Research Article Sliding Mode Control for Bearingless...

Research ArticleSliding Mode Control for Bearingless Induction Motor Based ona Novel Load Torque Observer

Zebin Yang1 Ling Wan1 Xiaodong Sun2 Lin Chen1 and Zheng Chen1

1School of Electrical and Information Engineering Jiangsu University Zhenjiang 212013 China2Automotive Engineering Research Institute Jiangsu University Zhenjiang 212013 China

Correspondence should be addressed to Xiaodong Sun xdsunujseducn

Received 11 June 2016 Revised 26 August 2016 Accepted 6 September 2016

Academic Editor Rafael Morales

Copyright copy 2016 Zebin Yang 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

For the problem of low control performance of Bearingless Induction Motor (BIM) control system in the presence of large loaddisturbance a novel load torque sliding mode observer is proposed on the basis of establishing sliding mode speed control systemThe load observer chooses the speed and load torque of the BIM control system as the observed objects uses the speed error todesign the integral sliding mode surface and adds the low-pass filter to reduce the torque observation error Meanwhile the outputof the load torque is used as the feedforward compensation for the control system which can provide the required current forload changes and reduce the adverse influence of disturbance on system performance Besides considering that the load changeslead to the varying rotational inertia the integral identification method is adopted to identify the rotational inertia of BIM andthe rotational inertia can be updated to the load observer in real time The simulation and experiment results all show that theproposed method can track load torque accurately improve the ability to resist disturbances and ameliorate the operation qualityof BIM control system The chattering of sliding mode also is suppressed effectively

1 Introduction

Based on the similarity principles of magnetic bearing andalternating current (AC) motor stator structure BIM isformed Two sets of windings are embedded in the stator slotof BIMwhich can separately produce electromagnetic torqueand radial levitation force BIM achieves the integration ofrapid rotation and stable suspension of rotor by changingthe currents in the windings and avoids the mechanicalbearing friction wear and tear and lubrication It breaksthe bottleneck of traditional asynchronous motor developingtowards the higher precision and higher speed direction [1ndash5] BIMhasmany better advantages than the traditional asyn-chronous motor such as simple structure uniform air gaphigh mechanical strength high speed and ultrahigh speedrunning in the corrosion or other special environmentsTherefore it shows broad development prospect in medicalequipment transportation national defense and so forth [6ndash9] However BIMhas the characteristics of nonlinearitymul-tivariability and strong couplingThe traditional PI controller

cannot acquire high-performance control for BIM when thecontrol system is disturbed by load torque [10]

Sliding mode variable structure control as a kind ofspecial nonlinear control can operate in accordance withthe trajectory designed by people and purposefully adjustoperation according to the system status which can gainexcellent control performance Due to the fact that the slidingmode control not only can be set by people but also doesnot need high precision mathematical model and has strongrobustness to disturbances it is becoming a hot researchtopic [11ndash15] and it is gradually applied in the AC servosystem In [16] a new reaching law was designed to improvethe operation quality of sliding mode At the same time itwas applied in the speed control which effectively enhancedthe robustness of permanent magnet synchronous motor(PMSM) system In [17] the sliding mode control combiningwith model reference adaptive was used to obtain the speedThe results showed that it increased the estimation precisionof rotor velocity for PMSM and decreased the chattering In[18] the sliding mode control was used in a generator based

Hindawi Publishing CorporationJournal of SensorsVolume 2016 Article ID 8567429 10 pageshttpdxdoiorg10115520168567429

2 Journal of Sensors

on the exercise equipment with nonlinear 119875-119881 characteristiccurves The amount of generator input current harmonicis greatly reduced In [19] the conventional sliding modecontrol was united with the adaptive fuzzy backsteppingscheme The simulation proved that this method improvedthe performance of mismatched uncertain system In [20]the sliding mode control dealt with the difficult problemof obtaining the counterelectromotive force and it finallyimplemented the direct torque control of brushless directcurrent motor In [21] the sliding mode control was used todetect the speed and position for PMSM The experimentalresults proved the validity of the proposed sliding modeobserver In [22] based on the nonsingular terminal slidingmode algorithm and backstepping method the sliding modeobserver and position controller were put forward which canestimate the torque accurately and track the position quicklyIn [23] the adaptive sliding mode control for uncertainsingularly perturbed nonlinear system was designed It notonly reduced the effects of uncertainty but also guaranteedthe control performance In [24 25] the load sliding modeobservers were proposedThey diminished the adverse effectsof load changes on PMSM and improved the antidisturbanceability of controlled system at some level However theyall ignored the problem that the load changes result in thedifferent rotational inertia and the controlled system hadlarge chattering Hence the system cannot achieve the bestdynamic performance

A novel sliding mode observer of load torque of whichthe state variables are the speed and load torque is proposedto suppress the impacts of the load torque changes on BIMcontrol system A low-pass filter used in the observer reducesthe observation error of torque Moreover the observer asfeedforward compensation for the given current alleviatesthe output pressure of sliding mode controller (SMC) Inaddition adopting the integral identification method validlyidentifies the rotational inertia and improves the precision ofBIM The simulation and experimental results show that theproposed method overcomes the disadvantageous effects onthe speed regulation system generated by load disturbancesand strengthens the antidisturbance ability of the system

2 The Dynamics Model of BIM

According to the electromagnetic field theory the radial lev-itation force of BIM in the 119889-119902 coordinates can be establishedas [6]

119865119909 = 119870 (12059511198891198941199042119889 + 12059511199021198941199042119902) 119865119910 = 119870 (12059511198891198941199042119902 minus 12059511199021198941199042119889) (1)

where 119870 = 119870119898 + 119870119897 119870119898 = 12058711987511198752119871119898118119897119903120583011987311198732 and119870119897 = 1198751119873221199031198731 119865119909 and 119865119910 are the components of theradial levitation force in 119909 and 119910 directions the subscript ldquo1rdquorepresents the torque windings the subscript ldquo2rdquo representsthe radial levitation force windings ldquo119904rdquo represents the statorand ldquo119903rdquo represents the rotor 1198751 and 1198752 separately representthe pole pairs of torque windings and suspension windings1198941199042119889 and 1198941199042119902 are the current components of the stator in

levitation force windings under the 119889-119902 axis 1198711198981 is mutualinductance of the levitation force windings 119897 is the effectivelength of the rotor 119903 is the stator inner diameter 1205830 is thepermeability of vacuum 1198731 and 1198732 respectively show theeffective number of turns of the torque windings and the lev-itation force windings and 1205951119889 and 1205951119902 are the componentsof flux linkage for the torque winding in the 119889-119902 coordinatesrespectively

With the torque windings and the levitation force wind-ings BIM is a nonlinear strongly coupled and complexsystem In order to simplify the analysis of BIM a hypothesisis given that the levitation force windings only create arotating magnetic field The rotor voltage equation can bedescribed as

1199061199031119889 = 11987711990311198941199031119889 + 1199011205951199031119889 minus (1205951119902 + 11987111990311198971198941199031119902) (1205961 minus 120596119903)= 0

1199061199031119902 = 11987711990311198941199031119902 + 1199011205951199031119902 minus (1205951119889 + 11987111990311198971198941199031119889) (1205961 minus 120596119903)= 0

(2)

where 1199061199031119889 and 1199061199031119902 are the rotor voltages of torque windingsin 119889-119902 coordinates 1198771199031 is the rotor resistance 1205961 and 120596119903 areseparately the air gap field speed and rotor speed and 119901 is thedifferential operator

The flux linkage can be expressed as

1205951119889 = (1198941199041119889 + 1198941199031119889) 11987111989811205951119902 = (1198941199041119902 + 1198941199031119902) 11987111989811205951199041119889 = 1205951119889 + 1198941199041119889119871 11990411198971205951199041119902 = 1205951119902 + 1198941199041119902119871 11990411198971205951199031119889 = 1205951119889 + 119894119903111988911987111990311198971205951199031119902 = 1205951119902 + 11989411990311199021198711199031119897

(3)

where119871 1199041119897 and1198711199031119897 are the stator leakage inductance and rotorleakage inductance of torque windings respectively

The electromagnetic torque equation is set up as

119879119890 = 1198751 (12059511198891198941199041119902 minus 12059511199021198941199041119889) (4)

The equation of motion is written as

119879119890 = 119879119871 + 1198691198751119901120596119903 (5)

where 119879119871 is the load torque and 119869 is the rotational inertiaAfter coordinate transforming the rotor flux in 119889-119902 axis

can be expressed as

1205951199031119889 = 11987111989811198941199041119889 + 119871119903111989411990311198891205951199031119902 = 11987111989811198941199041119902 + 11987111990311198941199031119902 (6)

Journal of Sensors 3

Making the axis of the rotating coordinates 119889 coincide withthe rotor flux linkage of torque windings it is written as1205951199031119889 = 1205951199031 Formula (6) can be simplified as

1198941199031119889 = 1205951199031 minus 119871119898111989411990411198891198711199031 1198941199031119902 = minus11987111989811198711199031 1198941199041119902

(7)

Putting Formula (7) into Formula (2) the excitationcurrent 1198941199041119889 and slip speed 120596119904 can be obtained as follows

1198941199041119889 = 1198791199031119901 + 11198711198981 1205951199031120596119904 = 119871119898111987911990311205951199031 1198941199041119902

(8)

where 120596119904 = 1205961 minus 120596119903 and 1198791199031 = 11987111990311198771199031 is the time constant ofrotor

The electromagnetic torque equation turns into

119879119890 = 1198751 11987111989811198711199031 11989411990411199021205951199031 (9)

where 1198711199031 is the rotor self-induction Figure 1 is the blockdiagram of rotor field-oriented decoupling control

3 The Speed SMC of BIM

The system state variables are defined as

1198901205961 = 120596lowast minus 1205961198901205962 = 11989010158401205961 = minus1205961015840 (10)

where 120596lowast is the given speed and 120596 is the actual speedAfter combining with (5) Formula (10) is described as

11989010158401205961 = 1198901205962 = minus119875211205951199031119869 11987111989811198711199031 1198941199041119902 +1198751119869 119879119871

11989010158401205962 = 119890101584010158401205961 = minus119875211205951199031119869 11987111989811198711199031 11989410158401199041119902

(11)

The sliding mode surface is chosen as

119904 = 1198901205961 + 11988811198901205962 (12)

This paper chooses the reaching law [16] to weaken theinherent chattering

119889119904119889119905 = minus120576 |119883|2 sgn (119904) minus 119896 |119883|2 119904lim119905rarrinfin

|119883| = 0 119886 ge 0 119887 ge 0 120576 gt 0 119896 gt 0 (13)

According to the Lyapunov stability theory the existencecondition of generalized sliding mode is written as [12]

119881 = 121199042 lt 0 (14)

is1q is1q

is1d

120596s 120579s1

120591r

120595r1 1

Lm1

1 + 120591rp

Figure 1 Rotor field-oriented decoupling control

Differentiating (14) with respect to time it becomes (15)by substituting (13)

= 119904 119904 = 119904 (minus120576 |119883|2 sgn (119904) minus 119896 |119883|2 119904)= minus120576 |119883|2 |119904| minus 119896 |119883|2 1199042 lt 0 (15)

Formula (15) always stands up Hence the system can arriveat sliding mode surface in limited time

Differentiating 119904 = 1198901205961 + 11988811198901205962 with respect to time it canbe gained as

1199041015840 = 11989010158401205961 + 119888111989010158401205962 = 1198901205962 minus 119888111987521120595119903111987111989811198691198711199031 11989410158401199041119902 (16)

Combining Formula (13) with Formula (16) gives thefollowing formula

1198901205962 minus 119888111987521120595119903111987111989811198691198711199031 11989410158401199041119902 = minus120576 |119883|2 sgn (119904) minus 119896 |119883|2 119904 (17)

Choosing119883 = 1198901205961 to avoid the differential interference in1198901205962 the sliding mode controller is designed as

1198941199041119902 = 119869119871119903111987521120595119903111987111989811198881sdot int (120576 1003816100381610038161003816119890120596110038161003816100381610038162 sat (119904) + 119896 1003816100381610038161003816119890120596110038161003816100381610038162 119904 + 1198901205962) 119889119905

(18)

From (18) it can be seen that the current can eliminate steady-state error and improve the accuracy of system

4 The Design of Novel Load Torque SlidingMode Observer

41 The Design of Load Torque Sliding Mode Observer Con-sidering the high switch frequency of the controller the loadtorque can be deemed to be a constant value in a controlcycle Considering the load torque as an extension the stateequation of BIM can be expressed as

119889120596119889119905 = 119875211205951199031119869 11987111989811198711199031 1198941199041119902 minus1198751119869 119879119871

119889119879119871119889119905 = 0(19)


Based on the equation above the extended load torqueobserver is written as

119889119889119905 = 119875211205951199031119869 11987111989811198711199031 1198941199041119902 minus1198751119869 119871 + 119881

119889119871119889119905 = 120578119881(20)

where 119881 = 120574 sgn (120596 minus ) 120574 is the sliding mode gain 120578 is thefeedback gain and and 119871 are the estimations of electricalangular velocity and load torque respectively

The estimation errors of speed and the load are defined as

[11990911199092] = [ 120596 minus 119879119871 minus 119871] (21)

After Formula (19) subtracts Formula (20) the observa-tion errors of sliding mode are obtained as

1198891199091119889119905 = minus1198751119869 1199092 minus 1198811198891199092119889119905 = minus120578119881

(22)

Because the torque change is expressed in the form ofspeed finally the designed sliding mode surface consists ofthe state variable 1199091 = 120596 minus The sliding mode surface isestablished as (23) to reduce the system overshoot

119904 = 1199091 + 119888int1199091119889119905 (23)

Differentiating the sliding mode surface and combiningwith (13) and (5) it can be acquired as

119871 = 1198691198751 [120574 sgn (120596 minus ) minus 119888 (120596 minus ) minus 119896 |120596 minus |2 119904minus 120576 |120596 minus |2 sgn (119904)] + 119879119871 (0)

(24)

Because the system load torque 119879119871 is an unknown variable119879119871(0) is recorded as the estimation of load torque at time zeroIn order to decrease the chattering in the sliding mode

the sign function sgn (119904) is replaced by saturation functionsat (119904 Δ) [25]

sat (119904 Δ) =

1 119904119894 (119909) gt minusΔ119904119894 (119909)Δ minusΔ lt 119904119894 (119909) lt Δminus1 119904119894 (119909) lt minusΔ

(25)

Formula (24) can be expressed as

119871 = 1198691198751 [120574 sat (120596 minus ) minus 119888 (120596 minus ) minus 119896 |120596 minus |2 119904minus 120576 |120596 minus |2 sat (119904 Δ)] + 119879119871 (0)

(26)

Speedregulation

Currentregulation BIM

Opticalencoder

Integral identificationof the rotational inertia

k

120596lowast

120596

120596

+

minusiq

Te

J

Figure 2 Diagram of inertia identification

For the purpose of improving the observation precisionof sliding mode observer a low-pass filter [7] shown in (27)is added to the observer

119871 = 120596119888119904 + 120596119888 119871 (27)

The output of load torque observer is as the feedforwarddisturbance compensation 1198941199022 and the given current 119894lowast119902 isdescribed as

119894lowast119902 = 1198941199041119902 + 1198941199022= 119869119871119903111987521120595119903111987111989811198881 int(120576 1003816100381610038161003816119890120596110038161003816100381610038162 sat (119904) + 119896 1003816100381610038161003816119890120596110038161003816100381610038162 119904 + 1198901205962) 119889119905

+ 119871120597 (28)

where 120597 gt 0 is the feedforward gain of torque observationFrom (18) and (28) it can be known that 120576 and 119896 in

Formula (18) need to be large enough for meeting the loaddisturbance while large 120576 and 119896 increase the amplitude ofdiscrete magnitude and result in big chattering However in(28) the observed disturbance is used to provide the requiredcurrent for disturbance changes and needs no large 120576 and 119896As a consequence the feedforward compensation scheme ofload torque reduces the gain amplitude of sliding mode andlowers the negative impacts on the control system caused bythe disturbances

42 The Load Torque Sliding Mode Observer with the Rota-tional Inertia Online Identification When a sliding modeobserver is designed the rotational inertia of system isusually regarded as a known quantity However in practiceapplication the load changes will lead to the inertia changesTherefore identifying the inertia online and timely updatingit to the observer will greatly improve the overall controlperformance of BIM system With the characteristics of highprecision and strong robustness the integral identificationalgorithm is used to identify the rotational inertia onlineThestructure diagram of BIMrsquos speed loop based on the integralidentification is shown in Figure 2

Equation (5) is rewritten as

119879119890 = 1198751119889120596119889119905 + 119903 (29)


where is the estimation of rotational inertia and 119903 is thecollection of disturbances and its specific expression is asfollows

119903 (119905) = Δ119869119889120596119889119905 + 119879119871 (30)

where Δ119869 is the inertia error 119869 minus = Δ119869By the formula above it can be found that the output

of disturbance is in the form of torque At the same timeobserving torque can obtain Δ119869 Using Δ119869 and constantrecursion correction based on selecting initial value of inertiathe recursion equation can be expressed as

(119896) = (119896 minus 1) + Δ119869 (31)

Thus the identification accuracy of inertia depends onΔ119869 In order to obtain high accuracy Δ119869 this paper choosesa periodic speed signal and uses the integral to eliminate theinfluences of torque disturbances on rotational inertia

Because of the high sample frequency the load torque119879119871 is regarded as 119879119871(119905) = 119879119871(119905 + 119879) and the speed signal 120596meets 120596(119905) = 120596(119905 + 119879) After on both sides of Formula (30)multiplying by (119905) and integrating it the equation is writtenas

int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905 = Δ119869int119896119879(119896minus1)119879

2 (119905) 119889119905+ int119896119879(119896minus1)119879

119879119871 (119905) (119905) 119889119905(32)

Due to the fact that the load torque is a constant withina cycle the definite integral of the load torque in (32) can becalculated as follows

int119896119879(119896minus1)119879

119879119871 (119905) (119905) 119889119905 = 119879119871120596 (119905)1003816100381610038161003816119896119879(119896minus1)119879= 119879119871120596 (119896119879) minus 119879119871120596 ((119896 minus 1) 119879)= 0

(33)

Hence (32) can be simplified as

int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905 = Δ119869int119896119879(119896minus1)119879

2 (119905) 119889119905 (34)

From (34) it can be found that the effects of loaddisturbance on the inertia have been solved

Equation (34) can be expressed as

Δ119869 = int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905int119896119879(119896minus1)119879

2 (119905) 119889119905 (35)

Combining with (35) the recursive equation of inertiaidentification is available as

(119896) = (119896 minus 1) + int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905int119896119879(119896minus1)119879

2 (119905) 119889119905 (36)

Table 1 Parameters of the Bearingless Induction Motor (BIM)

Parameters Torque winding Suspension windingRated power (Kw) 1 05Rated current (A) 286 286Stator resistance (Ω) 201 103Rotor resistance (Ω) 1148 0075Mutual inductance ofstator and rotor (H) 015856 000932

Stator leakageinductance (H) 016310 001199

Rotor leakageinductance (H) 016778 001474

Rotational inertia(kgsdotm2) 000769 000769

Rotor mass (kg) 285 285Stator inner diameter(mm) 98 98

Core length (mm) 105 105Pole pairs 1 2

Inertia identification is realized and can be updated to theload observer automatically

5 Simulation and Experimental Research

51 Results and Analysis of the Simulation To validate theeffectiveness of novel load torque observer with the functionof inertia identification online and feedforward compensa-tion scheme for BIM speed regulation system a simulationmode of control system is constructed The control blockdiagram of BIM includes two parts the rotation part and thesuspension part In rotation part the SMC outputs the givencurrent 119894lowast119902 by inputting the speed error Combining with thegiven air gap flux 120595lowast1 the excitation component is receivedWith the coordinate transformation method the two-phaseexcitation current is transformed into the three-phase givencurrent With the current regulation the three-phase currentis obtained which is used to control the rotation of rotor Insuspension part the radial levitation force of BIM is outputby PID controllers With 120595lowast1 the current in the levitationwindings can be calculated By the coordinate transformationand current regulation the required three-phase current isgained The steady suspension and rapid rotation of rotor arerealized finally The whole control block diagram of BIM isshown in Figure 3 and the specific parameters of BIM areshown in Table 1

In the simulation the rotational inertia is set to000769kgsdotm2 and the given speed is 119899 = 10000 rminBased on the tracking characteristics of integral identificationalgorithm a step signal is selected as the given speed whoseamplitude is 10000 rmin and the sampling cycle is definedas 119879 = 002 s Figure 4 shows the identification waveformof rotational inertia when the system suffers the load distur-bances It can be seen that the rotational inertia 119869 convergesto the given value within a sampling period at first When


BIMminus

+

+ minus

CRPWM

PID

PID

CRPWM23

yx

y

x

SMC

23

++

32

k

J

120596

120596

Te

Te

iq

120596lowast+

minus

is1q

xlowast

ylowast

Flowastx

Flowasty

ilowast2d

ilowast2q

ilowast2A

ilowast2B

ilowast2C

i2A

i2B

i2C

ilowastq

120595lowast1 ilowast1d

ilowast1q

ilowast1A

ilowast1B

ilowast1C

i1A

i1B

i1C

k1

iq2

Radialsuspension

forcemodel

Air-gapflux

orientedcontrolmodel

Load torquesliding mode

observerOnline

identificationof rotational

inertia

Figure 3 Control system diagram of Bearingless Induction Motor (BIM)

0 01 02 03 04 050004

0005

0006

0007

0008

t (s)

00076

000765

00077

02 022 024 026 028018

J(k

gmiddotm

2)

Figure 4 The identification result of rotational inertia

the system is attacked by the load disturbance (8Nsdotm) 119869 justhas small fluctuation and restores the stabilization quicklyTherefore the integral identification algorithm has bettertracking ability and shows good robustness for disturbance

Figures 5(a) and 5(b) show the estimation load 119871 and theactual load 119879119871 under load mutation As shown in (a) and (b)a sudden load (8Nsdotm) is added to the system at 119905 = 04 sThen the load drops to 0Nsdotm at 119905 = 06 s It can be seen fromFigure 5 that the sliding mode observer can accurately trackthe load torque and has strong robustness

Figure 6 shows the simulations of BIM in the presenceof sudden load (8Nsdotm) Figures 6(a) and 6(b) present theoutput currents from the SMC and the load torque observerrespectively Figure 6(c) shows the speed response of BIMunder the proposed control strategy in this paper and the con-ventional SMC Figure 6(a) shows that the current from SMCjust slightly increases and it returns back to the stable valuerapidly Figure 6(b) indicates that the compensation currentfrom load torque observer rises quickly which providesenough current for disturbance Figure 6(c) demonstratesthat the speed of BIM controlled by the conventional SMChas larger fluctuation and needs more time to operate at theoriginal speed value than the proposed method when it is

attacked by load mutation Based on the above analysis thefollowing two conclusions can be obtained

(1) The feedforward compensation scheme of load torqueobserver can provide the required current for loadchanges It can reduce the output pressure of SMCandmake the output of SMC almost invariant

(2) Based on the novel load observer and feedforwardcompensation strategy the speed of BIM under bigdisturbance has no fluctuation and can quickly con-verge to the original value The method weakensthe system chattering effectively and enhances thestability of system

Figures 7(a) and 7(b) show the rotor radial displacementat the speed of 119899 = 10000 rmin It can be known that therotor can arrive at the steady point rapidly with the proposedcontrol strategy in this paper It achieves the integration ofrapid rotation and stable suspensionThe systemhas excellentcontrol performance

52 Results and Analysis of the Experiment In order to fur-ther verify the effectiveness of the proposed control methodan experimental prototype with two degrees of freedomis used to build experimental platform Due to the limitsof photoelectrical encoder measuring speed the speed isset to 2000 rmin in the experiment The air gap of motorauxiliary bearing is 04mm Moreover the load mutation iscarried out to detect antijamming performance of BIM Theexperimental results are shown in Figure 8

Figure 8(a) shows the radial displacement when thesystem is controlled by the proposed method in this paperThe rotor is running around the equilibrium point and themaximum offset value is far less than the air gap of auxiliarybearing It indicates that the rotor is suspended steadily underthe proposed control strategy Figures 8(b)ndash8(e) show theresponses of BIM with the load mutation Figure 8(b) givesthe identification results of rotational inertia In view of theexcellent robustness of integral identification algorithm the


t (s)0 02 04 06 08 1

minus10

0

10

20

30

40

50T

(Nmiddotm

)

TL

(a)

t (s)0 02 04 06 08 1

minus10

0

10

20

30

40

50

T(N

middotm)

TL

(b)

Figure 5 The torque waveforms under load mutation (a) the waveform of the estimation torque (b) the waveform of the actual torque

t (s)0 02 04 06 08 1

minus5

0

5

10

15

20

i s1q

(A)

(a)

t (s)0 02 04 06 08 1

minus10

minus5

0

5

10

i q2

(A)

(b)

t (s)02 04 06 08 1

9800

9850

9900

9950

10000

10050

10100

The conventional SMC

The proposedcontrol strategy

n(r

min

)

(c)

Figure 6 The responses of current and speed under load mutation (a) the output current of SMC (b) the compensation current of loadtorque and (c) the speed response of BIM under sudden load

t (s)0 02 04 06 08 1

minus002

minus001

0

001

002

x (m

m)

(a)

t (s)0 02 04 06 08 1

minus003

minus002

minus001

0

001

002

y (m

m)

(b)

Figure 7 Waveforms of rotor radial displacement (a) the radial displacement on 119909-axis (b) the radial displacement on 119910-axis


x

y

80120583

m

80120583m

1

(a)

Time

J

5eminus

4kg

middotm2

20ms

1

(b)

T

Time 20ms

2Nmiddotm TL

TL

T

(c)

n

Time

n

20ms

2A

50

rm

in

1

is1q

(d)

n

Time 20ms

2A

50

rm

in 1

is1q

(e)

Figure 8 The experimental waveforms underload mutation (a) the radial displacement of rotor (b) the identification results of rotationalinertia under load mutation (c) the waveforms of torque (d) the responses of speed and current under the conventional SMC and (e) theresponses of speed and current under the proposed method

rotational inertia returns back to the stable value after a slightfluctuation The observing waveform of load torque is givenin Figure 8(c) It can be seen from the waveform that thesliding mode observer tracks the load precisely Figures 8(d)and 8(e) present the experimental results based on the controlstrategy of ordinary SMC and the proposed method in thispaper separately From Figure 8(d) it can be found that thecurrent has big undulation The speed decreases by 40 rminand needs 20ms to restore stability From Figure 8(e) theload mutation does not affect the output of SMC and thespeed is smooth By comparing (d) with (e) it can be foundthat using the new load torque sliding mode observer andthe feedforward compensation scheme canmake BIM exactlyand stably operate In addition it has low sensitivity withrespect to disturbances

6 Conclusions

A novel load torque sliding mode observer was proposedto eliminate the adverse impacts caused by load disturbancein BIM control system Owning to the application of a low-pass filter in the sliding mode observer the observation errorof load torque is effectively reduced At the same time theoutput of load torque observer as disturbance compensa-tion greatly diminishes the amplitude of discrete quantityand weakens chattering With the integral identificationmethod the proposed sliding mode observer can identifythe rotational inertia accurately and improve the robustnessof rotational inertia for disturbances The simulation andexperimental results all show that the proposed controlscheme in this paper effectively improves the dynamic and


static performance of BIM control system suppresses thesystem chattering and enhances the robustness of system

Competing Interests

The authors declare no conflict of interests

Authorsrsquo Contributions

Zebin Yang and Ling Wan proposed the new idea of slidingmode control for Bearingless Induction Motor based on anovel load torque observer Ling Wan derived the equationsXiaodong Sun established the simulation model Lin Chenwas in charge of analyzing the data Zheng Chen checkedthe language All authors were involved in preparing themanuscript

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Projects 51475214 61104016and 51305170 the China Postdoctoral Science FoundationFunded Project 2015T80508 the Natural Science Foundationof Jiangsu Province of China under Projects BK20130515BK20141301 and BK20150524 the Professional ResearchFoundation for Advanced Talents of JiangsuUniversity underProjects 12JDG057 and 14JDG076 Six Talent Peaks of JiangsuProvince under Projects ZBZZ-017 and 2015-XNYQC-003and the Priority Academic ProgramDevelopment (PAPD) ofJiangsu Higher Education Institutions

References

[1] T Hiromi T Katou A Chiba M A Rahman and T Fukao ldquoAnovel magnetic suspension-force compensation in bearinglessinduction-motor drive with squirrel-cage rotorrdquo IEEE Transac-tions on Industry Applications vol 43 no 1 pp 66ndash76 2007

[2] X Sun L Chen and Z Yang ldquoOverview of bearinglesspermanent-magnet synchronousmotorsrdquo IEEETransactions onIndustrial Electronics vol 60 no 12 pp 5528ndash5538 2013

[3] E F Rodriguez and J A Santisteban ldquoAn improved controlsystem for a split winding bearingless induction motorrdquo IEEETransactions on Industrial Electronics vol 58 no 8 pp 3401ndash3408 2011

[4] X Sun L Chen H Jiang Z Yang J Chen and W ZhangldquoHigh-performance control for a bearingless permanent mag-net synchronous motor using neural network inverse schemeplus internal model controllersrdquo IEEE Transactions on Indus-trial Electronics vol 63 no 6 pp 3479ndash3488 2016

[5] A T De Almeida F J T E Ferreira and A Q DuarteldquoTechnical and economical considerations on super high-efficiency three-phase motorsrdquo IEEE Transactions on IndustryApplications vol 50 no 2 pp 1274ndash1285 2014

[6] X Sun L Chen Z Yang and H Zhu ldquoSpeed-sensorless vectorcontrol of a bearingless induction motor with artificial neuralnetwork inverse speed observerrdquo IEEEASME Transactions onMechatronics vol 18 no 4 pp 1357ndash1366 2013

[7] Z Yang D Dong R Fan X Sun and R Jin ldquoRadialdisplacement-sensorless control for bearingless induction

motorrdquo Journal of Beijing University of Aeronautics andAstronautics vol 41 no 8 pp 1388ndash1395 2015

[8] T Schuhmann W Hofmann and R Werner ldquoImprovingoperational performance of active magnetic bearings usingKalman filter and state feedback controlrdquo IEEE Transactions onIndustrial Electronics vol 59 no 2 pp 821ndash829 2012

[9] Z Yang D Dong H Gao X Sun R Fan and H Zhu ldquoRotormass eccentricity vibration compensation control in bearinglessinduction motorrdquo Advances in Mechanical Engineering vol 7no 1 Article ID 168428 2015

[10] B-J Hou J-S Gao X-Q Li and Y-F Zhou ldquoStudy on repeti-tive PID control of linear motor in wafer stage of lithographyrdquoProcedia Engineering vol 29 pp 3863ndash3867 2012

[11] C-K Lai and K-K Shyu ldquoA novel motor drive design forincremental motion system via sliding-mode control methodrdquoIEEE Transactions on Industrial Electronics vol 52 no 2 pp499ndash507 2005

[12] Z Yang L Wan X Sun F Li and L Chen ldquoSliding modevariable structure control of a bearingless induction motorbased on a novel reaching lawrdquo Energies vol 9 no 6 p 4522016

[13] K J Lin ldquoSliding mode control design for uncertain singularsystemsrdquo Applied Mechanics amp Materials vol 145 no 8 pp 16ndash20 2011

[14] S J Zhu Y P He and J Ren ldquoDesign of vehicle activesuspension system using discrete-time sliding mode controlwith parallel genetic algorithmrdquo in Proceedings of the ASMEInternational Mechanical Engineering Congress and Exposition(IMECE rsquo13) 8 pages November 2013

[15] Y Chu and J Fei ldquoAdaptive global sliding mode control forMEMS gyroscope using RBF neural networkrdquo MathematicalProblems in Engineering vol 2015 Article ID 403180 9 pages2015

[16] X G Zhang K Zhao L Sun and Q An ldquoSliding mode controlof permanent magnet synchronous motor based on a novelexponential reaching lawrdquo Proceedings of the Chinese Society ofElectrical Engineering vol 31 no 15 pp 47ndash52 2011

[17] R Li G Y Zhao and S J Xu ldquoSensorless control of permanentmagnet synchronous motor based on extended sliding modeobserverrdquo Transactions of China Electrotechnical Society vol 3pp 79ndash85 2012

[18] M-H Wang M-L Huang and W-J Jiang ldquoMaximum powerpoint tracking and harmonic reducing control method forgenerator-based exercise equipmentrdquo Energies vol 9 no 2article 103 pp 1ndash13 2016

[19] H Q Hou Q Miao Q H Gao and B T Aorue ldquoFuzzybackstepping sliding mode control for mismatched uncertainsystemrdquo Journal of Engineering Science and Technology Reviewvol 7 no 2 pp 175ndash181 2014

[20] J J Zhu M Su X Z Wang and L Ma ldquoDirect-torque-controlof brushless DC motors based on segmented sliding-mode-variable-structurerdquo Chinese Journal of Scientific Instrument vol34 no 11 pp 2634ndash2640 2013

[21] Z-C Qiu J-L Guo BWang and J Xiao ldquoDeadbeat predictivecurrent control based on a sliding mode observer with Kalmanfilter for PMSMspeed and rotor positionrdquoElectricMachines andControl vol 18 no 4 pp 60ndash65 2014 (Chinese)

[22] Y-M Fang Z Li Y-YWu and X Yu ldquoBackstepping control ofPMSM position systems based on terminal-sliding-mode loadobserverrdquo Electric Machines and Control vol 18 no 9 pp 105ndash111 2014 (Chinese)


[23] K-J Lin ldquoAdaptive sliding mode control design for a classof uncertain singularly perturbed nonlinear systemsrdquo Interna-tional Journal of Control vol 87 no 2 pp 432ndash439 2014

[24] X G Zhang L Sun and K Zhao ldquoSliding mode control ofPMSM based on a novel load torque sliding mode observerrdquoProceedings of the Chinese Society of Electrical Engineering vol32 no 3 pp 111ndash116 2012 (Chinese)

[25] Z Li G D Hu J R Cui and G Cui ldquoSliding-mode variablestructure control with integral action for permanent magnetsynchronous motorrdquo Proceedings of the Chinese Society ofElectrical Engineering vol 34 no 3 pp 431ndash437 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



on the exercise equipment with nonlinear 119875-119881 characteristiccurves The amount of generator input current harmonicis greatly reduced In [19] the conventional sliding modecontrol was united with the adaptive fuzzy backsteppingscheme The simulation proved that this method improvedthe performance of mismatched uncertain system In [20]the sliding mode control dealt with the difficult problemof obtaining the counterelectromotive force and it finallyimplemented the direct torque control of brushless directcurrent motor In [21] the sliding mode control was used todetect the speed and position for PMSM The experimentalresults proved the validity of the proposed sliding modeobserver In [22] based on the nonsingular terminal slidingmode algorithm and backstepping method the sliding modeobserver and position controller were put forward which canestimate the torque accurately and track the position quicklyIn [23] the adaptive sliding mode control for uncertainsingularly perturbed nonlinear system was designed It notonly reduced the effects of uncertainty but also guaranteedthe control performance In [24 25] the load sliding modeobservers were proposedThey diminished the adverse effectsof load changes on PMSM and improved the antidisturbanceability of controlled system at some level However theyall ignored the problem that the load changes result in thedifferent rotational inertia and the controlled system hadlarge chattering Hence the system cannot achieve the bestdynamic performance

A novel sliding mode observer of load torque of whichthe state variables are the speed and load torque is proposedto suppress the impacts of the load torque changes on BIMcontrol system A low-pass filter used in the observer reducesthe observation error of torque Moreover the observer asfeedforward compensation for the given current alleviatesthe output pressure of sliding mode controller (SMC) Inaddition adopting the integral identification method validlyidentifies the rotational inertia and improves the precision ofBIM The simulation and experimental results show that theproposed method overcomes the disadvantageous effects onthe speed regulation system generated by load disturbancesand strengthens the antidisturbance ability of the system

2 The Dynamics Model of BIM

According to the electromagnetic field theory the radial lev-itation force of BIM in the 119889-119902 coordinates can be establishedas [6]

119865119909 = 119870 (12059511198891198941199042119889 + 12059511199021198941199042119902) 119865119910 = 119870 (12059511198891198941199042119902 minus 12059511199021198941199042119889) (1)

where 119870 = 119870119898 + 119870119897 119870119898 = 12058711987511198752119871119898118119897119903120583011987311198732 and119870119897 = 1198751119873221199031198731 119865119909 and 119865119910 are the components of theradial levitation force in 119909 and 119910 directions the subscript ldquo1rdquorepresents the torque windings the subscript ldquo2rdquo representsthe radial levitation force windings ldquo119904rdquo represents the statorand ldquo119903rdquo represents the rotor 1198751 and 1198752 separately representthe pole pairs of torque windings and suspension windings1198941199042119889 and 1198941199042119902 are the current components of the stator in

levitation force windings under the 119889-119902 axis 1198711198981 is mutualinductance of the levitation force windings 119897 is the effectivelength of the rotor 119903 is the stator inner diameter 1205830 is thepermeability of vacuum 1198731 and 1198732 respectively show theeffective number of turns of the torque windings and the lev-itation force windings and 1205951119889 and 1205951119902 are the componentsof flux linkage for the torque winding in the 119889-119902 coordinatesrespectively

With the torque windings and the levitation force wind-ings BIM is a nonlinear strongly coupled and complexsystem In order to simplify the analysis of BIM a hypothesisis given that the levitation force windings only create arotating magnetic field The rotor voltage equation can bedescribed as

1199061199031119889 = 11987711990311198941199031119889 + 1199011205951199031119889 minus (1205951119902 + 11987111990311198971198941199031119902) (1205961 minus 120596119903)= 0

1199061199031119902 = 11987711990311198941199031119902 + 1199011205951199031119902 minus (1205951119889 + 11987111990311198971198941199031119889) (1205961 minus 120596119903)= 0

(2)

where 1199061199031119889 and 1199061199031119902 are the rotor voltages of torque windingsin 119889-119902 coordinates 1198771199031 is the rotor resistance 1205961 and 120596119903 areseparately the air gap field speed and rotor speed and 119901 is thedifferential operator

The flux linkage can be expressed as

1205951119889 = (1198941199041119889 + 1198941199031119889) 11987111989811205951119902 = (1198941199041119902 + 1198941199031119902) 11987111989811205951199041119889 = 1205951119889 + 1198941199041119889119871 11990411198971205951199041119902 = 1205951119902 + 1198941199041119902119871 11990411198971205951199031119889 = 1205951119889 + 119894119903111988911987111990311198971205951199031119902 = 1205951119902 + 11989411990311199021198711199031119897

(3)

where119871 1199041119897 and1198711199031119897 are the stator leakage inductance and rotorleakage inductance of torque windings respectively

The electromagnetic torque equation is set up as

119879119890 = 1198751 (12059511198891198941199041119902 minus 12059511199021198941199041119889) (4)

The equation of motion is written as

119879119890 = 119879119871 + 1198691198751119901120596119903 (5)

where 119879119871 is the load torque and 119869 is the rotational inertiaAfter coordinate transforming the rotor flux in 119889-119902 axis

can be expressed as

1205951199031119889 = 11987111989811198941199041119889 + 119871119903111989411990311198891205951199031119902 = 11987111989811198941199041119902 + 11987111990311198941199031119902 (6)



1198941199031119889 = 1205951199031 minus 119871119898111989411990411198891198711199031 1198941199031119902 = minus11987111989811198711199031 1198941199041119902

(7)


1198941199041119889 = 1198791199031119901 + 11198711198981 1205951199031120596119904 = 119871119898111987911990311205951199031 1198941199041119902

(8)



119879119890 = 1198751 11987111989811198711199031 11989411990411199021205951199031 (9)






11989010158401205961 = 1198901205962 = minus119875211205951199031119869 11987111989811198711199031 1198941199041119902 +1198751119869 119879119871

11989010158401205962 = 119890101584010158401205961 = minus119875211205951199031119869 11987111989811198711199031 11989410158401199041119902

(11)


119904 = 1198901205961 + 11988811198901205962 (12)



|119883| = 0 119886 ge 0 119887 ge 0 120576 gt 0 119896 gt 0 (13)


119881 = 121199042 lt 0 (14)

is1q is1q

is1d

120596s 120579s1

120591r

120595r1 1

Lm1

1 + 120591rp






1199041015840 = 11989010158401205961 + 119888111989010158401205962 = 1198901205962 minus 119888111987521120595119903111987111989811198691198711199031 11989410158401199041119902 (16)




1198941199041119902 = 119869119871119903111987521120595119903111987111989811198881sdot int (120576 1003816100381610038161003816119890120596110038161003816100381610038162 sat (119904) + 119896 1003816100381610038161003816119890120596110038161003816100381610038162 119904 + 1198901205962) 119889119905

(18)




119889120596119889119905 = 119875211205951199031119869 11987111989811198711199031 1198941199041119902 minus1198751119869 119879119871

119889119879119871119889119905 = 0(19)



119889119889119905 = 119875211205951199031119869 11987111989811198711199031 1198941199041119902 minus1198751119869 119871 + 119881

119889119871119889119905 = 120578119881(20)



[11990911199092] = [ 120596 minus 119879119871 minus 119871] (21)



(22)


119904 = 1199091 + 119888int1199091119889119905 (23)



(24)



sat (119904 Δ) =


(25)



(26)

Speedregulation


Opticalencoder


k

120596lowast

120596

120596

+

minusiq

Te

J



119871 = 120596119888119904 + 120596119888 119871 (27)


119894lowast119902 = 1198941199041119902 + 1198941199022= 119869119871119903111987521120595119903111987111989811198881 int(120576 1003816100381610038161003816119890120596110038161003816100381610038162 sat (119904) + 119896 1003816100381610038161003816119890120596110038161003816100381610038162 119904 + 1198901205962) 119889119905

+ 119871120597 (28)





119879119890 = 1198751119889120596119889119905 + 119903 (29)



119903 (119905) = Δ119869119889120596119889119905 + 119879119871 (30)



(119896) = (119896 minus 1) + Δ119869 (31)



int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905 = Δ119869int119896119879(119896minus1)119879

2 (119905) 119889119905+ int119896119879(119896minus1)119879

119879119871 (119905) (119905) 119889119905(32)


int119896119879(119896minus1)119879


(33)


int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905 = Δ119869int119896119879(119896minus1)119879

2 (119905) 119889119905 (34)



Δ119869 = int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905int119896119879(119896minus1)119879

2 (119905) 119889119905 (35)


(119896) = (119896 minus 1) + int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905int119896119879(119896minus1)119879

2 (119905) 119889119905 (36)













BIMminus

+

+ minus

CRPWM

PID

PID

CRPWM23

yx

y

x

SMC

23

++

32

k

J

120596

120596

Te

Te

iq

120596lowast+

minus

is1q

xlowast

ylowast

Flowastx

Flowasty

ilowast2d

ilowast2q

ilowast2A

ilowast2B

ilowast2C

i2A

i2B

i2C

ilowastq


ilowast1q

ilowast1A

ilowast1B

ilowast1C

i1A

i1B

i1C

k1

iq2

Radialsuspension

forcemodel

Air-gapflux



observerOnline


inertia


0 01 02 03 04 050004

0005

0006

0007

0008

t (s)

00076

000765

00077

02 022 024 026 028018

J(k

gmiddotm

2)












t (s)0 02 04 06 08 1

minus10

0

10

20

30

40

50T

(Nmiddotm

)

TL

(a)

t (s)0 02 04 06 08 1

minus10

0

10

20

30

40

50

T(N

middotm)

TL

(b)


t (s)0 02 04 06 08 1

minus5

0

5

10

15

20

i s1q

(A)

(a)

t (s)0 02 04 06 08 1

minus10

minus5

0

5

10

i q2

(A)

(b)

t (s)02 04 06 08 1

9800

9850

9900

9950

10000

10050

10100



n(r

min

)

(c)


t (s)0 02 04 06 08 1

minus002

minus001

0

001

002

x (m

m)

(a)

t (s)0 02 04 06 08 1

minus003

minus002

minus001

0

001

002

y (m

m)

(b)



x

y

80120583

m

80120583m

1

(a)

Time

J

5eminus

4kg

middotm2

20ms

1

(b)

T

Time 20ms

2Nmiddotm TL

TL

T

(c)

n

Time

n

20ms

2A

50

rm

in

1

is1q

(d)

n

Time 20ms

2A

50

rm

in 1

is1q

(e)



6 Conclusions




Competing Interests




Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198941199031119889 = 1205951199031 minus 119871119898111989411990411198891198711199031 1198941199031119902 = minus11987111989811198711199031 1198941199041119902

(7)


1198941199041119889 = 1198791199031119901 + 11198711198981 1205951199031120596119904 = 119871119898111987911990311205951199031 1198941199041119902

(8)



119879119890 = 1198751 11987111989811198711199031 11989411990411199021205951199031 (9)






11989010158401205961 = 1198901205962 = minus119875211205951199031119869 11987111989811198711199031 1198941199041119902 +1198751119869 119879119871

11989010158401205962 = 119890101584010158401205961 = minus119875211205951199031119869 11987111989811198711199031 11989410158401199041119902

(11)


119904 = 1198901205961 + 11988811198901205962 (12)



|119883| = 0 119886 ge 0 119887 ge 0 120576 gt 0 119896 gt 0 (13)


119881 = 121199042 lt 0 (14)

is1q is1q

is1d

120596s 120579s1

120591r

120595r1 1

Lm1

1 + 120591rp






1199041015840 = 11989010158401205961 + 119888111989010158401205962 = 1198901205962 minus 119888111987521120595119903111987111989811198691198711199031 11989410158401199041119902 (16)




1198941199041119902 = 119869119871119903111987521120595119903111987111989811198881sdot int (120576 1003816100381610038161003816119890120596110038161003816100381610038162 sat (119904) + 119896 1003816100381610038161003816119890120596110038161003816100381610038162 119904 + 1198901205962) 119889119905

(18)




119889120596119889119905 = 119875211205951199031119869 11987111989811198711199031 1198941199041119902 minus1198751119869 119879119871

119889119879119871119889119905 = 0(19)



119889119889119905 = 119875211205951199031119869 11987111989811198711199031 1198941199041119902 minus1198751119869 119871 + 119881

119889119871119889119905 = 120578119881(20)



[11990911199092] = [ 120596 minus 119879119871 minus 119871] (21)



(22)


119904 = 1199091 + 119888int1199091119889119905 (23)



(24)



sat (119904 Δ) =


(25)



(26)

Speedregulation


Opticalencoder


k

120596lowast

120596

120596

+

minusiq

Te

J



119871 = 120596119888119904 + 120596119888 119871 (27)


119894lowast119902 = 1198941199041119902 + 1198941199022= 119869119871119903111987521120595119903111987111989811198881 int(120576 1003816100381610038161003816119890120596110038161003816100381610038162 sat (119904) + 119896 1003816100381610038161003816119890120596110038161003816100381610038162 119904 + 1198901205962) 119889119905

+ 119871120597 (28)





119879119890 = 1198751119889120596119889119905 + 119903 (29)



119903 (119905) = Δ119869119889120596119889119905 + 119879119871 (30)



(119896) = (119896 minus 1) + Δ119869 (31)



int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905 = Δ119869int119896119879(119896minus1)119879

2 (119905) 119889119905+ int119896119879(119896minus1)119879

119879119871 (119905) (119905) 119889119905(32)


int119896119879(119896minus1)119879


(33)


int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905 = Δ119869int119896119879(119896minus1)119879

2 (119905) 119889119905 (34)



Δ119869 = int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905int119896119879(119896minus1)119879

2 (119905) 119889119905 (35)


(119896) = (119896 minus 1) + int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905int119896119879(119896minus1)119879

2 (119905) 119889119905 (36)













BIMminus

+

+ minus

CRPWM

PID

PID

CRPWM23

yx

y

x

SMC

23

++

32

k

J

120596

120596

Te

Te

iq

120596lowast+

minus

is1q

xlowast

ylowast

Flowastx

Flowasty

ilowast2d

ilowast2q

ilowast2A

ilowast2B

ilowast2C

i2A

i2B

i2C

ilowastq


ilowast1q

ilowast1A

ilowast1B

ilowast1C

i1A

i1B

i1C

k1

iq2

Radialsuspension

forcemodel

Air-gapflux



observerOnline


inertia


0 01 02 03 04 050004

0005

0006

0007

0008

t (s)

00076

000765

00077

02 022 024 026 028018

J(k

gmiddotm

2)












t (s)0 02 04 06 08 1

minus10

0

10

20

30

40

50T

(Nmiddotm

)

TL

(a)

t (s)0 02 04 06 08 1

minus10

0

10

20

30

40

50

T(N

middotm)

TL

(b)


t (s)0 02 04 06 08 1

minus5

0

5

10

15

20

i s1q

(A)

(a)

t (s)0 02 04 06 08 1

minus10

minus5

0

5

10

i q2

(A)

(b)

t (s)02 04 06 08 1

9800

9850

9900

9950

10000

10050

10100



n(r

min

)

(c)


t (s)0 02 04 06 08 1

minus002

minus001

0

001

002

x (m

m)

(a)

t (s)0 02 04 06 08 1

minus003

minus002

minus001

0

001

002

y (m

m)

(b)



x

y

80120583

m

80120583m

1

(a)

Time

J

5eminus

4kg

middotm2

20ms

1

(b)

T

Time 20ms

2Nmiddotm TL

TL

T

(c)

n

Time

n

20ms

2A

50

rm

in

1

is1q

(d)

n

Time 20ms

2A

50

rm

in 1

is1q

(e)



6 Conclusions




Competing Interests




Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119889119889119905 = 119875211205951199031119869 11987111989811198711199031 1198941199041119902 minus1198751119869 119871 + 119881

119889119871119889119905 = 120578119881(20)



[11990911199092] = [ 120596 minus 119879119871 minus 119871] (21)



(22)


119904 = 1199091 + 119888int1199091119889119905 (23)



(24)



sat (119904 Δ) =


(25)



(26)

Speedregulation


Opticalencoder


k

120596lowast

120596

120596

+

minusiq

Te

J



119871 = 120596119888119904 + 120596119888 119871 (27)


119894lowast119902 = 1198941199041119902 + 1198941199022= 119869119871119903111987521120595119903111987111989811198881 int(120576 1003816100381610038161003816119890120596110038161003816100381610038162 sat (119904) + 119896 1003816100381610038161003816119890120596110038161003816100381610038162 119904 + 1198901205962) 119889119905

+ 119871120597 (28)





119879119890 = 1198751119889120596119889119905 + 119903 (29)



119903 (119905) = Δ119869119889120596119889119905 + 119879119871 (30)



(119896) = (119896 minus 1) + Δ119869 (31)



int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905 = Δ119869int119896119879(119896minus1)119879

2 (119905) 119889119905+ int119896119879(119896minus1)119879

119879119871 (119905) (119905) 119889119905(32)


int119896119879(119896minus1)119879


(33)


int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905 = Δ119869int119896119879(119896minus1)119879

2 (119905) 119889119905 (34)



Δ119869 = int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905int119896119879(119896minus1)119879

2 (119905) 119889119905 (35)


(119896) = (119896 minus 1) + int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905int119896119879(119896minus1)119879

2 (119905) 119889119905 (36)













BIMminus

+

+ minus

CRPWM

PID

PID

CRPWM23

yx

y

x

SMC

23

++

32

k

J

120596

120596

Te

Te

iq

120596lowast+

minus

is1q

xlowast

ylowast

Flowastx

Flowasty

ilowast2d

ilowast2q

ilowast2A

ilowast2B

ilowast2C

i2A

i2B

i2C

ilowastq


ilowast1q

ilowast1A

ilowast1B

ilowast1C

i1A

i1B

i1C

k1

iq2

Radialsuspension

forcemodel

Air-gapflux



observerOnline


inertia


0 01 02 03 04 050004

0005

0006

0007

0008

t (s)

00076

000765

00077

02 022 024 026 028018

J(k

gmiddotm

2)












t (s)0 02 04 06 08 1

minus10

0

10

20

30

40

50T

(Nmiddotm

)

TL

(a)

t (s)0 02 04 06 08 1

minus10

0

10

20

30

40

50

T(N

middotm)

TL

(b)


t (s)0 02 04 06 08 1

minus5

0

5

10

15

20

i s1q

(A)

(a)

t (s)0 02 04 06 08 1

minus10

minus5

0

5

10

i q2

(A)

(b)

t (s)02 04 06 08 1

9800

9850

9900

9950

10000

10050

10100



n(r

min

)

(c)


t (s)0 02 04 06 08 1

minus002

minus001

0

001

002

x (m

m)

(a)

t (s)0 02 04 06 08 1

minus003

minus002

minus001

0

001

002

y (m

m)

(b)



x

y

80120583

m

80120583m

1

(a)

Time

J

5eminus

4kg

middotm2

20ms

1

(b)

T

Time 20ms

2Nmiddotm TL

TL

T

(c)

n

Time

n

20ms

2A

50

rm

in

1

is1q

(d)

n

Time 20ms

2A

50

rm

in 1

is1q

(e)



6 Conclusions




Competing Interests




Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119903 (119905) = Δ119869119889120596119889119905 + 119879119871 (30)



(119896) = (119896 minus 1) + Δ119869 (31)



int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905 = Δ119869int119896119879(119896minus1)119879

2 (119905) 119889119905+ int119896119879(119896minus1)119879

119879119871 (119905) (119905) 119889119905(32)


int119896119879(119896minus1)119879


(33)


int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905 = Δ119869int119896119879(119896minus1)119879

2 (119905) 119889119905 (34)



Δ119869 = int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905int119896119879(119896minus1)119879

2 (119905) 119889119905 (35)


(119896) = (119896 minus 1) + int119896119879(119896minus1)119879

119903 (119905) (119905) 119889119905int119896119879(119896minus1)119879

2 (119905) 119889119905 (36)













BIMminus

+

+ minus

CRPWM

PID

PID

CRPWM23

yx

y

x

SMC

23

++

32

k

J

120596

120596

Te

Te

iq

120596lowast+

minus

is1q

xlowast

ylowast

Flowastx

Flowasty

ilowast2d

ilowast2q

ilowast2A

ilowast2B

ilowast2C

i2A

i2B

i2C

ilowastq


ilowast1q

ilowast1A

ilowast1B

ilowast1C

i1A

i1B

i1C

k1

iq2

Radialsuspension

forcemodel

Air-gapflux



observerOnline


inertia


0 01 02 03 04 050004

0005

0006

0007

0008

t (s)

00076

000765

00077

02 022 024 026 028018

J(k

gmiddotm

2)












t (s)0 02 04 06 08 1

minus10

0

10

20

30

40

50T

(Nmiddotm

)

TL

(a)

t (s)0 02 04 06 08 1

minus10

0

10

20

30

40

50

T(N

middotm)

TL

(b)


t (s)0 02 04 06 08 1

minus5

0

5

10

15

20

i s1q

(A)

(a)

t (s)0 02 04 06 08 1

minus10

minus5

0

5

10

i q2

(A)

(b)

t (s)02 04 06 08 1

9800

9850

9900

9950

10000

10050

10100



n(r

min

)

(c)


t (s)0 02 04 06 08 1

minus002

minus001

0

001

002

x (m

m)

(a)

t (s)0 02 04 06 08 1

minus003

minus002

minus001

0

001

002

y (m

m)

(b)



x

y

80120583

m

80120583m

1

(a)

Time

J

5eminus

4kg

middotm2

20ms

1

(b)

T

Time 20ms

2Nmiddotm TL

TL

T

(c)

n

Time

n

20ms

2A

50

rm

in

1

is1q

(d)

n

Time 20ms

2A

50

rm

in 1

is1q

(e)



6 Conclusions




Competing Interests




Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










BIMminus

+

+ minus

CRPWM

PID

PID

CRPWM23

yx

y

x

SMC

23

++

32

k

J

120596

120596

Te

Te

iq

120596lowast+

minus

is1q

xlowast

ylowast

Flowastx

Flowasty

ilowast2d

ilowast2q

ilowast2A

ilowast2B

ilowast2C

i2A

i2B

i2C

ilowastq


ilowast1q

ilowast1A

ilowast1B

ilowast1C

i1A

i1B

i1C

k1

iq2

Radialsuspension

forcemodel

Air-gapflux



observerOnline


inertia


0 01 02 03 04 050004

0005

0006

0007

0008

t (s)

00076

000765

00077

02 022 024 026 028018

J(k

gmiddotm

2)












t (s)0 02 04 06 08 1

minus10

0

10

20

30

40

50T

(Nmiddotm

)

TL

(a)

t (s)0 02 04 06 08 1

minus10

0

10

20

30

40

50

T(N

middotm)

TL

(b)


t (s)0 02 04 06 08 1

minus5

0

5

10

15

20

i s1q

(A)

(a)

t (s)0 02 04 06 08 1

minus10

minus5

0

5

10

i q2

(A)

(b)

t (s)02 04 06 08 1

9800

9850

9900

9950

10000

10050

10100



n(r

min

)

(c)


t (s)0 02 04 06 08 1

minus002

minus001

0

001

002

x (m

m)

(a)

t (s)0 02 04 06 08 1

minus003

minus002

minus001

0

001

002

y (m

m)

(b)



x

y

80120583

m

80120583m

1

(a)

Time

J

5eminus

4kg

middotm2

20ms

1

(b)

T

Time 20ms

2Nmiddotm TL

TL

T

(c)

n

Time

n

20ms

2A

50

rm

in

1

is1q

(d)

n

Time 20ms

2A

50

rm

in 1

is1q

(e)



6 Conclusions




Competing Interests




Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










t (s)0 02 04 06 08 1

minus10

0

10

20

30

40

50T

(Nmiddotm

)

TL

(a)

t (s)0 02 04 06 08 1

minus10

0

10

20

30

40

50

T(N

middotm)

TL

(b)


t (s)0 02 04 06 08 1

minus5

0

5

10

15

20

i s1q

(A)

(a)

t (s)0 02 04 06 08 1

minus10

minus5

0

5

10

i q2

(A)

(b)

t (s)02 04 06 08 1

9800

9850

9900

9950

10000

10050

10100



n(r

min

)

(c)


t (s)0 02 04 06 08 1

minus002

minus001

0

001

002

x (m

m)

(a)

t (s)0 02 04 06 08 1

minus003

minus002

minus001

0

001

002

y (m

m)

(b)



x

y

80120583

m

80120583m

1

(a)

Time

J

5eminus

4kg

middotm2

20ms

1

(b)

T

Time 20ms

2Nmiddotm TL

TL

T

(c)

n

Time

n

20ms

2A

50

rm

in

1

is1q

(d)

n

Time 20ms

2A

50

rm

in 1

is1q

(e)



6 Conclusions




Competing Interests




Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










x

y

80120583

m

80120583m

1

(a)

Time

J

5eminus

4kg

middotm2

20ms

1

(b)

T

Time 20ms

2Nmiddotm TL

TL

T

(c)

n

Time

n

20ms

2A

50

rm

in

1

is1q

(d)

n

Time 20ms

2A

50

rm

in 1

is1q

(e)



6 Conclusions




Competing Interests




Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Competing Interests




Acknowledgments


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Sliding Mode Control for Bearingless...

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Transcript of Research Article Sliding Mode Control for Bearingless...