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Evolving Fuzzy System Applied to Battery Charge
Capacity Prediction for Fault Prognostics
Murilo Camargosa, Iury Bessaa,b, Luiz Cordovil Juniora, Pedro
Coutinhoa, Daniel Leitec, Reinaldo Palharesa
a Federal University of Minas Gerais
b Federal University of Amazonas
c Federal University of Lavras
July 17, 2021

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Outline
Introduction and Motivation
Evolving data-driven model
Takagi-Sugeno representation
Prognostics with evolving fuzzy systems
Results and Discussion
Case study: battery capacity prediction
Conclusions
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General objectives and motivation
General objective
To improve reliability and safety
for critical systems through
condition based maintenance
(CBM).
How?
By proposing new approaches for
CBM in the context of prognostics
and health management systems.
Why?
Cost Number of Failure Events
Preventive
Maintenance
Condition-Based
Maintenance
Corrective
Maintenance
Total Cost
M
a
i
n
t
e
n
a
n
c
e
C
o
s
t
Operating
Cost
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Prognostics steps
Time
Raw
signal
Step 1: Data acquisition.
Time
Health
index
Step 2: Health index construction.
Time
Health
index
Step 3: Health stage division.
Time
Health
index
Step 4: RUL prediction.
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RUL prediction approaches
Physics-based
Rely on mathematical models
derived from Physics-of-Failure.
Pros
• It is the most accurate
approach for prognostics.
• Requires less (or no) training
data than other approaches.
Cons
• Its application can be very
restrict.
• Physics models of complex
systems are hard to obtain.
Data-driven
Use run-to-failure or past
experiments data to train models.
Pros
• No need for modeling complex
system’s relations.
• Can be reused for different
components or systems.
Cons
• Models are not related to any
physical meaning.
• Require lots of high quality
data for training.
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Data-driven RUL prediction
Statistical
Use data to fit empirical
probabilistic models.
Pros
• Effective on describing the
uncertainties inherent to the
prognostics process.
Cons
• Depicted, in general, as a
single degradation model
(stage).
Artificial Intelligence (AI)
Use data to learn the complex
input-output relationship.
Pros
• Almost complete abstraction
from any kind of degradation
model.
Cons
• The models are, in general,
black-boxes with no
explanatory capacity.
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Prognostics with Evolving Fuzzy Sytems (EFS)
Mitigate the disadvantages
• Requirement of large data sets from different operation conditions.
• Modeling the degradation as single stage phenomena.
• Non explanatory models with no physical meaning.
Explore the advantages
• Learning complex behavior through simple degradation models.
• Reuse for different units or components.
• Quantifying the uncertainty in RUL prediction.
Contributions
• Fault prognostics using EFS is performed on Li-ion battery dataset.
• An improved uncertainty quantification procedure for EFS prognostics.
• RUL’s confidence bounds are given as z-values of the normal distribution.
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Problem formulation
Fuzzy inference system - Takagi-Sugeno (TS)
Ri : if xk is Φi,k−1
| {z }
Antecedent
then ŷi,k =

1 x
k

θ̂i,k−1
| {z }
Consequent
Antecedent: modeling correlation
wi,k−1(xk ) = exp

−
1
2
xk − µ̂i,k−1
b
Σ
−1
i,k−1 xk − µ̂i,k−1

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Problem formulation
Final output
ŷk =
C
X
i=1
hi (xk ) ŷi,k , hi (xk ) =
wi,k−1(xk )
PC
m=1 wm,k−1(xk )
ŷk = (hk−1(xk ))

Θ̂

k−1

1 x
k

Where
hk−1(xk) = [h1,k−1(xk) · · · hC,k−1(xk)]
∈ RC
Θ̂k−1 =
h
θ̂1,k−1 · · · θ̂C,k−1
i
∈ Rnx +1×C
The number of rules C varies over time.
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Prognostics with evolving fuzzy systems
Objective
ˆ
rk = inf {N ∈ N : x̂k+N
| {z }
fk (vk+N,L)
≥ η}
N-steps ahead prediction
vk+N,L =



[xk xk−1 · · · xk−L+1]
, if N = 1
[x̂k+N−1 · · · x̂k+1 xk · · · xk+N−L]
, if 2 ≤ N ≤ L
[x̂k+N−1 · · · x̂k+N−L]
, if N L
Enabling fault prognostics in EBeTS
x̂k+N = fk (vk+N,L) = (hk (vk+N,L))
Θ̂

k ṽk+N,L, ∀N 0
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RUL prediction
Uncertainty quantification
Enabling fault prognostics in EBeTS
x̂+
k+N = (hk (zk+N))
Θ̂

k ṽ+
k+N,L, ∀N 0
N-steps ahead prediction
v+
k+N,L =





[xk xk−1 · · · xk−L+1]
, if N = 1

x̂+
k+N−1 · · · x̂+
k+1 xk · · · xk+N−L

, if 2 ≤ N ≤ L

x̂+
k+N−1 · · · x̂+
k+N−L

, if N L
Some definitions
zk+N , E[ṽk+N,L]
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RUL prediction
Uncertainty quantification
Considering the TS system output
x̂+
k+N = (hk (zk+N))
Θ̂

k ṽ+
k+N,L + k+N, k ∼ N 0, σ2

Error covariance online tracking
M,n = M,n−1 + (n − µ̂,n−1)(n − µ̂,n)
,
σ̂2
≈
M,k
k − 1
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RUL prediction
Uncertainty propagation
The TS system becomes
x̂+
k+N = (hk (zk+N))
Θ̂

k
| {z }
ΞN
ṽ+
k+N,L + k+N
Uncertainty N-steps ahead
λ2
N , Var x̂+
k+N

= ΞN ΛL
N Ξ
N + σ2

Where
ΛL
N , Cov ṽ+
k+N,L

=





0 0 · · · 0
0 λ2
N−1 · · · λN−LλN−1ρ̂L,1
.
.
.
.
.
.
...
.
.
.
0 λN−1λN−Lρ̂1,L · · · λ2
N−L





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RUL prediction
Uncertainty quantification - upper and lower bounds
RUL upper and lower bounds at a significance level of (α)(100)%
ˆ
rlower,k = inf {N ∈ N : x̂k+N + z1−α/2 λN ≥ η},
ˆ
rupper,k = inf {N ∈ N : x̂k+N + zα/2 λN ≥ η},
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Metrics used
Three metrics are used to evaluate the EFS degradation models for
prognostics:
MAPEk =
100
H
k+H
X
n=k+1

Evolving Fuzzy System Applied to Battery Charge Capacity Prediction for Fault Prognostics

,
RMSEk =
v
u
u
t 1
H
k+H
X
n=k+1
(xn − x̂n)2,
RAk = 1 −
|rk − ˆ
rk|
rk
.
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Case study: battery capacity prediction
The data set
• Provided by NASA Ames Prognostics Center of Excellence (PCoE).
• The cycle aging data sets of four Li-ion batteries.
• Charging in constant mode at 1.5 A until 4.2 V.
• Discharge at a constant current of 2 A until 2.7 V.
20 40 60 80 100 120 140 160
Discharge cycle
1
1.2
1.4
1.6
1.8
2
2.2
Charge
capacity
(Ah)
B0005 data
B0006 data
B0007 data
B0018 data
20 40 60 80 100 120 140 160
Discharge cycle
50%
60%
70%
80%
90%
100%
Health
index
B0005 data
B0006 data
B0007 data
B0018 data
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Evolving Fuzzy System Applied to Battery Charge Capacity Prediction for Fault Prognostics

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Evolving Fuzzy System Applied to Battery Charge Capacity Prediction for Fault Prognostics