Figure 2 where it givestwo kinds of controller construction the decentralized SISOcontroller and the centralized MIMO decoupling controllerThe former is the common control methodology in maglevsystem It utilizes only the sensors message of one levitationunit to realize its stable levitation which leads to the fact thatadjusting the movement of one levitation control unit affectsthe performance of the other one because of themodulersquos stiffstructure To cope with this problem a centralized MIMOdecoupling control scheme described in Figure 2 is presentedand will be discussed in detailThis kind of controller can useboth sensors messages of the levitation units in a module toproduce appropriate control laws







119894is







119888 =

(1205751+ 1205752)

2

120579 =

(1205751minus 1205752)

119897

1199111=

(31205751+ 1205752)

4

1199112=

(1205751+ 3120575

2)

4

(1)


119898119892 minus 1198651minus 1198652+ 119873

1+ 119873

2= 119898 119888

1198652

119897

4


119897

4

sdot cos 120579 + 1198731

119897

2


119897

2

sdot cos 120579 = 119869 120579(2)


119894= 119896

119890119868119894119911119894 119896119890= 120583

011987321198602 119868

119894is the coil




Track

Levitation moduleN1

1205751 z1 c z2 1205752 N2

O

F1 F2120579

l



119906119894= 119877119868

119894+

2119896119890

119911119894

119868119894minus

2119896119890

1199112

119894

119868119894119894 119894 = 1 2 (3)




119868119890119894= 119896119901(120575119894minus 119903119889minus 1205750) + 119896

119889120575119894 119894 = 1 2 (4)

where 119896119901and 119896


coefficients 119868119890119894




1198940


11986810= radic

(119898119892 + 31198731minus 119873

2) 1199112

0

12058301198732119860

11986820= radic

(119898119892 + 31198732minus 119873

1) 1199112

0

12058301198732119860

(5)


119906119894= 119896119888(119868119890119894minus 119868119894) 119894 = 1 2 (6)







[

1199101 (119904)

1199102 (119904)] = [

11989211 (119904) 11989212 (

119904)

11989221 (119904) 11989222 (

119904)] [

1199061 (119904)

1199062 (119904)] (7)




119870119889 (119904) = adj119866 (119904) (8)


minus2 (1198651+ 1198652) = 119898

1(1205751+1205752)

(

1

4

1198652minus

1

4

1198651) 1198972= 119869 (

1205751minus1205752)

119868119894=

1199110119894

2119896119890

119906119894minus

1198771199110119894

2119896119890

119868119894+

1198680119894

1199110119894

119894

119865119894= 119865119911119911119894minus 119865119868119868119894

119906119894= 119896119901119896119888(120575119894minus 119903119889) + 119896

119889119896119888120575119894

(9)

where 119877 = 119877 + 119896119888 119865119868119894= 2119896

11989011986811989401205752

0 and 119865

119911119894= 2119896

1198901198682

11989401205753

0




119866 = 1198660(119868 + 119866

0)minus1 (10)

1198660= 119866

1119866211986651198666+ 119866

111986621198664+ 119866

11198663 (11)



119870119889 (119904) = adj (119866

0 (119904) (119868 + 1198660 (

119904))minus1) (12)




119870119889 (119904) = (119868 + 1198660 (

119904)) adj1198660 (119904) (13)






119889(119904) as follows

119870119889 (119904) = (119868 + 1198660 (

119904)) adj1198660 (119904) 119866119886 (119904) (14)

Here119866119886(119904) = diag119892



119892119886119894 (119904) =

119904119903119894

(120582119894119904120596119899119894+ 1)

119903119894 119894 = 1 2 (15)



0(119904))adj119866

0(119904) The choice of (120582

119894119904 + 1)





119886119894(119904)


119894will be


119889(119904) acts on


119876 (119904) = 119866 (119904)119870119889 (119904) = det119866

0 (119904) sdot 119866119886 (

119904) (16)







r(s)C(s) Kd(s)

K(s)

Q(s)G(s) = G0(s)(I + G0(s))

minus1y(s)

minus



119870 (119904) = 119870119889 (119904) 119862 (119904)

119896119894119895 (119904) = 119888119894119894 (

119904) 119870119889119894119895(119904) 119894 119895 = 1 2

(17)


119867(119904) = 119866 (119904)119870119889 (119904) 119862 (119904) [119868 + 119866 (119904)119870119889 (

119904) 119862 (119904)]minus1

= det1198660 (119904) 119866119886 (

119904) sdot 119862 (119904) [119868 + det1198660 (119904) 119866119886 (

119904) sdot 119862 (119904)]minus1

= diag ℎ11 (119904) ℎ22 (

119904)

(18)


119862 (119904) = (119867 (119904)minus1minus 119868)

minus1

sdot det1198660 (119904) 119866119886 (

119904)

= diag ℎ11 (119904)

(1 minus ℎ11 (119904)) sdot 1199021 (

119904)

ℎ22 (119904)

(1 minus ℎ22 (119904)) sdot 1199022 (

119904)

(19)


ℎ119894119894 (119904) =

1205962

119899119894

(1199042+ 2120596

119899119894120585119894119904 + 120596

2

119899119894) ((120582

119894120596119899119894) 119904 + 1)

119897119894

prod119898119894

119896=1(119904 + 119911

119896)

prod119902119894

119895=1(119904 + 119901

119895)

119894 = 1 2

(20)

where 120582119894 120585119894 and 120596



The term (120582119894119904120596119899119894+ 1)


119889(119904) To make



119897119894= max 0 119903119889 (det (119866

0 (119904)))

minusmin 119903119889 (11986611989410(119904)) 119903119889 (119866

1198942

0(119904))

+ 119903119894+ 119898

119894minus 119902119894minus 1 119894 = 1 2

(21)













119878 (119904) = (119868 + 119866 (119904)119870 (119904))minus1

119879 (119904) = 119866 (119904)119870 (119904) (119868 + 119866 (119904)119870 (119904))minus1

(22)


119872st119894 = max120596(1003816100381610038161003816119878119894119894(119895120596)

10038161003816100381610038161003816100381610038161003816119879119894119894(119895120596)

1003816100381610038161003816) 119894 = 1 2 forall120596 isin 119877 (23)







IAE = intinfin

0

1003816100381610038161003816119903119894 (119905) minus 119910119894 (

119905)1003816100381610038161003816119889119905 119894 = 1 2 (24)



119894 120585119894 and 120596

119899119894 The



120596119894≜ 119904 isin 119862 | 119904 = 119895120596 120596

119904119894le 120596 le 120596

119898119894

The value of 120596119898119894


11990211989410 and 120596

119898119894= 10120596

119902119894 and 120596




119870119894119895 (119904) = 119860 (119904) =

119887119898119904119898+ sdot sdot sdot + 119887

1119904 + 119860 (0)

119886119899119904119899+ 119886119899minus1119904119899minus1

+ sdot sdot sdot + 1198861119904 + 1

(25)




119860119903 (119904) =

119873 (119904)

119863 (119904)

=

120573119903119904119903+ sdot sdot sdot + 120573

1119904 + 119860 (0)

120572119896119904119896+ 120572119896minus1119904119896minus1

+ sdot sdot sdot + 1205721119904 + 1

(26)

Denoting that 120579 = (1205721 120572

119896 1205731 120573




119869 = min119904isin119863120596119894

1

119873

119873

sum

119896=1

[1003816100381610038161003816119882 (119895119908

119896)1003816100381610038161003816

2sdot1003816100381610038161003816119860 (119895119908

119896) minus 119860

119903(119895119908

119896 120579)1003816100381610038161003816

2]

119882 (119895119908119896) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119863 (119895119908119896 120572119894)

119863 (119895119908119896 120572119894minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(27)



119903(119904) is



119870119891 (119904) =

1

1 + 119904119879119891+ 1199042(1198792

119891119873)

(28)


119870PID (119904) = 119870119891 (119904) (119870119901 +119879119894

119904

+ 119879119889119904)

=

1198791198891199042+ 119870

119901119904 + 119879

119894

119904 (1 + 119904119879119891+ 1199042(1198792

119891119873))

(29)






119875 (119904) asymp

12057321199042+ 1205731119904 + 119875 (0)

12057221199042+ 1205721119904 + 1

(30)



119870119901= 1205731 119879

119894= 119875 (0) 119879

119889= 1205732

119879119891= 1205721 119873 =

1205722

1

1205722

(31)


119866119901 (119904) = 119866 (119904) (119868 + Δ (119904)) (32)


1003816100381610038161003816119882119860(119895120596)

1003816100381610038161003816ge max119866119901isinΛ

120590 (119866minus1(119895120596) [119866

119901(119895120596) minus 119866 (119895120596)]) (33)




119872(119904) = minus119882119860 (119904) 119870PID (119904) [119868 + 119870PID (119904) 119866 (119904)]

minus1 (35)


4 Case Study






1198660 (119904) =

[

[

[

[

682713 (119904 + 4015)

1199042(119904 + 1135)

55329 (119904 minus 3285)

1199042(119904 + 1135)

55329 (119904 minus 3285)

1199042(119904 + 1135)

682713 (119904 + 4015)

1199042(119904 + 1135)

]

]

]

]

(36)



10 KN119860 00186m2

119897 265m119869 910 kgsdotm2

119873 320119896119888

3001205830

4120587 times 10minus7 Hm119896119901

4000119896119889

331205750

9mm119877 10Ω



119870119889 (119904) = (119868 + 1198660 (

119904)) adj1198660 (119904) 119866119886 (119904)

= [

1198961198891111989611988912

1198961198892111989611988922

]

11989611988911

=

6827131199041199031(119904 + 1074) (119904 + 1524) (119904

2+ 8604119904 + 3768)

1199044(119904 + 1135)

2(12058211199041205961198991+ 1)

1199031

11989611988912=

minus553291199041199031(119904 minus 3285)

1199042(119904 + 1135) (1205822

1199041205961198992+ 1)

1199032

11989611988921=

minus553291199041199031(119904 minus 3285)

1199042(119904 + 1135) (1205821

1199041205961198992+ 1)

1199031

11989611988922

=

6827131199041199032(119904 + 1074) (119904 + 1524) (119904

2+ 8604119904 + 3768)

1199044(119904 + 1135)

2(12058221199041205961198992+ 1)

1199032

(37)



1= 1199032= 4 After the




follows

119902119894 (119904) =

466 times 109(1199042+ 8072119904 + 9026)

(119904 + 1135)2(120582119894119904120596119899119894+ 1)

4 119894 = 1 2 (38)




02 03 04 05 06 07 08 09 11

12

14

16

18

2

22

24

26

28

3

120585

Mst


02 03 04 05 06 07 08 09 1004

0045

005

0055

006

0065

007

0075

008

0085

009

IAE

120585

120582 = 001

120582 = 005

120582 = 01






119888119894119894 (119904) =

1205962

119899119894

(1199042+ 2120596

119899119894120585119894119904 + 120596

2

119899119894) (120582

119894119904120596119899119894+ 1) minus 120596

2

119899119894

sdot

(119904 + 1135)2(120582119894119904120596119899119894+ 1)

4

466 times 109(1199042+ 8072119904 + 9026)

119894 = 1 2

(39)


119894 120585119894 and

120596119899119894




119899as a





119899











measurement noise


0 5 10 15 20 25 30 35 40 45 501

105

11

115

12

125

13

135

14M

st

120596n


0

005

01

015

02

025

03

IAE

0 5 10 15 20 25 30 35 40 45 50

120582 = 001

120582 = 005

120582 = 01

120596n




119899as follows




119903lt 01 s then we get 120596

119899gt 40



119899as 40


119894can serve as






119879119894

119879119889

119873 119879119891

1-12-2 391 438 00743 4927 010241-22-1 429 times 10minus6 0 00215 4927 01024




85

9

95

10

105

Gap

1 (m

m)

0 05 1 15 2 25 3Time (s)

(a)

0 05 1 15 2 25 385

9

95

10

105

Time (s)

Gap

2 (m

m)


(b)







119882119860 (119904) =

00318119904 + 04

(0031806) 119904 + 1

sdot 119868 (40)

Gap

1 (m

m)

85

9

95

10

105

0 05 1 15 2 25 3Time (s)

(a)

Gap

2 (m

m)

0 05 1 15 2 25 385

9

95

10

105

Time (s)


(b)









0

01

02

03

04

05

06

07

Frequency (rads)

Sing

ular

val

ues

10minus1 100 101 102 103


85

9

95

10

105Decoupling

0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 520

22

24

26

28

Time (s)

Curr

ent (

A)

Current 1Current 2

(b)




0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

85

9

95

10

105Nondecoupling

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Current 1Current 2

20

22

24

26

28

Curr

ent (

A)

(b)




5 Conclusions



0 05 1 15 2 25 3 35 42

4

6

8

10

12

14

16

18

Gap

(mm

)

Nondecoupling

Gap 1Gap 2


0 1 2 3 4 5 67

8

9

10

11

12

13

14

15

16

17

18

Gap

(mm

)

Decoupling

Gap 1Gap 2




Appendix


1198661=

[

[

[

[

[

minus

1

1198981

minus

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

minus

1198972

8119869

]

]

]

]

]

1198662= [

minus1198651198681

0

0 minus1198651198682

]

1198663=

[

[

[

[

31198651199111

4

1198651199111

4

1198651199112

4

31198651199112

4

]

]

]

]

1198666= [

119896119888(119896119901+ 119896119889119904) 0

0 119896119888(119896119901+ 119896119889119904)

]

1198664=

[

[

[

[

[

311989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

311989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

]

]

]

]

]

1198665=

[

[

[

[

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

0

0

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

]

]

]

]

(A1)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119894is







119888 =

(1205751+ 1205752)

2

120579 =

(1205751minus 1205752)

119897

1199111=

(31205751+ 1205752)

4

1199112=

(1205751+ 3120575

2)

4

(1)


119898119892 minus 1198651minus 1198652+ 119873

1+ 119873

2= 119898 119888

1198652

119897

4


119897

4

sdot cos 120579 + 1198731

119897

2


119897

2

sdot cos 120579 = 119869 120579(2)


119894= 119896

119890119868119894119911119894 119896119890= 120583

011987321198602 119868

119894is the coil




Track

Levitation moduleN1

1205751 z1 c z2 1205752 N2

O

F1 F2120579

l



119906119894= 119877119868

119894+

2119896119890

119911119894

119868119894minus

2119896119890

1199112

119894

119868119894119894 119894 = 1 2 (3)




119868119890119894= 119896119901(120575119894minus 119903119889minus 1205750) + 119896

119889120575119894 119894 = 1 2 (4)

where 119896119901and 119896


coefficients 119868119890119894




1198940


11986810= radic

(119898119892 + 31198731minus 119873

2) 1199112

0

12058301198732119860

11986820= radic

(119898119892 + 31198732minus 119873

1) 1199112

0

12058301198732119860

(5)


119906119894= 119896119888(119868119890119894minus 119868119894) 119894 = 1 2 (6)







[

1199101 (119904)

1199102 (119904)] = [

11989211 (119904) 11989212 (

119904)

11989221 (119904) 11989222 (

119904)] [

1199061 (119904)

1199062 (119904)] (7)




119870119889 (119904) = adj119866 (119904) (8)


minus2 (1198651+ 1198652) = 119898

1(1205751+1205752)

(

1

4

1198652minus

1

4

1198651) 1198972= 119869 (

1205751minus1205752)

119868119894=

1199110119894

2119896119890

119906119894minus

1198771199110119894

2119896119890

119868119894+

1198680119894

1199110119894

119894

119865119894= 119865119911119911119894minus 119865119868119868119894

119906119894= 119896119901119896119888(120575119894minus 119903119889) + 119896

119889119896119888120575119894

(9)

where 119877 = 119877 + 119896119888 119865119868119894= 2119896

11989011986811989401205752

0 and 119865

119911119894= 2119896

1198901198682

11989401205753

0




119866 = 1198660(119868 + 119866

0)minus1 (10)

1198660= 119866

1119866211986651198666+ 119866

111986621198664+ 119866

11198663 (11)



119870119889 (119904) = adj (119866

0 (119904) (119868 + 1198660 (

119904))minus1) (12)




119870119889 (119904) = (119868 + 1198660 (

119904)) adj1198660 (119904) (13)






119889(119904) as follows

119870119889 (119904) = (119868 + 1198660 (

119904)) adj1198660 (119904) 119866119886 (119904) (14)

Here119866119886(119904) = diag119892



119892119886119894 (119904) =

119904119903119894

(120582119894119904120596119899119894+ 1)

119903119894 119894 = 1 2 (15)



0(119904))adj119866

0(119904) The choice of (120582

119894119904 + 1)





119886119894(119904)


119894will be


119889(119904) acts on


119876 (119904) = 119866 (119904)119870119889 (119904) = det119866

0 (119904) sdot 119866119886 (

119904) (16)







r(s)C(s) Kd(s)

K(s)

Q(s)G(s) = G0(s)(I + G0(s))

minus1y(s)

minus



119870 (119904) = 119870119889 (119904) 119862 (119904)

119896119894119895 (119904) = 119888119894119894 (

119904) 119870119889119894119895(119904) 119894 119895 = 1 2

(17)


119867(119904) = 119866 (119904)119870119889 (119904) 119862 (119904) [119868 + 119866 (119904)119870119889 (

119904) 119862 (119904)]minus1

= det1198660 (119904) 119866119886 (

119904) sdot 119862 (119904) [119868 + det1198660 (119904) 119866119886 (

119904) sdot 119862 (119904)]minus1

= diag ℎ11 (119904) ℎ22 (

119904)

(18)


119862 (119904) = (119867 (119904)minus1minus 119868)

minus1

sdot det1198660 (119904) 119866119886 (

119904)

= diag ℎ11 (119904)

(1 minus ℎ11 (119904)) sdot 1199021 (

119904)

ℎ22 (119904)

(1 minus ℎ22 (119904)) sdot 1199022 (

119904)

(19)


ℎ119894119894 (119904) =

1205962

119899119894

(1199042+ 2120596

119899119894120585119894119904 + 120596

2

119899119894) ((120582

119894120596119899119894) 119904 + 1)

119897119894

prod119898119894

119896=1(119904 + 119911

119896)

prod119902119894

119895=1(119904 + 119901

119895)

119894 = 1 2

(20)

where 120582119894 120585119894 and 120596



The term (120582119894119904120596119899119894+ 1)


119889(119904) To make



119897119894= max 0 119903119889 (det (119866

0 (119904)))

minusmin 119903119889 (11986611989410(119904)) 119903119889 (119866

1198942

0(119904))

+ 119903119894+ 119898

119894minus 119902119894minus 1 119894 = 1 2

(21)













119878 (119904) = (119868 + 119866 (119904)119870 (119904))minus1

119879 (119904) = 119866 (119904)119870 (119904) (119868 + 119866 (119904)119870 (119904))minus1

(22)


119872st119894 = max120596(1003816100381610038161003816119878119894119894(119895120596)

10038161003816100381610038161003816100381610038161003816119879119894119894(119895120596)

1003816100381610038161003816) 119894 = 1 2 forall120596 isin 119877 (23)







IAE = intinfin

0

1003816100381610038161003816119903119894 (119905) minus 119910119894 (

119905)1003816100381610038161003816119889119905 119894 = 1 2 (24)



119894 120585119894 and 120596

119899119894 The



120596119894≜ 119904 isin 119862 | 119904 = 119895120596 120596

119904119894le 120596 le 120596

119898119894

The value of 120596119898119894


11990211989410 and 120596

119898119894= 10120596

119902119894 and 120596




119870119894119895 (119904) = 119860 (119904) =

119887119898119904119898+ sdot sdot sdot + 119887

1119904 + 119860 (0)

119886119899119904119899+ 119886119899minus1119904119899minus1

+ sdot sdot sdot + 1198861119904 + 1

(25)




119860119903 (119904) =

119873 (119904)

119863 (119904)

=

120573119903119904119903+ sdot sdot sdot + 120573

1119904 + 119860 (0)

120572119896119904119896+ 120572119896minus1119904119896minus1

+ sdot sdot sdot + 1205721119904 + 1

(26)

Denoting that 120579 = (1205721 120572

119896 1205731 120573




119869 = min119904isin119863120596119894

1

119873

119873

sum

119896=1

[1003816100381610038161003816119882 (119895119908

119896)1003816100381610038161003816

2sdot1003816100381610038161003816119860 (119895119908

119896) minus 119860

119903(119895119908

119896 120579)1003816100381610038161003816

2]

119882 (119895119908119896) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119863 (119895119908119896 120572119894)

119863 (119895119908119896 120572119894minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(27)



119903(119904) is



119870119891 (119904) =

1

1 + 119904119879119891+ 1199042(1198792

119891119873)

(28)


119870PID (119904) = 119870119891 (119904) (119870119901 +119879119894

119904

+ 119879119889119904)

=

1198791198891199042+ 119870

119901119904 + 119879

119894

119904 (1 + 119904119879119891+ 1199042(1198792

119891119873))

(29)






119875 (119904) asymp

12057321199042+ 1205731119904 + 119875 (0)

12057221199042+ 1205721119904 + 1

(30)



119870119901= 1205731 119879

119894= 119875 (0) 119879

119889= 1205732

119879119891= 1205721 119873 =

1205722

1

1205722

(31)


119866119901 (119904) = 119866 (119904) (119868 + Δ (119904)) (32)


1003816100381610038161003816119882119860(119895120596)

1003816100381610038161003816ge max119866119901isinΛ

120590 (119866minus1(119895120596) [119866

119901(119895120596) minus 119866 (119895120596)]) (33)




119872(119904) = minus119882119860 (119904) 119870PID (119904) [119868 + 119870PID (119904) 119866 (119904)]

minus1 (35)


4 Case Study






1198660 (119904) =

[

[

[

[

682713 (119904 + 4015)

1199042(119904 + 1135)

55329 (119904 minus 3285)

1199042(119904 + 1135)

55329 (119904 minus 3285)

1199042(119904 + 1135)

682713 (119904 + 4015)

1199042(119904 + 1135)

]

]

]

]

(36)



10 KN119860 00186m2

119897 265m119869 910 kgsdotm2

119873 320119896119888

3001205830

4120587 times 10minus7 Hm119896119901

4000119896119889

331205750

9mm119877 10Ω



119870119889 (119904) = (119868 + 1198660 (

119904)) adj1198660 (119904) 119866119886 (119904)

= [

1198961198891111989611988912

1198961198892111989611988922

]

11989611988911

=

6827131199041199031(119904 + 1074) (119904 + 1524) (119904

2+ 8604119904 + 3768)

1199044(119904 + 1135)

2(12058211199041205961198991+ 1)

1199031

11989611988912=

minus553291199041199031(119904 minus 3285)

1199042(119904 + 1135) (1205822

1199041205961198992+ 1)

1199032

11989611988921=

minus553291199041199031(119904 minus 3285)

1199042(119904 + 1135) (1205821

1199041205961198992+ 1)

1199031

11989611988922

=

6827131199041199032(119904 + 1074) (119904 + 1524) (119904

2+ 8604119904 + 3768)

1199044(119904 + 1135)

2(12058221199041205961198992+ 1)

1199032

(37)



1= 1199032= 4 After the




follows

119902119894 (119904) =

466 times 109(1199042+ 8072119904 + 9026)

(119904 + 1135)2(120582119894119904120596119899119894+ 1)

4 119894 = 1 2 (38)




02 03 04 05 06 07 08 09 11

12

14

16

18

2

22

24

26

28

3

120585

Mst


02 03 04 05 06 07 08 09 1004

0045

005

0055

006

0065

007

0075

008

0085

009

IAE

120585

120582 = 001

120582 = 005

120582 = 01






119888119894119894 (119904) =

1205962

119899119894

(1199042+ 2120596

119899119894120585119894119904 + 120596

2

119899119894) (120582

119894119904120596119899119894+ 1) minus 120596

2

119899119894

sdot

(119904 + 1135)2(120582119894119904120596119899119894+ 1)

4

466 times 109(1199042+ 8072119904 + 9026)

119894 = 1 2

(39)


119894 120585119894 and

120596119899119894




119899as a





119899











measurement noise


0 5 10 15 20 25 30 35 40 45 501

105

11

115

12

125

13

135

14M

st

120596n


0

005

01

015

02

025

03

IAE

0 5 10 15 20 25 30 35 40 45 50

120582 = 001

120582 = 005

120582 = 01

120596n




119899as follows




119903lt 01 s then we get 120596

119899gt 40



119899as 40


119894can serve as






119879119894

119879119889

119873 119879119891

1-12-2 391 438 00743 4927 010241-22-1 429 times 10minus6 0 00215 4927 01024




85

9

95

10

105

Gap

1 (m

m)

0 05 1 15 2 25 3Time (s)

(a)

0 05 1 15 2 25 385

9

95

10

105

Time (s)

Gap

2 (m

m)


(b)







119882119860 (119904) =

00318119904 + 04

(0031806) 119904 + 1

sdot 119868 (40)

Gap

1 (m

m)

85

9

95

10

105

0 05 1 15 2 25 3Time (s)

(a)

Gap

2 (m

m)

0 05 1 15 2 25 385

9

95

10

105

Time (s)


(b)









0

01

02

03

04

05

06

07

Frequency (rads)

Sing

ular

val

ues

10minus1 100 101 102 103


85

9

95

10

105Decoupling

0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 520

22

24

26

28

Time (s)

Curr

ent (

A)

Current 1Current 2

(b)




0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

85

9

95

10

105Nondecoupling

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Current 1Current 2

20

22

24

26

28

Curr

ent (

A)

(b)




5 Conclusions



0 05 1 15 2 25 3 35 42

4

6

8

10

12

14

16

18

Gap

(mm

)

Nondecoupling

Gap 1Gap 2


0 1 2 3 4 5 67

8

9

10

11

12

13

14

15

16

17

18

Gap

(mm

)

Decoupling

Gap 1Gap 2




Appendix


1198661=

[

[

[

[

[

minus

1

1198981

minus

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

minus

1198972

8119869

]

]

]

]

]

1198662= [

minus1198651198681

0

0 minus1198651198682

]

1198663=

[

[

[

[

31198651199111

4

1198651199111

4

1198651199112

4

31198651199112

4

]

]

]

]

1198666= [

119896119888(119896119901+ 119896119889119904) 0

0 119896119888(119896119901+ 119896119889119904)

]

1198664=

[

[

[

[

[

311989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

311989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

]

]

]

]

]

1198665=

[

[

[

[

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

0

0

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

]

]

]

]

(A1)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













[

1199101 (119904)

1199102 (119904)] = [

11989211 (119904) 11989212 (

119904)

11989221 (119904) 11989222 (

119904)] [

1199061 (119904)

1199062 (119904)] (7)




119870119889 (119904) = adj119866 (119904) (8)


minus2 (1198651+ 1198652) = 119898

1(1205751+1205752)

(

1

4

1198652minus

1

4

1198651) 1198972= 119869 (

1205751minus1205752)

119868119894=

1199110119894

2119896119890

119906119894minus

1198771199110119894

2119896119890

119868119894+

1198680119894

1199110119894

119894

119865119894= 119865119911119911119894minus 119865119868119868119894

119906119894= 119896119901119896119888(120575119894minus 119903119889) + 119896

119889119896119888120575119894

(9)

where 119877 = 119877 + 119896119888 119865119868119894= 2119896

11989011986811989401205752

0 and 119865

119911119894= 2119896

1198901198682

11989401205753

0




119866 = 1198660(119868 + 119866

0)minus1 (10)

1198660= 119866

1119866211986651198666+ 119866

111986621198664+ 119866

11198663 (11)



119870119889 (119904) = adj (119866

0 (119904) (119868 + 1198660 (

119904))minus1) (12)




119870119889 (119904) = (119868 + 1198660 (

119904)) adj1198660 (119904) (13)






119889(119904) as follows

119870119889 (119904) = (119868 + 1198660 (

119904)) adj1198660 (119904) 119866119886 (119904) (14)

Here119866119886(119904) = diag119892



119892119886119894 (119904) =

119904119903119894

(120582119894119904120596119899119894+ 1)

119903119894 119894 = 1 2 (15)



0(119904))adj119866

0(119904) The choice of (120582

119894119904 + 1)





119886119894(119904)


119894will be


119889(119904) acts on


119876 (119904) = 119866 (119904)119870119889 (119904) = det119866

0 (119904) sdot 119866119886 (

119904) (16)







r(s)C(s) Kd(s)

K(s)

Q(s)G(s) = G0(s)(I + G0(s))

minus1y(s)

minus



119870 (119904) = 119870119889 (119904) 119862 (119904)

119896119894119895 (119904) = 119888119894119894 (

119904) 119870119889119894119895(119904) 119894 119895 = 1 2

(17)


119867(119904) = 119866 (119904)119870119889 (119904) 119862 (119904) [119868 + 119866 (119904)119870119889 (

119904) 119862 (119904)]minus1

= det1198660 (119904) 119866119886 (

119904) sdot 119862 (119904) [119868 + det1198660 (119904) 119866119886 (

119904) sdot 119862 (119904)]minus1

= diag ℎ11 (119904) ℎ22 (

119904)

(18)


119862 (119904) = (119867 (119904)minus1minus 119868)

minus1

sdot det1198660 (119904) 119866119886 (

119904)

= diag ℎ11 (119904)

(1 minus ℎ11 (119904)) sdot 1199021 (

119904)

ℎ22 (119904)

(1 minus ℎ22 (119904)) sdot 1199022 (

119904)

(19)


ℎ119894119894 (119904) =

1205962

119899119894

(1199042+ 2120596

119899119894120585119894119904 + 120596

2

119899119894) ((120582

119894120596119899119894) 119904 + 1)

119897119894

prod119898119894

119896=1(119904 + 119911

119896)

prod119902119894

119895=1(119904 + 119901

119895)

119894 = 1 2

(20)

where 120582119894 120585119894 and 120596



The term (120582119894119904120596119899119894+ 1)


119889(119904) To make



119897119894= max 0 119903119889 (det (119866

0 (119904)))

minusmin 119903119889 (11986611989410(119904)) 119903119889 (119866

1198942

0(119904))

+ 119903119894+ 119898

119894minus 119902119894minus 1 119894 = 1 2

(21)













119878 (119904) = (119868 + 119866 (119904)119870 (119904))minus1

119879 (119904) = 119866 (119904)119870 (119904) (119868 + 119866 (119904)119870 (119904))minus1

(22)


119872st119894 = max120596(1003816100381610038161003816119878119894119894(119895120596)

10038161003816100381610038161003816100381610038161003816119879119894119894(119895120596)

1003816100381610038161003816) 119894 = 1 2 forall120596 isin 119877 (23)







IAE = intinfin

0

1003816100381610038161003816119903119894 (119905) minus 119910119894 (

119905)1003816100381610038161003816119889119905 119894 = 1 2 (24)



119894 120585119894 and 120596

119899119894 The



120596119894≜ 119904 isin 119862 | 119904 = 119895120596 120596

119904119894le 120596 le 120596

119898119894

The value of 120596119898119894


11990211989410 and 120596

119898119894= 10120596

119902119894 and 120596




119870119894119895 (119904) = 119860 (119904) =

119887119898119904119898+ sdot sdot sdot + 119887

1119904 + 119860 (0)

119886119899119904119899+ 119886119899minus1119904119899minus1

+ sdot sdot sdot + 1198861119904 + 1

(25)




119860119903 (119904) =

119873 (119904)

119863 (119904)

=

120573119903119904119903+ sdot sdot sdot + 120573

1119904 + 119860 (0)

120572119896119904119896+ 120572119896minus1119904119896minus1

+ sdot sdot sdot + 1205721119904 + 1

(26)

Denoting that 120579 = (1205721 120572

119896 1205731 120573




119869 = min119904isin119863120596119894

1

119873

119873

sum

119896=1

[1003816100381610038161003816119882 (119895119908

119896)1003816100381610038161003816

2sdot1003816100381610038161003816119860 (119895119908

119896) minus 119860

119903(119895119908

119896 120579)1003816100381610038161003816

2]

119882 (119895119908119896) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119863 (119895119908119896 120572119894)

119863 (119895119908119896 120572119894minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(27)



119903(119904) is



119870119891 (119904) =

1

1 + 119904119879119891+ 1199042(1198792

119891119873)

(28)


119870PID (119904) = 119870119891 (119904) (119870119901 +119879119894

119904

+ 119879119889119904)

=

1198791198891199042+ 119870

119901119904 + 119879

119894

119904 (1 + 119904119879119891+ 1199042(1198792

119891119873))

(29)






119875 (119904) asymp

12057321199042+ 1205731119904 + 119875 (0)

12057221199042+ 1205721119904 + 1

(30)



119870119901= 1205731 119879

119894= 119875 (0) 119879

119889= 1205732

119879119891= 1205721 119873 =

1205722

1

1205722

(31)


119866119901 (119904) = 119866 (119904) (119868 + Δ (119904)) (32)


1003816100381610038161003816119882119860(119895120596)

1003816100381610038161003816ge max119866119901isinΛ

120590 (119866minus1(119895120596) [119866

119901(119895120596) minus 119866 (119895120596)]) (33)




119872(119904) = minus119882119860 (119904) 119870PID (119904) [119868 + 119870PID (119904) 119866 (119904)]

minus1 (35)


4 Case Study






1198660 (119904) =

[

[

[

[

682713 (119904 + 4015)

1199042(119904 + 1135)

55329 (119904 minus 3285)

1199042(119904 + 1135)

55329 (119904 minus 3285)

1199042(119904 + 1135)

682713 (119904 + 4015)

1199042(119904 + 1135)

]

]

]

]

(36)



10 KN119860 00186m2

119897 265m119869 910 kgsdotm2

119873 320119896119888

3001205830

4120587 times 10minus7 Hm119896119901

4000119896119889

331205750

9mm119877 10Ω



119870119889 (119904) = (119868 + 1198660 (

119904)) adj1198660 (119904) 119866119886 (119904)

= [

1198961198891111989611988912

1198961198892111989611988922

]

11989611988911

=

6827131199041199031(119904 + 1074) (119904 + 1524) (119904

2+ 8604119904 + 3768)

1199044(119904 + 1135)

2(12058211199041205961198991+ 1)

1199031

11989611988912=

minus553291199041199031(119904 minus 3285)

1199042(119904 + 1135) (1205822

1199041205961198992+ 1)

1199032

11989611988921=

minus553291199041199031(119904 minus 3285)

1199042(119904 + 1135) (1205821

1199041205961198992+ 1)

1199031

11989611988922

=

6827131199041199032(119904 + 1074) (119904 + 1524) (119904

2+ 8604119904 + 3768)

1199044(119904 + 1135)

2(12058221199041205961198992+ 1)

1199032

(37)



1= 1199032= 4 After the




follows

119902119894 (119904) =

466 times 109(1199042+ 8072119904 + 9026)

(119904 + 1135)2(120582119894119904120596119899119894+ 1)

4 119894 = 1 2 (38)




02 03 04 05 06 07 08 09 11

12

14

16

18

2

22

24

26

28

3

120585

Mst


02 03 04 05 06 07 08 09 1004

0045

005

0055

006

0065

007

0075

008

0085

009

IAE

120585

120582 = 001

120582 = 005

120582 = 01






119888119894119894 (119904) =

1205962

119899119894

(1199042+ 2120596

119899119894120585119894119904 + 120596

2

119899119894) (120582

119894119904120596119899119894+ 1) minus 120596

2

119899119894

sdot

(119904 + 1135)2(120582119894119904120596119899119894+ 1)

4

466 times 109(1199042+ 8072119904 + 9026)

119894 = 1 2

(39)


119894 120585119894 and

120596119899119894




119899as a





119899











measurement noise


0 5 10 15 20 25 30 35 40 45 501

105

11

115

12

125

13

135

14M

st

120596n


0

005

01

015

02

025

03

IAE

0 5 10 15 20 25 30 35 40 45 50

120582 = 001

120582 = 005

120582 = 01

120596n




119899as follows




119903lt 01 s then we get 120596

119899gt 40



119899as 40


119894can serve as






119879119894

119879119889

119873 119879119891

1-12-2 391 438 00743 4927 010241-22-1 429 times 10minus6 0 00215 4927 01024




85

9

95

10

105

Gap

1 (m

m)

0 05 1 15 2 25 3Time (s)

(a)

0 05 1 15 2 25 385

9

95

10

105

Time (s)

Gap

2 (m

m)


(b)







119882119860 (119904) =

00318119904 + 04

(0031806) 119904 + 1

sdot 119868 (40)

Gap

1 (m

m)

85

9

95

10

105

0 05 1 15 2 25 3Time (s)

(a)

Gap

2 (m

m)

0 05 1 15 2 25 385

9

95

10

105

Time (s)


(b)









0

01

02

03

04

05

06

07

Frequency (rads)

Sing

ular

val

ues

10minus1 100 101 102 103


85

9

95

10

105Decoupling

0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 520

22

24

26

28

Time (s)

Curr

ent (

A)

Current 1Current 2

(b)




0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

85

9

95

10

105Nondecoupling

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Current 1Current 2

20

22

24

26

28

Curr

ent (

A)

(b)




5 Conclusions



0 05 1 15 2 25 3 35 42

4

6

8

10

12

14

16

18

Gap

(mm

)

Nondecoupling

Gap 1Gap 2


0 1 2 3 4 5 67

8

9

10

11

12

13

14

15

16

17

18

Gap

(mm

)

Decoupling

Gap 1Gap 2




Appendix


1198661=

[

[

[

[

[

minus

1

1198981

minus

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

minus

1198972

8119869

]

]

]

]

]

1198662= [

minus1198651198681

0

0 minus1198651198682

]

1198663=

[

[

[

[

31198651199111

4

1198651199111

4

1198651199112

4

31198651199112

4

]

]

]

]

1198666= [

119896119888(119896119901+ 119896119889119904) 0

0 119896119888(119896119901+ 119896119889119904)

]

1198664=

[

[

[

[

[

311989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

311989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

]

]

]

]

]

1198665=

[

[

[

[

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

0

0

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

]

]

]

]

(A1)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










r(s)C(s) Kd(s)

K(s)

Q(s)G(s) = G0(s)(I + G0(s))

minus1y(s)

minus



119870 (119904) = 119870119889 (119904) 119862 (119904)

119896119894119895 (119904) = 119888119894119894 (

119904) 119870119889119894119895(119904) 119894 119895 = 1 2

(17)


119867(119904) = 119866 (119904)119870119889 (119904) 119862 (119904) [119868 + 119866 (119904)119870119889 (

119904) 119862 (119904)]minus1

= det1198660 (119904) 119866119886 (

119904) sdot 119862 (119904) [119868 + det1198660 (119904) 119866119886 (

119904) sdot 119862 (119904)]minus1

= diag ℎ11 (119904) ℎ22 (

119904)

(18)


119862 (119904) = (119867 (119904)minus1minus 119868)

minus1

sdot det1198660 (119904) 119866119886 (

119904)

= diag ℎ11 (119904)

(1 minus ℎ11 (119904)) sdot 1199021 (

119904)

ℎ22 (119904)

(1 minus ℎ22 (119904)) sdot 1199022 (

119904)

(19)


ℎ119894119894 (119904) =

1205962

119899119894

(1199042+ 2120596

119899119894120585119894119904 + 120596

2

119899119894) ((120582

119894120596119899119894) 119904 + 1)

119897119894

prod119898119894

119896=1(119904 + 119911

119896)

prod119902119894

119895=1(119904 + 119901

119895)

119894 = 1 2

(20)

where 120582119894 120585119894 and 120596



The term (120582119894119904120596119899119894+ 1)


119889(119904) To make



119897119894= max 0 119903119889 (det (119866

0 (119904)))

minusmin 119903119889 (11986611989410(119904)) 119903119889 (119866

1198942

0(119904))

+ 119903119894+ 119898

119894minus 119902119894minus 1 119894 = 1 2

(21)













119878 (119904) = (119868 + 119866 (119904)119870 (119904))minus1

119879 (119904) = 119866 (119904)119870 (119904) (119868 + 119866 (119904)119870 (119904))minus1

(22)


119872st119894 = max120596(1003816100381610038161003816119878119894119894(119895120596)

10038161003816100381610038161003816100381610038161003816119879119894119894(119895120596)

1003816100381610038161003816) 119894 = 1 2 forall120596 isin 119877 (23)







IAE = intinfin

0

1003816100381610038161003816119903119894 (119905) minus 119910119894 (

119905)1003816100381610038161003816119889119905 119894 = 1 2 (24)



119894 120585119894 and 120596

119899119894 The



120596119894≜ 119904 isin 119862 | 119904 = 119895120596 120596

119904119894le 120596 le 120596

119898119894

The value of 120596119898119894


11990211989410 and 120596

119898119894= 10120596

119902119894 and 120596




119870119894119895 (119904) = 119860 (119904) =

119887119898119904119898+ sdot sdot sdot + 119887

1119904 + 119860 (0)

119886119899119904119899+ 119886119899minus1119904119899minus1

+ sdot sdot sdot + 1198861119904 + 1

(25)




119860119903 (119904) =

119873 (119904)

119863 (119904)

=

120573119903119904119903+ sdot sdot sdot + 120573

1119904 + 119860 (0)

120572119896119904119896+ 120572119896minus1119904119896minus1

+ sdot sdot sdot + 1205721119904 + 1

(26)

Denoting that 120579 = (1205721 120572

119896 1205731 120573




119869 = min119904isin119863120596119894

1

119873

119873

sum

119896=1

[1003816100381610038161003816119882 (119895119908

119896)1003816100381610038161003816

2sdot1003816100381610038161003816119860 (119895119908

119896) minus 119860

119903(119895119908

119896 120579)1003816100381610038161003816

2]

119882 (119895119908119896) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119863 (119895119908119896 120572119894)

119863 (119895119908119896 120572119894minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(27)



119903(119904) is



119870119891 (119904) =

1

1 + 119904119879119891+ 1199042(1198792

119891119873)

(28)


119870PID (119904) = 119870119891 (119904) (119870119901 +119879119894

119904

+ 119879119889119904)

=

1198791198891199042+ 119870

119901119904 + 119879

119894

119904 (1 + 119904119879119891+ 1199042(1198792

119891119873))

(29)






119875 (119904) asymp

12057321199042+ 1205731119904 + 119875 (0)

12057221199042+ 1205721119904 + 1

(30)



119870119901= 1205731 119879

119894= 119875 (0) 119879

119889= 1205732

119879119891= 1205721 119873 =

1205722

1

1205722

(31)


119866119901 (119904) = 119866 (119904) (119868 + Δ (119904)) (32)


1003816100381610038161003816119882119860(119895120596)

1003816100381610038161003816ge max119866119901isinΛ

120590 (119866minus1(119895120596) [119866

119901(119895120596) minus 119866 (119895120596)]) (33)




119872(119904) = minus119882119860 (119904) 119870PID (119904) [119868 + 119870PID (119904) 119866 (119904)]

minus1 (35)


4 Case Study






1198660 (119904) =

[

[

[

[

682713 (119904 + 4015)

1199042(119904 + 1135)

55329 (119904 minus 3285)

1199042(119904 + 1135)

55329 (119904 minus 3285)

1199042(119904 + 1135)

682713 (119904 + 4015)

1199042(119904 + 1135)

]

]

]

]

(36)



10 KN119860 00186m2

119897 265m119869 910 kgsdotm2

119873 320119896119888

3001205830

4120587 times 10minus7 Hm119896119901

4000119896119889

331205750

9mm119877 10Ω



119870119889 (119904) = (119868 + 1198660 (

119904)) adj1198660 (119904) 119866119886 (119904)

= [

1198961198891111989611988912

1198961198892111989611988922

]

11989611988911

=

6827131199041199031(119904 + 1074) (119904 + 1524) (119904

2+ 8604119904 + 3768)

1199044(119904 + 1135)

2(12058211199041205961198991+ 1)

1199031

11989611988912=

minus553291199041199031(119904 minus 3285)

1199042(119904 + 1135) (1205822

1199041205961198992+ 1)

1199032

11989611988921=

minus553291199041199031(119904 minus 3285)

1199042(119904 + 1135) (1205821

1199041205961198992+ 1)

1199031

11989611988922

=

6827131199041199032(119904 + 1074) (119904 + 1524) (119904

2+ 8604119904 + 3768)

1199044(119904 + 1135)

2(12058221199041205961198992+ 1)

1199032

(37)



1= 1199032= 4 After the




follows

119902119894 (119904) =

466 times 109(1199042+ 8072119904 + 9026)

(119904 + 1135)2(120582119894119904120596119899119894+ 1)

4 119894 = 1 2 (38)




02 03 04 05 06 07 08 09 11

12

14

16

18

2

22

24

26

28

3

120585

Mst


02 03 04 05 06 07 08 09 1004

0045

005

0055

006

0065

007

0075

008

0085

009

IAE

120585

120582 = 001

120582 = 005

120582 = 01






119888119894119894 (119904) =

1205962

119899119894

(1199042+ 2120596

119899119894120585119894119904 + 120596

2

119899119894) (120582

119894119904120596119899119894+ 1) minus 120596

2

119899119894

sdot

(119904 + 1135)2(120582119894119904120596119899119894+ 1)

4

466 times 109(1199042+ 8072119904 + 9026)

119894 = 1 2

(39)


119894 120585119894 and

120596119899119894




119899as a





119899











measurement noise


0 5 10 15 20 25 30 35 40 45 501

105

11

115

12

125

13

135

14M

st

120596n


0

005

01

015

02

025

03

IAE

0 5 10 15 20 25 30 35 40 45 50

120582 = 001

120582 = 005

120582 = 01

120596n




119899as follows




119903lt 01 s then we get 120596

119899gt 40



119899as 40


119894can serve as






119879119894

119879119889

119873 119879119891

1-12-2 391 438 00743 4927 010241-22-1 429 times 10minus6 0 00215 4927 01024




85

9

95

10

105

Gap

1 (m

m)

0 05 1 15 2 25 3Time (s)

(a)

0 05 1 15 2 25 385

9

95

10

105

Time (s)

Gap

2 (m

m)


(b)







119882119860 (119904) =

00318119904 + 04

(0031806) 119904 + 1

sdot 119868 (40)

Gap

1 (m

m)

85

9

95

10

105

0 05 1 15 2 25 3Time (s)

(a)

Gap

2 (m

m)

0 05 1 15 2 25 385

9

95

10

105

Time (s)


(b)









0

01

02

03

04

05

06

07

Frequency (rads)

Sing

ular

val

ues

10minus1 100 101 102 103


85

9

95

10

105Decoupling

0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 520

22

24

26

28

Time (s)

Curr

ent (

A)

Current 1Current 2

(b)




0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

85

9

95

10

105Nondecoupling

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Current 1Current 2

20

22

24

26

28

Curr

ent (

A)

(b)




5 Conclusions



0 05 1 15 2 25 3 35 42

4

6

8

10

12

14

16

18

Gap

(mm

)

Nondecoupling

Gap 1Gap 2


0 1 2 3 4 5 67

8

9

10

11

12

13

14

15

16

17

18

Gap

(mm

)

Decoupling

Gap 1Gap 2




Appendix


1198661=

[

[

[

[

[

minus

1

1198981

minus

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

minus

1198972

8119869

]

]

]

]

]

1198662= [

minus1198651198681

0

0 minus1198651198682

]

1198663=

[

[

[

[

31198651199111

4

1198651199111

4

1198651199112

4

31198651199112

4

]

]

]

]

1198666= [

119896119888(119896119901+ 119896119889119904) 0

0 119896119888(119896119901+ 119896119889119904)

]

1198664=

[

[

[

[

[

311989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

311989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

]

]

]

]

]

1198665=

[

[

[

[

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

0

0

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

]

]

]

]

(A1)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












IAE = intinfin

0

1003816100381610038161003816119903119894 (119905) minus 119910119894 (

119905)1003816100381610038161003816119889119905 119894 = 1 2 (24)



119894 120585119894 and 120596

119899119894 The



120596119894≜ 119904 isin 119862 | 119904 = 119895120596 120596

119904119894le 120596 le 120596

119898119894

The value of 120596119898119894


11990211989410 and 120596

119898119894= 10120596

119902119894 and 120596




119870119894119895 (119904) = 119860 (119904) =

119887119898119904119898+ sdot sdot sdot + 119887

1119904 + 119860 (0)

119886119899119904119899+ 119886119899minus1119904119899minus1

+ sdot sdot sdot + 1198861119904 + 1

(25)




119860119903 (119904) =

119873 (119904)

119863 (119904)

=

120573119903119904119903+ sdot sdot sdot + 120573

1119904 + 119860 (0)

120572119896119904119896+ 120572119896minus1119904119896minus1

+ sdot sdot sdot + 1205721119904 + 1

(26)

Denoting that 120579 = (1205721 120572

119896 1205731 120573




119869 = min119904isin119863120596119894

1

119873

119873

sum

119896=1

[1003816100381610038161003816119882 (119895119908

119896)1003816100381610038161003816

2sdot1003816100381610038161003816119860 (119895119908

119896) minus 119860

119903(119895119908

119896 120579)1003816100381610038161003816

2]

119882 (119895119908119896) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119863 (119895119908119896 120572119894)

119863 (119895119908119896 120572119894minus1)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(27)



119903(119904) is



119870119891 (119904) =

1

1 + 119904119879119891+ 1199042(1198792

119891119873)

(28)


119870PID (119904) = 119870119891 (119904) (119870119901 +119879119894

119904

+ 119879119889119904)

=

1198791198891199042+ 119870

119901119904 + 119879

119894

119904 (1 + 119904119879119891+ 1199042(1198792

119891119873))

(29)






119875 (119904) asymp

12057321199042+ 1205731119904 + 119875 (0)

12057221199042+ 1205721119904 + 1

(30)



119870119901= 1205731 119879

119894= 119875 (0) 119879

119889= 1205732

119879119891= 1205721 119873 =

1205722

1

1205722

(31)


119866119901 (119904) = 119866 (119904) (119868 + Δ (119904)) (32)


1003816100381610038161003816119882119860(119895120596)

1003816100381610038161003816ge max119866119901isinΛ

120590 (119866minus1(119895120596) [119866

119901(119895120596) minus 119866 (119895120596)]) (33)




119872(119904) = minus119882119860 (119904) 119870PID (119904) [119868 + 119870PID (119904) 119866 (119904)]

minus1 (35)


4 Case Study






1198660 (119904) =

[

[

[

[

682713 (119904 + 4015)

1199042(119904 + 1135)

55329 (119904 minus 3285)

1199042(119904 + 1135)

55329 (119904 minus 3285)

1199042(119904 + 1135)

682713 (119904 + 4015)

1199042(119904 + 1135)

]

]

]

]

(36)



10 KN119860 00186m2

119897 265m119869 910 kgsdotm2

119873 320119896119888

3001205830

4120587 times 10minus7 Hm119896119901

4000119896119889

331205750

9mm119877 10Ω



119870119889 (119904) = (119868 + 1198660 (

119904)) adj1198660 (119904) 119866119886 (119904)

= [

1198961198891111989611988912

1198961198892111989611988922

]

11989611988911

=

6827131199041199031(119904 + 1074) (119904 + 1524) (119904

2+ 8604119904 + 3768)

1199044(119904 + 1135)

2(12058211199041205961198991+ 1)

1199031

11989611988912=

minus553291199041199031(119904 minus 3285)

1199042(119904 + 1135) (1205822

1199041205961198992+ 1)

1199032

11989611988921=

minus553291199041199031(119904 minus 3285)

1199042(119904 + 1135) (1205821

1199041205961198992+ 1)

1199031

11989611988922

=

6827131199041199032(119904 + 1074) (119904 + 1524) (119904

2+ 8604119904 + 3768)

1199044(119904 + 1135)

2(12058221199041205961198992+ 1)

1199032

(37)



1= 1199032= 4 After the




follows

119902119894 (119904) =

466 times 109(1199042+ 8072119904 + 9026)

(119904 + 1135)2(120582119894119904120596119899119894+ 1)

4 119894 = 1 2 (38)




02 03 04 05 06 07 08 09 11

12

14

16

18

2

22

24

26

28

3

120585

Mst


02 03 04 05 06 07 08 09 1004

0045

005

0055

006

0065

007

0075

008

0085

009

IAE

120585

120582 = 001

120582 = 005

120582 = 01






119888119894119894 (119904) =

1205962

119899119894

(1199042+ 2120596

119899119894120585119894119904 + 120596

2

119899119894) (120582

119894119904120596119899119894+ 1) minus 120596

2

119899119894

sdot

(119904 + 1135)2(120582119894119904120596119899119894+ 1)

4

466 times 109(1199042+ 8072119904 + 9026)

119894 = 1 2

(39)


119894 120585119894 and

120596119899119894




119899as a





119899











measurement noise


0 5 10 15 20 25 30 35 40 45 501

105

11

115

12

125

13

135

14M

st

120596n


0

005

01

015

02

025

03

IAE

0 5 10 15 20 25 30 35 40 45 50

120582 = 001

120582 = 005

120582 = 01

120596n




119899as follows




119903lt 01 s then we get 120596

119899gt 40



119899as 40


119894can serve as






119879119894

119879119889

119873 119879119891

1-12-2 391 438 00743 4927 010241-22-1 429 times 10minus6 0 00215 4927 01024




85

9

95

10

105

Gap

1 (m

m)

0 05 1 15 2 25 3Time (s)

(a)

0 05 1 15 2 25 385

9

95

10

105

Time (s)

Gap

2 (m

m)


(b)







119882119860 (119904) =

00318119904 + 04

(0031806) 119904 + 1

sdot 119868 (40)

Gap

1 (m

m)

85

9

95

10

105

0 05 1 15 2 25 3Time (s)

(a)

Gap

2 (m

m)

0 05 1 15 2 25 385

9

95

10

105

Time (s)


(b)









0

01

02

03

04

05

06

07

Frequency (rads)

Sing

ular

val

ues

10minus1 100 101 102 103


85

9

95

10

105Decoupling

0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 520

22

24

26

28

Time (s)

Curr

ent (

A)

Current 1Current 2

(b)




0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

85

9

95

10

105Nondecoupling

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Current 1Current 2

20

22

24

26

28

Curr

ent (

A)

(b)




5 Conclusions



0 05 1 15 2 25 3 35 42

4

6

8

10

12

14

16

18

Gap

(mm

)

Nondecoupling

Gap 1Gap 2


0 1 2 3 4 5 67

8

9

10

11

12

13

14

15

16

17

18

Gap

(mm

)

Decoupling

Gap 1Gap 2




Appendix


1198661=

[

[

[

[

[

minus

1

1198981

minus

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

minus

1198972

8119869

]

]

]

]

]

1198662= [

minus1198651198681

0

0 minus1198651198682

]

1198663=

[

[

[

[

31198651199111

4

1198651199111

4

1198651199112

4

31198651199112

4

]

]

]

]

1198666= [

119896119888(119896119901+ 119896119889119904) 0

0 119896119888(119896119901+ 119896119889119904)

]

1198664=

[

[

[

[

[

311989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

311989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

]

]

]

]

]

1198665=

[

[

[

[

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

0

0

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

]

]

]

]

(A1)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119870119901= 1205731 119879

119894= 119875 (0) 119879

119889= 1205732

119879119891= 1205721 119873 =

1205722

1

1205722

(31)


119866119901 (119904) = 119866 (119904) (119868 + Δ (119904)) (32)


1003816100381610038161003816119882119860(119895120596)

1003816100381610038161003816ge max119866119901isinΛ

120590 (119866minus1(119895120596) [119866

119901(119895120596) minus 119866 (119895120596)]) (33)




119872(119904) = minus119882119860 (119904) 119870PID (119904) [119868 + 119870PID (119904) 119866 (119904)]

minus1 (35)


4 Case Study






1198660 (119904) =

[

[

[

[

682713 (119904 + 4015)

1199042(119904 + 1135)

55329 (119904 minus 3285)

1199042(119904 + 1135)

55329 (119904 minus 3285)

1199042(119904 + 1135)

682713 (119904 + 4015)

1199042(119904 + 1135)

]

]

]

]

(36)



10 KN119860 00186m2

119897 265m119869 910 kgsdotm2

119873 320119896119888

3001205830

4120587 times 10minus7 Hm119896119901

4000119896119889

331205750

9mm119877 10Ω



119870119889 (119904) = (119868 + 1198660 (

119904)) adj1198660 (119904) 119866119886 (119904)

= [

1198961198891111989611988912

1198961198892111989611988922

]

11989611988911

=

6827131199041199031(119904 + 1074) (119904 + 1524) (119904

2+ 8604119904 + 3768)

1199044(119904 + 1135)

2(12058211199041205961198991+ 1)

1199031

11989611988912=

minus553291199041199031(119904 minus 3285)

1199042(119904 + 1135) (1205822

1199041205961198992+ 1)

1199032

11989611988921=

minus553291199041199031(119904 minus 3285)

1199042(119904 + 1135) (1205821

1199041205961198992+ 1)

1199031

11989611988922

=

6827131199041199032(119904 + 1074) (119904 + 1524) (119904

2+ 8604119904 + 3768)

1199044(119904 + 1135)

2(12058221199041205961198992+ 1)

1199032

(37)



1= 1199032= 4 After the




follows

119902119894 (119904) =

466 times 109(1199042+ 8072119904 + 9026)

(119904 + 1135)2(120582119894119904120596119899119894+ 1)

4 119894 = 1 2 (38)




02 03 04 05 06 07 08 09 11

12

14

16

18

2

22

24

26

28

3

120585

Mst


02 03 04 05 06 07 08 09 1004

0045

005

0055

006

0065

007

0075

008

0085

009

IAE

120585

120582 = 001

120582 = 005

120582 = 01






119888119894119894 (119904) =

1205962

119899119894

(1199042+ 2120596

119899119894120585119894119904 + 120596

2

119899119894) (120582

119894119904120596119899119894+ 1) minus 120596

2

119899119894

sdot

(119904 + 1135)2(120582119894119904120596119899119894+ 1)

4

466 times 109(1199042+ 8072119904 + 9026)

119894 = 1 2

(39)


119894 120585119894 and

120596119899119894




119899as a





119899











measurement noise


0 5 10 15 20 25 30 35 40 45 501

105

11

115

12

125

13

135

14M

st

120596n


0

005

01

015

02

025

03

IAE

0 5 10 15 20 25 30 35 40 45 50

120582 = 001

120582 = 005

120582 = 01

120596n




119899as follows




119903lt 01 s then we get 120596

119899gt 40



119899as 40


119894can serve as






119879119894

119879119889

119873 119879119891

1-12-2 391 438 00743 4927 010241-22-1 429 times 10minus6 0 00215 4927 01024




85

9

95

10

105

Gap

1 (m

m)

0 05 1 15 2 25 3Time (s)

(a)

0 05 1 15 2 25 385

9

95

10

105

Time (s)

Gap

2 (m

m)


(b)







119882119860 (119904) =

00318119904 + 04

(0031806) 119904 + 1

sdot 119868 (40)

Gap

1 (m

m)

85

9

95

10

105

0 05 1 15 2 25 3Time (s)

(a)

Gap

2 (m

m)

0 05 1 15 2 25 385

9

95

10

105

Time (s)


(b)









0

01

02

03

04

05

06

07

Frequency (rads)

Sing

ular

val

ues

10minus1 100 101 102 103


85

9

95

10

105Decoupling

0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 520

22

24

26

28

Time (s)

Curr

ent (

A)

Current 1Current 2

(b)




0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

85

9

95

10

105Nondecoupling

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Current 1Current 2

20

22

24

26

28

Curr

ent (

A)

(b)




5 Conclusions



0 05 1 15 2 25 3 35 42

4

6

8

10

12

14

16

18

Gap

(mm

)

Nondecoupling

Gap 1Gap 2


0 1 2 3 4 5 67

8

9

10

11

12

13

14

15

16

17

18

Gap

(mm

)

Decoupling

Gap 1Gap 2




Appendix


1198661=

[

[

[

[

[

minus

1

1198981

minus

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

minus

1198972

8119869

]

]

]

]

]

1198662= [

minus1198651198681

0

0 minus1198651198682

]

1198663=

[

[

[

[

31198651199111

4

1198651199111

4

1198651199112

4

31198651199112

4

]

]

]

]

1198666= [

119896119888(119896119901+ 119896119889119904) 0

0 119896119888(119896119901+ 119896119889119904)

]

1198664=

[

[

[

[

[

311989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

311989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

]

]

]

]

]

1198665=

[

[

[

[

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

0

0

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

]

]

]

]

(A1)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










02 03 04 05 06 07 08 09 11

12

14

16

18

2

22

24

26

28

3

120585

Mst


02 03 04 05 06 07 08 09 1004

0045

005

0055

006

0065

007

0075

008

0085

009

IAE

120585

120582 = 001

120582 = 005

120582 = 01






119888119894119894 (119904) =

1205962

119899119894

(1199042+ 2120596

119899119894120585119894119904 + 120596

2

119899119894) (120582

119894119904120596119899119894+ 1) minus 120596

2

119899119894

sdot

(119904 + 1135)2(120582119894119904120596119899119894+ 1)

4

466 times 109(1199042+ 8072119904 + 9026)

119894 = 1 2

(39)


119894 120585119894 and

120596119899119894




119899as a





119899











measurement noise


0 5 10 15 20 25 30 35 40 45 501

105

11

115

12

125

13

135

14M

st

120596n


0

005

01

015

02

025

03

IAE

0 5 10 15 20 25 30 35 40 45 50

120582 = 001

120582 = 005

120582 = 01

120596n




119899as follows




119903lt 01 s then we get 120596

119899gt 40



119899as 40


119894can serve as






119879119894

119879119889

119873 119879119891

1-12-2 391 438 00743 4927 010241-22-1 429 times 10minus6 0 00215 4927 01024




85

9

95

10

105

Gap

1 (m

m)

0 05 1 15 2 25 3Time (s)

(a)

0 05 1 15 2 25 385

9

95

10

105

Time (s)

Gap

2 (m

m)


(b)







119882119860 (119904) =

00318119904 + 04

(0031806) 119904 + 1

sdot 119868 (40)

Gap

1 (m

m)

85

9

95

10

105

0 05 1 15 2 25 3Time (s)

(a)

Gap

2 (m

m)

0 05 1 15 2 25 385

9

95

10

105

Time (s)


(b)









0

01

02

03

04

05

06

07

Frequency (rads)

Sing

ular

val

ues

10minus1 100 101 102 103


85

9

95

10

105Decoupling

0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 520

22

24

26

28

Time (s)

Curr

ent (

A)

Current 1Current 2

(b)




0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

85

9

95

10

105Nondecoupling

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Current 1Current 2

20

22

24

26

28

Curr

ent (

A)

(b)




5 Conclusions



0 05 1 15 2 25 3 35 42

4

6

8

10

12

14

16

18

Gap

(mm

)

Nondecoupling

Gap 1Gap 2


0 1 2 3 4 5 67

8

9

10

11

12

13

14

15

16

17

18

Gap

(mm

)

Decoupling

Gap 1Gap 2




Appendix


1198661=

[

[

[

[

[

minus

1

1198981

minus

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

minus

1198972

8119869

]

]

]

]

]

1198662= [

minus1198651198681

0

0 minus1198651198682

]

1198663=

[

[

[

[

31198651199111

4

1198651199111

4

1198651199112

4

31198651199112

4

]

]

]

]

1198666= [

119896119888(119896119901+ 119896119889119904) 0

0 119896119888(119896119901+ 119896119889119904)

]

1198664=

[

[

[

[

[

311989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

311989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

]

]

]

]

]

1198665=

[

[

[

[

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

0

0

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

]

]

]

]

(A1)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 30 35 40 45 501

105

11

115

12

125

13

135

14M

st

120596n


0

005

01

015

02

025

03

IAE

0 5 10 15 20 25 30 35 40 45 50

120582 = 001

120582 = 005

120582 = 01

120596n




119899as follows




119903lt 01 s then we get 120596

119899gt 40



119899as 40


119894can serve as






119879119894

119879119889

119873 119879119891

1-12-2 391 438 00743 4927 010241-22-1 429 times 10minus6 0 00215 4927 01024




85

9

95

10

105

Gap

1 (m

m)

0 05 1 15 2 25 3Time (s)

(a)

0 05 1 15 2 25 385

9

95

10

105

Time (s)

Gap

2 (m

m)


(b)







119882119860 (119904) =

00318119904 + 04

(0031806) 119904 + 1

sdot 119868 (40)

Gap

1 (m

m)

85

9

95

10

105

0 05 1 15 2 25 3Time (s)

(a)

Gap

2 (m

m)

0 05 1 15 2 25 385

9

95

10

105

Time (s)


(b)









0

01

02

03

04

05

06

07

Frequency (rads)

Sing

ular

val

ues

10minus1 100 101 102 103


85

9

95

10

105Decoupling

0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 520

22

24

26

28

Time (s)

Curr

ent (

A)

Current 1Current 2

(b)




0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

85

9

95

10

105Nondecoupling

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Current 1Current 2

20

22

24

26

28

Curr

ent (

A)

(b)




5 Conclusions



0 05 1 15 2 25 3 35 42

4

6

8

10

12

14

16

18

Gap

(mm

)

Nondecoupling

Gap 1Gap 2


0 1 2 3 4 5 67

8

9

10

11

12

13

14

15

16

17

18

Gap

(mm

)

Decoupling

Gap 1Gap 2




Appendix


1198661=

[

[

[

[

[

minus

1

1198981

minus

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

minus

1198972

8119869

]

]

]

]

]

1198662= [

minus1198651198681

0

0 minus1198651198682

]

1198663=

[

[

[

[

31198651199111

4

1198651199111

4

1198651199112

4

31198651199112

4

]

]

]

]

1198666= [

119896119888(119896119901+ 119896119889119904) 0

0 119896119888(119896119901+ 119896119889119904)

]

1198664=

[

[

[

[

[

311989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

311989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

]

]

]

]

]

1198665=

[

[

[

[

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

0

0

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

]

]

]

]

(A1)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










85

9

95

10

105

Gap

1 (m

m)

0 05 1 15 2 25 3Time (s)

(a)

0 05 1 15 2 25 385

9

95

10

105

Time (s)

Gap

2 (m

m)


(b)







119882119860 (119904) =

00318119904 + 04

(0031806) 119904 + 1

sdot 119868 (40)

Gap

1 (m

m)

85

9

95

10

105

0 05 1 15 2 25 3Time (s)

(a)

Gap

2 (m

m)

0 05 1 15 2 25 385

9

95

10

105

Time (s)


(b)









0

01

02

03

04

05

06

07

Frequency (rads)

Sing

ular

val

ues

10minus1 100 101 102 103


85

9

95

10

105Decoupling

0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 520

22

24

26

28

Time (s)

Curr

ent (

A)

Current 1Current 2

(b)




0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

85

9

95

10

105Nondecoupling

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Current 1Current 2

20

22

24

26

28

Curr

ent (

A)

(b)




5 Conclusions



0 05 1 15 2 25 3 35 42

4

6

8

10

12

14

16

18

Gap

(mm

)

Nondecoupling

Gap 1Gap 2


0 1 2 3 4 5 67

8

9

10

11

12

13

14

15

16

17

18

Gap

(mm

)

Decoupling

Gap 1Gap 2




Appendix


1198661=

[

[

[

[

[

minus

1

1198981

minus

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

minus

1198972

8119869

]

]

]

]

]

1198662= [

minus1198651198681

0

0 minus1198651198682

]

1198663=

[

[

[

[

31198651199111

4

1198651199111

4

1198651199112

4

31198651199112

4

]

]

]

]

1198666= [

119896119888(119896119901+ 119896119889119904) 0

0 119896119888(119896119901+ 119896119889119904)

]

1198664=

[

[

[

[

[

311989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

311989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

]

]

]

]

]

1198665=

[

[

[

[

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

0

0

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

]

]

]

]

(A1)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

01

02

03

04

05

06

07

Frequency (rads)

Sing

ular

val

ues

10minus1 100 101 102 103


85

9

95

10

105Decoupling

0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 520

22

24

26

28

Time (s)

Curr

ent (

A)

Current 1Current 2

(b)




0 05 1 15 2 25 3 35 4 45 5Time (s)

Gap 1Gap 2

85

9

95

10

105Nondecoupling

Gap

(mm

)

(a)

0 05 1 15 2 25 3 35 4 45 5Time (s)

Current 1Current 2

20

22

24

26

28

Curr

ent (

A)

(b)




5 Conclusions



0 05 1 15 2 25 3 35 42

4

6

8

10

12

14

16

18

Gap

(mm

)

Nondecoupling

Gap 1Gap 2


0 1 2 3 4 5 67

8

9

10

11

12

13

14

15

16

17

18

Gap

(mm

)

Decoupling

Gap 1Gap 2




Appendix


1198661=

[

[

[

[

[

minus

1

1198981

minus

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

minus

1198972

8119869

]

]

]

]

]

1198662= [

minus1198651198681

0

0 minus1198651198682

]

1198663=

[

[

[

[

31198651199111

4

1198651199111

4

1198651199112

4

31198651199112

4

]

]

]

]

1198666= [

119896119888(119896119901+ 119896119889119904) 0

0 119896119888(119896119901+ 119896119889119904)

]

1198664=

[

[

[

[

[

311989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

311989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

]

]

]

]

]

1198665=

[

[

[

[

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

0

0

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

]

]

]

]

(A1)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 2 25 3 35 42

4

6

8

10

12

14

16

18

Gap

(mm

)

Nondecoupling

Gap 1Gap 2


0 1 2 3 4 5 67

8

9

10

11

12

13

14

15

16

17

18

Gap

(mm

)

Decoupling

Gap 1Gap 2




Appendix


1198661=

[

[

[

[

[

minus

1

1198981

minus

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

+

1198972

8119869

minus

1

1198981

minus

1198972

8119869

]

]

]

]

]

1198662= [

minus1198651198681

0

0 minus1198651198682

]

1198663=

[

[

[

[

31198651199111

4

1198651199111

4

1198651199112

4

31198651199112

4

]

]

]

]

1198666= [

119896119888(119896119901+ 119896119889119904) 0

0 119896119888(119896119901+ 119896119889119904)

]

1198664=

[

[

[

[

[

311989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986810119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

11989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

311989611989011986820119904

21205750((119877 + 119896

119888) 1205750+ 2119896

119890119904)

]

]

]

]

]

1198665=

[

[

[

[

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

0

0

1205750

(119877 + 119896119888) 1205750+ 2119896

119890119904

]

]

]

]

(A1)



Acknowledgments


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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