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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
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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
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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
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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
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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
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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
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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
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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
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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
Fig. 4. Waveforms of estimated (dashed line) and actual (solid line) steady-state
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
Fig. 2. Waveforms of estimated (dashed line) and actual (solid line) steady-state
phase-to-phase back-EMF for the proposed method under full-load at rated-speed.
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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]
Fig. 6. Waveforms of estimated speed (upper trace), measured speed (middle
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]
Fig. 7. Waveforms of estimated speed (upper trace), measured speed (middle
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.
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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.
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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.
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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
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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]
Fig. 22. 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 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]
Fig. 23. Waveforms of speed estimation error under full-load sensorless operation
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]
Fig. 24. Waveforms of speed estimation error under full-load sensorless operation
at rated-speed for resistance detuning. From top to bottom: proposed method with
25% deviation, sliding-mode method with 20% deviation, and Kalman filter
method with15% deviation
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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]
Fig. 25. Waveforms of speed estimation error under full-load sensorless operation
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.
496 S.A.KH. Mozaffari Niapour et al. / Energy Conversion and Management 64 (2012) 482498
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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