International audienceThe prediction of the gear behavior is becoming major concerns in many industries. For this reason, in this article, an electro-mechanical modeling is developed in order to simulate a gear element driven by an asynchronous motor. The electrical part, which is the induction motor, is simulated by using the Kron's model while the mechanical part, which is the single stage gear element, is accounted for by a torsional model. The mechanical model that simulates the pinion-gear pair is obtained by reducing the degree of freedom of the global spur or helical gear system. The electrical and mechanical state variables are combined in order to obtain a unique differential system that describes the dynamics of the elecro-mechanical system. The global coupled electro-mechanical model can be characterized by a unique set of non-linear state equations. The contribution of this work is to apply the control based on observers in order to supervise the electrical and mechanical behavior of the electro-mechanical system from only its inputs and its measurements outputs (sensors outputs). Some simulations on pinon/motor angular speed, electromagnetic torque, currents, are presented, which illustrate the system evolution (i.e., the electrical and mechanical quantities) and the good performances of the proposed observers

Abbes, Mohamed

Barbot, Jean,

Derbel, Syrine

Feki, Nabih

Haddar, Mohamed

Nicolau, Florentina

English

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Electro-mechanical system control based on observers
Syrine Derbel, Nabih Feki, Jean Barbot, Florentina Nicolau, Mohamed Abbes,
Mohamed Haddar
To cite this version:
Syrine Derbel, Nabih Feki, Jean Barbot, Florentina Nicolau, Mohamed Abbes, et al.. Electro-
mechanical system control based on observers. International Conference on Acoustics and Vibration,
Mar 2018, Hammamet, Tunisia. ￿hal-02014864￿
Electro-mechanical system control based on
observers
Syrine Derbel1,2, Nabih Feki2, Jean Pierre Barbot1, Florentina Nicolau1,
Mohamed Slim Abbes2, and Mohamed Haddar2
1 Quartz Laboratory / ENSEA, Cergy, France 95014,
syrina.derbel@hotmail.fr,
2 LA2MP Laboratory / ENIS, Sfax, Tunisia 3038
Abstract. The monitoring of the gear behavior is becoming major concerns in several
industries for improve production and safety. For this reason, in this article, an electro-
mechanical modeling is developed in order to simulate a gear element driven by an
asynchronous motor. The electrical part, which is an induction motor, is modeled
by using the Kron’s model while the mechanical part, which is a single stage gear
element, is accounted for by a torsional model. The mechanical model that simulates
the pinion-gear pair is obtained by reducing the degree of freedom of the global spur
or helical gear system. The electrical and mechanical state variables are coupling in
order to obtain a unique differential system that describes the dynamics of the elecro-
mechanical system. The global coupled electro-mechanical model can be characterized
by a unique set of non-linear state equation. The aim of this work is to apply the
control based on observers in order to supervise the electrical and mechanical behavior
of the electro-mechanical system from only its inputs and its measurements outputs
(sensors outputs).
Keywords: Gears transmission, Observers, Simulation, Asynchronous motor
1 Introduction
Electro-mechanical systems such as mechanical gear transmission driven by in-
duction motors are commonly used in many industrial applications. For this
reason, many techniques and tools of control and diagnostics, such as vibration
and sound signal analysis (Baydar & Ball 2001, Tan et al. 2007), analysis of
stator currents (Feki 2012), have traditionally been used to supervise the gear
element behavior and to detect faults (Chaari et al. 2008). These techniques are
advantageous by their reduced cost and present a high reliability. However, they
are sensitive to the positioning of the sensors and in some applications, they
present technical difficulties to implement sensors in rotating parts or hostile
environment. Although gear monitoring by vibration signal analysis and current
stator analysis are still widely used, a new method of electro-mechnical system
control and monitoring based on observers will be presented in this paper.
In general, for technical and economic reasons, the state of the system is
not completely accessible. Indeed, the complexity of the technical feasibility as
well as prohibitive costs for the implantation of several sensors can considerably
2reduce the number of states measured. In this case, the state vector size is greater
than the output vector size. However, under some conditions of existence, the
state can be reconstructed using an observer Larroque (2008).
The paper will be organized as follows: a) in section 2, a single stage gear
element is accounted for by using torsional model, b) in section 3, an electrical
modeling of the induction motor is presented, c) the electro-mechnical coupling
is developed in section 4, d) the section 5 is dedicated to implement the used
observer and e) the simulation results of the electro-mechanical system dynamic
behavior are presented in section 6.
2 Mechanical modeling
The modeling of the mechanical part (see Fig.1) contains only the gear element.
To do this, we are based on the modeling in (Feki et al. 2012) but reduce the
global model of 36 degrees of freedom to 2 degrees of freedom. A driving torque
Fig. 1: Electro-mechanical system
Cm is applied to the pinion gear and a load torque Cr is applied to the wheel gear
(attached to output shaft). The system modeling is accounted for by two degrees
of freedom, which correspond to the torsional components. Using Euler-Lagrange
equation, the motion equation of the torsional model of the gear element is
obtained as follows (1):
Mx¨+ Cx˙+K(t, x)x = F, (1)
where
– M : is the mass matrix composed by the polar moment of inertia for the gear
k,
– K(t, x): is stiffness matrix that depends on the state vector and on the time,
– C: is the Rayleigh model damping,
– F = [Cm Cr]
t ∈ R2: external forces,
3– x = [θ1 θ2]
t ∈ R2: represents the two degrees of freedom vector.
The stiffness matrix can be expressed in terms of a structural vector V (Mi)
(Maatar & Velex 1996) by using the following form K(t, x) = k(t)V (Mi)V (Mi)
t
,
with k(t) the stiffness simulated by a square waveform. Developing (1), we obtain
the space representation of the gear element:[
x˙
x¨
]
=
[
0 Id
−M−1K −M−1C
] [
x
x˙
]
+
[
0
M−1F
]
, (2)
with Id the 2nd order identity matrix.
3 Electrical modeling
The motor equations are obtained by using the Kron’s transformation model.
The key idea of this transformation is to translate the three-phases quantities
(abc) of the motor to an equivalent two-phases quantities (see Fig.2).
Fig. 2: Park transformation plan.
The stator variables are obtained by θ=θs=wst=(as, d) while the rotor vari-
ables are calculated by θ=θr=ωslt=(ar, d).xdxq
x0
 = T2/3(θs)
xaxb
xc
 , (3)
xdxq
x0
 = T2/3(θr)
xaxb
xc
 . (4)
The voltage equations of the three stator and rotor phases are given by (5) and
(6). vasvbs
vcs
 =
Rs 0 00 Rs 0
0 0 Rs
iasibs
ics
+ d
dt
φasφbs
φcs
 , (5)
4 varvbr
vcr
 =
Rr 0 00 Rr 0
0 0 Rr
iaribr
icr
+ d
dt
φarφbr
φcr
 . (6)
The flux equations can be written as follows:[
φabcs
]
= Lss
[
iabcs
]
+ Lsr
[
iabcr
]
, (7)[
φabcr
]
= Lrr
[
iabcr
]
+ Lrs
[
iabcs
]
, (8)
where
– Lss =
 ls ms msms ls ms
ms ms ls
: stator inductance matrix,
– Lrr =
 lr mr mrmr lr mr
mr mr lr
: rotor inductance matrix,
– Lsr = L
t
rs = msr
 cos(θ) cos(θ + 2Π3 ) cos(θ − 2Π3 )cos(θ + 2Π3 ) cos(θ) cos(θ + 2Π3 )
cos(θ + 2Π3 ) cos(θ − 2Π3 ) cos(θ)
: mutual induc-
tance matrix between stator and rotor,
with the constants parameters:
– ls (respectively lr): self-inductance of the stator (rotor),
– ms (respectively mr): mutual-inductance between the stator phases (the
rotor phases),
– msr: the maximum value of mutual inductances between stator and rotor
phases.
Applying the Kron’s transformation for (5),(6),(7) and (8), we obtain:{
vds = Rsids +
d
dtφds − ωsφds,
vqs = Rsiqs +
d
dtφqs + ωsφqs,
(9)
{
vdr = Rridr +
d
dtφdr − ωslφdr = 0,
vqr = Rriqr +
d
dtφqr + ωslφqr = 0,
(10){
φds = Lsids + Lmidr,
φqs = Lsiqs + Lmiqr,
(11){
φdr = Lridr + Lmids,
φqr = Lriqr + Lmiqs,
(12)
where Ls(respectively, Lr) represents the stator synchronous inductance ( respec-
tively, rotor synchronous inductance) and Lm is the magnetizing (synchronous)
inductance. The advantage of this transformation is the fact that we get a con-
stant mutual inductance and that along an axis, the fluxes depend only to the
rotor and stator currents. Relations (9) and (10) detail the electro-magnetic
5behavior of the asynchronous machine written in non a linear differential equa-
tions form. These equations can be described in matrix space representation by
choosing the space vector z(t) composed by the both stator currents and the
both rotor fluxes of the motor:
z˙(t) = Az(t) +BU(t), (13)
with z(t) = [ids iqs φdr φqr]
t ∈ R4 , U(t) = [vds vqs]t ∈ R2: the input
vector, A, B the state, input matrices given by:
A =

−( 1Tsσ + 1Tr 1−σσ ) ωs 1−σσ 1LmTr 1−σσ ωmLm
−ωs −( 1Tsσ + 1Tr 1−σσ ) − 1−σσ ωmLm 1−σσ 1LmTr
Lm
Tr
0 − 1Tr wsl
0 LmTr −wsl − 1Tr

, (14)
B =

1
σLs
0
0 1σLs
0 0
0 0
 , (15)
where the constants parameters are explained below:
– σ = 1− L2mLsLr , Ts = LsRs , Tr = LrRr .
The electromagnetic torque is represented by (16), with p the number of pole-
pairs.
Cem =
pLm
Lr
(φdriqs − φqrids). (16)
4 Electro-mechanical coupling
The aim of the electro-mechanical coupling is to obtain a space representation
that includes the mechanical system with two degrees of freedom and the in-
duction motor equations (see Fig.1). This modeling allows to implement control
methods to supervise the dynamic behavior of the system. The electro-magnetic
torque given by (16) is regarded as the input of the mechanical part and the
small vibrations caused by the gear element act on the speed rotation of the
asynchronous motor. The coupled system leads to a first order differential sys-
tem of the form:
ζ˙(t) = Ac(t, ζ)ζ(t) +Gu, (17)
where ζ(t) = [ids iqs φdr φqr θ1 θ2 θ˙1 θ˙2]
t : the global state vector,
Ac =
[
A 0
P H
]
is the state matrix, with A the state matrix associated to the
6electric part expressed in (14), P ∈ R(4×4) is the coupling matrix between the
electric and mechanical part depending to the electromagnetic torque donated by
(16) and H =

0 0 1 0
0 0 0 1
−M−1K −M−1C
 is the matrix associated to the gear modeling
, G =

1
σLs
0 0 0
0 1σLs 0 0
0 0 0 0
0 0 0 0
0 0 1 0
0 0 0 1
0 0
M−1
0 0

is the input matrix and u =

vds
vqs
0
Cr
 the input vector.
The evolution of the system is obtained by introducing the control based on
observers to monitor the dynamic behavior of the electro-mechanical system.
5 Observer form
A state observer is a control method (Perruquetti & Barbot 2002) that gives an
estimate of the internal state of the real system, only from the measurements
given by the sensors and the real input of the system. The observers are used
in order to control the behavior of systems, to detect the faults or to identify
the unknown parameters of systems (Oueder 2012). In our case, four differenti-
atiors (Ghanes et al. 2017) are used to estimate the drift of both currents of the
asynchronous motor and the displacements of the gear element. Assuming that
[s1, ..., s8] = [ids, i˙ds, iqs, i˙qs, θ1, θ˙1, θ2, θ˙2], the observer equations are written in
following form: 
˙ˆsi = sˆi+1 + k1µj |ei|αsign(ei),
˙ˆsi+1 = k2αµj
2|ei|2α−1sign(ei),
ei = si − sˆi,
(18)
where
– ei, i = {1, 3, 5, 7} are the output estimation errors,
– k1, k2 are constants acting on the stability of the system,
– µj , j = 1, 2 are positive constants, the first one associated to the electric
model and the second one is related to the mechanical part.
6 Simulation and Results
In this simulation, a spur gear system is considered. The main characteristics of
the gear are given in table 1 and the motor parameters are shown in table 2.
7Table 1: Gears parameters
Parameters value
Module (mm) 4
Tooth number of Pinion 21
Tooth number of wheel 31
Face width (mm) 10
Pressure angle 20
Table 2: Electric parameters
Parameters value
Stator resistance Rs (Ω) 9.163
Rotor resistance Rr (Ω) 5.398
Stator inductance Ls (H) 0.115
Rotor inductance Lr (H) 0.0943
Magnetizing inductance Lm (H) 0.0943
Number of pole-pairs p 1
As it can be noticed, the applied observer gives a good performances (see
Fig .3, 4, 5 and 6). In these figures, the estimated states converge, in finite
time, to the real quantities states. Fig .3 and Fig. 4 represent the two stator
currents expressed in (dq0) frame. They show that the periodic recurrence of
gears meshing frequency tm is regained in the electrical states.
Fig. 3: State ζ1 and its estimate
8Fig. 4: State ζ2 and its estimate
Fig. 5: State ζ7 and its estimate
9Fig. 5 illustrates that the rotational speed of the pinion is of the order to
300 rad/s. This value represents the average meshing speed added to the small
vibrations of the gear.
The last figure (Fig. 6) displays the evolution of the error transmission given
by the equation Rb1θ1 + Rb2θ2 (where Rb1, Rb2 are, respectively, the base radii
of the pinion and the wheel).
Fig. 6: Real and estimated transmission error
All results confirm the good convergence of the applied observer. This con-
vergence has appeared in the frequency content and the amplitude of the electro-
mechanical system behavior signals. Meaning that the evolution of the real states
and those of the estimated quantities are perfectly confused in the frequency
study. The observer gives a rapid and accurate convergence for all system states.
7 Conclusion
In this paper, an electro-mechanical coupling of the gear transmission system
driven by a asynchronous motor has been studied. The monitoring of this model
is obtained by using the control based on observers. Further work is in progress
in order to implement other types of observers with presence of gear faults and
variation of the sensors noise.
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Electro-mechanical system control based on observers

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