Multi-Output Dependent Observability Normal Form

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Multi-Output Dependent Observability Normal Form

Gang Zheng, Driss Boutat, Jean-Pierre Barbot

To cite this version:

Gang Zheng, Driss Boutat, Jean-Pierre Barbot. Multi-Output Dependent Observability Normal Form. Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2009, 70 (1), pp.404-418.

�10.1016/j.na.2007.12.012�. �inria-00192960v2�

Multi-Output Dependent Observability Normal Form

G. ZHENG

¹

, D. BOUTAT

²

and J.P. BARBOT

³

1

INRIA Rhone-Alpes, Inovall´ee, 655 avenue de l’Europe, Montbonnot Saint Martin, 38334 St. Ismier Cedex, France.

2

LVR/ENSI, 10 Boulevard de Lahitolle, 18020 Bourges, France.

3

ECS/ENSEA, 6 Avenue du Ponceau, 95014 Cergy-Pontoise, Project ALIEN, INRIA-Futures, France.

Abstract

This article gives the sufficient condition which guarantees the existence of a refer- ence frame in which a multi-output nonlinear system is linearizable with a linear part depending on its outputs. Our method is based on the design of a reference frame associated with nonlinear observable systems. Moreover, we give the general- ization of the result obtained in [5] and [19]. And some examples are given in order to highlight our thinking.

Key words: normal form, observer design, linearization, left invertibility

1 Introduction

Given a nonlinear system, it is clear that no general approach can be used to design an observer. However, at least there are two ways to try. The first way is to design an observer directly to the given nonlinear system, including Kalman-like observer [8], adaptive observer [3], sliding mode observer [13] and so on. Nevertheless, some extra conditions should be imposed for these direct approaches, such as Lipschitz, persistent excitation. The second method is based on the conception of normal form. A normal form represents a class of equivalent systems which possess the same properties (we focus on the observability in this paper). In other words, the appearance differences among this class of equivalent systems are not intrinsic, but because of the wrong choice of the coordinates.

Obviously one of the keys of this category is to find out a coordinate transfor-

mation with which a given system could be converted into the target observ-

able normal form. And then the existed nonlinear observer design techniques can be easily applied. In this category, [3,1,2] deal with the problem of trans- form a nonlinear system into a class of state affine ‘normal forms’. In fact, in their works, the sense of normal form is more general since no structure constraints were imposed for its linear part. However, it is also because of this merit, some existed techniques for linear systems cannot be easily applied.

And this leads to the raise of linearization problem of nonlinear system.

The linearization problem with output injection was firstly treated in [10]

for single output system. And it was generated in multi-output systems in [11,17]. Some other results about input-output injection for the linearization problem were stated in [6,14]. Moreover, [15,16] gave the sufficient and neces- sary geometrical conditions to transform a nonlinear system into the so-called output-dependent time scaling linear canonical form. In [7], the author gave independently the dual geometrical conditions of [15]. Furthermore, the geo- metrical conditions to guarantee the existence of a local diffeomorphism and an output injection to transform a nonlinear system in a ‘canonical’ normal form depending on its output was presented [18][19], called Single Output De- pendent Observability normal form (SODO). Moreover, an extension for this normal form with quadratic terms was also studied in [20]. As another natural extension for our previous study, we will extend our result for multi-outputs nonlinear system.

Motivated by this interest, this paper focuses on the analysis of nonlinear systems with multiple outputs as follows:







˙

x = f(x, u)

y = (h

₁

(x), ..., h

_m

(x))

^T

(1)

where x ∈ IR

ⁿ

, u ∈ IR

^p

, f : IR

ⁿ

× IR

^p

→ IR

ⁿ

and h :IR

ⁿ

→ IR

^m

are analytic, and we deal with the following problem: Find sufficient condition for the existence of a local diffeomorphism φ(x) = z to transform system (1) into







˙

z = A (y, u) z + β (y, u)

y = Cz (2)

where

z =













z

1

z

₂

...

z

_m













, β (y) =













β

1

(y

1

, u) β

₂

(y

₁

, y

₂

, u)

...

β

_m

(y

₁

, ..., y

_m

, u)













, C =













C

1

C

₂

...

C

_m













and

A (y, u) =













A

₁

(y

₁

, u) 0 · · · 0 0 A

₂

(y

₁

, y

₂

, u) · · · 0

... ... . .. ...

0 0 · · · A

_m

(y

₁

, ..., y

_m

, u)













with

z

_i

=













z

_i,1

z

i,2

...

z

_i,ki













, β

_i

(y

₁

, ..., y

_i

, u) =













β

_i,1

(y

₁

, ..., y

_i

, u) β

i,2

(y

1

, ..., y

i

, u)

...

β

_i,ki

(y

₁

, ..., y

_i

, u)













, C

_i

=

·

0 · · · 0 1

¸

1×ki

and

A

_i

(y

₁

, ..., y

_i

, u) =













0 · · · 0 0

α

_i,1

(y

₁

, ..., y

_i

, u) · · · 0 0

... . .. ... ...

0 · · · α

_i,ki−1

(y

₁

, ..., y

_i

, u) 0













with α

_i,j

6= 0 for all y and u in a certain studied neighborhood. For dynamical systems in the form 2, the high gain observer proposed in [4] can be applied directly (refer to [4] for more details).

In this paper, we will give the geometrical condition which is sufficient to

guarantee the existence of a local diffeomorphism and an output injection

to transform system (1) into the normal form (2). This kind of linearization

will be named Multi-Output Dependent Observability normal form (MODO

normal form) and generalizes the result obtained in [5,19] for nonlinear systems with single output. In section 2, we give some basic notations and a preliminary result is presented in Section 3 in order to introduce our method. In section 4, we propose our main result as a generalization of the result given in section 3.

2 Notations

Consider system (1), with a possible reordering of h

i

, we assume that there exist k

₁

≥ k

₂

≥ ... ≥ k

_m

≥ 1 and

^Pm

i=1

k

_i

= n such that

θ =

^³

θ

₁¹

, ..., θ

^k₁¹

, .., θ

_m¹

, .., θ

^k_m^m´T

(3)

where

θ

_i^j

= dL

^j−1_f

h

i

is a frame of the cotangent bundle T

^∗

U. Thus, system (1) is observable. Integers (k

_i

)

_1≤i≤m

are called observability indices of system (1). For a nice description and more details about this assumption, see [11]. Obviously the list of these integers is generally not unique. For 1 ≤ l ≤ m and 1 ≤ j ≤ k

_l

, let (τ

_i,1

)

_1≤i≤m

be the family of vector fields defined by:

θ

_l^j

(τ

i,1

) =







1, l = i, j = k

_i

0, otherwise (4)

and construct by induction the following family of vector fields:

τ

_i,r

= [τ

_i,r−1

, f ] = (−1)

^r−1

ad

^r−1_f

τ

_i,r−1

, for 2 ≤ r ≤ k

_i

. (5)

The family τ = (τ

_i,j

)

_1≤i≤m

and

1≤j≤ki

is a basis of the tangent bundle T U. The frame T was addressed firstly in [10] for p = 0 and m = 1 and it is well-known in [10] that system (1) can be transformed into the normal form (2) if and only if we have

[τ

1,i

, τ

1,j

] = 0, for 1 ≤ i, j ≤ n. (6)

In this case, τ is a frame with which system (1) is in the form (2).

3 Preliminary results

In this section, we will generalize the result stated in [5] and [19] to systems with multi-output.

Lemma 1 For a system in the form (2) we have for 1 ≤ l ≤ m, 1 ≤ t ≤ k

l

,

τ

_l,t

=

_π¹

l,t

∂

∂zl,t

+





 Pl r=1

³

A

^l,t_r,t−1

(y

₁

, ..., y

_l

, u) z

_r,kr−1^´

+η

_t−1^l,t

(y

₁

, ..., y

_l

, u)





∂zl,t−1^∂

+

^t−2P

i=1





 Pl r=1







A

^l,t_r,i

(y

₁

, ..., y

_l

, u) z

_r,kr−t+i

+

^krP−1

j=kr−t+i+1 krP−1

s=j

T

_r,j,s^l,t

(y

₁

, ..., y

_l

, u) z

_r,j

z

_r,s











∂z^∂l,i

+

^t−2P

i=1









 Pl r=1

à Pkr

j=kr−t+i+1

η

_r,i^l,t

(y

1

, ..., y

l

, u) z

r,j

!

+O

^[3]_{(y1,...,yl,u)}





z

_1,k1−t+i+1

, .., z

_1,k1−1

, ...., z

l,kl−t+i+1

, .., z

l,kl−1















∂

∂zl,i

,

(7)

where π

l,kl

= 1 and π

l,t−1

= π

l,t

α

l,t−1

for 2 ≤ t ≤ k

l

, η

_i^l,t

and T

_r,j,q^l,t

are some smooth functions of (y

₁

, ..., y

_l

, u), O

_(y^[3]_1,...,yl,u)





z

_1,k1−t+i+1

, .., z

_1,k1−1

, ...., z

l,kl−t+i+1

, .., z

l,kl−1





repre- sents the residue higher than order 2 with coefficient which is function of (y

₁

, ..., y

_l

, u) and

A

^l,t_r,i

(y

₁

, ..., y

_l

, u) = (−1)

^t−i+1

S

_r,t−i,1^{l,t ∂yr}_π2^πl,i

l,i

π

_r,kr−t+i

+ (−1)

^{t−i+1 t−1P}

m=t−i+1

S

r,t−i,m−t+i+1^l,t

∂_yrπl,t−m

π²_l,t−m

à Qm

j=t−i+1

α

_l,k−j

!

π

_r,kr−t+i

, (8)

where ∂

_yr

π

_l,i

represents

^∂πl,i(y_∂y^1,...,y_r ^l,u)

, S

_r,t−i,1^l,t

and S

r,t−i,m−t+i+1^l,t

are defined as follows

S

_r,j,1^l,t

= 1, S

_r,j,s^l,t

= S

_r,j−1,s^l,t−1

+ S

_r,j,s−1^l,t−1

, (9)

for 1 ≤ l ≤ m, 1 ≤ r ≤ l, 2 ≤ t ≤ k

_l

, 1 ≤ j ≤ t − 1 and 1 ≤ s ≤ t − j.

Proof 1 For a system in the (2) form, for 1 ≤ l ≤ m, τ

_l,1

=

_π¹

l,1

∂

∂zl,1

, then we

use equation (5) to obtain τ

_1,t

=

_π¹

1,t

∂

∂z1,t

+

^³

A

^1,t_1,t−1

(y

₁

, u) z

_1,k1−1

+ η

_t−1^1,t

(y

₁

, u)

^´_∂z^∂

1,t−1

+

^t−2P

i=1

Ã

A

^1,t_1,i

(y

₁

, u) z

_1,k1−t+i

+

^k1P−1

j=k1−t+i+1 k1P−1

l=j

T

_j,l^1,t

(y

₁

, u) z

_1,j

z

_1,l

!

∂

∂z1,i

+

^t−2P

i=1

à Pk1

j=k1−t+i+1

η

^1,t_i

(y

₁

, u) z

_1,j

+ O

_(y^[3]_1,u)

(z

_1,k1−t+i+1

, .., z

_1,k1−1

)

!

∂

∂z1,i

,

for 1 ≤ t ≤ k

1

and τ

_2,t

=

_π¹

2,t

∂

∂z2,t

+





A

^2,t_1,t−1

(y

₁

, y

₂

, u) z

_1,k1−1

+ A

^2,t_2,t−1

(y

₁

, y

₂

, u) z

_2,k2−1

+η

_t−1^2,t

(y

1

, y

2

, u)



_∂z2,t−1^∂

+

^t−2P

i=1















A

^2,t_1,i

(y

₁

, y

₂

, u) z

_1,k1−t+i

+

^k2P−1

j=k2−t+i+1 k2P−1

l=j

T

_1,j,l^2,t

(y

₁

, y

₂

, u) z

_1,j

z

_1,l

+ A

^2,t_2,i

(y

₁

, y

₂

, u) z

_2,k2−t+i

+

^k2P−1

j=k2−t+i+1 k2P−1

l=j

T

_2,j,l^2,t

(y

₁

, y

₂

, u) z

_2,j

z

_2,l















∂

∂z2,i

+

^t−2P

i=1







k1

P

j=k1−t+i+1

η

_1,i^2,t

(y

₁

, y

₂

, u) z

_1,j

+

^Pk2

j=k2−t+i+1

η

_2,i^2,t

(y

₁

, y

₂

, u) z

_2,j

+O

_(y^[3]_1,y2,u)

(z

_1,k1−t+i+1

, .., z

_1,k1−1

, z

_2,k2−t+i+1

, .., z

_2,k2−1

)





∂z^∂2,i

,

for 1 ≤ t ≤ k

₂

. Then by an induction, for 1 ≤ t ≤ k

_l

, we get

τ

_l,t

=

_π¹

l,t

∂

∂zl,t

+





 Pl r=1

³

A

^l,t_r,t−1

(y

₁

, ..., y

_l

, u) z

_r,kr−1^´

+η

_t−1^l,t

(y

₁

, ..., y

_l

, u)





∂zl,t−1^∂

+

^t−2P

i=1





 Pl r=1







A

^l,t_r,i

(y

₁

, ..., y

_l

, u) z

_r,kr−t+i

+

^krP−1

j=kr−t+i+1 krP−1

s=j

T

_r,j,s^l,t

(y

₁

, ..., y

_l

, u) z

_r,j

z

_r,s











∂z^∂l,i

+

^t−2P

i=1









 Pl r=1

à Pkr

j=kr−t+i+1

η

_r,i^l,t

(y

₁

, ..., y

_l

, u) z

_r,j

!

+O

^[3]_{(y1,...,yl,u)}





z

_1,k1−t+i+1

, .., z

_1,k1−1

, ...., z

l,kl−t+i+1

, .., z

l,kl−1















∂

∂zl,i

,

where

A

^l,t_r,i

(y

₁

, ..., y

_l

, u) = (−1)

^t−i+1

Ã

S

_r,t−i,1^l,t

∂

_yr

π

_l,i

π

²_l,i

+

t−1X

m=t−i+1

S

r,t−i,m−t+i+1^l,t

∂

_yr

π

_l,t−m

π

²_l,t−m



 Y^m

j=t−i+1

α

_l,k−j









π

_r,kr−t+i

,

with the coefficients S

_r,j,s^l,t

defined by 9.

In order to determine the α

_i,j

(y

₁

, ..., y

_i

, u) for 1 ≤ i ≤ m, 1 ≤ j ≤ k

_i

− 1, we also impose that for 1 ≤ i, l ≤ m,

∂

∂z

_i,j

h

_l

◦ φ =







1, if l = i and j = k

_i

, 0, otherwise.

Now, we are ready to state a set of partial differential equations which enables us to compute functions α

_i,j

(y

₁

, ..., y

_i

, u) for 1 ≤ i ≤ m, 1 ≤ j ≤ k

_i

− 1.

Proposition 1 If there exists a diffeomorphism which transforms system (1) into form (2) then for 1 ≤ l ≤ m, 1 ≤ t ≤ k

_l

− 1, 1 ≤ s ≤ l − 1,







[τ

_l,t

, τ

_s,ks

] = λ

^s_l,t

τ

_l,t

mod span {τ

_l,1

, ..., τ

_l,t−1

} [τ

l,t

, τ

l,kl

] = λ

^l_l,t

τ

l,t

+ G

^[1]_l,kl,t

+ R

l,t

where

G

^[1]_l,kl,t

=

"_t−1 X

i=1

Ã

1 π

l,t

T

_l,t,k^l,t _l−t+i

z

_l,kl−t+i

!

∂

∂z

l,i

#

+ 1 π

l,t

T

_l,t,t^l,t

z

_l,t

∂

∂z

l,2t−kl

,

and

R

_l,t

=

t−1X

i=1











kl

P

j=kl−t+i+1

η ¯

_i^l,t

(y

₁

, ..., y

_l

, u) +O

^[2]_{(y1,...,yl,u)}





z

_1,k1−t+i+1

, .., z

_1,k1−1

, ...., z

l,kl−t+i+1

, .., z

l,kl−1















∂

∂z

_l,i

and







λ

^s_l,t

= −

^∂ys_πl,t^πl,t

,

λ

^l_l,t

= diag{δ

₁^l,t

(y

₁

, ...y

_l

, u) , ..., δ

^l,t_i

(y

₁

, ...y

_l

, u) , ..., δ

_t^l,t

(y

₁

, ...y

_l

, u) , 0, ..., 0}, (10)

defining ∂

_ys

π

_l,t

:=

^∂

[

^{πl,t(y1,...yl,u)}

]

∂ys

, δ

^l,t_t

= A

^l,k_l,t^l

+

^∂yl_π^πl,t

l,t

and

δ

^l,t_i

= A

^l,k_l,i^l

− A

^l,k_l,k^l_l−t+i

− ∂

yl

³

A

^l,t_l,i^´

A

^l,t_l,i

for 1 ≤ i ≤ t − 1, and A

^l,t_l,i

is given in (8).

Proof 2 According to equation (7), for 1 ≤ l ≤ m, 1 ≤ t ≤ k

_l

− 1, we have

τ

_l,t

=

_π¹

l,t

∂

∂zl,t

+





 Pl r=1

³

A

^l,t_r,t−1

(y

₁

, ..., y

_l

, u) z

_r,kr−1^´

+η

_t−1^l,t

(y

₁

, ..., y

_l

, u)





∂zl,t−1^∂

+

^t−2P

i=1





 Pl r=1







A

^l,t_r,i

(y

₁

, ..., y

_l

, u) z

_r,kr−t+i

+

^krP−1

j=kr−t+i+1 krP−1

s=j

T

_r,j,s^l,t

(y

1

, ..., y

l

, u) z

r,j

z

r,s











∂z^∂l,i

+

^t−2P

i=1









 Pl r=1

à Pkr

j=kr−t+i+1

η

_r,i^l,t

(y

₁

, ..., y

_l

, u) z

_r,j

!

+O

^[3]_{(y1,...,yl,u)}





z

_1,k1−t+i+1

, .., z

_1,k1−1

, ...., z

_l,kl−t+i+1

, .., z

_l,kl−1















∂

∂zl,i

,

then we can calculate [τ

l,t

, τ

s,ks

] = − ∂

ys

π

l,t

π

_l,t

∂

∂z

_l,t

mod span {τ

l,1

, ..., τ

l,t−1

},

Defining λ

^s_l,t

= −

^∂ys_πl,t^πl,t

, we obtain

[τ

l,t

, τ

s,ks

] = λ

^s_l,t

τ

l,t

mod span {τ

l,1

, ..., τ

l,t−1

} Moreover, as

[τ

l,t

, τ

l,kl

] =

Ã

A

^l,k_l,t^l

+ ∂

_yl

π

_l,t

π

_l,t

!

1 π

_l,t

∂

∂z

_l,t

+

Xt−1

i=1





 µ

A

^l,k_l,i^l

− A

^l,k_l,k^l_l−t+i

−

^∂yl

(

^Al,tl,i

)

A^l,t_l,i

¶

A

^l,t_l,i

z

_l,kl−t+i

+

_π¹

l,t

T

_l,t,k^l,t _l−t+i

z

_l,kl−t+i







∂

∂z

_l,i

+ 1

π

_l,t

T

_l,t,t^l,t

z

_l,t

∂

∂z

_l,2t−kl

+

Xt−1

i=1











kl

P

j=kl−t+i+1

η ¯

_i^l,t

(y

₁

, ..., y

_l

, u) +O

^[2]_{(y1,...,yl,u)}





z

1,k1−t+i+1

, .., z

1,k1−1

, ...., z

_l,kl−t+i+1

, .., z

_l,kl−1















∂

∂z

l,i

.

Set

λ

^l_l,t

(y

₁

, ..., y

_l

, u) = diag{δ

^l,t₁

(y

₁

, ..., y

_l

, u) , ..., δ

_i^l,t

(y

₁

, ..., y

_l

, u) , ..., δ

_t^l,t

(y

₁

, ..., y

_l

, u) , ..., 0, ..., 0},

where δ

_t^l,t

= A

^l,k_l,t^l

+

^∂yl_π^πl,t

l,t

and

δ

^l,t_i

= A

^l,k_l,i^l

− A

^l,k_l,k^l_l−t+i

− ∂

_yl^³

A

^l,t_l,i^´

A

^l,t_l,i

for 1 ≤ i ≤ t − 1, then

[τ

_l,t

, τ

_l,kl

] = λ

^l_l,t

τ

_l,t

+ G

^[1]_l,kl,t

+ R

_l,t

. (11) Remark 1 In equation (11), λ

^l_l,t

could be uniquely determined since G

^[1]_l,kl,t

might be separated according to the coefficients of second-order terms in τ

l,kl

. Finally, the following result enables us to determine all the functions α

_l,i

(y

₁

, ..., y

_l

, u) for 1 ≤ l ≤ m and 1 ≤ i ≤ k

_l

− 1.

Proposition 2 If there exists a diffeomorphism which transforms system (1) into form (2), then α

_l,i

=

_π^πl,i

l,i+1

for 1 ≤ l ≤ m and 1 ≤ i ≤ k

_l

− 2, and

α

_l,kl−1

= π

_l,kl−1

, where the π

_l,i

for 1 ≤ i ≤ k

_l

− 1 is the solution of the following partial differential equations:















∂_ysπl,i

πl,i

= −λ

^s_l,i

(y

₁

, ..., y

_l

) , f or 1 ≤ s ≤ l − 1 and 1 ≤ i ≤ k

_l

− 1

∂_ylπl,i

πl,i

= exp

^{R ³}

δ

_i^l,i

− δ

_i^l,kl−1

− δ

_i+1^l,i+1´

dy

_l

− B ¯

_l,i^l,kl−1

, f or 1 ≤ i ≤ k

_l

− 2

∂_ylπ_l,kl−1

π_l,kl−1

=

^δl,kl

−1

kl−1 −A¯^l,kl_l,kl−1 2

(12)

with B ¯

_l,1^l,0

= 0 and for 1 ≤ i, t ≤ k

_l

− 1 B ¯

_l,i^l,t

=

t−1X

m=t−i+1

S

t−i,m−t+i+1^l,t

∂

yl

π

l,t−m

π

_l,t−m

. (13)

Proof 3 Obviously, according to (10), we have

∂

ys

π

l,i

π

_l,i

= −λ

^s_l,i

(y

₁

, ..., y

_l

, u) , f or 1 ≤ s ≤ l − 1 and 1 ≤ i ≤ k

_l

− 1

Define

B

_l,i^l,t

= ∂

_yl

π

_l,i

π

l,i

+ ¯ B

_l,i^l,t

. (14)

where 1 ≤ l ≤ m and 1 ≤ i, t ≤ k

_l

− 1.

According to (8), for 1 ≤ i, k ≤ k

_l

− 1,

∂

_yl^³

A

^l,t_l,i^´

A

^l,t_l,i

= ∂

_yl^³

B

_l,i^l,t´

B

^l,t_l,i

− ∂

_yl

π

_l,i

π

_l,i

+ ∂

_yl

π

_l,kl−t+i

π

_l,kl−t+i

.

As δ

_t^l,t

= A

^l,k_l,t^l

+

^∂yl_π^πl,t

l,t

, hence

δ

^l,k_i ^l−1

= A

^l,k_l,i^l

− A

^l,k_l,1+i^l

− ∂

_yl^³

B

_l,i^l,kl−1´

B

_l,i^l,kl−1

+ ∂

_yl

π

_l,i

π

_l,i

− ∂

_yl

π

_l,1+i

π

_l,1+i

= δ

_i^l,i

− δ

_1+i^l,1+i

− ∂

_yl

Ã

∂

_yl

π

_l,i

π

_l,i

+ ¯ B

_l,i^l,kl−1

!

/

Ã

∂

_yl

π

_l,i

π

_l,i

+ ¯ B

_l,i^l,kl−1

!

. which yields

∂

_yl

π

_l,i

π

_l,i

= exp

^{Z ³}

δ

^l,i_i

− δ

^l,k_i ^l−1

− δ

_i+1^l,i+1´

dy

_l

− B ¯

_l,i^l,kl−1

, f or 1 ≤ i ≤ k

_l

− 2

where B ¯

_l,i^l,kl−1

is defined in (13) and c

_l,i

6= 0. As δ

_k^l,k_l−1^l−1

= 2

^∂yl_π^πl,kl−1

l,kl−1

+ ¯ A

^l,k_l,k^l_l−1

, where A ¯

^l,k_l,k^l_l−1

=

^kPl−1

m=2

S

_1,m^l,kl∂yl_π^πl,kl−m

l,kl−m

, then we obtain

∂

_yl

π

_l,kl−1

π

_l,kl−1

= δ

^l,k_kl−1^l−1

− A ¯

^l,k_l,k^l_l−1

2

4 Main result

If there exists a diffeomorphism which transforms system (1) into form (2), then equation (12) of Proposition 2 gives all α

_l,i

(y

₁

, ..., y

_l

, u) for 1 ≤ l ≤ m and 1 ≤ i ≤ k

l

− 1. Therefore, let us consider a new family of vector fields defined as follows:

τ

e_l,1

= π

_l,1

(y

₁

, ..., y

_l

, u) τ

_l,1

(15)

τ

e_l,i

= 1

α

_l,i−1

[τ

_l,i−1

, f] for i = 2 : k

_l

where π

_l,1

(y

₁

, ..., y

_l

, u) =

^klQ−1

i=1

α

_l,i

(y

₁

, ..., y

_l

, u) . Set τ

^e

= ( τ

^e_i,j

)

_1≤i≤m

and

1≤j≤ki

and Λ =

^e

θ τ ,

^e

we can define the following multi 1-form:

ω = Λ

^e−1

θ. (16)

It is clear that ω τ

^e

= I

n×n

Then we are ready to state our main result.

Theorem 1 There exists a diffeomorphism which transforms system (1) into a MODO normal form (2) if

i) there exists a family of functions α

_l,i

(y

₁

, ..., y

_l

, u) for 1 ≤ l ≤ m and 1 ≤ i ≤ k

_l

− 1 such that the family of vector fields τ

^e_l,i

for 1 ≤ l ≤ m and 1 ≤ i ≤ k

_l

− 1 defined in (15) satisfies the following commutativity conditions

[ τ

^ei,j

, τ

^es,l

] = 0, f or 1 ≤ i, s ≤ m, 1 ≤ j ≤ k

i

and 1 ≤ l ≤ k

s

(17)

or

ii) there exists a family of functions α

l,i

(y

1

, ..., y

l

, u) for 1 ≤ l ≤ m and 1 ≤ i ≤ k

_l

− 1 such that the R

ⁿ

-valued form ω defined in (16) satisfies the following condition

dω = 0. (18)

Proof 4 Assume that there exist α

_l,i

(y

₁

, ..., y

_l

, u) > 0 for 1 ≤ l ≤ m and 1 ≤ i ≤ k

l

− 1 such that [ τ

^ei,j

, τ

^es,l

] = 0 for 1 ≤ i, s ≤ m, 1 ≤ j ≤ k

i

and 1 ≤ l ≤ k

s

, then it is well-known ([12], [9]) that we can find a local diffeomorphism φ = z such that

φ

∗

( τ

^el,i

) = ∂

∂z

_l,i

.

As φ

_∗

( τ

e_l,i

) =

_∂z^∂

l,i

is constant, hence

∂

∂z

_l,i

φ

_∗

(f) = φ

_∗

([ τ

^e_l,i

, f ]) = α

_l,i

φ

_∗

( τ

^e_l,i+1

) = α

_l,i

∂

∂z

_l,i+1

,