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1
Benchmark Problems
- Coupled Electric Drives
- F16 Ground Vibration
- Cascaded Tanks System
- Wiener-Hammerstein Process Noise System
- Bouc-Wen System
- Parallel Wiener Hammersterin
- Wiener Hammerstein
- SilverBox
Wiener Approaches
- Wiener Neural Identification [1]
- Iterative Wiener Identification [2]
[1] MM Arefi, A Montazeri, J Poshtan, MR Jahed Motlagh, “Wiener-neural identification
and predictive control of a more realistic plug-flow tubular reactor,” Chemical
Engineering Journal, vol 138, No 1-3, pp. 274-282, 2008.
[2] H Kazemi, MM Arefi, “A fast iterative recursive least squares algorithm for Wiener
model identification of highly nonlinear system,” ISA Transactions, vol. 67, pp. 382-388,
2017.
Page 2 of 27
An Investigation of the Wiener
Approach for Nonlinear System
Identification Benchmarks
Allahyar Montazeri
Mohammad Mehdi Arefi, Mehdi Kazemi
Lancaster University
Faculty of Science and Technology
E-mail: a.montazeri@lancaster.ac.uk
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Block Oreinted Approach
3
Physically insightful
Lower number of parameters
× The model is not linear in parameters
× Difficulty in finding a good initial condition
G q( )
f ( ) x
Wiener
u n( ) z n( ) y n( ) model structure
Can approximate almost any nonlinear system
with high accuracy
No specific assumption on the spectrum input
or static nonlinearity
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Wiener Neural Technique
NN x( ) u n( ) z n( ) y n( )
Step 1 Estimating the linear part using the given input/output data
Estimating the nonlinear part (Neural Network)
using the estimated linear model and the
measured output
Step 2
Step 3
Parametrising the estimated models in the
previous steps and optimising the overall
parameters of the Wiener model
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Wiener Neural Technique
Assume no non-linear part, input-output data
Calculate , ,
Step 1 NN x( ) u n( ) z n( ) y n( )
(,,,) ABCD
x(0)
n
Step 2
1 1
( ( )) ( , ) ( , , ) ( ) ( , ) ( , 1) ( )
l
j s
i j
NN z n s i s i j z n b s i b s n
2
1
1 1
1 1
1
( ) ( , ) ( , , )ˆ ( ) ( , ) ( , 1)
( ) (1, ) (1, , )ˆ ( ) (1, ) (1, 1)
min
N
k
i
l
j
l j
i
l
j
j
y
k l i l i j
z
k
b l i b l
y
k i i j
z
k
b i b
θ
(
s,i
)
(
s,i, j
)
b
(
s,i)
b
(
s,
1
) ((
2) 1) l l
θ
R
u n( ), ( ) : 1 y nn N
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Wiener Neural Technique
Parametrisation of the linear state-space model
Optimising the whole parameters of the system
Step 3 NN x( ) u n( ) z n( ) y n( )
Definition- The pair is in the output normal form if (,) A C
n
T
T
A
A
C
C
I
,
nn ln A C
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Benchmark Criteria
2
1
1 () () ˆ
N
rms
t
e
y
k
y
k
N
ˆ
(1 ) 100 y y FIT
y y
1
1 () () ˆ
N
mean
t
e
y
k
y
k
N
A plot with the modelled output and the simulation error in
time domain
A plot of simulation error in frequency domain
J. Schoukens, J. Suykens, L. Ljung, “Wiener-Hammerstein Benchmark,” 15th IFAC Symposium on
System Identification (SYSID 2009), July 6-8, St. Malo, France, 2009.
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Winer-Hammerstein
System
3rd order Chebyshev filter with cut
off 4.4 kHz
3rd order inverse Chebyshev filter
with cut off 5kHz
Transmission zero in the frequency
band of interest
1 G q( )
f ( ) x u n( ) x n( ) w n( )
2 G q( ) y n( )
Input/Output test data
Filtered Gaussian signal with cut off
10kHz
Sampling frequency 51.2kHz
SNR 70dB
Estimation
data
Test data
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Winer-Hammerstein
Frequency response plot of the
linear part
-1 -0.5 0 0.5 1 1.5 2 2.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Pole-Zero Map
Real Axis
Imaginary Axis
Pole-Zero map
of the linear part
NOTE: transmission zero of the second
filter is captured
Mag(dB)
P1= 0.89 േ i0.17
P2=0.7
േ i0.4
Z1= 0.6 േ i0.99
Z2= 0.8
േ i0.6
Page 23 of 27
Winer-Hammerstein
- Delay
- Transfer function order
- Static nonlinearity = exponential
2,
a
n
2
d
n
0 200 400 600 800 1000 1200 1400 1600 1800 2000
-1
-0.5
0
0.5
Test
FIT = 98.1439 %
Measured Output
Model Output
12345678
Samples 104
-0.01
0
0.01
Residu
rms= 0.0044405, mean = -4.884e-05
residu (dB)
2
b
n
Residual error spectrum