Sliding Mode Observer for Rotor Faults Diagnosis
Lane Maria Rabelo Baccarini
Universidade Federal de
São João del Rei
Minas Gerais, 36300-000
BRASIL
rabelo@cpdee.ufmg.br
Benjamim Rodrigues de Menezes,
Homero Nogueira Guimarães,
Walmir Matos Caminhas and
Leandro Henrique Batista
Universidade Federal de Minas Gerais
Belo Horizonte, MG, 31270.010
brm@cpdee.ufmg.br, hng@eee.ufmg.br caminhas@cpdee.ufmg.br
Abstract— Although induction motors are traditional thought
to be reliable and robust, the possibility of faults is
unavoidable once the machines can be exposed to different
hostile environments, misoperations, and manufacturing defects.
Therefore, motor monitoring incipient fault detection and
diagnosis are important topics. This paper presents a method for
on-line induction motor monitoring with the purpose of detecting
and locating a single rotor broken bar. The method avoids any
frequency analysis and observes instead the machine state with
the help of the two models. The torque difference between the
two models indicates a fault. The technique utilizes input signals
from standard transducers. An experimental setup has been
constructed to implement the new technique in on-line model.
I. I NTRODUCTION
The induction motor is the most commonly used motor
in industrial application because of its simplicity, rugged
construction, and relatively low manufaturing costs.
In the last decades, market improvement has been achieved
in the design and manufacture of stator windings. However,
cage rotor design and manufacturing have undergone little
change. As a result, rotor failures now accent for a large
percentage of total induction motor failures [1].
Broken rotor bars rarely cause immediate failures, especially
in large multi-pole (low-speed) motors. However, if there are
enough broken rotor bars, the motor may not start as it may not
be able to develop sufficient accelerating torque. Regarding,
the presence of broken rotor bars, it precipitates deterioration
in other components that can increase the time-consuming.
Replacement of the rotor core in large motors is costly;
therefore, by detecting broken rotor bars early, such secondary
deterioration can be avoided. The rotor can be repaired
at a fraction of the cost of motor replacement, not to
mention averting production revenue losses due to unplanned
downtime.
Different methods have been proposed for rotor fault
detection in the last years. The most well-known approaches
for diagnosis of broken rotor bars in induction machines are
based on monitoring the stator currents to detect side bands
around the fundamental. Meanwhile, the task of distinguishing
the fault conditions from the normal conditions based on the
resultant FFT spectrum is difficult. This is due the fact that
the stator current is a non-stationary signal whose properties
vary with the motor operation conditions [2].
The Vienna Monitoring Method (VMM) has been designed
for on-line application in variable-speed drives for faulty rotor
bar detection [3], [4]. This technique compares the estimated
machine states out of the two independents machine models
with different model structures. While the models respond
identically to the regular machine operation they diverge in
case of a structural machine asymmetry.
Caminhas et. all (1996) have developed an observer
to estimate the flux and torque of induction motor. This
model allows the observation of systems with non-matching,
non-linearities. Two formulations were presented: one for
rotor resistance disturbances rejection and another for stator
resistance disturbances rejection. The obtained observer is
suitable for real time implementation.
This paper presents a method for on-line induction motor
monitoring with the purpose of detecting and locating a single
broken rotor bar The method avoids any frequency analysis
and observes instead, the machine state with help of the two
models. One of the observers is designed for rejecting rotor
resistance disturbances [5]. The other estimates the rotor flux
by using the discrete model proposed by Bottura et. all (1993).
The technique utilizes input signals from standard transducers
and encoder.
The computed simulations results verify both sensitivity
and robustness of the method. An experimental setup has
been constructed to implement the new technique in on-line
mode. The results validate the proposed model and show the
reliability of the new method.
The paper is organized as follow: in the following section
the sliding models observer for discrete-time non-linear
systems is presented; in section 3 the state observer is designed
for unknown rotor resistance. The fault detection technique is
considered in section 4, Results are presented in section 5 and
conclusions are described in section 6.
II. S LIDING M ODES O BSERVER
The non-linear discrete-time system can be described by
(1). In this system, the vectors x, u, v, y represent, respectively,
the state vector, control inputs, unknown disturbances and the
measurements available.
x(k + 1) = F (x(k), u(k), k) + ∆F (x(k), u(k), k) + Dv(k)
y(k) = Cx(k)
(1)
The system dynamics is composed of a nominal part F and
of a disturbed part ∆F . The function F (., ., .) and the matrix
C are supposed to be known and the function ∆F (., ., .) and
the matrix D are supposed to be unknown, although obeying
the following perturbation matching condition:
R(H) = R(D) ∪ F
and F = Im(∆(F ))
(2)
Matrix H is supposed to be known, and must satisfy:
ρ(C.H) = ρ(H) = r
and ρ(C) = m ≥ r
(3)
where ρ(.) stands for the rank of the argument matrix.
The nominal plant without the perturbations must also be
stable and observable through the measurement matrix C in
the whole operation range. Then:
Hω = ∆F (x(k), u(k), k) + Dv(k)
(4)
This allows rewriting (1) as:
x(k + 1) = F (x(k), u(k), k) + Hω(k)
(5)
The discrete-time sliding modes observer is built by adding
to (5) a disturbances cancellation term:
x̂(k + 1) = F (x̂(k), u(k), k) + Hω(k) + Hd(k)
Consider the following output vector partition:
 1   1 
y
C
y(k) = Cx(k) =  ...  =  ...  x(k)
y2
C2
2) To choose the partition of C: It is easy to take
measurements of the stator currents, which are related
to the states in the following way:
»
iqs (k)
ids (k)
–
=
»
a1
0
0
a1
2
–6
0 6
0 6
4
0
−a2
−a2
0
(6)
λqs
λds
λqr
λdr
ωr
3
7
7
7 (13)
5
The constants terms in the equation are:
a1 =
(7)
lr
a0
a2 =
lm
a0
2
a0 = l s l r − l m
where ls , lr , lm are stator, rotor and mutual motor
inductance, λqs , λds , are the components d and q of
stator flux, and ωr is the angular rotor speed. Then:
The term d(k) is deducted from the constraint:
C 1 x̂(k + 1) − y 1 (k + 1) = L1 C 1 x̂(k) − y 1 (k)
(8)
+H(C 1 H)−L y 1 (k + 1)
(9)
which, once substituted into (6), leads to:
x̂(k + 1) = I − H(C 1 H)−L C 1 F (x̂(k), u(k), k) +
III. O BSERVER FOR U NKNOWN ROTOR R ESISTANCE
The observer for rejecting rotor resistance disturbances can
be obtained following the detailed procedure based on the
steps previously described, designing for a generic case.
1) To construct the matrix H: The number of important
systems variables measurement must be greater or
at least equal the number of states affected by
the parameter variation. Thus, the rejection of rotor
resistance requires at least two measurements, since it
influences the d and q components rotor flux estimation:
x3 (λqr ), x4 (λdr ). So, matrix H equation must be chosen
as:
T
0 0 1 0 0
H=
(12)
0 0 0 1 0
where (.) stands for any left inverse of the argument matrix.
The constraint given by (8) is equivalent to a system order
reduction, belonging the state vector to a surface in the state
space. This surface is called sliding surface, and the observer
is said to be in sliding modes in this surface.
In order to establish some freedom in the error dynamics
assigment, a proportional term is added to 9:
−L
x̂(k + 1) = I − H(C 1 H)−L C 1 F (x̂(k), u(k), k)+
+H(C 1 H)−L y 1 (k + 1) + L [C x̂(k) − y(k)]
(10)
Defining φ1 = I − φ2 C where φ2 = H(C 1 H)−L , leads to
the expression for the observer:
x̂(k + 1) = φ1 F (x̂, u, k) + φ2 y(k + 1) + L [C x̂ − y(k)] (11)
In order to project an observer in sliding-modes observer
is necessary: 1) to construct the matrix H; 2) to choose the
partition of C; 3) to find any left inverse for C 1 H; 4) to obtain
φ1 and φ2 ; 5) to choose gain observer matrix; 6) to construct
the observer final equation.
3 2 a1
C1
6 0
C = 4 .... 5 = 4
...
C2
0
0
a1
...
0
2
−a2
0
...
0
0
−a2
...
0
3
0
0 7
... 5
1
(14)
3) To find any left inverse for C1 H
(C1 H)−L = (C1 H)−1 =
»
0
−1/a2
−1/a2
0
–
(15)
4) To obtain φ1 and φ2 : As:
φ2 = H(C 1 H)−L
0
6
0
6
= 6 −1/a2
4
0
0
2
0
0
0
−1/a2
0
3
7
7
7
5
Then:
1
6 0
6
φ1 = I − φ 2 C = 6 α
4 0
0
2
0
1
0
α
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
3
7
7
7
5
where α = a1 /a2
5) To choose gain observer matrix: The observer error
dynamic depends on eigenvalues matrix φf . These
eigenvalues may be assigned through time-varying
feedback matrix L which are calculate at each sampling
time in order to fix a linear time-invariant error
dynamics.
As the system matrix A is known and let φf = φ1 A +
LC then the eigenvalues are choosen such that matrix
φf has the following structure:
k2
6 k1
6
φf = 6 αk2
4 αk
1
0
2
−k1
k2
−αk1
αk2
0
0
0
0
0
a53
0
0
0
0
a54
L13
L23
αL13
αL23
a55 + L53
3
7
7
7
5
L12 = −rs k6
L22 = L11
L32 = αL12
L42 = αL22
L52 = 0
(16)
L33 = αL13
L43 = αL23
The parameters in matrix φf are:
k1 = sin(ωh)
k5 = sin(ωh)
ω
a53 = −g2 λds (k)
hc2
g1 = e
k2 = cos(ωh)
k6 = 1−cos(ωh)
ω
a54 = −g2 λqs(k)
−3pc
g2 = 4c1 (g1 − 1)
a55 = g1
g3 = g1c−1
1
where c2 is the motor friction, Jm is the inertia force,
p is the pole number and h is the sampling period.
Three eigenvalues may be assigned through the choice
of the remaining parameters L13 , L23 e L53 . Note
that the rotor resistance does not influence matrix φf .
Those eigenvalues are determined through the algebraic
expression:
det (γI − φf ) = 0
The expression for parameters L13 , L23 and L53 is given
for the following linear system:
∆1 L13 + ∆2 L23 + ∆3 L53 = ∆3 (λi − a55 )
(17)
in which:
Rotor cage asymmetries of induction machines cause
disturbances of the air-gap flux pattern, which affect torque
and speeed as well as stator terminal voltages and currents.
Kral et. all (2000) proposed a rotor breakage fault detection
method, called The Vienna Monitoring Method. This technique
senses the actual machine state with help of real-time space
phasor models.
The key for the fault detection is thereby the comparison
of the estimated machine states out of two independents
machine models with different model structures. While the
models respond identically to the regular machine operation
they diverge in case of the structural machine asymmetry.
This paper presents an alternative approach to obtain the
torque deviation between two models. The VMM estimates
the machine states and our method observes the states. The
observer state is used to speed up convergence and to reduce
sensitivity to parameter variations. This model can be used in
advantage in simulation and real-time control applications due
to observer characteristics. The schemes are suitable for online monitoring and fault detection purposes. The two models
is described below.
A. Stator Flux Observer
The sliding modes observer designed for unknown rotor
resistance, described in section III, is used to obtain the stator
flux as in (18) and (19). The stator resistance is the only
required parameter for this model as the constants L11 , L12 ,
L21 and L22 in (18) and (19), depend on the resistance rs .The
observed torque (Tobs ) is calculated by the following equation:
3p
(x̂2 iqs − x̂1 ids )
22
(23)
B. Rotor Flux Observer
∆1 = α [a53 (k2 − λi ) − a54 k1 ]
∆2 = α [a54 (k2 − λi ) + a53 k1 ]
∆3 = λi (k2 − λi ) − (k2 λi − 1)
6) To obtain the observer final equation:
As x̂1 , x̂2 , x̂3 , x̂4 , and x̂5 are, respectively, the
component q of stator flux, component d of stator flux,
component q of rotor flux, component d of rotor flux,
and motor velocity, then the obsever final equations are:
The discrete model proposed by Bottura, Silvino and
Resende [6] is used to observer the rotor flux. At this way,
the rotor resistance influences the rotor flux estimate. The
torque obtained by discrete model (Tdisc ) is calculated by the
equation:
Tdisc =
3p
lm
(λdr iqs − λqr ids )
22
lr
(24)
C. Torque residue analysis
x̂1 (k + 1) = k2 x̂1 − k1 x̂2 + L13 x̂5 + k5 vqs −
−k6 vds − L11 iqs − L12 ids − L13 ωr
(18)
x̂2 (k + 1) = k1 x̂1 − k2 x̂2 + L23 x̂5 + k6 vqs +
+k5 vds − L21 iqs − L22 ids − L23 ωr
(19)
1
iqs (k + 1)
a2
(21)
x̂5 (k + 1) = −g2 x̂3 + g2 x̂2 + a55 )x̂5 + L53 (x̂5 − ωr ) (22)
Tobs =
x̂3 (k + 1) = αx̂1 (k + 1) −
1
ids (k + 1)
a2
IV. FAULT D ETECTION T ECHNIQUE
The eigenvalues are defining as:
L11 = rs k5
L21 = −L12
L31 = αL11
L41 = αL21
L51 = 0
x̂4 (k + 1) = αx̂2 (k + 1) −
(20)
Fig. 1 depicts the structure of the method proposed. The
torque difference between the two models is used to detect
and locate the broken rotor bar.
A broken rotor bars lead to a different model response in the
form of a modulated torque deviation ∆T = Tomd −Tdisc . The
frequency of the modulation is determined by two times the
slip frequency. This frequency modulation in the time domain
corresponds to a modulation with an angular period 2π
p in rotor
space. A minimum of torque difference indicates the faulty
rotor bar location.
u
1
0
1y
0
0
1
System
1
0
Discrete
Model
1
0
Tdisc
Torque
1
0
Observer
OMD
1
0
Fig. 1.
Tobs
T
Difference
Structure of the fault rotor bar detection method
V. E XPERIMENTAL T ESTS
A. The experimental setup description
Fig 5 illustrates the experimental setup. It consists of a 3
CV, 220/380 V, 60 Hz, four pole induction motor.
A mechanical load is provided by a separate dc generator
feeding a variable resistor. In order to allow tests to be
performed at different load levels, the dc excitation current
and load resistor are both controlable.
The data acquisition system consists of:
• three hall effect current sensor (LEM, LTA50P);
• three hall effect voltage sensor (LEM, LV 100-300)
• analog input board (National Instruments PCI 6013)
The computational implementation for detection and fault
diagnosis technique runs in a LabView environment.
1) Disconnect and remove the end rings.
2) Remove the rotor bars from the core.
3) Steam clean the rotor and all component parts with
proven methods and materials to avoid damage to the
insulation systems.
4) Oven dry the rotor and component parts to insure
optimum dryness.
5) Analyze the bar material and manufacture new bars.
6) Clean and prepare the core slots for installation of the
bars.
7) Manufacture new bars
8) Install the new bars. The bars protrude beyond the rotor
and are connected together using a screw. The screw
makes the end-ring
9) Dynamically balance the rotor.
The new rotor allows non-destructive tests of broken bars.
If one screw is taken away it seems the rotor has one broken
bar.
Fig 3 shows the clean rotor without the bars. In fig 4 we
have the rotor with the new bars connecting together using
screws.
Fig. 3.
Fig. 2.
The experimental setup
The original rotor windings are made up of rotor bars
passing through the rotor, from one end to the other, around
the surface of the rotor. The bars protrude beyond the rotor
and are connected together by a shorting ring at each end. We
removed the old bars, and installed new bars. The method for
to replace the bars was:
Fig. 4.
Rotor without the bars
Rotor with new bars.
B. Experiment Results
The motor was initially analysed without removing any
screw. The waveform voltage supply, waveform phase current
and the frequency voltage and current spectra is shown in
Fig 5, for a full load operation condition. It is interesting to
note that even the healthy machine has the (1 ± 2s)f pair
of sidebands around the fundamental. In original motor, the
presence of those components are probably due to inherent
rotor asymmetries. In our special motor, the dB difference
between the sideband magnitudes and the supply frequency
components is 28 dB which, by experience, is indicative of a
serious broken bar problem. We can reach the conclusion that
the contact bar-screw is not perfect, It seems that the rotor has
a numerous broken rotor bars.
magnitudes and the supply frequency component is greater
then 55 dB which is indicative of a healthy rotor.
Spectra 7(a) and 7(b) show the side bands around the
fundamental but in spectra 7(c) and 7(d), due to the small slip,
it is not possible to see the modulate components. Thus, the
distinguishing the fault conditions from the normal conditions
based on the FFT spectrum is a difficult task
(a) nominal load condition
(b) 89% of nominal load condition
(c) 72% of nominal load condition (d) 61% of nominal load condition
Fig. 5. Voltage supply waveform, phase current waveform, voltage frequency
spectrum and current frequency spectrum.
The torque residues for different operation conditions are
shown in Fig 6. As can been clearly seen, while the progressive
number of screws is removed, the torque residue increases.
Thus, this parameter is robust for rotor fault detection.
Fig. 7.
Current frequency spectra for healthy motor.
The torque residues for those four load situations are shown
in fig 8. The highest value residue is for the nominal load
motor operation. It is stablished to be the fault pattern. The
result for other tests will be compared to it for the motor’s
condition diagnose.
(a) small load
(b) full load
(a) nominal load condition
(b) 89% of nominal load conditiona
(c) one screw removed
(d) three screws removed
(c) 72% of nominal load condition
(d) 61% of nominal load condition
Fig. 6.
Torque residue.
In order to verify the accuracy of the proposed method, the
modified rotor was replaced by the same power original rotor
(WEG, 3 CV). The motor was initially set with intact cage.
The currents spectra for different operation load conditions
are shown in fig. 7. The dB difference between the sideband
Fig. 8.
Torque residue for healthy motor
The destructive rotor fault is simulated by drilling holes on
the aluminum bars. Fig 9 shows the rotor with one broken
rotor bar.
The torque residues are shown in Fig V-B for two operation
conditions: full operation and 61% of nominal load condition.
To validate the proposed method, a set of tests was prepared
for squirrel cage induction machine
The proposed technique is sensitive enough to detect and
locate even single bar faults under different load conditions. It
utilizes input signals from standard transducers and encoder
that are available in industrial drive. So the technique is
suitable for real time implementation.
Acknowledgments
The authors gratefully acknowledge the financial support
from FAPEMIG - Fundação de Amparo à Pesquisa do Estado
de Minas Gerais - and PICDT/CAPES - Programa Institucional
de Capacitação Docente e Técnica da Coordenação de
Aperfeiçoamento de Pessoal de Nı́vel Superior.
R EFERENCES
Fig. 9.
Rotor with one broken bar.
Comparing fig 8(a) with fig 10(a) and fig 8(d) with fig 10(b)
it can be clearly seen that the torque residue increases with
the progressive bar breakage.
Once the motor has 4 poles the minimum torque difference
occurs in four bars, shifted 90o from each other. Thus these
bars are: 2, 9, 16 and 23. It can be noticed that the proposed
method could find the broken bar even for a small load motor
operation.
The results for the methods are not affected by the low
frequency components around the fundamental frequency.
(a) nominal load condition
Fig. 10.
(b) 89% of nominal load condition
Torque residue with one broken rotor bar.
VI. C ONCLUSION
Machinery does not need to be taken out of service since
many tests can be done online, and in many cases very little
expertise is required for testing and data interpretation. This
enables the user to make well-informed decisions for planning
maintenance and repairs, which ultimately leads to increased
productivity.
This paper was concerned with rotor cage asymmetry.
Once a bar breaks, the neighboring bars also deteriorates
progressively due to increased stresses. To prevent such a
cumulative destructive process, the problem should be detected
early, when the bars are beginning to crack. Thus, a novel
approach has been proposed in this paper to detect and locate
one broken rotor bar in induction motors.
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