OBSERVABLE SYSTEMS TRANSFORMABLE INTO IMPLICIT AFFINE FORMS

Guillaume, D.^{} Rouchon, P.^{}

Centre Automatique et Systemes, Ecole des Mines de Paris, 60,boulevard Saint-Michel, 75272 Paris Cedex, FRANCE E-mail:guillaume@cas.ensmp.fr

idem.

E-mail:rouchon@cas.ensmp.fr

Abstract: A smooth observable system, _^{x} = ^{f}(^{x}), ^{y} = ^{h}(^{x}), that can be put,
after a change of state coordinates ^{x} ^{7!} ^{X}, into _^{X} = ^{A}(^{y})^{X} +^{b}(^{y}) with an
implicit observation equation ^{C}(^{y})^{X} = ^{d}(^{y}), is said to be transformable into an
implicit ane form. Characterization of such systems is an open problem. For single
output systems, a necessary constructive condition on the structure of the dierential
equation satised by the output is given. When dim(^{x}) = 2, this necessary condition,

y =^{f}^{0}(^{y}) +^{f}^{1}(^{y}) _^{y}+^{f}^{2}(^{y}) _^{y}^{2}+^{f}^{3}(^{y}) _^{y}^{3}, is shown to be sucient. Copyright ^{}c1998
IFAC

Keywords: observers, nonlinear control, ane systems, input injection 1. INTRODUCTION

Observers for nonlinear systems were much stud- ied in the last decade, and real advances were done during this period. One way of building observers is to nd new coordinates in which the system is in a special form. Two important classes of systems were successively studied with such techniques, systems linearizable by output injection (Krener and Respondek, 1985), and sys- tems which are ane regarding to the unmeasured states (Besancon and Bornard, 1997; Hammouri and Gauthier, 1992). In both classes, a part of the states can be considered as measured, e.g, the observation equation admits an explicit structure.

In this paper, a larger class of systems is consid- ered: systems transformable into an implicit ane form. This new class of systems, for which ane asymptotic observers can be designed, are systems

_

x = ^{f}(^{x}), ^{y} = ^{h}(^{x}), that can be put, after a
change of state coordinates ^{x} ^{7!} ^{X}, into _^{X} =

A(^{y})^{X} +^{b}(^{y}) with an implicit observation equa-
tion of the form^{C}(^{y})^{X} =^{d}(^{y}). For single output
systems, a necessary condition for arbitrary state
dimension is derived here. This necessary condi-
tion (see lemma 4) is shown to be sucient when
dim(^{x}) = 2 (see theorem 1). This indicates, even
for single output planar systems, that exist sys-
tems transformable into implicit ane forms that
are not ane regarding to the unmeasured states
in the sense of (Krener and Respondek, 1985; Be-
sancon and Bornard, 1997; Hammouri and Gau-
thier, 1992): taking implicit observation equations
enlarges the class of systems for which asymptotic
observers derived from linear and least squares
techniques can be designed.

This study of single output systems is local and
around "generic" states ^{x}: open subsets where

x can be expressed as a smooth function of
(^{h;}^{L}^{f}^{h;}^{:}^{:}^{:}^{;}^{L}^{n?1}^{f} ^{h}) = (^{y ;}^{y;}_ ^{:}^{:}^{:}^{;}^{y}^{(n?1)}) (^{n} =
dim(^{x})), are only considered here (i.e. open sub-
sets where the tangent manifold is observable).

More precisely, we consider here single output observable systems of the form

_

x=^{f}(^{x})^{;} ^{y} =^{h}(^{x}) (1)
where ^{f} and ^{h} are smooth and where ^{x} lies
an open subset of ^{I}^{R}^{n} such that exists a local
dieomorphism between ^{x} and (^{y ;}^{y;}_ ^{:}^{:}^{:}^{;}^{y}^{(n?1)}).

Such systems are called observable, in the sequel.

They are said to be transformable into an implicit
ane form, if, and only if, exists a local state dif-
feomorphism^{x}=^{}(^{X}) such that the observation
equation ^{y} = ^{h}(^{}(^{X})) denes locally an ane
subspace in the^{X}-space and the restriction of the
dynamics in the^{X}-space, _^{X} =^{F}(^{X}), to the ane
subspace ^{y}=^{h}(^{}(^{X})) is ane (the restriction of

F to ^{y} =^{h}(^{}(^{X})) is ane). In other words, this
means that in the^{X} coordinates (1) reads

_

X=^{A}(^{y})^{X} +^{b}(^{y})^{;} ^{C}(^{y})^{X} =^{d}(^{y}) (2)
where the matrices ^{A}, ^{b}, ^{C} and ^{d} are smooth
functions of^{y}.

Such systems admit an ane observer of the form:

_^

X=^{A}(^{y}) ^^{X} +^{b}(^{y}) (3)

?P

?1

C

0(^{y})(^{C}(^{y}) ^^{X} ^{?}^{d}(^{y}))
_

P=^{? P} ^{?}^{A}^{t}(^{y})^{P}^{?}^{P}^{A}(^{y})

+^{C}^{t}(^{y})^{C}(^{y}) (4)
which converges exponentially when ^{y} is per-
sistent (in the sense of (Guillaume and Rou-
chon, 1997), i.e.^{P} ^{>}^{I}^{d}for some^{}^{>}0).

The paper is organized as follows. In section 2, single output planar observable systems trans- formable into implicit ane forms are character- ized. Section 3 deals with the necessary condition for single output systems with an arbitrary state dimension.

2. SINGLE INPUT PLANAR SYSTEMS 2.1 Necessary condition

Assume that the observable system (1) is trans- formable into an implicit ane form. Set

A(^{y}) =

a

11(^{y}) ^{a}^{12}(^{y})

a

21(^{y}) ^{a}^{22}(^{y})

b(^{y}) =^{?}^{b}^{1}(^{y}) ^{b}^{2}(^{y})^{}^{t}

C(^{y}) =^{?}^{c}^{1}(^{y}) ^{c}^{2}(^{y})^{}

After derivation, (^{0} stands for derivation with
respect to^{y}):

CX=^{d}
(^{C}^{0}^{y}_+^{CA})^{X}+^{Cb}=^{d}^{0}^{y}_

d(^{CX})

dt

=^{P}^{X}+^{Q}=^{R}

with^{P} = (^{C}^{00}^{y}_^{2}+^{C}^{0}^{y}+ 2^{C}^{0}^{yA}_ +^{CA}^{0}^{y}_+^{CA}^{2}),

Q = 2^{C}^{0}^{b}^{y}_ +^{CAb}+^{Cb}^{0}^{y}_ and ^{R} = ^{d}^{00}^{y}_^{2}+^{d}^{0}^{y}.
Since

C

C

0^{y}_+^{CA}

is assumed invertible,^{X} can
be eliminated from the last equation. This yields a
second order dierential equation for ^{y}. Examine
the structure of this equation. Up to renumbering
the component of^{x}, assume that^{c}^{1}(^{y})^{6}= 0. Thus

X

1 = ^{d}(^{y})^{=c}^{1}(^{y})^{?}^{c}^{2}(^{y})^{=c}^{1}(^{y})^{X}^{2}. Without lost
of generality, assume also that the rst column of

A is zero (substitute ^{X}^{1} into the state equation)
and^{c}^{1}^{}1. This leads to the following simplied
structure:

A(^{y}) =

0^{a}^{1}(^{y})
0^{a}^{2}(^{y})

b(^{y}) = ^{?}^{b}^{1}(^{y}) ^{b}^{2}(^{y})^{}^{t}

C(^{y}) = ^{?}1^{c}(^{y})^{}^{:}

(5) Denote

1=^{a}^{1}+^{ca}^{2}^{;} ^{}^{1}=^{b}^{1}+^{cb}^{2}^{:} (6)
Elimination of^{X} leads to

g(^{y})^{y} =^{g}^{0}(^{y}) +^{g}^{1}(^{y}) _^{y}+^{g}^{2}(^{y}) _^{y}^{2}+^{g}^{3}(^{y}) _^{y}^{3}
with

g= (^{c}^{0}^{}^{1}+^{d}^{0}^{}^{1})

g

0= (^{?}^{1}^{}^{1}^{a}^{2}+^{}^{2}^{1}^{b}^{2})

g

1= (^{?}^{1}(^{}^{0}^{1}+^{c}^{0}^{a}^{2}) +^{d}^{0}^{}^{1}^{a}^{2}+ 2^{c}^{0}^{}^{1}^{b}^{2}+^{}^{1}^{}^{0}^{1})

g

2= (^{?d}^{00}^{}^{1}^{?}^{c}^{00}^{}^{1}+ (^{}^{0}^{1}+^{c}^{0}^{a}^{2})^{d}^{0}+ (^{}^{1}^{0}+^{c}^{0}^{b}^{2})^{c}^{0})

g

3= (^{d}^{0}^{c}^{00}^{?}^{c}^{0}^{d}^{00})^{:}

Observability of (1), means here that exists a
local dieomorphism between ^{X} and (^{y ;}^{y}_). This
implies that, locally, ^{g} cannot vanish. Thus the
following lemma is proved:

Lemma 1.(Necessary condition). If the observ-
able planar system (1) is transformable into an
implicit ane system (3) then, its output ^{y} sat-
ises a second order dierential equation of the
form

y=^{f}^{0}(^{y}) +^{f}^{1}(^{y}) _^{y}+^{f}^{2}(^{y}) _^{y}^{2}+^{f}^{3}(^{y}) _^{y}^{3}
with^{f}^{0},^{f}^{1},^{f}^{2},^{f}^{3} smooth functions.

2.2 Sucient condition

Assume that the output ^{y} of planar observable
system (1) satises a second order dierential
equation of the form

y=^{f}^{0}(^{y}) +^{f}^{1}(^{y}) _^{y}+^{f}^{2}(^{y}) _^{y}^{2}+^{f}^{3}(^{y}) _^{y}^{3}

with the ^{f}^{0},^{f}^{1}, ^{f}^{2} and ^{f}^{3} smooth function of ^{y}.
A dieomorphism^{x}=^{}(^{X}) that put the system
into an implicit ane form 3 can explicitely be
built.

From previous computation,it appears that the
problem to solve is equivalent to nd smooth ^{y}-
function^{c},^{d},^{}^{1},^{}^{1},^{b}^{2} and^{a}^{2}such that

f

0(^{c}^{0}^{}^{1}+^{d}^{0}^{}^{1}) = (^{?}^{1}^{}^{1}^{a}^{2}+^{}^{2}^{1}^{b}^{2})

f

1(^{c}^{0}^{}^{1}+^{d}^{0}^{}^{1}) = (^{?}^{1}(^{}^{0}^{1}+^{c}^{0}^{a}^{2}) +^{d}^{0}^{}^{1}^{a}^{2}
+2^{c}^{0}^{}^{1}^{b}^{2}+^{}^{1}^{}^{1}^{0})

f

2(^{c}^{0}^{}^{1}+^{d}^{0}^{}^{1}) = (^{?d}^{00}^{}^{1}^{?}^{c}^{00}^{}^{1}

+(^{}^{0}^{1}+^{c}^{0}^{a}^{2})^{d}^{0}+ (^{}^{1}^{0}+^{c}^{0}^{b}^{2})^{c}^{0})

f

3(^{c}^{0}^{}^{1}+^{d}^{0}^{}^{1}) = (^{d}^{0}^{c}^{00}^{?}^{c}^{0}^{d}^{00})^{:}

(7)

Basic algebraic manipulations of this system yield to the following three equations:

1

1

0

1=^{?}(^{c}^{0}^{}^{1}+^{d}^{0}^{}^{1})(2^{c}^{0}^{f}^{0}+^{}^{1}^{f}^{1})

?c 0

1

1 a

2+^{d}^{0}^{}^{2}^{1}^{a}^{2}+^{}^{2}^{1}^{}^{1}^{0}^{?}2^{c}^{0}^{}^{1}^{}^{1}^{a}^{2}

2

1

1(^{c}^{0}^{}^{1}+^{d}^{0}^{}^{1})^{c}^{00}=^{c}^{0}^{F}+^{}^{3}^{1}^{}^{1}^{f}^{3}(^{c}^{0}^{}^{1}+^{d}^{0}^{}^{1})

2

1

1(^{c}^{0}^{}^{1}+^{d}^{0}^{}^{1})^{d}^{00}=^{d}^{0}^{F}^{?}^{}^{2}^{1}^{}^{1}^{2}^{f}^{3}(^{c}^{0}^{}^{1}+^{d}^{0}^{}^{1}(8))
where

F=^{}^{1}(^{}^{2}^{1}(^{c}^{0}^{a}^{2}^{d}^{0}+^{}^{1}^{0}^{c}^{0})^{?}^{c}^{02}^{}^{1}^{}^{1}^{a}^{2})
+^{}^{1}^{d}^{0}(^{}^{1}(^{?}^{1}^{c}^{0}^{a}^{2}+^{d}^{0}^{}^{1}^{a}^{2}+^{}^{1}^{}^{1}^{0})

?(^{c}^{0}^{}^{1}+^{d}^{0}^{}^{1})(^{}^{1}^{d}^{0}(2^{c}^{0}^{f}^{0}+^{}^{1}^{f}^{1})
+^{}^{1}(^{c}^{02}^{f}^{0}+^{}^{2}^{1}^{f}^{2}))^{?}2^{c}^{0}^{}^{1}^{}^{1}^{a}^{2})^{:}
Denote by ^{x} the state around which the trans-
formation^{x}=^{}(^{X}) is looked for and denote by
(^{y;}^{y}_) the corresponding values of (^{y ;}^{y}_).

Set ^{}^{1}(^{y}) ^{} 1 and ^{a}^{2}(^{y}) ^{} 0. By the Cauchy-
Lipschitz theorem, there exists locally around ^{y},
a unique solution ^{y} ^{7!} (^{}^{1}^{;}^{c;}^{d}) of the dieren-
tial system (8) with the following initial values:

1(^{y}) = (_^{y})^{2} + 1, ^{c}(^{y}) = ^{d}(^{y}) = ^{d}^{0}(^{y}) = 0,

c

0(^{y}) = 1^{:} Such a solution yields to ^{b}^{2}(^{y}) via

2

1 b

2=^{f}^{0}(^{c}^{0}^{}^{1}+^{d}^{0}^{}^{1}) +^{}^{1}^{}^{1}^{a}^{2}.

For ^{A}, ^{b} and ^{C} derived from the above solution

y7!(^{}^{1}^{;}^{}^{1}^{;}^{a}^{2}^{;}^{b}^{2}^{;}^{c;}^{d}) of (7) via the formulae (5)
and (6), the relations

C

C

0^{y}_+^{CA}

X=

d

d
0^{y}_^{?}^{Cb}

(9)
dene a local dieomorphism between (^{y ;}^{y}_) and

X, that is between^{x}and^{X}. This results from the
following arguments:

since ^{c}^{0}(^{y})_^{y}+^{}^{1}(^{y}) = ^{y}^{2}+ ^{y}+ 1 ^{>} 0, the

matrix ^{}

C

C

0^{y}_+^{CA}

is locally invertible; thus ^{X} is a smooth
function of (^{y ;}^{y}_).

denote by ^{X} the solution of (9) for (^{y;}^{y}_);

since

(^{}^{1}(^{y}) +^{c}^{0}(^{y})_^{y}) ^{X}^{2}=^{d}^{0}(^{y})_^{y}^{?}^{}^{1}(^{y})^{;}

X

2

6= 0 and^{c}^{0}(^{y}) ^{X}^{2}^{?}^{d}^{0}(^{y})^{6}= 0; the implicit
function theorem for ^{X}^{1} +^{c}(^{y})^{X}^{2} = ^{d}(^{y})
ensures that^{y}can be expressed as a smooth
function of^{X};

similarly, _^{y} is also a smooth function of^{X}.
Thus, the following lemma is proved:

Lemma 2.(Sucient condition). Assume that the scalar output of the planar observable systems (1) satises a smooth second order dierential equation of the form

y=^{f}^{0}(^{y}) +^{f}^{1}(^{y}) _^{y}+^{f}^{2}(^{y}) _^{y}^{2}+^{f}^{3}(^{y}) _^{y}^{3}^{:}
Then, locally, there exists a change of state coor-
dinates putting the system into an implicit ane
form (3).

Lemmas 1 et 2 lead to the following

Theorem 1. A single output observable planar system (1) is transformable into an implicit ane form (3), if, and only if, the second-order deriva- tive of the output is a polynomial of third degree in the rst-order derivative of the output with coecients smooth function of the output.

Remark that:

planar observable systems transformable into implicit ane forms admit the following nor- mal forms

_

X

1 =^{a}(^{y})^{X}^{2}+^{b}^{1}(^{y})
_

X

2 =^{b}^{2}(^{y})

X

1+^{c}(^{y})^{X}^{2} =^{d}(^{y})^{:}

This results from the proof of the sucient condition.

The class of planar observable systems, ane regarding to the unmeasured states in the sense of (Krener and Respondek, 1985; Be- sancon and Bornard, 1997; Hammouri and Gauthier, 1992), admit the following output dierential equation:

y =^{k}^{0}(^{y}) +^{k}^{1}(^{y}) _^{y}+^{k}^{2}(^{y}) _^{y}^{2}^{:}
This former class is thus smaller than the
class considered here.

3. NECESSARY CONDITION FOR GENERAL SINGLE OUTPUT SYSTEMS

Denote by ^{n} the state dimension and take the
implicit ane form. After ^{n} derivations of the
observation equation, the following system:

CX=^{d}

d(^{CX})

dt

=^{d}^{0}^{y}_

d
2(^{CX})

dt

2 =^{d}"_^{y}^{2}+^{d}^{0}^{y} (10)

:::=^{:}^{:}^{:}

d
n(^{CX})

dt

n =^{d}^{0}^{y}^{(n)}+^{:}^{:}^{:}+^{d}^{(n)}^{y}_^{n}

is a linear overdetermined system with respect to

X. Eliminating of ^{X} yields to a polynomial im-
plicit scalar equation with respect to (_^{y;}^{:}^{:}^{:}^{;}^{y}^{(n)}).

A careful inspection of the dierent partial degrees of this polynomial equation yields to a necessary condition. Introduce the following denitions bor- rowed from (Krener and Respondek, 1985).

Denition 1. Note^{P} the ring of polynomials with
indeterminates the successive derivative ^{y}^{(k )} of

y and with coecients ^{C}^{1} functions of ^{y}. By
denition the degree of^{y}^{(k )} is^{k}and the degree of

y (j

1 )

:::y (j

r

)is^{j}^{1}+^{:}^{:}^{:}+^{j}^{r}. Note^{P}^{k} the subring
of polynomials, the degree of which is less than or
equal to^{k}and^{P}^{i}^{k} the subring of polynomials, the
degree of which is less than or equal to^{k}spanned
by^{P}^{i}, with ^{i}^{}^{k}.

Lemma 3. The system (10) can be written:

P

0

X+^{Q}^{0}(^{y}) =^{R}^{0}

:::=^{:}^{:}^{:}

P

n?1

X+^{Q}^{n?1}=^{R}^{n?1} (11)

P

n

X+^{Q}^{n}=^{R}^{n}

with^{P}^{i} and^{R}^{i} elements of^{P}^{i} for^{i}= 1^{:}^{:}^{:}^{n}and

Q

i element de^{P}^{i?1} for^{i}= 2^{:}^{:}^{:}^{n}.

Proof: Immediate for^{i}= 1^{;}2^{;}3^{:}^{:}^{:}The derivative
of a monomial of the form^{f}(^{y})^{y}^{(k )} is a monomial
of degree ^{k}+ 1 because ^{d(f(y )y}^{dt}^{(k)}^{)} =^{f}^{0}(^{y}) _^{yy}^{(k )}+

f(^{y})^{y}^{(k +1)}.

The structure of the (vector) polynomials can be more precisely described:

P= (^{p}^{ij}) =

0

B

B

B

@ P

0

P

...1 P

n?1 1

C

C

C

A

with

P

0=^{C}

P

1=^{C}^{0}^{y}_+^{CA}
...

P

n?1=^{C}^{(n?1)}^{y}_^{n?1}

+(^{n}^{?}1)^{C}^{(n?1)}^{y}^{y}_^{n?3}+^{:}^{:}^{:}+^{CA}^{n?2}

Q =

0

B

B

B

@ Q

0

Q

...1 Q

n 1

C

C

C

A

, ^{R}=

0

B

B

B

@ R

0

R

...1 R

n 1

C

C

C

A

, = det^{P}, ^{ij} the
cofactors of^{P} and^{M} = (^{ij}), the comatrix.

The elimination of^{X} from the previous equations
leads to:

P

n P

?1(^{R}^{?}^{Q}) +^{Q}^{n}=^{R}^{n}
that is:

P

n M

t(^{R}^{?}^{Q}) + ^{Q}^{n}= ^{R}^{n} (12)
The coecient^{D}of^{y}^{(n)}in (12) is:^{D}= (^{C}^{0}^{X}^{?}

d

0) =^{C}^{0}^{M}^{t}(^{R}^{?}^{Q})^{?}^{d}^{0} with^{R}and^{Q}elements
of ^{P}^{n?1}. ^{M} is an element of ^{P}^{n?1}^{n(n?1)=2?1}and
an element of ^{P}^{n?1}^{n(n?1)=2}. We have the following
degrees for the lines ^{M}(^{i}), ^{Q}(^{i}) and ^{R}(^{i}) of the
corresponding matrices

d

M(1) =^{n}(^{n}^{?}1)^{=}2

d

M(^{i}) =^{n}(^{n}^{?}1)^{=}2^{?}(^{i}^{?}1)^{;}^{i}= 2^{:}^{:}^{:}^{n}

d

Q(1) = 0

d

Q(^{i}) =^{i}^{?}2^{;}^{i}= 2^{:}^{:}^{:}^{n}

d

R(^{i}) =^{i}^{?}1^{;}^{i}= 1^{:}^{:}^{:}^{n}

Then,^{D}is an element of^{P}^{n?1}^{n(n?1)=2}. Isolating the
terms with^{d}^{0}in^{X} and using:

D=^{X}^{n}

k =1 c

0(^{k})(^{?}1)^{n+k}^{nk}^{d}^{0}^{y}^{(n?1)}^{?}^{d}^{0} +^{:}^{:}^{:}
with = ^{P}^{n}^{k =1}(^{?}1)^{n+k}^{p}^{nk}^{nk} and ^{p}^{nk} =

c

0(^{k})^{y}^{(n?1)}+^{:}^{:}^{:}, the term in ^{y}^{(n?1)} disappears.

Dis an element of a subring spanned by^{P}^{n?2}.
From the equation (^{C}^{0}^{X}^{?}^{d}^{0})_^{y}=^{?}(^{CAX}+^{Cb})
and after multiplication by , it appears that
the degree of the right term is less than equal to

n(^{n}^{?}1)^{=}2 (degree de ^{X} and of ) and the one
of the left term less than or equal^{n}(^{n}^{?}1)^{=}2+ 1.

This is possible only if the degree of (^{C}^{0}^{X}^{?}^{d}^{0})
is less than or equal to ^{n}(^{n}^{?}1)^{=}2^{?}1. There is
no term of degree ^{n}(^{n}^{?}1)^{=}2 in^{D}. Finally,^{D} is
an element of^{P}(n(n?1)=2?1)

n?2 .

Consider the degree of the remaining term^{N}. ^{N}
is got from the equation (12), in which the term
with^{y}^{(n)} is suppressed:

N = (^{P}^{n}(^{y ;}^{:}^{:}^{:}^{;}^{y}^{(n?1)})^{X}

+^{Q}^{n}(^{y ;}^{:}^{:}^{:}^{;}^{y}^{(n?2)})^{?}^{R}^{n}(^{y ;}^{:}^{:}^{:}^{;}^{y}^{(n?1)}))

P

nis an element of^{P}^{n?1}^{n} (polynomial of degree^{n}
without derivatives of order ^{n}) and ^{X} element
of^{P}^{n?1}^{(n(n?1)=2)}so ^{N} is an element of^{P}^{n?1}^{n(n+1)=2}.
Sum up these results in the following proposition:

Lemma 4. If the single output observable system
(1) is transformable into an ane system (3) then
the output ^{y} satises the following dierential
equation:

y

(n)=^{N}^{=D} (13)

with ^{N} element of ^{P}^{n?1}^{n(n+1)=2} and ^{D} element of

P

(n(n?1)=2?1)

n?2 (see denition 1).

For ^{n} = 3, the following output equation exists
necessarily:

y

(3) = ^{N}(^{y ;}^{y;}_ ^{y})

d

0(^{y}) +^{d}^{1}(^{y}) _^{y}+^{d}^{2}(^{y}) _^{y}^{2}
with

N=^{n}^{1}(^{y}) (^{y})^{3}+^{n}^{2}(^{y}) (^{y})^{2}^{y}_^{2}+^{n}^{3}(^{y}) ^{y}^{y}_^{4}+^{n}^{4}(^{y}) _^{y}^{6}
+^{n}^{5}(^{y}) (^{y})^{2}^{y}_+^{n}^{6}(^{y}) ^{y}^{y}_^{3}+^{n}^{7}(^{y}) _^{y}^{5}

+^{n}^{8}(^{y}) (^{y})^{2}+^{n}^{9}(^{y}) ^{y}^{y}_^{2}+^{n}^{10}(^{y}) _^{y}^{4}
+^{n}^{11}(^{y}) ^{y}^{y}_+^{n}^{12}(^{y}) _^{y}^{3}

+^{n}^{13}(^{y}) ^{y}+^{n}^{14}(^{y}) _^{y}^{2}
+^{n}^{15}(^{y}) _^{y}

+^{n}^{16}(^{y})

The fact that the denominator depends only on _^{y}
is not obvious.

Remark: the gap between the dimension 2 and 3
appears here. For the 3 dimensional case, there are
19 equations to satisfy for 12 functions (overdeter-
mined), when for the 2 dimensional case, there are
5 equations (4 if^{g}is set to 1) for 6 functions (un-
derdetermined). So we see that there are degrees
of freedoms for the 2 dimensional and relations to
satisfy in the 3 dimensional case.

4. CONCLUSION

Here is establishesd a necessary condition so that an nonlinear system is equivalent to an ane system (3) with an observation equation, ane regarding to the state and implicit regarding to the the measured output. A such class of systems contains linearizable systems by output injection, and also ane systems described in (Besancon and Bornard, 1997). An observer for the system described here can be found in (Guillaume and Rouchon, 1997). Our present work is on the in- trinsic characterization of such systems and also on the way to compute the coordinates change when it exists.

5. REFERENCES

Besancon, G. and G. Bornard (1997). On charac- terizing classes of observers forms for nonlin-

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Guillaume,D. and P. Rouchon (1997). Estimation d'etat et de parametres cinetiques pour une classe de bioreacteurs.Proc AGIS'97 1, 173{

Hammouri, H. and JP. Gauthier (1992). Global177.

time-varying linearization up to output injec- tion.SIAM J. Control Optim.30, 1295{1310.

Krener, A.J. and W. Respondek (1985). Nonlinear observers with linearizable error dynamics.

SIAM J. Control Optim.23, 1197{216.