Exact Feedback Linearization Control For a [623345]

Exact Feedback Linearization Control For a
Doubly Fed Induction Motor Using a Load
Torque Observer
BOUFADEN Mourad∗,BOUHENNA Abderrahmane∗,
CHENAFA Mohamed∗,MANSOURI Abdellah∗
∗LAAS Laboratory Department of Electrical Engineering, ENP O of
Oran, BP 1523 El M’Naouer, Oran, Algeria
e-mail: [anonimizat]
Abstract: This paper presents the Direct Field oriented control DFOC o f the doubly fed
induction machine DFIM with a robust nonlinear controller b ased on the input output feedback
linearization technique without including flux orientatio n ,associated with a Leuenberger load
torque observer using the speed ,rotor current and the stato r voltages ,where the proposed
controller shows a perfect speed and stator flux tracking,he nce simulation results have been
carried out under Simulink/Matlab that shows the control sy stem performance .
Keywords: DFIM,Exact Feedback Linearization,The load torque observ er ,Leunberger
Observer,FOC.
1. INTRODUCTION
The doubly fed induction machine DFIM has known since
1899 as a wound rotor asynchronous machine that is sup-
plied from stator and rotor sides with an external voltages
source (N.Akkari (2010)), the DFIM is highly nonlin-
ear and coupled ,multivariable system where it requires
more complex methods to control ,hence this machine
constitutes a theoretically challenging control problem
(A.Merabet (2012)).
The Wound rotor doubly fed asynchronous machine has
been the subject of most research primarily for its oper-
ation as a generator in applications of wind energy. this
work involves the operation in variable speed motor, for
improving the robustness of the control of the DFIM.
In the control structure shown in Figure (1) , the DFIM is
supplied to its stator by the network, while the rotor is fed
through a conversion system which comprises a rectifier, a
filter and an inverter.
Fig. 1. Diagram of the power of the DFIM for motor
application
The DFIM drives wound rotor has the advantage it can
be controlled from the stator or rotor as well as from
both (stator and rotor ) by various possible combinationscompared to the squirrel-cage machine .
The field oriented control technique that has been es-
tablished by (Blaschke F (1993)) is the most important
development in control area ,However this technique is
affected by unknown disturbances ,therefore an extensive
works have been established to find a new solution such as
sliding mode control ,feedback linearization control and
predictive control ((Chen,F (1999); Chiasson (1996);
Marino R (1993))), to improve the dynamic response and
reduce the complexity of the FOC.
In this paper, a MIMO robust nonlinear controller
(Shankar Sastry (1999)) based on the technique of lin-
earization to adjust the speed of the DFIM is introduced
and applied with a load torque observer ( ?) in order to
estimate the unmeasured load disturbance , this technique
madethelinearizedsysteminthecanonicalformwherethe
pole placement technique is applied to find the controller
parameters .
Finally ,the proposed control is tested using simulations
on the linearized model associated with the load torque
observer of the doubly fed induction motor
This paper is organized as follows ,we present in section
(2) Describes the Doubly fed induction motor with the a
null rotor flux model ,in section (3) Describes the design
of the robust feedback linearization controller ,in sectio n
(4) The load torque observer is presented , in section (5)
shows simulations results , in section (6) finally we end up
with a conclusion.
2. THE MATHEMATICAL MODEL OF DOUBLY FED
INDUCTION MOTOR
The doubly fed induction motor model can be described in
the synchronous frame with the stator flux aligned on the
d axis such that φsq= 0, by the following state equations:

Exact Feedback Linearization Control For a Doubly Fed Induc tion Motor Using a Load Torque Observer 2
˙x=f(x)+gu+dTL (1)
where the state vector xis defined as:
x= [ird,irq,φsd,Ω]T(2)
and the input vector is:
u=[urd, urq]T(3)
f(x) =
−γird+K
Tsφsd+Mi2
rq
Tsφsd−pΩirq+irq
φsdusq−Kusd
−γirq+Mirqird
Tsφsd−pΩird+ird
φsdusq+KpΩφsd−Kusq
M
Tsird−φsd
Ts+usd
−pM
JmLrφsdirq−fm
JmΩ
(4)
The control input matrix is defined by
g=
1
σLr0 0 0
01
σLr0 0
T
. (5)
d=/bracketleftbigg
0 0 0−1
Jm/bracketrightbiggT
. (6)
where the parameters σ, γ, K, T s, Trare defined as
follows
σ= 1−M2
LrLs, γ=1
Ts+1−σ
Ts1
σ
K=1−σ
σ, Ts=Ls
Rs, Tr=Lr
Rr(7)
σis the scattering coefficient, Tr,Tsare the time constant
of the rotor and stator dynamics, Jmis the rotor inertia,
fmis the mechanical viscous damping , p is the number of
pole pairs, cris the external load torque.
The state variables isα,isβ,φrα,φrβ,usα,usβ,urα,
urβ, are the stator currents, rotor flux linkages, stator
terminal voltage, rotor terminal voltage respectively and
Lr, Ls, M, R r, Rsare rotor inductance, stator induc-
tance,mutualinductance,statorresistanceandrotorresi s-
tance respectively. The state variables ird,irq,φsd,usd,usq,
urd,urq,are the rotor currents, stator flux linkages ,the
stator terminal voltage, the rotor terminal voltage respec –
tively. For the parameters Lr,Ls,M,Rr,Rsare respectively
rotor inductance, stator inductance, mutual inductance,
stator resistance, rotor resistance Where the estimated
stator pulsation is written as:
dθs
dt=ωs= (M
Tsird+usq)/(φsd) (8)
3. FEEDBACK LINEARIZATION CONTROLLER OF
DFIM
Mainly feedback linearization uses a state transformation
that enable us to express the system in a new coordinate
system that made it linear so that a state linearization
in the new coordinates is achieved ; theoretical back-
groundandproceduresforfindingsuchtransformationand
controller can be found in BOUFADEN Mourd (2012);
Shankar Sastry (1999); T.Orlowska (2012); A.Payma(2010) For the Doubly fed induction motor the objective
is to control the speed and the stator flux aligned with d
axis; hence the output vector is written as:
y=/bracketleftBiggh1(x)
h2(x)/bracketrightBigg
=/bracketleftBiggφsd
Ω/bracketrightBigg
(9)
The following notation used for the Lie derivatives of a
function
h(x) :ℜn→ ℜ (10)
along a vector field :
f(x) = (f1(x),…,f n(x)) (11)
Lfh(x) =n/summationdisplay
i=1∂h
∂xifi(x) (12)
Where the derviative of the output along the vector field
f(x) is defined Iteratively as
Lfh(x) =Lf(Li−1
fh) (13)
By appling the following change of coordinates yields

z1
z2
z3
z4
=
/hatwideh1
Lf/hatwideh1
/hatwideh2
Lf/hatwideh2
(14)
Where (/hatwide) symbol represents the estimated stator flux and
speed . To obtain the control law ; equation (14) need to
be differentiated as follows:

˙z1
˙z2
˙z3
˙z4
=
Lf/hatwideh1
L2
f/hatwideh1+Lg1Lf/hatwiderh1urd+Lg2Lf/hatwiderh1urd
Lf/hatwideh2
L2
f/hatwideh2+Lg1Lf/hatwiderh1urd+Lg2Lf/hatwiderh1urq+fm
J2m/hatwideTL
(15)
Where the number of differentiation reprsent the relative
degree of the system which is defined for both outputs by
r1,r2respectively and are equal to 2 then r1+r2= 4
,which is equal to the number of state ;therefore the
system admits an exact feedback linearization . Rewritten
equation ( 15) in matrix form gives :

˙z1
˙z2
˙z3
˙z4
=
z2
v1
z3
v2
(16)
So in order to construct the control vector ;equation ( 16)
is written in matrix form as follows:
/bracketleftBigg˙z2
˙z4/bracketrightBigg
=
L2
f/hatwideh1(x)
L2
f/hatwideh2(x)
+D(x)/bracketleftBiggurd
urq/bracketrightBigg
+
0
fm
Jm
/hatwideTL(17)

Exact Feedback Linearization Control For a Doubly Fed Induc tion Motor Using a Load Torque Observer 3
WhereD(x) denotes the decoupling matrix which is de-
fined as fllows:
D(x) =
Lg1Lf/hatwideh10
0Lg2/hatwideh2
 (18)
Where
Lg1Lf/hatwideh1=M
LrTsσ(19)
Lg2Lf/hatwideh2=−pM/hatwideφsd
JmL2
2σ(20)
L2h1(x)=/hatwideφsd
T2s−usd
Ts−(M
T2s+Mγ
s)ird
−KM
T2susd+KM
T2s/hatwideφsd+M2
T2s/hatwideφsdi2
rd(21)
+Musq
T2s/hatwideφsdirq−pM
TsΩirq
L2/hatwideh2(x)=(fm
Jm)(fm
Jm/hatwideΩ+pMirq/hatwideφsd
JmLr)
−(ird
JmLr)((pM/hatwideφsd)(usq
/hatwideφsd−p/hatwideΩ+Mirq
Ts/hatwideφsd))
+M2pirq
JmLrTs−pMirq
JmLr(usd−/hatwideφsd
Ts) (22)
+pM/hatwideφsd
JmLr(Kusd+γirq−Kp/hatwideΩ/hatwideφsd)
The matrix D(x) is nonsingular,since its determinant have
never been zero for any value of the statoric flux φsd
Therefore ,the control vector law [ urdurq]Tis drawn
from eqaution ( 17) :
/bracketleftBiggurd
urq/bracketrightBigg
= [D(x)]−1
−L2
f/hatwideh1(x)+ν1
−L2
f/hatwideh2(x)−fm
J2m/hatwideTL+ν2
(23)
Whereν1,ν2are the new input vectors that are going to
be calculated .
It is seen from equations (17) ,(23), that the problem of
controlling speed and stator flux is rendered to control a
double integrator for the rotor flux loop as well as for the
speed loop as shown in Figure
Fig. 2. The Linearized System
In order to track the reference trajectory of h1andh2so
the variation ν1andν2are calculated as follows:ν1=¨h1ref−kd1(˙h1−˙h1ref)−kp1(h1−h1ref) (24)
ν2=˙h2ref−kd2(˙h2−˙h2ref)−kp2(h2−h2ref)
So that the Closed-loop tracking error is given by:/braceleftbigg
¨e1+kd1˙e1+kp1e1= 0
¨e2+kd2˙e2+kp2e2= 0(25)
where by an appropriate choice of the positive constants
kp1andkp2,as well as kd1andkd2ensures the exponential
convergence of the tracking errors :
/braceleftBigge1=h1−h1ref
e2=h2−h2ref(26)
4. THE LEUNBERGER OBSERVER FOR LOAD
TORQUE AND SPEED
Since that the load torque is unknown and should be
measuredusingsensorstocompensateit,andsincesensors
are more expensive ; and therefore a good solution could
be the use of the load torque estimator; hence a second
order Luenberger observer is proposed with the rotor
currents as measured states ,the speed is estimated by
classical method. From equation ( 2) we have that the
electromagnetic torque and speed equations are simplified
to:
Cem=KTeirq (27)
˙/hatwideΩ =KTeirq−fm
Jm/hatwideΩ−/hatwideTL
Jm(28)
Where the KTeis the torque constant and defined by:
KTe=−pM
JmLr/hatwideφsd (29)
The estimated stator flux in the field oriented control is
given by :
˙/hatwideφsd+/hatwideφ
Ts=M
Tsird+usd (30)
Assuming that the load torque is bounded which means
that it changes at certain moments ,therefore the load
torque defined as :
˙TL= 0 (31)
Equations ( 28),( 31),could be written in state equations
as follows:/braceleftbigg
˙z=Az+Birq
y=Cz(32)
Where
z= [ΩTL]T(33)
A=/bracketleftBigg
−fm
Jm−1
Jm0 0/bracketrightBiggT
(34)
B= [KTe0]T(35)
The output system y is considered to be the speed such
that:
C= [1 0]T(36)

Exact Feedback Linearization Control For a Doubly Fed Induc tion Motor Using a Load Torque Observer 4
Fig. 3. Closed Lopp Observer
Then the Luenberger Observer that estimates the speed
and Load torque is written as follows:
˙/hatwidez=A/hatwidez+Birq+H(Cz−C/hatwidez)
= (A−HC)/hatwidez+Birq+HCz(37)
The differential equation describing the observer is ob-
tianed from equations (32),(37) as :
˙e= (A−HC)e (38)
Where the symbol ( /hatwide) denotes the estimated values, H=
[H1H2]Tis the observer gain matrix which has to
be chosen adequately such that the errors between the
estimated and real states converge to zero as t tends to
infinity.
5. SIMULATIONS AND RESULTS
Simulations have been performed with Matlab-Simulink
software. The doubly fed induction motor parameters are
given in Table 1, and the benchmark of Figure (5)
Table 1. Parameters of the DFIM
Designation Parameter Value
Rotor resistance Rr 3.805 Ω
Stator resistance Rs 4.85 Ω
Mutual inductance M 0.258 H
Stator cyclic inductance Ls 0.247 H
Rotor cyclic inductance Lr 0.247 H
Rotor inertia Jm0.031Kg/m3
Pole pair p 2
Viscous friction coefficient fm0.008 N.m.s/rd
Mechanical power Pm 15 KW
Nominal Stator Voltage Vs 220 V
Nominal Rotor Voltage Vr 12 V
Nominal Stator Current Is 3.46 A
Nominal Rotor Current Ir 6.31 A
Nominal speed Ωn1500 rev/min
0 1 2 3 4 5 6 7 8 9−200−1000100200The Speed (rad/s)
0 1 2 3 4 5 6 7 8 900.511.52Flux (Wb)
0 1 2 3 4 5 6 7 8 90510Torque (N.m)
Time (sec)
Fig. 4. Reference trajectoriesSpeed error tracking: We note that the speed error
tracking is canceld ,
0 1 2 3 4 5 6 7 8 9−0.025−0.02−0.015−0.01−0.00500.0050.010.0150.020.025The Speed Error(rad/s)
Time (sec)
Fig. 5. Non Linear Control Speed error tracking
The estimated Load Torque: We note from Figure 7 that
represents the error between the estimated and drived
load torque which shows that the error oscillates between
±0.0002N.m;therefore signify that the estimated torque
follows the drive torque as shown in Figure 6
0 1 2 3 4 5 6 7 8 9−5051015Real Load Tprque (N.m)
0 1 2 3 4 5 6 7 8 9−5051015Estimated Load Tprque (N.m)
Time (sec)
Fig. 6. The Real and ESstimated Load Torques (N.m)
0 1 2 3 4 5 6 7 8 9−1−0.500.511.5The Load Torque Error (N.m)
Time (sec)
Fig. 7. The Load Torque Error between estimated and real
Ce(N.m)

Exact Feedback Linearization Control For a Doubly Fed Induc tion Motor Using a Load Torque Observer 5
Torque: We note that the drive torque follows the load
torque when the speed is constant. During an increase or
decrease in the speed , a difference of almost ±5N.m
appears between the two torques, as shown in Figure (8).
0 1 2 3 4 5 6 7 8 9−6−4−2024681012The Load Torque (N.m)
Time (sec)Electromagnitic Torque
Estimated Load Torque
Fig. 8. The Torque Ce(N.m)
Stator Flux : We note a very good tracking, since the
stator flux is controlled with avery fast response time with
no overshoot in both transient and permanent regimes
because it is unaffected by the change of speed as well
as the load torque as shown in Figure 10 ,where we note
from figure 9 that φrq= 0Wb,φrd= 1Wbas stated by the
field oriented control theory as well.
0123456789−0.200.20.40.60.811.2The Stator Flux Error φsq (Wb)
Time (sec)φsqφsd
Fig. 9. The rotor flux φr(Wb)
0 1 2 3 4 5 6 7 8 9−0.200.20.40.60.811.2The Stator Flux Error φsd (Wb)
Time (sec)
Fig. 10. The rotor flux error φr(Wb)Rotor currents: we note that the current isdhas a peak
at the start up and after a short period of time the current
swings between ±50 ; for the isqcurrent we notice that
the current is inversely proportional to the load torque as
shown in Figure 11
0 1 2 3 4 5 6 7 8 9−100−50050100Rotor current ird (A)
0 1 2 3 4 5 6 7 8 9−15−10−5051015Rotor current irq (A)
Time (sec)
Fig. 11. The Rotor currents ( A)
6. CONCLUSION
Inthispaper,theexactinputoutputfeedbacklinearizatio n
of the doubly fed induction machine is applied with the
orientation of the rotoric flux ,associated with a luen-
beger observer to componsate the unmeasered load torque
(disturbance) .According to the simulation results we see
that the application of nonlinear control based on the
input output linearization of DFIM has helped highlight
the static and dynamic properties of linearizing control.I t
appears in the results an excellent decoupling between
speed and stator flux.We hope to validate the obtained
results in real time.
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