Reservoir Computing Methods for Prognostics and Health ... · •260 run-to-failure trajectories...

Reservoir Computing Methods for

Prognostics and Health Management (PHM)

Piero Baraldi, Mingjing Xu

Energy Department

Politecnico di Milano

Italy

In this presentaton

• Recurrent Neural Network (RNN)

• Reservoir Computing

• Echo State Network

• Application 1: Prediction of Turbofan Engine RUL

• Application 2: Prediction of ALSTOM Fast Train Brake system RUL

2

Recurrent NN: General Idea 3

𝑅𝑈𝐿 𝑡𝑝𝒖 1: 𝑡𝑝

𝑢1

𝑢𝑁Time

trajectory

𝑡 = 1

𝑡 = 𝑡𝑝

PROGNOSTIC MODEL

𝒖 1: 𝑡𝑝


𝑥2

𝑥1

𝑥𝑀

Linear

Regression

Non Linear

Expansion

𝑅𝑈𝐿 𝑡𝑝 = 𝑾𝒐𝒖𝒕𝒙(𝑡𝑝)𝒖 1: 𝑡𝑝

𝑢1

𝑢𝑁Time

trajectory

𝑡 = 1

𝑡 = 𝑡𝑝𝑟𝑢𝑙

𝑀 ≫ 𝑁

𝒙 𝑡𝑝 = 𝑓 𝒖 1: 𝑡𝑝

𝒖 1: 𝑡𝑝

𝒙 𝑡𝑝

𝑅𝑈𝐿 𝑡𝑝


𝑥2

𝑥1

𝑥𝑀

Linear

Regression

Non Linear

Expansion

𝑅𝑈𝐿 𝑡𝑝 = 𝑾𝒐𝒖𝒕𝒙(𝑡𝑝)𝒖 1: 𝑡𝑝

𝑢1

𝑢𝑁Time

trajectory

𝑡 = 1

𝑡 = 𝑡𝑝𝑟𝑢𝑙

𝑀 ≫ 𝑁

𝒙 𝑡𝑝 = 𝑓 𝒖 1: 𝑡𝑝 = 𝑓 𝒖 1: 𝑡𝑝 − 1 , 𝒖 𝑡𝑝

𝑥 𝑡𝑝 − 1 = 𝑓 𝒖 1: 𝑡𝑝 − 1

𝒙 𝑡𝑝 = 𝑓 𝒙(𝑡𝑝 − 1), 𝒖 𝑡𝑝

𝒖 1: 𝑡𝑝

𝒙 𝑡𝑝


Recurrent NN 6

𝒖 1: 𝑡𝑝 𝑥1(𝑡𝑝) = 𝑓

𝑖=1

𝑁

𝑤𝑖1𝑖𝑛 𝑢𝑖(𝑡𝑝) +

𝑖=1

𝑀

𝑤𝑖1 𝑥𝑖 (𝑡𝑝 − 1)

𝑾𝒊𝒏

𝑾Non Linear Expansion

𝑢3(𝑡𝑝)

𝑢2(𝑡𝑝)

𝑢1(𝑡𝑝)

Recurrent NN 7

𝒖 1: 𝑡𝑝 𝒙(𝑡𝑝) = 𝑓 𝑾𝒊𝒏𝒖(𝑡𝑝) +𝑾𝒙(𝑡𝑝 − 1) 𝑅𝑈𝐿 𝑡𝑝 = 𝑊𝑜𝑢𝑡𝒙(𝑡𝑝)

𝑾Linear RegresionNon Linear Expansion

𝑢3(𝑡𝑝)

𝑢2(𝑡𝑝)

𝑢1(𝑡𝑝)

𝑾𝒊𝒏

𝑾𝒐𝒖𝒕


RNN: Training 8

𝑊𝑖𝑛,𝑊,𝑊𝑜𝑢𝑡

TRAINING SET

𝒖 1 , 𝑅𝑈𝐿𝐺𝑇 1 = 𝑡𝑓 − 1

…𝒖 5 , 𝑅𝑈𝐿𝐺𝑇 5 = 𝑡𝑓 − 5

…

𝒖 𝑡𝑓 − 1 , 𝑅𝑈𝐿𝐺𝑇 𝑡𝑓 − 1 = 1

Run-to-failure

degradation trajectory

t

𝒖

5 𝑡𝑓

𝒖(5)

𝑅𝑈𝐿𝐺𝑇 5 = 𝑡𝑓 − 5

RNN: Training 9


Training Objective: minimize the error function

𝐸 𝑅𝑈𝐿, 𝑅𝑈𝐿𝐺𝑇 =RMSE=

𝑡=1

𝑡𝑓−11

𝑡𝑓 − 1𝑅𝑈𝐿(𝑡) − 𝑅𝑈𝐿𝐺𝑇(𝑡) 2

TRAINING SET

𝒖 1 , 𝑅𝑈𝐿𝐺𝑇 1 = 𝑡𝑓 − 1

𝒖 2 , 𝑅𝑈𝐿𝐺𝑇 2 = 𝑡𝑓 − 2

…

𝒖 𝑡𝑓 − 1 , 𝑅𝑈𝐿𝐺𝑇 𝑡𝑓 − 1 = 1

RNN: Training 10


Training Objective: minimize the error function

𝐸 𝑅𝑈𝐿, 𝑅𝑈𝐿𝐺𝑇 =RMSE=

𝑡=1

𝑡𝑓−11

𝑡𝑓 − 1𝑅𝑈𝐿(𝑡) − 𝑅𝑈𝐿𝐺𝑇(𝑡) 2

TRAINING SET

𝒖 1 , 𝑅𝑈𝐿𝐺𝑇 1 = 𝑡𝑓 − 1

𝒖 2 , 𝑅𝑈𝐿𝐺𝑇 2 = 𝑡𝑓 − 2

…

𝒖 𝑡𝑓 − 1 , 𝑅𝑈𝐿𝐺𝑇 𝑡𝑓 − 1 = 1

Training Methods:

• Gradient-descent-based methods


Gradient-descent-based methods for RNN

RNN are difficult to train using gradient-descent-based methods:

• Bifurcations

• Many updating cycles → Too long training times

• Hard to obtain long range memory

11

𝑾𝒊𝒏 𝑾𝒐𝒖𝒕

𝑾

-

𝑅𝑈𝐿𝐺𝑇(𝑡)𝑅𝑈𝐿(𝑡)

Error(t)

𝒖(𝑡)

In this presentaton






12

Reservoir Computing (RC): Terminology

What is it? Purpose

ReservoirNon-linear temporal

expansion function

Expand the input history 𝒖 1: 𝑡𝑝 into a rich-enough

reservoir space 𝒙(𝑡𝑝)

readout Linear function

Combine the neuron signals 𝒙(𝑡𝑝) into the desired output

signal target 𝑅𝑈𝐿 𝑡𝑝

Reservoir Computing (RC): Basic Idea 14

What is it? Purpose

ReservoirNon-linear temporal

expansion function

Expand the input hystory 𝒖 1: 𝑡𝑝 into a rich-enough

reservoir space 𝒙(𝑡𝑝)

readout Linear function

Combine the neuron signals 𝒙(𝑡𝑝) into the desired output

signal target 𝑅𝑈𝐿 𝑡𝑝

Reservoir and readout

serve different purposes

They can be separately

trained

In this presentaton






15

Generate the reservoir

...

...

Readout

Readout

Linear regression

Traditional RNN ESN

-

Training: Traditional RNN VS ESN 18

error

-

error

random

The Echo State Property

In this presentaton






20

Prognostics: What is the Problem?

Aircraft Turbofan Engine

N Monitored Signals

• Signal 1

• Signal 2

• ………..

• Signal N

Aircraft Engine

RUL Prediction

TimeR

UL

Prognostic

Model

Tem

per

atu

re

Time

• 260 run-to-failure trajectories

• 21 measured signals + 3 signals representative of the operating

conditions

• 6 different operating conditions

Data

Preprocessing**

The C-MAPPS dataset*

* A. Saxena, K. Goebel, D. Simon, N. Eklund, Damage propagation modeling for aircraft engine run-to-failure simulation, PHM2008

**M. Rigamonti, P. Baraldi, E. Zio, I. Roychoudhury, K. Goebel, S. Poll, Echo State Network for Remaining Useful Life Prediction of a

Turbofan Engine, PHM 2016, Bilbao

ESN Architecture Optimization

Network Architecture Optimization: Parameters

1) Network Dimensions

2) Spectral Radius

3) Connectivity

4) Input Scaling

5) Output Scaling

RUL(t)



RUL(t)


2) Spectral Radius

3) Connectivity

4) Input Scaling

5) Output Scaling

RUL(t)



2) Spectral Radius

3) Connectivity

4) Input Scaling/Shifting

5) Output Scaling/Shifting

6) Output Feedback


RUL(t)



2) Spectral Radius

3) Connectivity

4) Input Scaling

5) Output Scaling


RUL(t)



2) Spectral Radius

3) Connectivity

4) Input Scaling

5) Input Shifting


RUL(t)



2) Spectral Radius

3) Connectivity

4) Input Scaling

5) Input Shifting


Sigmoidal Activation Function


• Optimization Algorithm

• Population-based:

• Evolutionary-based

CHROMOSOME

Network

DimensionsConnectivity

Spectral

Radius

Input

Scaling

Inout

Shifting

Initialization Mutation Crossover Selection

• Experience + trial & errors→ difficult, good performance not guaranteed

• Differential evolution

Objective function: 𝑅𝐴 =σ 𝑅𝑈𝐿𝐺𝑇−𝑅𝑈𝐿

𝑅𝑈𝐿𝐺𝑇

Optimal Architecture



2) Spectral Radius

3) Connectivity

4) Input Scaling

5) Input Shifting

RUL(t)

Network

DimensionsConnectivity

Spectral

Radius

Input

Scaling

Output

Scaling

385 0.17 0.67 0.45 -0.05

ESN for Prognostics: Results (I)

0 20 40 60 80 100 1200

20

40

60

80

100

120

Time (Cycle)

RU

L (

Cyc

le)

RUL Prediction for Tansient 157

True RUL

ESN

FS

ELM

ESN = Echo State Network

FS = Fuzzy Similarity-based Prognosti Method

ELM = Extreme Learning Machine

32

Cumulative Relative Accuracy Steadiness

Extreme

Learning

Machine

0.42 ± 0.03 15.3 ± 2.2

Fuzzy

Similarity-based

Method

Echo State

Network

➢ Results – Prognostic Metrics (70 test trajectories)

RUL

RULLURRA

GT−=

ˆ,)var( :)( tttt TSI −=

ESN for Prognostics: Results (II)

In this presentaton






33

What is the problem?

Problem: Use Prognostic model to predict RUL of Fast Train Brake System

Fast Train Brake System

Fast Train Brake System

Brake systems of Fast Train

Component 1 Component 2 Component 3 Component 4

Case Study

Run-to-Fail Trajectory 170

Signals 6

➢ Available Dataset

Artificial data that mimic the complexity of the industrial data.


0 2000 4000 6000 8000 10000 12000 14000

time

350

360

370

380

390

Tem

pe

ratu

re [

K]

0 2000 4000 6000 8000 10000 12000 14000

time

0

2

4

6

8

10

Op

era

tio

n M

od

e

Case Study

Run-to-Fail Trajectory 170

Signals 6


0 5000 10000 15000

time

0.09

0.1

0.11

0.12

0.13

Degra

da

tion I

nde

x

Component 1

0 5000 10000 15000

time

0.08

0.1

0.12

0.14

0.16

0.18

Degra

da

tion I

nde

x

Component 2

0 5000 10000 15000

time

0.08

0.1

0.12

0.14

0.16

0.18D

egra

da

tion I

nde

x

Component 3

0 5000 10000 15000

time

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Degra

da

tion I

nde

x

Component 4

➢ Available Dataset

Artificial data that mimic the complexity of the industrial data.

Case Study

Simulated Data

Acquisition

➢ Animation: Data Collection Process

Case Study

Simulated Data

Acquisition

➢ Difficulty

• Signals acquisition only when

event occurs

• If no Event occurs, no measurement signals will be collected.

Case Study

Simulated Data

Acquisition

➢ Difficulty


event occurs

Case Study

Simulated Data

Acquisition

➢ Difficulty


event occurs

Stop

data acquisition

Stop

data acquisition

Stop

data acquisition

Stop

data acquisition

Stop

data acquisition

Stop

data acquisition

Case Study 42

➢ i) incomplete data

➢ ii) dynamic and non stationary signal behavouir

➢ iii) continuous modification of industrial equipment operating conditions

Dataset Characteristic

43

➢ RUL prediction result

• prognostic metrics computed on 100 test

trajectories

Metrics CRA [-inf,1] 𝛼 − 𝜆 [0,1]

0.711±0.140 0.635±0.276

• RUL prediction examples

0 1 2 3 4

time 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

RU

L

104

true RUL vs. predicted RUL-- trajectory #120

CRA=0.89778 - =0.92581 SI=95.3245

0 1 2 3 4

time 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

RU

L

104

true RUL vs. predicted RUL-- trajectory #99

CRA=0.75715 - =0.45984 SI=86.79

Events

Events

ESN for Prognostics: Results

Conclusions

Recurrent Neural Network

Training: Reservoir Computing

Echo State Network

• Accurate RUL prediction

• Short Training Time

• Able to catch the system dynamics

Time

RU

L

Dynamic problem

Thank you very much

for your attention!

45

Reservoir Computing Methods for Prognostics and Health ... · •260 run-to-failure trajectories...

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