Benchmark Problems Wiener Approaches - Coupled Electric Drives - - - PowerPoint PPT Presentation

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.

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

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

( ) u n ( ) z n ( ) y n

Wiener model structure

• Can approximate almost any nonlinear system

with high accuracy

• No specific assumption on the spectrum input
• r static nonlinearity

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

Wiener Neural Technique

• Assume no non-linear part, input-output data
• Calculate , ,

Step 1

( ) NN x

( ) u n ( ) z n ( ) y n

( , , , ) A B C D (0) x 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

) 1 , ( ) , ( ) ( ˆ ) , , ( ) , ( ) ( ) 1 , 1 ( ) , 1 ( ) ( ˆ ) , , 1 ( ) , 1 ( ) ( min 

   

    

                                       

N k i l j j l i l j j

l b i l b k z j i l i l k y b i b k z j i i k y

 

       

θ

) , ( i s 

) , , ( j i s 

) , ( i s b ) 1 , (   s b

) 1 ) 2 ((  



 l l

R θ

 

( ), ( ) : 1 u n y n n N  

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

I C C A A  

,

n n l n  

  A C

Wiener Neural Technique

            

n n I

T T T A C ...

2 1

) ( ) ( 1 l n l n k k k n k

R

    

            I U I T

      

T k k k k k

s r s S U

. 1 , 1 ,

T k k k k l k k k k T k k

s s t r I t r s s t       S

( 1) ( 1) l l k   

 U

1

1,

l k k

s s



 

k

s

parameter vector The parameters in matrices

• n

θ

2 (0), , 1

ˆ min ( ) ( , , )

• n

N

• n

x k

y k y k







θ θ

θ θ

( , ) B D

θ

NN parameters Initial condition

(0) x

2 2

) ( ˆ min Θ

Θ

y y 

) ( ) ( ) 1 ( t t t ΔΘ Θ Θ   

)) ( ( ) ) ( ) ( ( t e t t

T T

Θ J Θ I J J      ˆ ( ( )) : , 1: , 1: ( ).

i ij j

y t i N j length      Θ J θ Θ

            

l l

L J L J L J J        

2 2 1 1

Wiener Neural Technique

ˆ ( ( )) : , 1: , ( ).

i ij j

y t i N j length      Θ L θ Θ

 

 Θ θ θ

Nonlinear SYS ID in SLICOT Software

• Levenberg-Marquardt to optimize the whole parameters

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.

Input/output data

• First part: filtered white Gaussian signal

with cut off frequency 200Hz.

• Second part: odd harmonic multi-sine

with uniformly distributed random phase.

• High SNR.

SilverBox

Input Output

Filtered Gaussian Noise 10 realisation of multi-sine signal

System

• A second order mechanical system

with nonlinear stiffness.

( ) my dy k y y u     

2

( ) k y a by   Estimation data

• ID1 filtered white Gaussian (1:40,000)
• ID2 multi-sine (80,000:126,000)
• ID3 mixed of ID1 and ID2 (1:100,000)

Test data

• TST1 130,000:188,000
• TST2 126,000:188,000

Residu

SilverBox

Parameters Neural network order = 10 State space model order = 5 Parameter s = 20 ID2-TST2 Silver Box System TEST ESTM TST1 TST2

ID1 MEAN

2.4e‐4 2.5e‐5

RMS

3e‐3 4.5e‐3

FIT

66.2% 84.1%

ID2 MEAN

1.2e‐4 1.3e‐4

RMS

2.2e‐3 3.1e‐3

FIT

74% 89.1%

ID3 MEAN

2e‐4 2.1e‐4

RMS

2.4e‐3 3.3e‐3

FIT

71.8% 88.4%

Residu (dB)

Residual error spectrum

Winer-Hammerstein

System

• 3rd order Chebyshev filter with cut
• ff 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

Winer-Hammerstein

Neural network order = 15 State space model order = 6 Parameter s = 15

Test Residu

Residu (dB)

Residual error spectrum

Winer-Hammerstein

Frequency response plot of the linear part

• 1
• 0.5

0.5 1 1.5 2 2.5

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1

Pole-Zero Map Real Axis Imaginary Axis

Pole-Zero map

• f 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

Wiener-Hammerstein with Noise

Input/Output estimation data

• Multi-sine input up to 15kHz (2 periods,

1 amplitude, 4096 points per period)

• Additive process noise (filtered white

Gaussian, cut off 20kHz )

• Additive measurement noise (white

Gaussian noise) Test data

• Noiseless multi-sine and swept sine

( )

x

e n

1( )

G q

( ) f x

( ) u n ( ) x n ( ) w n

2( )

G q

( ) y n ( )

m

u n

( )

m

y n ( )

u

e n ( )

y

e n Estimation data Test data

input

• utput

input

• utput

Winer-Hammerstein with Noise

Neural network order = 15 State space model order = 5 Parameter s = 15

200 400 600 800 1000 1200 1400 1600 1800 2000

• 2
• 1

1 2

Test FIT = -0.89747 %

Measured Output Model Output

2000 4000 6000 8000 10000 12000 14000 16000

Samples

• 2
• 1

1 2

Residu rms= 0.7169, mean = 0.017049

Residu (dB)

Residual error spectrum

Winer-Hammerstein with Noise

Imaginary Axis

Frequency response plot of the linear part Pole-Zero map

• f the linear part

Mag(dB)

P= 0.94 i0.11 Z1= 0.58 i0.84 Z2= 1 i0.36

Iterative Least Square Wiener

( ) ( )

d

q B q A q



( ) f x

( ) u n ( ) x n ( ) m n

( ) ( ) C q A q

( ) n  ( ) y n ( ) ( ) ( ) ( ) ( ) C q y n n m n A q    ( ) ( ( )) m n f x n 

( ) ( ) ( ) ( )

d

q B q x n u n A q





1 1

( ) 1

a a

n n

A q a q a q



  

1 1

( ) 1

c c

n n

C q c q c q



  

1 1

( )

b b

n n

B q b b q b q



  

2 1 2

( )

f f

n n

f x x x x        

Iterative Least Square Wiener

1 1 2

( ) ( ) ( ) ( ) ,

a a

T n n

n a n a n a n       θ 

2 1 2

( ) ( ) ( ) ( ) ,

b b

T n n

n b n b n b n       θ 

4 1 2

( ) ( ) ( ) ( ) ,

c c

T n n

n c n c n c n       θ 

1 1 1 3 1 1

( )

f f f a f a a

T n n n n n n n

n f f f f f f       θ   

1 1 1 2 1 2 1

( ) (1 ( )) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ).

f f

n d n

y n A q y n q B q u n A q x n A q x n C q n   

     

       

1 2 3 4

 

1( )

( 1) ( )

a

T n a

n y n y n n     r 

 

2( )

( 1) ( )

b

T n b

n u n u t n     r 

4( )

( 1) ( )

c c

T n n

n n n n           r 

2 2 3( )

( ) ( ) ( ) ( )

f f a f

T n n n n a a

n x n x n n x n x n n         r   

1 2 3 4

Iterative Least Square Wiener

1 2 3 4

( ) ( ) ( ) ( ) ( )

T T T T T n

n n n n n       r r r r r

unknown unknown

1 2 3 4

( ) ( ) ( ) ( ) ( )

T T T T T n

n n n n n       θ θ θ θ θ

( ) ( )

d

q B q A q



( ) f x ( ) u n ( ) x n

( ) m n

( ) ( ) C q A q ( ) n 

( ) y n

( ) ( ) ( )

T

y n n n  r θ

0(0),

P ˆ (1), r

ˆ (0) θ

unknown

1, n k  

Step 2: set and as the initial value for the next time step, i.e.

Iterative Least Square Wiener

Step 1: keep fix and update

n

If

k k 

1 1 1 1 1 1 1 1 1 1

ˆ ( 1) ( ) ( ) ˆ ˆ 1 ( ) ( 1) ( ) ˆ ˆ ˆ ˆ ( ) ( 1) ( )[ ( ) ( ) ( 1)] ˆ ( ) ( ( ) ( )) ( 1).

k k T k k k T k k k k T k k k

n n n n n n n n n y n n n n n n n

         

            P r k r P r θ θ k r θ P I k r P ˆ ˆ ˆ ( ) ( ) ( ) ( )

k k k

n y n n n    r θ

ˆ ( , ) ˆ ( ) ( ) ˆ ( , )

d k k k

q B q n x n u n A q n





update the parameter vector update the regression vector ˆ ( )

k n

θ

1 k k  

ˆ ( )

k n

r

Step 3: add the new samples to the regression vector from step 2, Update , and go to step 1 1 n n   ˆ ( ),

k n

θ

If

k k 

 

( ), ( ) y n u n

ˆ ( ) n r

k 

ˆ ( )

k n

r ˆ ( ) n r

ˆ ( ), n θ

SilverBox

TEST ESTM TST1 TST2

ID1 MEAN

1.6e‐4 1.3e‐4

RMS

1e‐3 4.e‐3

FIT

86.6% 86.4%

ID2 MEAN

1.4e‐4 1.1e‐4

RMS

1.1e‐3 3.9e‐3

FIT

86.8% 86.4%

ID3 MEAN

1.6e‐4 1.3e‐4

RMS

1.1e‐3 4e‐3

FIT

86.7% 86.4% ID2-TST2 Silver Box System Transfer function order

2,

a

n  2

b

n 

Delay

2

d

n 

Static nonlinearity = polynomial

Test Residu

Winer-Hammerstein

• Delay
• Transfer function order
• Static nonlinearity = exponential

2,

a

n  2

d

n 

200 400 600 800 1000 1200 1400 1600 1800 2000

• 1
• 0.5

0.5

Test FIT = 98.1439 %

Measured Output Model Output

1 2 3 4 5 6 7 8

Samples

104

• 0.01

0.01

Residu rms= 0.0044405, mean = -4.884e-05

residu (dB)

2

b

n 

Residual error spectrum

Winer-Hammerstein

Frequency response plot of the linear part Pole-Zero map

• f the linear part

Imaginary Axis

P= 0.98i0.13 Z=0.85i0.84

Winer-Hammerstein with Noise

Delay Transfer function order Static nonlinearity = exponential

3,

a

n  4

b

n  2,

d

n 

Residual error spectrum

Test Residu

2

c

n 

residu (dB)

Winer-Hammerstein with Noise

Frequency response plot of the linear part Pole-Zero map

• f the linear part

Different map Different amplitude

Imaginary Axis

Further Discussions?