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Design and analysis of speed-sensorless robust stochastic L1-induced observer

for high-performance brushless DC motor drives with diminished torque ripple

S.A.KH. Mozaffari Niapour a,, M. Tabarraie a, M.R. Feyzi b

a Private Research Laboratory, 51668 Tabriz, Iranb Faculty of Electrical & Computer Engineering, University of Tabriz, 51664 Tabriz, Iran

a r t i c l e i n f o

Article history:

Received 17 January 2012

Accepted 11 May 2012

Available online 10 October 2012

Keywords:

L1-induced observer

Brushless DC motor

Sensorless control

High-performance drive

Torque ripple

a b s t r a c t

This paper aims to present an analysis anddesign of a high-performance speed-sensorless control scheme

for a three-phase brushless DC (BLDC) motor drive by means of a novel observer techniquein the induced

L1norm setting, named robust stochasticL1-induced observer, with the purpose of reducing torque rip-

ple and increasing system robustness. The proposed observer is used for estimating the phase-to-phase

trapezoidal back-electromotive-force (back-EMF) for the BLDC motor merely via utilizing measured line

stator currents and voltages in such a way that by estimating the back-EMF, position and speed of the

rotor is readily obtained. In contrast to the conventional back-EMF sensing methods, this strategy of uti-

lized drive requires no filtering of current andvoltage; furthermore, it does not suffer from any sensitivity

to switching noises. Owing to that high-speed operation is vital for a motor, the varying input voltage

method is used for realizing the minimization of commutation-torque-ripple in a parallel way to the pro-

posed method since drive performance intensely degrades in this mode. Apart from analytic investigation

of the proposed method, two other types of observers, namely, the sliding-mode observer and Kalman

filter are compared with the proposed method for the aim of determining steady-state accuracy, dynamic

performance, parameter and noise sensitivity, low-speed-operation performance, and computational

complexity. Finally, the proposed system has been simulated in different operating conditions of the

BLDC motor by computer simulation, and the effects of the proposed speed-sensorless control schemehas been assessed by comparative studies and simulation results. Simulation results authenticate that

the proposed method is of excellent robustness and high precision estimation in comparison with

sliding-mode and Kalman filter methods under different operating conditions in spite of the existence

of measurement noise and electric parameter uncertainty. Therefore, the proposed method with its

strong robustness makes it possible for the drive to enable the motor to undergo a stable tensionless

operation without facing any problem at high-and low-speeds.

2012 Elsevier Ltd. All rights reserved.

1. Introduction

Brushless DC (BLDC) motor drives according to their applica-

tions require position sensors such as Hall-effect, resolver, or abso-

lute encoder for accurate implementation of current commutation

in stator windings and/or empowerment of appropriate desired

control. However, installation of these sensors in the motor for

meeting the control needs will make the motor-drive system

encounter several problems. The main drawbacks are the increased

cost and size of the motor, and a special arrangement needs to be

made for mounting the sensors. Moreover, Hall sensors are tem-

perature sensitive and hence the operation of the motor is limited,

which could reduce the system reliability because of the extra

components and wiring. Thus, considering the disadvantages

mentioned above and powerful and economical accessibility of to-

days microprocessors, it is worthwhile to replace sensorless con-

trol methods with rotor speed- and position-sensor.

In the two recent decades, considerable efforts have been made

for optimizing sensorless control techniques from the viewpoints

of the BLDC motor drive[110]. In reference[1], the terminal volt-

age sensing method which is based on float phase voltage sensing

with respect to virtual neutral point was originally proposed in or-

der to detect zero-crossing point (ZCP) of the back-electromotive-

force (back-EMF) waveform. However, when using techniques of

chopping drive in this method, neutral point is no longer a stand-

still point andthis points potential varies between zero and dc-bus

voltage. A compensation for the introduced phase delay of LPF in

[2]has been reported by using frequency-independent phase shif-

ter which can shift ZCP of input signal by a known phase delay. In

[3], the direct back-EMF detection approach which is not in need of

sensing or reconstructing motor neutral point and uses voltage

0196-8904/$ - see front matter 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.enconman.2012.05.011

Corresponding author. Tel./fax: +98 411 33 13962.

E-mail address: [email protected] (S.A.KH. MozaffariNiapour).

Energy Conversion and Management 64 (2012) 482498

Contents lists available at SciVerse ScienceDirect

Energy Conversion and Management

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e n c o n m a n
http://dx.doi.org/10.1016/j.enconman.2012.05.011mailto:[email protected]://dx.doi.org/10.1016/j.enconman.2012.05.011http://www.sciencedirect.com/science/journal/01968904http://www.elsevier.com/locate/enconmanhttp://www.elsevier.com/locate/enconmanhttp://www.sciencedirect.com/science/journal/01968904http://dx.doi.org/10.1016/j.enconman.2012.05.011mailto:[email protected]://dx.doi.org/10.1016/j.enconman.2012.05.011


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difference of unexcited phase and power ground of dc-link voltage

for direct back-EMF information elicitation has been analyzed. In

this method, sensing circuit can only operate during freewheeling

period (off-time of PWM) with a minimum off-time 3 ls samplingwhich results in that the maximum duty cycle of PWM be lower

than 100%. Another direct back-EMF detection approaches to ex-

tend duty cycle control from 5% to 95% has been proposed in [4]

by means of measurement of line voltages without considering

the back-EMF. Under an ideal assumption that there exists no free-

wheeling current in non-conducted phase, recently a simple posi-

tion-sensorless technique for detecting the back-EMF ZCPs in [5]

and starting of the motor in [6] are presented. This method empha-

sizes on the issue that by measuring difference of line voltages in

the motor terminals, it will be possible to create amplified version

of back-EMF in order to extend its ZCPs detection at lower speeds.

Unfortunately, the considered assumptions in [5,6]methods can-

not always come true; in fact, using these methods, there may be

a possibility that freewheeling currents in non-conducted phase

exist both during normal commutation period and during un-com-

mutated period in such a way that their amplitude, duration, and

location of effectivity can differ according to the type of switching

method. In[7] a method based on proper PWM strategy (PWM-

ON-PWM) is offered in order that overcome the disadvantages in

[5,6]. Although by using this method can realize good motor per-

formance over a much speed range, there is no wonder it results

in a tiny variety in application of BLDC motor drives. In [8] a

speed-independent new physical concept has been proposed to de-

tect commutation instants by utilizing speed-independent position

function. However, since this function depends on calculations of

current derivatives, this method, firstly, requires digital implemen-

tation, and, secondly, due to the extreme sensitivity of the method

mentioned to measurement noises and machine parameters, this

issue inevitably leads to a disorder in the determination of commu-

tation points.

Nevertheless, the strategies above-mentioned operate only in a

bounded speed range and are considered to be among open-loop

speed-sensorless methods, but observer-based methods are mainlyconsidered to be among closed-loop speed-sensorless techniques

which are more robust and are of high-accuracy with respect to

uncertainty in parameters and disturbances. Therefore, observer-

based drives for high-performance applications can be the best

and safest choice. In[9], an extended Kalman filter (EKF) has been

used for instantaneous estimation of system state variables and

stator resistance by using line measured voltages and currents

and utilizing complete model of the BLDC motor. Unfortunately,

the most basic problem for EKF is that its robustness against

parameter detuning is too weak. In addition, determining the val-

ues of noise covariance matrices is difficult in them, and as this

method is based on having accurate knowledge of practical system

noises, the parameters determined by simulation should still be

adjusted in practical system which increases the inconveniencesfor EKF. In [10] a sliding-mode observer has been presented by

means of the stator line voltages and currents and electrical motor

model to estimate the phase-to-phase back-EMF of the BLDC mo-

tor. In this respect, it should be pointed that a continuous approx-

imation has been used for switching sign function by applying

sliding-mode observer to drive system in order to reduce chatter-

ing effect in the method mentioned, which results in that, on the

one side, it reduces the accuracy of observer in estimating state

variables, and, on the other side, the applied approximation is no

longer effective in the reduction of chattering effect when a high-

level noise exists in the system output.

The main aim of this paper is to develop a novel observer ap-

proach based on the stochastic L1-induced filter [11] which re-

cently has been designed for state-multiplicative stochasticsystems. This innovative observer is the first known study that

extends state estimation to the case where the stochastic system

contains a known input as well as input-dependent noise. Such

an observer, which we refer to as the robust stochastic L1-induced

observer, improves the robustness and accuracy of the conven-

tional aforementioned methods for sensorless BLDC motor drives.

The proposed method in comparison with Kalman filter has supe-

riority from robustness point of view against parameter uncer-

tainty although it has some more computational complexity than

Kalman filter and requires adjusting more additional parameters.

Furthermore, compared to sliding-mode approach, the proposed

method not only excludes any chattering but also takes advantage

of excellent robustness against external disturbance of course in

turn of accepting more computational complexity and adjusting

covariance matrices in addition to more additional tuning param-

eters. In this observer, apart from deterministic parameter uncer-

tainties, stochastic uncertainties have been considered as well.

Additionally, it has a more realistic viewpoint in comparison with

the sliding-mode observer and Kalman filter because its estimation

error variance or energy should not necessarily be minimized;

rather, its peak value is bounded. The proposed observer has been

designed for estimating phase-to-phase trapezoidal back-EMF of

the BLDC motor by utilizing measured line voltages and currents

so that rotor speed and position can easily be obtained by the

back-EMF estimation. In order to overcome the big commuta-

tion-torque-ripple created under high-speed operation, the varying

input voltage method has been utilized in parallel with the pro-

posed method for reducing commutation-torque-ripple. Likewise,

in this paper, basic principles of designing the proposed method

has analytically been studied along with the two types of conven-

tional observers, namely, the sliding-mode and Kalman filter meth-

ods. At the end, the proposed system has been simulated in

different operating conditions of the BLDC motor by computer sim-

ulation, and the effects of proposed sensorless control method have

been evaluated from five perspectives including steady-state accu-

racy, dynamic performance, parameter and noise sensitivity, low-

speed-operation performance, and computational complexity via

a comparative study of two conventional observersaforementioned.

2. Modeling of BLDC motor

General voltage equation of each BLDC motor active phase is ob-

tained by means of Kirchhoffs voltage law as the following [8]:

vxRixXnk1

dwkxh; ixdt

1

where vx, R, ix, and wkx(h, ix) are active phase voltage, resistance,current, and total flux-linkage respectively, h is rotor position,

and n is number of motor phases. The flux-linkage in active

phase includes both self and mutual flux-linkages. For a three-phase BLDC motor, the total flux-linkage of the phase a includes

[8,12]

waLaah; iaiaLabh; ibibLach; icickarh 2

where first termrepresents self flux-linkage, second and third terms

represent mutual flux-linkage between phaseb andcand phasea,

and the fourth term represents the flux-linkage of the permanent-

magnet on the rotor. Supposing that the saturation effect to be neg-

ligible and the inductance variation dramatically to be small (Ld-

Lq),(2)can be expressed as follows:

waLaaiaLabibLacickarh 3

Substituting (3) into(1)and its extension for all three phases, wehave

S.A.KH. Mozaffari Niapour et al. / Energy Conversion and Management 64 (2012) 482498 483


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vaRaiaddt

LaaiaLabibLacic dkarhdt

vbRbibddt

LbaiaLbbibLbcic dkbrh 2p=3dt

vcRcicddt

LcaiaLcbibLccic dkcrh 2p=3dt

4

In balanced three-phase BLDC motors we have

RaRbRcRLaaLbbLccLsLabLbaLcaLacLbcLcbLm

5

whereLs andLm represent self inductance and mutual inductance

respectively. Substituting(5)into(4)gives

vaRaiaddt

LsiaLmibLmic dkarhdt

vbRbibddt

LmiaLsibLmic dkbrh 2p=3dt

vcRcicddt

LmiaLmibLsic dkcrh 2p=3dt

6

For a balanced star-connected BLDC motor, three phase currentswill still provide the following equation:

iaibic 0 7Using(7), (6)can be summarized as

vaRaia LsLm diadt

dkarhdt

RaiaL diadt

dkarhdt

vbRbib LsLm dibdt

dkbrh 2p=3dt

RbibL dibdt

dkbrh 2p=3dt

vcRcic LsLm dicdt

dkcrh 2p=3dt

RcicL dicdt

dkcrh 2p=3dt

8

whereL = Ls Lm is defined in the name of phase inductance under

balanced conditions. Last term is considered as back-EMF in each of

the above-mentioned voltage equations, and can be extended in thefollowing manner:

va Raia Ldiadt

dkarhdt

Raia Ldiadt

ke dhdt

dfarhdh

vb Rbib Ldibdt

dkbrh2p=3dt

Rbia Ldibdt

ke dh2p=3dt

dfbrh2p=3dh2p=3

vcRcicLdicdt

dkcrh2p=3dt

Rcia Ldicdt

ke dh2p=3dt

dfcrh2p=3dh2p=3

9

where theke is called back-EMF constant. From Eq. (9)it could be

realized that kar(h), kbr (h), and kcr(h) are a constant values which

is a function of flux-linkage and only varies according to rotor posi-

tion. Thefar(h),fbr(h), andfcr(h) are a flux-linkage form functions that

are a functions of rotor position. Owing to that stator winding neu-tral point is floaty, and is not generally accessible, it leads to the fea-

sibility of direct measurement of the phase voltages in practice.

Therefore,(9) can be rewritten in the following matrix form accord-

ing to phase-to-phase state variables and in terms of phase-to-

phase currents by substituting e instead ofdkrh

dt .

d

dt

iaibibicicia

264375

1

L 0 0

0 1L

0

0 0 1L

264375 R 0 00 R 0

0 0 R

264375 iaibibic

icia

264375 eaebebec

ecea

264375

0B@

vavbvbvcv va

2643751CA 10

ea, eb, ec (in volts) are stator phase winding back-EMFs. As regards tothe fact that equations related to phases (a b), (b c), and (c a)

are similar, the observer is designed for phase (a b) and then uti-

lized for phase (b c) in a similar way for facility purposes.

Since the sampling period is significantly smaller than the sys-

tem time constants, the rotor speed and position may be assumed

to remain constant during each sampling period. Thus, the dy-

namic of the back-EMF term can be assumed to zero, i.e.,dea;b;c

dt 0. It should be noted that since the system has been consid-

ered in a balanced way and for achieving back-EMF between the

two phasescanda (eca), we can readily utilize the equation eca= -

(eab+ ebc). The phase (a b) equations in the state space repre-

sentation are written as follows by considering process and

measurement noises:

_x1 RL

x1 1L

x2 1L

u1x1_x2x2

yx1f11

wherex1= ia ibandx2= ea ebare the state variables.u1= va vbis the input variable, andyrepresents the phase current (a b) that

is corruptedwith the white noisef. In addition,x1, x2, andf are theuncorrelated zero-mean white noises that satisfy

Efx1tx1tsg Q1dsEfx2tx2tsg Q2dsEfftftsg Rds

12

where Q1, Q2, and R are the covariances of noisesx1, x2, and frespectively.

3. Kalman filter design

By appearance of the Kalman filter in early 1960s, this estimator

gained significant application in the estate estimation. The Kalman

filter is an optimal estimator minimizes the estimation error vari-

ance in presence of the noises in measurement and inside the sys-

tem [13]. Designing the Kalman filter requires an accurate

knowledge of the dynamic model of the system under consider-

ation, and the Kalman filter performance deteriorates significantly

in presence of parameter uncertainties. In order to design such

estimator, the state space model(11)is expressed in the following

matrix form:

_xAxB1xB2u1yC2xf

13

where

A RL 1

L

0 0

" #; B1

1 0

0 1

; B2

1

L

0

" #; C2 1 0

wherex x1 x2 T

is the process noise with covariance matrix

Q= diag(Q1, Q2), and f is the measurement noise with covariance

R. The steady-state Kalman (KalmanBucy) filter is given by [14]

_xA^xB2u1KyC2^x 14where K PC

12 R

1. P is the covariance of the estimation error

which is obtained via the following algebraic Riccati equation:

APPAT PCT2

R1C2PB1QBT10and noting that (A, C2) is observable and (A,B1) is controllable, the

equation has a unique positive definite solution. As a result, the Kal-

man filter is asymptotically stable.

A crucial stage in the Kalman filter implementation is choosing

values of matrices R andQthat are very affective on the perfor-

mance of the Kalman filter. Qrepresents inaccuracy in the systemmodel. Hence, ifQis reduced, the Kalman filter will assume that

484 S.A.KH. Mozaffari Niapour et al. / Energy Conversion and Management 64 (2012) 482498


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the system model is more accurate, and consequently the filter

gains will be reduced. On the contrary, ifQis increased, the Kalman

filter increases its gains. R represents inaccuracy in the measure-

ments. Therefore, the filter gain will be increased when R is re-

duced and vice versa. Since the covariance matrices are not

available in practice, the values of the covariance matrix elements

are used as tuning parameters. For simplicity and avoiding compu-

tational complexity, the covariance matrices are chosen diagonal

and constant, and they are tuned by trial and error [15] to maintain

the filter stability and to achieve a desired compromise between

transient-state behavior and steady accuracy of estimated back-

EMF. On the condition that the exact knowledge of the motor mod-

el and the statistics of the noise signals are available, the filter per-

formance which is minimization of the variance of the estimation

error is optimal. However, the statistics of practical system noises

are different with ones tuned by trial and error. Consequently, the

filter performance degrades.

4. Sliding-mode observer design

Along with extension of introducing the effects of discontinuous

control term in dynamic systems, the concept of sliding-mode was

introduced in USSR in 1950s. Due to its inherent robustness a

noticeable attention was created in the area of sliding-mode con-

trol across the world. This idea was then extended to state estima-

tion issue. Nowadays, the sliding-mode observers have received

widespread attention in motor drives. They use a prediction based

on the model and a nonlinear discontinuous function, which de-

pends on the output estimation error, as a correction term [16].

The principal idea in sliding-mode observer is that a sliding motion

takes place on the surface in the error space for which the output

error is forced to zero in finite time[17].

The sliding-mode observer is known for its robustness against

parameter uncertainties and disturbances. Although this observer

is robust against noises in the system input, it does not operate

well in the presence of output noise [18]. From a different point

of view, the sliding-mode observers contain inherent high-gainstructure. The high-gain observer [19] is able to quickly recon-

struct the state variables and remove model uncertainties. None-

theless, high gain brings about an undesirable amplification of

the measurement noise which the observer performance. Another

main drawback for the sliding-mode observer is undesired high-

frequency oscillations with small amplitude which is known as

chattering. Usually parasitic dynamics that reflect the rapid ne-

glected actuator and sensor dynamics and the time delay due to

the digital implementation of the sliding-mode observer are the

main cause of the chattering in such type of observers [20]. Chat-

tering causes a reduction in accuracy of the observer, wear of mov-

ing mechanical parts, and high heat losses in electric power

circuits. A common approach for chattering reduction in sliding-

mode observer is using a continuous approximation for the discon-tinuous switching term the idea of which has been taken from the

boundary layer approach [21] in sliding-mode control. However,

this scheme has its disadvantages. Firstly, the observers accuracy

reduces in the state estimation; secondly when there is a high-le-

vel output noise, this approach is not effective enough in chattering

reduction. Another scheme to reduce chattering is the use of

smoothing filters [22], but these filters increase phase shift be-

tween the real and estimated state variables, which leads to an er-

ror in estimation of speed and position.

In order to estimate of the states of the model (11), sliding-

mode observer takes the following form[10]:

_x1 RL

x1 1L

^x2 1L

u1k1signx1x1_x2k2signx1x1

15

wherek1 and k2 are the observer gains. Sliding surface r(t) is de-fined as

rt e1t x1t x1t 0wheree1 is the error between the real and estimated currents.

For simplicity in analysis of the observer performance and how

to regulate its gains, we neglect the process and measurement

noises in(11). Thus, the error dynamic is given by

_e1 1L

e2k1v_e2 k2v

16

wheree2 :x2 ^x2 and v = sign(e1). For the observers convergence,

the following sliding conditionr _r< 0 should be satisfied. From theerror dynamic we obtain

r _rr 1L

e2k1v

61

Ljrjje2j k1rv 1

Ljrjje2j k1jrj

jrj k1 1Lje2j

17

If k1 is large enough fulfillingk1 >1L

je2tjmax , we can guarantee

the convergence to the sliding surfacer(t) = 0 in a finite time. Usingthe equivalent control concept (see [18]), during sliding motion

r= 0 and _r 0 take place. Therefore(16)becomes

0 1L

e2k1v) v veq 1k1L

e2 18

which leads to

_e2 k2v 1L

k2k1

e2 19

Thus,e2(t) converges to zero asymptotically provided that the con-

dition k 2k1


5/17

and the disturbances should only have bounded energy. Robust H1filter ensures that the worst-case energy gain from the input dis-

turbances to the estimation error is less than a prescribed value

for all the acceptable uncertainties. A robust H1 filter has been

investigated with the Riccati equation-based method in [24] and

with LMI approach in[25].

Two types of uncertainties have been considered in literature.

The first type is the deterministic uncertainties which are usually

posed in two forms: norm-bounded uncertainty and convex-poly-

topic uncertainty. Polytopic uncertainty is utilized exhaustively in

robust control and estimation of uncertain systems. In these types

of uncertainties, deterministic parameters of these systems are not

known thoroughly, and it is assumed that they lie in a given poly-

topic. The second type is the stochastic uncertainties which have

been considered to be multiplicative noise or Markov jump pertur-

bations. In the case of stochastic systems with multiplicative noise,

the parameter uncertainties are modeled as white noise processes

(see [26,27]). Markov jump systems are efficiently used to model

the systems which sudden variations occur in their structures.

For a comprehensive survey of linear stochastic systems related

to these perturbations, future information is available in [28]. In

the references mentioned above concerning robust H2 and H1 fil-

ters only the deterministic uncertainties had been taken into ac-

count. In the case of state-multiplicative noisy systems, a robust

H1 filter in [26] and a robust Kalman filter in [29] have been

presented.

Stochastic uncertainty in the system under study can be consid-

ered according to inductance model. Motions of magnetic materi-

als which are close to each other can induce rapid changes in the

inductance value. To be more precise, inductance Lsto= Lsto(t) can

be modeled in the form ofL1sto L1

L1r _gt [27]which _gt is azero-mean Gaussian white noise with unity covariance. Further-

more,L is the phase inductance and the value ofLris obtained by

estimation of reciprocal inductance covariance. The model recalled

for the inductance creates state-multiplicative (state-dependent)

and input-dependent noise in state space model.

Although in designingH1filter the exogenous disturbances aresupposedly ofL2 type (energy-bounded signals), in practice, they

often have a bounded peak (of L1 type). These disturbances are

known as persistent bounded disturbances which are studied in

theL1 control theory[30]. For such types of disturbances, the in-

duced operator norm is the induced L1 or peak-to-peak norm of

the system under investigation (or the L1 norm of its impulse re-

sponse). In[11] inducedL1 estimation for a linear stochastic sys-

tem with state-multiplicative noise has been investigated. This

estimator has also been called stochastic peak-to-peak filter be-

cause the proportion between the peak value of the mean-square

of estimation error and the peak value of the mean-square of exog-

enous disturbances is bounded by a prescribed value. From practi-

cal applications perspective, significant advantage of this estimator

is that there is no need to minimize the estimation error energy orvariance; rather, the peak value of estimation error should be lim-

ited instead. This estimator is also used in the case in which the

deterministic component of state space model matrices and the

covariance matrices of multiplicative noises are uncertain but re-

side in a convex-bounded polytopic domain.

In designing filters such as Kalman filter and H1 filter the

known input signal is added to the estimator due to complete elim-

ination of its effect in estimation error, but this important charac-

teristic is no longer valid in the presence of parameter

uncertainties[24]. Owing to this fact and also since the underlying

system contains a known input signal (phase-to-phase voltage) as

well as an input-dependent noise term which have not been taken

into consideration in the design of the stochastic peak-to-peak fil-

ter; thus, we develop a novel observer technique based on the sto-chastic peak-to-peak filter [11], which as mentioned in the

introduction is referred to as the robust stochastic L1-induced

observer.

Notation. The superscript T shows matrix transposition. Rn

determines the n-dimensional Euclidean space, and k k is the

Euclidean vector norm, and Rnm is a set of all the n m real

matrices. The notation P> 0 for Pnn means that Pis symmetric and

positive definite.E{

} stands for expectation. The symbol is used

for the symmetric terms in a symmetric matrix. By L#Rk we

denote the space of bounded Rk-valued functions on the proba-

bility space (X, #, W), where X is the sample space, # is an r-algebra of subsets of the sample space, and W is a probability

measure on #. By (#t)t>0 we denote an increasing family of r-algebras #t #. Likewise, let L

1#t

Rk denote the space of non-

anticipative stochastic processf() = (f(t))t2[0, 1)in Rk with respect

to (#t)t2[0, 1) which satisfies kfk1:suptP0 [E{kf(t)k2}1/2] < 1. It

should be mentioned that stochastic differential equations are of

Ito type.

5.1. Upper bound on induced L1norm of linear stochastic systems

The main tool that we will use in the design of the stochasticL1-induced observer is an extension of important Lemma 1 in

[11]to the case where input-dependent noise is considered. In or-

der to describe it we consider the following linear stochastic sys-

tem with state-dependent and input-dependent noise:

dxt Axt B1xtdt G1xt G2xtdbt;x0 x0zt C1xt D11x

20wherex 2 Rn is the system state vector, andx0represents the initial

state. xt 2 L1#tRk

is the exogenous disturbance vector, and

z2 Rm is the objective vector.A,B1,C1,G1,G2, andD11 are constant

matrices with appropriate dimensions. b(t) is a zero-mean real sca-

lar Wiener process which satisfies

Efdbtg 0; Efdbt2g dt 21In fact, G1_b and G2_bcan be interpreted as white noise parameter

perturbations in the matricesAand B1respectively by adopting the

fact that white noise signals are formally the derivatives of Wiener

processes.

The following performance index is considered:

JE : kzk1ckxk1 22which c > 0 is a given scalar. In thispart of the paper, we use the fol-lowing definition.

Definition 1 [28]. The system(20) with x(t) = 0 is called expo-nentially stable in mean square (ESMS) if there exista > 0 and bP 1 such thatE{kx(t)k2}6 beatkx0k

2 for alltP 0 andx0 2 Rn.

In the following theorem, necessary and sufficient condition for

exponential stability in the mean square sense is given.

Theorem 1 [28]. The system (20)is ESMS if and only if there exists

Q> 0 such that ATQ QA GT1QG1 < 0.

Using Theorem 1, we make the following lemma, which is the

extension ofLemma 1of[11], for linear stochastic system.

Lemma 1. The system (20)is ESMS, and JEof(22) is negative for all

nonzero xt 2 L1#t Rk if there exist Q> 0, l> 0, and k> 0 which

satisfy the following two LMIs:

C1 :

AT

Q

QA

kQ

GT

1QG1 QB1

GT

1QG2

BT1

QGT2

QG1 lIGT2QG2" #< 0 23



6/17

C2 :kQ 0 C

T1

0 c lI DT11

C1 D11 cIm

264375> 0 24

Proof. This Lemma can be proven through a trend similar to that

of the ItLemma 1in[11]and by applying the Ito formula[28]to

evaluate differential of the quadratic formx

T

Qx, so we remove itsproof here. h

Remark 1. Similar to the Remark 2 of[11]where D11= 0, we, from

(24), come to the clear conclusion that the optimal value ofl isl= c in solution to(23) and (24).

5.2. Design of stochastic L1-induced observer

Now we consider the following ESMS system with stochastic

uncertainty and a known input signal:

dxt Axt B1xt B2rtdt G1xt G2rtdbt;x0 x0dy

t

C2x

t

D21x

t

dt

zt C1xt25a-c

whose description is similar to that of system (20), in addition,

r2 Rp is the known deterministic input signal, y 2 Rr is the mea-

surement vector, and D21 is constant matrix with appropriate

dimensions. Furthermore, the objective signal z2 Rm here is the

combination of the states to be estimated.

Now we consider the following estimator to estimate z(t):

d^xAf^xdtB1fdyB2frdt; ^x0 0;^zCf^x 26

wherex2 Rn is the estimate of the state vector x andz2 Rm is the

estimate of the objective signalz. Since r(t) is a peak-bounded signal

in practice, in(25a)we substitute the disturbance vectorx(t) withthe augmented disturbance vector ~xt xtT rtT

T, then

have

dx AxB ~xdt G1xG2 ~xdb 27

whereB B1 B2 ;G2 0 G2 .

In the same way, in(26)by substitutingdy of(25b)we obtain

d^x Af^xB1fC2xdtBf~xdt 28

whereBf B1fD21 B2f .

Denoting

~zt zt ^zt 29

and for a given scalar c > 0, the following cost function is defined:

JS : k~zk1ck ~xk1 30

The aim of stochastic L1-induced observer is to seek for estima-

tion ^ztfrom thez(t) over the infinite time horizon [0, 1) in sucha

way thatJSof(30)is negative for all nonzero ~xt 2 L1#t

Rkp. Con-

sidering the Eqs.(27) and (28)and denotingn xT ^xT T

, the fol-

lowing augmented system, which shows the observation error

dynamic, will be obtained:

dneAndteB ~xdteG1neG2 ~xh idb; ~zeCn 31

where

eA A 0B1fC2 Af

" #; eB B

Bf

" #

B1 B2

B1fD21 B2f

" #; eG1 G1 0

0 0

" #

eG2

G2

0

" #

0 G2

0 0

" #;

eC C1 Cf

32

Theorem 2. We consider the system (25ac)and the observer(26).

Forc > 0 the following results hold:

(a) The system (31) is ESMS, and JS is negative for all nonzero~xt 2 L1#tR

kp, if there exist R RT 2 Rnn;

W WT 2 Rnn; Z2 Rnr; Z2 Rnp; S2 Rnn; T2 Rmn,

and a positive tuning scalark such thatX1

R;W;Z;Z; S< 0;X

2

R;W; T> 0 33a; b

which is shown the following:

X1

:

RAATRkR ATWCT2ZTST RB1 RB2 GT1R GT1W

SSTkW WB1 ZD21 WB2 Z 0 0 cIk 0 0 0 cIp GT2R GT2W R 0 W

26666666664

37777777775

X2

:kR 0 kW

C1T T cIm

264375

(b) If (33a,b) is satisfied, a mean square exponential stabilizing

observer in the form of(26), which providesJS< 0, is specified

by

Af W1S; B1f W1Z; B2f W1Z; Cf T34

Proof.

(a) According toLemma 1andRemark 1, system(31)is ESMS,

and JS is negative for all nonzero ~xt 2 L1#t

Rkp, if there

exist Q> 0 andk > 0 that satisfy the following LMIs:

eATQQeAkQeGT1QeG1 QeBeGT1QeG2eBTQeGT2

QeG1 lIkpeGT2QeG2" #< 0 35kQ eCTeC cIm" #

> 0 36

Applying Schur complement, we obtain

eATQQeAkQ QeB eGT1

QeBTQ lIkp eGT2QQeG1 QeG2 Q

26643775< 0 37

kQ> c1eCTeC 38



7/17

Q and Q1 are partitioned in the form of Q : X MMT U

and

Q1 : Y N

NT V

, where we require that X> Y1. Defining

J : Y In

NT 0

and eJ : diagJ; Ikp;J,(37)is pre-and post-multiplied

by

eJT and

eJ, and (38) is pre-and post-multiplied by JT and J,

respectively. Carrying out some multiplications and through thesubstitution of

Z:MB1f; Z:MB2f; eZ :CfNT; bZ :MAfNT 39(40), which is shown the following, is obtained.

Defining! and! as below,

! :diag R 0R In

; Ik; Ip;

R 0

R In

;

! :

R 0 0

R In 0

0 0 Im

264

375

41a;b

and substituting

S:bZR; T :eZR; R:Y1; WXR 42with pre-and post-multiplying the(40a)by! and!T, and(40b)by

! and!T, respectively,(33a) and (33b)are achieved.

(b) If there exists a solution to(33a,b), from (39)we obtain that

Af M1bZNT; B1f M1Z; B2f M1Z; CfeZNT43

Applying (43)in the transfer function matrix of the observer, which

is obtained of(26),we find that

Hstop2ps Hzys

Hzrs

CfsIAf1 B1f

B2f

eZNTsIM1bZNT1M1 Z

Z

eZsMNT bZ1 Z

Z

eZsInXY

bZ1 Z

Z

44

Now considering(42),Hstop2p(s) is obtained as the following:

Hstop2ps TsRX S1 Z

Z

TsI RX1S1 RX

1Z

RX1Z

" # 45

Considering the relation above,(34)is obtained. h

Remark 2. Similar to the Remark 4 of [11] the tuning scalar k in

Theorem 2 is bounded in the open interval (0, 2 max

(real{eig(A)})).

Due to the fact that LMIs are affine in the system parameters,

Theorem 2can be extended for the case which these parameters

are uncertain. We assume that A, B1, B2, C2, D21, G1, andG2 residein the polytopic as follows:

where Xi:(Ai,B1i,B2i,C2i,D21,i,G1i,G2i),i= 1, . . .,s are the polytopic

vertices.

Corollary 1. Consider the system (25ac) and the observer(26). For a

givenc > 0 and for all nonzero ~xt 2 L1#t Rkpand for all (A, B1, B2,

C2, D21, G1, G2) 2X, JSis negative if(33a,b) is satisfied by a single set of

R;W;Z;Z; S; T; k for all the polytopic vertices. In the latter case, the

observer matrices are obtained via (34).

In order to design the stochasticL1-induced observer, the state

space model(11)including state-and input-dependent noise term

resulted from inductance model (L1sto L1

L1r _gt can be writtenin the form of(25ac)as the following:

X :

A;B1;B2;C2;D21;G1;G2

jA; B1; B2; C2;D21;G1;G2

Xs

i1li

Ai;B1i;B2i;C2i;D21;i;G1i;G2i

;li P 0;X

s

i1li

1( ) 46

AYYAT kY A

T XAYZC2YkIbZ XAATXkXCT2ZT ZC2 BT

1 BT

1XDT

21ZT cIk

BT2

BT2XZT 0 cIp

G1Y G1 0 G2 Y XG1Y XG1 0 XG2 I X

26666666664

37777777775

< 0

kY kI kX

C1YeZ C1 cIm264

375> 0

40a; b



8/17

dx AxB1xB2u1dt G1xG2u1dgdy C2xD21xdt

zC1x47

where

A RL 1

L

0 0

" #;B1

ffiffiffiffiffiffiQ1

p 0 0

0ffiffiffiffiffiffi

Q2p

0

" #;B2

1

L

0

" #;

G1 RLr

1Lr

0 0" #;G21

Lr

0" #;C1 0 1 ;C2 1 0 ;D21 0 0

ffiffiffiR

p andx x1 x2 f

Twith E{x(t)x(t s)} = I3 d(s), namely, we

have embedded the covariance matrices of noises in B1 andD21. In

addition, _gt is not correlated with the other noise signals. Sincethe matrix A is not stable, a very small negative perturbation term

(106) is added to element (2,2) inmatrixA. It is worthy tomention

that the matrixC1 implies thatthe objective signalz(t)isthe eab here.

Duo to the covariance matrices of the additive noises, i.e.,Q1,Q2andR are unknown in practice, similar to the tuning gains in the

Kalman filter they are used as weighting factors by trial and error

method. Since the peak value of the estimated back-EMF error and

consequently its ripple can directly be regulated by a suitable c, theproposed method is appealing from practical viewpoint. For mini-mizingc , Scherer and Weiland [31]have presented a method byperforming a line-search over 0 < k< 2 max (real{eig(A)}) to min-

imizec(k) which is the minimal value ofc ifk is held fixed. How-ever, there are two drawbacks. First, c is just an upper bound onthe real peak-to-peak norm of the system [31], and it is not indi-

cated how close c is to the true operator norm [11]. Second, theoptimized value ofc does not guarantee the best behavior in thesimulation. In fact,c should be adjusted for the proposed observerin a perfect harmony with the whole simulated system.

6. Description of the overall drives system configuration

The overall block diagram of the speed-sensorless controllingdrive has been shown in Fig. 1. In this controlling strategy the

proposed observer block outputs provide an estimation of the

phase-to-phase back-EMFs for detecting rotor position and rotor

speed estimation in such a way that matrices B1f andB 2fas well

as matrixAf in this block are obtained via solving matrix inequality

according to(34). In order to detect commutation points and 120

electrical conduction mode signals for each active phase the

following trend could be followed. If the desired phase is

considered x1, and the motor sequence cycle direction x1

a;b; c!n1

x2 b; c; a!n2

x3 c; a;bis focused; then it can be claimed

that the conditions exnxn1 >0 and exn2xn


9/17

Hysteresis current control with current shaping is one of the

strategy types that is utilized for reducing torque ripple in

high-performance application for the BLDC motors. In this

strategy, three hysteresis controllers are utilized due to their

simplicity, accuracy, and fast response. Afterwards appropriate

reference currents are generated and then compared according

to the pre-stored back-EMF waveforms for reducing torque ripple

in reference current shaping block by using rotor position

resulted from the proposed observer. Next real values of the sta-

tor currents are measured and compared with current references

and generate current error. Finally, utilizing rotor position

information which is detected for ensuring commutation and

appropriate pattern of switching, all the three hysteresis control-

lers independently issue the required order for the power

switches to regulate each of phase currents with the aim of

two-phase conduction mode.

In this way, with harmonic and simultaneous operation of each

of these controllers, the BLDC motor can experience a drive with

successful performance. It is important to note that this paper does

not aim to implement the proposed system experimentally; rather,

it focuses on a new approach of the observer utilized in the speed-

sensorless BLDC motor drive. It should be noted as well that in

practice all the controlling algorithms utilized in this overall struc-

ture of the proposed speed-sensorless drive can be implemented

by using fast digital signal processors (DSPs) that are nowadays

commercially available.

7. Simulation results and comparative studies

In order to reach an insight distinct from the whole system

performance and emphasize on the advantages of the proposed

sensorless control scheme in comparison with other relevant ap-

proaches, the motor operation should be evaluated under different

conditions. The proposed control scheme has been simulated un-

der different operating conditions of the motor. To set the gating

signals of the power switches easily and represent the real condi-tions in simulation as close as possible the electrical model of the

actual BLDC motor withRLelements and the inverter with power

semiconductor switches considering the snubber circuit, the

simulation model has been designed in Matlab/Simulink using

the SimPower System toolbox. Moreover, the dead-time of the in-

verter and non-ideal effects of the BLDC motor are neglected in the

simulation model.

In this simulation, the sampling interval and the magnitude of

the currents hysteresis band are 10ls and 0.2 A, respectively. Sim-ulation parameters of a standard BLDC motor for testing the pro-

posed sensorless drive technique performance are as follow:

Vdc= 300V, TeN= 3 N m, R= 0.4 X, L= 0.013 H, nN= 1500 r/min,

p= 2, j = 0.004 kg m2, ke = 0.4 V/(rad/s). In this section, the effects

of the proposed observer will comparatively be analyzed withtwo types of the conventional observers, namely, the Kalman filter

and sliding-mode observer from five aspects including: steady-

state accuracy, dynamic performance, parameter and noise

sensitivity, low-speed-operation performance, and computation

complexity. The control parameters that have been selected for

all the accomplished simulations by a fine tuning are as follows:

Q1= 3 1010, Q2= 1 10

2,R1= 3 1014,g(t) = 0 , c = 2.5, and

k= 1.5 106 for the proposed method; it should be noted that

the system under study is modeled by a four-vertex polytope

due to the uncertainties resulting from the resistance (50% R)

and inductance (+30% and25% L).

Q1= 3 108, Q2= 1, and R1= 3 10

9 for the Kalman filter

method; k1= 1300 andk2= 23000 for the sliding-mode method.

7.1. Steady-state accuracy analysis

In this part, the first category of simulations under investigation

has been brought for the purpose of emphasizing on the behavior

effects of the proposed sensorless control drive based on the motor

steady-state accuracy analysis in nominal operating conditions.

Figs. 24 illustrate the measured and estimated phase-to-phase

back-EMFs (eab) under full-load and rated-speed respectively for

the proposed, Kalman filter, and sliding mode methods. As Fig. 2

makes evident, accuracy of the estimated back-EMF is very high

with the proposed method, so that the distinction between the

estimated back-EMF and the actual one is extremely difficult,

and also the estimated back-EMF error has oscillations of small

amplitude with the maximum peak 1.3 V imposed on top and

bottom of zero. This exists whereas the estimated back-EMF with

the Kalman filter method does not match with the actual back-EMF, and its maximum peak is 14 V. Again, even worse than that,

2 2. 015 2. 03 2 .04 5 2 .06 2 .0 75 2 .0 9 2. 10 5 2. 12 2 .1 35 2 .1 5

-130

-100

-70

-40

-10

20

50

80

110

130

Time [sec]

Ba

ck-EMF[V]

Measured

Estimated

Fig. 3. Waveforms of estimated (dashed line) and actual (solid line) steady-state

phase-to-phase back-EMF for the Kalman filter method under full-load at rated-

speed.

2 2 .0 15 2. 03 2 .04 5 2 .0 6 2. 07 5 2 .09 2. 10 5 2 .12 2. 13 5 2 .15-130

-100

-70

-40

-10

20

50

80

110

130

Time [sec]

Back-EM

F[V]

Estimated

Measured

Phase Shift

Phase Shift


phase-to-phase back-EMF for the sliding-mode method under full-load at rated-

speed.

2 2. 01 5 2. 03 2 .04 5 2 .06 2 .07 5 2 .09 2 .1 05 2 .1 2 2. 13 5 2. 15-130

-100

-70

-40

-10

20

50

80

110

130

Time [sec]

Back-EMF[V]

Estimated

Measured


phase-to-phase back-EMF for the proposed method under full-load at rated-speed.



10/17

inFig. 4the back-EMF estimation error accuracy with the sliding-

mode (23 V) is less than Kalman filter method. The main reason

for the generation of these back-EMF big values of estimation error

for the sliding-mode approach is the big phase shift which hasbeen created between the estimation values and actual ones (at

steady-state). This point implicitly means a phase difference be-

tween the quasi-square current and trapezoidal back-EMF wave-

forms of the motor, and regarding the electromagnetic torque

equation Tem P2eaiaeb ib ecic

xe

h i , it inevitably leads to an increase

in the torque ripple and consequently causes a degrade in the drive

system. It is worth to be expressed that this problem could to some

extent be solved via selecting bigger gains of this estimator, yet the

result of such an issue in this case will be the intense increase of

the estimation error in the flat portion of back-EMF due to chatter-

ing effect increase.

Figs. 57in nominal operating condition of the motor show the

estimated speeds, actual speeds, and also speed estimation errors

by using the proposed, Kalman filter, and sliding-mode methods

respectively in which the white dashed lines represent reference

values of each quantity. InFig. 5, estimated speed replicates well

the measured one with very small ripple in such a way that

maximum estimation error in it reaches 0.2 r/min, that is,

%0.027 of the rated-speed. By comparingFig. 5withFigs. 6 and 7

it could be concluded that speed estimation error in Kalman filter

and sliding-mode methods is by far more than in the proposed

method so that the obtained speed estimation error in them reach

2.5 r/min and 2 r/min, respectively. Generally speaking, the

speed estimation error includes a direct relationship with the tor-

que ripple. Electromagnetic torque waveforms have been brought

for the proposed, the Kalman filter, and the sliding-mode methods

respectively in Figs. 810. Inferring these results obtained, it can be

1499.5

1499.7

1499.9

1500.1

1500.3

1500.5

Speed[rpm]

1499.51499.7

1499.9

1500.1

1500.3

1500.5

Speed[rpm]

0.5 1 1.5 2 2.5 3 3.5 4

-0.5

-0.3

-0.1

0.1

0.3

0.5

Time [sec]

Erorr[rpm]

Fig. 5. Waveforms of estimated speed (upper trace), measured speed (middle

trace), and speed estimation error (lower trace) for the proposed method at rated-

speed.

1495

1497

1499

15011503

1505

Speed[rpm

]

1499

1499.5

1500

1500.5

1501

Speed[rpm]

0.5 1 1.5 2 2.5 3 3.5 4

-3

-1

1

3

Time [sec]

Erorr[rpm]


trace), and speed estimation error (lower trace) for the Kalman filter method at

rated-speed.

1495

1497

1499

1501

1503

1505

Speed[rpm]

1498

1499

1500

1501

Speed

0.5 1 1.5 2 2.5 3 3.5 4-5

-3

-1

1

3

5

Time [sec]

Erorr[rpm]

[rpm]


trace), and speed estimation error (lower trace) for the sliding-mode method at

rated-speed and full-load.

0.5 1 1.5 2 2.5 3 3.5 4

2.9

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

Time [sec]

Torque[N.m]

Fig. 8. Waveform of electromagnetic torque for the proposed method at rated

speed.

0.5 1 1.5 2 2.5 3 3.5 4

2.4

2.7

3

3.3

3.6

3.9

4.2

Time [sec]

Torque[N.m]

Fig. 9. Waveform of electromagnetic torque for the Kalman filter method at rated-

speed.

0.5 1 1.5 2 2.5 3 3.5 4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

Time [sec]

Torque[N.m]

Fig. 10. Waveform of electromagnetictorque for the sliding-mode method at rated-

speed.



11/17

concluded that torque ripple (19.7%) for the proposed method has

the lowest value in comparison with 30.3% and 42.42% torque rip-

ple for the Kalman filter and sliding-mode methods. The reason of

torque ripples bigness of the sliding-mode method compared theKalman filter method is, as expected, the phase shift created

between the estimated back-EMF and the actual one, which is an-

other cause for the effect on torque ripple in addition to speed esti-

mation error.

7.2. Dynamic performance analysis

The second category of the simulations under investigation in

this section has been taken into account to emphasize on the ef-

fects of the proposed sensorless control drive behavior based on

dynamic performance analysis of the motor according to torque

and speed profiles applied to it is as follows.

Speed profile: during interval 03 s. the motor starts up in nom-

inal operating conditions at 1500 r/min;t= 36 s: acceleration

and following that high-speed operation (Vdc< 4E) at 2500 r/

min;t= 69 s: deceleration and low-speed operation (Vdc> 4E)

at 200 r/min.

Load torque profile: during interval 01.5 s. the motor starts to

operate under half of the full-load (1.5 N m); t= 1.58 s:increase of the applied load to the motor and following that

0

400

800

1200

1600

2000

2400

2600

MeasuredSpeed[rpm]

0

400

800

1200

1600

2000

2400

2600

EstimatedSpeed[rpm]

0 1 2 3 4 5 6 7 8 9

-20

-10

0

10

20

Time [sec]

Erorr[rpm]

200 rpm

200 rpm

1500 rpm

1500 rpm

2500 rpm

2500 rpm

Fig. 11. Dynamic responses of the proposed method when load torque and speed

reference change. From topto bottom: measuredspeed, estimatedspeed, andspeed

estimation error.

0

1000

2000

3000

4000

5000

6000

7000

8000

MeasuredSpeed[rpm]

0 1 2 3 4 5 6 7 8 9

-5000

-3500

-2000

-500

1000

2000

Time [sec]

Estimatedspeed[rpm]

1500 rpm

Unstable Operation

Unstable Operation

1500 rpm

Fig. 13. Dynamic responses of the sliding-mode method when load torque andspeed referencechange. From topto bottom: measuredspeed, andestimated speed.

0

500

1000

1500

2000

2500

2700

Speed[rpm]

0 1 2 3 4 5 6 7 8 9-30-20-10

0102030

Time [sec]

Erorr[rp

m]

Estimated

Measured

Fig. 14. Dynamic responses of the proposed method when load torque and speed

reference change without using LPF. From top to bottom: estimated and measured

speed, and speed estimation error.

0

400

800

1200

1600

2000

2400

2600

MeasuredSpeed[rpm]

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

-2

6

14

22

30

Time [sec]

Erorr[rpm

]

0

400

800

1200

1600

2000

2400

2600

EstimtedSpeed[rpm]

1500 rpm

200 rpm

1500 rpm

2500 rpm

2500 rpm

200 rpm

Fig. 12. Dynamic responses of the Kalman filter method when load torque and

speed referencechange. From topto bottom: measured speed, estimatedspeed, and

speed estimation error.



12/17

full-load operation (3 N m); t= 89 s: decrease of the applied

load to the motor and on track of it half of the full-load

operation.

Figs. 1113 depict the dynamic responses of the measured

speed, estimated speed, and speed estimation error (except for

sliding-mode method) under the above-mentioned conditions for

the proposed, Kalman filter, and sliding-mode methods respec-

tively. Three common significant points can be concluded from

the speed estimation error viewpoint out of these figures. First,

the error observed in operating transient instants; acceleration

and deceleration, has been created due to the phase delay created

by LPF, which in strategy of the sensorless drive scheme in order to

smooth the estimated speed has been utilized as a feedback signal

in speed and current controllers. Another similar test has been car-

ried out for the proposed method with establishing the stated con-

ditions in this part without utilizing LPF to corrborate this claim

(seeFig. 14). Thus, as it can be understood from this figure, there

is no phase delay between the estimated speed and the measured

one in this state, of course, in return of a little increase in estima-

tion error of the steady-state. Therefore, LPF can be eliminated

where speed transient state has very high significance. The origin

of the observed high-frequency ripples in this figure are the sameas commutation notches whose frequency is six times as much as

the electric frequency of the motor. The frequency of these com-

mutation notches results from the multiplication of the numbers

of motors poles by one-twentieth of the motor speed in rpm, and

the relative width and depth of these notches will increase as

speed increases. In addition to this fact,Fig. 15 verifies a good tran-

sient response for the proposed method in the back-EMF estima-

tion in transient instants of the motor operation. As the figure

demonstrates, the distinction between the estimated back-EMF

and the measured one is difficult, and this issue is another verifica-

tion for the claim made above. Second, although a load torque step-

change has been applied to the motor in 1.5 and 8 s for testing the

sensorless drive response, and the motor experiences a undershoot

less than-12 r/min approximately within a short period of time,according toFigs. 1113,it should be evident that no error of stea-

dy-state is found in motor speeds after passing the undershoot.

This issue means that the selected sensorless drive strategy has a

good stability against un-modeled mechanical disturbances of

the motor. The last point is that a few overshoots, which emanate

from a big rapid transient response in the back-EMF estimation, are

observed in transient state (see Figs. 1113) in response to speed

profile. In explanation for that, it should be pointed out that the

observers under consideration require a high enough gain for con-

vergence, which forces it to peak to big values before the transient

response rapidly decays towards zero. This impulsive-like response

is recognized as the peaking phenomenon[33].

As it was expected, the proposed method acts much more suc-

cessfully than Kalman filter and sliding-mode methods in responseto high-speeds, Figs. 1113 prove the case. In such a state, the

speed estimation error is restricted between 0.75 r/min and

14 r/min ranges for the proposed and Kalman filter methods

respectively. Nonetheless, the sliding-mode method is unable in

response to high-speed demand of the motor and experiences an

unstable operation. Due to the fact that commutation currentsform in an inharmonic and distorted way in high-speed operation

mode, this effect might be justified as that the motor is affected by

a current strong noise at high-speed, and the sliding-mode method

is unable in face of it. It is not surprising that due to such the com-

mutation currents situation and speed estimation error, the torque

ripple at 2500 r/min increases to 23.67% and 140.85% values in

comparison with rated-speed for the proposed and the Kalman fil-

ter methods respectively. The method presented in [34]has been

utilized in order to overcome this commutation-torque-ripple or

torque dips at high-speed. This approach, known as a varying input

voltage control method, is a complementary and very effective

method for reducing commutation-torque-ripple at high-speed

using circuit analysis in Laplace domain by means of dividing

commutation interval into freewheeling and build-up regions.Realization of this scheme is typically feasible via varying dc-bus

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

-140

-105

-70

-35

0

35

70

105

140

Time [sec]

Back-EMF[V]

Measured

Estimated

Fig. 15. Waveforms of estimated (dashed line) and actual (solid line) transient

behavior of the phase-to-phase back-EMF for the proposed method during start-up

under full-load at rated-speed.

-5

-4

-3

-2

-10

1

2

3

4

300

360

420

480

540

600

660

Currents[A]

Voltage[V]

4.5 4.505 4.51 4.515 4.52 4.525

2.5

3

3.5

4

Time [sec]

Torque[N.m]

Vdc=300 Volt

Commutation Torque Ripple

icibia

Fig. 16. Voltage waveform of dc bus (upper dashed line trace), phase currents

(middle trace), electromagnetic torque (lower trace): without using varying input

voltage at 2500 rpm.

-5

-4

-3

-2

-1

0

1

2

3

4

300

360

420

480

540

600

660

Currents[A]

Voltage[V]

4.5 4.505 4.51 4.515 4.52 4.52

3

3.2

3.4

3.6

3.8

4

Time [sec]

Torque[N.m]

ia icib

Vdc

Commutation Torque Ripple

Fig. 17. Voltage waveform of dc-bus (upper dashed line trace), phase currents

(middle trace), electromagnetic torque (lower trace): using varying input voltage

method at 2500 rpm.



13/17

voltage which is in practice implemented either by the dc choppers

or by the fire angles controlling of rectifier switches.

Figs. 16 and 17exhibit the simulation results for waveforms ofdc-bus voltage, phase currents, and electromagnetic torque with-

out and with utilizing varying input voltage control method at

2500 r/min respectively. As it can be seen inFig. 16, the controller

has not been able to confront high-speed commutation-current-

ripple of the motor in commutation interval without applying

the desirable scheme of varying input voltage even by utilizing

hysteresis current control with current shaping method in such a

way that this issue has led to an undesirable commutation-tor-

que-ripple for the motor (23.67%). On the contrary, Fig. 17 confirms

that it can intensely decrease to 7% value of load average torque by

applying the complementary method of varying input voltage con-

trol in the commutation interval and the current control approach

mentioned. As a result, the torque ripple of the proposed method

can approximately decrease to one-twentieth of the Kalman filtermethod by utilizing the complementary method of varying input

voltage and by using the proposed sensorless drive scheme. It

should be noted that the complementary method of varying input

voltage requires smooth and non-distorted currents in un-commu-tated region to be able to have accurate effective operation, this

process which itself requires accurate speed estimation and

appropriate controller. Since same current control method has

been utilized in all the three methods under study, the speed esti-

mation accuracy is what should be the determinant of current dis-

tortion in un-commutated region. Consequently, noting the high-

speed performance observed for the Kalman filter and sliding-

mode methods, parallel and effective operation of the varying in-

put voltage control method with these two methods is impossible

for reducing commutation-torque-ripple.

In this section, another test has been allocated in order to eval-

uate the ability of the systemto operate in all four quadrants of the

torque-speed plane under parameter variations for the proposed

(+50% R and +15% L), sliding-mode (+30% R and +10% L), andKalman filter (+30% R and +10% L) methods as illustrated in

-1800

-1400

-1000

-600

-200

200

600

1000

1200

EstimaedSpeed[rpm]

2.5 2.75 3 3.25 3.5 3.75 4-15

-10

-5

0

5

10

15

Time [sec]

Torque[N.m]

I III IV

Forward Motoring Reverse Motoring Reverse Generating

II

0

200

400

600

800

1000

1200

3 3.0025 3.005 3.0075 3.013.0-100

-80

-60

-40

-20

0

20

Forward Regenerating

II

Fig. 18. Estimatedspeed(upper trace) andelectromagnetic torque(lower trace)responses for four-quadrantoperation using theproposed method under +50% deviation of R

and +15% deviation of L

-5000

-4000

-3000

-2000

-1000

0

1000

2000

EstimatedSpeed[rpm]

2.5 2.75 3 3.25 3.5 3.75 4

-100

-50

0

50

100

Time [sec]

Torque[N.m

]

Forward Motoring

Unstable Operation

Unstable Operation

I

Fig. 19. Estimated speed (upper trace) and electromagnetic torque (lower trace)

responses for four-quadrant operation using the sliding-mode method under +30%

deviation of R and +10% deviation of L

-3500

-2000

-500

1000

2000

EstimatedSpeed[rpm]

2.5 2.75 3 3.25 3.5 3.75 4

-100

-50

0

50

100

Time [sec]

Torque[N.m]

Unstable operation

Forward Motoring

Unstable operation

I

Fig. 20. Estimated speed (upper trace) and electromagnetic torque (lower trace)

responses for four-quadrant operation using the Kalman filter method under +30%

deviation of R and +10% deviation of L



14/17

Figs. 1820, respectively. In this test, the machine is operated for

3 s. at 1000 r/min under full constant load torque (3 N m), and then

the speed reference and load torque values are reversed to-1000 r/

min for 1 s and-3 N m for 0.5 s, respectively. Afterwards while the

speed reference has been kept constant, the load torque is changed

to 3 N m for 0.5 s. In the considered test as it is clear fromFigs. 19

and 20 a stable operation for sliding-mode and Kalman filter meth-

ods cannot establish in quadrants IIIV while the proposed method

in response to four-quadrant operation (see Fig. 18) show perfectly

satisfactory performance.

FromFig. 18, four possible modes or quadrants of operation of

speed versus torque are observable. In quadrant I (2.53 s) ma-

chine undergoes forward speed and torque. In fact, the torque is

propelling the motor in the forward direction. Next in quadrant II

(33.011 s) where the motor is spinning in the forward direction,

but torque is being applied in reverse. In this condition, the torque

is being used to brake the motor, in the other word, during this

time speed rapidly decreases toward standstill and the machine

is now generating power as a result. After passing a short transient

time causing due to the peaking phenomenon and fast change in

torque and speed, quadrant III (3.093.5 s) that is exactly the oppo-

site of quadrant I is started. Simultaneously, the machine experi-

ences motoring state in the reverse direction, spinning

backwards with the reverse torque. In fact, in this stage the motor

is affected by the reverse state to some extent and consequently

torque and speed ripples increase. Finally, in quadrant IV (3.5-

4 s) the machine is spinning in the reverse direction, but the torque

is being applied in the forward direction. Again, power is being

generated by the machine and under this situation the machine

operates in sustained regeneration mode. Furthermore, in this

stage owing to applying forward torque, torque ripple in compari-

son with pervious quadrant decreases. The obtained results from

Figs. 1820 authenticate that the proposed method by having its

strong robustness is inherently capable of four-quadrant operation

and has the benefit of the sliding-mode and Kalman filter methods.

7.3. Parameter and noise sensitivity analysis

Since parameter detuning, modeling error, and measurement

noise are among the most important disturbances which affect

the observers accuracy, in this section, another test based on sen-

sitivity to parameters and noise has been carried out on the system

for analyzing the robustness of the proposed system against the

disturbances. In sensorless BLDC motor drives based on the obser-

ver, stator phase resistance R and phase inductance L are used as

constant parameters of model for estimating state variables. Nev-

ertheless, the stator resistance can be deviated a lot from theirnominal values due to skin-effect, temperature variation, and the

-1

-0.5

0

0.5

1

Erorr[rpm]

-5

-2.5

0

2.5

5

Erorr[rpm]

0.5 1 1.5 2 2.5 3 3.5 4

-30

-15

0

15

30

Time [sec]

Erorr[rpm]

Fig. 21. Waveforms of speed estimation error under full-load sensorless operation

at rated-speed for resistance detuning. From top to bottom: proposed method with

+50% deviation, sliding-mode method with +50% deviation, and Kalman filtermethod with +30% deviation.

-1

-0.5

0

0.5

1

Erorr[rpm]

-4-2

0

2

4

Ero

rr[rpm]

0.5 1 1.5 2 2.5 3 3.5 4-6

-3

0

3

6

Time [sec]

Erorr[rpm]



50% deviation, sliding-mode method with 25% deviation, and Kalman filter

method with25% deviation.

-2

-1

0

1

2

Erorr[rpm

]

-20

-10

0

10

20

Erorr[rpm]

0.5 1 1.5 2 2.5 3 3.5 4-40

-20

0

20

40

Time [sec]

Erorr[rpm]


at rated-speed for inductance detuning. From topto bottom: proposed method with

+30% deviation, sliding-mode method with +25% deviation, and Kalman filter

method with +20% deviation.

-4

-2

0

2

4

Erorr[rpm]

-50

-25

0

25

50

Ero

rr[rpm]

0.5 1 1.5 2 2.5 3 3.5 4

-150

-100

-50

0

50

100

150

Time [sec]

Erorr[rpm]



25% deviation, sliding-mode method with 20% deviation, and Kalman filter

method with15% deviation



15/17

inductance as a result of flux-saturation, demagnetization effect,

and other disturbances.Fig. 21illustrates the speed estimation er-

ror at full-load and rated-speed for resistance detuning (resistance

increase) for the proposed method (+50% R), the sliding-mode

(+50% R), and the Kalman filter (+30% R) respectively. As it can

clearly be seen from the figure, the proposed and the sliding-mode

methods are insensitive to resistance increase, yet the Kalman fil-

ter method is severely sensitive to it. Under such a test the speed

estimation error oscillates between 0.3 r/min, 2.3 r/min, and

25 to 15 r/min for the proposed, sliding-mode, and Kalman filter

methods respectively. A stable operation in the Kalman filter is not

possible for bigger speed errors. It should be pointed that the rest

of the tests brought in this part have been carried out for the sim-

ilar conditions, namely, under full-load and rated-speed as well.

The proposed method is insensitive to resistance decrease, but

the sliding-mode and Kalman filter methods are both sensitive to

it, as it can be recognized from Fig. 22, which shows speed estima-

tion error waveforms for resistance detuning (resistance decrease)

for the proposed (50% R), sliding-mode (25% R), and Kalman fil-

ter methods (25% R) respectively. Such being the case, the maxi-

mum speed estimation error reaches 0.5 r/min, 2.85 r/min, and

4 to 2.9 r/min for the proposed, sliding-mode, and Kalman filter

methods respectively (seeFig. 22).

Similarly, Figs. 23 and 24 report the results of the sensorless

drive sensitivity tests against inductance uncertainty (increase

and decrease respectively) for the proposed, sliding-mode, and Kal-

man filter methods respectively. As it can be seen, the proposed

method is robust against overestimation (+30%) of L and underes-

timation (25%) of L, and the parameter variations mentioned ear-

lier create very small disturbances in speed estimation, so that

speed estimation error in this state remains bounded 1 r/min

and 2 r/min limits respectively. In spite of the very good behavior

of the drive in response to speed for the proposed method, the slid-

ing-mode and Kalman filter methods are intensely affected by

inductance variations, and under these conditions no acceptable

behavior of the motor drive can be expected.Figs. 23 and 24 reveal

that the speed estimation error disturbances is much bigger for theKalman filter than for sliding-mode method in such a way that if

numerical analogy is required, the maximum of speed estimation

error in face of underestimation (20%) and (15%) of L varies be-

tween 25 r/min and 80 r/min for the sliding-mode method and

Kalman filter respectively. What seems to have considerable

importance about selecting sliding-mode observer gains is that a

reconciliation should be established between the oscillation de-

crease and the stability increase. Although the decrease of these

gains increases the sensitivity of the system to electrical parame-

ters variations, it decreases the oscillations of the system and the

speed of convergence of the whole system. With regards to the re-

sults obtained in this section, it can be claimed that the proposed

method can create a satisfactory compromise between the exis-

tence of uncertainty in electrical parameters and speed accuracy

required for the system in comparison with the two methods un-

der study. This point emphasizes that the proposed method in con-

trast to the two conventional methods does not require an online

estimation of electrical parameters for achieving a good speed

accuracy; which will definitely increase the complexity of the cal-

culation power of the controllers. It is worth mentioning that the

variations considered in this category may not be realistic, but they

have been used only for studying the estimation performance.

The second category of the tests accomplished in this section

has been brought for analyzing capability of the proposed method

under the applying conditions of current measurement noise along

with a comparison with the sliding-mode and Kalman filter meth-

ods. In order to realize such a test a uniform random noise with

zero-mean and intensity of 0.4 have been added to currents iaandi b separately, and its results have been exhibited under full-

load and rated-speed inFig. 25. According to this figure, as it was

expected, the proposed method shows to be a strong robustness

against measurement noises, so that the speed estimation error

created in this state has approximately grown to one-fourth in size

in comparison with the Kalman filter method, and this can be an-

other emphasis on the features of the proposed method. If a com-

parison is required from system robustness viewpoint against

measurement noise, it could be claimed that the Kalman filter

method should be prioritized after the proposed method, and the

sliding-mode method will come last. The speed estimation error,

as it can easily be observed from Fig. 25, varies between 4 r/

min, 15 r/min, and 30 to 20 r/min, under the conditions ex-

pressed, for the proposed, Kalman filter, and sliding-mode methods

respectively. Taking into account these big ripples produced by the

Kalman filter and sliding-mode methods under the aforemen-

tioned test, there is no surprise that the acoustic noises generated

by the drive be noticeable.

7.4. Low-speed-operation performance analysis

The fourth category of the simulations under study in this sec-

tion has been allocated to low-speed-operation performance since

it is one of the most challenging tests and the most critical situa-

tions for a sensorless BLDC motor drive. One of the most important

reasons that make the estimation of the state variables problem-

atic in very low speeds where the excitation amplitude is low, is

that in these conditions the observers sensitivity to the parameter

variations especially the resistance increases. However, resistive

-8

-4

0

4

8

Erorr[rpm]

-20

-10

0

10

20

Erorr[rpm]

0.5 1 1.5 2 2.5 3 3.5 4

-30

-15

0

15

30

Time [sec]

Erorr[rpm]


at rated-speed for 40% noisy currents measurement. From top to bottom: proposedmethod, Kalman filter method, and sliding-mode method.

9.85

9.95

10.05

10.15

Speed[rp

m]

9.85

9.95

10.05

10.15

Speed[rpm]

0 .2 5 2 .2 5 4 .2 5 6 .2 5 8 .2 5 1 0.25 1 2.25 1 4 .2 5 1 6.25 1 8 .2 5 20 .0

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Time [sec]

Erorr[rpm]

Fig. 26. Full-loadsensorless operation at 10 r/min with proposed method. From topto bottom: estimated speed, measured speed, and speed estimation error.



16/17

voltage drop is smaller than stator voltage in high-speed operation

range; consequently, the back-EMF and speed estimation can be

performed with high accuracy. In contrast, in low-speed regime,

the observer faces the serious problem of considerable effect of

the resistive voltage drop against low stator voltage or increase

in the proportion of noise to measured signals. In order to achieve

an accurate estimation in very low speeds regime, for the sensor-

less scheme under analysis in this paper, it is self-evident that

the observer will act more successfully having a high robustness

against sensitivity to electrical parameters. Fig. 26 depicts the

full-load sensorless operation with the proposed method at 10 r/

min (about 1-Hz electrical reference) for the actual and estimated

speeds, and speed estimation error. As the figure clearly and

expectedly shows, the estimated speed replicates the actual speed

by a very small ripple in such a way that the oscillation ripple ofthe speed estimation error restricts between 0.08 r/min, i.e.,

1.6% of the speed under study. This small speed ripple results from

the very good estimation of the back-EMF. The speed estimation

error has been exhibited in Fig. 27 under the same conditions by

the proposed method for the sliding-mode and Kalman filter meth-

ods. A speed ripple of 11% for the sliding-mode method and speed

ripple of 13% for the Kalman filter method, on the one hand, repre-

sents the priority of the sliding-mode method over the Kalman fil-

ter method at low-speed, and, on the other hand, verifies the

remarks expressed about the observers success with much robust-

ness against electrical parameters uncertainty. The big ripples ob-

served in Fig. 27 indicate the worthwhile point that these two

observers face severe oscillation in zero-crossing instants, in the

other words, the two observers are unable in accurate estimationof the back-EMF in commutation instants. It seems interesting to

note that the ripple of the system decreases for all the three

observers as the load applied to the motor decreases.

Since the sensorless schemes are not self-starting, speed-sens-

orless methods cannot be applied well when the motor is at stand-

still. Thus, a starting procedure is needed to start the motor from

standstill. Most of these starting strategies are based on arbitrarily

energizing the two or three windings and expecting the rotor to

align to a certain definite position [1,6,3538]. Consequently, for

the motor starting, one of the already known procedures would

have to be applied. Among the simplest of them, for instance two

phases can be excited to result in the rotor to rotate and lock into

position. If the rotor is not in the desired position, the forcing tor-

que from the excited phases causes it to rotate and stop at the de-sired position. After prepositioning, the next commutation signal

advancing the switching pattern by 60 electrical degrees is applied.

Then instantaneously the proposed speed-sensorless scheme can

be taken over to detect the next commutation instant. With regard

to the reasons mentioned about the proposed method in the sec-

tion, the 60 electrical degrees rotor movement is enough to detect

the commutation instants and speed of the motor. After the first

detection of the commutation instant, both current and speed con-

trol is possible using the estimated speed.

7.5. Comments on computational complexity

For the Kalman filter and sliding-mode observer, there are three

and two parameters that should be regulated respectively, but for

proposed method the number of these parameters is five. Alterna-

tively, there are only sumand product for sliding-mode while com-

putation process also consists of matrix product and inverse for

Kalman filter method. However, for proposed method the condi-

tion will exacerbate in comparison with Kalman filter method be-

cause in addition to multiplying matrix by its inverse it needs to be

solved by LMI. The LMI can be solved rapidly and user-friendly

manner by using efficient numerical algorithm and software pack-

age such as MATLAB. It should be noted that the calculation timefor proposed method can dramatically be decreased by optimiza-

tion of LMIs algorithm and programmers ability which is easy to

obtain with nowadays microcontrollers of DSP systems. One more

thing that deserves to be mentioned is that the use of DSP for sim-

ilar on-line complicated mathematical computations has been re-

ported in different references such as[39,40].

Nonetheless, what can have more effect on computation time is

on-line structure of estimator. In this respect, the sliding-mode and

proposed methods suffer from using saturation function and extra

matrix product (see Fig. 1), respectively with respect to Kalman fil-

ter method. If a numerical comparison from computation time

point of view is required,Table 2can be considered. For this aim,

each simulation has been run separately in Matlab/Simulink under

tests considered inFigs. 57. FromTable 2, it can be inferred that

the Kalman filter has superiority over the sliding-mode and pro-

posed methods; however, the proposed control scheme enhances

robustness and accuracy of the driv

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