INTRODUCTION
Sensorless control methods have been making remarkable developments in the
most recent year, due to lower cost and greater reliability without mounting
problems (Mohanty et al., 2002; Menaa
et al., 2008). Flux estimation methods are being used that avoid
the flux measurement setup and commercial sensorless vectorcontrolled drives
are already available.
This study presents an innovative strategy to the problem of nonlinear estimation of states for Induction Motors (IM) sensorless control.
The naturally structure of nonlinear multivariable state of IM models induces
the use of robust feedback linearization strategy in order to permit a decoupling
and good dynamic stability of the IM variables in a fieldoriented (d, q) coordinate
so that rotor flux and speed can be separately and independently controlled
(Asseu et al., 2008; Mohanty
and Patra, 2005; Yazdanpanah et al., 2008).
However, this feedback control strategy requires the knowledge of rotor fluxes which are not usually measurable in practice. Also, a variation of the rotor time constant in the IM can induce a lack of field orientation and a statespace coupling, which can involve a degradation of the system. Thus, In order to achieve better dynamic performance, an estimation of rotor time constant and flux is necessary.
An approach, proposed by Hilairet et al. (2009),
Murat et al. (2007) and Shi
et al. (2002) to estimate with success the state variables in an
IM in the presence of modeling uncertainty, measurement and system noises (stochastic
estimation), is the use of the fullorder EKF. This latter provides not only
the unmeasurable state variable estimation (fluxes) but also the estimation
of the measurable parameters (currents and speed). However, the determination
of the measurable parameters estimation imposes some estimation algorithms very
long and usually sophisticated. Therefore, in order to reduce the computation
rate of the estimation algorithms, the measured parameters estimation is not
necessary.
In this study, in order to compare with the full order EKF and respect to the
rotor time constant variations in the presence of measurement and system noises,
a new approach using a reduced order EKF (REFK) is presented to solve only and
specially the problem of the unmeasurable parameters estimation (rotor time
constant and flux). Consequently, practical and important improvements are achieved
with respect to the well known drawbacks associated to the EKF, like the computational
effort for realtime applications, the complexity and the hard tuning of the
covariance matrices. In fact, with the 3rd order EKF that is obtained, the dimension
of all matrices of the algorithm becomes small enough from a practical point
of view.
After a brief review of the IM model, the simulation results for a 1.8 kW induction motor drive system are presented to validate the high robustness of the proposed REKF approach against parameter variations, measurement and system noises.
MATERIALS AND METHODS
Induction motor model: This research project, conducted in the Laboratory of Applied Electrical and Electronic (INPHB Yamoussoukro, Côte d’Ivoire) from May 2009 to December 2009 by a theoretical work, has been confirmed by simulations results on an induction machine.
By assuming that the saturation of the magnetic parts and the hysteresis phenomenon
are neglected, the classical dynamic model of the induction motor in a (d, q)
synchronous reference frame can be described by De Fornel and
Louis (2007):
The load mechanical equation is:
The application of (1ac) returns a system of fifthorder nonlinear differential equation, with as state variables the stator currents (I_{ds}, I_{qs}), the rotor fluxes (Φ_{dr}, Φ_{qr}) and the speed (ω_{r}). Assume that among the state variable, only the stator currents and the speed are measurable.
Thus, the IM model can be rewritten as:
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, it is well known that this rotor field orientation in a rotating synchronous reference frame realizes:
From the expressions Eq. 1c, 2 and 3,
one can write:
This relation (Eq. 4) shows that the IM dynamic model can
be represented as a nonlinear function 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 and robustness stability under
the rotor time constant variation, a robust inputoutput linearization via feedback
control, proposed by Asseu et al. (2009) is used
to provide a good regulation and convergence of the rotor flux (Φ_{r
}) and speed (ω_{r}) for the IM drive. However, since, the
resolution of the feedback control for the IM requires the knowledge of the
rotor flux value that is not measurable, an online estimation of rotor fluxes
is necessary.
Thus, in order to take into account the rotor time constant variations, measurement and system noises (stochastic estimation) and then reduce the execution time of the estimation algorithm, this study uses a reducedorder EKF method to provide only the estimation of rotor fluxes and rotor time constant (considering that the currents and speed are already measured).
Reducedorder extended kalman filter model: Let us consider the dynamic model of the IM given by the system Eq. 2. The currents (I_{ds}, I_{qs}) and speed (ω_{r}) estimation is not necessary since, they are measurable. Thus, in order to estimate only the rotor flux (Φ_{dr}, Φ_{qr}) and rotor time constant (σ_{r} = R_{r}/L_{r}), a reduced dimensional state vector extended to rotor time constant defined by x = [Φ_{dr} Φ_{qr} σ_{r} ]^{T} =[ x_{1} x_{2} x_{3}]^{T} has been introduced. The corresponding threedimensional extended state space equation obtained is:
where, v(t) = [I_{ds}, I_{qs}]^{T} is the new input vector.
where, ε presents the slow variation of σ_{r}. The fact that the state vector only consists of the rotor time constant and flux offers an advantage namely the reduction of the computational volume and complexity. Thus, the rotor time constant and flux can be more easily and rapidly estimated.
For parameter estimation using a REKF, the model structure given by Eq.
5 is directly discretized by means of Euler’s approximation (2nd order)
proposed in (Lewis, 1992). Thus, the new discretetime
and stochastic nonlinear reduced order model is given by:
where,
where, y(k) is the output equation 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.
The 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.
Let us assume that the output vector, without the measurement noise, is given by:
It results:
with,
From the electrical equations Eq. 1a of the IM and the expression
Eq. 7, an approximate discretetime relation (1st order) of
the fluxes is given by:
The Eq. 8 is composed of recurrence relations (between the rotor fluxes at the k+1th instant and their value at the kth instant) which can be obtained by means of the measurements of the voltage, stator currents and speed, as shown in Eq. 9.
Thus, for the output vector given by: y(k+1) = z(k+1) + r(k), the state vector
is estimated in order to minimize the prediction error: where,
with:
As said in Eq. 6, r(k) represents the noises (white Gaussian noise) present on the currents, voltage or speed transducers, or the errors of discretization and measurement.
Finally, the proposed REKF algorithm, which a general form is presented in the appendix, applied to the nonlinear model described by Eq. 6 can be defined as follows:
where, the prediction vector is:
with .
L(k) is the Jacobian matrix of partial derivatives of J(•) with respect
to .
From Eq. 12, L(k) is given by:
Thus, once the fluxes and rotor time constant are estimated, from the Eq.1c,
we can deduce the estimated torque given by:
RESULTS
The proposed REKF algorithm, controlled by a robust feedback linearization
strategy (Fig. 1), has been investigated with simulation tests
carried out for a 1.8 kW IM by means of SIMULINK in order to illustrate its
effectiveness against measurement noise and parameter variation. The nominal
electrical parameters of the IM, estimated by means of the identification techniques
proposed by Leite et al. (2003) and
De Fornel and Louis (2007) are shown in the Table 1.
Thus, the REKF algorithm is implanted in a S_function using C language. In order to evaluate its performances, the comparisons between the observed state variables and the simulated ones have been realized for several operating conditions in the presence of about 15% white noise on the measured currents (Fig. 2) and with a load torque (C_{l}>1 N.m).
Thus, the simulations are obtained at first in the nominal case with the nominal parameters of the IM (Table 1) used to realize vector control orientation and the feedback linearization and then in the second case, with 50% of the nominal rotor time constant (σ_{r} = 1.5σ_{rn}) in
order to verify the behavior of the proposed REKF algorithms with respect to rotor time constant variation.
Initialization and tuning of the reduced order EKF algorithm: The reduced
order EKF is initialized as follows (j, i = 1, 2, 3): P_{3x3}(0) = diag{p_{ij}}
with p_{11} = p_{22} = 10^{6}, p_{33} =1; x(0)
= [ 0 0 σ_{rn} ]^{T} with σ_{rn} = R_{rn}
/ L_{rn.}
The system covariance matrix can be adjusted by: Q_{3x3}= diag{q_{ij}} with q_{11} = q_{22} = 10^{3}, q_{33} = 1 and the measurement noise covariance matrix has been fixed as follows: R_{2x2} = diag{r =10}.
The three positive gains (p_{,} q and r) must be adequately tuned in order to have a good performance, convergence and considerable rapidity of the reduced order EKF. Our proposed Feedback control and REKF operate at 1 m sec sampling period using Euler approximation.
Simulation results: Figure 3 and 4
show the simulation results for a step variation of the rotor flux and speed
(Φ_{drRef} and ω_{rRef}). One can see that in both
nominal (Fig. 3) and nonnominal cases (Fig.
4), the estimated values of fluxes, torque and rotor time constant converge
very well to their simulated values. The observed fluxes (Fig.
3a, 4a) indicate the good orientation (Φ_{dr}
is constant and Φ_{qr} converges to zero) due to a favorable rotor
time constant estimation (Fig. 3b, 4b).
Also, we can see an absence or a rejection of noises on the fluxes.
Furthermore these results show the good uncoupling between the flux (Φ_{dr})
and the speed (ω_{r}) because a step variation in ω_{r}
(50π to 35π rad sec^{1}), in order to generate a torque change,
can not influence on the flux response that remains acceptable (the field orientation
is well maintained).

Fig. 2:  Presence
of white noises on the measured current I_{ds} 

Fig. 3:  (ad)
Nominal case (R_{r} = R_{rn}): Comparison between estimated
and simulated values 

Fig. 4:  (ad)
Non nominal case (R_{r} = 1.5*R_{rn}): Comparison between
estimated and simulated values 
These waveforms illustrate the fast convergence and high performance of the robust feedback decoupling control and REKF algorithm against rotor time constant variations and measurement noises.
CONCLUSION
In this study, a robust feedback linearization strategy and a REKF algorithm
are used not only to decouple and then control independently the rotor flux
and the speed (or the generated torque) of the IM in a fieldoriented (d, q)
coordinate but also to provide the unmeasurable state variable estimation (flux,
rotor time constant and torque). A series of simulations tests have been achieved
on the induction motor. The results obtained have demonstrated a good performance
of this robust decoupling control and REKF algorithm against rotor time constant
variations, measured noise and load torque. The main conclusion and contribution
of this work is that the wellknown drawbacks of the full order EKF, like heavy
computational effort for realtime applications, complexity and hard tuning
of noise covariance matrices are widely overcome using the proposed reduced
order EKF. In fact, the execution time of the REKF algorithm is about half of
the full order Extended KF (Asseu et al., 2010).
Thus, in the industrial applications, one will appreciate very well the experimental
implement of this robust estimator for the reconstitution of the fluxes and
the torque as well as the rotor resistance.
APPENDIX
Steps of the EKF algorithm (Blanchard et al., 2007):
The EKF algorithm consists of repeated use of step (19) for each measurement.
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 rad sec^{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 sec^{1}) 

: 
=the estimate of x(k); =
the prediction vector 

: 
The state covariance matrix of prediction error and estimation error 
G(k + 1) 
: 
Kalman filter gain 

: 
The measurement of prediction error; ε(k+1) 