Transcript Development of Node-Decoupled Extended Kalman Network Diagnostic/Prognostic Reasoners

Development of Node-Decoupled Extended Kalman
Filter (NDEKF) Training Method to Design Neural
Network Diagnostic/Prognostic Reasoners
EE645 Final Project
Kenichi Kaneshige
Department of Electrical Engineering
University of Hawaii at Manoa
2540 Dole St.
Honolulu, HI 96822
Email: [email protected]
Contents
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
Motivation
What is Kalman Filter?
–
Linear Kalman Filter
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–
–
–
Kalman Filter and Neural Network
Extended Kalman Filter (EKF)
Node-Decoupled Extended Kalman Filter (NDEKF)
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Diagnosis/prognosis
Simulation
–
–
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Result
Detecting the fault condition
–

Node-decoupling
NDEKF Algorithm
Training the network
–

Simulation with Linear KF
Create a situation
Train the neural network
Result
Conclusion and possible future work
Reference
Motivation
 The detection of the system condition in a
real time manner
 The node-decoupled extended kalman filter
(NDEKF) algorithm
 Should work even when the input changes
(robustness)
 The strength of the neural network
What is Kalman Filter?
 Kalman filter = an optimal recursive data
processing algorithm
 Used for stochastic estimation from noisy
sensor measurements
 Predictor-Corrector type estimator
 Optimal because it incorporates all the
provided information (measurements)
regardless of their precision
Linear Kalman Filter
 Used for linear system model
 Time update equation (predictor equation)
– Responsible for projecting
 the current state
 the error covariance estimates
– Obtain a priori estimates
 Measurement update equation (corrector
equation)
– Responsible for feedback
 Incorporate a new measurement into the a priori estimate
– Obtain a posteriori estimates
Linear Kalman Filter Algorithm
Time Update (predict)
Measurement Update (correct)
(1) Project the state ahead
xˆ

k
(1) Compute the Kalman gain
Pk  APk 1 AT  Q
 Ax
ˆ k 1  BU k
(2) Update estimate with measurement Zk
(2) Project the error covariance ahead
K k  Pk H T ( HPk H T  R) 1
xˆ k  xˆk  k k ( zk  Hxˆk )
(3) Update the error covariance

k
Pk  ( I  K k H ) P
Initial estimates for
xˆ k 1
and
Pk 1
Simulation with Linear KF
Linear Kalman Filter Performance
80
70
60
50
Position(feet)
40
30
20
10
0
Actual Altitude of the Aircraft
Linear KF Estimate
Sensor Data
-10
-20
0
5
10
15
Time(sec)
20
25
30
The downside of LKF
 LKF seems to be working fine. What’s
wrong?
– Works only for the linear model of a dynamical
system
 When the system is linear, we may extend
Kalman filtering through a linearization
procedure
Kalman Filter and Neural Network
 Want better training methods in
–
–
–
–
Training speed
Mapping accuracy
Generalization
Overall performance
 The most promising training methods to satisfy above.
– weight update procedures based upon second-order derivative information
(Standard BP is based on first derivative)
 Popular second-order methods
– Quasi-Newton
– Levenburg-Marquardt
– Conjugate gradient techniques
 However, these often converge to local optima because of the lack of a
stochastic component in the weight update procedures
Extended Kalman Filter (EKF)
 Second-order neural network training method
 During training, not only the weights, but also an error
covariance matrix that encodes second-order information is
also maintained
 Practical and effective alternative to the batch oriented
 Developed to enable the application of feedfoward and
recurrent neural network (late 1980s)
 Shown to be substantially more effective than standard BP
(epochs)
 Downside of EKF: computational complexity
– Because of second-order information that correlates every pair of
network weights
Decoupled Extended Kalman Filter (DEKF)
 EKF: develops and maintains correlations
between each pair of network weights
 DEKF: develops and maintains secondorder information only between weights that
belong to mutually exclusive groups
 Family of DEKF:
– Layer Decoupled EKF
– Node Decoupled EKF
– Fully Decoupled EKF
Neural Network’s Behavior
Process Equation:
wk 1  wk  k
Measurement Equation:
yk  hk ( wk , uk , vk 1 )  k
Process Noise:
k ~ N [0, Q]
E[klT ]   k ,l Qk
Measurement Noise:
 k ~ N [0, R]
E[ k lT ]   k ,l Rk
wk
Weight Parameter Vector
yk
Desired Response Vector
hk
Nonlinear Function
uk
Input Vector
Node-decoupling
 Perform the Kalman recursion on a smaller part of
the network at a time and continue until each part
of the network is updated.
 Reduces the P() matrix to diagonal matrix
 State Variable Representation is the following
xk 1  xk
yk  h( xk , uk )  k
NDEKF Algorithm
K ki  Pki H ki Ak
xˆ () Estimated value of x ()
xˆki 1  xki  K ki  k
P() Approximate conditional error
Pki1  Pki  K ki H 'ik Pki  Qk
K () Kalman filter gain matrix
N
Ak  [ Rk   H 'ik Pki H ki ]1
i 1
covariance matrix of
x ()
 () Error between desired and actual outputs
R() Measurement noise covariance matrix
Q () Process noise covariance matrix
H 'ik
Weight update equation (take partial derivatives)
(For details, please refer to the paper)
NDEKF with Neural Network
 1-20-50-1 MLFFNN is used. (1 input, 20
nodes in the first layer, 50 nodes in the
second layer)
 Used bipolar activation function for the two
hidden layers
 Linear activation function at the output node
Training the Network
Input
Actual System
Neural Network
System in normal
condition
System in normal
condition
System in failed
condition 1
System in failed
condition 1
System in failed
condition 2
System in failed
condition 2
System in failed
condition n
System in failed
condition n
Simulation
 Assume the system or the plant has
characteristics with following nonlinear equation
y (k  1)  0.3 y (k )  0.6 y (k  1)  f [u (k )]
f [u(k )]  u 3  0.3u 2  0.4u
 Inputs are the following
2k
 For k=1:249
u (k )  sin(
)
250
 For k=250:500
u (k )  0.8 sin(
2k
2k
)  0.2 sin(
)
250
25
Why Neural Network?
 Input independency
 Robustness
 High accuracy for fault detection and
identification
Result for normal condition and
failure condition 1
The Neural Network Output for Normal Condition
10
Amplitude
Input
Desired Output
NN Output
5
0
-5
0
50
100
150
200
250
Time
300
350
400
450
500
The Neural Network Output for Failure Condition 1
10
Amplitude
Input
Desired Output
NN Output
5
0
-5
0
50
100
150
200
250
Time
300
MSE = 4.7777
350
400
450
500
Result for failure condition 2 and 3
The Neural Network Output for Failure Condition 2
10
Amplitude
Input
Desired Output
NN Output
5
0
-5
0
50
100
150
200
250
Time
300
350
400
450
500
The Neural Network Output for Failure Condition 3
10
Amplitude
Input
Desired Output
NN Output
5
0
-5
0
50
100
150
200
250
Time
300
MSE=7.8946
350
400
450
500
The actual outputs of the system
with its inputs.
The Actual Output of the System
10
Amplitude
Input
Actual Output
5
0
-5
0
50
100
150
200
250
Time
300
350
400
450
500
Diagnosis of the system in actual condition
 Create an actual situation using one of the
conditions (normal, failed1, failed2, failed3)
 Take MSE with the neural network of each
of the conditions with the same input to the
actual system.
 Take minimum of the MSE (MMSE), and it is
the most probable condition of the actual
system
Result
Here, the actual condition was tested with failed condition 2
Inputs 1 (Time 1 to 249)
Inputs 2 (Time 250 to 500)
Normal Condition
3.2703
1.3738
Failure Condition 1
5.4838
3.3171
Failure Condition 2
0.0123
0.1010
Failure Condition 3
5.4671
3.4935
From above, the MMSE shows the actual system is most probably be
in failed condition 2
Conclusion and possible future work
– The downside is there have to be a priori knowledge
about the fault conditions
– Work in frequency domain (FFT)
– Implement with different algorithm and compare
 SVM, BP, Perceptron, etc…
– Work with huge noise
– Work with an actual model
– OSA/CBM (Open Systems Architecture / Condition
Based Maintenance)
 Using XML and let the real time report available on the internet
References
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