research article fault reconstruction based on...

Research ArticleFault Reconstruction Based on Sliding Mode Observer forCurrent Sensors of PMSM

Changfan Zhang Huijun Liao Xiangfei Li Jian Sun and Jing He

College of Electrical and Information Engineering Hunan University of Technology Zhuzhou Hunan 412007 China

Correspondence should be addressed to Jing He hejing263net

Received 6 January 2015 Revised 20 April 2015 Accepted 20 April 2015

Academic Editor Mehmet Karakose

Copyright copy 2016 Changfan Zhang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper deals with a method of phase current sensor fault reconstruction for permanent magnet synchronous motor (PMSM)drives A new state variable is introduced so that an augmented system can be constructed to treat PMSM sensor faults as actuatorfaultsThismethod uses the PMSM two-phase stationary reference frame faultmodel and a slidingmode variable structure observerto reconstruct fault signals A logic algorithm is built to isolate and identify the faulty sensor for a stator phase current fault afterreconstructing the two-phase stationary reference frame fault signals which allows the phase fault signals to be reconstructedSimulation results are presented to illustrate the functionality of the theoretical developments

1 Introduction

Permanent magnet synchronous motors (PMSMs) are animportant category of electric machines in which the rotormagnetization is created by permanent magnets attached tothe rotor Due to their high efficiency high ratio of torqueto weight high power factor faster response and ruggedconstruction PMSMs are widely used for high performancevariable speed motors in many industry applications Withthe development of permanent magnet materials especiallythe neodymium-iron-boron (Nd-Fe-B) which has highmag-netic energy high coercive force and low price the applica-tions of PMSMs have become more extensive in recent years

High performance systems usually have demanding req-uirements on availability reliability and survivability [1]Thereliable operation of PMSM is a primary concern for theentire ship system Early detection of abnormalities in thePMSM drives will help to avoid failures Indeed detectionlocation and analysis of faults play a very important rolein reliable operation of electrical machines and are essentialfor major concerns such as efficiency and performance ofapplications involving PMSM drives Sensor failure is one ofseveral faults occurring in drive systems Sensors are of greatimportance in the installation of monitoring and controlsystems However in most publications detection of inverter

faults or physical damage of the electrical machine is consid-ered rather than sensor faultsThis paper proposes a methodof phase current sensor fault detection and reconstruction forPMSM control systems

In recent years fault-tolerant control has been developedwhich can be realized by reconstructing phase currents tosubstitute current feedback after the fault occurrence Forexample Lu et al [2] reconstructed three-phase current basedonDCbus sensors for fault-tolerant control for electricmach-ines Salmasi and Najafabadi [3] observed faulty phase statorcurrents based on an adaptive observer when there is onlyone normal phase current sensor However these designs donot separate fault signals from faulty sensors Therefore nofurther information can be obtained on the sensor state itself

Due to the simplicity of the two-axis 119889-119902-model it hasbecome themost widely usedmodel in electricmachine engi-neering controller design The 119889-119902-model offers significantconvenience for control system design by transforming sta-tionary symmetrical AC variables to DC variables in a rotat-ing reference frame Therefore for electric machine sensorfault detection the 119889-119902 axis currents 119894119889 and 119894119902 are assumedto be measured directly through sensors Najafabadi et al [4]detected 119889-119902 axis current sensor faults based on an adaptiveobserver Liu [5] used a nonlinear parity relation methodfor detection of additive faults for virtual sensors for 119889-119902

Hindawi Publishing CorporationJournal of SensorsVolume 2016 Article ID 9307560 9 pageshttpdxdoiorg10115520169307560

2 Journal of Sensors

axis currents Huang [6] an unknown input observer wasdesigned for detection of 119902 axis stator current residuals forinductionmotor However the two currents 119894119889 and 119894119902 are notpractically measurable These two virtual sensing signals arecalculated from the measured phase current 119894119886119887119888 by applyinga linear Clarke transformation Abnormal changes in 119894119889 and 119894119902may indicate a fault appearing in the phase current sensorsbut this design will not provide more specific informationFurthermore as the calculations of 119894119889 and 119894119902 are coupled ifa fault occurred in the measurement devices for 119894119886119887119888 conseq-uential faults in 119894119889 and 119894119902 will appear simultaneously Henceit is relatively difficult to detect the abnormal signals in themeasurement devices for 119894119886119887119888

Sliding mode techniques are known for their robustnessand insensitivity to the so-calledmatched uncertaintyThere-fore these techniques offer great potential for robust faultdetection and isolation (FDI) [7ndash10] Faults are classifiedaccording to their physical locations into system actuatorand sensor faults Compared with actuators sensors are pass-ive elements in the sense that they only provide operationalinformation about the system and do not affect the systembehavior directlyThus they have been less studied comparedto actuator fault detection and isolation This paper presentsa method in which using a newly designed filter the sensorfaults can be modeled as pseudoactuator faults Then usingthe transformed system structure and characteristics of thedesigned filter a sliding mode observer is proposed to recon-struct the sensor fault in the system [11ndash13]

The paper is organized as follows Section 2 describes asurface-mounted PMSMwith current sensor faults Section 3proposes the method for reconstruction of faults Section 4presents simulation results and conclusions in Section 5 fol-low

2 System Formulation

In the 120572-120573 rotor reference frame a surface-mounted PMSMwith current sensors faults can be expressed as the followingdynamic model

(119905) = 119860119909 (119905) + 119861119906 (119905) + 119864120595

119910 (119905) = 119862119909 (119905) + 119873119891119904 (119909 119906 119905)

(1)

where 119909(119905) 119906(119905) 119910(119905) 120595 and 119891119904(119909 119906 119905) are the state variablesinputs outputs rotor flux and sensor faults respectivelyThen the matrices are defined as

119860 =[[[

[

minus119877119904

119871120572

0

0 minus119877119904

119871120573

]]]

]

119861 =[[[

[

1

119871120572

0

01

119871120573

]]]

]

119862 = [1 0

0 1]

119864 =[[

[

minus120596

119871120572

0

0 minus120596

119871120573

]]

]

119873 = [1 0

0 1]

119909 = [119894120572

119894120573

]

119906 = [119906120572

119906120573

]

120595 = [120595120572

120595120573

] = [minus120595119891 sin 120579119903120595119891 cos 120579119903

]

119910 = [119894120572

119894120573

]

119891119904 = [119891119904120572

119891119904120573

]

(2)

with

119871120572 = 1198710 + 1198711 cos (2120579119903)

119871120573 = 1198710 minus 1198711 cos (2120579119903)

1198710 =119871119889 + 119871119902

2

1198711 =119871119889 minus 119871119902

2

(3)

where 119894120572 119894120573 119906120572 and 119906120573 are the 120572-120573 axis currents and voltagesrespectively 120595119891 is the rotor permanent magnet flux linkage120579119903 is the electrical rotor angular position 120595120572 120595120573 119891119904120572 and119891119904120573 are the 120572-120573 axis stator flux linkages and current sensorfaults respectively 119877119904 is the stator resistance 119871120572 119871120573 and120596 are the 120572-120573 axis stator inductances and electrical rotorspeed respectively and 119871119889 and 119871119902 are the stator 119889 and 119902 axisinductances respectivelyThe surface-mounted PMSM statorwinding shows the same electrical inductance on both119889 and 119902axes that is 119871119889 = 119871119902 = 119871 119904 where 119871 119904 is the stator inductance

3 Sensor Faults Reconstruction Based onSliding Mode Variable Structure

The section proposes a novelmethod to reconstruct the phasecurrent sensor faults for PMSM drives Before dealing withthe faults an extended 120572-120573 axis fault model of PMSM is refor-mulated by using a transformation filter which increases thesystemrsquos stateThen a slidingmode observer is constructed to

Journal of Sensors 3

reconstruct 120572-120573 fault signals A logic algorithm is built toconvert the reconstructed 120572-120573 fault signal into 119886119887119888-phase soas to reconstruct phase current sensor faults

31 Extended PMSM 120572-120573 Axis Fault Model and Sensor FaultsReconstruction For system (1) a low-pass linear filter isintroduced [11]

(119905) = 119860 119904119911 (119905) + 119861119904119910 (4)

where 119911 is the new vector state 119860 119904 and 119861119904 are constantmatrices which are design parameters to be defined later and119910 is the output

Combining systems (1) and (4) we have

(119905) = 119860 119904119911 (119905) + 119861119904119862119909 (119905) + 119861119904119873119891119904 (119909 119906 119905) (5)

Then the following augmented system can be obtained

[

] = [

119860 0

119861119904119862 119860 119904

][119909

119911] + [

119861

0] 119906 + [

0

119861119904119873]119891119904 + [

119864

0]120595

119911 = [0 119868] [119909

119911]

(6)

where 119891119904 is the sensor faults of system (1) The new statevariables and matrices in a compatible way with system (6)are as follows

119860 = [119860 0

119861119904119862 119860 119904

]

119861 = [119861

0]

119872 = [0

119861119904119873]

119864 = [119864

0]

119862 = [0 119868]

119909 (119905) = [119909 (119905)

119911 (119905)]

119910 (119905) = 119911

(7)

Therefore following this transformation the system isextended and the initial sensor fault problem has becomean actuator fault problem The corresponding extended faultmodel is

(119905) = 119860119909 (119905) + 119861119906 (119905) + 119872119891119904 + 119864120595

119910 (119905) = 119862119909 (119905)

(8)

This paper focuses only on sensor faults which have beentransformed into pseudoactuator faults in system (8)

Assumption 1 ((119860 119862) is observable) When 119861119904 is a nonsin-gular matrix if (119860 119862) in system (1) is observable (119860 119862) insystem (8) will still be observable [14]

Therefore there is a matrix 119871 which makes 1198600 = 119860 minus 119871119862

a stable matrix and satisfies the Lyapunov function

119860119879

0119875 + 1198751198600 = minus119876 (9)

where matrices 119875 119876 are symmetric positive definite

Assumption 2 There is a matrix 119865 such that 119875119872 = 119862119879

119865119879

Assumption 3 A fault in the system is a bounded functionsuch that 119891119904 le 1205881 where 1205881 is a selected scalar larger than0

The slidingmode observer for system (8) can be designedas

119909 (119905) = 119860 (119905) + 119861119906 (119905) + 119871 (119910 (119905) minus 119862 (119905)) + 119872V

+ 119864120595

(119905) = 119862 (119905)

(10)

where (119905) and (119905) are the estimated states and outputsrespectively and V is the sliding mode signal defined as

V =

minus120588119865119890119910 (119905)

10038171003817100381710038171003817119865119890119910 (119905)

10038171003817100381710038171003817

if 119890119910 = 0

0 if 119890119910 = 0

(11)

where 119865 is the observer gain and 120588 is a positive scalar Both ofthese need to be designed

If the state estimation errors are defined as 119890(119905) = (119905) minus

119909(119905) and 119890119910(119905) = 119862(119905) minus 119910(119905) = 119862119890(119905) then from (8) and (10)the state estimation error dynamical system is

119890 (119905) = 1198600119890 (119905) + 119872(V minus 119891119904) (12)

The convergence of the above observer is guaranteed bythe following proposition

Proposition 4 Considering the system described by (8) andits observers described by (10) if Assumptions 1ndash3 hold theparameter of the observer is selected according to the followingcriteria

120588 gt 1205881 (13)

and thus 119890 is asymptotically convergent that is lim119905rarrinfin119890(119905) =0

Proof For system (12) consider a Lyapunov function can-didate 1198811 = 119890

119879119875119890 The time derivative of 1198811 along the

trajectories of system (12) is

1 = 119890119879(1198751198600 + 1198600119875) 119890 + 2119890

119879119875119872V minus 2119890

119879119875119872119891119904 (14)


and it follows that

1 le minus119890119879119876119890 + 2 (119865119862119890)

119879

(V + 119891119904)

1 le minus119890119879119876119890 minus 2 (119865119890119910)

119879

12058811986511989011991010038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817

+ 210038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817

10038171003817100381710038171198911199041003817100381710038171003817

1 le minus119890119879119876119890 minus 2

10038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817(120588 minus

10038171003817100381710038171198911199041003817100381710038171003817)

(15)

Then under Assumption 3 the following can be obtainedfrom (15)

1 le minus119890119879119876119890 minus 2

10038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817(120588 minus 1205881) le minus119890

119879119876119890 (16)

from which 1 is negative definite From the Lyapunov theo-rem observer (10) is designed asymptotically stable

It can be known fromabove that 119890(119905)willmake asymptoticconvergence to zero that is lim119905rarrinfin119890(119905) = 0

This completes the proof

Proposition 4 implies that 119890 is bounded that is 1199050 existsand when 119905 gt 1199050

119890 le 1205751 (17)

where 1205751 is a finite positive scalar

Remark 5 The obtained control algorithm of a sliding modeobserver is simple and easy to implement Because of theexcellent robustness of the sliding mode variable techniquesthe dependence on the precise mathematical model can beeffectively reduced The performance of the observer can beensured in the case of a system modeling error parameterperturbation and the unknown inputs such as external noiseand disturbance Therefore it has very strong engineeringpracticability

Consider a sliding mode surface

119904 = 119865119890119910 (119905) (18)

Proposition 4 implies that the sliding mode dynamics ofthe error system (12) associated with the sliding surface (18) isstable According to the sliding mode theory observer stabil-ity will be guaranteed upon proving that the error system canbe driven to the sliding mode surface in finite time by choos-ing an appropriate gain of 120588 for the input signals (11) In viewof this the conclusion is presented by the following proposi-tion

Proposition 6 If Assumptions 1ndash3 hold and 120588 is sufficientlylarge then the error system (12) will be driven to the slidingmode surface in finite time

Proof Selecting the Lyapunov function

1198812 =1

2119904119879119904 (19)

then the time derivative of1198812 along the trajectories of system(12) is

2 = 119904119879(119865119862 119890) = 119904

119879119865119862 (1198600119890 +119872(V minus 119891119904))

= 1199041198791198651198621198600119890 + 119904

119879119865119862119872(V minus 119891119904)

= 1199041198791198651198621198600119890 + 119904

119879119872119879119875119872(V minus 119891119904)

= 1199041198791198651198621198600119890 minus 119904

119879119872119879119875119872119891119904 + 119904

119879119872119879119875119872V

= 1199041198791198651198621198600119890 minus 119904

119879119872119879119875119872119891119904 minus 120588119904

119879119872119879119875119872

119904

119904

(20)

Thus from (17) there is

2 le 119904100381710038171003817100381710038171198651198621198600

10038171003817100381710038171003817119890 + 119904

10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

10038171003817100381710038171198911199041003817100381710038171003817

minus 120588120582min (119872119879119875119872) 119904 le 119904 (

100381710038171003817100381710038171198651198621198600

100381710038171003817100381710038171205751

+10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

10038171003817100381710038171198911199041003817100381710038171003817 minus 120588120582min (119872

119879119875119872)) le minus 119904

sdot 120582min (119872119879119875119872)(120588

minus (

10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

120582min (119872119879119875119872)

10038171003817100381710038171198911199041003817100381710038171003817 +

100381710038171003817100381710038171198651198621198600

10038171003817100381710038171003817

120582min (119872119879119875119872)

1205751))

(21)

The variables can be defined as

119870 = 120582min (119872119879119875119872)(120588

minus (

10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

120582min (119872119879119875119872)

10038171003817100381710038171198911199041003817100381710038171003817 +

100381710038171003817100381710038171198651198621198600

10038171003817100381710038171003817

120582min (119872119879119875119872)

1205751))

(22)

Since 119872119879119875119872 ge 120582min(119872

119879119875119872) gt 0 it follows that

(119872119879119875119872120582min(119872

119879119875119872))119891119904+(1198651198621198600120582min(119872

119879119875119872))1205751 ge

119891119904 So the scalar 120588 in (11) satisfies 120588 gt 1205881 and 120588 can beselected to be large enough to satisfy

120588 ge

10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

120582min (119872119879119875119872)

1205881 +

100381710038171003817100381710038171198651198621198600

10038171003817100381710038171003817

120582min (119872119879119875119872)

1205751(23)

then

2 le minus119870 119904 (24)

This shows that the slidingmode reachability condition issatisfied As a consequence according to slidingmode princi-ple [15] an ideal sliding motion will take place on the surface119904 after some finite time


When the system reaches the sliding mode surface 119904 =

119904 = 0 according to the sliding mode equivalent principle [9]This implies

119865119862119890 (119905) = 119865119890119910 (119905) = 0

119865119862 119890 = 119865 119890119910 (119905) = 119865119890119910 (119905) = 0

(25)


and from (12) and (25) the fault reconstruction equation canbe got in the form of

119891119904 = V (26)

that is the fault is now equivalent to the sliding mode signal

Remark 7 Unlike many other methods which use residualsto diagnose the occurrence of sensor fault qualitatively themethod of fault reconstruction presented in this paper esti-mates the current sensor fault quantitatively By this way notonly the original appearance of a fault can be reflected vividlybut also more specific fault information can be obtainedIt actually becomes the important basis of adopting moretargeted measures to eliminate the effect of fault on PMSMdrives or achieving the active fault-tolerant control proposalas presented in this paper

32 Reconstructing the Phase Current Sensors Fault SignalsAs mentioned previously a fault on one of the 120572-120573 axis cur-rent sensors can be reconstructed But this design will notprovide more specific information whether the faulty sensoris in phase ldquo119886rdquo or ldquo119887rdquoTherefore the phase current sensor faultisolation and identification become somewhat complicatedTo overcome this problem a logic algorithm is constructeddevoted to transforming the 120572-120573 axis fault signals in 119886119887119888-phase fault signals of PMSM Thus the phase current sensorfaults will be reconstructed

Generally for electric motors only two phase sensors areusedThat is based on considerations of cost and the fact thatthe three phase currents make a vector sum to zero in a starconnected system without a neutral line So in actual controlsystems of PMSM the two phase stator currents 119894119886 and 119894119887 aremeasured by the phase current sensors respectively and thethird phase current 119894119888 is calculated using 119894119886 + 119894119887 + 119894119888 = 0

A linear (32) Clarke transformation is applied to trans-form the three-phase plane coordinate system ABC to thetwo-phase plane rectangular coordinate system 120572120573 the trans-formation equation is as below

[[

[

119894120572

119894120573

1198940

]]

]

= radic2

3

[[[[[[[[

[

1 minus1

2minus1

2

0radic3

2minusradic3

2

1

radic2

1

radic2

1

radic2

]]]]]]]]

]

[[

[

119894119886

119894119887

119894119888

]]

]

(27)

where 1198940 is a variable introduced distinct from 119894119886 and 119894119887 usedfor construction of the 120572-120573-0 coordinate system called thezero-axis current The zero-axis is vertical to both 120572 and 120573

axes and so it will have no influence on 120572 and 120573 axesFor star connected systems without a neutral line 119894119886+119894119887+

119894119888 = 0 that is 119894119888 = minus(119894119886 + 119894119887) Equation (27) becomes

[119894120572

119894120573

] =

[[[[

[

radic3

20

radic2

2radic2

]]]]

]

[119894119886

119894119887

] (28)

It can be seen that decoupling can be realized for currentin this equation

As discussed above practically currents 119894120572 and 119894120573 arecalculated from phase currents 119894119886 and 119894119887 Then from (28) theeffects of the stator 120572-120573 axis currents are related to the errorsin phase 119886 and 119887 current sensors outputs

[119894120572 + 119891119904120572

119894120573 + 119891119904120573

] =

[[[[

[

radic3

20

radic2

2radic2

]]]]

]

[119894119886 + 119891119904119886

119894119887 + 119891119904119887

] (29)

where 119891119904120572 and 119891119904120573 are the 120572-120573 axis sensor faults respectivelyand 119891119904119886 and 119891119904119887 are the 119886 and 119887 phase sensor faults respec-tively

From (28) and (29) the phase sensor faults119891119904119886 and119891119904119887 are

119891119904119886 =radic2

3119891119904120572

119891119904119887 =1

radic2119891119904120573 minus

radic1

6119891119904120572

(30)

where 119891119904120572 119891119904120573 are derived by reconstruction of the stateobserver in the previous section

In addition the actual phase current is

119894119886 =radic2

3119894120572

119894119887 =1

radic2119894120573 minus

radic1

6119894120572

(31)

Thus the phase current sensor fault signals can be recon-structed by transformation of the 120572-120573 axis fault signals of aPMSM

Remark 8 In the previous literature [5 6] the current sensorfault detection was usually corresponding to 119889-119902 axis whichis a virtual axis In contrast this paper proposes a new currentsensor FDI algorithm which is directly corresponding to119886119887119888-phase the actual mounting position of current sensorThis innovation can significantly improve the practicabilityof traditional FDI algorithm as well as bring many potentialapplications

4 Simulation Study

To verify the effectiveness of the method proposed in thispaper the drive tests with respect to two types of faults havebeen carried out One tested fault is the incipient fault and theother is the gain sensor fault The parameters for the PMSMof this study are given in Table 1

The matrixes 119860 119904 and 119861119904 in (4) are then

119860 119904 = [minus200 0

0 minus200]

119861119904 = [200 0

0 200]

(32)


0 005 01 015 02

0

5

10

15

Time (s)

i aan

di a

(A)

iaia

minus5

minus10

minus15

(a) 119894119886 and its estimation 119886

0 005 01 015 02

0

5

10

15

Time (s)

ibib

i ban

di b

(A)

minus5

minus10

minus15

(b) 119894119887 and its estimation 119887

Figure 1 Case 1 current estimation

Table 1 Nominal value for PMSM

Parameters Unit ValuesStator resistance (119877

119904) Ω 2875

Number of pole pairs (119899119901) Pairs 4Stator inductance (119871 119904) H 00085Rotor PM flux (120595119891) Wb 0175Rotor moment of inertia (119869) kgsdotm2 0008

The matrixes in system (8) are

119860 =

[[[[[

[

minus33823 0 0 0

0 minus33823 0 0

200 0 minus200 0

0 200 0 minus200

]]]]]

]

119861 =

[[[[[

[

11765 0

0 11765

0 0

0 0

]]]]]

]

119862 = [0 0 1 0

0 0 0 1]

119864 =

[[[[[

[

minus11765120596 0

0 minus11765120596

0 0

0 0

]]]]]

]

119872 =

[[[[[

[

0 0

0 0

200 0

0 200

]]]]]

]

(33)

and matrix (119860119862) is observable since 119861119904 is nonsingularThe matrix 119871 in (10) is

119871 = [100 0 200 0

0 100 0 100]

119879

(34)

And so the corresponding 1198600 matrix is

1198600 =

[[[[[

[

minus33823 0 minus100 0

0 minus338 23 0 minus100

200 0 minus400 0

0 200 0 minus100

]]]]]

]

(35)

and using linear matrix inequality techniques we have thefollowing solution

119875 =

[[[[[

[

7244 0 0 0

0 7595 0 0

0 0 5674 0

0 0 0 7455

]]]]]

]

119865 = [113480 0

0 149100]

(36)


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsa

(A)

Estimation

(a) The fault 119891119904119886 and its estimation

0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsb

(A)

Estimation

(b) The fault 119891119904119887 and its estimation

Figure 2 Case 1 current sensor faults estimation

0 005 01 015 02

0

50

100

150

200

250

300

350

400

Time (s)

Spee

d (r

pm)

minus50

Figure 3 Case 1 the motor speed waveform during current sensor faults on 119886 and 119887 phases

Remark 9 The following saturation function can be used tosubstitute for the sliding mode signal V to reduce chatteringand eliminate high-frequency interference caused by thechattering

V = minus120588119865119890119910

10038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817+ 1205752

(37)

where 1205752 is a scalar of relatively small value

The following parameters are chosen 120588 = 46 1205752 = 01initial simulation conditions are all 5A output limiting valuesof state variable are chosen as plusmn10A fault reconstructionlimiting values are chosen as plusmn7A simulation time is 022 sand the given speed is 300 rpm

Case 1 A sinusoidal signal is added with amplitude 5A andfrequency 150Hz for 119886-phase from 0063 s simulating anincipient fault and a step signal with amplitude 3A from0135 s for 119887-phase simulating a gain sensor fault

The simulation (Figures 1-2) shows that the fault signalscan be accurately reconstructed The faults produce motorspeed abnormalities in the driversquos operation as shown inFigure 3

Case 2 Astep signal is addedwith amplitude 5A from0063 sfor 119886-phase simulating a gain sensor fault and a sinusoidalsignal with amplitude 3A and frequency 150Hz for 119887-phasefrom 0135 s simulating an incipient fault

The simulation (Figures 4-5) shows that the fault signalscan be accurately reconstructed The faults produced motor


0 005 01 015 02

0

5

10

15

Time (s)

i aan

di a

(A)

iaia

minus5

minus10

minus15


0 005 01 015 02

0

5

10

15

Time (s)

i ban

di b

(A)

ibib

minus5

minus10

minus15



0 005 01 02015

0

2

4

6

8

10

Time (s)

FaultEstimation

minus10

minus8

minus6

minus4

minus2

fsa

(A)


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsb

(A)

Estimation



speed abnormalities in the driversquos operation as shown inFigure 6

The simulations show that the fault reconstruction isrealized The fault signal can be estimated to determine thesize location and time of occurrence of a fault on phase 119886 or119887 current sensor of a PMSM intuitively This method will beideal for directly isolating the faulty current sensor

5 Conclusions

This paper presents a PMSM phase current sensor faultreconstructionmethod based on slidingmode variable struc-ture observer An 120572-120573 axis fault model of PMSM is firstlydefined Based on this a first-order low-pass filter is intro-duced for constructing an augmented system with which thePMSM sensor faults can be considered as actuator faults


0 005 01 015 02

0

50

100

150

200

250

300

350

400

Time (s)

Spee

d (r

pm)

minus50


The design of a sliding mode variable structure observerfollows to achieve fault reconstruction by using sliding modeequivalent principle Then it comes to the design of logicalgorithm with which the reconstructed 120572-120573 axis fault signalcan be converted into 119886119887119888-phase and then the detection andreconstruction of actual fault of phase current sensor can beimplemented It can be seen from the proof and simulationresults that with the proposed fault reconstruction schemethere is almost no restriction on the fault type In other wordsthe scheme is applicable for abrupt fault incipient fault or anyother types of faults

Conflict of Interests

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

Acknowledgments

This work was supported by the Natural Science Foundationof China (nos 61273157 and 61473117) Hunan ProvincialNatural Science Foundation of China (nos 14JJ5024 and2015JJ5011) andHunanProvince EducationDepartment Pro-ject of China (nos 12A040 and 13CY018)

References

[1] I AydinMKarakose and E Akin ldquoAn approach for automatedfault diagnosis based on a fuzzy decision tree and boundaryanalysis of a reconstructed phase spacerdquo ISA Transactions vol53 no 2 pp 220ndash229 2014

[2] H Lu X Cheng W Qu S Sheng Y Li and Z Wang ldquoAthree-phase current reconstruction technique using single DCcurrent sensor based on TSPWMrdquo IEEE Transactions on PowerElectronics vol 29 no 3 pp 1542ndash1550 2014

[3] F R Salmasi and T A Najafabadi ldquoAn adaptive observer withonline rotor and stator resistance estimation for inductionmotors with one phase current sensorrdquo IEEE Transactions onEnergy Conversion vol 26 no 3 pp 959ndash966 2011

[4] T A Najafabadi F R Salmasi and P Jabehdar-Maralani ldquoDet-ection and isolation of speed- DC-link voltage- and current-sensor faults based on an adaptive observer in induction-motordrivesrdquo IEEE Transactions on Industrial Electronics vol 58 no5 pp 1662ndash1672 2011

[5] L Liu Robust fault detection and diagnosis for permanentmagnet synchronous motors [A Thesis Presented for the Degreeof Doctor of Philosophy] The Florida State University SummerSemester 2006

[6] Y Huang Fault diagnosis method based on observer and applica-tion research [MS thesis]TheHunanUniversity of TechnologySummer Semester 2011

[7] A Sabanovic ldquoVariable structure systems with sliding modesin motion controlmdasha surveyrdquo IEEE Transactions on IndustrialInformatics vol 7 no 2 pp 212ndash223 2011

[8] Z Gao B Jiang P Shi M Qian and J Lin ldquoActive fault tolerantcontrol design for reusable launch vehicle using adaptive slidingmode techniquerdquo Journal of the Franklin Institute vol 349 no4 pp 1543ndash1560 2012

[9] B Jiang M Staroswiecki and V Cocquempot ldquoFault estima-tion in nonlinear uncertain systems using robustsliding-modeobserversrdquo IEE Proceedings Control Theory and Applicationsvol 151 no 1 pp 29ndash37 2004

[10] C Edwards S K Spurgeon and R J Patton ldquoSliding modeobservers for fault detection and isolationrdquo Automatica vol 36no 4 pp 541ndash553 2000

[11] C P Tan andC Edwards ldquoSlidingmode observers for detectionand reconstruction of sensor faultsrdquo Automatica vol 38 no 10pp 1815ndash1821 2002

[12] H K Khalil Nonlinear Systems Prentice-Hall EnglewoodCliffs NJ USA 3rd edition 2002

[13] X-G Yan and C Edwards ldquoSensor fault detection and isolationfor nonlinear systems based on a sliding mode observerrdquoInternational Journal of Adaptive Control and Signal Processingvol 21 no 8-9 pp 657ndash673 2007

[14] J He J Qiu and C Zhang ldquoA multiple fault diagnosis methodbased on sliding mode observerrdquo Journal of Dynamics andControl vol 7 no 1 pp 84ndash91 2009

[15] F Y Wang Sliding Mode Variable Structure Control TsinghuaUniversity Press Beijing China 2005

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



axis currents Huang [6] an unknown input observer wasdesigned for detection of 119902 axis stator current residuals forinductionmotor However the two currents 119894119889 and 119894119902 are notpractically measurable These two virtual sensing signals arecalculated from the measured phase current 119894119886119887119888 by applyinga linear Clarke transformation Abnormal changes in 119894119889 and 119894119902may indicate a fault appearing in the phase current sensorsbut this design will not provide more specific informationFurthermore as the calculations of 119894119889 and 119894119902 are coupled ifa fault occurred in the measurement devices for 119894119886119887119888 conseq-uential faults in 119894119889 and 119894119902 will appear simultaneously Henceit is relatively difficult to detect the abnormal signals in themeasurement devices for 119894119886119887119888

Sliding mode techniques are known for their robustnessand insensitivity to the so-calledmatched uncertaintyThere-fore these techniques offer great potential for robust faultdetection and isolation (FDI) [7ndash10] Faults are classifiedaccording to their physical locations into system actuatorand sensor faults Compared with actuators sensors are pass-ive elements in the sense that they only provide operationalinformation about the system and do not affect the systembehavior directlyThus they have been less studied comparedto actuator fault detection and isolation This paper presentsa method in which using a newly designed filter the sensorfaults can be modeled as pseudoactuator faults Then usingthe transformed system structure and characteristics of thedesigned filter a sliding mode observer is proposed to recon-struct the sensor fault in the system [11ndash13]

The paper is organized as follows Section 2 describes asurface-mounted PMSMwith current sensor faults Section 3proposes the method for reconstruction of faults Section 4presents simulation results and conclusions in Section 5 fol-low

2 System Formulation

In the 120572-120573 rotor reference frame a surface-mounted PMSMwith current sensors faults can be expressed as the followingdynamic model

(119905) = 119860119909 (119905) + 119861119906 (119905) + 119864120595

119910 (119905) = 119862119909 (119905) + 119873119891119904 (119909 119906 119905)

(1)

where 119909(119905) 119906(119905) 119910(119905) 120595 and 119891119904(119909 119906 119905) are the state variablesinputs outputs rotor flux and sensor faults respectivelyThen the matrices are defined as

119860 =[[[

[

minus119877119904

119871120572

0

0 minus119877119904

119871120573

]]]

]

119861 =[[[

[

1

119871120572

0

01

119871120573

]]]

]

119862 = [1 0

0 1]

119864 =[[

[

minus120596

119871120572

0

0 minus120596

119871120573

]]

]

119873 = [1 0

0 1]

119909 = [119894120572

119894120573

]

119906 = [119906120572

119906120573

]

120595 = [120595120572

120595120573

] = [minus120595119891 sin 120579119903120595119891 cos 120579119903

]

119910 = [119894120572

119894120573

]

119891119904 = [119891119904120572

119891119904120573

]

(2)

with

119871120572 = 1198710 + 1198711 cos (2120579119903)

119871120573 = 1198710 minus 1198711 cos (2120579119903)

1198710 =119871119889 + 119871119902

2

1198711 =119871119889 minus 119871119902

2

(3)

where 119894120572 119894120573 119906120572 and 119906120573 are the 120572-120573 axis currents and voltagesrespectively 120595119891 is the rotor permanent magnet flux linkage120579119903 is the electrical rotor angular position 120595120572 120595120573 119891119904120572 and119891119904120573 are the 120572-120573 axis stator flux linkages and current sensorfaults respectively 119877119904 is the stator resistance 119871120572 119871120573 and120596 are the 120572-120573 axis stator inductances and electrical rotorspeed respectively and 119871119889 and 119871119902 are the stator 119889 and 119902 axisinductances respectivelyThe surface-mounted PMSM statorwinding shows the same electrical inductance on both119889 and 119902axes that is 119871119889 = 119871119902 = 119871 119904 where 119871 119904 is the stator inductance

3 Sensor Faults Reconstruction Based onSliding Mode Variable Structure

The section proposes a novelmethod to reconstruct the phasecurrent sensor faults for PMSM drives Before dealing withthe faults an extended 120572-120573 axis fault model of PMSM is refor-mulated by using a transformation filter which increases thesystemrsquos stateThen a slidingmode observer is constructed to




(119905) = 119860 119904119911 (119905) + 119861119904119910 (4)



(119905) = 119860 119904119911 (119905) + 119861119904119862119909 (119905) + 119861119904119873119891119904 (119909 119906 119905) (5)


[

] = [

119860 0

119861119904119862 119860 119904

][119909

119911] + [

119861

0] 119906 + [

0

119861119904119873]119891119904 + [

119864

0]120595

119911 = [0 119868] [119909

119911]

(6)


119860 = [119860 0

119861119904119862 119860 119904

]

119861 = [119861

0]

119872 = [0

119861119904119873]

119864 = [119864

0]

119862 = [0 119868]

119909 (119905) = [119909 (119905)

119911 (119905)]

119910 (119905) = 119911

(7)


(119905) = 119860119909 (119905) + 119861119906 (119905) + 119872119891119904 + 119864120595

119910 (119905) = 119862119909 (119905)

(8)





119860119879

0119875 + 1198751198600 = minus119876 (9)



119865119879



119909 (119905) = 119860 (119905) + 119861119906 (119905) + 119871 (119910 (119905) minus 119862 (119905)) + 119872V

+ 119864120595

(119905) = 119862 (119905)

(10)


V =

minus120588119865119890119910 (119905)

10038171003817100381710038171003817119865119890119910 (119905)

10038171003817100381710038171003817

if 119890119910 = 0

0 if 119890119910 = 0

(11)




119890 (119905) = 1198600119890 (119905) + 119872(V minus 119891119904) (12)



120588 gt 1205881 (13)





1 = 119890119879(1198751198600 + 1198600119875) 119890 + 2119890

119879119875119872V minus 2119890

119879119875119872119891119904 (14)


and it follows that

1 le minus119890119879119876119890 + 2 (119865119862119890)

119879

(V + 119891119904)

1 le minus119890119879119876119890 minus 2 (119865119890119910)

119879

12058811986511989011991010038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817

+ 210038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817

10038171003817100381710038171198911199041003817100381710038171003817

1 le minus119890119879119876119890 minus 2

10038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817(120588 minus

10038171003817100381710038171198911199041003817100381710038171003817)

(15)


1 le minus119890119879119876119890 minus 2

10038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817(120588 minus 1205881) le minus119890

119879119876119890 (16)





119890 le 1205751 (17)




119904 = 119865119890119910 (119905) (18)




1198812 =1

2119904119879119904 (19)


2 = 119904119879(119865119862 119890) = 119904

119879119865119862 (1198600119890 +119872(V minus 119891119904))

= 1199041198791198651198621198600119890 + 119904

119879119865119862119872(V minus 119891119904)

= 1199041198791198651198621198600119890 + 119904

119879119872119879119875119872(V minus 119891119904)

= 1199041198791198651198621198600119890 minus 119904

119879119872119879119875119872119891119904 + 119904

119879119872119879119875119872V

= 1199041198791198651198621198600119890 minus 119904

119879119872119879119875119872119891119904 minus 120588119904

119879119872119879119875119872

119904

119904

(20)


2 le 119904100381710038171003817100381710038171198651198621198600

10038171003817100381710038171003817119890 + 119904

10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

10038171003817100381710038171198911199041003817100381710038171003817

minus 120588120582min (119872119879119875119872) 119904 le 119904 (

100381710038171003817100381710038171198651198621198600

100381710038171003817100381710038171205751

+10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

10038171003817100381710038171198911199041003817100381710038171003817 minus 120588120582min (119872

119879119875119872)) le minus 119904

sdot 120582min (119872119879119875119872)(120588

minus (

10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

120582min (119872119879119875119872)

10038171003817100381710038171198911199041003817100381710038171003817 +

100381710038171003817100381710038171198651198621198600

10038171003817100381710038171003817

120582min (119872119879119875119872)

1205751))

(21)


119870 = 120582min (119872119879119875119872)(120588

minus (

10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

120582min (119872119879119875119872)

10038171003817100381710038171198911199041003817100381710038171003817 +

100381710038171003817100381710038171198651198621198600

10038171003817100381710038171003817

120582min (119872119879119875119872)

1205751))

(22)

Since 119872119879119875119872 ge 120582min(119872


(119872119879119875119872120582min(119872

119879119875119872))119891119904+(1198651198621198600120582min(119872

119879119875119872))1205751 ge


120588 ge

10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

120582min (119872119879119875119872)

1205881 +

100381710038171003817100381710038171198651198621198600

10038171003817100381710038171003817

120582min (119872119879119875119872)

1205751(23)

then

2 le minus119870 119904 (24)





119865119862119890 (119905) = 119865119890119910 (119905) = 0

119865119862 119890 = 119865 119890119910 (119905) = 119865119890119910 (119905) = 0

(25)



119891119904 = V (26)






[[

[

119894120572

119894120573

1198940

]]

]

= radic2

3

[[[[[[[[

[

1 minus1

2minus1

2

0radic3

2minusradic3

2

1

radic2

1

radic2

1

radic2

]]]]]]]]

]

[[

[

119894119886

119894119887

119894119888

]]

]

(27)




[119894120572

119894120573

] =

[[[[

[

radic3

20

radic2

2radic2

]]]]

]

[119894119886

119894119887

] (28)



[119894120572 + 119891119904120572

119894120573 + 119891119904120573

] =

[[[[

[

radic3

20

radic2

2radic2

]]]]

]

[119894119886 + 119891119904119886

119894119887 + 119891119904119887

] (29)



119891119904119886 =radic2

3119891119904120572

119891119904119887 =1

radic2119891119904120573 minus

radic1

6119891119904120572

(30)



119894119886 =radic2

3119894120572

119894119887 =1


radic1

6119894120572

(31)



4 Simulation Study



119860 119904 = [minus200 0

0 minus200]

119861119904 = [200 0

0 200]

(32)


0 005 01 015 02

0

5

10

15

Time (s)

i aan

di a

(A)

iaia

minus5

minus10

minus15


0 005 01 015 02

0

5

10

15

Time (s)

ibib

i ban

di b

(A)

minus5

minus10

minus15





119904) Ω 2875



119860 =

[[[[[

[

minus33823 0 0 0

0 minus33823 0 0

200 0 minus200 0

0 200 0 minus200

]]]]]

]

119861 =

[[[[[

[

11765 0

0 11765

0 0

0 0

]]]]]

]

119862 = [0 0 1 0

0 0 0 1]

119864 =

[[[[[

[

minus11765120596 0

0 minus11765120596

0 0

0 0

]]]]]

]

119872 =

[[[[[

[

0 0

0 0

200 0

0 200

]]]]]

]

(33)


119871 = [100 0 200 0

0 100 0 100]

119879

(34)


1198600 =

[[[[[

[



200 0 minus400 0

0 200 0 minus100

]]]]]

]

(35)


119875 =

[[[[[

[

7244 0 0 0

0 7595 0 0

0 0 5674 0

0 0 0 7455

]]]]]

]

119865 = [113480 0

0 149100]

(36)


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsa

(A)

Estimation


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsb

(A)

Estimation



0 005 01 015 02

0

50

100

150

200

250

300

350

400

Time (s)

Spee

d (r

pm)

minus50



V = minus120588119865119890119910

10038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817+ 1205752

(37)








0 005 01 015 02

0

5

10

15

Time (s)

i aan

di a

(A)

iaia

minus5

minus10

minus15


0 005 01 015 02

0

5

10

15

Time (s)

i ban

di b

(A)

ibib

minus5

minus10

minus15



0 005 01 02015

0

2

4

6

8

10

Time (s)

FaultEstimation

minus10

minus8

minus6

minus4

minus2

fsa

(A)


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsb

(A)

Estimation





5 Conclusions



0 005 01 015 02

0

50

100

150

200

250

300

350

400

Time (s)

Spee

d (r

pm)

minus50





Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












(119905) = 119860 119904119911 (119905) + 119861119904119910 (4)



(119905) = 119860 119904119911 (119905) + 119861119904119862119909 (119905) + 119861119904119873119891119904 (119909 119906 119905) (5)


[

] = [

119860 0

119861119904119862 119860 119904

][119909

119911] + [

119861

0] 119906 + [

0

119861119904119873]119891119904 + [

119864

0]120595

119911 = [0 119868] [119909

119911]

(6)


119860 = [119860 0

119861119904119862 119860 119904

]

119861 = [119861

0]

119872 = [0

119861119904119873]

119864 = [119864

0]

119862 = [0 119868]

119909 (119905) = [119909 (119905)

119911 (119905)]

119910 (119905) = 119911

(7)


(119905) = 119860119909 (119905) + 119861119906 (119905) + 119872119891119904 + 119864120595

119910 (119905) = 119862119909 (119905)

(8)





119860119879

0119875 + 1198751198600 = minus119876 (9)



119865119879



119909 (119905) = 119860 (119905) + 119861119906 (119905) + 119871 (119910 (119905) minus 119862 (119905)) + 119872V

+ 119864120595

(119905) = 119862 (119905)

(10)


V =

minus120588119865119890119910 (119905)

10038171003817100381710038171003817119865119890119910 (119905)

10038171003817100381710038171003817

if 119890119910 = 0

0 if 119890119910 = 0

(11)




119890 (119905) = 1198600119890 (119905) + 119872(V minus 119891119904) (12)



120588 gt 1205881 (13)





1 = 119890119879(1198751198600 + 1198600119875) 119890 + 2119890

119879119875119872V minus 2119890

119879119875119872119891119904 (14)


and it follows that

1 le minus119890119879119876119890 + 2 (119865119862119890)

119879

(V + 119891119904)

1 le minus119890119879119876119890 minus 2 (119865119890119910)

119879

12058811986511989011991010038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817

+ 210038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817

10038171003817100381710038171198911199041003817100381710038171003817

1 le minus119890119879119876119890 minus 2

10038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817(120588 minus

10038171003817100381710038171198911199041003817100381710038171003817)

(15)


1 le minus119890119879119876119890 minus 2

10038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817(120588 minus 1205881) le minus119890

119879119876119890 (16)





119890 le 1205751 (17)




119904 = 119865119890119910 (119905) (18)




1198812 =1

2119904119879119904 (19)


2 = 119904119879(119865119862 119890) = 119904

119879119865119862 (1198600119890 +119872(V minus 119891119904))

= 1199041198791198651198621198600119890 + 119904

119879119865119862119872(V minus 119891119904)

= 1199041198791198651198621198600119890 + 119904

119879119872119879119875119872(V minus 119891119904)

= 1199041198791198651198621198600119890 minus 119904

119879119872119879119875119872119891119904 + 119904

119879119872119879119875119872V

= 1199041198791198651198621198600119890 minus 119904

119879119872119879119875119872119891119904 minus 120588119904

119879119872119879119875119872

119904

119904

(20)


2 le 119904100381710038171003817100381710038171198651198621198600

10038171003817100381710038171003817119890 + 119904

10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

10038171003817100381710038171198911199041003817100381710038171003817

minus 120588120582min (119872119879119875119872) 119904 le 119904 (

100381710038171003817100381710038171198651198621198600

100381710038171003817100381710038171205751

+10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

10038171003817100381710038171198911199041003817100381710038171003817 minus 120588120582min (119872

119879119875119872)) le minus 119904

sdot 120582min (119872119879119875119872)(120588

minus (

10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

120582min (119872119879119875119872)

10038171003817100381710038171198911199041003817100381710038171003817 +

100381710038171003817100381710038171198651198621198600

10038171003817100381710038171003817

120582min (119872119879119875119872)

1205751))

(21)


119870 = 120582min (119872119879119875119872)(120588

minus (

10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

120582min (119872119879119875119872)

10038171003817100381710038171198911199041003817100381710038171003817 +

100381710038171003817100381710038171198651198621198600

10038171003817100381710038171003817

120582min (119872119879119875119872)

1205751))

(22)

Since 119872119879119875119872 ge 120582min(119872


(119872119879119875119872120582min(119872

119879119875119872))119891119904+(1198651198621198600120582min(119872

119879119875119872))1205751 ge


120588 ge

10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

120582min (119872119879119875119872)

1205881 +

100381710038171003817100381710038171198651198621198600

10038171003817100381710038171003817

120582min (119872119879119875119872)

1205751(23)

then

2 le minus119870 119904 (24)





119865119862119890 (119905) = 119865119890119910 (119905) = 0

119865119862 119890 = 119865 119890119910 (119905) = 119865119890119910 (119905) = 0

(25)



119891119904 = V (26)






[[

[

119894120572

119894120573

1198940

]]

]

= radic2

3

[[[[[[[[

[

1 minus1

2minus1

2

0radic3

2minusradic3

2

1

radic2

1

radic2

1

radic2

]]]]]]]]

]

[[

[

119894119886

119894119887

119894119888

]]

]

(27)




[119894120572

119894120573

] =

[[[[

[

radic3

20

radic2

2radic2

]]]]

]

[119894119886

119894119887

] (28)



[119894120572 + 119891119904120572

119894120573 + 119891119904120573

] =

[[[[

[

radic3

20

radic2

2radic2

]]]]

]

[119894119886 + 119891119904119886

119894119887 + 119891119904119887

] (29)



119891119904119886 =radic2

3119891119904120572

119891119904119887 =1

radic2119891119904120573 minus

radic1

6119891119904120572

(30)



119894119886 =radic2

3119894120572

119894119887 =1


radic1

6119894120572

(31)



4 Simulation Study



119860 119904 = [minus200 0

0 minus200]

119861119904 = [200 0

0 200]

(32)


0 005 01 015 02

0

5

10

15

Time (s)

i aan

di a

(A)

iaia

minus5

minus10

minus15


0 005 01 015 02

0

5

10

15

Time (s)

ibib

i ban

di b

(A)

minus5

minus10

minus15





119904) Ω 2875



119860 =

[[[[[

[

minus33823 0 0 0

0 minus33823 0 0

200 0 minus200 0

0 200 0 minus200

]]]]]

]

119861 =

[[[[[

[

11765 0

0 11765

0 0

0 0

]]]]]

]

119862 = [0 0 1 0

0 0 0 1]

119864 =

[[[[[

[

minus11765120596 0

0 minus11765120596

0 0

0 0

]]]]]

]

119872 =

[[[[[

[

0 0

0 0

200 0

0 200

]]]]]

]

(33)


119871 = [100 0 200 0

0 100 0 100]

119879

(34)


1198600 =

[[[[[

[



200 0 minus400 0

0 200 0 minus100

]]]]]

]

(35)


119875 =

[[[[[

[

7244 0 0 0

0 7595 0 0

0 0 5674 0

0 0 0 7455

]]]]]

]

119865 = [113480 0

0 149100]

(36)


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsa

(A)

Estimation


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsb

(A)

Estimation



0 005 01 015 02

0

50

100

150

200

250

300

350

400

Time (s)

Spee

d (r

pm)

minus50



V = minus120588119865119890119910

10038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817+ 1205752

(37)








0 005 01 015 02

0

5

10

15

Time (s)

i aan

di a

(A)

iaia

minus5

minus10

minus15


0 005 01 015 02

0

5

10

15

Time (s)

i ban

di b

(A)

ibib

minus5

minus10

minus15



0 005 01 02015

0

2

4

6

8

10

Time (s)

FaultEstimation

minus10

minus8

minus6

minus4

minus2

fsa

(A)


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsb

(A)

Estimation





5 Conclusions



0 005 01 015 02

0

50

100

150

200

250

300

350

400

Time (s)

Spee

d (r

pm)

minus50





Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










and it follows that

1 le minus119890119879119876119890 + 2 (119865119862119890)

119879

(V + 119891119904)

1 le minus119890119879119876119890 minus 2 (119865119890119910)

119879

12058811986511989011991010038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817

+ 210038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817

10038171003817100381710038171198911199041003817100381710038171003817

1 le minus119890119879119876119890 minus 2

10038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817(120588 minus

10038171003817100381710038171198911199041003817100381710038171003817)

(15)


1 le minus119890119879119876119890 minus 2

10038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817(120588 minus 1205881) le minus119890

119879119876119890 (16)





119890 le 1205751 (17)




119904 = 119865119890119910 (119905) (18)




1198812 =1

2119904119879119904 (19)


2 = 119904119879(119865119862 119890) = 119904

119879119865119862 (1198600119890 +119872(V minus 119891119904))

= 1199041198791198651198621198600119890 + 119904

119879119865119862119872(V minus 119891119904)

= 1199041198791198651198621198600119890 + 119904

119879119872119879119875119872(V minus 119891119904)

= 1199041198791198651198621198600119890 minus 119904

119879119872119879119875119872119891119904 + 119904

119879119872119879119875119872V

= 1199041198791198651198621198600119890 minus 119904

119879119872119879119875119872119891119904 minus 120588119904

119879119872119879119875119872

119904

119904

(20)


2 le 119904100381710038171003817100381710038171198651198621198600

10038171003817100381710038171003817119890 + 119904

10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

10038171003817100381710038171198911199041003817100381710038171003817

minus 120588120582min (119872119879119875119872) 119904 le 119904 (

100381710038171003817100381710038171198651198621198600

100381710038171003817100381710038171205751

+10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

10038171003817100381710038171198911199041003817100381710038171003817 minus 120588120582min (119872

119879119875119872)) le minus 119904

sdot 120582min (119872119879119875119872)(120588

minus (

10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

120582min (119872119879119875119872)

10038171003817100381710038171198911199041003817100381710038171003817 +

100381710038171003817100381710038171198651198621198600

10038171003817100381710038171003817

120582min (119872119879119875119872)

1205751))

(21)


119870 = 120582min (119872119879119875119872)(120588

minus (

10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

120582min (119872119879119875119872)

10038171003817100381710038171198911199041003817100381710038171003817 +

100381710038171003817100381710038171198651198621198600

10038171003817100381710038171003817

120582min (119872119879119875119872)

1205751))

(22)

Since 119872119879119875119872 ge 120582min(119872


(119872119879119875119872120582min(119872

119879119875119872))119891119904+(1198651198621198600120582min(119872

119879119875119872))1205751 ge


120588 ge

10038171003817100381710038171003817119872119879119875119872

10038171003817100381710038171003817

120582min (119872119879119875119872)

1205881 +

100381710038171003817100381710038171198651198621198600

10038171003817100381710038171003817

120582min (119872119879119875119872)

1205751(23)

then

2 le minus119870 119904 (24)





119865119862119890 (119905) = 119865119890119910 (119905) = 0

119865119862 119890 = 119865 119890119910 (119905) = 119865119890119910 (119905) = 0

(25)



119891119904 = V (26)






[[

[

119894120572

119894120573

1198940

]]

]

= radic2

3

[[[[[[[[

[

1 minus1

2minus1

2

0radic3

2minusradic3

2

1

radic2

1

radic2

1

radic2

]]]]]]]]

]

[[

[

119894119886

119894119887

119894119888

]]

]

(27)




[119894120572

119894120573

] =

[[[[

[

radic3

20

radic2

2radic2

]]]]

]

[119894119886

119894119887

] (28)



[119894120572 + 119891119904120572

119894120573 + 119891119904120573

] =

[[[[

[

radic3

20

radic2

2radic2

]]]]

]

[119894119886 + 119891119904119886

119894119887 + 119891119904119887

] (29)



119891119904119886 =radic2

3119891119904120572

119891119904119887 =1

radic2119891119904120573 minus

radic1

6119891119904120572

(30)



119894119886 =radic2

3119894120572

119894119887 =1


radic1

6119894120572

(31)



4 Simulation Study



119860 119904 = [minus200 0

0 minus200]

119861119904 = [200 0

0 200]

(32)


0 005 01 015 02

0

5

10

15

Time (s)

i aan

di a

(A)

iaia

minus5

minus10

minus15


0 005 01 015 02

0

5

10

15

Time (s)

ibib

i ban

di b

(A)

minus5

minus10

minus15





119904) Ω 2875



119860 =

[[[[[

[

minus33823 0 0 0

0 minus33823 0 0

200 0 minus200 0

0 200 0 minus200

]]]]]

]

119861 =

[[[[[

[

11765 0

0 11765

0 0

0 0

]]]]]

]

119862 = [0 0 1 0

0 0 0 1]

119864 =

[[[[[

[

minus11765120596 0

0 minus11765120596

0 0

0 0

]]]]]

]

119872 =

[[[[[

[

0 0

0 0

200 0

0 200

]]]]]

]

(33)


119871 = [100 0 200 0

0 100 0 100]

119879

(34)


1198600 =

[[[[[

[



200 0 minus400 0

0 200 0 minus100

]]]]]

]

(35)


119875 =

[[[[[

[

7244 0 0 0

0 7595 0 0

0 0 5674 0

0 0 0 7455

]]]]]

]

119865 = [113480 0

0 149100]

(36)


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsa

(A)

Estimation


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsb

(A)

Estimation



0 005 01 015 02

0

50

100

150

200

250

300

350

400

Time (s)

Spee

d (r

pm)

minus50



V = minus120588119865119890119910

10038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817+ 1205752

(37)








0 005 01 015 02

0

5

10

15

Time (s)

i aan

di a

(A)

iaia

minus5

minus10

minus15


0 005 01 015 02

0

5

10

15

Time (s)

i ban

di b

(A)

ibib

minus5

minus10

minus15



0 005 01 02015

0

2

4

6

8

10

Time (s)

FaultEstimation

minus10

minus8

minus6

minus4

minus2

fsa

(A)


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsb

(A)

Estimation





5 Conclusions



0 005 01 015 02

0

50

100

150

200

250

300

350

400

Time (s)

Spee

d (r

pm)

minus50





Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119891119904 = V (26)






[[

[

119894120572

119894120573

1198940

]]

]

= radic2

3

[[[[[[[[

[

1 minus1

2minus1

2

0radic3

2minusradic3

2

1

radic2

1

radic2

1

radic2

]]]]]]]]

]

[[

[

119894119886

119894119887

119894119888

]]

]

(27)




[119894120572

119894120573

] =

[[[[

[

radic3

20

radic2

2radic2

]]]]

]

[119894119886

119894119887

] (28)



[119894120572 + 119891119904120572

119894120573 + 119891119904120573

] =

[[[[

[

radic3

20

radic2

2radic2

]]]]

]

[119894119886 + 119891119904119886

119894119887 + 119891119904119887

] (29)



119891119904119886 =radic2

3119891119904120572

119891119904119887 =1

radic2119891119904120573 minus

radic1

6119891119904120572

(30)



119894119886 =radic2

3119894120572

119894119887 =1


radic1

6119894120572

(31)



4 Simulation Study



119860 119904 = [minus200 0

0 minus200]

119861119904 = [200 0

0 200]

(32)


0 005 01 015 02

0

5

10

15

Time (s)

i aan

di a

(A)

iaia

minus5

minus10

minus15


0 005 01 015 02

0

5

10

15

Time (s)

ibib

i ban

di b

(A)

minus5

minus10

minus15





119904) Ω 2875



119860 =

[[[[[

[

minus33823 0 0 0

0 minus33823 0 0

200 0 minus200 0

0 200 0 minus200

]]]]]

]

119861 =

[[[[[

[

11765 0

0 11765

0 0

0 0

]]]]]

]

119862 = [0 0 1 0

0 0 0 1]

119864 =

[[[[[

[

minus11765120596 0

0 minus11765120596

0 0

0 0

]]]]]

]

119872 =

[[[[[

[

0 0

0 0

200 0

0 200

]]]]]

]

(33)


119871 = [100 0 200 0

0 100 0 100]

119879

(34)


1198600 =

[[[[[

[



200 0 minus400 0

0 200 0 minus100

]]]]]

]

(35)


119875 =

[[[[[

[

7244 0 0 0

0 7595 0 0

0 0 5674 0

0 0 0 7455

]]]]]

]

119865 = [113480 0

0 149100]

(36)


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsa

(A)

Estimation


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsb

(A)

Estimation



0 005 01 015 02

0

50

100

150

200

250

300

350

400

Time (s)

Spee

d (r

pm)

minus50



V = minus120588119865119890119910

10038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817+ 1205752

(37)








0 005 01 015 02

0

5

10

15

Time (s)

i aan

di a

(A)

iaia

minus5

minus10

minus15


0 005 01 015 02

0

5

10

15

Time (s)

i ban

di b

(A)

ibib

minus5

minus10

minus15



0 005 01 02015

0

2

4

6

8

10

Time (s)

FaultEstimation

minus10

minus8

minus6

minus4

minus2

fsa

(A)


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsb

(A)

Estimation





5 Conclusions



0 005 01 015 02

0

50

100

150

200

250

300

350

400

Time (s)

Spee

d (r

pm)

minus50





Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 005 01 015 02

0

5

10

15

Time (s)

i aan

di a

(A)

iaia

minus5

minus10

minus15


0 005 01 015 02

0

5

10

15

Time (s)

ibib

i ban

di b

(A)

minus5

minus10

minus15





119904) Ω 2875



119860 =

[[[[[

[

minus33823 0 0 0

0 minus33823 0 0

200 0 minus200 0

0 200 0 minus200

]]]]]

]

119861 =

[[[[[

[

11765 0

0 11765

0 0

0 0

]]]]]

]

119862 = [0 0 1 0

0 0 0 1]

119864 =

[[[[[

[

minus11765120596 0

0 minus11765120596

0 0

0 0

]]]]]

]

119872 =

[[[[[

[

0 0

0 0

200 0

0 200

]]]]]

]

(33)


119871 = [100 0 200 0

0 100 0 100]

119879

(34)


1198600 =

[[[[[

[



200 0 minus400 0

0 200 0 minus100

]]]]]

]

(35)


119875 =

[[[[[

[

7244 0 0 0

0 7595 0 0

0 0 5674 0

0 0 0 7455

]]]]]

]

119865 = [113480 0

0 149100]

(36)


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsa

(A)

Estimation


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsb

(A)

Estimation



0 005 01 015 02

0

50

100

150

200

250

300

350

400

Time (s)

Spee

d (r

pm)

minus50



V = minus120588119865119890119910

10038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817+ 1205752

(37)








0 005 01 015 02

0

5

10

15

Time (s)

i aan

di a

(A)

iaia

minus5

minus10

minus15


0 005 01 015 02

0

5

10

15

Time (s)

i ban

di b

(A)

ibib

minus5

minus10

minus15



0 005 01 02015

0

2

4

6

8

10

Time (s)

FaultEstimation

minus10

minus8

minus6

minus4

minus2

fsa

(A)


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsb

(A)

Estimation





5 Conclusions



0 005 01 015 02

0

50

100

150

200

250

300

350

400

Time (s)

Spee

d (r

pm)

minus50





Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsa

(A)

Estimation


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsb

(A)

Estimation



0 005 01 015 02

0

50

100

150

200

250

300

350

400

Time (s)

Spee

d (r

pm)

minus50



V = minus120588119865119890119910

10038171003817100381710038171003817119865119890119910

10038171003817100381710038171003817+ 1205752

(37)








0 005 01 015 02

0

5

10

15

Time (s)

i aan

di a

(A)

iaia

minus5

minus10

minus15


0 005 01 015 02

0

5

10

15

Time (s)

i ban

di b

(A)

ibib

minus5

minus10

minus15



0 005 01 02015

0

2

4

6

8

10

Time (s)

FaultEstimation

minus10

minus8

minus6

minus4

minus2

fsa

(A)


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsb

(A)

Estimation





5 Conclusions



0 005 01 015 02

0

50

100

150

200

250

300

350

400

Time (s)

Spee

d (r

pm)

minus50





Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 005 01 015 02

0

5

10

15

Time (s)

i aan

di a

(A)

iaia

minus5

minus10

minus15


0 005 01 015 02

0

5

10

15

Time (s)

i ban

di b

(A)

ibib

minus5

minus10

minus15



0 005 01 02015

0

2

4

6

8

10

Time (s)

FaultEstimation

minus10

minus8

minus6

minus4

minus2

fsa

(A)


0 005 01 02015

0

2

4

6

8

10

Time (s)

Fault

minus10

minus8

minus6

minus4

minus2

fsb

(A)

Estimation





5 Conclusions



0 005 01 015 02

0

50

100

150

200

250

300

350

400

Time (s)

Spee

d (r

pm)

minus50





Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 005 01 015 02

0

50

100

150

200

250

300

350

400

Time (s)

Spee

d (r

pm)

minus50





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









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