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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 3, MARCH 2014
Sensorless Control of Induction-Motor Drive
Based on Robust Kalman Filter and
Adaptive Speed Estimation
Francesco Alonge, Member, IEEE, Filippo D’Ippolito, Member, IEEE, and
Antonino Sferlazza, Student Member, IEEE
Abstract—This paper deals with robust estimation of rotor flux
and speed for sensorless control of motion control systems with
an induction motor. Instead of using sixth-order extended Kalman
filters (EKFs), rotor flux is estimated by means of a fourth-order
descriptor-type robust KF, which explicitly takes into account motor parameter uncertainties, whereas the speed is estimated using
a recursive least squares algorithm starting from the knowledge of
the rotor flux itself. It is shown that the descriptor-type structure
allows for a direct translation of parameter uncertainties into variations of the coefficients appearing in the model, and this improves
the degree of robustness of the estimates. Experimental findings,
carried out on a closed-loop system consisting of a low-power
induction-motor-load system, a proportional–integral-type controller, and the proposed estimator, are shown with the aim of
verifying the goodness of the whole closed-loop control system.
Index Terms—Adaptive speed estimation, induction motor, robust Kalman filter, sensorless control.
N OMENCLATURE
usd , usq
isd , isq
ψsd , ψsq
Ls (Lr )
Lm
Rs (Rr )
Tr = Lr /Rr
σ
σs
σr
ω
ωr
Ts
In
Stator voltages in a fixed reference frame [V].
Stator currents in fixed reference frame [A].
Rotor fluxes in fixed reference frame [Wb].
Stator (rotor) inductance [H].
Mutual inductance [H].
Stator (rotor) resistance [Ω].
Rotor time constant [s].
Total leakage factor.
Stator leakage factor.
Rotor leakage factor.
Electrical angular rotor speed [el.rad/s].
Mechanical angular rotor speed [rad/s].
Sampling time.
Identity matrix (nth order).
I. I NTRODUCTION
N
OWADAYS, field-oriented control of induction-motor
electrical drives is widely used when high performances
Manuscript received July 17, 2012; revised December 20, 2012 and
March 20, 2013; accepted March 24, 2013. Date of publication April 5, 2013;
date of current version August 23, 2013.
The authors are with the Department of Energy, Information Engineering,
and Mathematical Models, University of Palermo, 90128 Palermo, Italy
(e-mail: francesco.alonge@unipa.it; filippo.dippolito@unipa.it; antonino.
sferlazza@unipa.it).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2013.2257142
must be obtained. However, performance is greatly affected
by parameter uncertainty; thus, the behavior of both the controller and the estimator, which are designed using a modelbased approach, rapidly deteriorates in the presence of these
uncertainties. Obviously, the behavior of the whole control
system is particularly sensitive to that of the state estimator.
To cope with these uncertainties, online parameter identification, i.e., adaptive or robust design techniques, can be employed for designing either the estimator or the controller,
or both.
To this regard, in [1] and [2], both rotor and stator resistances are estimated online using neural networks (NNs) or two
extended Kalman filters (EKFs), respectively, whereas in [3],
only stator resistance is estimated using an EKF; all of these
works assume that stator, rotor, and mutual inductances are
well known. In [1], rotor and stator resistances are estimated
by two different schemes involving two different NNs that are
constructed starting from a model of the induction motor, and
speed is estimated using the same approach described in [4],
in which the authors state that estimated speed is sensitive to
noise, thereby requiring filtering; experimental results obtained
by processing the data acquired from a closed-loop drive show
that, in the presence of load torque, the speed does not track
the measured one. In [2], rotor and stator resistances are estimated by two different seventh-order EKFs that estimate the
augmented state of the induction-motor-load system, consisting of stator current and rotor flux components, speed, load
torque and, alternatively, stator resistance and rotor resistance.
However, the braided EKF, consisting of two seventh-order
KFs, is too complex; moreover, in practical applications it is
not possible to know the existence of the persistent excitation
condition a priori, which is a mandatory condition for an
exact parameter estimation. This appears clearly in the results
shown in [2], aimed at proving the need of identifying both
rotor and stator resistances. In [5], it is described the real-time
implementation of a biinput EKF estimator, which deals with
the estimation of the whole state of the induction motor together
with stator and rotor resistances, in the wide speed range.
In [3], a conventional EKF is studied, and the corresponding
results are given in the presence of various scenarios, but
sensitivity analysis in the presence of parameter variations is
not carried out, whereas in [6], it is shown that EKF is sensitive
to parameter variations, particularly at low speeds. Moreover,
the experimental results shown in [2] and [3] are obtained
by processing data acquired from voltage/frequency-controlled
0278-0046 © 2013 IEEE
ALONGE et al.: SENSORLESS CONTROL OF INDUCTION-MOTOR DRIVE
drives. Note that the EKF is applied also for sensorless control
of synchronous ac drives [7].
An alternative solution to EKFs is presented in [8], where
the design and implementation of unscented KFs (UKFs) for
induction-motor sensorless drives is investigated. This filter
requires higher computational effort.
In [4], rotor flux and stator currents are estimated by means of
a sliding mode observer (SMO), processed by the difference of
the measured and observed stator currents. For those systems
affine with respect to the input, the SMO is robust against all
disturbances, including parameter uncertainties that belong to
the space generated by the column of the forcing matrix, but
the produced estimates are affected by chattering, as shown
in [4].
In [9]–[11], model reference adaptive control techniques
are proposed for designing observers, whereas in [12], the
same techniques are employed for designing a controller. Other
model reference adaptive system (MRAS) observers are described in the literature, and some of them are compared in [13].
From the control engineering point of view, the approach
proposed in this paper explicitly assumes the objective of
the robustness of the estimator against variations of all the
parameters of the motor, without the need of estimating some of
them. The first step to reach this objective is that of formulating
the state estimation problem in two steps: In the first step, for a
given speed, the rotor flux and stator currents are estimated by
means of a linear fourth-order robust descriptor KF (RDKF),
starting from measured stator currents and the supplied voltages
computed by the controller; in the second step, the speed is
estimated by solving a total least squares problem starting
from the dynamic equations of the rotor flux components (cf.
also [10] and [11]). The descriptor form of the KF is used
here because the coefficients of the model are functions of the
physical electromagnetic parameters that are simpler than those
appearing in the conventional form. Consequently, physical
parameter variations can be directly translated into variations
of the coefficients appearing in the model, and this intrinsically
leads to a certain degree of robustness of the DKF. Moreover,
in order to take explicitly into account parameter uncertainties,
the RDKF is designed according to [14] and [15].
The advantages of the described procedure is that the mechanical equation is not included in the state estimation procedure, thus avoiding the use of nonlinear estimation methods,
such as EKF, and the connected lack of an observability property of the model in certain operating conditions [16]; instead,
only two linear least squares problems must be solved for
estimating the state of the system. Moreover, neither load torque
estimation nor parameter estimation is required, guaranteeing,
in any case, the robustness of the estimation. Finally, all of the
parameter variations are simultaneously taken into account, and
this occurs independently of the causes of their variation. The
controller designed for the experiments consists of four simple
proportional–integral (PI) control loops; only the speed controller is equipped with an antiwind-up scheme because speed
is subject to a large range of variations, and its variations are
slower than other variables. Experimental results are carried out
on a closed-loop control system that uses the state estimation as
feedback variables for computing the PI-type control law.
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In Section II, a mathematical model of the motor and a description of the uncertainties are shown. The formulation of the
problem is considered in Section III. Then, in Sections IV and
V, the RDKF and the speed estimator are shown. In Section VI,
a procedure is described to determine those system parameters
useful for tuning the estimator. In Section VII, closed-loop
experimental results are shown in order to validate the approach
previously described. Finally, Section VIII deals with some
conclusions.
II. M ATHEMATICAL M ODEL OF THE M OTOR AND
U NCERTAINTY D ESCRIPTION
The mathematical model of an induction motor in the descriptor form is given by
σLs
Lr dψrd
isd
+
= −Rs isd + usd
dt
Lm dt
(1)
σLs
Lr dψrq
isq
+
= −Rs isq + usq
dt
Lm dt
(2)
dψrd
Lm
1
=
isd − ψrd − ωψrq
dt
Tr
Tr
(3)
dψrq
Lm
1
=
isq − ψrq + ωψrd
dt
Tr
Tr
(4)
J
dωr
= −fv ωr + kt (ψrd isq − ψrq isd ) − tl
dt
z = [isd
isq ]T
(5)
(6)
where kt = 2pLm /(3Lr ), J is the inertia moment, fv is the
viscous friction coefficient, and z(t) is the output vector.
The model (1)–(6) is nonlinear and multivariable, and is
affected by parametric uncertainties. Moreover, the load torque
tl is unknown. For estimating speed, two approaches could
be employed. The first approach is based on the assumption
that speed varies slowly with respect to the electromagnetic
variables; this suggests that ω̇ = 0 should substitute (5), thus
obtaining the fifth-order model. The second approach leads to
a sixth-order model, in which the load torque is assumed as
slowly varying, and consequently, the five-order state of the
system (1)–(6) is augmented by the variable tl whose dynamics
is expressed by ṫl = 0. The resulting model is nonlinear and
requires an EKF.
As stated in the introduction, in this paper, we use an alternative procedure for estimating speed based on the assumption
that speed is a parameter in (3) and (4) of the model. This is
the reason why we consider only the model consisting of the
linear equations (1)–(4) and (6) for designing the RDKF. The
mathematical model (1)–(4) and (6) in the compact matricial
form, including stochastic uncertainties, can be written in the
following descriptor form:
Ẽ ẋ(t) = F̃ (t)x(t) + B̃u(t) + Q̃w(t)
z(t) = Hx(t) + Rv(t)
(7)
(8)
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 3, MARCH 2014
where x(t) and u(t) are the state and input vectors, respectively, given by x = [isd isq ψrd ψrq ]T , u=[usd usq ]T ;
w(t) and v(t) are the system and measurement noise assumed
zero-mean white noise uncorrelated between them and with
the other variables, having covariance matrices equal to I 4
and I 2 , respectively; Q̃ and R are diagonal square matrices
of suitable dimensions; and matrices Ẽ, B̃, F̃ (t), and H are
given by
⎡
0
σLs
⎢ 0
σLs
Ẽ = ⎢
⎣ 0
0
0
0
⎡
0
−Rs
⎢ 0
−R
s
F̃ (t) =⎢
⎣ LTm
0
r
Lm
0
Tr
Lr
Lm
0
1
0
0
0
− T1r
ω(t)
⎤
0
⎥
⎥,
0 ⎦
1
Lr
Lm
⎡
1
⎢0
B̃ = ⎣
0
0
⎤
0
1⎥
⎦
0
0
⎤
0
0 ⎥
⎥, H= 1
−ω(t)⎦
0
0
1
whereas (10) remains unchanged because H and R are constant. Matrices δE k and δF k are given by
⎡
⎤
δ(σLs )
0
δ LLmr
0
⎥
⎢
Lr
⎢ 0
⎥
)
0
δ
δ(σL
s
δE k = ⎢
Lm ⎥
⎣
⎦
0
0
0
0
0
0
0
0
δF k = δE k + Ts δ F̃ k
where
0
0
0
.
0
− T1r
Remark 1: The coefficients appearing in the model (1)–(4)
and, consequently, in (7) and (8), have simple expressions in
terms of the physical parameters of the motor. For example, in
looking at model (1)–(4), a variation in σLs produces variations
in the first two terms of (1) and (2), whereas in considering the
usual model of the motor, the same variation in parameter σLs
produces variations in all of the coefficients of the differential
equations expressing the dynamics of the stator currents and in
two terms of the equations expressing the dynamics of the rotor
flux components. Although this introduces robustness into the
descriptor form of the KF with respect to the conventional one,
in this paper, a RDKF is designed for estimating stator currents
and rotor flux.
Remark 2: Matrices B̃ and H are not affected by
uncertainties.
Remark 3: Matrix F̃ (t) is time-varying because it depends
on the speed ω(t). Matrix Ẽ is always nonsingular.
Starting from model (7) and (8), the following discrete-time
stochastic model is obtained by using the Euler method:
Exk+1 = F k xk + Buk + Qwk
z k = Hxk + Rv k
(9)
(10)
where k := kTs (k ∈ Z) is the current discrete time in which
Ts is the sampling time, and E = Ẽ, F k = Ẽ + Ts F̃ k , F̃ k =
F̃ (kTs ), B = Ts B̃, and Q = Ts Q̃.
A. Uncertainty Description
Assuming that the values of the electromagnetic parameters
are different from the nominal ones, denoting with δE k and
δF k the corresponding variations of matrices E and F k , (9)
becomes
(E + δE k )xk+1 = (F k + δF k )xk + Buk + Qwk
(11)
⎡ δ(R )
s
⎢ 0 ⎢
δ F̃ k = ⎢ δ Lm
Tr
⎣
0
0
δ(Rs )
δ
0
Lm
Tr
0
0 δ
0
0
⎥
⎥
.
0 ⎥
⎦
1
Tr
0
⎤
1
Tr
δ
Generally, uncertainties for descriptor-type models are represented in the following standard form:
−δF k δE k+1
M f,k
−N f,k N e,k+1
0
=
Δk
0
δH k
0
N h,k
0
M h,k
(12)
where Δk is a bounded arbitrary contraction with Δk ∞ ≤
1, and M f,k , M h,k , N f,k , N e,k+1 , and N h,k are known
matrices. In our case, since δH k = 0 ∀k ∈ Z, we have
M f,k = I 4 ,
M h,k = I 2 ,
N e,k+1 = max δE k ,
k
N f,k = max δF k k
N h,k = 0.
Now, with the aim of avoiding the formulation of problems
that are too complex, it is convenient to analyze the parameter
variations due to the increase in temperature and magnetic saturation. Temperature variation produces variation in rotor and
stator resistances. However, rotor resistance varies also with the
slip and, consequently, with load. All of these variations must
be taken into account. In order to take into account magnetic
saturation effects with accuracy, very complex mathematical
models must be constructed [17], but these models generally
lead to complex controllers. In many cases, saturation is taken
into account, assuming that it leads to a reduction of the mutual
inductance Lm . Consequently, saturation affects the values of
the rotor and stator inductances given by
Lr = Lσr + Lm
Ls = Lσs + Lm
(13)
(14)
where Lσr = σr Lm and Lσs = σs Lm are the rotor and stator
leakage inductances, respectively, which are assumed to be
constant because we are interested in the saturation of the flux
main path. Because the values of the given leakage inductances are small with respect to Lm , we have δ(σLs ) σLs ,
δ(Lm /Lr ) Lm /Lr and δ(Lm /Tr ) ∼
= (Lm /Lr )δ(Rr ).
It follows that:
E + δEk ∼
= E,
Fk + δFk ∼
= E + Ts (F˜k + δ F̃k ).
In order to set magnetic parameter variations, after the
computation of Lσr and Lσs from the nominal values of the
ALONGE et al.: SENSORLESS CONTROL OF INDUCTION-MOTOR DRIVE
parameters, for a given percentage variation of Lm , the values
of Lr and Ls are obtained using (13) and (14).
III. P ROBLEM F ORMULATION
The problem we deal with in this paper is that of estimating
the rotor flux components and the speed of an induction-motorload system, by solving two linear least squares subproblems.
More precisely, assuming at instant k the knowledge of the
speed, denoted by ω̂k , the first subproblem is that of estimating
the state of the model (9) and (10) by means of a DKF, by
solving the following minimization problem:
for k = 0
(15)
min x0 2P −1 +z 0 −Hx0 2R−1
x0
0
min xk − x̂k|k 2P −1 + Exk+1
xk ,xk+1
k|k
− (F k xk + Buk )2Q−1 + z k+1 − Hxk 2R−1 ,
for k > 0.
where w1 = 1 − (1/Tr )Ts and w2 = (Lm /Tr )Ts , and the values of the rotor flux components are given from the solution of
the first subproblem (see Section V). From (17), the formulation
of the second subproblem is as follows:
for k ≥ 0
ω̂k
where
Φ̂k =
ŷ k =
−Ts ψ̂rq (k)
Ts ψ̂rd (k)
A. DKF
Let us suppose that ωk is known. In addition, in (9) and (10),
matrix [E T H T ]T is full column rank, the recursive filtered
estimate x̂k|k of state xk , i.e., the solution of the problem (15)
and (16), is given by the following algorithm [14].
At instant k = 0, the algorithm is initialized with
T −1
−1
P 0|0 = [P −1
0 +H R H]
Since our objective is to take explicitly uncertainties [cf.
model (11)], we need to modify the formulation (15) and (16)
into the following robust version:
for k = 0
(19)
min x0 2P −1 + z 0 − Hx0 2R−1
x0
0
min
max xk − x̂k|k 2P −1
k|k
+ (E + δE k )xk+1
− ((F k + δF k )xk + Buk )2Q−1
+ z k+1 − Hxk 2R−1 ,
for k > 0.
(20)
Equation (18) can be solved using least squares methods, as
will be shown in Section V.
x̂0|0 = P 0|0 + H T R−1 z 0 .
Then, at step k, update {x̂k|k , P k|k } to {x̂k+1|k+1 , P k+1|k+1 }
as follows:
−1
P k+1|k+1 = E T (Q + F k P k|k F Tk )−1 E + H T R−1 H
(21)
−1
x̂k+1|k+1 =P k+1|k+1 E T Q + F k P k|k F Tk
−1
× (F k x̂k|k + Buk )+H R z k+1 .
T
(22)
Algorithms (21) and (22) can be obtained as a solution of the
following regularized least squares problem:
min = xT Qx + (Ax − b)T W (Ax − b)
(23)
x
where xT Qx is a regularization term, and Q = QT > 0 and
W = W T ≥ 0 are weight matrices; x ∈ Rn is the unknown
vector; A ∈ Rn×n is the data matrix; and b ∈ Rn×1 is the
observation vector. The solution of (23) is
x̂ = [Q + AT W A]−1 AT W b.
(18)
ψ̂rd (k + 1) − w1 ψ̂rd (k) − w2 isd (k)
.
ψ̂rq (k + 1) − w1 ψ̂rq (k) − w2 isq (k)
xk ,xk+1 δF k ,δE k
IV. K ALMAN FILTERING
(16)
With reference to the second subproblem, using the Eulero
method, the last two equations of (9), expressing the dynamics
of the rotor flux components, can be rewritten in the following
matrix form:
ψrd (k + 1) − w1 ψrd (k) − w2 isd (k)
−Ts ψrq (k)
ωk =
Ts ψrd (k)
ψrq (k + 1) − w1 ψrq (k) − w2 isq (k)
(17)
min Φ̂k ω̂k − ŷ k 2
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(24)
The problem (16) can be rewritten in the regularized least
squares form (23) with the following identifications:
−F k E
F k x̂k|k + Buk
A←
, b←
z k+1
0
H
−P −1
−Q−1
0
0
k|k
,
Q
←
W ←
0
R−1
0
0
−xk − x̂k|k
x←
.
(25)
xk+1
Consequently, (24) with the identification (25) leads to the
time and measurement update of the KF (21) and (22).
B. RDKF
Consider the following robust version of the optimization
problem (23):
(26)
min max = x2Q + (A + δA)x − (b + δb)2W
x
δA,δb
where {δA, δb} are uncertainties modeled by
[δA
δb] = M Δ[δN a
δN b ]
(27)
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 3, MARCH 2014
where Δ∞ ≤ 1. The solution of (26) is given by
x̂ = [Q̂ + AT Ŵ A]−1 AT Ŵ b + λ̂N Ta N Tb
(28)
P k+1|k+1
−1
T
T −1
T
−1
= Ê k+1 Q̂k + F̂ k P k|k F̂ k
Ê k+1 + Ĥ R̂k+1 Ĥ
where {Q̂, Ŵ } are defined as follows:
Q̂ = Q − λ̂−1 N Ta N a
(29)
Ŵ = W + W M (λ̂I − M T W M )M T W
(30)
λ̂ is a nonnegative scalar parameter obtained by solving the
following optimization problem [14]:
λ̂ = arg
min
Then, at step k, update {x̂k|k , P k|k } to {x̂k+1|k+1 ,
P k+1|k+1 } as follows:
λ≥M T W M G(λ)
(33)
x̂k+1|k+1
T
T −1
= P k+1|k+1 Ê k+1 Q̂k + F̂ k P k|k F̂ k
T
−1
× F̂ k x̂k|k + B̂uk + Ĥ R̂k+1 z k+1
(31)
where
G(λ) = x(λ)Q2 −λN a x(λ) − N b 2 +Ax(λ)−b2W (λ)
−1 T
A W (λ)b + λN Ta N b
x(λ) := Q(λ) + AT W (λ)A
Q(λ) := Q − λ−1 N Ta N a
W (λ) := W + W M (λI − M T W M )M T W .
Now, the minimax problem (20) can be rewritten as problem
(26) by means of the following identifications:
−F k E
F k x̂k|k + Buk
A←
, b←
z k+1
0
H
−Q−1
−P −1
0
0
k|k
W ←
,
Q
←
0
R−1
0
0
−δF k δE
δF k x̂k|k
δA ←
, δb ←
0
0
0
−N f,k N e
N f,k x̂k|k
Na ←
,
Nb ←
0 0
0
−xk − x̂k|k
0
−M f
, x←
(32)
M←
0
Nf
xk+1
and the initial conditions are
A ← H, b ← z 0 , δA ← 0, δb ← 0, Q ← P −1
0 ,
M ← M h , N a ← 0, N b ← 0.
From (28) and the identifications (32), the filtered robust
optimum estimate x̂k|k is obtained from the following recursive
algorithm.
At instant k = 0, the algorithm is initialized with
(34)
where
Q̂k 0
0 I4
E
Fk
Ê k+1 =
, F̂ k =
λ̂−1
λ̂−1
k Ne
k N f,k
H
B
Ĥ =
, B̂ =
02×4
04×2
R̂k+1 0
−1
R̂k+1 = R − λ̂k I 2 R̂k+1 =
0
I2
Q̂k = Q − λ̂−1
k I 4,
Q̂k =
(35)
and λ̂k is obtained by minimizing the function G(λ) [cf.
(31)] with the identification (32) over the interval λ̂k > λl =
diag{Q−1 , R−1 }.
Computation of λ̂k can be carried out by means of the given
optimization procedure. However, in [15], it is proposed to
choose λ̂k as follows:
(1 + 0, 5)λl , for λl = 0
λ̂k = λ̂ =
(36)
0,
for λl = 0
which allows the offline computation of several of the given
matrices.
Remark 4: From (33) and (34), it is easy to verify
that, for descriptor systems without uncertainties (M f,k =
0, M h,k = 0, N f,k = 0, N e,k+1 = 0, N h,k = 0), this algorithm collapses to the DKF (21) and (22).
Remark 5: The main difference between the robust filter
and the standard one is that, in the robust algorithm, the new
recursion operates on system and noise covariance matrices,
modified with respect to the given nominal values. More
precisely, the algorithm updates these matrices to the values
necessary for obtaining robust estimation.
−1
T
−1
P 0|0 = [P −1
0 + H R̂ H]
−1
x̂0|0 = P 0|0 + H T R̂ z 0
R̂ = R − λ̂−1
−1 I 2
where λ̂−1 is obtained by minimizing the function G(λ) [cf.
(31)] with the identification (32) over the interval λ̂−1 > λl =
R−1 .
V. S PEED E STIMATION
The problem (18) could be solved by means of the ordinary
least squares (OLS) method as follows:
T
−1
−1
= τ Pω,k
+ Φ̂k Φ̂k
Pω,k+1
T
(37)
ω̂k+1 = ω̂k + Pω,k+1 Φ̂k Φ̂k ω̂k − ŷ k
(38)
ALONGE et al.: SENSORLESS CONTROL OF INDUCTION-MOTOR DRIVE
Fig. 1.
cal model of the electromagnetic circuit of the motor, expressed
in complex state variables:
dis
Ls − σLs
1
= − Rs +
is − jω ψ̃ r + ψ̃ r + us
σLs
dt
Tr
Tr
(41)
Ls − σLs
dψ̃ r
1
=
(42)
is + jω ψ̃ r + ψ̃ r
dt
Tr
Tr
Block diagram of the whole estimator.
where τ < 1 is the forgetting factor, and the algorithm is
initialized with Pω,0 = P0 for some P0 > 0 and ω̂0 = 0.
Remark 6: Forgetting factor τ is introduced in (37), in order
−1
increases linearly, which implies
to avoid that, for τ = 1, Pω,k
that Pω,k converges to zero for k → ∞; this in turn implies that,
during speed transients, the estimated speed tracks the actual
one very slowly.
As it is well known, the OLS algorithm assumes that Φ̂k is
not affected by errors, and errors are confined to ŷ k . However,
this hypothesis does not correspond to our case because estimation and modeling errors cause errors also in Φ̂k . Therefore, in
this application, the employment of total least squares (TLS) is
better because it also takes into account the errors in the data
matrix. In fact, estimated rotor flux present in Φ̂k is affected
both by modeling errors and noise, similar to the observation
vector. Consequently, instead of (18), a TLS problem is solved
by minimizing the following modified cost function:
min
ω̂k
Φ̂k ω̂k − ŷ k 2
.
1 + ω̂k2
(39)
T
ω̂k+1 = ω̂k − αk ΓTk Φ̂k + αk Φ̂k Φ̂k ω̂k
(40)
where αk is a positive constant, and Γk is given by
Δk
,
1 + ω̂k2
where is = isd + jisq , ψ̃ r = ψ̃rd + j ψ̃rq , and us = usd +
jusq .
In order to identify the parameters of model (41) and (42),
the Levenberg–Marquardt algorithm (LMA) has been used
[19], which provides a numerical solution to the problem of
minimizing a nonlinear function over a space of parameters of
the function.
For the application of the LMA, the aforementioned nonlinear function consists of
N 1 isd,k − îsd,k 2 + isq,k − îsq,k 2 (43)
S(β) = N
k=1
where (isd,k , isq,k ) are the direct and in-quadrature stator currents measured at instant k, and (îsd,k , îsq,k ) are the corresponding values computed by solving model (41) and (42), for
a given set of stator voltages and the actual parameter vector β
defined as follows:
β = [Rs
The adaptation law that minimizes (39) is
Γk =
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Δk = Φ̂k ω̂k − ŷ k .
In [18], it is proven that the origin ω̂ = 0 always belongs to
the convergence domain of TLS. Hence, the use of a null initial
condition is the best choice if no prior information is given. The
block diagram of the complete estimator is shown in Fig. 1.
Ls
σLs
T r ]T .
The algorithm is stopped when the parameter vector β is such
that function (43) is less than the chosen step-size tolerance.
In order to obtain all of the electromagnetic parameters, an
often used procedure consists on the assumption Lr = Ls [20];
then, it is possible to compute rotor resistance from the rotor
time constant Tr . Alternatively, we can assume σr = 2σs [21],
to compute σ from σLs and Ls , and then compute σr and σs as
follows:
9
3
σ
+
−
(45)
σs =
2(1 − σ) 16 4
σr = 2σs
VI. PARAMETER E STIMATION P ROCEDURE FOR
E STIMATOR T UNING
For the tuning of the estimator, a set of parameters is identified offline using the procedure described in the following.
The identified parameters are assumed as nominal parameters
for designing the DKF, the speed estimator, and the controller.
These parameters together with their hypothesized variation
ranges are taken into account for designing the RDKF. Consequently, the parameters are not updated online.
Since the direct and in-quadrature components of the induced
part flux cannot be measured, only some parameters appearing
in standard model (1)–(4) can be estimated. In order to obtain a
mathematical model in which only identifiable parameters appear, the expression of the scaled rotor flux ψ̃r = (Lm /Lr )ψr
is replaced. This allows for obtaining the following mathemati-
(44)
(46)
and finally, we compute Lm and Lr by the equations
Ls
1 + σs
(47)
Lr = Lm (1 + σr ).
(48)
Lm =
In this paper, the last method is employed.
VII. E XPERIMENTAL R ESULTS
In order to validate the estimator described earlier, closedloop experiments are shown, which were carried out on a
system consisting of a 750-W induction motor with two pole
pairs and a powder brake system shown in Fig. 2, a three-phase
ac/dc converter–filter–source voltage inverter unit supplying the
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 3, MARCH 2014
Fig. 3. Speed response of the system with feedback from the RDKF at high
reference speed and no load. The machine is fluxed at zero reference speed up
to 0.5 s, and then, it is started with a trapezoidal reference speed of 150 rad/s.
Fig. 2. Motor brake system.
TABLE I
M OTOR C HARACTERISTIC
TABLE II
M OTOR PARAMETERS
motor, and a cascade controller consisting of four PI control
loops, two inner current loops, and two outer rotor flux and
speed loops [22], [23]. These PI controllers are designed as
described in [23], to obtain a bandwidth of 10 Hz for speed
and rotor flux loops, and 40 Hz for current loops. An antiwindup scheme is designed for a speed control loop [24]. The
module of stator current and voltage vectors are constrained
to Is,max = 10 A in order to avoid damage of the machine,
Vs,max = 0, 866VBU S V, i.e., the maximum modulus of the
rotating voltage vector that the inverter can generate according
to the pulsewidth modulation technique employed. The motor,
fluxed at 0.5 Wb at t = 0, starts at t = 0.5 s. The whole
controller, including the proposed estimator, is implemented on
a DS1103 microcontroller that processes the controller itself at
12 kHz, and allows data acquisition of the measured variables
and their visualization on the cockpit provided by dSPACE
software.
The measured variables are the speed computed starting from
data acquired by means of a 1024 ppr incremental encoder
useful for comparing estimated and measured speeds, the latter
filtered using a phase-locked loop (PLL) scheme and the two
stator currents given by two Hall effect transducers.
The rated data of the motor are shown in Table I. The
parameters of the motor, identified as described in Section VI,
are given in Table II.
Fig. 4. Current estimation error of the system with feedback from the RDKF
at high reference speed. Same conditions of Fig. 3.
Fig. 5. Speed response of the system with feedback from the DKF at high
reference speed. Same conditions of Fig. 3.
In order to analyze robustness, both DKF and RDKF estimators are designed assuming the following uncertainties: 50%
for Rr and Rs , and 30% for Lm . Note that neither cause nor
rate of variation are needed for designing the estimator. Both
estimators were initialized assuming P 0 = 50I 4 , x0 = 0, R =
I 2 , and Q = diag{2 × 10−2 , 2 × 10−2 , 2 × 10−3 , 2 × 10−3 }.
Figs. 3–6 show the responses of the closed-loop system in the
presence of either robust or standard estimators, corresponding
to a trapezoidal reference speed with a maximum speed of
150 rad/s, at no load and with speed reversal.
Figs. 9–12 show the responses of the closed-loop system at
no load, during low speed tests (3 rad/s). Both estimators are
able to track the reference speed with a mean speed equal to
ALONGE et al.: SENSORLESS CONTROL OF INDUCTION-MOTOR DRIVE
Fig. 6. Current estimation error of the system with feedback from the DKF at
high reference speed. Same conditions of Fig. 3.
Fig. 7. Speed response of the system with feedback from the RDKF at high
reference speed. A 5-N · m load torque is applied at 2 s and removed at 13 s.
1451
Fig. 9. Speed response of the system with feedback from the RDKF at no
load and low reference speed. The machine is fluxed at zero reference speed. At
1 s, it is started with a step of 3 rad/s; and a further step of −3 rad/s, applied at
15 s, leads the reference speed to zero.
Fig. 10. Current estimation error of the system with feedback from the RDKF
at low-speed reference. Same operating conditions of Fig. 9.
Fig. 8. Speed response of the system with feedback from the DKF at high
reference speed. Same conditions of Fig. 7.
zero, but the RDKF displays better dynamic properties and is
slightly more noisy than the DKF.
An examination of Figs. 3 and 5 shows that both the estimators give good results. In fact, in both cases, the speed tracks
the reference one, the mean error is zero, and the maximum
difference between measured and estimate speeds is less than
±1 rad/s. Spikes are due to the resolution of the encoder. Figs. 4
and 6 show that both estimators are able to reproduce measured
currents. Obviously, acting on the elements of matrix Q, it is
possible to conveniently filter these currents.
Figs. 7 and 8 show the closed-loop responses corresponding
to a trapezoidal reference speed in the presence of a load torque
of 5 N · m applied at 2 s and removed at 13 s. A comparison
of that figures shows that the RDKF works better than DKF at
Fig. 11. Speed response of the system with feedback from the DKF at low
reference speed. Same operating conditions of Fig. 9.
load; in fact, the maximum difference between measured and
estimates speeds is in the interval [−2.5,0] rad/s, with a mean
displacement of about −1 rad/s for RDKF, whereas it is in
the interval [−5, −3] rad/s, with a mean displacement of about
−4 rad/s for the DKF. The estimated current behavior is quite
similar to that of the previous test and, then, are not reported
here for the sake of brevity.
Figs. 13 and 14 show the responses at 3 rad/s, also in presence
of a 4-N · m step load torque applied at 2 s and removed at
13 s. A comparison of that figures show that the RDKF has a
good behavior also at load, with a maximum difference between
measured and estimate speeds in the interval [−2, 2] rad/s, with
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 3, MARCH 2014
Fig. 12. Current estimation error of the system with feedback from the DKF
at low speed reference. Same operating conditions of Fig. 9.
Fig. 13. Speed response of the system with feedback from the RDKF at low
reference speed. The machine is fluxed at zero reference speed. At 1 s, it is
started with a step of 3 rad/s. A load torque of 4 N · m is applied at 2 s and
removed at 13 s. Finally, the reference speed is put to zero.
Fig. 15. Speed response of the system with feedback from the RDKF at step
reference speed and no load. The machine is fluxed at zero reference speed. At
0.5 s, it is started with a step of 70 rad/s.
Fig. 16. Speed response of the system with feedback from the DKF at step
reference speed and no load. Same operating conditions of Fig. 15.
RDKF works better then the DKF for step reference speed variations, particularly during transients. We want to point out again
that our experiments are carried out on an induction-motor
drive in which the estimated variables are employed for
closing the control loops. In this paper, we show results at
3 rad/s at no load and load, even if we also reach lower speeds
(1–2 rad/s) but with a worse speed waveform. In our opinion,
this is due to the nonlinear behavior of the brake, particularly at
low speed, which in certain conditions blocks the motor.
VIII. C ONCLUSION
Fig. 14. Speed response of the system with feedback from the DKF at low
speed reference. Same operating conditions of Fig. 13.
a mean displacement of about −1 rad/s. In addition, in these
difficult operating conditions, the RDKF appears better on the
dynamic point of view but, at the same time, more noisily with
respect to the DKF.
Finally, Figs. 15 and 16 show the speed responses of the
system with feedback from the RDKF and the DKF at step reference speed and no load. Only in this experiment, both estimators are designed assuming the nominal resistances increased
by 30%, and the mutual inductance decreased by 20%, with respect to the values obtained with the previously described identification process. Examination of these figures shows that the
In this paper, speed and rotor flux estimators are designed
for sensorless control of motion control systems with induction
motors. More precisely, the estimators consist of an interconnection of an adaptive speed estimation scheme and either a
robust or standard descriptor-type KF. It is shown that the
descriptor structure of the KF allows for a direct translation
of parameter variations into coefficient variations of the system model, which leads to simplifications in the describing
uncertainties. The use of a speed estimator separate from flux
one allows us to design a fourth-order linear KF estimator.
The DKF displays intrinsic robustness properties with respect
to the conventional KF. However, the design of the DKF,
including explicitly robustness requirements, leads to better
results during load tests in both low and high speed ranges and
during transients, for step reference speed, but at the expense
ALONGE et al.: SENSORLESS CONTROL OF INDUCTION-MOTOR DRIVE
of an increased noise level of the estimate. Note that if a great
accuracy in full-load conditions or low-speed operations are not
required, the DKF can be conveniently used. The whole estimator scheme is suitable for implementation on a digital signal
processor. Experiments carried out on a prototype show that the
estimator scheme proposed in this paper is particularly suitable
for sensorless control of induction-motor drive applications.
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Francesco Alonge (M’02) was born in Agrigento,
Italy, in 1946. He received the Laurea degree in electronic engineering from the University of Palermo,
Palermo, Italy, in 1972.
Since then, he has been with the University of
Palermo, where he is currently a Full Professor of
automatic control with the Department of Energy,
Information Engineering, and Mathematical Models.
His research interests include electrical drive control
(including linear and nonlinear observers, stochastic
observers, and parametric identification), robot control, parametric identification and control in power electronics, and motion
control of unmanned aerial vehicles in aeronautics.
Filippo D’Ippolito (M’00) was born in Palermo,
Italy, in 1966. He received the Laurea degree in
electronic engineering and the Research Doctorate
degree in systems and control engineering from the
University of Palermo, Palermo, Italy, in 1991 and
1996, respectively.
He is currently a Research Associate with the
Department of Energy, Information Engineering, and
Mathematical Models, University of Palermo. His
research interests include control of electrical drives,
adaptive and visual/force control of robot manipulators, and control of electrical power converters.
Dr. D’Ippolito received the 2000 Kelvin Premium from the Institution of
Electrical Engineers, for the paper Parameter identification of induction motor
model using genetic algorithms.
Antonino Sferlazza (S’12) was born in Palermo,
Italy, in November 1987. He received the Master’s
degree in automation engineering from the University of Palermo, Palermo, Italy, in 2011. He is currently working toward the Ph.D. degree in system
and control engineering in the Department of Energy,
Information Engineering, and Mathematical Models,
University of Palermo.
His research interests include the development of
feedback control algorithms for nonlinear dynamical systems, optimization techniques, estimation of
stochastic dynamical systems, and applications of control of electrical drives,
power converters, and mechanical systems.