DesignofRobustAdaptiveObserveragainstMeasurement ...downloads.hindawi.com/journals/jece/2020/6570942.pdf · 2020. 8. 1. · ResearchArticle DesignofRobustAdaptiveObserveragainstMeasurement

Research ArticleDesign of Robust Adaptive Observer against MeasurementNoise for Sensorless Vector Control of Induction Motor Drives

Asma Boulmane Youssef Zidani Mohammed Chennani and Driss Belkhayat

Faculty of Sciences and Technologies Cadi Ayyad University Marrakesh Morocco

Correspondence should be addressed to Asma Boulmane asmaboulmanegmailcom

Received 30 March 2020 Revised 26 June 2020 Accepted 13 July 2020 Published 1 August 2020

Academic Editor Luca Maresca

Copyright copy 2020 Asma Boulmane et al ampis 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 isproperly cited

Position control in electrical drives is a challenging problemwhich is complicated by sensor noise and unknown disturbancesampispaper proposes a new cascade sensorless speed control technique for induction motor drives suitable for electric vehicle ap-plications using the full-order adaptive Luenberger observer that is insensitive to measurement noise and parametric variationampe adaptive speed law is obtained by the Lyapunov method using the estimated currents and fluxes ampis technique ensures thestability of the induction motor considered as nonlinear dynamic system Since the Luenberger observer works on deterministicenvironment and it is most effective when sensor noise is limited the present study aims to design a robust observer insensitive tomeasurement noise and parametric variation integrated in a cascade structure ampe observer allows the filtering of the measuredcurrents To highlight the advantages of the new scheme a comparative study and spectrum analysis will be presented ampeproposed structure is verified using MATLABSimulink

1 Introduction

Motor drive technology is a very complex and multidisci-plinary field and it has gone through a dynamic evolutionover the last several decades by way of many inventions inpower electronics (semiconductor devices converters andPWM techniques) electrical machines and advancedcontrol and simulation techniques

Due to its robustness and rugged structure the inductionmotor (IM) is widely used in industrial applications espe-cially for electric and hybrid electric vehicles

Control techniques of IMs can be divided into two maincategories scalar and vector controls Scalar control is basedon the steady-state model of the IM Although this control issimple easy to implement and offers a good satisfactorysteady-state response this control is not suitable for ap-plications requiring high dynamic performance Vector (orfield-oriented) control offers good satisfactory performancein terms of both steady-state and transient response It al-lows separated control in a DC motor which has drawbackscaused by the brushes [1 2]

Vector control is based on having speed measurementHowever physical sensors have shortcomings that can de-grade the control system Indeed sensor cost can substan-tially raise the total cost of a control system sensors (andtheir associated wiring) reduce the reliability of the controlsystem Also sensors can induce significant errors such asstochastic noise cyclical errors and limited responsiveness[3] ampis has generated interest in sensorless control

Sensorless control has received considerable attentionduring the last decades both in research context and also inapplication domain on real processes ampis technique en-sures reduced hardware complexity lower cost reduced sizeof the drive machine elimination of the sensor cables betternoise immunity increased reliability and lower mainte-nance requirements [1 3 4]

ampere are a great variety of rotor speed estimationtechniques in the literature ampey can be classified into twomain categories (a) signal injection method and (b) state-observer methods ampe first category suffers from compu-tational complexity and the requirement of external hard-ware for signal injection ampose in the second category are

HindawiJournal of Electrical and Computer EngineeringVolume 2020 Article ID 6570942 12 pageshttpsdoiorg10115520206570942

based on mathematical models of the IM and are simple androbust against disturbances [2] ampese techniques includeopen-loop estimators the Kalman filter the Luenbergerobserver and Model Reference Adaptive System [5 6]

Authors of [7] combined MRAS and sliding mode forindirect vector control to improve the dynamic performanceof the speed estimation while authors of [8] designed ahybrid observer based on current sliding mode for low speedand flux linkage sliding mode for high speed and optimizedusing Adaptive Neural Fuzzy Interference System (ANFIS)and fuzzy PID ampis system has advantages of robustnessand noise reduction is possible through the minimization oftorque ripple and high precision of speed ampe extendedKalman filter is tested in [9] under noisy current mea-surement for direct vector control

ampe observer combines sensed signals with otherknowledge of the control system to produce observed signalsthat are more accurate less expensive to produce and morereliable than sensed signals [3] ampis observed state is used asa feedback signal in the control section

Many techniques can be used to reduce noise sensitivityreducing the observer bandwidth filtering the observeddisturbance or modifying the observer compensatorstructure As discussed in [10] lowering observer bandwidthwill reduce noise susceptibility but it also reduces the abilityof the observer to improve the system For example

reducing observer bandwidth reduces the accuracy of theobserved disturbance signal

In this paper the adaptive full-order Luenberger ob-server is used to estimate stator currents and rotor fluxesampe observed states are then used to estimate the rotor speedbased on the Lyapunov theory

To reduce the noise measurements sensitivity of theLuenberger observer a robust reduced-order observer isdeveloped It improves the response of the Luenbergerobserver in the presence of measurements noise

amperefore the paper is structured as follows after theIntroduction section the mathematical dynamic model isdescribed then the Luenberger observer design and stabilityare detailed ampe design of the reduced observer is shownfollowed by simulations results and conclusion

2 Mathematical Model and ControlTechnique of IM Drives

21 Dynamic Model ampe mathematical dynamic model ofthe IM consists of the differential equations describing theelectromagnetic relationships of the stator and rotor as wellas the equation of motion [1]

In this paper the Γ-model is used It consists of usingonly one leakage inductance LL instead of stator and rotorleakage inductances Lsl and Lrl used in the T-model ampefollowing equations describe the dynamic model of the IM

disx

dt minus

Rs LM + LL( 11138572

+ RrL2M

LMLL LM + LL( 11138571113888 1113889isx +

Rr

LL LM + LL( 1113857Φrx +

p

LL

ωΦry +1

LM

+1

LL

1113888 1113889usx

disy

dt minus

Rs LM + LL( 11138572

+ RrL2M

LMLL LM + LL( 11138571113888 1113889isy minus

p

LL

ωΦrx +Rr

LL LM + LL( 1113857Φry +

1LM

+1

LL

1113888 1113889usy

dΦrx

dt

RrLM

LM + LL

isx minusRr

LM + LL

Φrx minus pωΦry

dΦry

dt

RrLM

LM + LL

isy + pωΦrx minusRr

LM + LL

Φry

dωdt

32

p

JΦsxisy minus Φsyisx1113872 1113873 minus

TL

J

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

22 Vector Control As mentioned before the vector controlis based on imitating the DC motor behavior which allowsthe separated control of the flux and the torque by adjustingrespectively the direct and quadrature component of thestator current

ampis technique is based on orientation of the dq-refer-ence in such a way to eliminate the quadrature componentampus for the Rotor Field-Oriented Control (RFOC) thequadrature component of the flux is considered zeroΦrq 0 so the flux will be carried out entirely on the directcomponent Φr Φrd [11]

ampis control allows the determination of the statorvoltages in the dq-reference frame which attacks the three-phase voltage source inverter (VSI) illustrated in Figure 1 Itis composed of three arms each has two complementaryswitches based on IGBT transistor ampe control of theseswitches is ensured by different modulation techniques ofwhich the PWM is the simplest to implement As shown inFigure 2 the switchesrsquo states are determined from the ref-erence voltage and the high-frequency carrier [12] ampusFigure 3 presents the control structure Indeed the vectorcontrol allows the calculation of stator voltages which are

2 Journal of Electrical and Computer Engineering

used as reference voltages for the modulation techniqueampis allows the generation of the VSI switch states And thenthe inverter voltage output supplies the machine

Different studies have analyzed the harmful impact ofdead times on system performance As known this deadtime is introduced to avoid the simultaneous conduction oftwo power devices (IGBT) in the same arm of the inverterampis solution causes system performance degradation(voltage output drop disturbances and distortions of thesignal appearance of undesirable harmonic componentsand reduction of the fundamental of the voltage) Howevervarious dead time compensation methods are discussed inthe literature Some researches proposed different tech-niques to implement without any changes at the hardwarelevel only the software part is slightly modified

In some cases as presented in [13] it is recommended toadd an observer to the system to compensate the voltagemagnitude which has been affected due to dead times

Authors of [14] propose a dead-time compensationmethod for a vector-controlled induction motor ampismethod includes the effect of dead time turn-onoff time ofswitching devices and voltage drops of switching devicesand freewheeling diodes It allows improvement of thesystem response and suppression of the current distortion inthe motor ampen the dead-time effect is eliminated

Dead-time effects can be also greatly reduced using theproposed method in [15] ampis method is tested for thesensorless vector control applications and it has beenverified that the speed error under different dead timedurations can be reduced Only slight modification of the

controller software is required and thus no additionalhardware is needed in this method ampey used a simpleequation to calculate the real output voltage and thereforethe proposed method can effectively improve the controllerperformance without degrading its operating speed

Moreover it was verified in [16] that the compensationmethod improves the current waveform and improves thestability of the motor operation

3 Sensorless Vector Control UsingAdaptive Observer

31e Full-OrderObserver Luenberger Observer Design andStability Based on the dynamic model of the inductionmotor the full-order Luenberger observer is described bythe following state presentation

_1113954x(t) A middot 1113954x(t) + B middot u(t) + L middot (y minus 1113954y)

1113954y(t) C middot 1113954x(t)

⎧⎨

⎩ (2)

ampe state vector and the state matrix are 1113954x 1113954is1113954Φr1113872 1113873

T

A a1I a2I minus a3J

a4I minus a5I minus a6J1113888 1113889

where a1 minus ((Rs(LM + LL)2 + RrL2M)LMLL(LM + LL))

a2 minus a5LL a3 minus a6LL a4 minus Rra5 a5 minus (RrLM + LL)and a6 minus pω

ampe input and output matrices are

B (1LM) + (1LL) 0

0 (1LM) + (1LL)1113888 1113889

T

C 1 00 11113888 1113889

ampe input and output vectors are u usx usy1113872 1113873T

y isx isy1113872 1113873T

ampe observer gain matrix is L L1I + L2J

L3I + L4J1113888 1113889

To determine the gain matrix L the pole-placementmethod is used ampis technique allows the determination ofthe observer dynamics It consists in calculating the full-order observer eigenvalues which are the characteristicpolynomial roots ampey are given by

det λLOI(A minus LC)( 1113857 λ2LO + ALOλLO + BLOλLO1113872 1113873I

+ CLOλLO + DLO( 1113857J 0(3)

where

ALO L1 minus a1 minus a5BLO a5(a1 minus L1) minus a6L2 + a2L3 minus a2a4 + a3L4CLO a6 + L2DLO a6L1 minus a1a6 minus a5L2 + a2L4 + a3a4 minus a3L3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

In the same manner the IM eigenvalues are given asfollows

det λIMI(A minus LC)( 1113857 λ2IM + AIMλIM + BIMλIM1113872 1113873I

+ CIMλIM + DIM( 1113857J 0(4)

where

AIM minus a1 minus a5BIM a1a5 minus a2a4CIM a6DIM minus a1a6 + a3a4

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

ampe poles of the observer and the motor must satisfy theequation λLO KLλIM where KL gt 1 [17] ampus the iden-tification of both characteristic polynomials (3) and KL lowast(4)

leads to the following results

Vdc

S1 S2 S3

Sprime1 Sprime2 Sprime3

vavbvc

Figure 1 Voltage source inverter structure

Reference signal Carrier signal

Switch state

Figure 2 Pulse-width modulation principle

Journal of Electrical and Computer Engineering 3

L1 KL minus 1( 1113857 minus a1 minus a5( 1113857

L2 KL minus 1( 1113857a6

L3 minus KL minus 1( 1113857Rr

a26 minus a2

5a26 + a2

5

L4 minus KL minus 1( 1113857pLLω

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

ampe poles of the observer are chosen to accelerate itsconvergence But they must stay slow in comparison to themeasurement noise so the constant KL is usually set small

32 Rotor Speed Estimation To determine the rotor speedadaptive mechanism the Lyapunov theorem is used ampistechnique is suitable for determining the stability of nonlineardynamic systems such as IM drives ampe main idea is to in-troduce a generalized energy function called the Lyapunovfunctionwhich is zero at the equilibriumandpositive elsewhereampe equilibrium will be stable if we can show that the Lyapunovfunction decreases along the trajectories of the system [18]ampusthe Lyapunov function is defined as follows [17]

Vω eTe +Δω2

λω (6)

where e X minus 1113954X is the estimation errorIts derivative is defined as _e 1113954A middot e + ΔA middot 1113954X and

ΔA A(ω) minus A(1113954ω) 0 (pLL)ΔωJ

0 pΔωJ1113888 1113889

amperefore the derivative of Vω is given bydVω

dt e

T 1113954AT

+ 1113954A1113874 1113875e minus 2p

LL

1113954ΦrβeIsαminus 1113954ΦrαeIsβ

1113874 1113875Δω

minus 2Δωλω

d1113954ωdt

+ εω

(7)

where εω 2p( 1113954ΦrαeΦrβminus 1113954ΦrβeΦrα

)Δωampen the adaptive law is obtained as

1113954ω Kp1113954ΦrβeIsα

minus 1113954ΦrαeIsβ1113874 1113875 + Ki 1113946 1113954ΦrβeIsα

minus 1113954ΦrαeIsβ1113874 1113875

(8)

33 Noise in the Luenberger Observer ampe Luenberger ob-servers are most effective when the sensor produces limitednoise Indeed they often can exacerbate sensor noise whichis frequently a problem in motion control systems ampisnoise comes from two major sources electromagnetic in-terference (EMI) generated by power converters that is thentransmitted to the control section and resolution limitationsin sensors EMI can be reduced through appropriate wiringpractices and through the selection of components that limitnoise generation whereas resolution noise from sensors ismore difficult to address [19 20]

In this study the position sensor is not used Howeverthe noise could be transmitted via the observer In fact thespeed estimation is based on the current measurementsFigure 4 shows the structure of the designed observer in thepresence of the measurements noise To improve the systemperformance a reduced-order observer is designed

34 Design of the Reduced-Order Observer ampe reduced-order observer aims to filter the measured currents beforereaching the Luenberger observer as can be explained inFigure 5

ampe actual reduced observer tracks only a subset of thestate vector contrary to the first observer referred to as anidentity observer as it tracks the entire state vector It takesthe following form [21]

_1113954xe(t) Ae middot 1113954xe(t) + Be middot u(t) + K middot (y minus 1113954y)

1113954y(t) Ce middot 1113954xe(t)

⎧⎨

⎩ (9)

where the state vector is given by

1113954xe 1113954Φs1113954Φr

1113888 1113889 1113954Φsx + j 1113954Φsy

1113954Φrx + j 1113954Φry

1113888 1113889

And the state matrix is

Ae minus Rse((1LMe) + (1LLe)) RseLLe

RreLLe minus (RreLLe) + jpω1113888 1113889

ampe input and the output matrices are Be 101113888 1113889

Ce (1LMe) + (1LLe) minus (1LLe)( 1113857ampe observer gain matrix is

K Re middot k Rse 00 Rre

1113888 1113889ksx + jksy

krx + jkry1113888 1113889

dq

abc

Vector control

Pulse-width modulation

Va

Vb

Vc

Vlowastsq

Vlowastsd

vp

S1 Sprime1S2 Sprime2S3 Sprime3

Figure 3 Induction machine control


ampe estimated fluxes ared1113954Φsdt us minus Rse((1 + ks)

1113954is minus ksis)

d1113954Φrdt jpω 1113954Φr minus Rre1113954ir

1113896

ampe estimated currents are1113954is ( 1113954ΦsLMe) minus ( 1113954Φr minus 1113954ΦsLLe)1113954ir kr(

1113954is minus is) + ( 1113954Φr minus 1113954ΦsLLe)1113896

As the observer poles are the eigenvalues of the matrix(Ae minus RekCe) then the characteristic polynomial of theobserver is given by

P(s) s2

+ c1s + c2 s2

minus p1 + p2( 1113857s + p1p2 (10)

where p1 and p2 are the characteristic polynomial rootsampe determination of the gain matrix k in the observer is

the same mathematical problem as the problem of deter-mining the feedback matrix in the pole-placement problemampe selection of the observer poles is a compromise betweensensitivity to measurement errors and rapid recovery ofinitial errors A fast observer will converge quickly but it willalso be sensitive to measurement errors ampe resolution is

based on Ackermannrsquos formula k Rminus 1e P(Ae)Θminus 1 0

11113890 1113891 [18

21]

k

1Rse

0

01

Rre

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

A2e + c1Ae + c2I1113872 1113873

LMeLLe Rre + jpω LMe + LLe( 1113857( 1113857

R2re + p2ω2 LMe + LLe( 1113857

2

L2Le + LMeLLe( 1113857 Rre + jpω LMe + LLe( 1113857( 1113857

R2re + p2ω2 LMe + LLe( 1113857

2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(11)

Induction motor(physical system)

+ +

Observer gain+

ndash

+ +

Sensor noiseN (s) Actual

sensor outputZ (s)

Observed outputY (s)

ObservederrorE (s)

MeasuredoutputY (s)

Excitation inputU (s)

Observeddisturbance

Observed stateX (s)

Modeled system

Figure 4 Structure of the full-order Luenberger observer in the presence of sensor noise


+ +

Observer gain+

ndash

+ +

N (s)

Y (s)U (s) Z (s)

Observeddisturbance

Observed stateX (s)

Modeled system

Reduced-orderobserver

Y (s)

Z (s)

ObservederrorE (s)

Figure 5 Proposed structure using the reduced-order observer


where Θ Ce

CeAe

1113888 1113889 is the observability matrix of system(9) ampis matrix is invertible when the system is observableampen the gain matrix is given by

k

c2 LMeLLeRse( 1113857 Rre + jpω LMe + LLe( 1113857( 1113857

R2re + p2ω2 LMe + LLe( 1113857

2 minus 1

LLe LMe + LLe( 1113857Rre( 1113857( 1113857 Rre + jpω LMe + LLe( 1113857( 1113857

R2re + p2ω2 LMe + LLe( 1113857

2 + 1 minus c1 + jpω( 1113857

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(12)

Components of the gain matrix are obtained by iden-tification as follows

ksx LMeLLe( 1113857Rse( 1113857 R c2( 1113857Rre minus I c2( 1113857pω LMe + LLe( 1113857( 1113857

R2re + p2ω2 LMe + LLe( 1113857

2 minus 1

ksy LMeLLeRse( 1113857 I c2( 1113857Rre + R c2( 1113857pω LMe + LLe( 1113857( 1113857

R2re + p2ω2 LMe + LLe( 1113857

2 minus 1

krx LLe LMe + LLe( 1113857Rre( 1113857( 1113857 R c2( 1113857Rre minus I c2( 1113857pω LMe + LLe( 1113857( 1113857

R2re + p2ω2 LMe + LLe( 1113857

2 +1 minusLLe

Rre

I c1( 1113857

kry LLe LMe + LLe( 1113857Rre( 1113857( 1113857 I c2( 1113857Rre + R c2( 1113857pω LMe + LLe( 1113857( 1113857

R2re + p2ω2 LMe + LLe( 1113857

2 minusLLe

Rre

I c1( 1113857pω( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

To summarize the actual study combines the full-orderLuenberger observer and the reduced-order observer toaccurize the response system ampe proposed structure allowsa noise immunity and it is insensitive to parametric vari-ations Figure 6 describes the bloc diagram used to verify theproposed structure

4 Simulation Results

ampe proposed structure is verified usingMATLABSimulinkampe induction motor parameters are shown in Table 1

ampe following simulations are carried out under aparametric variation of the stator resistance (+20 of therated value) ampe rated load torque is applied at 3 s andremoved at 7 s ampe speed is kept constant at 600 rpm

By integrating measurement noise increasing the statorresistance and applying the load torque the system responseis influenced In fact speed oscillations around the finalvalue appear smaller when the reduced-order observer is

used (Figure 7(a)) ampis leads to an error that exceeds 75when only the Luenberger observer is used ampe integrationof the filter reduces this error to 20 (Figure 7(b)) ampeelectromagnetic torque (Figure 7(c)) and q-axis statorcurrent (Figure 7(d)) are affected too especially when therated load torque is applied

Even though the stator resistance is increased the re-duced-order observer maintains the flux orientation(Figures 8(a) and 8(b)) and the error is negligible contraryto the use of Luenberger observer only where the errorreaches 25 (Figure 8(d))

ampe reduced-order observer allows α-axis rotor flux(Figure 9(a)) and the stator currents (Figure 9(b)) to keep theirsinusoidal forms despite the disturbances caused by the noisemeasurements while the Luenberger observer alone does notmanage to reject these disturbances which distort signals

Spectrum analysis shows how the reduced-order ob-server decreases THD and increases the fundamental am-plitude for both current (Figures 10 and 11) and torque(Figures 12 and 13)


Currentcontroller

Speed controller

Fluxcontroller

Field weakening control

Reduced-order observer

Full-order Luenberger observer

+adaptation mechanism

for rotor speed estimation

Rotor speed estimation

Rand

omno

ise

M1 + Trs

LrpMФlowast

Isd

Isd Isq

Isβ

Vsβ

Isα

Vsα VaVbVc

Tlowaste

Ilowastsq Vlowastsq

Vlowastsd

Isq

+

+ ndash

++

ndash

ndashndash

Ф

ω Is

Control law

dq

dq αβ

αβ

αβ

αβ abc

abc

PWM

ωlowast

Figure 6 Sensorless vector control using a cascade structure of the full-order Luenberger and the reduced-order observers

Table 1 Induction motor parameters

T-model Γ-model UnitsStator resistance RT

s 68 Stator resistance Rs 68 ΩRotor resistance RT

r 543 Rotor resistance Rr 677 ΩStator inductances LT

s M + Lsl 03973 Magnetizing inductance LM 03973 HRotor inductances LT

r M + Lrl 03558 Leakage inductance LL 00463 HMutual inductance M 03558

Rs RTs Rr RT

r k2c LM Mkc

LL (Lslkc) + (Lrlk2c) kc M(M + Lsl)

HTotal inertia J 210minus 2 kgm2

Friction coefficient F 2 510minus 4 Nmsradminus 1

Pole pairs number P 2

12001000

800600400200

0

ω (r

pm)

0 1 2 3 4 5 6 7 8 9 10

LuenbergerLuenberger + filter

Reference speed ωlowast

(a)

150

100

50

0

ndash50

ndash100

ndash150

Spee

d er

ror (

)

0 1 2 3 4 5 6 7 8 9 10


(b)

Figure 7 Continued


As known the system control is sensitive to the load torquedisturbance the parametric variation and noise measure-ments All these parameters affect the system response

Indeed the stator resistance and the load torque affectespecially the flux orientation while the noise measurementsdisturb the speed and torque responses

According to the simulations the reduced-order ob-server designed is insensitive to the parametric variation andallows filtering the noise measurements which reduces thetotal harmonic distortion (THD)

To not restrict the study to medium speed and to validatethe proposed structure for a wide range of speeds the

0 1 2 3 4 5 6 7 8 9 100

05

1

15

Reference flux Фlowast


Фrd

(Wb)

(a)

0 1 2 3 4 5 6 7 8 9 10ndash1

ndash05

0

05

1


Фrq

(Wb)

(b)

ndash20

ndash10

0

10

20

30

I sd (A

)

0 1 2 3 4 5 6 7 8 9 10LuenbergerLuenberger + filter

(c)

ndash100

ndash50

0

50

100

Flux

erro

r (

)


(d)

Figure 8 (a) d-axis rotor flux (b) q-axis rotor flux (c) d-axis stator current (d) Flux error

0 1 2 3 4 5 6 7 8 9 10

50

0

ndash50

T e (N

m)


(c)

20

10

ndash10

0

ndash20

I sq (A

)

0 1 2 3 4 5 6 7 8 9 10


(d)

Figure 7 (a) Rotor speed (b) Speed error (c) Electromagnetic torque (d) q-axis stator current


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

15 16 17 18 19 2 21 22 23 24 25ndash1

ndash05

0

05

1


Фrα

(Wb)

(a)

I sa (A

)

0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

15 16 17 18 19 2 21 22 23 24 25ndash15

ndash10

ndash5

0

5

10

15


(b)

Figure 9 (a) α-axis rotor flux (b) Stator current

100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000

Fundamental (50Hz) = 0811 THD = 1245

Figure 10 Spectrum analysis of stator current using only the full-order observer


following simulations show the obtained results at lowspeed

As can be noticed in Figure 14(a) the estimated speedfollows the reference when the filter is used while with theLuenberger observer alone ripples are very great ampe same

effect is observed for the electromagnetic torque inFigure 14(b) Also the flux orientation is no longer main-tained (Figure 14(c)) and the flux error is bigger(Figure 14(d))ampe disturbances appear for both α-axis rotorflux and d-axis stator current in Figure 15

100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000


Figure 12 Spectrum analysis of electromagnetic torque using onlythe full-order observer

100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000


Figure 13 Spectrum analysis of electromagnetic torque using theLuenberger and the reduced-order observer

100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000


Figure 11 Spectrum analysis of stator current using the Luenberger and the reduced-order observer


5 Conclusion

ampe purpose of this study is to design a robust reduced-orderobserver insensitive to the parametric variation and noise

measurements for IM drives ampe control system involves acascade structure that combines two observers ampe full-order adaptive Luenberger observer ensures the sensorlessvector control of the induction motor It allows the

0 1 2 3 4 5 6 7 8 9 10ndash400

ndash200

0

200

400

600Es

timat

ed ω

(rpm

)



(a)

0 1 2 3 4 5 6 7 8 9 10ndash30

ndash20

ndash10

0

10

20

30

T e (N

m)


(b)

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

Фrd

(Wb)



(c)


0 1 2 3 4 5 6 7 8 9 10ndash50

0

50

100

Flux

erro

r (

)

(d)

Figure 14 (a) Estimated speed (b) Electromagnetic torque (c) d-axis rotor flux (d) Flux error

Фrα

(Wb)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

3 32 34 36 38 4 42 44 46 48 5ndash1

ndash05

0

05

1

(a)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

I sa (A

)

3 32 34 36 38 4 42 44 46 48 5ndash15ndash10

ndash505

1015

(b)



estimation of stator current and rotor flux used to design therotor speed adaptive mechanism ampe stability is proved bythe Lyapunov criteria ampis observer is more effective whenthe noise measurement is limited To accurize the systemresponse in the presence of this disturbances including theparametric variation the flux reduced-order observer isinserted between the measured currents and the full-orderobserver It allows the filtering of stator current ampe esti-mated current technique is enhanced using a new techniqueto avoid the impact of the noise in the observer ampespectrum analysis of stator current and electromagnetictorque were presented using the Luenberger and the re-duced-order observer ampe actual structure is insensitive tonoise measurements and parametric variation

Nomenclature

is us Stator current and voltageΦs Φr Stator and rotor fluxω Rotor speedvp vabc Carrier and reference three-phase voltageVdc DC bus voltageS123 S123prime Switch statesTe Tl Electromagnetic torque and load torqueXe Filter parameter1113954X Estimated parameterXlowast References Laplace variableR(X) Real part of XI(X) Imaginary part of X

Data Availability

ampere are no data to provide as part of the supplementaryinformation to the article

Conflicts of Interest

ampe authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] M Cirrincione M Pucci and G Vitale Power Converters andAC Electrical Drives with Linear Neural Networks CRC PressBoca Raton FL USA 2016

[2] T Ramesh A Kumar Panda and S Shiva Kumar ldquoType-2fuzzy logic control based MRAS speed estimator for speedsensorless direct torque and flux control of an inductionmotor driverdquo ISA Transactions vol 57 pp 262ndash275 2015

[3] G Ellis ldquoControl systems and the role of observersrdquo inObservers in Control Systems pp 1ndash4 Academic PressCambridge MA USA 2002

[4] G Ellis ldquoControl-system backgroundrdquo in Observers inControl Systems pp 5ndash39 Academic Press Cambridge MAUSA 2002

[5] G Wang G Zhang and D Xu Position Sensorless ControlTechniques for Permanent Magnet Synchronous MachineDrives Springer Singapore 2020

[6] A A Z Diab A-H M Al-Sayed H H Abbas Mohammedand Y S Mohammed Development of Adaptive Speed

Observers for Induction Machine System StabilizationSpringer Singapore 2020

[7] S Yang X Li Z Xie and X Zhang ldquoA combined speedestimation scheme for indirect vector-controlled inductionmotorsrdquo Electrical Engineering vol 100 no 4 pp 2243ndash22522018

[8] L S Murugan and P Maruthupandi ldquoSensorless speedcontrol of 64mdashpole switched reluctance motor with ANFISand fuzzymdashPIDmdashbased hybrid observerrdquo Electrical Engi-neering vol 102 no 2 pp 831ndash844 2020

[9] Y Mederharhet O Bennis F Benchabane A Titaouine andA Guettaf ldquoCharacteristics of speed sensorless vector con-trolled induction motor with high efficiency taking core lossinto accountrdquo Procedia Computer Science vol 83 pp 824ndash831 2016

[10] G Ellis ldquoNoise in the luenberger observerrdquo in Observers inControl Systems pp 141ndash172 Academic Press CambridgeMA USA 2002

[11] K Zeb ldquoAdaptive fuzzy logic controller for indirect fieldoriented controlled induction motorrdquo in Proceedings of the2019 2nd International Conference on Computing Mathe-matics and Engineering Technologies (iCoMET) pp 1ndash6Sukkur Pakistan 2019

[12] Z Guo J Zhang Z Sun and C Zheng ldquoIndirect field ori-ented control of three-phase induction motor based oncurrent-source inverterrdquo Procedia Engineering vol 174pp 588ndash594 2017

[13] H-S Kim H-T Moon and M-J Youn ldquoOn-line dead-timecompensation method using disturbance observerrdquo IEEETransactions on Power Electronics vol 18 no 6 pp 1336ndash1345 2003

[14] R Guedouani B Fiala and M S Boucherit ldquoImplementationof multi-carrier PWM using a DSP TMS320F28335 Appli-cation to series multicellular single-phase inverterrdquo in Pro-ceedings of the 2017 10th International Conference on Electricaland Electronics Engineering (ELECO) Bursa Turkey December2017

[15] Y-H Liu and C-L Chen ldquoNovel dead time compensationmethod for induction motor drives using space vectormodulationrdquo IEE ProceedingsmdashElectric Power Applicationsvol 145 no 4 pp 387ndash392 1998

[16] Z Miao J Wei T Guo and M Zheng ldquoDead-time com-pensation method based on field oriented control strategyrdquoIOP Conference Series Earth and Environmental Sciencevol 358 no 4 pp 0ndash8 2019

[17] H Kubota K Matsuse and T Nakano ldquoDSP-based speedadaptive flux observer of induction motorrdquo IEEE Transactionson Industry Applications vol 29 no 2 pp 344ndash348 1993

[18] K Astrom and B Wittenmark Computer Controlled Systemseory and Design Prentice-Hall Upper Saddle River NJUSA 1997

[19] G Ellis ldquoUsing the luenberger observer in motion controlrdquo inControl System Design Guide pp 407ndash429 Butterworth-Heinemann Oxford UK 4th edition 2012

[20] D Xu B Wang G Zhang G Wang and Y Yu ldquoA review ofsensorless control methods for AC motor drivesrdquo CESTransactions on Electrical Machines and Systems vol 2 no 1pp 104ndash115 2019

[21] B Peterson Induction Machine Speed Estima-tionmdashObservations on Observers Lund Institute of Technol-ogy-Sweden Lund Sweden 1996


based on mathematical models of the IM and are simple androbust against disturbances [2] ampese techniques includeopen-loop estimators the Kalman filter the Luenbergerobserver and Model Reference Adaptive System [5 6]

Authors of [7] combined MRAS and sliding mode forindirect vector control to improve the dynamic performanceof the speed estimation while authors of [8] designed ahybrid observer based on current sliding mode for low speedand flux linkage sliding mode for high speed and optimizedusing Adaptive Neural Fuzzy Interference System (ANFIS)and fuzzy PID ampis system has advantages of robustnessand noise reduction is possible through the minimization oftorque ripple and high precision of speed ampe extendedKalman filter is tested in [9] under noisy current mea-surement for direct vector control

ampe observer combines sensed signals with otherknowledge of the control system to produce observed signalsthat are more accurate less expensive to produce and morereliable than sensed signals [3] ampis observed state is used asa feedback signal in the control section

Many techniques can be used to reduce noise sensitivityreducing the observer bandwidth filtering the observeddisturbance or modifying the observer compensatorstructure As discussed in [10] lowering observer bandwidthwill reduce noise susceptibility but it also reduces the abilityof the observer to improve the system For example

reducing observer bandwidth reduces the accuracy of theobserved disturbance signal

In this paper the adaptive full-order Luenberger ob-server is used to estimate stator currents and rotor fluxesampe observed states are then used to estimate the rotor speedbased on the Lyapunov theory

To reduce the noise measurements sensitivity of theLuenberger observer a robust reduced-order observer isdeveloped It improves the response of the Luenbergerobserver in the presence of measurements noise

amperefore the paper is structured as follows after theIntroduction section the mathematical dynamic model isdescribed then the Luenberger observer design and stabilityare detailed ampe design of the reduced observer is shownfollowed by simulations results and conclusion

2 Mathematical Model and ControlTechnique of IM Drives

21 Dynamic Model ampe mathematical dynamic model ofthe IM consists of the differential equations describing theelectromagnetic relationships of the stator and rotor as wellas the equation of motion [1]

In this paper the Γ-model is used It consists of usingonly one leakage inductance LL instead of stator and rotorleakage inductances Lsl and Lrl used in the T-model ampefollowing equations describe the dynamic model of the IM

disx

dt minus

Rs LM + LL( 11138572

+ RrL2M

LMLL LM + LL( 11138571113888 1113889isx +

Rr

LL LM + LL( 1113857Φrx +

p

LL

ωΦry +1

LM

+1

LL

1113888 1113889usx

disy

dt minus

Rs LM + LL( 11138572

+ RrL2M

LMLL LM + LL( 11138571113888 1113889isy minus

p

LL

ωΦrx +Rr

LL LM + LL( 1113857Φry +

1LM

+1

LL

1113888 1113889usy

dΦrx

dt

RrLM

LM + LL

isx minusRr

LM + LL

Φrx minus pωΦry

dΦry

dt

RrLM

LM + LL

isy + pωΦrx minusRr

LM + LL

Φry

dωdt

32

p

JΦsxisy minus Φsyisx1113872 1113873 minus

TL

J

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

22 Vector Control As mentioned before the vector controlis based on imitating the DC motor behavior which allowsthe separated control of the flux and the torque by adjustingrespectively the direct and quadrature component of thestator current

ampis technique is based on orientation of the dq-refer-ence in such a way to eliminate the quadrature componentampus for the Rotor Field-Oriented Control (RFOC) thequadrature component of the flux is considered zeroΦrq 0 so the flux will be carried out entirely on the directcomponent Φr Φrd [11]

ampis control allows the determination of the statorvoltages in the dq-reference frame which attacks the three-phase voltage source inverter (VSI) illustrated in Figure 1 Itis composed of three arms each has two complementaryswitches based on IGBT transistor ampe control of theseswitches is ensured by different modulation techniques ofwhich the PWM is the simplest to implement As shown inFigure 2 the switchesrsquo states are determined from the ref-erence voltage and the high-frequency carrier [12] ampusFigure 3 presents the control structure Indeed the vectorcontrol allows the calculation of stator voltages which are












1113954y(t) C middot 1113954x(t)

⎧⎨

⎩ (2)


T

A a1I a2I minus a3J





B (1LM) + (1LL) 0

0 (1LM) + (1LL)1113888 1113889

T

C 1 00 11113888 1113889


y isx isy1113872 1113873T


L3I + L4J1113888 1113889



+ CLOλLO + DLO( 1113857J 0(3)

where


⎧⎪⎪⎪⎨

⎪⎪⎪⎩



+ CIMλIM + DIM( 1113857J 0(4)

where


⎧⎪⎪⎪⎨

⎪⎪⎪⎩



Vdc

S1 S2 S3


vavbvc



Switch state




L2 KL minus 1( 1113857a6


a26 minus a2

5a26 + a2

5


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)



Vω eTe +Δω2

λω (6)



0 pΔωJ1113888 1113889


dt e

T 1113954AT

+ 1113954A1113874 1113875e minus 2p

LL


1113874 1113875Δω

minus 2Δωλω

d1113954ωdt

+ εω

(7)





minus 1113954ΦrαeIsβ1113874 1113875

(8)






1113954y(t) Ce middot 1113954xe(t)

⎧⎨

⎩ (9)


1113954xe 1113954Φs1113954Φr

1113888 1113889 1113954Φsx + j 1113954Φsy

1113954Φrx + j 1113954Φry

1113888 1113889







1113888 1113889ksx + jksy

krx + jkry1113888 1113889

dq

abc

Vector control


Va

Vb

Vc

Vlowastsq

Vlowastsd

vp







1113896




P(s) s2

+ c1s + c2 s2

minus p1 + p2( 1113857s + p1p2 (10)




11113890 1113891 [18

21]

k

1Rse

0

01

Rre

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

A2e + c1Ae + c2I1113872 1113873


R2re + p2ω2 LMe + LLe( 1113857

2


R2re + p2ω2 LMe + LLe( 1113857

2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(11)


+ +

Observer gain+

ndash

+ +


sensor outputZ (s)


ObservederrorE (s)

MeasuredoutputY (s)


Observeddisturbance

Observed stateX (s)

Modeled system



+ +

Observer gain+

ndash

+ +

N (s)

Y (s)U (s) Z (s)

Observeddisturbance

Observed stateX (s)

Modeled system


Y (s)

Z (s)

ObservederrorE (s)



where Θ Ce

CeAe


k


R2re + p2ω2 LMe + LLe( 1113857

2 minus 1


R2re + p2ω2 LMe + LLe( 1113857

2 + 1 minus c1 + jpω( 1113857

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(12)



R2re + p2ω2 LMe + LLe( 1113857

2 minus 1


R2re + p2ω2 LMe + LLe( 1113857

2 minus 1


R2re + p2ω2 LMe + LLe( 1113857

2 +1 minusLLe

Rre

I c1( 1113857


R2re + p2ω2 LMe + LLe( 1113857

2 minusLLe

Rre

I c1( 1113857pω( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)











Currentcontroller

Speed controller

Fluxcontroller







Rand

omno

ise

M1 + Trs

LrpMФlowast

Isd

Isd Isq

Isβ

Vsβ

Isα

Vsα VaVbVc

Tlowaste

Ilowastsq Vlowastsq

Vlowastsd

Isq

+

+ ndash

++

ndash

ndashndash

Ф

ω Is

Control law

dq

dq αβ

αβ

αβ

αβ abc

abc

PWM

ωlowast








Rs RTs Rr RT

r k2c LM Mkc





12001000

800600400200

0

ω (r

pm)

0 1 2 3 4 5 6 7 8 9 10



(a)

150

100

50

0

ndash50

ndash100

ndash150

Spee

d er

ror (

)

0 1 2 3 4 5 6 7 8 9 10


(b)

Figure 7 Continued






0 1 2 3 4 5 6 7 8 9 100

05

1

15



Фrd

(Wb)

(a)

0 1 2 3 4 5 6 7 8 9 10ndash1

ndash05

0

05

1


Фrq

(Wb)

(b)

ndash20

ndash10

0

10

20

30

I sd (A

)


(c)

ndash100

ndash50

0

50

100

Flux

erro

r (

)


(d)


0 1 2 3 4 5 6 7 8 9 10

50

0

ndash50

T e (N

m)


(c)

20

10

ndash10

0

ndash20

I sq (A

)

0 1 2 3 4 5 6 7 8 9 10


(d)



0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

15 16 17 18 19 2 21 22 23 24 25ndash1

ndash05

0

05

1


Фrα

(Wb)

(a)

I sa (A

)

0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

15 16 17 18 19 2 21 22 23 24 25ndash15

ndash10

ndash5

0

5

10

15


(b)


100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000







100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000




5 Conclusion



0 1 2 3 4 5 6 7 8 9 10ndash400

ndash200

0

200

400

600Es

timat

ed ω

(rpm

)



(a)

0 1 2 3 4 5 6 7 8 9 10ndash30

ndash20

ndash10

0

10

20

30

T e (N

m)


(b)

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

Фrd

(Wb)



(c)


0 1 2 3 4 5 6 7 8 9 10ndash50

0

50

100

Flux

erro

r (

)

(d)


Фrα

(Wb)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

3 32 34 36 38 4 42 44 46 48 5ndash1

ndash05

0

05

1

(a)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

I sa (A

)

3 32 34 36 38 4 42 44 46 48 5ndash15ndash10

ndash505

1015

(b)




Nomenclature


Data Availability




References


































1113954y(t) C middot 1113954x(t)

⎧⎨

⎩ (2)


T

A a1I a2I minus a3J





B (1LM) + (1LL) 0

0 (1LM) + (1LL)1113888 1113889

T

C 1 00 11113888 1113889


y isx isy1113872 1113873T


L3I + L4J1113888 1113889



+ CLOλLO + DLO( 1113857J 0(3)

where


⎧⎪⎪⎪⎨

⎪⎪⎪⎩



+ CIMλIM + DIM( 1113857J 0(4)

where


⎧⎪⎪⎪⎨

⎪⎪⎪⎩



Vdc

S1 S2 S3


vavbvc



Switch state




L2 KL minus 1( 1113857a6


a26 minus a2

5a26 + a2

5


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)



Vω eTe +Δω2

λω (6)



0 pΔωJ1113888 1113889


dt e

T 1113954AT

+ 1113954A1113874 1113875e minus 2p

LL


1113874 1113875Δω

minus 2Δωλω

d1113954ωdt

+ εω

(7)





minus 1113954ΦrαeIsβ1113874 1113875

(8)






1113954y(t) Ce middot 1113954xe(t)

⎧⎨

⎩ (9)


1113954xe 1113954Φs1113954Φr

1113888 1113889 1113954Φsx + j 1113954Φsy

1113954Φrx + j 1113954Φry

1113888 1113889







1113888 1113889ksx + jksy

krx + jkry1113888 1113889

dq

abc

Vector control


Va

Vb

Vc

Vlowastsq

Vlowastsd

vp







1113896




P(s) s2

+ c1s + c2 s2

minus p1 + p2( 1113857s + p1p2 (10)




11113890 1113891 [18

21]

k

1Rse

0

01

Rre

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

A2e + c1Ae + c2I1113872 1113873


R2re + p2ω2 LMe + LLe( 1113857

2


R2re + p2ω2 LMe + LLe( 1113857

2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(11)


+ +

Observer gain+

ndash

+ +


sensor outputZ (s)


ObservederrorE (s)

MeasuredoutputY (s)


Observeddisturbance

Observed stateX (s)

Modeled system



+ +

Observer gain+

ndash

+ +

N (s)

Y (s)U (s) Z (s)

Observeddisturbance

Observed stateX (s)

Modeled system


Y (s)

Z (s)

ObservederrorE (s)



where Θ Ce

CeAe


k


R2re + p2ω2 LMe + LLe( 1113857

2 minus 1


R2re + p2ω2 LMe + LLe( 1113857

2 + 1 minus c1 + jpω( 1113857

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(12)



R2re + p2ω2 LMe + LLe( 1113857

2 minus 1


R2re + p2ω2 LMe + LLe( 1113857

2 minus 1


R2re + p2ω2 LMe + LLe( 1113857

2 +1 minusLLe

Rre

I c1( 1113857


R2re + p2ω2 LMe + LLe( 1113857

2 minusLLe

Rre

I c1( 1113857pω( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)











Currentcontroller

Speed controller

Fluxcontroller







Rand

omno

ise

M1 + Trs

LrpMФlowast

Isd

Isd Isq

Isβ

Vsβ

Isα

Vsα VaVbVc

Tlowaste

Ilowastsq Vlowastsq

Vlowastsd

Isq

+

+ ndash

++

ndash

ndashndash

Ф

ω Is

Control law

dq

dq αβ

αβ

αβ

αβ abc

abc

PWM

ωlowast








Rs RTs Rr RT

r k2c LM Mkc





12001000

800600400200

0

ω (r

pm)

0 1 2 3 4 5 6 7 8 9 10



(a)

150

100

50

0

ndash50

ndash100

ndash150

Spee

d er

ror (

)

0 1 2 3 4 5 6 7 8 9 10


(b)

Figure 7 Continued






0 1 2 3 4 5 6 7 8 9 100

05

1

15



Фrd

(Wb)

(a)

0 1 2 3 4 5 6 7 8 9 10ndash1

ndash05

0

05

1


Фrq

(Wb)

(b)

ndash20

ndash10

0

10

20

30

I sd (A

)


(c)

ndash100

ndash50

0

50

100

Flux

erro

r (

)


(d)


0 1 2 3 4 5 6 7 8 9 10

50

0

ndash50

T e (N

m)


(c)

20

10

ndash10

0

ndash20

I sq (A

)

0 1 2 3 4 5 6 7 8 9 10


(d)



0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

15 16 17 18 19 2 21 22 23 24 25ndash1

ndash05

0

05

1


Фrα

(Wb)

(a)

I sa (A

)

0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

15 16 17 18 19 2 21 22 23 24 25ndash15

ndash10

ndash5

0

5

10

15


(b)


100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000







100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000




5 Conclusion



0 1 2 3 4 5 6 7 8 9 10ndash400

ndash200

0

200

400

600Es

timat

ed ω

(rpm

)



(a)

0 1 2 3 4 5 6 7 8 9 10ndash30

ndash20

ndash10

0

10

20

30

T e (N

m)


(b)

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

Фrd

(Wb)



(c)


0 1 2 3 4 5 6 7 8 9 10ndash50

0

50

100

Flux

erro

r (

)

(d)


Фrα

(Wb)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

3 32 34 36 38 4 42 44 46 48 5ndash1

ndash05

0

05

1

(a)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

I sa (A

)

3 32 34 36 38 4 42 44 46 48 5ndash15ndash10

ndash505

1015

(b)




Nomenclature


Data Availability




References

























L2 KL minus 1( 1113857a6


a26 minus a2

5a26 + a2

5


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)



Vω eTe +Δω2

λω (6)



0 pΔωJ1113888 1113889


dt e

T 1113954AT

+ 1113954A1113874 1113875e minus 2p

LL


1113874 1113875Δω

minus 2Δωλω

d1113954ωdt

+ εω

(7)





minus 1113954ΦrαeIsβ1113874 1113875

(8)






1113954y(t) Ce middot 1113954xe(t)

⎧⎨

⎩ (9)


1113954xe 1113954Φs1113954Φr

1113888 1113889 1113954Φsx + j 1113954Φsy

1113954Φrx + j 1113954Φry

1113888 1113889







1113888 1113889ksx + jksy

krx + jkry1113888 1113889

dq

abc

Vector control


Va

Vb

Vc

Vlowastsq

Vlowastsd

vp







1113896




P(s) s2

+ c1s + c2 s2

minus p1 + p2( 1113857s + p1p2 (10)




11113890 1113891 [18

21]

k

1Rse

0

01

Rre

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

A2e + c1Ae + c2I1113872 1113873


R2re + p2ω2 LMe + LLe( 1113857

2


R2re + p2ω2 LMe + LLe( 1113857

2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(11)


+ +

Observer gain+

ndash

+ +


sensor outputZ (s)


ObservederrorE (s)

MeasuredoutputY (s)


Observeddisturbance

Observed stateX (s)

Modeled system



+ +

Observer gain+

ndash

+ +

N (s)

Y (s)U (s) Z (s)

Observeddisturbance

Observed stateX (s)

Modeled system


Y (s)

Z (s)

ObservederrorE (s)



where Θ Ce

CeAe


k


R2re + p2ω2 LMe + LLe( 1113857

2 minus 1


R2re + p2ω2 LMe + LLe( 1113857

2 + 1 minus c1 + jpω( 1113857

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(12)



R2re + p2ω2 LMe + LLe( 1113857

2 minus 1


R2re + p2ω2 LMe + LLe( 1113857

2 minus 1


R2re + p2ω2 LMe + LLe( 1113857

2 +1 minusLLe

Rre

I c1( 1113857


R2re + p2ω2 LMe + LLe( 1113857

2 minusLLe

Rre

I c1( 1113857pω( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)











Currentcontroller

Speed controller

Fluxcontroller







Rand

omno

ise

M1 + Trs

LrpMФlowast

Isd

Isd Isq

Isβ

Vsβ

Isα

Vsα VaVbVc

Tlowaste

Ilowastsq Vlowastsq

Vlowastsd

Isq

+

+ ndash

++

ndash

ndashndash

Ф

ω Is

Control law

dq

dq αβ

αβ

αβ

αβ abc

abc

PWM

ωlowast








Rs RTs Rr RT

r k2c LM Mkc





12001000

800600400200

0

ω (r

pm)

0 1 2 3 4 5 6 7 8 9 10



(a)

150

100

50

0

ndash50

ndash100

ndash150

Spee

d er

ror (

)

0 1 2 3 4 5 6 7 8 9 10


(b)

Figure 7 Continued






0 1 2 3 4 5 6 7 8 9 100

05

1

15



Фrd

(Wb)

(a)

0 1 2 3 4 5 6 7 8 9 10ndash1

ndash05

0

05

1


Фrq

(Wb)

(b)

ndash20

ndash10

0

10

20

30

I sd (A

)


(c)

ndash100

ndash50

0

50

100

Flux

erro

r (

)


(d)


0 1 2 3 4 5 6 7 8 9 10

50

0

ndash50

T e (N

m)


(c)

20

10

ndash10

0

ndash20

I sq (A

)

0 1 2 3 4 5 6 7 8 9 10


(d)



0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

15 16 17 18 19 2 21 22 23 24 25ndash1

ndash05

0

05

1


Фrα

(Wb)

(a)

I sa (A

)

0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

15 16 17 18 19 2 21 22 23 24 25ndash15

ndash10

ndash5

0

5

10

15


(b)


100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000







100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000




5 Conclusion



0 1 2 3 4 5 6 7 8 9 10ndash400

ndash200

0

200

400

600Es

timat

ed ω

(rpm

)



(a)

0 1 2 3 4 5 6 7 8 9 10ndash30

ndash20

ndash10

0

10

20

30

T e (N

m)


(b)

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

Фrd

(Wb)



(c)


0 1 2 3 4 5 6 7 8 9 10ndash50

0

50

100

Flux

erro

r (

)

(d)


Фrα

(Wb)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

3 32 34 36 38 4 42 44 46 48 5ndash1

ndash05

0

05

1

(a)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

I sa (A

)

3 32 34 36 38 4 42 44 46 48 5ndash15ndash10

ndash505

1015

(b)




Nomenclature


Data Availability




References



























1113896




P(s) s2

+ c1s + c2 s2

minus p1 + p2( 1113857s + p1p2 (10)




11113890 1113891 [18

21]

k

1Rse

0

01

Rre

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

A2e + c1Ae + c2I1113872 1113873


R2re + p2ω2 LMe + LLe( 1113857

2


R2re + p2ω2 LMe + LLe( 1113857

2

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(11)


+ +

Observer gain+

ndash

+ +


sensor outputZ (s)


ObservederrorE (s)

MeasuredoutputY (s)


Observeddisturbance

Observed stateX (s)

Modeled system



+ +

Observer gain+

ndash

+ +

N (s)

Y (s)U (s) Z (s)

Observeddisturbance

Observed stateX (s)

Modeled system


Y (s)

Z (s)

ObservederrorE (s)



where Θ Ce

CeAe


k


R2re + p2ω2 LMe + LLe( 1113857

2 minus 1


R2re + p2ω2 LMe + LLe( 1113857

2 + 1 minus c1 + jpω( 1113857

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(12)



R2re + p2ω2 LMe + LLe( 1113857

2 minus 1


R2re + p2ω2 LMe + LLe( 1113857

2 minus 1


R2re + p2ω2 LMe + LLe( 1113857

2 +1 minusLLe

Rre

I c1( 1113857


R2re + p2ω2 LMe + LLe( 1113857

2 minusLLe

Rre

I c1( 1113857pω( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)











Currentcontroller

Speed controller

Fluxcontroller







Rand

omno

ise

M1 + Trs

LrpMФlowast

Isd

Isd Isq

Isβ

Vsβ

Isα

Vsα VaVbVc

Tlowaste

Ilowastsq Vlowastsq

Vlowastsd

Isq

+

+ ndash

++

ndash

ndashndash

Ф

ω Is

Control law

dq

dq αβ

αβ

αβ

αβ abc

abc

PWM

ωlowast








Rs RTs Rr RT

r k2c LM Mkc





12001000

800600400200

0

ω (r

pm)

0 1 2 3 4 5 6 7 8 9 10



(a)

150

100

50

0

ndash50

ndash100

ndash150

Spee

d er

ror (

)

0 1 2 3 4 5 6 7 8 9 10


(b)

Figure 7 Continued






0 1 2 3 4 5 6 7 8 9 100

05

1

15



Фrd

(Wb)

(a)

0 1 2 3 4 5 6 7 8 9 10ndash1

ndash05

0

05

1


Фrq

(Wb)

(b)

ndash20

ndash10

0

10

20

30

I sd (A

)


(c)

ndash100

ndash50

0

50

100

Flux

erro

r (

)


(d)


0 1 2 3 4 5 6 7 8 9 10

50

0

ndash50

T e (N

m)


(c)

20

10

ndash10

0

ndash20

I sq (A

)

0 1 2 3 4 5 6 7 8 9 10


(d)



0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

15 16 17 18 19 2 21 22 23 24 25ndash1

ndash05

0

05

1


Фrα

(Wb)

(a)

I sa (A

)

0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

15 16 17 18 19 2 21 22 23 24 25ndash15

ndash10

ndash5

0

5

10

15


(b)


100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000







100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000




5 Conclusion



0 1 2 3 4 5 6 7 8 9 10ndash400

ndash200

0

200

400

600Es

timat

ed ω

(rpm

)



(a)

0 1 2 3 4 5 6 7 8 9 10ndash30

ndash20

ndash10

0

10

20

30

T e (N

m)


(b)

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

Фrd

(Wb)



(c)


0 1 2 3 4 5 6 7 8 9 10ndash50

0

50

100

Flux

erro

r (

)

(d)


Фrα

(Wb)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

3 32 34 36 38 4 42 44 46 48 5ndash1

ndash05

0

05

1

(a)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

I sa (A

)

3 32 34 36 38 4 42 44 46 48 5ndash15ndash10

ndash505

1015

(b)




Nomenclature


Data Availability




References
























where Θ Ce

CeAe


k


R2re + p2ω2 LMe + LLe( 1113857

2 minus 1


R2re + p2ω2 LMe + LLe( 1113857

2 + 1 minus c1 + jpω( 1113857

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(12)



R2re + p2ω2 LMe + LLe( 1113857

2 minus 1


R2re + p2ω2 LMe + LLe( 1113857

2 minus 1


R2re + p2ω2 LMe + LLe( 1113857

2 +1 minusLLe

Rre

I c1( 1113857


R2re + p2ω2 LMe + LLe( 1113857

2 minusLLe

Rre

I c1( 1113857pω( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)











Currentcontroller

Speed controller

Fluxcontroller







Rand

omno

ise

M1 + Trs

LrpMФlowast

Isd

Isd Isq

Isβ

Vsβ

Isα

Vsα VaVbVc

Tlowaste

Ilowastsq Vlowastsq

Vlowastsd

Isq

+

+ ndash

++

ndash

ndashndash

Ф

ω Is

Control law

dq

dq αβ

αβ

αβ

αβ abc

abc

PWM

ωlowast








Rs RTs Rr RT

r k2c LM Mkc





12001000

800600400200

0

ω (r

pm)

0 1 2 3 4 5 6 7 8 9 10



(a)

150

100

50

0

ndash50

ndash100

ndash150

Spee

d er

ror (

)

0 1 2 3 4 5 6 7 8 9 10


(b)

Figure 7 Continued






0 1 2 3 4 5 6 7 8 9 100

05

1

15



Фrd

(Wb)

(a)

0 1 2 3 4 5 6 7 8 9 10ndash1

ndash05

0

05

1


Фrq

(Wb)

(b)

ndash20

ndash10

0

10

20

30

I sd (A

)


(c)

ndash100

ndash50

0

50

100

Flux

erro

r (

)


(d)


0 1 2 3 4 5 6 7 8 9 10

50

0

ndash50

T e (N

m)


(c)

20

10

ndash10

0

ndash20

I sq (A

)

0 1 2 3 4 5 6 7 8 9 10


(d)



0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

15 16 17 18 19 2 21 22 23 24 25ndash1

ndash05

0

05

1


Фrα

(Wb)

(a)

I sa (A

)

0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

15 16 17 18 19 2 21 22 23 24 25ndash15

ndash10

ndash5

0

5

10

15


(b)


100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000







100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000




5 Conclusion



0 1 2 3 4 5 6 7 8 9 10ndash400

ndash200

0

200

400

600Es

timat

ed ω

(rpm

)



(a)

0 1 2 3 4 5 6 7 8 9 10ndash30

ndash20

ndash10

0

10

20

30

T e (N

m)


(b)

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

Фrd

(Wb)



(c)


0 1 2 3 4 5 6 7 8 9 10ndash50

0

50

100

Flux

erro

r (

)

(d)


Фrα

(Wb)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

3 32 34 36 38 4 42 44 46 48 5ndash1

ndash05

0

05

1

(a)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

I sa (A

)

3 32 34 36 38 4 42 44 46 48 5ndash15ndash10

ndash505

1015

(b)




Nomenclature


Data Availability




References
























Currentcontroller

Speed controller

Fluxcontroller







Rand

omno

ise

M1 + Trs

LrpMФlowast

Isd

Isd Isq

Isβ

Vsβ

Isα

Vsα VaVbVc

Tlowaste

Ilowastsq Vlowastsq

Vlowastsd

Isq

+

+ ndash

++

ndash

ndashndash

Ф

ω Is

Control law

dq

dq αβ

αβ

αβ

αβ abc

abc

PWM

ωlowast








Rs RTs Rr RT

r k2c LM Mkc





12001000

800600400200

0

ω (r

pm)

0 1 2 3 4 5 6 7 8 9 10



(a)

150

100

50

0

ndash50

ndash100

ndash150

Spee

d er

ror (

)

0 1 2 3 4 5 6 7 8 9 10


(b)

Figure 7 Continued






0 1 2 3 4 5 6 7 8 9 100

05

1

15



Фrd

(Wb)

(a)

0 1 2 3 4 5 6 7 8 9 10ndash1

ndash05

0

05

1


Фrq

(Wb)

(b)

ndash20

ndash10

0

10

20

30

I sd (A

)


(c)

ndash100

ndash50

0

50

100

Flux

erro

r (

)


(d)


0 1 2 3 4 5 6 7 8 9 10

50

0

ndash50

T e (N

m)


(c)

20

10

ndash10

0

ndash20

I sq (A

)

0 1 2 3 4 5 6 7 8 9 10


(d)



0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

15 16 17 18 19 2 21 22 23 24 25ndash1

ndash05

0

05

1


Фrα

(Wb)

(a)

I sa (A

)

0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

15 16 17 18 19 2 21 22 23 24 25ndash15

ndash10

ndash5

0

5

10

15


(b)


100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000







100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000




5 Conclusion



0 1 2 3 4 5 6 7 8 9 10ndash400

ndash200

0

200

400

600Es

timat

ed ω

(rpm

)



(a)

0 1 2 3 4 5 6 7 8 9 10ndash30

ndash20

ndash10

0

10

20

30

T e (N

m)


(b)

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

Фrd

(Wb)



(c)


0 1 2 3 4 5 6 7 8 9 10ndash50

0

50

100

Flux

erro

r (

)

(d)


Фrα

(Wb)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

3 32 34 36 38 4 42 44 46 48 5ndash1

ndash05

0

05

1

(a)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

I sa (A

)

3 32 34 36 38 4 42 44 46 48 5ndash15ndash10

ndash505

1015

(b)




Nomenclature


Data Availability




References




























0 1 2 3 4 5 6 7 8 9 100

05

1

15



Фrd

(Wb)

(a)

0 1 2 3 4 5 6 7 8 9 10ndash1

ndash05

0

05

1


Фrq

(Wb)

(b)

ndash20

ndash10

0

10

20

30

I sd (A

)


(c)

ndash100

ndash50

0

50

100

Flux

erro

r (

)


(d)


0 1 2 3 4 5 6 7 8 9 10

50

0

ndash50

T e (N

m)


(c)

20

10

ndash10

0

ndash20

I sq (A

)

0 1 2 3 4 5 6 7 8 9 10


(d)



0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

15 16 17 18 19 2 21 22 23 24 25ndash1

ndash05

0

05

1


Фrα

(Wb)

(a)

I sa (A

)

0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

15 16 17 18 19 2 21 22 23 24 25ndash15

ndash10

ndash5

0

5

10

15


(b)


100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000







100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000




5 Conclusion



0 1 2 3 4 5 6 7 8 9 10ndash400

ndash200

0

200

400

600Es

timat

ed ω

(rpm

)



(a)

0 1 2 3 4 5 6 7 8 9 10ndash30

ndash20

ndash10

0

10

20

30

T e (N

m)


(b)

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

Фrd

(Wb)



(c)


0 1 2 3 4 5 6 7 8 9 10ndash50

0

50

100

Flux

erro

r (

)

(d)


Фrα

(Wb)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

3 32 34 36 38 4 42 44 46 48 5ndash1

ndash05

0

05

1

(a)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

I sa (A

)

3 32 34 36 38 4 42 44 46 48 5ndash15ndash10

ndash505

1015

(b)




Nomenclature


Data Availability




References
























0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

15 16 17 18 19 2 21 22 23 24 25ndash1

ndash05

0

05

1


Фrα

(Wb)

(a)

I sa (A

)

0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

15 16 17 18 19 2 21 22 23 24 25ndash15

ndash10

ndash5

0

5

10

15


(b)


100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000







100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000




5 Conclusion



0 1 2 3 4 5 6 7 8 9 10ndash400

ndash200

0

200

400

600Es

timat

ed ω

(rpm

)



(a)

0 1 2 3 4 5 6 7 8 9 10ndash30

ndash20

ndash10

0

10

20

30

T e (N

m)


(b)

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

Фrd

(Wb)



(c)


0 1 2 3 4 5 6 7 8 9 10ndash50

0

50

100

Flux

erro

r (

)

(d)


Фrα

(Wb)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

3 32 34 36 38 4 42 44 46 48 5ndash1

ndash05

0

05

1

(a)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

I sa (A

)

3 32 34 36 38 4 42 44 46 48 5ndash15ndash10

ndash505

1015

(b)




Nomenclature


Data Availability




References



























100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000



100

90

80

70

60

50

40

30

20

10

0

Mag

( o

f fun

dam

enta

l)

0 1000 2000 3000 4000 5000Frequency (Hz)

6000 7000 8000 9000 10000




5 Conclusion



0 1 2 3 4 5 6 7 8 9 10ndash400

ndash200

0

200

400

600Es

timat

ed ω

(rpm

)



(a)

0 1 2 3 4 5 6 7 8 9 10ndash30

ndash20

ndash10

0

10

20

30

T e (N

m)


(b)

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

Фrd

(Wb)



(c)


0 1 2 3 4 5 6 7 8 9 10ndash50

0

50

100

Flux

erro

r (

)

(d)


Фrα

(Wb)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

3 32 34 36 38 4 42 44 46 48 5ndash1

ndash05

0

05

1

(a)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

I sa (A

)

3 32 34 36 38 4 42 44 46 48 5ndash15ndash10

ndash505

1015

(b)




Nomenclature


Data Availability




References
























5 Conclusion



0 1 2 3 4 5 6 7 8 9 10ndash400

ndash200

0

200

400

600Es

timat

ed ω

(rpm

)



(a)

0 1 2 3 4 5 6 7 8 9 10ndash30

ndash20

ndash10

0

10

20

30

T e (N

m)


(b)

0 1 2 3 4 5 6 7 8 9 100

02

04

06

08

1

12

Фrd

(Wb)



(c)


0 1 2 3 4 5 6 7 8 9 10ndash50

0

50

100

Flux

erro

r (

)

(d)


Фrα

(Wb)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash1

ndash05

0

05

1

3 32 34 36 38 4 42 44 46 48 5ndash1

ndash05

0

05

1

(a)


0 1 2 3 4 5 6 7 8 9 10Time (s)

ndash20

ndash10

0

10

20

I sa (A

)

3 32 34 36 38 4 42 44 46 48 5ndash15ndash10

ndash505

1015

(b)




Nomenclature


Data Availability




References

























Nomenclature


Data Availability




References
























Documents

DesignofRobustAdaptiveObserveragainstMeasurement ...downloads.hindawi.com/journals/jece/2020/6570942.pdf · 2020. 8. 1. · ResearchArticle DesignofRobustAdaptiveObserveragainstMeasurement