NONPARAMETRIC APPROACH TO IDENTIFICATION

OF BLOCK-ORIENTED SYSTEMS

Grzegorz Mzyk, Zygmunt Hasiewicz

The Institute of Computer Engineering, Control and Robotics

Wrocław University of Technology

Wrocław, Poland

Statement of the problem

k

u

k

y

k

z

k

w

( )

µ

{ }

∞

=

0

i

i

γ

k

v

Fig.1. Hammerstein system

( )

{

|

}

( )

k

k

R u

y

u

u

c u

s

µ

=

=

=

+

E

∑

∞

=

−

+

=

0

)

(

i

k

i

k

i

k

z

u

y

µ

γ

NONPARAMETRIC METHODS

Kernel regression estimation

1

1

ˆ ( )

( )

( )

M

M

i

i

M

i

i

i

u u

u u

u

y K

K

h M

h M

µ

=

=

−

−

=

























∑

∑

Theorem 1. If

( )

0

h M

→

and

( )

Mh M

→ ∞

as

M

→ ∞

then

ˆ ( )

( )

M

u

u

µ

µ

→

in probability as

M

→ ∞

in each continuity point of ()

µ

and the input probability density.

Theorem 2. If

()

µ

is twice differentiable in the point

u

then for

1/ 5

( )

(

)

h M

O M

−

=

it holds that

2/ 5

ˆ ( )

( )

(

)

M

u

u

O M

µ

µ

−

−

=

Orthogonal expansion

( )

( )

( )

u

g u

f u

µ

=

, where

( )

( )

( )

u

g u

f u

µ

=

0

0

( )

( ),

( )

( )

i i

i i

i

i

g u

a

u

f u

b

u

φ

φ

∞

∞

=

=

=

=

∑

∑

( ),

( )

i

k

i

k

i

i

k

a

y

u

b

u

φ

φ

=

=

E

E

1

1

1

1

ˆ

ˆ

( ),

( )

M

M

i

k i

k

i

i

k

k

k

a

y

u

b

u

M

M

φ

φ

=

=

=

=

∑

∑

(

)

0

( )

0

ˆ

( )

ˆ ( )

ˆ ( )

q M

i i

i

M

q M

i i

i

u

a

u

b

u

µ

φ

φ

=

=

=

∑

∑

The convergence conditions:

trigonometric series

2

( )

lim

0

M

q M

M

→∞

=

Legendre series

2

( )

lim

0

M

q M

M

→∞

=

Laguerre series

6

( )

lim

0

M

q M

M

→∞

=

Hermite series

5/ 3

( )

lim

0

M

q

M

M

→∞

=

Daubechies wavelets

2 ( ) 2

2

lim

0

q M

M

M

+

→∞

=

COMBINED PARAMETRIC-

NONPARAMETRIC ALGORITHMS

Estimation of the static nonlinearity

( )

( , )

k

k

u

u c

µ

µ

=

1

2

( )

sin(

)

c u

k

k

u

e

c u

µ

=

+

(

)

2

,

,

1

ˆ

ˆ

arg min

( , )

N

N M

c

k M

k

k

c

w

u c

µ

=

=

−

∑

(

)

1

,

,

ˆ

ˆ

T

T

N M

N

N

N

N M

c

W

−

= Φ Φ

Φ

1

2

( ( ), ( ),...,

( ))

T

N

k

k

N

k

u

u

u

φ

φ

φ

Φ =

(

)

1

2

( )

( ), ( ),...,

( )

T

k

k

k

m

k

u

f u

f u

f u

φ

=

Identification of the linear dynamics

(

)

1

,

,

,

,

ˆ

ˆ

ˆ

ˆ

T

T

N M

N M

N M

N M N

Y

θ

−

= Ψ

Θ

Ψ

0

1

1

2

(

, ,...,

, ,

,...,

)

T

s

p

θ

α α

α β β

β

=

T

k

k

k

y

z

ϑ θ

=

+

1

1

2

(

,

,...,

,

,

,...,

)

T

k

k

k

k s

k

k

k p

w w

w

y

y

y

ϑ

−

−

−

−

−

=

1

2

( ,

,...,

)

T

N

N

Y

y y

y

=

,

1

2

ˆ

( ,

,...,

)

T

N M

N

ϑ ϑ

ϑ

Θ

=

,

,

0,

ˆ

ˆ

ˆ

/

i M

i M

M

γ

χ

χ

=

,

1

1

ˆ

(

)(

)

M i

i M

k i

k

k

y

y u

u

M

χ

−

+

=

=

−

−

∑

1

1

1

1

,

M

M

k

k

k

k

y

y

u

u

M

M

=

=

=

=

∑

∑

Nonparametric instrumental variables

*

1

(

,...,

,

,...,

)

T

k

k

k s

k

k p

w

w

y

y

ψ

−

−

−

=

#

1

(

,...,

,

,...,

)

T

k

k

k s

k

k p

w

w

y

y

ψ

−

−

−

=





,

,

0

ˆ

ˆ

APR

k

i M

k i M

i

y

w

γ

−

=

=

∑



-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

-1,2

-0,8

-0,4

0,0

0,4

0,8

1,2

( )

u

µ

nonparametric

kernel estimate

parametric-nonparametric

estimate

true

Example

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

1,2

-1

1

3

5

7

9

11

impulse response

true

nonparametric

estimate

Conclusions

• Each part is identified separately
• The convergence is strictly proved

Hasiewicz, Z. and G. Mzyk (2004). Combined parametric-nonparametric identification
of Hammerstein systems. IEEE Transactions on Automatic Control, vol. 49, pp. 1370-
1376.
Hasiewicz, Z. and G. Mzyk (2006). Nonparametric instrumental variables for
Hammerstein system identification. IEEE Transactions on Automatic Control
(submitted to)

• The method works under existence of correlated random

noise