INTRODUCTION
During this last decade, the Induction Motor (IM) is widely used
in industrial applications due to its reasonable cost, robust qualities
and simple maintenance. To effectively control the torque dynamics of
an Induction Motor, it is necessary to use more elaborate control strategies.
The control usually used is the Field Oriented Control (FOC) (Blaschke,
1972) which allows the control of the torque transient. Since the first
work of Blaschke at the beginning of the 70′s, improvements and alternatives
of the FOC appeared (Montanari et al., 2000; RonceroSanchez et
al., 2007). Moreover, the naturally structure of nonlinear and multivariable
state of IM models induced the use of the nonlinear control methods and
in particular the techniques of inputoutputs linearization and decoupling
(Isidori, 1989). Many results have been published such as (Marino et
al., 1993; Mohanty et al., 2002). However, this technique required
the knowledge of the rotor fluxes which are not usually measurable in
practice. An observer for estimation of the rotor fluxes is necessary.
Furthermore, the control quality depends on the model accuracy. A variation
of the rotor resistance (which varies with temperature) can induce a statespace
coupling which can induce a performance degradation of the system. In
order to achieve better system dynamic performance, the approach proposed
in this paper consists in, on the one hand, synthesizing robust controllers
combined with inputoutput decoupling and on the other hand, to design
extended observers allowing an online estimation of rotor time constant.
The Extended Kalman Filter presented in El Moucary et al. (1999),
Said et al. (2000) and Chbeb et al. (2006) can be used for
realtime estimation of rotor fluxes and resistance. Unfortunately the
initialization and the optimal choice of covariance and gain matrix are
delicate. These matrixes play a critical role in robustness of the Extended
Kalman Filter (EFK).
Another approach proposed in Benchaib et al. (1999), Derdiyok
(2005) and Amuliu and Ali (2007) to estimate the state variables in an
IM is the use of Sliding Mode Observer (SMO). This continuousmode approach
observer, based on the variable structure system theory, has been known
to produce excellent functional performances and robustness. Some gains
of this observer can be easily adjusted compared with the EKF.
This research proposes a sixdimensional Discretetime Extended Sliding
Mode Observer (DESMO) to provide not only rotor fluxes estimation but
also the estimations of the rotor time constant and torque in the induction
motor.
ROBUST DECOUPLING CONTROL
Model of Induction Motor
By assuming that the saturation of the magnetic parts and the hysteresis
phenomenon are neglected, the dynamic model of the induction motor in
a (d, q) synchronous reference frame can be described by a fifthorder
nonlinear differential equation, with as state variables the stator currents
(I_{ds}, I_{qs}), the rotor fluxes (Φ_{dr},
Φ_{qr}) and the rotor pulsation (ω_{r}):
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:
Thus the dynamic model of the IM, completed with the output equation,
can be rewritten as:
with 

(3) 


From the expressions (1) and (2), one can write:
This relation (4) shows that the dynamic model of the IM 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 orientation, performance and
stability. Thus the next section uses a feedback linearization strategy
and a robust controller to regulate the motor states with respect to the
parameter variations and disturbances.
Robust Inputoutput Linearization via Feedback for a Nonlinear System
The aim of this research is to show how we can analyze the synthesis
of feedback control for the nonlinear dynamic model of the IM given by
the system (3) and (4). Thus, we can see that the system (3) has relative
degree r_{1} = r_{2} = 2 and can be transformed into a
linear and controllable system by chosen:
• 
A suitable change of coordinates z = Ψ (x) given by:
z_{1} = h_{1}(x); z_{2} = L_{f} h_{1}(x);
z_{3} = h_{2}(x); z_{4} = L_{f} h_{2}(x); 
• 
And the feedback control having the following form: 
where, v_{1} and v_{2} are the new inputs of the obtained
decoupled systems.
The Jacobian matrix of the transformation thus defined as:
is nonsingular for all x such that Φ_{r} ≠ 0 and in the
new coordinates, the system appears as:
which is linear and controllable. The block diagram of system (5) is
as follows :
In order to impose, after a closed loop, a second order dynamic behaviour
defined by H(s):
the two new block diagram structure for the control of (Φ_{r},
ω_{r}) can be chosen by:
with i = 1, 2 corresponding respectively to Φ_{r }and ω_{r}
On the other hand, to preserve the reliability and stability of the system
under parameters variation and noises, a second loop using robust control
approach has been added on the motor drives. This control algorithm uses
H_{∞} synthesis and Doyle method presented in (Doyle et
al., 1992; Asseu, 2000), to define two robust controllers C(s) in
order to realize the regulation of the rotor flux and speed (Φ_{r},
ω_{r}).
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 rotor flux and speed.
However the control of an induction motor 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. Here, a sixdimensional extended
sliding mode observer is proposed for online estimation of rotor fluxes
(Φ_{dr, }Φ_{qr}), torque (C_{em}), speed
(ω_{r}) and rotor time constant (σ_{r} = 1/T_{r}
= R_{r}/L_{r}).
EXTENDED SLIDING MODE OBSERVER
Classical Sliding Mode Observer
Let us consider the dynamic model of the IM given by the system (1). Assume
that among the state variable, only the stator currents noted z_{1},
z_{2} and the rotor speed ω_{r} are measurable. Denote
and
the estimates of the fluxes Φ_{dr} and Φ_{qr}. Consider
that and
are
the estimates of the stator currents I_{ds}, I_{qs} and rotor
speed ω_{r}. The SMO is a copy of the model (1) by adding corrector
gains with switching terms:
where, Γ_{1}, Γ_{2} and Λ_{1},
Λ_{2, }Λ_{3} are the observer gains. The switching
I_{s} is defined as:
Setting
the estimation error dynamics is given by:
The condition for convergence is verified by chosen the following observer
gain matrices:
where, q and n are two positive adjusting parameters which play a critical
role in the stability and the velocity of the observer convergence. From
the fluxes estimation, it is easy to deduce the estimated torque defined
by:
As previously underline, a variation of the rotor resistance can induce performance
degradation of the system.
This research will present an Extended Sliding Mode Observer for the IM to
solve at the same time the problem of the rotor fluxes and rotor time constant
estimations.
Extended Sliding Mode Observer
In order to estimate the rotor time constant, a sixdimensional extended
state vector defined by X_{e} = [I_{ds} I_{qs}
Φ_{dr} Φ_{qr} ω_{r} σ_{r}
]^{t} = [z_{1 }z_{2} x_{1} x_{2 }z_{3}
x_{3}]^{t} has been introduced with σ_{r}
= R_{r}/L_{r}. The corresponding extended state space
equation become:
where, ε presents the slow variation of σ_{r}. The
proposed ESMO has the following form:
where, is, Γ_{1}, Γ_{2} and Λ_{1}, Λ_{2,
}Λ_{3} are, respectively defined by Eq. 8
and 10.
To determine observer gain Γ_{3}, it can be supposed that the
observation errors of the fluxes converge to zero.
The estimation error dynamics of the fluxes
are then given by:
By replacing the expressions of Γ_{1} and Γ_{2},
we obtain:
The estimation error dynamics of the rotor time constant is given by:
We can see that this error dynamics is locally and exponentially stable
by chosen:
The proposed extended sliding mode observer has been implemented using
Euler approximation.
The Discretetime Extended Sliding Mode Observer (DESMO) should be written
as:
where, k means the kth sampling time, i.e., t = k.Te with T_{e}
the sampling period and
From the expression s defined in Eq. 10 and 14,
it can be seen that there are three positive adjusting gains: q, n and
m which play a critical role in the potential stability of the scheme
with respect to rotor time constant estimation. These three adjusting
gainsmust be chosen so that the estimator satisfies robustness properties,
global or local stability, good accuracy and considerable rapidity.
SIMULATED RESULTS
In order to verify the feasibility of the proposed DESMO, the simulation
on SIMULINK from Mathwork has been carried out for a 1.8 kW induction
motor controlled with a robust linearization via feedback algorithm (Fig.
1). The nominal parameters of the induction motor are shown in the
Table 1.
The DESMO 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 15% noise on the simulated currents or speed.

Fig. 1: 
Stimulation scheme 





Fig. 2 (a, b, c, d, e) : 
Nominal case (Rr = Rrn) 





Fig. 3(a, b, c, d, e) : 
Non Nominal case (Rr = 1.5Rrn) 
Thus, using a sampling period Te = 2 ms, the simulations are realized at first
in the nominal case with the nominal parameters of the induction motor (Table
1) 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.
Figure 2 and 3 shows the simulation
results for a step input of the rotor speed and flux. One can see that
in both nominal (Fig. 2ae) or nonnominal
(Fig. 3ae) cases, the estimated
values of currents, fluxes, torque and speed converge very well to their
simulated values.
The observed fluxes (Fig. 2d) indicate the good orientation
(Φ_{dr} is constant and Φ_{qr} converges to
zero) which is due to a favorable rotor time constant estimation (Fig.
2a, 3a). The estimated torque (Fig.
2b) is in good agreement with the simulated value.
Also the waveforms show the good uncoupling between the flux and the
speed because a step variation in Φ_{dr} (Fig.
3d) can not generate a speed ω_{r} change (Fig.
3e). Thus the field orientation and the synthesis of robust linearization
and decoupling control are well verified.
All those results show the satisfying tuning, the excellent performance
of the robust decoupling control and DESMO against rotor resistance variations
and perturbations or noises.
CONCLUSION
This research has proposed a feedback linearization strategy and
a robust controller to permit a decoupling and regulation for the Induction
motor states in order to assure a good dynamic performance and stability
of the global system. Also a DESMO has been realized. It is based on SMO
principle but extended for the reconstruction of the fluxes, the rotor
time constant and the torque estimation. The parameter tuning, the choice
of initial conditions are easier compared to EKF.
The interesting simulation results obtained on the induction motor show
the effectiveness, the convergence and the stability of this robust decoupling
control and DESMO against rotor resistance variations measured noise and
load. 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.
NOMENCLATURE
C_{em}, C_{l} 
= 
Electromagnetic and load torques (N.m) 
I_{ds}, I_{qs, }I_{mr} 
= 
Stationary frame (d, q)axis stator currents and rotor magnetizing current
(A) 
p, J, f 
= 
Pole pair number, Inertia (kg m^{2}) and Friction coefficient
(Nm.s/rad) 
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} = 1/ ó_{r}; T_{s} = L_{s}/R_{s})
(sec) 
V_{ds}, V_{qs} 
= 
Stationary frame d and qaxis stator voltage (V) 
Φ_{dr}, Φ_{qr} 
= 
dq components of rotor fluxes (Wb) 
ω_{s}, ω_{r}, ω_{sl} 
= 
Stator, rotor and slip pulsation (or speed) (rad sec^{1}) 