INTRODUCTION
Due to lower cost and greater reliability, the Induction Motors (IM) become
more and more popular for motion control applications. However, the control
of IM is proved very difficult because the dynamic model of the IM is non linear,
multidimensional and complex where some parameters vary with temperature, skin
effect or saturation. This nonlinear dynamic behavior induces the use of nonlinear
feedback control strategy (Yazdanpanah et al., 2008;
Mohanty et al., 2002; Asseu
et al., 2008) in order to permit a decoupling of the IM variables
in a fieldoriented (d, q) coordinate so, that stator currents can be separately
controlled and then to control independently the generated torque and the rotor
flux. On the other hand, to preserve and improve the reliability under parameters
variation and noises injected by the inverter (which can induce a statespace
“coupling„ and degradation of the system), a robust control approach
has been made on the motor drives. This control algorithm uses Hinfinity synthesis
of currents correctors in order to insure robust stability and performances
of the inner current loop.
To guarantee good performances in presence of parameters variations (more specifically
the rotor time constant variation) and whereas, advanced induction machine control
strategies require knowledge of the instantaneous flux (which is not usually
measurable or difficult to access), the technique based on the state observer
allowing an online estimation of the fluxes and the rotor time constant is
necessary.
Accurate estimation of flux in the presence of measurement and system noise
and parameter variations is a challenging task. Kalman Filter (KF) named after
Rudolph E. Kalman (Kalman, 1960) is one of the most well
known and often used tools for stochastic estimation. The KF is essentially
a set of mathematical equations (Mohanty and Patra, 2005;
Blanchard et al., 2007) that implement a predictorcorrector
type estimator that is optimal in the sense that it minimizes the estimated
error covariance when some presumed conditions are met. For the flux estimation
problem of IM, where parameter variation and measurement noise is present, KF
is the ideal one.
Many literatures on the KF technique and its applications, essentially extended
for the estimation of the speed, have been published (Hilairet
et al., 2009; Murat et al., 2007;
Shi et al., 2002). However, this Extended KF
(EKF) technique doesn’t take into account the rotor time constant variation.
In the present research, after a brief review of the induction motor model, on the one hand a robust inputoutput linearization and decoupling scheme is developed and on the other hand, a fifthorder discretetime EKF, based on KF principle, is proposed to estimate the fluxes, currents and extended for the rotor time constant reconstruction.
Finally, the proposed combination nonlinear feedback control and EKF approach are confirmed by simulations and experimental results carried out on IM drive system in the presence of measurement noise and parameter variations.
MATERIALS AND METHODS
General Test Bench Description
This research project, conducted in the Laboratory of applied Electrical
and Electronic (INPHB Yamoussoukro, Côte d’Ivoire) from July 2007
to June 2009 by a theoretical work, has been implemented and validated in realtime
on a test bench (Fig. 1). Globally the test bench is composed
of electrical fittings, as follows:
• 
A PCboard Pentium 4 which contains the Dspace software (Cockpit, Trace
…) and MatlabSimulink environment 
• 
A 5.5 kW IM coupled with a powder brake completed by current and voltage
sensors 
• 
An ASIC card used as communication interface between the PC and a 15 kW
three phase static inverter supplied by a voltage source which provides
about 0400V with current limitation of about 6 A (Fig. 2) 

Fig. 1: 
A global view of the test bench 

Fig. 2: 
A view of the inverter and ASIC card 
Induction Motor Model
The classical dynamic model of the IM in a (d, q) synchronous reference
frame can be represented as a timevarying fourth order system given by De
Fornel and Louis (2007), with as state variables the stator currents (I_{ds},
I_{qs}) and the rotor fluxes (Φ_{dr} Φ_{qr}):
The electromagnetic torque defined in terms of x is:
Moreover, by choosing a rotating reference frame (d, q) so that the direction
of axe d is always coincident with the direction of the rotor flux representative
vector (field orientation), it is well known that this rotor field orientation
in a rotating synchronous reference frame realizes:
Φ_{dr} = Φ_{r} = Constant
and Φ_{qr }= 0 
Thus, the dynamic model of the IM, completed with the output equation, can
be rewritten as:
This Eq. 3 shows that the dynamic model of the IM can be
represented as a nonlinear funct ion of the rotor time constant. A variation
of this parameter can induce, for the IM, a lack of field orientation, performance
and stability. Thus, to preserve the reliability, robustness performance and
stability of the system under parameters variation (in particular the rotor
time constant variations) and measurement noise, we can use a robust feedback
linearization strategy to regulate the motor states.
Robust Feedback Control
In order to control stator currents and then control independently the generated
torque and the rotor flux, an inputoutput linearization approach is used in
this study. We can see that the system Eq. 3 has relative
degree r_{1} = r_{2} = 1 and can be transformed into a linear
and controllable system by chosen the feedback control given by the following
form:
The matrix Δ(x) is nonsingular, so the linearizing control Eq.
4 is effective and leads the system Eq. 3 to two decoupled
subsystems, each one is composed of an integrator.
The new inputs of the obtained decoupled systems are v_{1} and v_{2}.
It can be determined by chosen a first order dynamic behaviour with a time constant
T = 1/K. Two block diagrams for the control of I_{ds} and I_{qs}
can be summarized as follows:
where, i = 1, 2 corresponding, respectively to I_{ds} and I_{qs} with P(s) = 1/ (1+ T.s) .
In order to take into account the physical parameter variations and noises,
a second loop using robust control approach has been added. The new block diagram
structure is given by:
where, the controller C(s), using H_{∞} synthesis and Doyle method
(Doyle et al., 1992) is defined as:
The real t_{0} is an adjusting positive parameter, chosen adequately small (t_{0} < 1), in order to satisfy the robustness performance, to have a good regulation and convergence of the currents.
However, the control of an IM generally required the knowledge of the instantaneous
flux of the rotor that is not measurable. Also, a variation of the rotor resistance
can induce a lack of field orientation. In order to achieve better dynamic performance,
an online estimation of rotor fluxes and resistance is necessary. Thus, an
Extended Sliding Mode Observer (ESMO) is widely used (Asseu
et al., 2008; Amuliu and Ali, 2007) with success
for online estimation of rotor fluxes (Φ_{dr, }Φ_{qr}),
rotor time constant (σ_{r} =1/T_{r} = R_{r}/L_{r})
and torque (C_{em}).
In this study, in order to compare with the ESMO and respect to the rotor time constant variations in the presence of measurement and system noise (stochastic estimation), an EKF’s algorithm for flux estimation and extended for the rotor time constant reconstruction is presented and explained.
Model of Extended Kalman Filter
For parameter estimation using a full order EKF, the model structure given
by Eq.1 is discretized directly using Euler approximation
(1^{st} order) proposed by Lewis (1992). Furthermore,
the state vector is extended to rotor time constant. Thus, the new discretetime
and stochastic fifthorder nonlinear model is given by:
where:
where, x(k) and y(k) are the state vector and output, respectively at the kth
sampling instant, i.e., t = k.Te with T_{e} the adequate sampling period
chosen without failing the stability and the accuracy of the discretetime model.
n(k) represents the random disturbance input; it is the sum of modeling uncertainty,
the discretization errors and the system noise. r(k) is the measurement noise.
Both n(k) and r(k), are assumed to be white Gaussian noise with zero mean and
covariance matrix Q and R, respectively. Consider that:
• 

: 
The estimate of x_{e}(k) and K(k + 1) = EKF gain 
• 

: 
The linear minimum mean square estimate of x_{e}(k+1) 
• 
P(k +1 k) 
: 
State prediction covariance error 
• 
P(k+1k+1) 
: 
State estimation covariance error 
• 
Initialization givens 
: 
and P(00)=P(0) 
The steps of the proposed fifthorder EKF algorithm are as follows:
The EKF algorithm consists of repeated use of step (18) for each measurement.
F(k) is the Jacobian matrix of partial derivatives of f(•) with respect
to x_{e}(k). From Eq. 6, we obtain:
RESULTS
In order to verify the feasibility of the proposed EKF, the simulation on SIMULINK
from Mathwork has been carried out for a 5.5 kW induction motor controlled with
a linearization via feedback algorithm (Fig. 3). The nominal
electrical parameters of the IM, estimated by means of the identification techniques
proposed by the authors in the references (Leite et al.,
2003; De Fornel and Louis, 2007), are shown in the
Table 1.
The EKF algorithm is implanted in a S_function using C language. In order to
evaluate its performances and effectiveness, the comparisons between the observed
state variables and the simulated ones have been realized for several operating
conditions with the presence of about 10% noise on the simulated currents.
Table 1: 
Nominal parameters of the induction motor 


Fig. 3: 
Simulation scheme 
Thus, the simulations are obtained at first in the nominal case with the nominal
parameters of the IM used to realize vector control orientation and then, in
the second case, with 50% variation of the nominal rotor time constant (σ_{r}
= 1.5σ_{rn}) in order to verify the rotor time constant tracking
and flux estimation.
Initialization and Tuning of the EKF Algorithms
The full order EKF is initialized as follows: P_{5x5}(0) = diag
{10¯^{6}} and x_{e}(0) = [ 0 0 0 0 σ_{rn}
]^{T} with σ_{rn} = R_{rn}/L_{rn.}. Furthermore,
the measurement noise covariance matrix has been fixed as follows: R_{2x2}
= diag {10^{3}} and the system covariance matrix can be adjusted by:
Q_{5x5} = diag {q_{i}} with q_{i}
∈ [0 1] , i = 1, .., 5 
The real q_{i} must be tuned adequately small (q_{i} < 1) in order to have a good performance, convergence and considerable rapidity of the EKF. Our proposed Feedback control and EKF algorithm operate with a sampling period Te = 1 ms and using Euler approximation.
Simulation Results
Figure 4 and 5 show the simulation results
for a step variation of the currents reference (I_{dsRef} and I_{qsRef}).
One can see that in both nominal (Fig. 4) and nonnominal
cases (Fig. 5), the estimated values of currents, fluxes,
torque and rotor time constant converge very well to their simulated values.
The observed fluxes (Fig. 4b, 5b) indicate
the good orientation (Φ_{dr} is constant and Φ_{qr}
converges to zero) due to a favorable rotor time constant estimation (Fig.
4d, 5d). In the case of rotor resistance variation where
Rr = 1.5R_{rn }(Fig. 5), we can see a light perturbation
on the flux orientation (Fig. 5b) tied to the starting of
the IM (at the instant t =2 sec) when a step variation in I_{qs} (0
to 6 A) is applied in order to generate the required torque. However, after
the IM starting of the motor (at the instant t>3 sec), the flux orientation
and currents responses remain acceptable despite a step change in the current
I_{qs} (from 6 to 4 A at the instant t = 5 sec).
These waveforms illustrate the fast convergence and high performance of the robust decoupling control and EKF against rotor time constant variations and noises.
Experimental Results
Finally, the implementation in realtime of the proposed scheme is carried
out on the testing bench given by the Fig. 1. Figure
6 shows the experimental setup.

Fig. 4: 
Nominal case (Rr = Rrn): Comparison between estimated and
simulated values 

Fig. 5: 
Non Nominal case (Rr = 1.5HRrn) with the presence of noises 

Fig. 6: 
Experimental configuration diagram 
This Fig. 6 is composed of a 5.5 kW IM, a powder brake with
load torque measurements, three LEM current sensors and a 2000 point incremental
encoder. A PCboard and a Dspace1102 combined a TMS320C31/40Mhz are used to
implement PWM function and control algorithms. The PWM and the position measurement
work at 10 kHz.
Two kinds of tests have been performed (without and with load torque) in order to compare the simulation and experimental results and show the robustness of the nonlinear control and EKF algorithm against modeling uncertainty and measurement noise.
The first one (Fig. 7) illustrates the results where the
motor speed is regulated at Ω_{n} = 800 rpm without load torque
(C_{l }≅ 0 N.m ) and with a stator current reference I_{dsRef}
= 4 A.
In the second one (Fig. 8), a step change in the stator current
reference I_{qsRef} (from 3.5 to 2.5 A) has been made for a motor speed
regulated at 1000 rpm with a load torque C_{l} = 1.5 Nm.
For each test, measured and observed values have been registered and have been
compared to simulation values. Better estimation performance yielded by the
proposed EKF is obvious from the experimental results. Thus, it can be seen
that the experimental waves are quite similar to the simulation ones. The experimental
observed fluxes (Fig.7e, 8e) indicate a
good orientation (the flux Φ_{qr} converges well to zero) which
is due to a favourable rotor time constant estimation (Fig. 7f).
Here, the rotor time constant effectively drifts with the overheating of the
IM because its estimated value is inferior to the nominal one (σ_{rn}
= 10.16 sec¯^{1}). The experimental estimated torque (Fig.
7g, 8f) is in good agreement with the simulated value.
The weak perturbations on the experimented fluxes or torque are probably tied
to position noises and the inverter.
Moreover, Fig. 8 proves the effectiveness of the proposed
feedback linearization strategy in order to decouple the stator currents and
then control independently the generated torque and the rotor flux. The currents
I_{ds} and I_{qs} are controlled at 2 and 3.5 A, respectively.
A step variation in I_{qs} (3.5 to 2.5A) is applied (Fig.
8a, b) in order to generate a torque change (Fig.
8f). The waveforms show the good uncoupling between the currents I_{ds}
and I_{qs}. Thus, the field orientation and the synthesis of robust
linearization and decoupling control are well verified. The agreement between
the experimental dynamic performance and the simulated ones is demonstrated.
The performance of the EKF algorithm has proved to be as good as the one obtained
with the ESMO against parameter variation and noise.


Fig. 7: 
ah) Results for regulating the speed motor to 800 rpm without
load torque C1 @ 0 N.m 


Fig. 8: 
(af) Results for regulating the speed motor to 1000 rpm with a load torque
C1 @ 1.5 N.m and a step change of current Iqs (from 3.5 to 2.05 A) 
CONCLUSION
The case study that was presented in this research was chosen as a proof
of concept and shows that the combination feedback control and EKF approach
works well and is quite robust. For this purpose, rotor flux, torque and rotor
time constant in the IM are estimated using Extended Kalman filter. Torque and
rotor flux are decoupled and the IM model is linearized using a robust feedback
linearization approach. Sensorless control of the linearized and decoupled drive
using estimated flux and rotor time constant is simulated and carried out on
the testing bench. The good simulation and experimental results obtained on
the Induction motor show the effectiveness and the stability of this robust
decoupling control and EKF algorithm is found to be very good and fast for flux
and rotor time constant estimation in the presence of system and measurement
noise. Thus, the dynamic response of IM sensorless drive control using an EKF
flux estimation technique is as fast as that of drives with physical sensors.
NOMENCLATURES
C_{em}, C_{l} 
: 
Electromagnetic and load torques (N.m) 
I_{ds}, I_{qs} 
: 
Stationary frame (d, q)axis stator currents (A) 
I_{dr}, I_{qr,} I_{mr} 
: 
Stationary frame (d, q)axis rotor currents and rotor magnetizing current
(A) 
p, J, f: p 
: 
Pole pair No.; J : Inertia (kg m¯^{2}); f : friction coefficient
(Nm.s rad¯^{1}) 
L_{r}, L_{s}, L_{m}, L_{f} 
: 
Rotor, stator, mutual and leakage inductances (H) 
R_{s}, R_{r} 
: 
Stator and rotor referred resistance (Ω) 
T_{e} , T_{r} , T_{s} 
: 
Sampling period, rotor and stator time constant: T_{r} = L_{r}
/R_{r} ; T_{s} = L_{s}/R_{s} (s) 
V_{ds}, V_{qs} 
: 
Stationary frame d and qaxis stator voltage (V) 
Φ_{dr}, Φ_{qr}, Φ_{ds},
Φ_{qs} 
: 
dq components of rotor fluxes (Φ_{dr}, Φ_{qr})
and stator fluxes (Φ_{ds}, Φ_{qs}), (Wb) 
ω_{s}, ω_{r}, ω_{sl} 
: 
Stator, rotor and slip pulsation (or speed), (rad s¯^{1}) 