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About Non Linear Identification Using Volterra Model
ABOUT NON LINEAR IDENTIFICATION USING VOL TERRA MODEL J.-P. Sagaspe Ecole Nationale Supen"eure d'Arts et Metiers, Esplanade de l'Universite, 33405 Talence-Cedex, France
Abstract. A physical nonlinear system, with a single input and a single output can be rough estimated by the Volterra development. The favoured tool for describing its kernels seems to be the multidimensional transforms. The multivariable Laguerre transform allows to associate to each kernel a restricted set of numbers. It allows to identify the kernels of an unknown system from spectra of input system and from channel representation output. Keywords. Nonlinear systems. FUNCTIONAL REPRESENTATION OF A NONLINEAR SYSTEM V {h ,Au(t)}=A~ {h ,u(t)} n n n n
The behaviour of any continuous system, with a single input u (t) and a single output y (t) can be described by the following development
MULTI VARIABLE LAGUERRE TRANSFORM
N
y( t)
The monovariable Laguerre transform has been introduced by Steiglitz and Bozzo, and it allows to associate to every function f(t) E LZ, a set at numbers {f }, computed
Vi (t)
L
i=1 with ttt Vi (t)
ffi
~
n
i hi(TI'TZ,···'Ti)k~_u(t-Tk)dTk
from a decomposition on a Laguerre functions basis. This transform allows to build the isomorphism beetween the space of continuous signals and the space of discrete signals. To describe multivariable functions, we introduce multi variable Laguerre transforms. We shall give the description in the case of two variables (without restricting its obvious generalization to n variables).
i
Vi(t) is a causal i-th order functional,which can be regarded as a direct generalization of the convolution integral for linear systems ; hi (TI, ••. ,T ) is a function of i i variables and is called i-th order kernel. Kernels can be arranged symmetrically in connection with Ti variables, their share in
The Laguerre functions A.(t) build an orthonormal system of continu6us functions on the time interval {O,m {:
the functional remaining unaltered.
i k .( )k A. (t)=(-I)i/2 exp(-t) L (-I) C1~ 1 k=o k kl
For instance, for the second kernel, we can replace hZ(T , T ) by the s~etrical kernel: I Z
so, the Ai(t ) Aj(t ) functions build a mulZ l tivariable orthonormal system on {O,oo{x {O,w{. Volterra's functionals are not orthogonal they change if one modifies the order N of the expansion. Wiener defined orthogonal functionals ; but as functionals are related to the input signal u(t), they will be made orthogonal only for a type of input signal. Wiener chose the case where u(t) is a gaussian white noise with a given spectral density.
A function f (tl,t ) admits a un1que expansion Z
Volterra's functionals are homogeneous; if the input signal is multiplied by any scalar value A, the n-th order functional is multiplied by An :
We shall call multivariable Laguerre transform of f(tl,t ) the following serie Z
f( t I' t
Z) =
L
i=o i=o the f." numbers are given by the following 1J
relation : f
SDI
ij
= J~ f~f(tl,tz) Ai(t l ) Aj(tz)dt l dt Z
J-P. Sagaspe
502
If we impose the assumption that
f ..
1J
with ~I and '~2 belonging to complex numbers set C. We shall call "Laguerre spectrum of f(t ,t )" the set of {f } numbers. l 2 ij We can show that :
~
=1=5
f
=
~+I
for k
.
r,1
And this entails :
yS(£I'~2) /:2~' i~ j~O
where the sum is made along the unit circle of the complex map. We can show that ~ variable and Laplace variable are tied by the following relations I+s and s ~-I
~=O
we can write
h ij
a~O
fa,i
£~a
f . -S S~oS.J£2
E E a=O S=O yaS Then
CALCULATION OF THE OUTPUT OF A NON LINEAR ELEMENT SUBJECT TO A GIVEN INPUT Let a non linear system, reduced to a kernel of order two, h(T ,T ) with Laguerre spectrum 1 2 {h .. }, subject to an input signal u(t) ,with La~~erre spectrum {u.}, the output being y 1
with Laguerre spectrum {y.}. We have:
YNO = E ~"
E
i=O j=O
E E i=O j=O We compute now the spectrum {Yk} of yS (£) S
from the set of coefficients YaS of Y (£1'£2)
1
y(t) =
J~J~(TI"2)
y(tl=t, t =t) = y(t) 2
u(t-'I) u(t-'2)d'l dT2
We build the following multi variable function
the Laguerre spectrum of which is {y .. }. Its multi variable Laguerre transform is ~iven by s s s s I+~ 1 1+~2 Y (~I'~2) = H (~I'~2)U (~I)U (~2)-=-- ' ~
fie: 1 ,t2~2
S H et US being the respective transforms of h('I"2) and u(t). "" "" s -i -j -k Y (~1'~2)= L L hij£1 ~2' E ~~I k=O i=O j=0
We have
Yk=
y = k
E
E
a=o S=oYaS
r 0
Aa ( t) AS (t) Ak (t) dt
S(a,S,k)
L L YaS a=o S=o'
with: S(i,j,k)
(_I)i+j+k 212
L
a=o S=o y=o (a+S+y)1 3a+S+y+ 1 So the spectrum {y }of the output y(t) is very easy to compu~e from the multi variable spectrum of the generalized output. This calculus needs no calculus in complex map as usual Laplace variable association calculus :
We write k
1
E
r=O with
Y(s) = 2ITj
r < i
f
r = i
f
r > i
f
r,i r,i r,i
u
j ""
j ""
Y(s-w,w) dw
APPLICATION TO THE IDENTIFICATION OF A SECOND ORDER KERNEL
0
u
L
The founded relationships give a linear relations set between the hij spectrum coefficient~
o r-i
+ u
r-i-I
y = E E S(a,S,k) L E hij k a=oS=o 1=0 j=o
About non linear identification using Volterra model
and with a finite number of terms, and with symmetrised kernels : I:
P I:
1/1=0
fj=o
P
y
k
•
M
i
S(a,fj,k) I: i~o
I:
j-o
hij fa,i ffj,j
12
/2
This set is completely defined if we have at least (M+I)(M+2)/2 equations i and the identification problem is solved. CONCLUSIONS The multivariable Laguerre transform seems to be more practical for an easy identifica tion of nonlinear systems than Laplace and z multivariable transforms.
503
REFERENCES Steiglitz, K. (1965). Rational transform approximation via the Laguerre spectrum. J. Franklin Inst. (U.S.A.), 280, 5, 387,394. Bozzo, c. (1970). Etude d'un isomorphisme entre l'espace des signaux continus et l'espace des signaux discrets. Differentes applications. These de Doctorat es-Sciences physiques, Marseille. Sagaspe, J-P. (1976). Contributions a l'identification non lineaire par la representation fonctionnelle et la transformee de Laguerre. These de Doctorat es-Sciences, Bordeaux.
Abstract. A physical nonlinear system, with a single input and a single output can be rough estimated by the Volterra development. The favoured tool for describing its kernels seems to be the multidimensional transforms. The multivariable Laguerre transform allows to associate to each kernel a restricted set of numbers. It allows to identify the kernels of an unknown system from spectra of input system and from channel representation output. Keywords. Nonlinear systems. FUNCTIONAL REPRESENTATION OF A NONLINEAR SYSTEM V {h ,Au(t)}=A~ {h ,u(t)} n n n n
The behaviour of any continuous system, with a single input u (t) and a single output y (t) can be described by the following development
MULTI VARIABLE LAGUERRE TRANSFORM
N
y( t)
The monovariable Laguerre transform has been introduced by Steiglitz and Bozzo, and it allows to associate to every function f(t) E LZ, a set at numbers {f }, computed
Vi (t)
L
i=1 with ttt Vi (t)
ffi
~
n
i hi(TI'TZ,···'Ti)k~_u(t-Tk)dTk
from a decomposition on a Laguerre functions basis. This transform allows to build the isomorphism beetween the space of continuous signals and the space of discrete signals. To describe multivariable functions, we introduce multi variable Laguerre transforms. We shall give the description in the case of two variables (without restricting its obvious generalization to n variables).
i
Vi(t) is a causal i-th order functional,which can be regarded as a direct generalization of the convolution integral for linear systems ; hi (TI, ••. ,T ) is a function of i i variables and is called i-th order kernel. Kernels can be arranged symmetrically in connection with Ti variables, their share in
The Laguerre functions A.(t) build an orthonormal system of continu6us functions on the time interval {O,m {:
the functional remaining unaltered.
i k .( )k A. (t)=(-I)i/2 exp(-t) L (-I) C1~ 1 k=o k kl
For instance, for the second kernel, we can replace hZ(T , T ) by the s~etrical kernel: I Z
so, the Ai(t ) Aj(t ) functions build a mulZ l tivariable orthonormal system on {O,oo{x {O,w{. Volterra's functionals are not orthogonal they change if one modifies the order N of the expansion. Wiener defined orthogonal functionals ; but as functionals are related to the input signal u(t), they will be made orthogonal only for a type of input signal. Wiener chose the case where u(t) is a gaussian white noise with a given spectral density.
A function f (tl,t ) admits a un1que expansion Z
Volterra's functionals are homogeneous; if the input signal is multiplied by any scalar value A, the n-th order functional is multiplied by An :
We shall call multivariable Laguerre transform of f(tl,t ) the following serie Z
f( t I' t
Z) =
L
i=o i=o the f." numbers are given by the following 1J
relation : f
SDI
ij
= J~ f~f(tl,tz) Ai(t l ) Aj(tz)dt l dt Z
J-P. Sagaspe
502
If we impose the assumption that
f ..
1J
with ~I and '~2 belonging to complex numbers set C. We shall call "Laguerre spectrum of f(t ,t )" the set of {f } numbers. l 2 ij We can show that :
~
=1=5
f
=
~+I
for k
.
r,1
And this entails :
yS(£I'~2) /:2~' i~ j~O
where the sum is made along the unit circle of the complex map. We can show that ~ variable and Laplace variable are tied by the following relations I+s and s ~-I
~=O
we can write
h ij
a~O
fa,i
£~a
f . -S S~oS.J£2
E E a=O S=O yaS Then
CALCULATION OF THE OUTPUT OF A NON LINEAR ELEMENT SUBJECT TO A GIVEN INPUT Let a non linear system, reduced to a kernel of order two, h(T ,T ) with Laguerre spectrum 1 2 {h .. }, subject to an input signal u(t) ,with La~~erre spectrum {u.}, the output being y 1
with Laguerre spectrum {y.}. We have:
YNO = E ~"
E
i=O j=O
E E i=O j=O We compute now the spectrum {Yk} of yS (£) S
from the set of coefficients YaS of Y (£1'£2)
1
y(t) =
J~J~(TI"2)
y(tl=t, t =t) = y(t) 2
u(t-'I) u(t-'2)d'l dT2
We build the following multi variable function
the Laguerre spectrum of which is {y .. }. Its multi variable Laguerre transform is ~iven by s s s s I+~ 1 1+~2 Y (~I'~2) = H (~I'~2)U (~I)U (~2)-=-- ' ~
fie: 1 ,t2~2
S H et US being the respective transforms of h('I"2) and u(t). "" "" s -i -j -k Y (~1'~2)= L L hij£1 ~2' E ~~I k=O i=O j=0
We have
Yk=
y = k
E
E
a=o S=oYaS
r 0
Aa ( t) AS (t) Ak (t) dt
S(a,S,k)
L L YaS a=o S=o'
with: S(i,j,k)
(_I)i+j+k 212
L
a=o S=o y=o (a+S+y)1 3a+S+y+ 1 So the spectrum {y }of the output y(t) is very easy to compu~e from the multi variable spectrum of the generalized output. This calculus needs no calculus in complex map as usual Laplace variable association calculus :
We write k
1
E
r=O with
Y(s) = 2ITj
r < i
f
r = i
f
r > i
f
r,i r,i r,i
u
j ""
j ""
Y(s-w,w) dw
APPLICATION TO THE IDENTIFICATION OF A SECOND ORDER KERNEL
0
u
L
The founded relationships give a linear relations set between the hij spectrum coefficient~
o r-i
+ u
r-i-I
y = E E S(a,S,k) L E hij k a=oS=o 1=0 j=o
About non linear identification using Volterra model
and with a finite number of terms, and with symmetrised kernels : I:
P I:
1/1=0
fj=o
P
y
k
•
M
i
S(a,fj,k) I: i~o
I:
j-o
hij fa,i ffj,j
12
/2
This set is completely defined if we have at least (M+I)(M+2)/2 equations i and the identification problem is solved. CONCLUSIONS The multivariable Laguerre transform seems to be more practical for an easy identifica tion of nonlinear systems than Laplace and z multivariable transforms.
503
REFERENCES Steiglitz, K. (1965). Rational transform approximation via the Laguerre spectrum. J. Franklin Inst. (U.S.A.), 280, 5, 387,394. Bozzo, c. (1970). Etude d'un isomorphisme entre l'espace des signaux continus et l'espace des signaux discrets. Differentes applications. These de Doctorat es-Sciences physiques, Marseille. Sagaspe, J-P. (1976). Contributions a l'identification non lineaire par la representation fonctionnelle et la transformee de Laguerre. These de Doctorat es-Sciences, Bordeaux.