decoupling control for dual-winding bearingless switched...

Research ArticleDecoupling Control for Dual-Winding Bearingless SwitchedReluctance Motor Based on Improved Inverse System Method

Zhiying Zhu1 Yukun Sun1 and Ye Yuan2

1School of Electric Power Engineering Nanjing Institute of Technology Nanjing 211167 China2School of Electrical and Information Engineering Jiangsu University Zhenjiang 212013 China

Correspondence should be addressed to Zhiying Zhu zyzhunjiteducn

Received 12 October 2016 Revised 4 January 2017 Accepted 11 January 2017 Published 16 February 2017

Academic Editor Luis J Yebra

Copyright copy 2017 Zhiying Zhu 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

Dual-winding bearingless switched reluctance motor (BSRM) is a multivariable high-nonlinear system characterized by strongcoupling and it is not completely reversible In this paper a new decoupling control strategy based on improved inverse systemmethod is proposed Robust servo regulator is adopted for the decoupled plants to guarantee control performances and robustnessA phase dynamic compensation filter is also designed to improve system stability at high-speed In order to explain the advantagesof the proposed method traditional methods are compared The tracking and decoupling characteristics as well as disturbancerejection and robustness are deeply analyzed Simulation and experiments results show that the decoupling control of dual-windingBSRM in both reversible and irreversible domains can be successfully resolved with the improved inverse system method Thestability and robustness problems induced by inverse controller can be effectively solved by introducing robust servo regulator anddynamic compensation filter

1 Introduction

Switched reluctance motors (SRM) are favored in harshconditions and some high-speed driving applications owingto their rugged structure fault tolerance and robustness[1ndash4] The drawback is that SRM running at high-speedusually suffer from the mechanical friction between shaftand bearing Magnetic bearings (MB) present the advantagesof no lubrication and wear during high-speed operationbut MB need extra axial space thus the shaft length ofmagnetic-bearing SRM is usually increased and its criticalrotating speed is limited Bearingless switched reluctancemotor (BSRM) that combines MB with SRM is becominga promising alternative to the traditional SRM because ofits inherent superior features such as zero friction nolubrication no wear short rotor shaft high critical speedlong life and adjustable bearing stiffness and damping [5]

Recently several types of BSRM have been proposedfor example dual-winding [6] single-winding [7 8] hybrid-rotor [9] hybrid-stator [10] double-stator [11] and per-manent magnet-biased [12 13] types Among these typesdual-winding BSRM possesses double saliency with two

kinds of concentrated stator windings Dual-winding BSRMoffers simpler structure and clearer winding function andthus receives more attention than other types of BSRMConversely the double saliency leads to complex nonlinearcharacteristics and dual-winding results in mutual couplingbetween the torque and radial forceTherefore dual-windingBSRM is amultivariable high-nonlinearmotor characterizedby strong coupling which presents a challenging controlproblem

Regarding the control of dual-winding BSRM the typ-ical method is square-wave current control proposed byTakemoto et al In [14 15] the square-wave current controlhas realized the stable operation of dual-winding BSRMfrom no load to full load and the influences of magneticsaturation and coupling effects on the torque and radialforce are considered [16 17] Furthermore an independentcontrol strategy of average torque and radial force waspresented [18] In this method the current calculating algo-rithm is deduced to minimize the magnitude of instanta-neous torque in the levitation region In addition the leastmagnetomotive force strategy was investigated to enhancethe availability of winding currents and to decrease the

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 5853423 17 pageshttpsdoiorg10115520175853423

2 Mathematical Problems in Engineering

torque ripple and stator vibration [19] Recently interest inthe study on decoupling control has been increasing [20ndash24] mainly including the differential geometric method [2021] and inverse system method [22ndash25] Compared withthe former the latter needs neither to use the complexnonlinear coordinate transformation nor to transfer thenonlinear control problems into geometric ones Thereforethe inverse system method is relatively simple to implementin practice In detail the inverse system method includesanalytic inverse system [22 23] and intelligent ones [24 25]Compared with analytic inverse system intelligent ones neednot rely on the precise mathematical model but they usuallyrequire large allocations of computer resources that are oftentypically restricted in high-speed system Hence analyticinverse system method has attracted more attention thanintelligent ones in high-speed magnetically suspended field[26 27]

The implementation of analytic inverse system methodrequires the controlled object to be reversible However itis difficult to satisfy the reversibility in practical systemAccording to [23] the dual-winding BSRM is not completelyreversible and its working area includes two parts reversibleand irreversible domains Traditional analytic inverse sys-tem method can only realize the decoupling control inthe reversible domain while the decoupling control inthe irreversible domain cannot be realized To solve theseissues a modified inverse decoupling control method forBSRM operating in irreversible domain is proposed byauthors in [23] However it is known that the analyticinverse controller usually affects the robustness and stabilityof system because uncertainties and model errors alwaysexist in reality Especially at high-speed the mathemati-cal model is not equivalent to the actual system becausethe former does not consider the amplifiers bandwidthand computation delay and these dynamics can degradedecoupling performance and even endanger system stabilityTo combat these adverse effects proportion-integration-differentiation (PID) control [22ndash25] and internal modelcontrol [26] have been typically employed for the decoupledplants Nevertheless nearly all these methods experiencedifficulty in realizing tracking and robustness independently[27]

This study presents a novel decoupling control strat-egy based on improved inverse system method to realizethe decoupling control of dual-winding BSRM in boththe reversible and irreversible domains Robust servo reg-ulator is adopted for the decoupled plants to guaranteecontrol performances and robustness A phase dynamiccompensation filter (DCF) is also designed to improvesystem stability at high-speed One main contribution ofthis study is to demonstrate that the decoupling control ofdual-winding BSRM at high-speed (up to 20000 rmin) inboth reversible and irreversible domains can all be success-fully resolved with the improved inverse system methodThe other contribution is to show that the stability androbustness problems induced by inverse controller can beeffectively solved by introducing robust servo regulator andDCF

Torque winding

Suspending winding

F

ima+

isa1minus

isa1+

imaminus

Nma

Nsa2

Nsa1

Ψma

Ψsa1

Air gap-2 Air gap-1

isa2minus

isa2+

Nma

Nma

Nsa2

Figure 1 Principle of radial force production of dual-windingBSRM

2 Principle and Characteristic Analysis ofthe Dual-Winding BSRM

21 Radial Force Production Principle of Dual-Winding BSRMFigure 1 shows the principle of radial force production ofdual-winding BSRM with A-phase windings The torquewinding 119873119898119886 consists of four coils connected in seriesand the suspending windings 1198731199041198861 and 1198731199041198862 consist of twocoils each When the torque winding 119873119898119886 and suspendingwinding 1198731199041198861 conduct the currents 119894119898119886 and 1198941199041198861 respectivelythe symmetrical four-pole main fluxes Ψ119898119886 and two-polesuspending fluxes Ψ1199041198861 should be produced The flux densityin air gap-1 increases but decreases in air gap-2 Thereforethis superimposed magnetic field results in the radial force119865120572 acting on the rotor in the 120572-axis The radial force 119865120573 in the120573-axis can also be produced in the samemanner Radial forcein any desired direction can be produced by generating thetwo radial forces This principle can be similarly applied tothe B-phase and C-phase windings

22 Mathematical Model of Dual-Winding BSRM Neglectingthe leakage flux and saturation effects the theoretical formu-lae of the torque and radial forces can be derived from thederivatives of the stored magnetic energy 119882 with respect todisplacements 120572 and 120573 and rotor position 120579 as [17]

119865120572 = 120597119882120597120572 = 119894119898119886 (1198701198911 sdot 1198941199041198861 minus 1198701198912 sdot 1198941199041198862) 119865120573 = 120597119882120597120573 = 119894119898119886 (1198701198912 sdot 1198941199041198861 + 1198701198911 sdot 1198941199041198862) 119879119890 = 120597119882120597120579 = 119870119905 (211987321198981198942119898119886 + 1198732119904 11989421199041198861 + 1198732119904 11989421199041198862)

(1)

where 119873119898 and 119873119904 are the number of turns of torquewinding and suspending windings 119894119898119886 1198941199041198861 and 1198941199041198862 are the

Mathematical Problems in Engineering 3

currents inA-phase torquewinding and suspendingwindingrespectively 119865120572 and 119865120573 are the radial force acting on rotorin 120572- and 120573-axis 119879119890 is the electromagnetic torque acting onrotor and 1198701198911 1198701198912 and 119870119905 are the proportional coefficientsof radial force and torque and they are the functions of therotor position and motor dimensions [17]

1198701198911 = 119873119898119873119904 (1205830119897119903 (120587 minus 12 |120579|)61205752+ 321205830119897119903119888 |120579|120587 (4119903119888 |120579| 120575 + 1205871205752))

1198701198912 = 119873119898119873119904(1205830119897119903 (120587 minus 12 |120579|) |120579|121205752 minus 21205830119897120575+ 161205830119897119888 (119903 |120579|2 + 2120575)120587 (4119903119888 |120579| 120575 + 1205871205752) )

119870119905 =

1205830119897119903120575 minus 161205830119897119903 (120575 minus 119903 |120579|)(4120575 minus 120587119903 |120579|)2 minus 12058712 le 120579 le 0minus1205830119897119903120575 + 161205830119897119903 (120575 minus 119903 |120579|)(4120575 minus 120587119903 |120579|)2 0 le 120579 le 12058712

(2)

where 120575 is the air-gap length 119903 is the radius of the rotor pole119897 is the axial stack length 120579 is the rotor position from thealigned position of exciting phase 1205830 is the permeability ofvacuum (4120587 times 10minus7) and 119888 is a constant of 149

The actual control performance of the inverse systemmethod largely depends on the precision of the mathematicalmodelThemajor factors that affect the accuracy of themodelare the torque and radial forces Fortunately the theoreticalrelationships (1) are verified with experimental results in [17]by considering cross coupling and fringing fluxes The testvalues show good agreements with those from the modelconfirming that formulae (1) are reasonable Mathematicalmodel (1) is used in this study for its convenience andaccuracy

According to Newtonrsquos second law and rotor dynamicsthe dynamic model of the rotor can be described as

119898 = 119865120572119898 120573 = 119865120573 minus 119898119892119869 = 119879119890 minus 119879119871

(3)

where 119898 is the mass of the rotor 119892 is the acceleration ofgravity 119869 is the moments of inertia of the rotor 120572 and 120573are the linear displacements of the rotor in 120572- and 120573-axisrespectively 120596 is the mechanical angular velocity of the rotorand119879119871 refers to the load torque Substituting (1) into (3) yields

= 1119898 [119894119898119886 (1198701198911 sdot 1198941199041198861 minus 1198701198912 sdot 1198941199041198862)] 120573 = 1119898 [119894119898119886 (1198701198912 sdot 1198941199041198861 + 1198701198911 sdot 1198941199041198862)] minus 119892

= 1119869 [119870119905 (211987321198981198942119898119886 + 1198732119904 11989421199041198861 + 1198732119904 11989421199041198862) minus 119879119871] (4)

23 Nonlinearity and Coupling Characteristics of Dual-Winding BSRM According to the first equation of the equa-tions set in (1) 119865120572 is the function of 1198941199041198861 1198941199041198862 and 119894119898119886Therefore 119865120572 is coupling with 119865120573 and 119879119890 The couplingof 119865120573 and 119879119890 can be drawn similarly Dynamic couplingalways exists among 119865120572 119865120573 and 119879119890 Also from equationsset in (1) 119879119890 is quadratic to 119894119898119886 1198941199041198861 and 1198941199041198862 and 119865120572 and119865120573 are proportional to the product of 119894119898119886 1198941199041198861 and 1198941199041198862Strong nonlinearity exists between F = 119865120572 119865120573 119879119890 and i =119894119898119886 1198941199041198861 1198941199041198862 and the greater the current value the strongerthe nonlinearity

The coupling and nonlinearity characteristics can be seenclearly with finite element analysis (FEA) Figure 2 shows theFEA results of 119879119890 119865120573 and 119865120572 at different 120579 with different 1198941199041198862under the conditions 120572 = 120573 = 0mm 119894119898119886 = 10A and 1198941199041198862 =0A

As shown in Figure 2(a) torque 119879119890 is nonlinear to rotorposition 120579 and the greater the current 1198941199041198862 the greater thetorque 119879119890 Thus current 1198941199041198862 can produce torque 119879119890 thatis torque 119879119890 is coupling with radial force 119865120573 Figure 2(b)illustrates that radial force 119865120573 is nonlinear to rotor position120579 and current 1198941199041198862 Figure 2(c) demonstrates that radial force119865120572 continues to exist when current 1198941199041198861 = 0 and the greaterthe current 1198941199041198862 the greater the radial force 119865120572 At the samecurrent 1198941199041198862 radial force 119865120572 is greatest at 120579 = minus15∘

According to the aforementioned analysis we con-clude that the dual-winding BSRM is a multivariable high-nonlinear motor with strong coupling not only betweenthe radial forces but also between the torque and radialforces To realize the rotor translation and motion controlthe decoupling control between the torque and radial forcesshould be achieved

3 Decoupling Control of Dual-Winding BSRM

31 Reversibility of Dual-Winding BSRM According to (4)state variables x input variables u and output variables y canbe defined as follows

x = [1199091 1199092 1199093 1199094 1199095 1199096]T = [120572 120573 120579 120573 120596]T (5)

u = [1199061 1199062 1199063]T = [1198941199041198861 1198941199041198862 119894119898119886]T (6)

y = [1199101 1199102 1199103]T = [120572 120573 120596]T (7)Then the corresponding state-variable equation of the non-linear system (4) can be rewritten as

x = 119891 (x u)

=

[[[[[[[[[[[[[[[[

1199094119909511990961119898 [1199063 (1198701198911 sdot 1199061 minus 1198701198912 sdot 1199062)]1119898 [1199063 (1198701198912 sdot 1199061 minus 1198701198911 sdot 1199062)] minus 119892

1119869 [119870119905 (2119873211989811990623 + 1198732119904 11990621 + 1198732119904 11990622) minus 119879119871]

]]]]]]]]]]]]]]]]

(8)


0Rotor position (deg)

isa2 = 2Aisa2 = 6A

isa2 = 10A

minus25 minus5minus20 minus15 minus10minus2

0

2

4

6

8

10

12To

rque

Te

(Nmiddotm

)

(a)


minus25 minus5minus20 minus15 minus100

1000

2000

3000

4000

5000

6000

Radi

al fo

rceF

(N

)

isa2 = 2Aisa2 = 6A

isa2 = 10A

(b)



minus150

minus100

minus50

0

50

Radi

al fo

rceF

(N

)

isa2 = 2Aisa2 = 6A

isa2 = 10A

(c)

Figure 2 FEA results of torque 119879119890 and radial forces 119865120573 and 119865120572 at different rotor positions with different values of 1198941199041198862 under the conditions120572 = 120573 = 0mm 119894119898119886 = 10A and 1198941199041198861 = 0A (a) FEA results of 119879119890 (b) FEA results of 119865120573 (c) FEA results of 119865120572

From (5)ndash(8) the state equation of dual-winding BSRM isa six-order nonlinear system with three inputs and threeoutputs and its reversibility must be justified

When analyzing the reversibility of the system the firststep is to take the derivative of output y = [1199101 1199102 1199103]T =[120572 120573 120596]T with respect to time until the components of u =[1199061 1199062 1199063]T = [1198941199041198861 1198941199041198862 119894119898119886]T are explicitly included [28]Then we can obtain

J (u) = [[[

119910111991021199103]]]

=[[[[[[[[


1119869 [119870119905 (2119873211989811990623 + 1198732119904 11990621 + 1198732119904 11990622) minus 119879119871]

]]]]]]]]

(9)

Taking the derivative of J(u) we further obtain the Jacobimatrix A as follows

A =[[[[[[[[[

120597 11991011205971199061120597 11991011205971199062

120597 11991011205971199063120597 11991021205971199061120597 11991021205971199062

120597 11991021205971199063120597 11991031205971199061120597 11991031205971199062

120597 11991031205971199063

]]]]]]]]] (10)

Including (9) into (10) the Jacobi matrixA can be resolved as

A =[[[[[[[[[[

11987011989111199063119898minus11987011989121199063119898

11987011989111199061 minus 119870119891211990621198981198701198912119906311989811987011989111199063119898

11987011989121199061 + 1198701198911119906211989821198701199051198732119904 1199061119869 21198701199051198732119904 1199062119869 411987011990511987321198981199063119869

]]]]]]]]]]

(11)

Hencedet (A)= 2119870119905 (11987021198911 + 11987021198912)1198691198982 1199063 (2119873211989811990623 minus 1198732119904 11990621 minus 1198732119904 11990622)

(12)


Obviously 2119870119905(11987021198911 + 11987021198912)1198691198982 = 0 Therefore if 1199063 = 0and 2119873211989811990623 minus 1198732119904 11990621 minus 1198732119904 11990622 = 0 that is if 119894119898119886 = 0 and1198942119898119886 = (1198732119904 11989421199041198861+1198732119904 11989421199041198862)(21198732119898) then the inequality det(A) = 0always holds and the relative order is 120572 = [1205721 1205722 1205723] =[2 2 1] which satisfies1205721+1205722+1205723 = 5 le 119899 (119899 is the number ofthe state variables defined in (5)) According to inverse systemtheory [28] the system is reversible Contrarily if 119894119898119886 = 0 or1198942119898119886 = (1198732119904 11989421199041198861+1198732119904 11989421199041198862)(21198732119898) then the system is irreversibleThus the mathematical model of dual-winding BSRM is notcompletely reversible The working area can be divided intoreversible domainD and irreversible domain D

D = 119894119898119886 = 0 1198942119898119886 = 1198732119904 11989421199041198861 + 1198732119904 1198942119904119886221198732119898 D = 119894119898119886 = 0 or 1198942119898119886 = 1198732119904 11989421199041198861 + 1198732119904 1198942119904119886221198732119898

(13)

To realize the decoupling control of dual-winding BSRM thecontrol scheme should be designed in both reversible domainD and irreversible domain D

32 Inversion in the Reversible Domain From (13) theinequality of 119894119898119886 = 0 always holds when dual-winding BSRMis working in reversible domainD According to the first twoequations of the equations set in (4) we can obtain

1198941199041198861 = 119898(1198701198911 + 1198701198912 120573 + 1198921198701198912)(11987021198911+ 11987021198912) 119894119898119886

1198941199041198862 = 119898(1198701198911 120573 minus 1198701198912 + 1198921198701198911)(11987021198911+ 11987021198912) 119894119898119886

(14)

Then integrating (14) into the third equation of the equationsset in (4) gives

211987321198981198944119898119886 minus 119869 + 119879119871119870119905 1198942119898119886 + 1198732119904119898211987021198911+ 11987021198912

[2 + (119892 + 120573)2]= 0

(15)

Accordingly the discriminant of the quadratic equation (15)can be given by

Δ = (119869 + 119879119871119870119905 )2 minus 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 120573)2] (16)

Considering that (15) is a unary quadratic equation and itsdiscriminant Δ ge 0 always holds its roots exist undoubtedlyaccording to the implicit function theorem The two roots of(15) can be described as follows

119894211989811988612 = 141198732119898 (119869 + 119879119871119870119905 plusmn radicΔ) (17)

In an actual system if the value of torque winding current 119894119898119886is small then the bias magnetic field in dual-winding BSRM

will be weak which is unsatisfactory to generate continuousradial force Hence the value of torque winding current 119894119898119886is selected as

119894119898119886 = 12119873119898radic119869 + 119879119871119870119905 + radicΔ (18)

Then substituting (18) into (14) the values of suspendingwinding currents 1198941199041198861 and 1198941199041198862 can be calculated as follows

1198941199041198861 = 2119898119873119898 (1198701198911 + 1198701198912 120573 + 1198921198701198912)(11987021198911+ 11987021198912)radic[(119869 + 119879119871) 119870119905] + radicΔ

1198941199041198862 = 2119898119873119898 (1198701198911 120573 minus 1198701198912 + 1198921198701198911)(11987021198911+ 11987021198912)radic[(119869 + 119879119871) 119870119905] + radicΔ

(19)

According to inverse system theory [28] we define the newinput variables [1205931 1205932 1205933]T = [ 1199101198891 1199101198892 1199101198893]T here 119910119889119894 (119894 =1 2 3) denotes the desired outputs By substituting variables[1205931 1205932 1205933]T for [ 1199101 1199102 1199103]T we can obtain the current-modeinversion of dual-winding BSRM in reversible domainD as

1199061 = 2119898119873119898 (11987011989111205931 + 11987011989121205932 + 1198921198701198912)(11987021198911+ 11987021198912)radic[(1198691205933 + 119879119871) 119870119905] + radicΔ

1199062 = 2119898119873119898 (11987011989111205932 minus 11987011989121205931 + 1198921198701198911)(11987021198911+ 11987021198912)radic[(1198691205933 + 119879119871) 119870119905] + radicΔ

1199063 = radic[(1198691205933 + 119879119871) 119870119905] + radicΔ(2119873119898)

(20)

33 Improved Inversion in the Irreversible Domain Accordingto (13) the irreversible domain of dual-winding BSRMincludes the following two parts

D = D1 = 119894119898119886 = 0D2 = 1198942119898119886 = 1198732119904 11989421199041198861 + 1198732119904 1198942119904119886221198732119898 (21)

(I) In the first part of irreversible domain D1 substituting theequality of 119894119898119886 = 0 into (4) yields

= 0120573 = minus119892 (22)

From (22) the rotor will be in free fall when 119894119898119886 = 0 becauseno bias magnetic field is present to generate radial force forbalancing rotor gravity From the radial force productionprinciple of dual-winding BSRM a bias magnetic field mustexist in the dual-winding BSRM to produce radial force andthus current 119894119898119886 cannot be zero The first part of irreversibledomain deduced by the theoretical analysis does not exist in


the actual systemTherefore it does not have to be consideredin decoupling control

(II) In the second part of irreversible domain D2 equality1198942119898119886 = (1198732119904 11989421199041198861 + 1198732119904 11989421199041198862)(21198732119898) always holds Therefore thediscriminant of (16) equals zero (ie Δ = 0)

(119869 + 119879119871119870119905 )2 = 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 120573)2] (23)

From the deduction of (23) equality (23) always holdswhen the dual-winding BSRM is working in the irreversibledomain from the inverse deduction of (23) if equality(23) holds then the dual-winding BSRM will work in theirreversible domainThe dual-winding BSRMworking in theirreversible domain is equivalent to equality (23) holdingthat is equality 1198942119898119886 = (1198732119904 11989421199041198861 + 1198732119904 11989421199041198862)(21198732119898) is equivalentto (23) Based on this equivalence the improved inversesystem method can be proposed as the following two stepsFirstly we can multiply modifying factors the value of whichapproximately is equal to one to the feedback variables of 120573 and in (23) respectively to make Equality (23) nothold that is to make the working area from irreversibledomain to reversible domain Secondly the inverse systemmethod can be adopted in the irreversible domain In thefirst step modifying three feedback variables of 120573 and simultaneously is difficult To avoid this problem 120573 isselected tomultiplymodified factor119870120573 to build the improvedinversion in this study By multiplying modified factor119870120573 by120573 we can obtain the following inequality

(119869 + 119879119871119870119905 )2 = 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 119870120573 120573)2] (24)

Hence the working area of dual-winding BSRM is changedfrom the irreversible domain to the reversible domain Thensubstituting119870120573 120573 for 120573 to (19) that is substituting1198701205731205932 for 1205932to (20) the improved inversion of the dual-winding BSRM inthe irreversible domain can be given as

1199061 = 2119898119873119898 (11987011989111205931 + 11987011989121198701205731205932 + 1198921198701198912)(11987021198911+ 11987021198912)radic[(1198691205933 + 119879119871) 119870119905] + radicΔ1015840

1199062 = 2119898119873119898 (11987011989111198701205731205932 minus 11987011989121205931 + 1198921198701198911)(11987021198911+ 11987021198912)radic[(1198691205933 + 119879119871) 119870119905] + radicΔ1015840

1199063 = radic[(1198691205933 + 119879119871) 119870119905] + radicΔ1015840(2119873119898)

(25)

where Δ1015840 = ((1198691205933 +119879119871)119870119905)2 minus(8119898211987321198981198732119904 (11987021198911 +11987021198912))[12059321 +(119892 + 1198701205731205932)2]In comparing the improved inversion (25) in the irre-

versible domain with the inversion (20) in the reversibledomain the improved inverse system method involveschanging the working area of dual-winding BSRM from theirreversible domain to the reversible domain by modifying

the feedback values of the control object and then adoptingthe traditional inverse system to realize the decouplingcontrol of the dual-winding BSRM in irreversible domain Inaddition inversion (20) is a special case of 119870120573 = 1 of theimproved inversion (25) Hence the improved inverse systemmethod is more universal and efficient than the traditionalone and it can be applied to the irreversible system

34 Design of Robust Servo Regulator According to inversesystem theory [28] by connecting the improved inversion(25) before dual-winding BSRM it can be linearized anddecoupled to two second-order integral type displacementsubsystems and one first-order integral type speed subsystemThe transfer functions are as follows

G (s) = [119866120572 (119904) 119866120573 (119904) 119866120596 (119904)]T = [ 11199042 11199042 1119904 ]T (26)

However these transfer functions (26) are the nominalmodelof three pseudo-linear subsystems Practically consideringthe uncertainty of parameters and model errors the compo-sition of the dual-winding BSRM and its improved inversion(25) is not exactly equivalent to the linear subsystem (26)Decoupled plants should be combinedwith robust controllersbecause the remaining coupling and nonlinearity alwaysexist Robust servo regulator is employed in this studygiven its excellent tracking and robust performance [21]which consists of servo compensator 119879(119904) and stabilizingcompensator119870(119904)

As for the displacement subsystems their transfer func-tions can be described as 119866120572(119904) = 119866120573(119904) = 119904minus2 According tothe design method of robust servo regulator we let 1198791(119904) =1198792(119904) = (1198860 + 1198861119904)119904 and 1198701(119904) = 1198702(119904) = 1198960 + 1198961119904 Then theclosed-loop transfer function of the displacement subsystemscan be described as

Φ (119904) = 1198861119904 + 1198860119904 (1199042 + 1198961119904 + 1198960) (27)

To simplify the selection of the controller parameters we candesign the system that comprises a pair of complex-numberdominant poles and the other poles are far away from theimaginary axis [21] that is

Φ (119904) = 1205962119899 (119904 + 120575)(119904 + 120575) (1199042 + 2120585120596119899119904 + 1205962119899) (28)

To improve the system response speed this study selects 120575 =6 120585 = radic22 and 120596119899 = 800 rads Combining (27) and (28)yields the regulator parameters

1198861 = 1205962119899 = 6400001198860 = 1205962119899120575 = 38400001198961 = 2120585120596119899 + 120575 asymp 113721198960 = 1205962119899 + 2120585120596119899120575 asymp 6467872

(29)

Similarly for the speed subsystem 119866120596(119904) = 119904minus1 we let 1198793(119904) =1198862(119904 + 1205752)119904 and 1198963(119904) = 0 here the value of 1198862 and 1205752


Impr

oved

inve

rse s

yste

m

(25)

Robust servoregulator

Filter PWMamplifier

PWMamplifier

PWMamplifier

Decouplingcontroller Controlled object

Dua

l-win

ding

BSR

M sy

stem

DCF

x

is1

im

is2

u1

u2

u3

1

2

3

K2(s)

K1(s)

T2(s)

T1(s)

T3(s)

lowast

lowast

lowast

++

++

+

minusminus

minusminus

minus

Filter

Filter

Figure 3 Closed-loop compound control system based on robust servo regulator

is selected by simulations as 1198862 = 1200 and 1205752 = 6 Theclosed-loop transfer function of the speed subsystems can bedescribed as

Φ (119904) = 1200119904 + 72001199042 + 1200119904 + 7200 (30)

The schematic of the closed-loop compound control systembased on the improved inverse system and robust servoregulator is shown in Figure 3 According to Figure 3 itis divided into four parts such as robust servo regulatordecoupling controller DCF and controlled object In thecontrolled object pulse width modulation (PWM) amplifiersare applied to the torque and suspending windings of thedual-winding BSRM for the torque drive and suspendingforce In decoupling controller by means of current-modeimproved inversion equation (25) it is designed and con-nected before the controlled object so as to realize thelinearization and decoupling control Robust servo regulatorconsists of servo compensator 119879(119904) and stabilizing com-pensator 119870(119904) is described as (27)ndash(30) DCF between thedecoupling controller and PWM amplifier in Figure 3 isphase lag dynamic compensation filter which is designed toimprove system stability to achieve 5 high-speed operationand the design of DCF is described in the next subsection

35 Design of Dynamic Compensation Filter Mathematicalmodel (4) is not dynamically equivalent to actual systembecause it does not consider the amplifier bandwidth andcomputation delay These dynamics can deteriorate decou-pling performance and even endanger system stability espe-cially at high-speed [26 27] One method to solve this issueis introducing compensation filters [29] The rated speedof dual-winding BSRM is 20000 rmin that is the controlbandwidth employed is approximately 333Hz To resolveits compensation filter the frequency response of PWMamplifier is measured via a sine sweep test the blue thinline in Figure 4 shows the positive frequency phase response

curve drawn by an Agilent 35670A dynamic signal analyzerThe phase lag at 333Hz is nearly 5∘ Generally the desiredphase lag is 45∘ and thus the phase lag that should becompensated at the rated speed frequency is approximately40∘

Based on the aforementioned analysis and consideringthe simplicity of filter realization a second-order filter isdesigned and the transfer function of the designed filter isgiven as

Φ (119904) = 211199042 + 3400119904 + 48 times 1061199042 + 2080119904 + 48 times 106 (31)

The red thick line in Figure 4 shows the positive frequencyphase response curve after phase compensation The phaselag at the rated speed frequency around 333Hz has increasedfrom 5∘ to nearly 45∘ This trend demonstrates that therelative stability of dual-winding BSRM system can be greatlyincreased by employing DCF

4 Simulation and Experimental Results

41 Experimental Setup A test machine of a dual-windingBSRM is illustrated in Figure 5 and many extensive experi-ments have been carried out on this testmachine where dual-winding BSRM uses a 3-degree of freedom (DOF) hybridmagnetic bearing (HMB) to realize 5-DOF active control andit is controlled by a separate controller We take 2-DOF dual-winding BSRM as the experimental subject Two assistantbearings are installed in the test machine and the average airgap between the rotor shaft and assistant bearing is 02mmThe specifications of dual-winding BSRM are presented inTable 1

The simulation is developed based on the Sim-Power-System and the Simulink of MATLAB The proposed decou-pling control algorithm is implemented in the digital signalprocessor (DSP) chip TMS320F28335 and the analog-to-digital converter 1674 is employed Both the sampling and


Before compensationAfter compensation

Frequency (Hz)103102101100

minus90

minus45

0

45

90

Phas

e (de

g)

Figure 4 Positive frequency phase response before and after phasecompensation

Dual-winding BSRM 3-DOF-HMB

Displacement sensor Optical electrical rotary encoder

Figure 5 Experimental test machine of the dual-winding BSRM

servo frequency are set to 67 kHz and the PWM frequencyis 20 kHz In the proposed motor an incremental opticalelectrical encoder is employed to be angular position sensorThe displacement sensor is made based on the eddy currentprinciple Each channel has two displacement sensors Thesetools are used to supply rotor eccentric displacement and thephase commutation logic signals and calculate the rotationalspeed for displacement and speed closed-loop control

The structure of the entire control system and theflowchart of decoupling control are shown in Figures 3 and6 respectively

According to Figure 3 the radial displacements androtational speed are compared with the reference values andthe errors are fed to robust servo regulator The robust servoregulator outputs are then fed to improved inverse systemto realize the decoupling and through the compositionof the DCF the duty cycles of the PWM are generatedThen the control of dual-winding BSRM can be dividedinto two independent parts that is the torque drive andsuspending force In the part of torque drive the directcontrol currents im from PWM amplifier are applied on thetorque Consequently the torque 119879119890 is generated By meansof the rotor motion equation when the generated torque is

Table 1 Specifications of dual-winding BSRM

Parameter ValueRated power of torque winding kW 2Rated power of suspending winding kW 1Rated voltage of torque winding V 110Rated voltage of suspending winding V 110Rated speed rmin 20000Rated frequency Hz 333Torque winding turns turns 17Suspending winding turns turns 15Stator outer diameter mm 130Stator pole arc deg 15Stator yoke width mm 10Rotor diameter mm 60Rotor pole arc deg 15Rotor yoke width mm 10Air-gap length mm 025Axial stack length mm 70Mass of the rotor Kg 1Moments of inertia of the rotor kgsdotm2 9 times 10minus3

Statorrotor core material DW360 50

Table 2 Computer run time of different control methods

Method Computer run timeSquare-wave currents control [14] 1454 120583sAnalytic inverse plus PID control [22] 988120583sNeural network inverse control [24] 4258120583sSupport vector machine inverse control [25] 3085 120583sProposed control method 994 120583s

larger than the load torque 119879119871 the rotor can speed up Inthe part of suspending force control currents is1 and is2 aregenerated by the PWM amplifiers which are applied on thesuspendingwindingsThen the suspending control flux fromthe control currents is1 and is2 is composited with bias fluxfrom the torque winding current im Therefore resultant fluxgenerated the suspending force

As shown in Figure 6 the decoupling control flowchartmainly includes four steps The first step is to establishthe mathematical model and analyze its reversibility Thesecond one is to divide the work area into reversible andirreversible domain based on the model reversibility analysisand construct the inversion and the improved inversionbefore the controlled object to realize the linearization anddecoupling The third is to design the robust servo regulatorand DCF so as to build the closed-loop compound controlsystemThe fourth step that is the last step is the simulationand experiment research and in this step the controllerparameters should bemodified until the control performancerequirements are met

The computer run time of different methods is tested (seeTable 2) The computer run time of the proposed method isshorter than those of the square-wave current control muchshorter than those of the neural network inverse and support


start

Reversible

Resolve improved inversion (25)

Irreversible domainReversible domain

Resolve inversion (20)

Irreversible domain analysis

Make up the displacement and speed subsystems (26)

Design robust servo regulator

Test the frequency response before compensation

Design compensation filter (31)

Test the frequency response after compensation

Satisfy the demandReslove regulator parameters

Make up the closed-loop compound control system Figure 3

Simulation and experiments

Satisfy the demand

Over

Yes

No

No

Yes

Yes

No

Modeling and Reversibility analysis ofDual-winding BSRM (4)

Robust servo regulator Dynamic compensation filter

K

Design n and

Design T1(s) T2(s) T3(s)

K1(s) K2(s) and K3(s)

Figure 6 Flowchart of decoupling control

vector machine inverse control methods and similar to thatof the analytic inverse system plus PID control methodConsistentwith the analysis in Section 1 this finding indicatesthat the proposed control method can greatly simplify theindustrial realization comparedwith the square-wave currentcontrol and intelligent inverse system decoupling controlstrategy

42 Decoupling Performance To further verify the decou-pling control performance between the proposedmethod and

the analytic inverse plus PID control [22] hereafter referredto as traditional method many comparative simulationsand experiments have been developed In the traditionalmethod as shown in [22] the values of coefficients fordisplacement-PID and speed-PID are given as 119896119901 = 032119896119894 = 02 and 119896119889 = 005 and 119896119903119901 = 43 119896119903119894 = 16and 119896119903119889 = 04 respectively In this study the methodof choice for PID coefficients in traditional method isskipped Interested readers are referred to [22] for detailedinformation


0 1 2 3 4 5

0 1 2 3 4 5

081

1214

0 1 2 3 4 5

t (s)

times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

4 531 20

4 531 20

4 531 20t (s)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)

Figure 7 Simulation and experimental comparison of decoupling performance in the reversible domain based on two different controlmethods (a) Simulation results by traditional method (b) Simulation results by the proposedmethod (c) Experimental results by traditionalmethod (d) Experimental results by the proposed method

421 Decoupling in the Reversible Domain In the reversibledomain comparative simulation and experiments betweenthe two different methods have been developed with variousrotational speeds and displacements At time 119905 = 05 s thereference translation displacement120572 steps from0 tominus01mmat time 119905 = 2 s and the reference rotational speed 120596 stepsfrom 10000 to 12000 rmin Figure 7 shows the results ofcomparative simulation and experiments

The step of 120572 does not bring a marked effect on 120573and 120596 through both the traditional and proposed methodsHowever as shown in Figure 7(a) the step of 120596 results indistinct fluctuations of approximately 80 and 100120583mon120572 and120573 respectively with the traditional method whereas it hasslight effect on 120572 and 120573 with the proposed control strategy asshown in Figure 7(b) Similar conclusions can be drawn fromthe experimental results as shown in Figures 7(c) and 7(d)Although the experimental and simulation results slightly

vary because of noise and dynamic imbalances they agreewell overall This conclusion can also be observed in latercomparisons

These results indicate that the traditional method canrealize the decoupling between two translation motions butit cannot completely eliminate the coupling between the rota-tion and translation motions That is the traditional methodcan only realize the decoupling between the two translationmotions but not the decoupling among the rotation andtranslation motions

Comparedwith the traditionalmethod the proposed onepossesses more decoupling DOF

422 Decoupling in the Irreversible Domain To further vali-date the decoupling performance of the proposed algorithmin the irreversible domain comparative simulation andexperiments of the traditional method and the proposed one


0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)

Figure 8 Simulation and experimental comparison of decoupling performance in the irreversible domain based on two different controlmethods (a) Simulation results by traditional method (b) Simulation results by the proposedmethod (c) Experimental results by traditionalmethod (d) Experimental results by the proposed method

have been carried out under the conditions 120596 = 10000 rminand 120572 = 120573 = 0

The simulation and experimental results by the twomethods are shown in Figure 8 As shown in Figure 8(a)when dual-winding BSRM works from reversible domain toirreversible domain at time 119905 = 1 s significant fluctuationsexist as large as approximately 120120583m and 100 120583m on dis-placement 120572 and displacement 120573 as well as roughly 400 rminon speed 120596 with the traditional method Strong fluctuationsalso appear repeatedly as the adjusted time lasts as longas nearly 3 s By contrast according to Figure 8(b) whenthe proposed control method is adopted the above valuesof fluctuation are reduced to 80 120583m 50 120583m and 100 rminrespectively Strong fluctuation only appears once because theadjusted time is shortened to 15 s which is roughly 50 ofthat in the traditional method Similar conclusions can be

drawn from the experimental results (see Figures 8(c) and8(d))

These results indicate that the proposed method has real-ized the decoupling control of dual-windings BSRM in theirreversible domain and the control precision of the rotationand translation motions has been improved evidently byadopting the presented strategy

43 Tracking Performance To demonstrate the tracking per-formance of the proposed strategy three different shapesignals are utilized to test the tracking precision In detailthe reference displacement 120572 tracks the sine wave signal01 sin(2120587119905 + 02120587)mm the reference displacement 120573 tracksthe sawtooth wave signal 01sawtooth(2120587119905)mm and thereference speed 120596 steps from 10000 to 12000 rmin at time119905 = 05 s and then back to 11000 rmin at time 119905 = 35 s


0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(a)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(d)

Figure 9 Simulation and experimental comparison of tracking performance based on two different control methods (a) Simulation resultsby traditional method (b) Simulation results by the proposed method (c) Experimental results by traditional method (d) Experimentalresults by the proposed method

Figure 9 shows the comparative simulation and experi-mental results between the traditional and the proposedmethods

Figures 9(a) and 9(b) illustrate that both the traditionaland the proposed methods do not bring any marked over-shoot on speed 120596 when reference 120596 tracks the step signaland the accelerating time from 10000 to 12000 rmin is lessthan 2min This finding indicates that the proposed methodexhibits a good tracking performance aswell as the traditionalone

However as Figure 9(a) shows with the traditionalmethod pulse disturbances exist on 120572 and 120573 when 120596 tracksthe step signal and the peak amplitude of disturbances on 120572and 120573 are roughly 90 and 100120583m By contrast Figure 9(b)demonstrates that such issue is nonexistent when employing

the proposed method because the decoupling between thetranslation and rotation motions has been realized by thismethod as verified in the last subsection

Figures 9(a) and 9(b) also show that at the valley pointsof the sin wave and the sawtooth wave signals the proposedmethod has higher tracking accuracy than the traditionalmethod

These comparative results further confirm that the pre-sentedmethod show great improvements in terms of trackingperformance Similar conclusions can be drawn from theexperimental results (see Figures 9(c) and 9(d))

44 Disturbance Rejection and Robustness Performances Tofurther verify the robustness performances of the proposedcontrol algorithm the reference steps external disturbances


0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

t (s)

t (s)

minus02

0

02

(m

m)

minus02

0

02

(mm

)

081

1214times104

(r

min

)

kd = 003kd = 005

kd = 003kd = 005

krd = 01krd = 04

(a)

0 1 2 3 4 5t (s)

0 1 2 3 4 5

0 1 2 3 4 5minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)

Figure 10 Comparative simulation and experimental results of disturbance rejection and robustness performances of the traditional methodwith different controller parameters (a) Comparative simulation results with different values of 119896119889 and 119896119903119889 (b) Experimental results with 119896119889 =005 and 119896119903119889 = 04 (c) Experimental results with 119896119889 = 003 and 119896119903119889 = 01

and parameter variations are imposed on the system At time119905 = 05 s the reference displacement 120572 and displacement 120573step from 0 to 01mm and the reference speed 120596 steps from10000 to 12000 rmin At time 119905 = 3 s a radial disturbingforce 10N and a load torque minus08Nsdotm are imposed on therotor and at time 119905 = 4 s we assume that the coefficients

of radial force 1198701198912 increase by 25 and the coefficients oftorque 119869119905 decrease by 30 Comparative simulations andexperiments with different values of 119896119889 and 119896119903119889 in thetraditional controller and different values of 120585 and 1205752 inthe proposed one are also performed in Figures 10 and 11respectively


0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

2 = 6

2 = 4

= 0707 = 05

= 0707 = 05

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)

Figure 11 Comparative simulation and experimental results of disturbance rejection and robustness performances of the proposed methodwith different controller parameters (a) Comparative simulation results with different values of 120585 and 1205752 (b) Experimental results with 120585 = 05and 1205752 = 4 (c) Experimental results with 120585 = 0707 and 1205752 = 6

From Figure 10(a) for the traditional controller theexternal disturbance and parameter variations result inroughly 80 120583m 50120583m and 800 rmin disturbances on 120572120573 and 120596 respectively Although this outcome has beengreatly improved by decreasing the differential coefficients119896119889 (from 005 to 003) and 119896119903119889 (from 04 to 01) its responsetime prolongs when tracking the reference displacement or

speed That is the traditional method cannot adjust theperformances of the tracking disturbance rejection androbustness independently Concretely speaking the decreaseof 119896119889 and 119896119903119889 can improve disturbance rejection and robust-ness performances but it sacrifices the tracking performance

By contrast according to Figure 11(a) with the proposedstrategy the external disturbance and parameter variations


do not bring obvious disturbances into 120572 120573 and 120596 Inaddition the increase of 120585 (from 05 to 0707) and 1205752 (from4 to 6) can decrease the values of disturbances without anyeffects on the tracking performance That is the proposedcontrol strategy can adjust tracking disturbance rejectionand robustness performances independentlyThe experimen-tal results as shown in Figures 10(b) 10(c) 11(b) and 11(c)further verify the correctness of the simulation

In conclusion the traditional control method cannotadjust the control performances of the tracking disturbancerejection and robustness independently Seeking its optimalcoefficients is rather difficult Often even if the coefficientshave been selected the traditional method cannot satisfy thecontrol performances of the tracking disturbance rejectionand robustness simultaneously Contrarily the proposedcontrol strategy can adjust the control performances ofthe tracking and robustness independently Therefore thismethod not only simplifies the adjustment of parameters butalso improves tracking disturbance rejection and robustnessperformances simultaneously

5 Conclusion

In this study the model and reversibility analysis of dual-winding BSRM were explored and a new decoupling controlmethod based on improved inverse system and robust servoregulator was proposedThe following five conclusions can bedrawn from this study

(i) Dual-winding BSRM is a multivariable high-nonlin-ear motor with strong coupling not only between theradial forces but also between the torque and radialforces It is also an incomplete reversible systemwhoseworking area can be divided into reversible andirreversible domains

(ii) To generate the radial force for balancing rotor grav-ity themain winding current of conducting phase 119894119898119886of dual-winding BSRM cannot be zero so 119894119898119886 = 0 inthe irreversible domain analysis does not exist in thereal system The decoupling control of dual-windingBSRM in the irreversible domain need not considerthe case of 119894119898119886 = 0

(iii) The proposed improved inverse system methodinvolves modifying the state feedback of the con-trolled system The traditional inverse method isequivalent to the special case of the modified factor119896120573 = 1 in the improved inverse system so theimproved inverse system is more universal and gen-eralized than the traditional inverse system method

(iv) Simulation and experimental results confirm that therobustness and stability problems induced by theinverse controller can been successfully solved byintroducing robust servo regulator and DCF and theproposed method exhibits more decoupling DOFhigher precision better tracking and stronger dis-turbance rejection and robustness performances thanthe traditional one

(v) The improved inverse system method can realize thedecoupling control of dual-winding BSRM in bothreversible and irreversible domains with a satisfiedstability and the static as well as dynamic perfor-mance Hence the working area of dual-windingBSRM is expended from the reversible domain toboth the reversible and irreversible domains by adopt-ing proposed method

Nomenclature

119865120572 119865120573 Radial forces acting on rotor in 120572- and120573-axis respectively119873119898119873119904 Number of turns of torque windingand suspending winding respectively120579 Rotor position from the alignedposition of exciting phase119897 Axial stack length of the iron core119903 Radius of the rotor pole119888 Constant 149119898 Mass of the rotor120572 120573 Rotor displacements in 120572- and 120573-axisrespectively119879119871 Load torque119896119901 119896119894 and 119896119889 Proportional integral and differentialcoefficients of the displacement-PIDcontroller in traditional method119879119890 Electromagnetic torque acting on rotor119894119898119886 1198941199041198861 and 1198941199041198862 Currents in A-phase torque windingand suspending winding respectively

im = (119894119898119886 119894119898119887 119894119898119888) Control currents in three-phase torquewindings

is1 = (1198941199041198861 1198941199041198871 1198941199041198881) Control currents in three-phasesuspending windings in the 120572-axis

is2 = (1198941199041198862 1198941199041198872 1198941199041198882) Control currents in three-phasesuspending windings in the 120573-axis1198701198911 1198701198912 and119870119905 Proportional coefficients of radial forceand torque respectively120575 Air-gap length1205830 Permeability of vacuum119892 Acceleration of gravity119869 Moments of inertia of the rotor

DD Reversible and irreversible domainrespectively120596 Mechanical angular velocity of therotor119896119903119901 119896119903119894 and 119896119903119889 Proportional integral and differentialcoefficients of the speed-PID controllerin traditional method

Competing Interests

The authors declare no conflict of interests regarding thepublication of this paper

Acknowledgments

The work described in this paper was supported bythe National Natural Science Foundation of China (no


51507077 and no 51377074) the Natural Science Researchof Higher Education Institutions of Jiangsu Province (no15KJB470005) Qing Lan Project of Jiangsu Province (Grantno [2016] 15) and The Open Research Fund of Jiangsu Col-laborative InnovationCenter for Smart DistributionNetwork(Grant no XTCX201601) as well as foundations sponsored byNanjing Institute of Technology (no CKJA201407)

References

[1] D-H Lee T H Pham and J-W Ahn ldquoDesign and operationcharacteristics of four-two pole high-speed SRM for torqueripple reductionrdquo IEEE Transactions on Industrial Electronicsvol 60 no 9 pp 3637ndash3643 2013

[2] S-M Yang and J-Y Chen ldquoControlled dynamic braking forswitched reluctance motor drives with a rectifier front endrdquoIEEE Transactions on Industrial Electronics vol 60 no 11 pp4913ndash4919 2013

[3] H-Y Yang Y-C Lim andH-C Kim ldquoAcoustic noisevibrationreduction of a single-phase SRMusing skewed stator and rotorrdquoIEEE Transactions on Industrial Electronics vol 60 no 10 pp4292ndash4300 2013

[4] F L M dos Santos J Anthonis F Naclerio J J C GyselinckH Van der Auweraer and L C S Goes ldquoMultiphysics NVHmodeling simulation of a switched reluctance motor for anelectric vehiclerdquo IEEETransactions on Industrial Electronics vol61 no 1 pp 469ndash476 2014

[5] B B Choi and M Siebert ldquoA bearingless switched reluctancemotor for high specific power applicationsrdquo in Proceedings ofthe 42nd AIAAASMESAEASEE Joint Propulsion Conference ampExhibit Sacramento Calif USA July 2006

[6] M Takemoto K Shimada and A Chiba ldquoA design andcharacteristics of switched reluctance type bearingless motorsrdquoin Proceedings of the 4th International Symposium on MagneticSuspension Technology pp 49ndash63 1998

[7] F-C Lin and S-M Yang ldquoAn approach to producing controlledradial force in a switched reluctance motorrdquo IEEE Transactionson Industrial Electronics vol 54 no 4 pp 2137ndash2146 2007

[8] L Chen and W Hofmann ldquoSpeed regulation technique of onebearingless 86 switched reluctance motor with simpler singlewinding structurerdquo IEEE Transactions on Industrial Electronicsvol 59 no 6 pp 2592ndash2600 2012

[9] C R Morrison M W Siebert and E J Ho ldquoElectromag-netic forces in a hybrid magnetic-bearing switched-reluctancemotorrdquo IEEE Transactions on Magnetics vol 44 no 12 pp4626ndash4638 2008

[10] Z Xu D H Lee and J W Ahn ldquoControl characteristics of 810and 1214 bearingless switched reluctancemotorrdquo in Proceedingsof the International Power Electronics Conference pp 994ndash999Hiroshima Japan October 2014

[11] W Peng F Zhang and J-WAhn ldquoDesign and control of a novelbearingless SRM with double statorrdquo in Proceedings of the 21stIEEE International Symposium on Industrial Electronics (ISIErsquo12) pp 1928ndash1933 IEEE Hangzhou China May 2012

[12] H Wang J Liu J Bao and B Xue ldquoA novel bearinglessswitched reluctance motor with biased permanent magnetrdquoIEEE Transactions on Industrial Electronics vol 61 no 12 pp6947ndash6955 2014

[13] H Wang J Bao B Xue and J Liu ldquoControl of suspendingforce in novel permanent-magnet-biased bearingless switched

reluctance motorrdquo IEEE Transactions on Industrial Electronicsvol 62 no 7 pp 4298ndash4306 2015

[14] M Takemoto A Chiba and T Fukao ldquoA new control methodof bearingless switched reluctance motors using square-wavecurrentsrdquo in Proceedings of the IEEE Power Engineering SocietyWinter Meeting vol 1 pp 375ndash380 Singapore June 2000

[15] M Takemoto A Chiba and T Fukao ldquoA method of determin-ing the advanced angle of square-wave currents in a bearinglessswitched reluctance motorrdquo IEEE Transactions on IndustryApplications vol 37 no 6 pp 1702ndash1709 2001

[16] M Takemoto A Chiba H Akagi and T Fukao ldquoRadialforce and torque of a bearingless switched reluctance motoroperating in a region ofmagnetic saturationrdquo IEEETransactionson Industry Applications vol 40 no 1 pp 103ndash112 2004

[17] M Takemoto H Suzuki A Chiba T Fukao and M A Rah-man ldquoImproved analysis of a bearingless switched reluctancemotorrdquo IEEE Transactions on Industry Applications vol 37 no1 pp 26ndash34 2001

[18] X Cao Z Deng G Yang and X Wang ldquoIndependent controlof average torque and radial force in bearingless switched-reluctance motors with hybrid excitationsrdquo IEEE Transactionson Power Electronics vol 24 no 5 pp 1376ndash1385 2009

[19] Y Yang Z Deng G Yang X Cao and Q Zhang ldquoA con-trol strategy for bearingless switched-reluctance motorsrdquo IEEETransactions on Power Electronics vol 25 no 11 pp 2807ndash28192010

[20] L Zhang and Y-K Sun ldquoVariable-structure control of bearing-less switched reluctancemotors based on differential geometryrdquoProceedings of the Chinese Society of Electrical Engineering vol26 no 19 pp 121ndash126 2006 (Chinese)

[21] Y Ren and J Fang ldquoHigh-precision and strong-robustnesscontrol for anMSCMGbased onmodal separation and rotationmotion decoupling strategyrdquo IEEE Transactions on IndustrialElectronics vol 61 no 3 pp 1539ndash1551 2014

[22] JWu Y Sun X Liu L Zhang G Liu and J Chen ldquoDecouplingcontrol of radial force in bearingless switched reluctancemotorsbased on inverse systemrdquo in Proceedings of the 6th WorldCongress on Intelligent Control and Automation (WCICA rsquo06)pp 1100ndash1104 June 2006

[23] Z Zhu and Y Sun ldquoModified inverse decoupling control forbearingless switched reluctancemotors operating in irreversibledomainrdquo in Proceedings of the 17th International Conference onElectrical Machines and Systems (ICEMS rsquo14) pp 1245ndash1250IEEE Hangzhou China October 2014

[24] Y Sun Y Zhou and X Ji ldquoDecoupling control of bearinglessswitched reluctance motor with neural network inverse systemmethodrdquo Zhongguo Dianji Gongcheng XuebaoProceedings ofthe Chinese Society of Electrical Engineering vol 31 no 30 pp117ndash123 2011

[25] Y Zhou Y Sun and Y Huang ldquoFull decoupling control ofbearingless switched reluctance motor based on support vectormachine inverse systemrdquo Journal of Jiangsu University (NaturalScience Edition) vol 33 no 1 pp 60ndash64 2012 (Chinese)

[26] J Fang and Y Ren ldquoDecoupling control of magnetically sus-pended rotor system in control moment gyros based on aninverse system methodrdquo IEEEASME Transactions on Mecha-tronics vol 17 no 6 pp 1133ndash1144 2012

[27] J Fang and Y Ren ldquoHigh-precision control for a single-gimbalmagnetically suspended control moment gyro based on inversesystem methodrdquo IEEE Transactions on Industrial Electronicsvol 58 no 9 pp 4331ndash4342 2011


[28] X Dai D He X Zhang and T Zhang ldquoMIMO systeminvertibility and decoupling control strategies based on ANN120572th-order inversionrdquo IEE Proceedings Control Theory andApplications vol 148 no 2 pp 125ndash136 2001

[29] Y Ren and J Fang ldquoCurrent-sensing resistor design to includecurrent derivative in PWMH-bridge unipolar switching poweramplifiers for magnetic bearingsrdquo IEEE Transactions on Indus-trial Electronics vol 59 no 12 pp 4590ndash4600 2012

Submit your manuscripts athttpswwwhindawicom

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


torque ripple and stator vibration [19] Recently interest inthe study on decoupling control has been increasing [20ndash24] mainly including the differential geometric method [2021] and inverse system method [22ndash25] Compared withthe former the latter needs neither to use the complexnonlinear coordinate transformation nor to transfer thenonlinear control problems into geometric ones Thereforethe inverse system method is relatively simple to implementin practice In detail the inverse system method includesanalytic inverse system [22 23] and intelligent ones [24 25]Compared with analytic inverse system intelligent ones neednot rely on the precise mathematical model but they usuallyrequire large allocations of computer resources that are oftentypically restricted in high-speed system Hence analyticinverse system method has attracted more attention thanintelligent ones in high-speed magnetically suspended field[26 27]

The implementation of analytic inverse system methodrequires the controlled object to be reversible However itis difficult to satisfy the reversibility in practical systemAccording to [23] the dual-winding BSRM is not completelyreversible and its working area includes two parts reversibleand irreversible domains Traditional analytic inverse sys-tem method can only realize the decoupling control inthe reversible domain while the decoupling control inthe irreversible domain cannot be realized To solve theseissues a modified inverse decoupling control method forBSRM operating in irreversible domain is proposed byauthors in [23] However it is known that the analyticinverse controller usually affects the robustness and stabilityof system because uncertainties and model errors alwaysexist in reality Especially at high-speed the mathemati-cal model is not equivalent to the actual system becausethe former does not consider the amplifiers bandwidthand computation delay and these dynamics can degradedecoupling performance and even endanger system stabilityTo combat these adverse effects proportion-integration-differentiation (PID) control [22ndash25] and internal modelcontrol [26] have been typically employed for the decoupledplants Nevertheless nearly all these methods experiencedifficulty in realizing tracking and robustness independently[27]

This study presents a novel decoupling control strat-egy based on improved inverse system method to realizethe decoupling control of dual-winding BSRM in boththe reversible and irreversible domains Robust servo reg-ulator is adopted for the decoupled plants to guaranteecontrol performances and robustness A phase dynamiccompensation filter (DCF) is also designed to improvesystem stability at high-speed One main contribution ofthis study is to demonstrate that the decoupling control ofdual-winding BSRM at high-speed (up to 20000 rmin) inboth reversible and irreversible domains can all be success-fully resolved with the improved inverse system methodThe other contribution is to show that the stability androbustness problems induced by inverse controller can beeffectively solved by introducing robust servo regulator andDCF

Torque winding

Suspending winding

F

ima+

isa1minus

isa1+

imaminus

Nma

Nsa2

Nsa1

Ψma

Ψsa1

Air gap-2 Air gap-1

isa2minus

isa2+

Nma

Nma

Nsa2

Figure 1 Principle of radial force production of dual-windingBSRM

2 Principle and Characteristic Analysis ofthe Dual-Winding BSRM

21 Radial Force Production Principle of Dual-Winding BSRMFigure 1 shows the principle of radial force production ofdual-winding BSRM with A-phase windings The torquewinding 119873119898119886 consists of four coils connected in seriesand the suspending windings 1198731199041198861 and 1198731199041198862 consist of twocoils each When the torque winding 119873119898119886 and suspendingwinding 1198731199041198861 conduct the currents 119894119898119886 and 1198941199041198861 respectivelythe symmetrical four-pole main fluxes Ψ119898119886 and two-polesuspending fluxes Ψ1199041198861 should be produced The flux densityin air gap-1 increases but decreases in air gap-2 Thereforethis superimposed magnetic field results in the radial force119865120572 acting on the rotor in the 120572-axis The radial force 119865120573 in the120573-axis can also be produced in the samemanner Radial forcein any desired direction can be produced by generating thetwo radial forces This principle can be similarly applied tothe B-phase and C-phase windings

22 Mathematical Model of Dual-Winding BSRM Neglectingthe leakage flux and saturation effects the theoretical formu-lae of the torque and radial forces can be derived from thederivatives of the stored magnetic energy 119882 with respect todisplacements 120572 and 120573 and rotor position 120579 as [17]

119865120572 = 120597119882120597120572 = 119894119898119886 (1198701198911 sdot 1198941199041198861 minus 1198701198912 sdot 1198941199041198862) 119865120573 = 120597119882120597120573 = 119894119898119886 (1198701198912 sdot 1198941199041198861 + 1198701198911 sdot 1198941199041198862) 119879119890 = 120597119882120597120579 = 119870119905 (211987321198981198942119898119886 + 1198732119904 11989421199041198861 + 1198732119904 11989421199041198862)

(1)

where 119873119898 and 119873119904 are the number of turns of torquewinding and suspending windings 119894119898119886 1198941199041198861 and 1198941199041198862 are the



1198701198911 = 119873119898119873119904 (1205830119897119903 (120587 minus 12 |120579|)61205752+ 321205830119897119903119888 |120579|120587 (4119903119888 |120579| 120575 + 1205871205752))

1198701198912 = 119873119898119873119904(1205830119897119903 (120587 minus 12 |120579|) |120579|121205752 minus 21205830119897120575+ 161205830119897119888 (119903 |120579|2 + 2120575)120587 (4119903119888 |120579| 120575 + 1205871205752) )

119870119905 =


(2)




119898 = 119865120572119898 120573 = 119865120573 minus 119898119892119869 = 119879119890 minus 119879119871

(3)



= 1119869 [119870119905 (211987321198981198942119898119886 + 1198732119904 11989421199041198861 + 1198732119904 11989421199041198862) minus 119879119871] (4)







x = [1199091 1199092 1199093 1199094 1199095 1199096]T = [120572 120573 120579 120573 120596]T (5)

u = [1199061 1199062 1199063]T = [1198941199041198861 1198941199041198862 119894119898119886]T (6)


x = 119891 (x u)

=

[[[[[[[[[[[[[[[[


1119869 [119870119905 (2119873211989811990623 + 1198732119904 11990621 + 1198732119904 11990622) minus 119879119871]

]]]]]]]]]]]]]]]]

(8)



isa2 = 2Aisa2 = 6A

isa2 = 10A


0

2

4

6

8

10

12To

rque

Te

(Nmiddotm

)

(a)



1000

2000

3000

4000

5000

6000

Radi

al fo

rceF

(N

)

isa2 = 2Aisa2 = 6A

isa2 = 10A

(b)



minus150

minus100

minus50

0

50

Radi

al fo

rceF

(N

)

isa2 = 2Aisa2 = 6A

isa2 = 10A

(c)




J (u) = [[[

119910111991021199103]]]

=[[[[[[[[


1119869 [119870119905 (2119873211989811990623 + 1198732119904 11990621 + 1198732119904 11990622) minus 119879119871]

]]]]]]]]

(9)


A =[[[[[[[[[

120597 11991011205971199061120597 11991011205971199062

120597 11991011205971199063120597 11991021205971199061120597 11991021205971199062

120597 11991021205971199063120597 11991031205971199061120597 11991031205971199062

120597 11991031205971199063

]]]]]]]]] (10)


A =[[[[[[[[[[

11987011989111199063119898minus11987011989121199063119898

11987011989111199061 minus 119870119891211990621198981198701198912119906311989811987011989111199063119898

11987011989121199061 + 1198701198911119906211989821198701199051198732119904 1199061119869 21198701199051198732119904 1199062119869 411987011990511987321198981199063119869

]]]]]]]]]]

(11)


(12)



D = 119894119898119886 = 0 1198942119898119886 = 1198732119904 11989421199041198861 + 1198732119904 1198942119904119886221198732119898 D = 119894119898119886 = 0 or 1198942119898119886 = 1198732119904 11989421199041198861 + 1198732119904 1198942119904119886221198732119898

(13)



1198941199041198861 = 119898(1198701198911 + 1198701198912 120573 + 1198921198701198912)(11987021198911+ 11987021198912) 119894119898119886

1198941199041198862 = 119898(1198701198911 120573 minus 1198701198912 + 1198921198701198911)(11987021198911+ 11987021198912) 119894119898119886

(14)


211987321198981198944119898119886 minus 119869 + 119879119871119870119905 1198942119898119886 + 1198732119904119898211987021198911+ 11987021198912

[2 + (119892 + 120573)2]= 0

(15)


Δ = (119869 + 119879119871119870119905 )2 minus 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 120573)2] (16)


119894211989811988612 = 141198732119898 (119869 + 119879119871119870119905 plusmn radicΔ) (17)



119894119898119886 = 12119873119898radic119869 + 119879119871119870119905 + radicΔ (18)


1198941199041198861 = 2119898119873119898 (1198701198911 + 1198701198912 120573 + 1198921198701198912)(11987021198911+ 11987021198912)radic[(119869 + 119879119871) 119870119905] + radicΔ


(19)


1199061 = 2119898119873119898 (11987011989111205931 + 11987011989121205932 + 1198921198701198912)(11987021198911+ 11987021198912)radic[(1198691205933 + 119879119871) 119870119905] + radicΔ


1199063 = radic[(1198691205933 + 119879119871) 119870119905] + radicΔ(2119873119898)

(20)


D = D1 = 119894119898119886 = 0D2 = 1198942119898119886 = 1198732119904 11989421199041198861 + 1198732119904 1198942119904119886221198732119898 (21)


= 0120573 = minus119892 (22)





(119869 + 119879119871119870119905 )2 = 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 120573)2] (23)


(119869 + 119879119871119870119905 )2 = 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 119870120573 120573)2] (24)


1199061 = 2119898119873119898 (11987011989111205931 + 11987011989121198701205731205932 + 1198921198701198912)(11987021198911+ 11987021198912)radic[(1198691205933 + 119879119871) 119870119905] + radicΔ1015840


1199063 = radic[(1198691205933 + 119879119871) 119870119905] + radicΔ1015840(2119873119898)

(25)





G (s) = [119866120572 (119904) 119866120573 (119904) 119866120596 (119904)]T = [ 11199042 11199042 1119904 ]T (26)



Φ (119904) = 1198861119904 + 1198860119904 (1199042 + 1198961119904 + 1198960) (27)


Φ (119904) = 1205962119899 (119904 + 120575)(119904 + 120575) (1199042 + 2120585120596119899119904 + 1205962119899) (28)


1198861 = 1205962119899 = 6400001198860 = 1205962119899120575 = 38400001198961 = 2120585120596119899 + 120575 asymp 113721198960 = 1205962119899 + 2120585120596119899120575 asymp 6467872

(29)



Impr

oved

inve

rse s

yste

m

(25)


Filter PWMamplifier

PWMamplifier

PWMamplifier


Dua

l-win

ding

BSR

M sy

stem

DCF

x

is1

im

is2

u1

u2

u3

1

2

3

K2(s)

K1(s)

T2(s)

T1(s)

T3(s)

lowast

lowast

lowast

++

++

+

minusminus

minusminus

minus

Filter

Filter



Φ (119904) = 1200119904 + 72001199042 + 1200119904 + 7200 (30)





Φ (119904) = 211199042 + 3400119904 + 48 times 1061199042 + 2080119904 + 48 times 106 (31)








minus90

minus45

0

45

90

Phas

e (de

g)

















start

Reversible













Satisfy the demand

Over

Yes

No

No

Yes

Yes

No



K

Design n and








0 1 2 3 4 5

0 1 2 3 4 5

081

1214

0 1 2 3 4 5

t (s)

times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

4 531 20

4 531 20

4 531 20t (s)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)









0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)








0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(a)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(d)










0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

t (s)

t (s)

minus02

0

02

(m

m)

minus02

0

02

(mm

)

081

1214times104

(r

min

)

kd = 003kd = 005

kd = 003kd = 005

krd = 01krd = 04

(a)

0 1 2 3 4 5t (s)

0 1 2 3 4 5

0 1 2 3 4 5minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)





0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

2 = 6

2 = 4

= 0707 = 05

= 0707 = 05

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)








5 Conclusion







Nomenclature






Competing Interests


Acknowledgments




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198701198911 = 119873119898119873119904 (1205830119897119903 (120587 minus 12 |120579|)61205752+ 321205830119897119903119888 |120579|120587 (4119903119888 |120579| 120575 + 1205871205752))

1198701198912 = 119873119898119873119904(1205830119897119903 (120587 minus 12 |120579|) |120579|121205752 minus 21205830119897120575+ 161205830119897119888 (119903 |120579|2 + 2120575)120587 (4119903119888 |120579| 120575 + 1205871205752) )

119870119905 =


(2)




119898 = 119865120572119898 120573 = 119865120573 minus 119898119892119869 = 119879119890 minus 119879119871

(3)



= 1119869 [119870119905 (211987321198981198942119898119886 + 1198732119904 11989421199041198861 + 1198732119904 11989421199041198862) minus 119879119871] (4)







x = [1199091 1199092 1199093 1199094 1199095 1199096]T = [120572 120573 120579 120573 120596]T (5)

u = [1199061 1199062 1199063]T = [1198941199041198861 1198941199041198862 119894119898119886]T (6)


x = 119891 (x u)

=

[[[[[[[[[[[[[[[[


1119869 [119870119905 (2119873211989811990623 + 1198732119904 11990621 + 1198732119904 11990622) minus 119879119871]

]]]]]]]]]]]]]]]]

(8)



isa2 = 2Aisa2 = 6A

isa2 = 10A


0

2

4

6

8

10

12To

rque

Te

(Nmiddotm

)

(a)



1000

2000

3000

4000

5000

6000

Radi

al fo

rceF

(N

)

isa2 = 2Aisa2 = 6A

isa2 = 10A

(b)



minus150

minus100

minus50

0

50

Radi

al fo

rceF

(N

)

isa2 = 2Aisa2 = 6A

isa2 = 10A

(c)




J (u) = [[[

119910111991021199103]]]

=[[[[[[[[


1119869 [119870119905 (2119873211989811990623 + 1198732119904 11990621 + 1198732119904 11990622) minus 119879119871]

]]]]]]]]

(9)


A =[[[[[[[[[

120597 11991011205971199061120597 11991011205971199062

120597 11991011205971199063120597 11991021205971199061120597 11991021205971199062

120597 11991021205971199063120597 11991031205971199061120597 11991031205971199062

120597 11991031205971199063

]]]]]]]]] (10)


A =[[[[[[[[[[

11987011989111199063119898minus11987011989121199063119898

11987011989111199061 minus 119870119891211990621198981198701198912119906311989811987011989111199063119898

11987011989121199061 + 1198701198911119906211989821198701199051198732119904 1199061119869 21198701199051198732119904 1199062119869 411987011990511987321198981199063119869

]]]]]]]]]]

(11)


(12)



D = 119894119898119886 = 0 1198942119898119886 = 1198732119904 11989421199041198861 + 1198732119904 1198942119904119886221198732119898 D = 119894119898119886 = 0 or 1198942119898119886 = 1198732119904 11989421199041198861 + 1198732119904 1198942119904119886221198732119898

(13)



1198941199041198861 = 119898(1198701198911 + 1198701198912 120573 + 1198921198701198912)(11987021198911+ 11987021198912) 119894119898119886

1198941199041198862 = 119898(1198701198911 120573 minus 1198701198912 + 1198921198701198911)(11987021198911+ 11987021198912) 119894119898119886

(14)


211987321198981198944119898119886 minus 119869 + 119879119871119870119905 1198942119898119886 + 1198732119904119898211987021198911+ 11987021198912

[2 + (119892 + 120573)2]= 0

(15)


Δ = (119869 + 119879119871119870119905 )2 minus 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 120573)2] (16)


119894211989811988612 = 141198732119898 (119869 + 119879119871119870119905 plusmn radicΔ) (17)



119894119898119886 = 12119873119898radic119869 + 119879119871119870119905 + radicΔ (18)


1198941199041198861 = 2119898119873119898 (1198701198911 + 1198701198912 120573 + 1198921198701198912)(11987021198911+ 11987021198912)radic[(119869 + 119879119871) 119870119905] + radicΔ


(19)


1199061 = 2119898119873119898 (11987011989111205931 + 11987011989121205932 + 1198921198701198912)(11987021198911+ 11987021198912)radic[(1198691205933 + 119879119871) 119870119905] + radicΔ


1199063 = radic[(1198691205933 + 119879119871) 119870119905] + radicΔ(2119873119898)

(20)


D = D1 = 119894119898119886 = 0D2 = 1198942119898119886 = 1198732119904 11989421199041198861 + 1198732119904 1198942119904119886221198732119898 (21)


= 0120573 = minus119892 (22)





(119869 + 119879119871119870119905 )2 = 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 120573)2] (23)


(119869 + 119879119871119870119905 )2 = 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 119870120573 120573)2] (24)


1199061 = 2119898119873119898 (11987011989111205931 + 11987011989121198701205731205932 + 1198921198701198912)(11987021198911+ 11987021198912)radic[(1198691205933 + 119879119871) 119870119905] + radicΔ1015840


1199063 = radic[(1198691205933 + 119879119871) 119870119905] + radicΔ1015840(2119873119898)

(25)





G (s) = [119866120572 (119904) 119866120573 (119904) 119866120596 (119904)]T = [ 11199042 11199042 1119904 ]T (26)



Φ (119904) = 1198861119904 + 1198860119904 (1199042 + 1198961119904 + 1198960) (27)


Φ (119904) = 1205962119899 (119904 + 120575)(119904 + 120575) (1199042 + 2120585120596119899119904 + 1205962119899) (28)


1198861 = 1205962119899 = 6400001198860 = 1205962119899120575 = 38400001198961 = 2120585120596119899 + 120575 asymp 113721198960 = 1205962119899 + 2120585120596119899120575 asymp 6467872

(29)



Impr

oved

inve

rse s

yste

m

(25)


Filter PWMamplifier

PWMamplifier

PWMamplifier


Dua

l-win

ding

BSR

M sy

stem

DCF

x

is1

im

is2

u1

u2

u3

1

2

3

K2(s)

K1(s)

T2(s)

T1(s)

T3(s)

lowast

lowast

lowast

++

++

+

minusminus

minusminus

minus

Filter

Filter



Φ (119904) = 1200119904 + 72001199042 + 1200119904 + 7200 (30)





Φ (119904) = 211199042 + 3400119904 + 48 times 1061199042 + 2080119904 + 48 times 106 (31)








minus90

minus45

0

45

90

Phas

e (de

g)

















start

Reversible













Satisfy the demand

Over

Yes

No

No

Yes

Yes

No



K

Design n and








0 1 2 3 4 5

0 1 2 3 4 5

081

1214

0 1 2 3 4 5

t (s)

times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

4 531 20

4 531 20

4 531 20t (s)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)









0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)








0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(a)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(d)










0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

t (s)

t (s)

minus02

0

02

(m

m)

minus02

0

02

(mm

)

081

1214times104

(r

min

)

kd = 003kd = 005

kd = 003kd = 005

krd = 01krd = 04

(a)

0 1 2 3 4 5t (s)

0 1 2 3 4 5

0 1 2 3 4 5minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)





0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

2 = 6

2 = 4

= 0707 = 05

= 0707 = 05

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)








5 Conclusion







Nomenclature






Competing Interests


Acknowledgments




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











isa2 = 2Aisa2 = 6A

isa2 = 10A


0

2

4

6

8

10

12To

rque

Te

(Nmiddotm

)

(a)



1000

2000

3000

4000

5000

6000

Radi

al fo

rceF

(N

)

isa2 = 2Aisa2 = 6A

isa2 = 10A

(b)



minus150

minus100

minus50

0

50

Radi

al fo

rceF

(N

)

isa2 = 2Aisa2 = 6A

isa2 = 10A

(c)




J (u) = [[[

119910111991021199103]]]

=[[[[[[[[


1119869 [119870119905 (2119873211989811990623 + 1198732119904 11990621 + 1198732119904 11990622) minus 119879119871]

]]]]]]]]

(9)


A =[[[[[[[[[

120597 11991011205971199061120597 11991011205971199062

120597 11991011205971199063120597 11991021205971199061120597 11991021205971199062

120597 11991021205971199063120597 11991031205971199061120597 11991031205971199062

120597 11991031205971199063

]]]]]]]]] (10)


A =[[[[[[[[[[

11987011989111199063119898minus11987011989121199063119898

11987011989111199061 minus 119870119891211990621198981198701198912119906311989811987011989111199063119898

11987011989121199061 + 1198701198911119906211989821198701199051198732119904 1199061119869 21198701199051198732119904 1199062119869 411987011990511987321198981199063119869

]]]]]]]]]]

(11)


(12)



D = 119894119898119886 = 0 1198942119898119886 = 1198732119904 11989421199041198861 + 1198732119904 1198942119904119886221198732119898 D = 119894119898119886 = 0 or 1198942119898119886 = 1198732119904 11989421199041198861 + 1198732119904 1198942119904119886221198732119898

(13)



1198941199041198861 = 119898(1198701198911 + 1198701198912 120573 + 1198921198701198912)(11987021198911+ 11987021198912) 119894119898119886

1198941199041198862 = 119898(1198701198911 120573 minus 1198701198912 + 1198921198701198911)(11987021198911+ 11987021198912) 119894119898119886

(14)


211987321198981198944119898119886 minus 119869 + 119879119871119870119905 1198942119898119886 + 1198732119904119898211987021198911+ 11987021198912

[2 + (119892 + 120573)2]= 0

(15)


Δ = (119869 + 119879119871119870119905 )2 minus 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 120573)2] (16)


119894211989811988612 = 141198732119898 (119869 + 119879119871119870119905 plusmn radicΔ) (17)



119894119898119886 = 12119873119898radic119869 + 119879119871119870119905 + radicΔ (18)


1198941199041198861 = 2119898119873119898 (1198701198911 + 1198701198912 120573 + 1198921198701198912)(11987021198911+ 11987021198912)radic[(119869 + 119879119871) 119870119905] + radicΔ


(19)


1199061 = 2119898119873119898 (11987011989111205931 + 11987011989121205932 + 1198921198701198912)(11987021198911+ 11987021198912)radic[(1198691205933 + 119879119871) 119870119905] + radicΔ


1199063 = radic[(1198691205933 + 119879119871) 119870119905] + radicΔ(2119873119898)

(20)


D = D1 = 119894119898119886 = 0D2 = 1198942119898119886 = 1198732119904 11989421199041198861 + 1198732119904 1198942119904119886221198732119898 (21)


= 0120573 = minus119892 (22)





(119869 + 119879119871119870119905 )2 = 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 120573)2] (23)


(119869 + 119879119871119870119905 )2 = 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 119870120573 120573)2] (24)


1199061 = 2119898119873119898 (11987011989111205931 + 11987011989121198701205731205932 + 1198921198701198912)(11987021198911+ 11987021198912)radic[(1198691205933 + 119879119871) 119870119905] + radicΔ1015840


1199063 = radic[(1198691205933 + 119879119871) 119870119905] + radicΔ1015840(2119873119898)

(25)





G (s) = [119866120572 (119904) 119866120573 (119904) 119866120596 (119904)]T = [ 11199042 11199042 1119904 ]T (26)



Φ (119904) = 1198861119904 + 1198860119904 (1199042 + 1198961119904 + 1198960) (27)


Φ (119904) = 1205962119899 (119904 + 120575)(119904 + 120575) (1199042 + 2120585120596119899119904 + 1205962119899) (28)


1198861 = 1205962119899 = 6400001198860 = 1205962119899120575 = 38400001198961 = 2120585120596119899 + 120575 asymp 113721198960 = 1205962119899 + 2120585120596119899120575 asymp 6467872

(29)



Impr

oved

inve

rse s

yste

m

(25)


Filter PWMamplifier

PWMamplifier

PWMamplifier


Dua

l-win

ding

BSR

M sy

stem

DCF

x

is1

im

is2

u1

u2

u3

1

2

3

K2(s)

K1(s)

T2(s)

T1(s)

T3(s)

lowast

lowast

lowast

++

++

+

minusminus

minusminus

minus

Filter

Filter



Φ (119904) = 1200119904 + 72001199042 + 1200119904 + 7200 (30)





Φ (119904) = 211199042 + 3400119904 + 48 times 1061199042 + 2080119904 + 48 times 106 (31)








minus90

minus45

0

45

90

Phas

e (de

g)

















start

Reversible













Satisfy the demand

Over

Yes

No

No

Yes

Yes

No



K

Design n and








0 1 2 3 4 5

0 1 2 3 4 5

081

1214

0 1 2 3 4 5

t (s)

times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

4 531 20

4 531 20

4 531 20t (s)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)









0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)








0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(a)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(d)










0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

t (s)

t (s)

minus02

0

02

(m

m)

minus02

0

02

(mm

)

081

1214times104

(r

min

)

kd = 003kd = 005

kd = 003kd = 005

krd = 01krd = 04

(a)

0 1 2 3 4 5t (s)

0 1 2 3 4 5

0 1 2 3 4 5minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)





0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

2 = 6

2 = 4

= 0707 = 05

= 0707 = 05

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)








5 Conclusion







Nomenclature






Competing Interests


Acknowledgments




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











D = 119894119898119886 = 0 1198942119898119886 = 1198732119904 11989421199041198861 + 1198732119904 1198942119904119886221198732119898 D = 119894119898119886 = 0 or 1198942119898119886 = 1198732119904 11989421199041198861 + 1198732119904 1198942119904119886221198732119898

(13)



1198941199041198861 = 119898(1198701198911 + 1198701198912 120573 + 1198921198701198912)(11987021198911+ 11987021198912) 119894119898119886

1198941199041198862 = 119898(1198701198911 120573 minus 1198701198912 + 1198921198701198911)(11987021198911+ 11987021198912) 119894119898119886

(14)


211987321198981198944119898119886 minus 119869 + 119879119871119870119905 1198942119898119886 + 1198732119904119898211987021198911+ 11987021198912

[2 + (119892 + 120573)2]= 0

(15)


Δ = (119869 + 119879119871119870119905 )2 minus 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 120573)2] (16)


119894211989811988612 = 141198732119898 (119869 + 119879119871119870119905 plusmn radicΔ) (17)



119894119898119886 = 12119873119898radic119869 + 119879119871119870119905 + radicΔ (18)


1198941199041198861 = 2119898119873119898 (1198701198911 + 1198701198912 120573 + 1198921198701198912)(11987021198911+ 11987021198912)radic[(119869 + 119879119871) 119870119905] + radicΔ


(19)


1199061 = 2119898119873119898 (11987011989111205931 + 11987011989121205932 + 1198921198701198912)(11987021198911+ 11987021198912)radic[(1198691205933 + 119879119871) 119870119905] + radicΔ


1199063 = radic[(1198691205933 + 119879119871) 119870119905] + radicΔ(2119873119898)

(20)


D = D1 = 119894119898119886 = 0D2 = 1198942119898119886 = 1198732119904 11989421199041198861 + 1198732119904 1198942119904119886221198732119898 (21)


= 0120573 = minus119892 (22)





(119869 + 119879119871119870119905 )2 = 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 120573)2] (23)


(119869 + 119879119871119870119905 )2 = 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 119870120573 120573)2] (24)


1199061 = 2119898119873119898 (11987011989111205931 + 11987011989121198701205731205932 + 1198921198701198912)(11987021198911+ 11987021198912)radic[(1198691205933 + 119879119871) 119870119905] + radicΔ1015840


1199063 = radic[(1198691205933 + 119879119871) 119870119905] + radicΔ1015840(2119873119898)

(25)





G (s) = [119866120572 (119904) 119866120573 (119904) 119866120596 (119904)]T = [ 11199042 11199042 1119904 ]T (26)



Φ (119904) = 1198861119904 + 1198860119904 (1199042 + 1198961119904 + 1198960) (27)


Φ (119904) = 1205962119899 (119904 + 120575)(119904 + 120575) (1199042 + 2120585120596119899119904 + 1205962119899) (28)


1198861 = 1205962119899 = 6400001198860 = 1205962119899120575 = 38400001198961 = 2120585120596119899 + 120575 asymp 113721198960 = 1205962119899 + 2120585120596119899120575 asymp 6467872

(29)



Impr

oved

inve

rse s

yste

m

(25)


Filter PWMamplifier

PWMamplifier

PWMamplifier


Dua

l-win

ding

BSR

M sy

stem

DCF

x

is1

im

is2

u1

u2

u3

1

2

3

K2(s)

K1(s)

T2(s)

T1(s)

T3(s)

lowast

lowast

lowast

++

++

+

minusminus

minusminus

minus

Filter

Filter



Φ (119904) = 1200119904 + 72001199042 + 1200119904 + 7200 (30)





Φ (119904) = 211199042 + 3400119904 + 48 times 1061199042 + 2080119904 + 48 times 106 (31)








minus90

minus45

0

45

90

Phas

e (de

g)

















start

Reversible













Satisfy the demand

Over

Yes

No

No

Yes

Yes

No



K

Design n and








0 1 2 3 4 5

0 1 2 3 4 5

081

1214

0 1 2 3 4 5

t (s)

times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

4 531 20

4 531 20

4 531 20t (s)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)









0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)








0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(a)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(d)










0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

t (s)

t (s)

minus02

0

02

(m

m)

minus02

0

02

(mm

)

081

1214times104

(r

min

)

kd = 003kd = 005

kd = 003kd = 005

krd = 01krd = 04

(a)

0 1 2 3 4 5t (s)

0 1 2 3 4 5

0 1 2 3 4 5minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)





0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

2 = 6

2 = 4

= 0707 = 05

= 0707 = 05

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)








5 Conclusion







Nomenclature






Competing Interests


Acknowledgments




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(119869 + 119879119871119870119905 )2 = 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 120573)2] (23)


(119869 + 119879119871119870119905 )2 = 811989821198732119898119873211990411987021198911+ 11987021198912

[2 + (119892 + 119870120573 120573)2] (24)


1199061 = 2119898119873119898 (11987011989111205931 + 11987011989121198701205731205932 + 1198921198701198912)(11987021198911+ 11987021198912)radic[(1198691205933 + 119879119871) 119870119905] + radicΔ1015840


1199063 = radic[(1198691205933 + 119879119871) 119870119905] + radicΔ1015840(2119873119898)

(25)





G (s) = [119866120572 (119904) 119866120573 (119904) 119866120596 (119904)]T = [ 11199042 11199042 1119904 ]T (26)



Φ (119904) = 1198861119904 + 1198860119904 (1199042 + 1198961119904 + 1198960) (27)


Φ (119904) = 1205962119899 (119904 + 120575)(119904 + 120575) (1199042 + 2120585120596119899119904 + 1205962119899) (28)


1198861 = 1205962119899 = 6400001198860 = 1205962119899120575 = 38400001198961 = 2120585120596119899 + 120575 asymp 113721198960 = 1205962119899 + 2120585120596119899120575 asymp 6467872

(29)



Impr

oved

inve

rse s

yste

m

(25)


Filter PWMamplifier

PWMamplifier

PWMamplifier


Dua

l-win

ding

BSR

M sy

stem

DCF

x

is1

im

is2

u1

u2

u3

1

2

3

K2(s)

K1(s)

T2(s)

T1(s)

T3(s)

lowast

lowast

lowast

++

++

+

minusminus

minusminus

minus

Filter

Filter



Φ (119904) = 1200119904 + 72001199042 + 1200119904 + 7200 (30)





Φ (119904) = 211199042 + 3400119904 + 48 times 1061199042 + 2080119904 + 48 times 106 (31)








minus90

minus45

0

45

90

Phas

e (de

g)

















start

Reversible













Satisfy the demand

Over

Yes

No

No

Yes

Yes

No



K

Design n and








0 1 2 3 4 5

0 1 2 3 4 5

081

1214

0 1 2 3 4 5

t (s)

times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

4 531 20

4 531 20

4 531 20t (s)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)









0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)








0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(a)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(d)










0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

t (s)

t (s)

minus02

0

02

(m

m)

minus02

0

02

(mm

)

081

1214times104

(r

min

)

kd = 003kd = 005

kd = 003kd = 005

krd = 01krd = 04

(a)

0 1 2 3 4 5t (s)

0 1 2 3 4 5

0 1 2 3 4 5minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)





0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

2 = 6

2 = 4

= 0707 = 05

= 0707 = 05

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)








5 Conclusion







Nomenclature






Competing Interests


Acknowledgments




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Impr

oved

inve

rse s

yste

m

(25)


Filter PWMamplifier

PWMamplifier

PWMamplifier


Dua

l-win

ding

BSR

M sy

stem

DCF

x

is1

im

is2

u1

u2

u3

1

2

3

K2(s)

K1(s)

T2(s)

T1(s)

T3(s)

lowast

lowast

lowast

++

++

+

minusminus

minusminus

minus

Filter

Filter



Φ (119904) = 1200119904 + 72001199042 + 1200119904 + 7200 (30)





Φ (119904) = 211199042 + 3400119904 + 48 times 1061199042 + 2080119904 + 48 times 106 (31)








minus90

minus45

0

45

90

Phas

e (de

g)

















start

Reversible













Satisfy the demand

Over

Yes

No

No

Yes

Yes

No



K

Design n and








0 1 2 3 4 5

0 1 2 3 4 5

081

1214

0 1 2 3 4 5

t (s)

times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

4 531 20

4 531 20

4 531 20t (s)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)









0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)








0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(a)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(d)










0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

t (s)

t (s)

minus02

0

02

(m

m)

minus02

0

02

(mm

)

081

1214times104

(r

min

)

kd = 003kd = 005

kd = 003kd = 005

krd = 01krd = 04

(a)

0 1 2 3 4 5t (s)

0 1 2 3 4 5

0 1 2 3 4 5minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)





0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

2 = 6

2 = 4

= 0707 = 05

= 0707 = 05

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)








5 Conclusion







Nomenclature






Competing Interests


Acknowledgments




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












minus90

minus45

0

45

90

Phas

e (de

g)

















start

Reversible













Satisfy the demand

Over

Yes

No

No

Yes

Yes

No



K

Design n and








0 1 2 3 4 5

0 1 2 3 4 5

081

1214

0 1 2 3 4 5

t (s)

times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

4 531 20

4 531 20

4 531 20t (s)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)









0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)








0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(a)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(d)










0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

t (s)

t (s)

minus02

0

02

(m

m)

minus02

0

02

(mm

)

081

1214times104

(r

min

)

kd = 003kd = 005

kd = 003kd = 005

krd = 01krd = 04

(a)

0 1 2 3 4 5t (s)

0 1 2 3 4 5

0 1 2 3 4 5minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)





0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

2 = 6

2 = 4

= 0707 = 05

= 0707 = 05

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)








5 Conclusion







Nomenclature






Competing Interests


Acknowledgments




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










start

Reversible













Satisfy the demand

Over

Yes

No

No

Yes

Yes

No



K

Design n and








0 1 2 3 4 5

0 1 2 3 4 5

081

1214

0 1 2 3 4 5

t (s)

times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

4 531 20

4 531 20

4 531 20t (s)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)









0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)








0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(a)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(d)










0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

t (s)

t (s)

minus02

0

02

(m

m)

minus02

0

02

(mm

)

081

1214times104

(r

min

)

kd = 003kd = 005

kd = 003kd = 005

krd = 01krd = 04

(a)

0 1 2 3 4 5t (s)

0 1 2 3 4 5

0 1 2 3 4 5minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)





0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

2 = 6

2 = 4

= 0707 = 05

= 0707 = 05

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)








5 Conclusion







Nomenclature






Competing Interests


Acknowledgments




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5

0 1 2 3 4 5

081

1214

0 1 2 3 4 5

t (s)

times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

4 531 20

4 531 20

4 531 20t (s)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)









0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)








0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(a)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(d)










0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

t (s)

t (s)

minus02

0

02

(m

m)

minus02

0

02

(mm

)

081

1214times104

(r

min

)

kd = 003kd = 005

kd = 003kd = 005

krd = 01krd = 04

(a)

0 1 2 3 4 5t (s)

0 1 2 3 4 5

0 1 2 3 4 5minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)





0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

2 = 6

2 = 4

= 0707 = 05

= 0707 = 05

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)








5 Conclusion







Nomenclature






Competing Interests


Acknowledgments




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

095

1

105times104

(r

min

)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

(d)








0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(a)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(d)










0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

t (s)

t (s)

minus02

0

02

(m

m)

minus02

0

02

(mm

)

081

1214times104

(r

min

)

kd = 003kd = 005

kd = 003kd = 005

krd = 01krd = 04

(a)

0 1 2 3 4 5t (s)

0 1 2 3 4 5

0 1 2 3 4 5minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)





0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

2 = 6

2 = 4

= 0707 = 05

= 0707 = 05

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)








5 Conclusion







Nomenclature






Competing Interests


Acknowledgments




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(a)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5minus02

0

02

(m

m)

0 1 2 3 4 5

t (s)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(mm

)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(d)










0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

t (s)

t (s)

minus02

0

02

(m

m)

minus02

0

02

(mm

)

081

1214times104

(r

min

)

kd = 003kd = 005

kd = 003kd = 005

krd = 01krd = 04

(a)

0 1 2 3 4 5t (s)

0 1 2 3 4 5

0 1 2 3 4 5minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)





0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

2 = 6

2 = 4

= 0707 = 05

= 0707 = 05

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)








5 Conclusion







Nomenclature






Competing Interests


Acknowledgments




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

t (s)

t (s)

minus02

0

02

(m

m)

minus02

0

02

(mm

)

081

1214times104

(r

min

)

kd = 003kd = 005

kd = 003kd = 005

krd = 01krd = 04

(a)

0 1 2 3 4 5t (s)

0 1 2 3 4 5

0 1 2 3 4 5minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)





0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

2 = 6

2 = 4

= 0707 = 05

= 0707 = 05

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)








5 Conclusion







Nomenclature






Competing Interests


Acknowledgments




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

2 = 6

2 = 4

= 0707 = 05

= 0707 = 05

(a)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(b)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

t (s)

minus02

0

02

(m

m)

minus02

0

02

(m

m)

081

1214times104

(r

min

)

(c)








5 Conclusion







Nomenclature






Competing Interests


Acknowledgments




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












5 Conclusion







Nomenclature






Competing Interests


Acknowledgments




References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









decoupling control for dual-winding bearingless switched...

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