Sliding-Mode Observer with Resistances or Speed Adaptation for ...

Sliding-Mode Observer with Resistances or Speed Adaptation
for Field-Oriented Induction Motor Drives
Ciro Picardi
Francesco Scibilia
Dept. of Electronics, Computer and System Science
University of Calabria
Dept. of Electronics, Computer and System Science
University of Calabria
Via P. Bucci, 42C
Via P. Bucci, 42C
87036 Rende (CS), ITALY 87036 Rende (CS), ITALY
picardi@deis.unical.it
francesco.scibilia@gmail.com
Abstract – A sliding-mode observer working in parallel with a
particular adaptive scheme is proposed for field-oriented
control of induction motor. The observer detects the rotor flux
components in the two-phase stationary reference frame. The
adaptive scheme is able to on-line estimate either the motor
resistive parameters or the rotor speed. In the former case, the
adaptive scheme allows to have continuously the exact values of
the stator and rotor resistances, parameters usually subjected to
variations from their nominal values. In the second one, the
scheme can be used to realize an induction motor sensorless
speed control. The analytical development of the sliding mode
observer and the adaptive scheme is fully explained. Simulation
results, based on a Matlab/Simulink/Real-Time-Workshop
model designed on purpose, are presented to verify the validity
of the proposed adaptive observer structure.
I. INTRODUCTION
The induction motor has found very wide industrial
applications due to its well-known advantages as simple
construction, reliability and low cost. The most popular high
performance induction motor control method is that one
known as Field-Oriented Control. It is based on a d-q
reference frame rotating synchronously with the rotor flux
vector, which allows to achieve a decoupled control between
the flux and the produced torque, likewise to a separately
excited DC motor [1].
In Direct Field-Oriented Control, both the instantaneous
magnitude and position of the rotor flux vector are supposed
to be precisely known. However, as the rotor flux cannot be
directly measured, efforts have been made to estimate the
rotor flux using various kinds of observers, based on the
measurements of the stator currents, the stator voltages and
the motor speed.
An important problem is that the exact values of the motor
parameters, from which the observer and some high
performance control systems depend, are different from
nominal values and change with respect to the temperature
and the operating conditions. Another question is the need of
a speed sensor to provide the rotor speed measurement,
necessary to regulation purpose as well as observer
operation. The presence of this sensor increases the drive
cost and can reduce the robustness of the overall system;
moreover, in some cases a speed sensor cannot be mounted,
such as motor drives in a hostile environment and very highspeed
motor drives.
The extended Kalman filters for simultaneous estimation
of the rotor flux, the speed and some motor parameters
(frequently only the rotor resistance, the parameter subjected
to the widest variation) have been proposed as a potential
solution to the above problems [2], [3]. Unfortunately, this
approach has some inherent disadvantages, such as the
influence of noise and computation burden.
In the last years, the sliding-mode observer has
represented an attractive choice for its being quite robust to
disturbances, parameter deviations and system noise [4], [5].
Adaptive flux sliding-mode observers, in which the motor
speed is estimated by additional equations, have been
designed trying to reduce the influence of parameters
variations [6], [7], [8].
This paper presents a sliding-mode observer working in
parallel with a particular adaptive scheme. This scheme is
able to on-line estimate either the motor resistive parameters
or the rotor speed. Thus, the adaptive sliding-mode observer
allow to obtain a robust rotor flux estimation. Moreover, the
configuration with the adaptive scheme for the on-line
estimation of stator and rotor resistances values allows to
implement high accurate speed controls, where there is the
need to have constantly the right values of these parameters
to preserve high level performances. On the other hand, the
configuration with the rotor speed estimation can be used for
the implementation of a sensorless control.
The paper explains the analytical development of the
sliding-mode observer and the adaptive scheme.
Finally, the validity of the proposed algorithms is verified
by means of some simulation results, obtained using a
Simulink model designed on purpose.
II. FLUX OBSERVER-BASED FIELD-ORIENTED
INDUCTION MOTOR DRIVES
A. Induction Machine Model
The induction motor model, with the stator currents and
rotor fluxes defined as state variables, in the stationary α, β
coordinate system can be written as:
⎡ p is
⎤ ⎡ A11
⎢ ⎥ = ⎢
⎣ p φr
⎦ ⎣ A21
where p=d/dt
R s + Rrp
A11
= − I
σLs
A12
⎤ ⎡is
⎤ ⎡B1
⎤
vs
A
⎥ ⎢ ⎥ + ⎢
22 φr
0
⎥
⎦ ⎣ ⎦ ⎣ ⎦
A
12
1 ⎛ Rrp
⎞
= ⎜ I − ω J ⎟
σL
s ⎝ M ⎠
A21 = Rrp
I A − R rp
22 = I + ω J
M
(1)

s =
[ i i ] T
i stator current
r
sα
LM
Lr
s β
ϕ = φ , φ = [ φ φ ] T rotor flux
s =
r
[ v v ] T
sα
s β
r
rα
r β
v stator voltage
R
rp
= R
r
⎛ L
⎜
⎝ L
M
r
⎞
⎟
⎠
2
2
M
L
M =
L
r
σ = 1 −
L 2
M
⎡1
0⎤
⎡0
− 1⎤
1
I = ⎢ ⎥ J =
⎣0
1
⎢ ⎥ B1
= I
⎦ ⎣1
0 ⎦ σL s
ω is the rotor electrical speed, subscripts α and β are used
for α-axis and β-axis components. L M
, L r
, R r
, L s
and R s
are
mutual inductance, rotor inductance and resistance, and stator
inductance and resistance, respectively.
B. Direct Field-Oriented Induction Motor Drive System
Fig. 1 shows the block diagram of a direct field-oriented
induction motor drive.
The three desired stator currents for the current-controlled
voltage-source-inverter (CR-VSI) are calculated by the
following transformation equations from the two-phase d-q
rotating reference frame to the three-phase stationary a-b-c
system:
isad = − isqd
sin θˆ + isdd
cosθˆ
1
( isqd
+ 3 isdd
) − cosθˆ ( isdd
− isqd
)
2
1
( i − 3 i ) − cosθˆ ( i 3 i )
1
isbd
sinθˆ
3
2
1
i scd = sinθˆ
sqd sdd
sdd + sqd
2
2
= (2)
where i sqd is the torque current component, i sdd is the field
current component and θˆ is the instantaneous angle between
d-axis and a-axis.
Since d-axis has to be aligned with the rotor flux space
vector and α-axis of the α−β stationary reference frame is
aligned with a-axis, θˆ can be obtained as follows:
cos θˆ
where
ˆ
/ ˆ
= φrα
φr
φr
β φr
φ r α
sinθˆ
= ˆ / ˆ
(3)
ˆ and ˆ φ r β are the components of the estimate
2
vector ˆφ r of the rotor flux and ˆ 2
φ ˆ ˆ
r = φ rα + φrβ
.
Taking into account the induction motor model used in (1),
the vector ˆφ r can be obtained as:
ˆ φ = ˆ (L /L )
(4)
r ϕ r
r
M
Keeping in mind (4), in the following, the vector
used referring as the rotor flux vector.
L
r
L
s
ˆϕ r will be
Fig. 1 Block diagram of direct field-oriented induction motor drive.
III. CURRENT AND FLUX SLIDING-MODE
OBSERVERS DESIGN
Assuming that the speed is measured and the parameters
are exactly known, the sliding-mode observer can be seen as
composed by two parts working in parallel: a current slidingmode
observer and a rotor flux sliding-mode observer.
The current sliding-mode observer is defined by the
following equation and it uses the measured stator voltage
and current vectors as inputs:
pˆi s = A11is
+ A12
ˆ ϕ r + B1v
s + K sgn( ei
)
(5)
where ˆι s is the estimation of i s ,
e ˆ
i is
− is
surface, and
⎡k1
0 ⎤
K = ⎢ ⎥ is the current gain matrix
⎣ 0 k2
⎦
ˆϕ r is the estimation of ϕ r ,
= is the stator current error, chosen as sliding-mode
The rotor flux sliding-mode observer, using the measured
stator current vector as input, is defined by the equation:
p ˆ ϕr
= A 21 is
+ A22
ˆ ϕr
+ Η sgn( ei
)
(6)
⎡h11
h12
⎤
where Η = ⎢ ⎥ is the rotor flux gain matrix
⎣h21
h22
⎦
Defining the rotor flux error as e ϕ = ˆ ϕr
−ϕr
, we obtain
the following simple error equations:
p e i = A12 eϕ
+ K sgn( ei
)
(7)
p eϕ = A22 eϕ
+ Η sgn( ei
)
(8)
The design of the current gain matrix K can be derived by
using the Lyapunov’s stability theorem. We choose the
candidate Lyapunov function as follows:
V1 = ( e T
i e i ) / 2
(9)
This function is positive definite, satisfying the first
Lyapunov stability condition. Taking into account (7), we
can write the time derivative of V 1
as follows:

T
T
pV1 = e i (p ei
) = ei
[ f + K sgn( ei
)]
(10)
with f = [ f 1 f ]
T
2 = A12eϕ
From the condition of (10) being definite negative, we
obtain the following conditions for the gain matrix K:
k < , k 0 and k 1 > f1
, k2
> f 2
(11)
1 0 2 <
According to the equivalent control method, in sliding
mode the system constituted by the two observers behaves as
if the term sgn( e i ) is replaced by its equivalent value
x = sgn( e )]
, calculated by (7) assuming e p e = 0 :
[ i
eq
i = i
−1
x = −K
A12e ϕ
(12)
Substituting (12) into (8) and taking into account that
A = − σL , we obtain:
22 s A 12
p eϕ = −Qe ϕ with Q [ I HK
−1 = σ Ls
+ ] A 12 (13)
Then, we can make e ϕ → 0 by choosing
Q = q I with q > 0
(14)
that leads to the following relation to determine H :
−1
H = [ QA12 − σLs
I]
K
IV. CURRENT AND FLUX SLIDING-MODE
OBSERVERS WITH ADAPTIVE SCHEME
(15)
In order to take into account the parameter variations and
the absence of the speed sensor, the authors proposes an
adaptive scheme working in parallel with the sliding-mode
observers described in the previous section (Fig. 2).
The equations of the two observers become:
pi ˆ ˆ ˆ
s = A11is
+ A12
ˆ ϕ r + B1v
s + K sgn( ei
)
(16)
p ˆ ϕ ˆ ˆ
r = A 21 is
+ A22
ˆ ϕ r + Η sgn( ei
)
(17)
where ˆι s , ˆϕ r , e i , K , H are already defined and
A ˆ ij = Αij
+ ∆Aij
(i, j = 1,2) with ∆ A ij denoting the estimation
of the parameter and speed variations from their real values.
In particular, we consider that this adaptive scheme is able to
operate in two different ways. In the former, it gives the
estimated values of the resistive parameters Rˆ s and Rˆ rp ;
then, it can be seen as on-line identification scheme allowing
the realization of high performance drive systems, for which
the actual values of stator and rotor resistances are required.
In the latter way, the adaptive scheme provides the
estimation ωˆ of the rotor speed and allows the realization of
a sensorless speed control.
Fig. 2 Configuration of current and flux sliding-mode observers
working in parallel with the adaptive scheme
Therefore, in the case considered in this paper, the matrices
are specified as:
Âij
Rˆ
s + Rˆ
⎛
⎞
rp
Aˆ
11 = − I
⎜
Rˆ
Aˆ
1 rp ⎟
12 = I − ωˆ J
σL
⎜
⎟
s
σLs
M
⎝
⎠
A ˆ Rˆ
21 = Rˆ
rp I
Aˆ
rp
22 = − I + ωˆ J (18)
M
where Rˆ
s = Rs
+ ∆ Rs
, Rˆ
rp = Rrp
+ ∆ Rrp
and ˆ ω = ω + ∆ω
.
A. Adaptive Scheme for Resistive Parameters Identification
If a rotor speed measure is available ( ∆ ω = 0 ), from (1),
(16) and (18) we obtain the following current error equation:
pe i = ∆ A11is
+ A12eϕ
+ ∆A12
ˆ ϕr
+ K sgn( ei
) (19)
where
∆ Rs
+ ∆ Rrp
1 ⎛ ∆ Rrp
⎞
∆A11
= −
I ∆A
12 = ⎜ I − ω J ⎟
σL
s
σLs
⎝ M ⎠
The adaptive scheme can be derived by using the
Lyapunov’s stability theorem. We choose the following
positive definite Lyapunov function:
T
V2 = ( i ei
) / 2
2
Rs
/ 2ks
2
Rrp
/ 2krp
e + ∆ + ∆
(20)
where k s
and k rp
are positive constants.
The time derivative of V 2
becomes:
T
pV2 = ei
(p ei
) + ∆ Rs(p Rˆ
s /ks
) + ∆ Rrp(pRˆ
rp /krp
) (21)
Taking into account (19), we obtain
⎡ Rˆ
s 1 T
⎤
pV2
= pV1
+ ∆ Rs
⎢ p − ei
is
⎥ +
⎢⎣
ks
σLs
⎥⎦
(22)
⎡ Rˆ
rp 1 ˆ ⎤
T ϕ
+ R ⎢
r
∆ rp p − ei
( is
− ) ⎥
⎢ k σL M
⎣ rp s
⎥⎦
where the function pV 1 is given by (10).
Let the second term null and the third term null in (22), we
can find the following adaptive scheme for the resistive

parameters identification:
Rˆ
−1
ks
T
= p ( ei
is
)
(23)
σL
s
s
1 krp
T ˆ ϕr
Rˆ
rp = p
− [ i ( is
− )]
σLs
M
e (24)
By means of (23) and (24) the time derivative pV 2 is equal
to pV 1 and thus it is negative definite choosing the conditions
given in (11). In other words, the matrix K for the adaptive
current sliding-mode observer defined by (16) can be
designed on the basis of conditions (11).
Under the assumption that the sliding mode is attained and
the resistive parameters identification is completed, the
design strategy of the rotor flux gain matrix H is the same
described in the previous section, that led to relation (15).
B. Adaptive Scheme for Rotor Speed Estimation
In order to derive an adaptive scheme for rotor speed
estimation we define the following definite positive
Lyapunov function:
T 2
W = ( e ϕ eϕ
) / 2 + ∆ω
/ 2kω
(25)
where k ω is a positive constant.
The time derivative of W is expressed as
T
pW = e ϕ (p eϕ
) + ∆ω(p
ˆ ω /kω
)
(26)
By considering only the rotor speed as variable, we obtain:
∆ A11 = ∆A21
= 0, ∆A22
= − σLs
and ∆A12
= ∆ωJ.
So, the
current and flux error equations can be written as:
pe i = A12 eϕ
−(
∆ω /σ Ls
) J ˆ ϕr
+ K sgn( ei
) (27)
pe ϕ = − σLs
A12 eϕ
+ ∆ωJ
ˆ ϕr
+ H sgn( ei
)
(28)
In according with the equivalent control method, if the
current trajectories reach the sliding manifold, i.e. p e i = 0 ,
from (27) and (28) we obtain:
definite and the following equation for the speed estimation
is obtained:
1 qk
ˆ − ω T T
ω = p [ x K J ˆ ϕr
]
(32)
dσ L
s
V. SIMULATION RESULTS
In this section, the performance of the proposed observer is
presented via simulation results. Fig. 3 shows the Simulink
block diagram designed to simulate the drive system.
As shown in Fig. 1, the field-oriented controller is based on
a CR-VSI structure: the output of the speed regulator
represents the q-axis desired current i sqd
, while the field
weakening block gives the d-axis desired current i sdd;
a (d,q)
to (a,b,c) transformation provides the components i sad
, i sbd
and
i scd
necessary for the current regulators. These are fixed
frequency hysteresis regulators and give the inputs to the
VSI. Standard proportional plus integral controllers (with
anti-windup system) are used for speed and flux regulators.
The adaptive sliding-mode observer provides the rotor flux
position θˆ , necessary for the field orientation, and the flux
magnitude ˆφ r , used also to close the flux control loop as
described in relations (2) and (3).
The adaptive scheme can operate either as resistive
parameter identification or as rotor speed estimation.
The whole control system is discretized with a 10 µ s time
step. In order to simulate a microcontroller device
environment the “Adaptive SM Observer” uses a 20 µ s
sample time, the “Speed Controller” uses a 60 µ s sample
time and the “F.O.C” uses a 20 µ s sample time.
To further on investigate the implementation feasibility,
two standalone real-time applications for a 32-bit Generic
Embedded Processor has been developed from the Simulink
drive system model using Real-Time-Workshop (RTW).
The motor parameter nominal values are the following:
PW = 370 W P = 2 poles
V = 230 Volts I s = 1. 7 Amps
s
R s = 24.6 Ohm R = 16. 1Ohm
L = 39.8 mH L = 26. 1mH L = 1. 46 mH
s
r
r
M
1
eϕ = A12 − [( ∆ω /σ Ls
) J ˆ ϕr
− Kx]
(29)
−1
pe
ϕ = [ σLs
K + H]
x = QA12
Kx
(30)
where x = sgn( e )]
and Q is the matrix given in (14).
[ i
−1
T
12 ]
eq
Since [ A = A /d
12
with d = det [ A 12 ] > 0 and
Q = q I with q > 0 , the time derivative of W becomes:
q T T
q T T pωˆ
pW = − x K Kx + ∆ ω[
− x K J ˆ ϕr
+ ] (31)
d
dσ L
k
s
ω
The first term in (31) is negative definite; thus, imposing the
second term null, the whole function pW is negative
Fig. 3 High level block diagram of the drive system considered

A. Sliding- Mode Observer without Adaptive Scheme
The sliding mode observer is used to calculate the rotor
flux components in order to implement the field oriented
control. The motor operates under nominal torque load and it
is driven to a desired rotor speed of 200 rpm.
Fig. 4 shows the real and the estimated values of a
component of rotor flux when the parameters are assumed
exactly known. Fig. 5 shows the real and the estimated
values of the same component of rotor flux, when we
suppose the stator and the rotor resistances are affected by
+5% and +10% variations form their nominal values
respectively. It can be seen that the estimated flux is accurate
and robust with respect to the resistive parameters variations.
Fig. 6 Stator resistance identification, with 5% variation
from nominal value
Fig. 4 Real and estimated
φ r α
Fig. 7 Rotor resistance identification with 10% variation
from nominal value
Fig. 5 Real and estimated
φ r α under parameters variation
Fig. 8 Identification of a unknown stator resistance
B. Sliding Mode Observer with Adaptive Scheme
for Resistive Parameters Identification
The adaptive scheme, working in parallel with the sliding
mode observer, allows to obtain the correct value of resistive
parameters, which can change from the nominal ones
according to motor temperature and operating condition.
Fig. 6 and 7 show the results of the identification adaptive
scheme when a step parameter variation of the two resistive
parameters is considered. Fig. 8 and 9 show how the
proposed adaptive scheme is able to estimate the correct
values of the stator and rotor resistances also when they are
supposed completely unknown.
Fig. 9 Identification of a unknown rotor resistance

C. Sliding Mode Observer with Adaptive Scheme
for Rotor Speed Estimation
This adaptive scheme allows to eliminate the speed sensor
using an estimation of the rotor speed in place of the
measured one.
Fig. 10 and 11 show the transient behaviour of the system
in two particular operating conditions: speed reversion and
motor braking, when the motor is under nominal load torque.
Finally, Fig. 12 refers to a real-time simulation done using
the standalone application developed from the Simulink
model of this scheme.
VI. CONCLUSION
In this paper a particular adaptive scheme working in
parallel with a sliding-mode observer has been developed.
The Lyapunov’s stability theorem has been used in the
development, granting the stability of the adaptive slidingmode
observer.
This adaptive scheme cannot estimate simultaneously the
resistive parameters and the speed, because the estimation is
done using only the stator electrical quantities and so the
error signal suitable to be used as “engine” for the estimation
mechanism, is not enough. Thus, the proposed configuration
can be used either when the speed is measured but the exact
values of stator and rotor resistances are required as in high
performance drives or to realize a sensorless control where
the characteristics of robustness and parameter insensitivity
of the sliding-mode approach are considered adequate.
The paper has shown some simulation results, based on the
development of a Matlab/Simulink/Real-Time-Workshop
model, which have proved the validity of the proposed
scheme in different significant operating conditions.
V. REFERENCES
Fig. 10 Reference and estimated speeds
Fig. 11 Reference and estimated speeds
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Fig. 12 Reference and estimated speed in real-time simulation