Nonlinear observer based on observable cascade form

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Nonlinear observer based on observable cascade form

Mariem Sahnoun, Hassan Hammouri

To cite this version:

Mariem Sahnoun, Hassan Hammouri. Nonlinear observer based on observable cascade form. 2014.

�hal-00982014�

Nonlinear observer based on observable cascade form

Mariem Sahnoun and Hassan Hammouri

Abstract— In this paper, the error observer linearization is extended to a class of observable cascade systems which contains state affine systems up to output injection. First, we give a theoretical result which states necessary and sufficient conditions. Next, we give an algorithm permitting to calculate a system of coordinates in which a nonlinear system takes the desired cascade observable form.

Index Terms— Nonlinear systems, output injection, nonlinear observer.

I. INTRODUCTION

The implementation of linear or nonlinear observers in control systems design, fault detection and other domains is well understood by now.

To design an observer for nonlinear systems, many ap- proaches have been developed. Among them, the geometric approaches consist in characterizing nonlinear systems which can be transformed by a change of coordinates to a special class of systems for which a simple observer can be designed.

The observer error linearization problem consists of trans- form a nonlinear system into a linear one plus a nonlinear term depending only on the known inputs and outputs. For such systems, a Luenberger observer can be designed. This problem has attracted a good deal of attention, since its formulation by [9] (see for instance [2], [3], [10]–[13]. Using immersion technics, an extension of this problem has been stated in [8] in the single output case. In the same spirit as for the error linearization problem, the authors in [4]–[7]

characterized nonlinear systems which can be steered by a change of coordinates to state affine systems up to output injection. For these systems, a Kalman-like observer can be designed.

In this paper, we will characterize nonlinear systems which can be transformed by local coordinate systems into the following cascade form:









˙z=A(u)z+ψ(u,y) e˙

z=A(u)e ez+ψe(u,z,ey) Y=

y e y

= Cz

e Cez

(1)

For these systems, an observer structure may take the

M. Sahnoun and H. Hammouri are with Universit´e de Lyon, F-69622, Lyon, France; Universit´e Lyon 1, Villeurbanne;

CNRS, UMR 5007, LAGEP (Laboratoire d’Automatique et de G´enie des Proc´ed´es). 43 bd du 11 novembre, 69100 Villeurbanne, France sahnoun@lagep.univ-lyon1.fr, hammouri@lagep.univ-lyon1.fr

following form:









 b˙

z=A(u)bz+ψ(u,y)−S⁻¹C^TR(Cbz−y) be˙z=A(u)e bez+ψe(u,bz,ey)−Se⁻¹Ce^TR(eCebez−y)e

S˙=−θ^S−A^T(u)S−SA(u) +C^TRC

˙e

S=−θe^Se−Ae^T(u)eS−SeA(u) +e Ce^TReCe

(2)

where S(0), S(0), R ande R are symmetric positive definitee matrices, θ >0, θe>0 are parameters. The proof of the convergence of this observer has been stated in [1].

This paper is organized as follows:

In section II, the problem under consideration is formalized and an existence theorem is stated. In section III, an al- gorithm permitting to calculate a system of coordinates in which a nonlinear system takes the desired cascade form is proposed.

II. PRELIMINARY RESULTS AND EXISTENCE THEOREM

A. Preliminary results

For the sake of simplicity, we only consider the case where the outputs y and ey are scalars. The following classes of nonlinear systems will be considered:





˙

x=f(u,x) y=h(x) e y=eh(x)

(3)

where x∈Rⁿ, the input u(t)∈R^m and the outputs y(t)and e

y(t)are belong toR. f , h andeh are assumed to be of class C^∞.

We adopt the following definition.

Definition 1: System (1) is said to be cascade- observable, if system (1) together with its associated reduced system in z are observable.

The following geometric notions will be used in the sequel.

In the system of coordinates(x1, . . . ,x_n), let X=∑ⁿ_i=1αi ∂

∂xi

be a vector field and let ω =∑ⁿ_i=1a_idx_i a one-differential form, then the following operations will be considered:

• Lie derivative action: L_X(ω) = ∑ⁿ_i=1αiL_X(ai)dxi+

∑ⁿ_i=1aidαi

• The duality product:ω(X) =∑ⁿ_i=1αiai

The above duality product can be extended to k- differential forms as follow:

If ω = ∑1≤i1<...<ik≤na_(i

1,...,ik)dx_i1 ∧. . .∧dx_ik is a k-differential form and X = (X1, . . . ,X_k) is a k- tuple of vector fields, with X_i =∑ⁿ_l=1αil ∂

∂x_l, then

ω(X) =∑1≤i1<...<ik≤na_(i1,...,ik)

α1i1 . . . αki1

. . . . α1i_k . . . αkik

.

• Inner product: Let X = (X1, . . . ,X_l) be a l-tuple of vector fields, with l≤k. Then iX(ω) is the (k−l)- differential form defined by:

i_X(ω)(Y1, . . . ,Y_k−l) =ω(X1, . . . ,Xl,Y1, . . . ,Yk−l).

In particular, if k=l, then iX(ω)is a function (a 0−differential form).

Let f_ube the vector field defined by f_u(x) =f(u,x)and let X be a vector field onRⁿ. We define the family of real vector spacesΩ^X_k of 2-differential forms as follows:

• Ω^X₀ =0 andΩ^X₁ =S pan{dLfu(h)∧dh; u∈R^m}. Notic- ing that these two spaces do not depend on X ,

• for k≥1, we set Ω^X_k+1=S pan{Lfu(iX(ω))∧dh; u∈ R^m; ω∈Ω^X_k}+Ω^X_k.

Now settingπ=dϕ1∧. . .∧dϕq, whereϕkareC^∞functions, and letXe= (Xe₁, . . . ,Xe_q+1)be a(q+1)-tuple of vector fields.

As above, we define the vector spaces Ω^X_k,π^e of (q+2)- differential forms as follows:

• Ω^X_0,π^e =0 andΩ^X_1,π^e =S pan{dLfu(eh)∧deh∧π^{; u}∈R^m},

• for k≥1, Ω^X_k+1,π^e =S pan{Lfu(i_Xe(ωe))∧deh∧π^{; u}∈ R^m; ωe∈Ω^X_k,π^e }+Ω^X_k,π^e .

B. Existence theorem

In the single output case (see [4], [6]), ( [5] for the the multi-output case) the authors gave necessary and sufficient conditions under which nonlinear systems can be trans- formed in a state affine system up to output injection.

The following theorem states an existence theorem which extends those stated in [4], [5]:

Theorem 1:

Observable system (3) can be transformed by a local change of coordinates around some x⁰∈Rⁿ to a cascade- observable system (1) in which C andC are of rank 1, if ande only if, the following conditions hold on some neighborhood of x⁰:

1) It exists a vector field X satisfying the following con- ditions:

1-i) L_X(h) =1.

1-ii) The algebraic sumΩ^X=

∑

k≥1

Ω^X_k is a real vector space of dimension q−1.

1-iii) For everyω∈Ω^X, d(iX(ω)) =0.

1-iv) The dimension of [^Vq−1(iX(Ω^X))∧dh]|_x0 is equal to 1, where[^Vq−1(iX(Ω^X))∧dh]|_x0 ={iX(ω1)∧. . .∧ i_X(ωq−1)∧dh(x⁰); ωi∈Ω^X ,1≤i≤q−1}.

2) Consider the following functions ϕ1, . . . ,ϕq+1 defined by:

ϕ1=h ϕq+1=eh

(dϕ1, . . . ,dϕq)forms a basis of i_X(Ω^X) +Rdh (4)

Settingπ=dϕ1∧. . .∧dϕq, then there exists a(q+1)- tuple of vector fields Xe= (Xe₁, . . . ,Xe_q+1) satisfying the following conditions on some neighborhood of x⁰: 2-i) L_Xe

i(ϕj) =δi j, whereδi j=1 if i=j and 0 otherwise.

2-ii) The algebraic sum Ω^X_π^e=∑k≥1Ω^X_k,π^e is a real vector space of dimension n−q−1.

2-iii) For everyωe∈Ω^X_π^e, d(i_Xe(ωe)) =0.

2-iv) The dimension of [^Vn−q−1(i_Xe(Ω^X_π^e))∧dϕ1∧. . .∧ dϕq+1]|_x0 is equal to 1.

The proof of theorem 1 can be obtained by following the same approach as the one proposed in the works [4], [5].

The outline of the proof is summarized as follows:

1) Sufficient condition: i_X(Ω^X) and i_Xe(Ω^X_π^e) are vector spaces of dimension q−1 and n−q−1 respectively, and (iX(ω1), . . . ,i_X(ωq−1)), (i_Xe(ωe1), . . . ,i_Xe(ωen−q−1)) are their respective bases. Setting dz₁ =dh, dz_i = i_X(ωi+1), dez₁=deh and dez_i=i_Xe(ωei+1). It can be shown that L_fu(zi) =

∑

a_{i j}(u)zj+ψi(u,z₁) and L_fu(ez_i) =

n−q

∑

j=2

e

a_{i j}(u)ez_j+ψi(u,z,ez₁). Consequently, in the(z,ez)sys- tem of coordinates system (3) takes the cascade form (1).

2) Necessary condition: Since conditions 1), 2) of theorem 1 are intrinsic (they do not depend on the system of coordinates), it suffices to show them for the cascade observable system (1). After a simple linear change of coordinates, we can assume that y=Cz=z₁ and ey= Ceez=ez₁, and it can be shown that X= ∂

∂^z1

and Xe= (eX₁, . . . ,Xe_q+1) = ( ∂

∂^z1

, . . . , ∂

∂^zq

, ∂

∂ez₁)satisfy conditions 1) and 2) of theorem 1.

In the following, we focus on the development of an algorithm permitting to calculate vector fields X,Xe₁, . . . ,Xe_q+1 which meet conditions 1)and 2)of theorem 1.

III. PROCEDURE OF CALCULATION OF VECTOR FIELDS

X,Xe₁, . . . ,Xe_q+1 A. Preliminary results

The following notations will be used in the sequel:

• Let V be a vector space, and W a subspace of V , then forξ,ξ^′∈V , the notation ξ =ξ^{′ modulo} (W) means thatξ =ξ^′+w, for some w∈W .

• Setting F (resp. V) to be a set of one-differential form (resp. of vector fields). D=S pan(F) (resp.∆= S pan(V)) will denote the co-distribution (resp. the distribution) spanned byF (resp. byV).

• The orthogonal of a co-distribution D is the distribution

∆=Ker(D) =S pan({X ; ω(X) =0, ∀ω∈F}), where ω(X) is the duality product between one-form and vector fields. In particular, ifF is spanned by a family of one-exact form {dϕ^; ϕ ∈Ff}, then ∆=Ker(D)

is the distribution spanned by the set of vector fields {X ; LX(ϕ) =0, ∀ϕ∈Ff}.

• Let D, D^′be two co-distributions, with D^′⊂D, then the quotient D/D^′will denote the set of equivalent class of differential forms[ω] =ω+D^′={ω+ω^′; ω^′∈D^′}, whereω∈D. Similarly, if∆⊂∆^′are two distributions, elements of the quotient∆^′/∆will be denoted by[X] = X+∆ where X∈∆^′.

If [ω]∈D/D^′ andχ∈D such that[ω] = [χ], then we setω=χ ^{modulo (D′).}

Finally, if X , Z are two vector fields,[X,Z]will denote the Lie bracket of these vector fields.

The following flag of co-distributions and distributions will be considered:

D₀⊂. . .⊂D_k⊂. . .

∆0⊃. . .⊃∆_k⊃. . . De₀⊂. . .⊂De_k⊂. . . e∆0⊃. . .⊃∆ek⊃. . .

(5)

Where,

• D₀=0 the null co-distribution, D₁=S pan({dh}), by induction D_k+1=D_k+S pan({dLf_uk. . .L_fu1(h); u1, . . . , u_k∈R^m}), and D♯=∑k≥1D_k.

• De₀=D_♯, De₁=De₀+S pan({deh}), for k≥1, De_k+1= e

D_k+S pan({dLf_uk. . .L_fu1(eh); u1, . . . ,u_k ∈ R^m}), and e

D_♯=∑k≥1De_k.

• ∆_k=Ker(D_k), and∆_♯=Ker(D_♯).

• e∆k=Ker(De_k), and∆e♯=Ker(eD_♯).

• The quotient co-distribution D_k/D_k−1 (resp.De_k/De_k−1) is the dual of the quotient distribution ∆_k−1/∆k (resp.

e∆_k−1/e∆k). The duality product[ω]([X]) =ω(X)is well defined.

In the two following claims, f_u=

∑

(Ai(u)z+ψi(u,y)) ∂

∂^zi

+

n−q

∑

i=1

(Ae_i(u)ez+ψei(u,z,ey)) ∂

∂ez_i, and the outputs h,eh are respec- tively y=Cz=z₁,ey=Ceez=ez₁.

Considering the ringsH_k,Hf_k such that:

• H₀=C^∞{z1} (resp. Hf₀=C^∞{z1, . . . ,z_q,ez₁}) is the ring ofC^∞-functionsϕ(z1)(resp.ϕ(z1, . . . ,z_q,ez₁)).

• C^∞{z} (resp. C^∞{z,ez}) denotes the ring of C^∞- functionsϕ(z1, . . . ,z_q)(resp.ϕ(z1, . . . ,zq,ez₁, . . . ,ez_n−q)).

Then for k≥1, H_k (resp. Hf_k) is the smallest sub- ring of C^∞{z} (resp. of C^∞{z,ez}) containingH_k−1∪ {CA(u1). . .A(uk)z; u1, . . . ,u_k ∈ R^m} (resp. Hf_k−1∪ {CeeA(u1). . .A(ue k)ez; u₁, . . . ,u_k∈R^m}).

Then we have:

Claim 1:

i) L_fuk. . .Lf_u1(Cz) =CA(u1). . .A(uk)z modulo(H_k−1).

ii) L_fuk. . .Lf_u1(Ceez)) =CeA(ue 1). . .A(ue k)ez modulo(Hf_k−1).

The following claim can be deduced from the above one.

Claim 2:

• The flags of co-distributions D₀ ⊂. . . ⊂D_k ⊂. . .;

e

D₀/D_♯⊂. . .⊂De_k/D_♯⊂. . . are of constant dimensions and

defined as follows:

a) D₁=S pan(dCz), and for k≥2, D_k is spanned by the set of one-forms{dCz} ∪ {dCA(u1). . .A(ul)z; 1≤l≤ k−1, uj∈R^m}.

b) Similarly, De₁/D♯ can be identified with the co- distribution S pan(dCeez), and for k≥2,De_k/D♯is isomor- phic to the co-distribution spanned by the set of one- forms {dCeez} ∪ {dCeA(ue 1). . .A(ue l)ez; 1≤l≤k−1, uj∈ R^m}.

• System (1) is cascade observable iff: dim D_♯=q (q is the dimension of the z-space), and dimDe_♯/D♯=n−q (n−q is the dimension ofez-space).

In the sequel, we set ν (resp νe) to be the smallest integer such that D_ν=D_♯ (resp.De_νe/D_♯=De_♯/D_♯):

D₀⊂. . .⊂D_ν=D_ν+1

De₀/D♯⊂. . .⊂De_eν/D♯=De_eν+1/D♯

(6) This subsection will be ended by the two following technical results:

Lemma 1:

If dϕ∈D_k−1(resp. dϕe∈De_k−1) and X∈∆k−1(resp.Xe∈ e∆k−1), then dϕ([fu,X]) =−d(Lfu(ϕ))(X) = −LX(Lfu(ϕ)) (resp. dϕ([f_u,X]) =e −d(Lfu(eϕ))(Xe) =−L_Xe(Lfu(ϕe))).

Proof of lemma 1.

Let dϕ ∈D_k−1 and X ∈∆k−1, the equality dϕ([f_u,X]) =

−d(Lfu(ϕ))(X)follows from the following facts:

• dϕ([f_u,X]) =L_fu(LX(ϕ))−L_X(Lfu(ϕ))

=d(LX(ϕ))(f_u)−d(Lfu(ϕ))(X),

• X∈∆k⊂∆k−1=Ker(Dk−1),

• L_X(ϕ) =dϕ(X) =0

Similar argument can be used to prove dϕ([f_u,X]) =e

−d(Lfu(eϕ))(Xe).

Claim 3:

Let Z = (Z1, . . . ,Z_k) be a k-tuple of vector fields, let

g, ϕ1, . . . ,ϕk be C^∞-functions such that dϕ1∧. . .∧dϕk is

nowhere vanish and that L_Zj(ϕi) =δi j, then:

i_Z(dg∧dϕ1∧. . .∧dϕk) =dg−

∑

L_Zj(g)dϕj. More precisely, we have:

i_Z(dg∧dϕ1∧. . .∧dϕk) = (−1)^q(dg−

∑

L_Zj(g)dϕj).

B. Algorithm

In this subsection, we will give an algorithm permitting to calculate the vector fields X,Xe₁, . . . ,Xe_q+1, which meet conditions of theorem 1. This algorithm will be obtained in three steps:

1) The first step consists to calculate X using only f(u,x) and h(x).

2) The knowledge of f(u,x), h, eh(x) and X allows to calculateXe_q+1.

3) Finally,Xe₁, . . . ,Xe_qcan be computed based on the knowl- edge of f(u,x), h,eh(x), X andXe_q+1.

Assuming that the flags of co-distributions:

0=D₀⊂. . .⊂D_ν=D_ν+1

0=De₀/D_ν⊂. . .⊂De_eν/D_ν=De_eν+1/D_ν (7)

are of constant dimensions and that dim(D_ν) = q, dim(De_eν/D_ν) =n−q.

For k≥1, we define the bases B_k and Be_k of D_k/Dk−1 and e

D_k/De_k−1as follows:

B₁={[dh]}, Be₁={[deh]}

for k≥2 :

B_k={[d(Lf_uk−1. . .L_fu1(h))]; (u1, . . . ,u_k−1)∈U_k−1} e

B_k={[d(Lf_uk−1e . . .L_f

eu1(eh))]; (ue₁, . . . ,eu_k−1)∈Uf_k−1} (8) for some subsetsU_k−1 andUf_k−1 of(R^m)^k−1.

The symbol[(.)]stands for the equivalent class of(.).

Now, let B^∗_ν,Be^∗_eνbe the respective dual bases of B_ν andBe_eν (B^∗_ν,Be^∗_eν are bases of ∆_ν−1/∆_ν and∆e_eν−1/e∆_eν), the following vector fields will be required in theorem 2 below :

• The vector fields[Zu1...u_ν−1],[eZ_ue1...eu

eν−1]:

Let (u1, . . . ,u_ν−1), (resp. (ue₁, . . . ,ue_ν−1e )) be fixed elements ofU_ν−1 (resp. ofUf

eν−1), then[Y] = [Zu₁...u_ν−1](resp.[eY] = [eZ_ue1...ue

eν−1]) is the element of B^∗_ν (resp. of Be^∗_eν) defined by:

for(v1, . . . ,v_ν−1)∈U_ν−1, d(Lfvν−1. . .L_fv1(h))(Y) =1, if (u1, . . . ,u_ν−1) = (v1, . . . ,v_ν−1), and 0 otherwise for(ev₁, . . . ,ev_eν−1)∈Uf_ν−1e , d(Lf_ev

eν−1. . .L_fe

v1(eh))(Ye) =1, if (eu₁, . . . ,ue_eν−1) = (ve₁, . . . ,ve_eν−1), and 0 otherwise

(9)

• The vector fields[Y^u1...uν−1],[eY^{ue1...ueeν−1}]:

Setting [Y] = [Zu₁...u_ν−1]and[eY] = [eZ_ue1...ue

eν−1], then:

Y^u1...uν−1 = [fu_ν−1,[. . . ,[fu₁,Y]. . .]]]

e

Y^{ue1...ueeν−1} = [f_ue

eν−1,[. . . ,[f_ue1,Y]. . .]]] (10) In order to state lemma 2 below, the following notations will be required:

• Let(dϕ1, . . . ,dϕq)be a basis of D_ν and dϕq+1=deh.

• Settingπe=dϕ1∧. . .∧dϕq+1.

• LetXe= (Xe₁, . . . ,Xe_q+1)be a(q+1)-tuple of vector fields satisfying L_Xe

i(ϕj) =δi j.

• Foreu₁∈Uf₁, we set ωe_eu₁=dL_fe

u1(eh)∧πe^.

• For k ≥2 and (eu₁, . . . ,ue_k)∈Uf_k, we set ωeu_e1...eu_k = L_fe

uk(i_Xe(ωe_eu₁...eu_k−1))∧πe^. Thus we have:

Lemma 2:

For 1≤k≤eν−1; for every(ue₁, . . . ,eu_k)∈Uf_kthe following

properties hold:

ωeu_e1...uek=dL_f

uke . . .Lf_u1e (eh)∧πe +

k−1

∑

l=1

∑

(ue₁,...,ue_l)∈Uf_l

g_ue1...ue

l(x)dLf_eul. . .L_f

eu1(eh)∧πe ⁽¹¹⁾ i_Xe(ωeu_e1...euk) =dLf_euk. . .Lf_eu1(eh)−

∑

L_Xe

jLf_euk. . .Lf_u1e(eh)dϕj+Θk

Θk=Θek−

∑

k−1

∑

l=1

∑

(eu₁,...,eu_l)∈Uf_l

g_eu1...uel(x)L_Xe

jL_fe

ul. . .L_fe

u1(eh)dϕj

Θek=

k−1

∑

l=1

∑

(ue₁,...,ue_l)∈Uf_l

g_eu1...eul(x)dLf_eul. . .Lf_u1e(eh) +g_k(x)dϕq+1

(12) with the property that g_ue1...uel(.), gk(.) are C^∞-functions which do not depend on(eX₁, . . . ,Xe_q).

Proof of lemma 2.

• For k=1:

Let u₁∈U₁, by definition ωeu_e1 =dL_fe

u1(eh)∧πe, and from claim 3, we know that i_Xe(ωe_eu1) = dL_f

eu1(eh)−

q+1

∑

j=1

L_Xe

jL_fe

u1(eh)dϕj=dL_fe

u1(eh)−

∑

L_Xe

jL_fe

u1(eh)dϕj+Θ1, hereΘ1=L_Xe

q+1L_fe

u1(eh)dϕq+1. Hence (11), (12) are true for k=1.

• Assuming that (11), (12) hold for 1≤l≤k−1, and let us show them for k. Using the definition ofωe_eu1...uek and applying (12) for k−1, we get:

ωeu_e1...ue_k=dL_f

uke . . .Lf_eu1(eh)∧πe

−Lf_uke[

∑

L_Xe

jL_fe

uk−1. . .Lf_u1e (eh)dϕj]∧πe+Lf_euk(Θk−1)∧πe Θk−1=Θek−1−

∑

k−2

∑

l=1

∑

(eu₁,...,eu_l)∈Uf

l

g_ue1...eul(x)L_Xe

jL_f

eul. . . L_f

u1e (eh)dϕj

Θe_k−1=

k−2

∑

l=1

∑

(ue₁,...,ue_l)∈Uf_l

g_ue1...eul(x)dLf_eul. . .Lf_eu1(eh)

+ge_k−1(x)dϕq+1

(13) and g_ue1...eul, ge_k−1 do not depend on(eX₁, . . . ,Xe_q).

Using the fact that dϕi∈D_ν, for 1≤i≤q, and that L_fu(D_ν)⊂D_ν, then the following equality holds for every smooth functions a₁(x), . . . ,a_q(x):

L_fu(

∑

a_j(x)dϕj)∧πe=0 (14) Combining (14) with expressions of Θ_k−1, Θe_k−1, we get:

ωeu_e1...eu_k=dL_fe

uk. . .L_fe

u1(eh)∧πe+

k−2

∑

l=1

∑

(ue1,...,ue_l)∈Uf

l

L_fe

uk

[eg_ue1...eul(x)dLf_eul. . .L_f

eu1(eh) +ge_k−1(x)dϕq+1]∧πe (15)

By construction L_fe

uk[ge_ue1...uel(x)dLf_eul. . .L_fe

u1(eh)] and L_f

uke(eg_k−1(x)dϕq+1)∧πe do not depend on (Xe₁, . . . ,Xe_q) and{dϕ1, . . . ,dϕq+1} ∪{dLf_eul. . .Lf_u1e (eh); (ue₁, . . . ,eu_l)∈

f

U_l, 1≤l≤k−1} forms a basis of De_k, hence the last term of the right hand expression (15) takes the form

k−1

∑

l=1

∑

(ue1,...,uel)∈Uf_l

g_ue1...uel(x)dLf_eul. . .L_fe

u1(eh)∧πe^{, where} the g_ue1...ue

l(x)’s are C^∞-functions which do not depend on(Xe₁, . . . ,Xe_q). Consequently, expression (11) is satis- fied.

In order to end the proof of lemma 2, it remains only to check (12).

Applying claim 3 to expression (11), we get:

i_Xe(ωeu_e1...ue_k) =dL_f

euk. . .L_f

eu1(eh) +

k−1

∑

l=1

∑

(eu₁,...,eu_l)∈Uf

l

g_eu1...eul(x)

dL_f

eul. . .L_f

eu1(eh)−

q+1

∑

j=1

L_Xe

jL_f

uke . . .Lf_eu1(eh)dϕj

−

q+1

∑

j=1 k−1

∑

l=1

∑

(ue₁,...,ue_l)∈Uf_l

g_ue1...ue

l(x)L_Xe

jL_f

eul. . .L_f

eu1(eh)dϕj

(16) Finally, expression (12) follows from (16) in which we introduce:

Θek=

k−1

∑

l=1

∑

(eu₁,...,eu_l)∈Uf_l

g_ue1...eul(x)dL_fe

ul. . .L_fe

u1(eh) +g_k(x)dϕq+1

where g_k(x) =−L_Xe

q+1Lf_uke . . .Lf_eu1(eh)−

k−1

∑

l=1

∑

(ue1,...,uel)∈Uf_l

g_ue1...eul(x)L_Xe

q+1Lf_ule. . .Lf_eu1(eh) Θk=Θek−

∑

k−1

∑

l=1

∑

(ue₁,...,eu_l)∈Uf_l

g_ue1...eul(x)L_Xe

jL_feul. . .L_feu1(eh)dϕj

(17) Moreover, by construction g_eu1...uel(x) and g_k do not depend on (Xe₁, . . . ,Xeq). This ends the proof of lemma 2.

Now we can state the algorithm which allows to calculate vector fields X ,Xe₁, . . . ,Xe_q,Xe_q+1satisfying conditions 1) and 2) of theorem 1.

Theorem 2: (Algorithm)

System (3) can be steered by a local change of coordinates around some x⁰ to a cascade-observable system (1), if, and only if, the following conditions hold:

a) The flag of co-distributions D₀⊂. . .⊂D_ν =D_ν+1, De₀/D_ν⊂. . .⊂De_eν/D_ν=De_eν+1/D_ν are of constant di- mension on some neighborhood of x⁰, and dim(D_ν) =q, dim(De_eν/D♯) =n−q

b) Let B_ν and Be_eν be any fixed bases of D_ν/D_ν−1 and De_eν/De_ν−1e (see the construction (8)). Let Y and Y bee any fixed vector fields of the form[Y] = [Z_u0

1...u⁰_ν−1]∈B^∗_ν and[eY] = [eZ_eu0

1...eu⁰_eν−1]∈Be^∗_eν, then the following properties hold:

1) The vector X= (−1)^ν−1Y^{u01...u0ν−1} satisfies condition 1) of theorem 1.

2) Setting Xe_q+1 = (−1)^eν−1Ye^{eu01...eu0eν−1} and considering C^∞-functionsϕ1, . . . ,ϕq+1such thatϕ1=h,ϕq+1=eh and that (dϕ1, . . . ,dϕq) forms a basis of iX(Ω^X) + Rdh. Let Xe₁, . . . ,Xe_q be vector fields satisfying L_Xe

j(ϕi) =δi j, 1≤j≤q, 1≤i≤q+1, and such that for every(ue₁, . . . ,eu_k)∈Uf_k, 1≤k≤eν−1, we have:

∑

d(L_Xe

jL_f

uke . . .Lf_u1e (eh))∧dϕj=dΘ_ue1...uek (18) where Θ_ue1...euk is the one-differential form stated in (12). ThenXe₁, . . . ,Xe_q+1 satisfy condition 2) of theo- rem 1.

Remark 1: According to expression (12) of lemma 2, expression (18) is then equivalent to d(i_Xe(ωeu_e1...ue_k)) =0.

Some comments on the procedure of calculation of vector fields X , Xe₁, . . . ,Xe_q,Xe_q+1:

1) The calculation of the vector field X requires only the knowledge of expressions of fuand h.

2) Xe_q+1can be directly computed from the knowledge of X , f_u, h andeh.

3) For 1≤i≤q+1, the functionsϕi can be deduced from X , f_uand h andeh.

4) Finally, we end these comments by giving the algorithm of computation of(Xe₁, . . . ,Xeq):

Computation of(Xe₁, . . . ,Xeq):

Based on the construction of Be_k and the functions ϕ1, . . . ,ϕq+1, the set {ϕ1, . . . ,ϕq+1} ∪ {Lf_euk. . .Lf_eu1(eh); 1≤

k≤ eν−1, (ue₁, . . . ,ue_k) ∈ Uf_k} forms a local system of

coordinates, which we denote by(ξ,ξe), and where ξ= (ϕ1, . . . ,ϕq+1) = (ξ1, . . . ,ξq+1) ξe= (ξe1, . . . ,ξe_eν−1), ξek= (ξek1, . . . ,ξe_k,de

k) where {dξek1, . . . ,dξe_k,de

k}={dLf_euk. . .L_f

eu1(eh); (ue₁, . . . ,eu_k)∈ f

U_k}, and{[dLf_euk. . .Lf_u1e(eh)]; (ue₁, . . . ,ue_k)∈Uf_k}=Be_k+1. Therefore, we adopt the following notations:

ωe_eu₁...eu_k=ωeki=dξeki

L_Xe

j(ξeki) =L_Xe

jL_f

euk. . .L_f

eu1(eh) =Xe_ki^j Using the fact that L_Xe

j(ϕj) = L_Xe

j(ξj) =δi j, we obtain e

X_j= ∂

∂ξj

+

eν−1 k=1

∑

dek

i=1

∑

e X_ki^j ∂

∂ξ^eki

. Thus, the expression (12) can be rewritten:

i_e

X(ωeki) =dξeki−

∑

Xe_ki^jdξj+Θki (19) where theΘki’s are one-differential forms depending at most onXe_li^j, 1≤l≤k−1, 1≤j≤q+1.

The calculation ofXe_ki^j’s follows from the following recursive procedure: