Reservoir Computing Methods for Prognostics and Health ... · Reservoir Computing Methods for Prognostics and Health Management (PHM) Piero Baraldi Energy Department Politecnico di

Reservoir Computing Methods for

Prognostics and Health Management (PHM)

Piero Baraldi

Energy Department

Politecnico di Milano

Italy

2 Piero Baraldi

Industry 4.0 2

20172012 Time

Da

ta

Available data

• Digitalization2.8 Trillion GD (ZD)

generated in 2016

• Analytics

AnalyticsData

3 Piero Baraldi

Industry 4.0 3

20172012 Time

Da

ta

Available data

• Digitalization2.8 Trillion GD (ZD)

generated in 2016

• Analytics

AnalyticsData Predictive

Maintenance

4 Piero Baraldi

4Predictive Maintenance 4

Monitored

Signals

Ambient &

Operating

Conditions

Prognostic

ModelIndustry 4.0

Remaining Useful Life

(RUL)

tp

𝑢1

𝑢2

𝑢𝑁

Analytics

Present time

Failure time

5 Piero Baraldi

5Predictive Maintenance 5

• Future demand

• Logistics options

• …

Monitored

Signals

Ambient &

Operating

Conditions

Prognostic

Model

Optimal

Maintenance DecisionsSafety

improvement,

Cost Saving,

New Business

• Maximum Availability

• Business continuity

• Warehouse savings

• Zero-defect production

• Zero-waste production

Industry 4.0

Remaining Useful Life

(RUL)

tpPresent

timeFailure

time

𝑢1

𝑢2

𝑢𝑁

Analytics

6 Piero Baraldi

In this Presentaton

• Prognostics

• Recurrent Neural Network (RNN)

• Reservoir Computing

• Echo State Network

• Application to the Prediction of Turbofan Engine RUL

6

7

Prognostics: What is the Problem?

Aircraft Turbofan Engine

N Monitored Signals

• Signal 1

• Signal 2

• ………..

• Signal N

Time

Te

mp

era

ture

8



N Monitored Signals

• Signal 1

• Signal 2

• ………..

• Signal N

Aircraft Engine

RUL Prediction

Time Time

RU

L

Te

mp

era

ture

Prognostic

Model

9 Piero Baraldi

Prognostics: the Challenge 9

• 2 identical components

• Same measurements at

time 𝑡𝑝: 𝑢 𝑡𝑝 ≡ 𝑢(𝑡𝑝)

• System evolution depends on present and past signal values

(the memory of the history)

Monitored

Signal

𝑡𝑝

𝑢

𝑡𝑓0 𝑡𝑝

𝑡𝑝 𝑡𝑝 𝑡𝑓

𝑢

Monitored

Signal

10

• Connection:

• Computational unit (neurons):

Prognostics: Methods

Feedforward Neural Network

𝑤11𝑢1

𝑢1𝑤11

𝑢1𝑤11

𝑢2𝑤21

𝑢3𝑤31

𝑓

𝑖=1

3

𝑢𝑖 𝑤𝑖1

𝑅𝑈𝐿 𝑡𝑝

11


• connections only "from left

to right", no connection

cycle

• no memory



12

• connections only "from left

to right", no connection

cycle

• no memory



• at least one connection

cycle

• activation can

"reverberate", persist even

with no input

• system with memory

Recurrent Neural Network

𝑅𝑈𝐿 𝑡𝑝𝑅𝑈𝐿 𝑡𝑝

13 Piero Baraldi

In this presentaton

• Prognostics





13

14 Piero Baraldi

Recurrent NN: General Idea 14

𝑅𝑈𝐿 𝑡𝑝𝒖 1: 𝑡𝑝

𝑢1

𝑢𝑁Time

trajectory

𝑡 = 1

𝑡 = 𝑡𝑝

PROGNOSTIC MODEL

𝒖 1: 𝑡𝑝

15 Piero Baraldi

Recurrent NN: General Idea 15

𝑥2

𝑥1

𝑥𝑀

Linear

Regression

Non Linear

Expansion

𝑅𝑈𝐿 𝑡𝑝 = 𝑾𝒐𝒖𝒕𝒙(𝑡𝑝)𝒖 1: 𝑡𝑝

𝑢1

𝑢𝑁Time

trajectory

𝑡 = 1

𝑡 = 𝑡𝑝

𝑟𝑢𝑙

𝑀 ≫ 𝑁

𝒙 𝑡𝑝 = 𝑓 𝒖 1: 𝑡𝑝

𝒙 𝑡𝑝 = 𝑓 𝒙(𝑡𝑝 − 1), 𝒖 𝑡𝑝

Recursive

definition

𝒖 1: 𝑡𝑝

𝒙 𝑡𝑝


16 Piero Baraldi

Recurrent NN 16

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)

17 Piero Baraldi

Recurrent NN 17

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


18

RNN: Training

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

19

RNN: Training19


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

20

RNN: Training20


Training Objective: minimize the error function


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


21

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

21

𝑾𝒊𝒏 𝑾𝒐𝒖𝒕

𝑾

-

𝑅𝑈𝐿𝐺𝑇(𝑡)

𝑅𝑈𝐿(𝑡)

Error(t)

𝒖(𝑡)

22 Piero Baraldi

In this presentaton

• Prognostics





22

23 Piero Baraldi

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

24 Piero Baraldi

Reservoir Computing (RC): Basic Idea 24

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

25 Piero Baraldi

Reservoir Methods

• Echo State Networks

• Liquid State Machines

• Evolino

• Backpropagation-Decorrelation

• Temporal Recurrent Networks

• …

25

26 Piero Baraldi

In this presentaton

• Prognostics





26

27

Generate the reservoir

• Purpose: obtain a rich enough reservoir space 𝒙(𝑛)

• Recipe:

➢ Big reservoir (𝑀 up to 104) rich enough reservoir space

➢ Sparsely connected 𝑊 is sparse (no more than 20% of possible

connections)

➢ Randomly connected weights of the connections are randomly

generated from a uniform distribution symmetric around the zero value

...

...

𝑾

28

Readout

• Purpose: learn the weights 𝑊𝑜𝑢𝑡 which minimize:


𝑡=1

𝑡𝑓−11

𝑡𝑓 − 1𝑊𝑜𝑢𝑡𝒙(𝑡) − 𝑅𝑈𝐿𝐺𝑇(𝑡) 2

𝑾𝒊𝒏

𝑾𝒐𝒖𝒕

𝑾

(𝑡𝑝)

𝑢1(𝑡)

𝑢2(𝑡)

𝑢𝑁(𝑡)

𝑅𝑈𝐿 𝑡 = 𝑊𝑜𝑢𝑡𝑥(𝑡)

Readout𝑊𝑜𝑢𝑡𝒙 𝑡 = 𝑅𝑈𝐿𝐺𝑇(𝑡)

Linear regression

29

Traditional RNN ESN

𝑾𝒊𝒏 𝑾𝒐𝒖𝒕

𝑾

-

Training: Traditional RNN VS ESN 29

error

𝑾𝒊𝒏

𝑾𝒐𝒖𝒕

𝑾

-

𝑅𝑈𝐿𝐺𝑇

𝑅𝑈𝐿

error


𝑅𝑈𝐿

random

𝒖𝒖

30

The Echo State Property

• The effect of 𝒙 𝑡 and 𝒖 𝑡 on a future state 𝒙 𝑡 + 𝑘 should vanish

gradually as time passes (i.e., 𝑘 → ∞ ) and not persist or even get

amplified.

• For most practical purposes:

𝜌 𝑊 : spectral radius of 𝑊 = largest absolute eigenvalue of W < 1

Echo State Property is satisfied

31

In this presentaton

• Prognostics:

• Recurrent Neural Networks (RNN)



• Application to the prediction of Turbofan Engine RULs

31

32



N Monitored Signals

• Signal 1

• Signal 2

• ………..

• Signal N

Aircraft Engine

RUL Prediction

TimeR

UL

Prognostic

Model

Te

mp

era

ture

Time

33

• 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

34

ESN Architecture Optimization

Network Architecture Optimization: Parameters

1) Network Dimensions

2) Spectral Radius

3) Connectivity

4) Input Scaling

5) Output Scaling

RUL(t)

35



RUL(t)


2) Spectral Radius

3) Connectivity

4) Input Scaling

5) Output Scaling

36

RUL(t)



2) Spectral Radius

3) Connectivity

4) Input Scaling/Shifting

5) Output Scaling/Shifting

6) Output Feedback


37

RUL(t)



2) Spectral Radius

3) Connectivity

4) Input Scaling

5) Output Scaling


38

RUL(t)



2) Spectral Radius

3) Connectivity

4) Input Scaling

5) Input Shifting


39

RUL(t)



2) Spectral Radius

3) Connectivity

4) Input Scaling

5) Input Shifting


Sigmoidal Activation Function

40


• Optimization Algorithm

• Population-based:

• Evolutionary-based

CHROMOSOME

Network

DimensionsConnectivity

Spectral

Radius

Input

Scaling

Inout

Shifting

Initialization Mutation Crossover Selection

Objective function: 𝑅𝐴 =σ 𝑅𝑈𝐿𝐺𝑇−𝑅𝑈𝐿


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

• Differential evolution

41

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

42

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

43

43

Cumulative Relative Accuracy Alpha-Lambda

𝜶 = 𝟎. 𝟐

Steadiness

Extreme

Learning

Machine

0.42 ± 0.03 0.31 ± 0.04 15.3 ± 2.2

Fuzzy

Similarity-based

Method

0.48 ± 0.04 0.34 ± 0.04 12.7 ± 0.7

Echo State

Network 0.37 ± 0.03 0.38 ± 0.04 12.4 ± 1.2

➢ Results – Prognostic Metrics (70 test trajectories)

RUL

RULLURRA

GT

ˆ,)var( :)( tttt TSI

ESN for Prognostics: Results (II)

44

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

45

Acknowledgments

• Dr. Sameer Al-Dahidi

• Francesco Cannarile

• Dr. Michele Compare

• Dr. Francesco Di Maio

• Dr. Marco Rigamonti

• Mingjing Xu

• Zhe Yang

• Prof. Enrico Zio

45

46

Thank

You!

Documents

Reservoir Computing Methods for Prognostics and Health ... · Reservoir Computing Methods for Prognostics and Health Management (PHM) Piero Baraldi Energy Department Politecnico di