C Califano

L A Marquez-Martinez

C H Moog

On the observer canonical form for Nonlinear Time-Delay Systems

Keywords: Nonlinear time-delay systems, differential geometry, observer canonical form, output injection

Necessary and sufficient geometric conditions for the equivalence of a nonlinear timedelay system with one output, under bicausal change of coordinates and output transformation, to a linear weakly observable time-delay system up to output injection are given. These conditions are derived through the use of the Extended Lie Bracket operator recently introduced in the literature for dealing with time-delay systems. The results presented show how this operator is useful in the analysis of this class of nonlinear systems.

INTRODUCTION

Conditions are derived under which a given nonlinear time-delay system is equivalent under bicausal change of coordinates and output transformation and up to output injection, to a linear weakly observable time-delay system. As it is the case for delay-free nonlinear systems, such an equivalence has the immediate benefit to transpose the problem of the reconstruction of the state of a nonlinear system, to a linear problem. In fact, it is possible to define an observer candidate such that the error between the state and its estimate is solution of a linear time-delay differential equation. Thus, the stability results known for such linear time-delay systems will be the tools to design an asymptotic observer.

In the delay-free case, this problem has been widely studied both for continuous-time and discrete-time systems (see for example Besancon et al. (1998[START_REF] Califano | Canonical observer forms for multi-output systems up to coordinate and output transformations in discrete time[END_REF], [START_REF] Glumineau | New algebrogeometric conditions for the linearization by inputoutput injection[END_REF], [START_REF] Hammouri | A new procedure for timevarying linearization up to output injection[END_REF] ÷ Huijberts (1999), [START_REF] Monaco | Linearization by Output Injection under Approximate Sampling[END_REF], [START_REF] Xia | Nonlinear observer design by observer error linearization[END_REF], and the references therein). For time-delay systems the problem of the design of an observer has been addressed through several approaches. More precisely, in [START_REF] Marquez-Martinez | Observability and observers for nonlinear systems with time delay[END_REF], sufficient conditions for the equivalence to the observer canonical form up to output injection are given through an algebraic approach while in [START_REF] Marquez-Martinez | InputOutput Feedback Linearization of Time-Delay Systems[END_REF] necessary and sufficient existence conditions are given still through the use of one forms; in [START_REF] Germani | A new approach to state observation of Nonlinear Systems with delayed output[END_REF], the problem is solved through a chain observer by admitting delays on the output function only; finally in [START_REF] Anguelova | On analytic and algebraic observability of nonlinear delay systems[END_REF] a new approach for the definition of the conditions under which a weakly observable system admits a unique state corresponding to a given input-output behavior is proposed.

In the present paper the analysis is performed following the geometric approach proposed in Califano et al. (2010) with respect to the linear equivalence problem and used in Califano et al. (2010b) for the analysis of time-delay systems. It is shown that such a geometric framework allows to define a constructive procedure for the computation of the desired bicausal change of coordinates. The results obtained not only may be generalized to the multi-output case as done in [START_REF] Califano | Canonical observer forms for multi-output systems up to coordinate and output transformations in discrete time[END_REF] with reference to nonlinear discrete time systems, but enlighten the importance of the Extended Lie bracket operator for addressing control problems.

The paper is organized as follows. Some technical arguments concerning the geometric framework are given in Section 2. In Section 3 the main results are presented. An example displays the technical computations.

PRELIMINARIES

The class of nonlinear systems is considered, where the state and control variables are affected by possible multiple constant commensurate delays. The given dynamics can be modeled, without loss of generality through the equations Σ :

     ẋ(t) = F (x [s] ) + s j=0 G j (x [s] )u(t -j) y = H(x [s] ) (1) where x [s] = (x(t), • • •x(t -s)), with x ∈ IR n , u ∈ IR, y ∈ IR.
Throughout the paper we will denote by x [s] (-p) = (x(t-p), • • •x(t-s-p)). When no confusion is possible the subindex will be omitted so that x and x(-p) will stand for x [s] and x [s] (-p) respectively. Furthermore, K denotes the field of meromorphic functions of a finite number of symbols in {x(t -i), u(t -i), u(ti), . . . , u (k) (t -i), i, k ∈ IN }; E is the vector space spanned by the symbols {dx(t -i), du(t -i), d u(t -i), . . . , du (k) (ti), i, k ∈ IN } over K. The elements of this space are called 1-forms; d is the standard differential operator that maps elements from

K to E := span K {dx(t -i), du(t - i), d u(t -i), . . . , du (k) (t -i), i, k ∈ IN }; δ represents the backward time-shift operator, that is for a(•), f(•) ∈ K: δ[ a(t) df (t)] = a(t -1)δdf(t) = a(t -1)df (t -1); K(δ]
is the (left) ring of polynomials in δ with coefficients in K. Every element of K(δ] may be written as

α(δ] = α 0 (t) + α 1 (t) δ + • • • + α rα (t) δ rα , α i ∈ K,
where r α = deg(α(δ]) the polynomial degree in δ of its argument. Addition and multiplication on this ring are defined by α [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over non-commutative rings[END_REF]; ∆(δ] = span K(δ] {r 1 , . . . , r s }, is the right module spanned over K(δ] by the column elements r 1 , . . . , r s ∈ K n (δ].

(δ] + β(δ] = max{rα, rβ } i=0 (α i (t) + β i (t))δ i and α(δ]β(δ] = rα i=0 rβ j=0 α i (t) β j (t -i)δ i+j ; this ring is a non-commutative, Euclidean ring,
Σ L , the differential form representation of Σ, is given by

Σ L : d ẋ = f(x [s] , u [s] , δ)dx + g(x [s] , δ)du dy = h(x [s] , δ)dx (2) with f(x [s] , u [s] , δ) = s i=0 ∂F (x [s] ) ∂x(t -i) δ i + s j=0 u(t -j) s i=0 ∂G j (x [s] ) ∂x(t -i) δ i g(x [s] , δ) = s j=0 G j (x [s] )δ j , h(x [s] , δ) = s i=0 ∂H(x [s] ) ∂x(t -i) δ i .
In this context we will thus consider bicausal change of coordinates as defined in [START_REF] Marquez-Martinez | Observability and observers for nonlinear systems with time delay[END_REF].

We have Definition 1. Consider the dynamics Σ with state coor-

dinates x. z = φ(x [α] ), φ ∈ K n is a bicausal change of coordinates for Σ if there exist an integer ∈ IN and a function φ -1 (z [ ] ) ∈ K n such that x(t) = φ -1 (z [ ] ).
Denoting by dz = T x [γ] , δ dx its associated differential form representation, one has that the following properties hold true:

P1) T x [γ] , δ = α i=0 ∂φ(x [α] ) ∂x(t -i) δ i = α i=0 T i (x [γ] )δ i , with γ ≤ α is unimodular 1 1 A polynomial matrix T (x, δ) is called unimodular if it has a polynomial inverse.
P2) The inverse T -1 (z, δ) has polynomial degree ≤ (n-1) α and is given by T

-1 (z, δ) = i=0 ∂φ -1 (z [ ] ) ∂z(t -i) δ i . Under the bicausal change of coordinates z(t) = φ(x [α] ), (2) is transformed into d ż(t) = f(z, u, δ)dz + g(z, δ)du (3) dy = h(z, δ)dz with f (z, u, δ) = = T (x, δ)f(x, u, δ)+ Ṫ (x, δ) T -1 (x, δ) |x=φ -1 (z) (4) g(z, δ) = (T (x, δ)g(x, δ)) |x=φ -1 (z) , (5) h(z, δ) = h(x, δ)T -1 (x, δ) |x=φ -1 (z) . (6)

The Extended Lie bracket and its properties

The Extended Lie Bracket has been recently introduced for dealing with time-delay systems in Califano et al. (2010) where the linear equivalence problem has been addressed and solved. Let

r i (x, δ) = s j=0 r j i (x)δ j , and set r s+j i (x) = 0
for any j > 0. We have:

Definition 2. Let r i (x, δ) = s j=0 r j i (x)δ j , i = i, 2. For any k, l ≥ 0, the Extended Lie bracket [r k 1 (•), r l 2 (•, )] Ei , on R (i+1)n , is recursively defined as [r k 1 (•), r l 2 (•)] E0 = (7) k i=0 ∂r l 2 (x) ∂x(t -i) r k-i 1 (x(-i)) - l i=0 ∂r k 1 (x) ∂x(t -i) r l-i 2 (x(-i)). r k 1 (•), r l 2 (•) Ei = (8) min(k,l,i) j=0 [r k-j 1 (•), r l-j 2 (•)] E0 T |x(-j) ∂ ∂x(t -j) .
As in the delay free case it is convenient to introduce an Extended Lie derivative whose definition is given below and is slightly different from the one given in [START_REF] Oguchi | A finite spectrum assignment for retarded nonlinear systems and its solvability condition[END_REF]. Definition 3. Given the function τ (x [s] ) and the submod-

ule element r i (x, δ) = s j=0 r j i (x)δ j , the Extended Lie derivative L r j i (x) τ (x [s] ) L r j i (x) τ (x [s] ) = j l=0 ∂τ (x [s] ) ∂x(t -l) r j-l i (x(-l)) (9) Accordingly r k 1 (•), r l 2 (•) Ei = = min(k,l,i) j=0 L r k-j 1 (x) r l-j 2 (x)-L r l-j 2 (x) r k-j 1 (x) T |x(-j) ∂ ∂x(t -j) .
thus recovering the definitions of Lie derivative and Lie bracket in the delay free case.

From (1), consider now the module element

F(x, δ) = ns j=0 F j (x)δ j = ns j=0 F (x)δ j .
Thus, the i-th derivative of τ (x) computed for u = 0, is given by τ

(i) (x, 0) = L i F ns (x) τ (x)
, for any i. It follows that the observability rank condition can be easily checked by considering the matrix O(x, δ) defined by

O(x, δ)dx =      dH(x) dL F ns (x) H(x)
. . .

dL n-1 F ns (x) H(x)      (10)
We have the following notions of observability. Definition 4. System (1) is weakly observable locally around u = 0 if O(x, δ) computed according to (10) has full rank n locally around x 0 , is strongly observable if O(x, δ) is unimodular.

We can now state the definition of nilpotent submodule, which generalizes that of nilpotent distribution and which is a key tool in many control problems. Definition 5. A submodule ∆ = span{r 1 (x, δ), • • • , r j (x, δ)} nonsingular locally around x 0 , with r l (x, δ) =

s k=0 r k l (x)δ k is nilpotent of order 0 if [r k l , r p i ] E,2s = 0, ∀k ≤ p ∈ [0, 2s]. Definition 6. Consider the bicausal change of coordinates z = φ(x [α]
), with dz = T (x, δ)dx. In the new coordinates the submodule element r(x, u, δ) is transformed as

r(z, u, δ) = [T (x, δ)r(x, u, δ)] |x=φ -1 (z) . ( 11
)
From ( 11), setting T j = 0 for j > α = deg(T (x, δ)) and

r j = 0 for j > deg(r(x, u, δ)) one has rl (z, u) = l p=0 T p (x)r l-p (x(-p), u(-p)) |x=φ -1 (z) . ( 12
)
As a consequence one has the following result concerning the action of a bicausal change of coordinates on the extended Lie bracket (Califano et al. (2010)).

Lemma 1. Let r i (x, u, δ) = s j=0 r j i (x, u)δ j , i = 1, 2. Under the bicausal change of coordinates z(t) = φ(x [α] ), charac- terized by dz = T (x, δ)dx, T (x, δ) = α j=0 T j (x)δ j one has, for k ≤ l, [r k 1 (z, u), rl 2 (z, u)] Ek = = Γ l-k (x)[r k 1 (x, u), r l 2 (x, u)] Ek |x=φ -1 (z)
where, setting T j = 0 for j > α,

Γ l-k (x) =    T 0 (x) • • • T l-k (x) . . . 0 T 0 (x(-l + k))    .
Next theorem (Califano et al. (2010b)), enlightens the conditions under which a set of n one-forms are exact and can be used to define a bicausal change of coordinates.

Theorem 1. Let P n (x, δ) = [r 1 (x, δ), • • • , r n (x, δ)] be a full rank matrix, with r i = s j=0 r j i (x [β]
)δ j which can be factorized as

P n (x, δ) = T (x, δ)Q(δ), with T (x [β] , δ) = s i=0 T i (x [β] )δ i unimodular and such that β ≤ s. Then T -1 (x, δ) defines a bicausal change of coordinates if and only if ∀l, j ∈ [1, n]; and ∀i ≤ k ∈ [0, 2s] [r i j (x), r k l (x)] Ei = 0, (13)
or equivalently ∀l, j ∈ [1, n]; and ∀i ≤ k ∈ [0, 2s]

[r i j (x), r k l (x)] E0 = 0. ( 14)

MAIN RESULTS

One major drawback encountered when dealing with nonlinear delay systems, is that they are infinite dimensional systems, so that the direct application of Frobenius Theorem is not possible. It is shown hereafter that thanks to the mathematical preliminaries in Section 2, it is possible to fully characterize those nonlinear time-delay systems which are equivalent under bicausal change of coordinates to the weakly observable linear canonical form up to nonlinear output injections. The constructive proof allows to compute the required change of coordinates.

Problem Statement (The Problem of the Equivalence to the Observer Form up to output injection): Given system (1) find, if possible, a bicausal change of coordinates z = φ(x [α] ), such that in the new coordinates the given system is weakly observable and reads

ż(t) = s j=0 A j z(t -j) + ψ(y [s] , u [s] , ) (15) y = s j=0 C j z(t -j)
To solve the problem, search for a submodule element

r 1 (x, δ) solution of O(x, δ)r 1 (x, δ) = (0 • • • 0 a(x, δ)) T (16)
with a(x, δ) of minimal degree. If such an element exists, we can also compute iteratively, for i = 2,

• • • n + 1 r i (x, u, δ) = f(x, u, δ)r i-1 (x, u, δ) -ṙi-1 (x, u, δ). (17)
Immediate consequences are the following: Proposition 1. If r 1 (x, δ) satisfying ( 16) exists, then the observability matrix has full rank loc. around u = 0, x 0 .

The proof is omitted for reasons of space and is based on the consideration that the matrix O(x, δ) satisfies

O(x, δ) (r 1 (x, 0, δ) • • •r n (x, 0, δ)) =    0 • • • a(x, δ) . . . • • • * a(x, δ) * *   
thus proving the result. Proposition 2. Under any bicausal change of coordinates z = φ(x [α] ), with dz = T (x, δ)dx, the submodule elements r i (x, u, δ) for i ≥ 1 defined by ( 16), ( 17) are transformed according to (11).

Proof. We first have to show that ( 11) is satisfied for i = 1. In fact we have that in the new coordinates H(z) =

H (φ(x)) and F (z) = α i=0 ∂φ(x) ∂x(t -i) F (x(-i)) | x=φ -1 (z)
. Iteratively, assume that L i-1

F ns (z) H(z) = L i-1 F ns (x) H(x) | x=φ -1 (z)
which is true for i = 1 then we have that,

L i F ns (z) H(z) = L F ( ns z) L i-1 F ns (x) H(x) | x=φ -1 (z) = ns j=0 ns l=0 ∂ L i-1 F ns (x) H(x) ∂x(t -l) x=φ -1 (z) ∂φ -1 (z(-l)) ∂z(t -j) F (z(-j)) = ns l=0 ∂ L i-1 F (x) H(x) ∂x(t -l) F (x(-l)) x=φ -1 (z) = L i F ns (x) H(x) | x=φ -1 (z)
.

It follows that O(x, δ)r 1 (x, δ) = Õ(z, δ) z=φ(x) T (x, δ)r 1 (x, δ) = (0 • • • 0 a(x, δ)) T which implies that r1 (z, δ) = [T (x, δ)r 1 (x, δ)] x=φ -1 (z)
, due to the bicausality of the change of coordinates. Assume now that ( 11) is true for i -1. According to (4) we have

ri (z, u, δ) = f(z, u, δ)r i-1 (z, u, δ) -ṙi-1 (z, u, δ) = [T (x, δ)r i (x, u, δ)] x=φ -1 (z)
which ends the proof.

Linear equivalence under bicausal change of coordinates up to output injection

In the present section necessary and sufficient conditions for the equivalence to the weakly observable canonical form up to output injection under bicausal change of coordinates are given. The solution uses the submodule elements r i (x, u, δ), i ∈ [1, n], computed according to ( 16) and ( 17). We have: Theorem 2. System (1) is equivalent, under a bicausal change of coordinates, to a linear weakly observable delay system up to output injection if and only if there exist r 1 (x, δ) solution of ( 16) with a(x, δ) = a(δ) and two matrices T -1 (x, δ) unimodular, and

Q(δ) of full rank such that a) O n (x, u) = (r 1 (x, δ), • • • r n (x, u, δ)) = T -1 (x, δ)Q(δ) b) for i, j ∈ [1, n] and l ≤ k ∈ [0, 2s], the following relation are satisfied [r l i (x), r k j (x)] E0 = 0. c) r n+1 (x, u, δ) = T -1 (x, δ)q 2 (y, u, δ)
Proof. Necessity. First consider the time-delay system (15) which is already in the canonical observer form up to output injection. Is is easily verified that for this system r 1 (x, δ) = r 1 (δ) solution of ( 16) with a(x, δ) = a(δ) exists and conditions a)÷ c) are satisfied since by construction for i ∈ [1, n], r i (•, δ) = r i (δ) and according to Proposition 1 they are linearly independent. Furthermore

r n+1 = f(z, u, δ)r n (δ) = A(δ)r n (δ) + η(y, u, δ) = q 2 (y, u, δ). ( 18
)
From Proposition 2 and Lemma 1, conditions a)÷b) are invariant under a bicausal change of coordinates. Condition c) is an immediate consequence of (18) and Proposition 2.

Sufficiency. Assume that the conditions are satisfied. According to Theorem 1, since T -1 (x, δ) is unimodular and b) is satisfied, we can consider the bicausal change of coordinates z = φ(x [α] ) such that dz = T (x, δ)dx. According to a) the r i (x, u, δ)'s do not depend on the control variable and thus in the new coordinates, due to Proposition 2, (r

1 (z, δ), • • • rn (z, δ)) = = [T (x, δ)(r 1 (x, δ), • • •r n (x, δ))] x=φ -1 (z) = Q(δ) It follows that the output H(z [s] ) is such that s i=0 ∂H ∂z(t -i) δ i (r 1 (z, δ), • • • rn (z, δ)) = h(z, δ)(r 1 (z, δ), • • • rn (z, δ)) = h(x, δ)T -1 (x, δ) x=φ -1 (z) (r 1 (z, δ), • • • rn (z, δ)) = (0 0 • • • 0 a(δ))
Multiplying on the right by adj[Q(δ)], one gets

h(z, δ)[det(Q(δ)] = a(δ) (q n1 qn2 • • • qnn ) = C(δ)[det(Q(δ)] (19)
where (q n1 qn2 • • • qnn ) denotes the last row of the matrix adj [Q(δ)]. Relation (19) shows in fact the linearity of the output function in the new coordinates, due to the identity of polynomials. As for the forced dynamics, similarly, we have that due to c),

f (z, u, δ)(r 1 (δ), • • • rn (δ)) = (r 2 (z, δ), • • • , rn (z, δ), rn+1 (z, u, δ)) = [Q 1 (δ), q 2 (y, u, δ)].
Multiplying again on the right by adj[Q(δ)]a(δ), one gets

f(z, u, δ)[det(Q(δ)]a(δ) = = Q 2 (δ) + q 2 (y, u, δ) (q n1 qn2 • • • qnn ) a(δ) = A(δ)[det(Q(δ)]a(δ) + q2 (y, u, δ)C(δ)[det(Q(δ)]a(δ)
that is, the system in the z-coordinates reads (15

) since f (z, u, δ) = A(δ) + q2 (y, u, δ)C(δ).
An immediate consequence is the following. Corollary 1. System (1) is equivalent, under a bicausal change of coordinates, to a linear weakly observable delay system, if and only if there exist two matrices T -1 (x, δ) unimodular, and Q(δ) of full rank such that conditions a)÷ c) of Theorem 2 are satisfied and d')

r n+1 (x, u, δ) ∈ span K(δ] {r 1 (x, δ), • • • , r n (x, δ)}
Remark. Under appropriate assumptions, necessary and sufficient existence conditions equivalent to Theorem 2 were given in [START_REF] Marquez-Martinez | InputOutput Feedback Linearization of Time-Delay Systems[END_REF]. However the conditions given in Theorem 2 are constructive, since the module elements r i (x, δ) characterize the change of coordinates. In particular if the system is strongly observable, then O n (x, δ) is exactly the inverse of the bicausal change of coordinates differential form representation.

An Example

Consider the following nonlinear delay system

ẋ1 (t) = x 2 (t -1) + 2x 2 (t) + [x 1 (t) -x 2 2 (t -1)] 2 +2x 2 (t -1)([x 1 (t -2) -x 2 2 (t -3)] 2 + u(t -1)) ẋ2 (t) = [x 1 (t -1) -x 2 2 (t -2)) 2 + u(t) y(t) = x 1 (t) -x 2 2 (t -1) We have that L F H(x) = x 2 (t -1) + 2x 2 (t) + [x 1 (t) - x 2 2 (t -1)] 2 , so that, setting c(x) = 2(x 1 (t) -x 2 2 (t -1)), the observability matrix is O = 1 -2x 2 (t -1)δ c(x) 2 + δ -c(x)2x 2 (t -1)δ)
which is of full rank but not unimodular. Let us thus compute, if it exists, r 1 (x, δ) solution of ( 16), that is s.t.

1

-2x 2 (t -1)δ c(x) 2 + δ -c(x)2x 2 (t -1)δ) r 1 (x, δ) = 0 a(δ)

Standard computations show that r 1 (x, δ) = 2x 2 (t -1)δ 1 = 0 1 + 2x 2 (t -1) 0 δ
To check the conditions of Theorem 2, we have now to compute r 2 (x, u, δ), and r 3 (x, u, δ). We have that O n = (r 1 (x, δ), r 2 (x, δ)) = 2x 2 (t -1)δ δ + 2 1 0 = 2x 2 (t -1)δ 1 1 0 1 0 0 δ + 2 so that it is easily verified that they are satisfied with T -1 (x, δ) = 2x 2 (t -1)δ 1 1 0 , Q(δ) = 1 0 0 δ + 2 , and q 2 (y, u, δ) = 2y(t -1)δ 2y(t) .

Finally we have to test condition c) that is the nilpotency of ∆ = span K(δ] {r 1 (x, δ), r 2 (x, δ)}, where the maximum delay s = 1. The nilpotency can be easily verified by referring to the following matrix constructed starting form r 1 (x, δ) and r 2 (x, δ)

         R 0 1 R 0 2 R 1 1 R 1 2 R 2 1 R 2 2 0 2 2x 2 (t -1) 1 0 1 0 0 0 0 0 2 2x 2 (t -2) 1 1 0 0 0 0 0 0 2 0 0 1 0          which shows that [r 0 1 , r 0 2 ] E0 = [r 0 1 , r 1 1 ] E0 = [r 0 1 , r 1 2 ] E0 = [r 0 1 , r 2 1 ] E0 = [r 0 1 , r 2 2 ] E0 = [r 1 1 , r 2
1 ] E0 = 0 as well as [r 0 2 , r 2 2 ] E0 = [r 1 2 , r 2 2 ] E0 = [r 2 1 , r 2 2 ] E0 = 0. It follows that the bicausal change of coordinates z 1 (t) = x 2 (t), z 2 (t) = x 1 (t) -x 2 2 (t -1) transforms the given dynamics into

  of Theorem 2 is thus satisfied. As for condition b) and d) we readily have that


D Maalouf

C H Moog

Y Aoustin

S J Li

Maximum feedback linearization with internal stability of 2-DOF underactuated mechanical systems

Keywords: Stability, Equilibrium, Nonlinear control systems, Mechanical systems, Feedback linearization

Maximum feedback linearization with internal stability is considered. A classification of popular mechanical systems is derived, depending on the achievable closed loop system under the constraint of internal stability. It is shown that the angular momentum plays a crucial role for this class of systems. The study includes the special case of a paraglider which consists essentially of two articulated bodies.

INTRODUCTION

Exact feedback linearization lies in the heart of the success of modern nonlinear control theory for over three decades. Input-output linearization generalizes the socalled computed torque method which has been popularized in robotics. However, some major drawbacks are mentioned from time to time, amongst the existence of singularities, the cancelation of useful nonlinear terms [START_REF] Ortega | Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment[END_REF]) or the nonobservability of the socalled zero dynamics which may destabilize the closed loop system whenever the system is non minimum phase.

Herein, the problem of exact linearization with internal stability is addressed. The core consists in the design of a dummy output, whenever it exists, such that the underlying system becomes minimum phase. This goal is of major importance by its own as it is a standing assumption for many control schemes including sliding modes or PID feedback... A general theory for this issue is not available yet. The goal in this paper is to consider a class of underactuated systems whose stabilization is a challenging issue. A general frame is displayed to design a suitable control law for three subclasses of systems according to their structure. One class corresponds to fully linearizable (or flat) systems. It includes the inertia wheel pendulum [START_REF] Ortega | Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment[END_REF], [START_REF] Beznos | Formal'sky. A pendulum controlled by a flywheel[END_REF]). The second class includes the celebrated Acrobot which has already been worked out by [START_REF] Cambrini | Stable trajectory tracking for biped robots[END_REF] and shown to be linearizable up to order 3 with stable, critically stable or unstable internal dynamics, depending on the choice of output coordinates. The third class is much more challenging and contains systems as the Pendubot [START_REF] Aoustin | Pendubot: Combining of energy and intuitive approaches to swing up, stabilization in erected pose[END_REF]) and a model of paraglider on which a special attention is focused. Such a classification of simple mechanical systems was considered for different purposes in [START_REF] Olfati-Saber | Nonlinear Control of underactuated mechanical systems with application to robotics and aerospace vehicles[END_REF].

The paraglider is considered to be equivalent to a double pendulum. The control input is the thrust force applied at the joint between both links. It is proven that the system is linearizable up to order 3, with a critical internal stability. The main contribution in that respect is to find a suitable output function with relative degree 2 such that the feedback linearization yields asymptotic stability of the full state. The importance of the generalized angular momentum in the stability of the system is argued and the physical interpretation of the functions obtained from the control algorithms is discussed as well.

In Section 2, some standard results are recalled from [START_REF] Marino | On the largest feedback linearizable subsystem[END_REF], [START_REF] Conte | Algebraic Methods for Nonlinear Control Systems[END_REF] on input-output linearization. The main results are given in Section 3. A general underactuated double pendulum is modeled, so that the Acrobot and the Pendubot become some special cases. Three subclasses of systems are identified and characterized. Section 4 is devoted to the paraglider system for which worked out computations are completed, including some simulation results in Section 5 which make profit of the stability results. Concluding remarks and perspectives a r eo ff e r e di nS e c t i o n6 .

MAXIMUM FEEDBACK LINEARIZATION WITH STABILITY

Consider a single-input nonlinear system

Λ: ẋ = f (x)+g(x)u (2.1)
where the state x ∈ R n ,t h ec o n t r o lu ∈ R and the entries f , g are meromorphic vector fields on R n .L e tC be the infinite set of real indeterminates given by C = {x, u k ; k ≥ 0} and denote by K, a field of meromorphic functions depending on a finite subset of indeterminates of C.W e define the time derivative of a function ϕ ∈Kas follows k+1) .

φ = dϕ dt = ∂ϕ ∂x (f (x)+g(x)u)+ k≥0 ∂ϕ ∂u (k) u (
Let E denote the vector space spanned over K by the elements of dC,n a m e l yE =span K {dx, du (k) ; k ≥ 0}. Any element of E is in the form

ω = a dx + k≥0 b k du (k)
which is called a differential one-form and its time derivative is defined by

ω =(ȧ dx + a dẋ)+ k≥0 ( ḃk du (k) + b k du (k+1) ).
The relative degree of a one-form ω is defined as the least integer r such that ω (r) ∈ span K {dx}.I fi td oe sn ote xi st , we say that r = ∞. Similarly, the relative degree of a function ϕ ∈Kis defined as the least integer such that dϕ (r) ∈ span K {dx}. We have the following result (see [START_REF] Conte | Algebraic Methods for Nonlinear Control Systems[END_REF]) Proposition 2.1. The function ϕ ∈Kand the one-form dϕ have the same relative degree.

Introduce a sequence of subspace

{H k } of E by H 0 =s p a n K {dx, du} H k+1 = {ω ∈H k | ω ∈H k },k≥ 1.
(2.2)

Each subspace H k denotes the set of all one-forms with relative degree at least k. Clearly, sequence (2.2) is decreasing, i.e., E⊃H 0 ⊃H 1 ⊃H 2 •••, and we have

H 1 =span K {dx}.N o t et h a tH 2 =s p a n K {g} ⊥ . Denote k * the least integer such that H 0 ⊃H 1 ⊃ •••H k * ⊃H k * +1 = H k * +2 = •••H ∞ .
T h e ns y s t e mΛ is strongly accessible if and only if it satisfies H ∞ =0 (see [START_REF] Conte | Algebraic Methods for Nonlinear Control Systems[END_REF]). In this paper, all the mechanical systems with which we work satisfy H ∞ =0.

Let y = h(x) ∈ R be an output of Λ, where h is a meromorphic function, and consider the corresponding SISO (single-input single-output) nonlinear system

Σ: ẋ = f (x)+g(x)u y = h(x). (2.3)
Assume that Σ has a relative degree r ≤ n at some point x 0 , i.e., the relative degree of the output function y = h(x)e q u a lr, then locally there exist a regular static state feedback u = α(x)+β(x)v and a state transformation (z, w)=φ(x), where z =( z 1 ,...,z r ), w =( w 1 ,...,w n-r )a n dφ is a diffeomorphism, such that in the (z, w)-coordinates, system Σ reads, around

x 0 , ż1 = z 2 ż2 = z 3 . . . żr = v ẇ = η(z, w) y = z 1 .
(2.4)

The system description (2.4) is called the normal form for the SISO system Σ. Consequently, it can be easily seen that Σ is fully state linearizable if and only if there exists an output function y = h(x) that has relative degree n at x 0 . Clearly the variables w are unobservable since z do not depend on w at all and the equation ẇ = η(z, w) represents the "internal dynamics".

Definition 2.2. The zero dynamics of system Σ, given by (2.3), is defined by the dynamics ẇ = η(0,w) which are the internal dynamics consistent with the constraint that y(t) ≡ 0.

In this paper, we will study the maximal feedback linearization with internal stability of the 2 DOF underactuated mechanical system. More precisely, given a single-input nonlinear system Λ : ẋ = f (x)+g(x)u,w ew a n tt o find an output y = h(x) ∈ R with the maximal relative degree r and having the following property: there exist a regular static state feedback u = α(x)+β(x)v and a state transformation (z, w)=φ(x), where z =(z 1 ,...,z r ), w =( w 1 ,...,w n-r )a n dφ is a diffeomorphism, such that in the (z, w)-coordinates the corresponding SISO system Σ can be transformed into the normal form (2.4) and simultaneously the zero dynamics is (asymptotically) stable.

A GENERAL CLASS OF 2-DOF SYSTEMS

Consider the double link pendulum shown in Figure 1. It is assumed to be attached to a fixed pivot point

O 1 F G 2 G 1 O 1 O 2 C 1 C 2 O 3 q 1 q 2
Fig. 1. Scheme of a double link pendulum. creating angle q 1 .Angleq 2 is the joint angle between links. Let C 1 and C 2 be the centers of mass of the first and second link respectively. The center of mass

C 1 is located on line O 1 O 2 . Let the following lengths be O 1 O 2 = l 1 , O 2 O 3 = l 2 , O 1 C 1 = r 1 and O 2 C 2 = r 2
.L e tm 1 and m 2 be the masses of the first and second links. The moment of inertia of the first link about joint O 1 is denoted I 1 , the moment of inertia of the second link about joint O 2 is denoted I 2 . At the tip of the second link O 2 an external force

-→ F =[F x ,F y ] t is applied.
The expressions for the kinetic energy T and the potential energy Π of the two-link pendulum are well known:

2T = a 11 q2 1 +2a 21 cosq 2 q2 1 + a 22 q2 1 + (a 22 + a 21 cosq 2 )q 1 q2 + a 22 q2 2 Π=-b 1 cosq 1 -b 2 cos(q 1 + q 2 ), with a 11 = I 1 +m 2 l 2 1 , a 21 = m 2 r 2 l 1 , a 22 = I 2 , b 1 =(m 1 r 1 + m 2 l 1 )g, b 2 = m 2
r 2 g where g is the gravity acceleration.

Lagrangian L = T -Π yields the following well known matrix equation of motion: D(q)q + C(q, q)q + G(q)=J t F (3.1) where q =[q 1 ,q 2 ] t D(q)= a 11 + a 22 +2a 21 cosq 2 a 22 + a 21 cosq 2 a 22 + a 21 cosq 2 a 22 , C(q, q)= -a 21 q2 sinq 2 -a 21 (q 1 +q 2 )sinq 2 a 21 q1 sinq 2 0 ,

G(q)= b 1 sinq 1 + b 2 sin(q 1 + q 2 ) b 2 sin(q 1 + q 2 ) .
The matrix J is the Jacobian of the forward kinematics.

The arbitrarily directed force makes a torque at each joint such that:

Γ= Γ 1 Γ 2 = J t F = l 2 cos(q 1 + q 2 )+l 1 cosq 1 l 2 sin(q 1 + q 2 )+l 1 sinq 1 l 2 cos(q 1 + q 2 ) l 2 sin(q 1 + q 2 ) F x F y .
A double link pendulum can define a family of underactuated mechanical systems categorized according to the orientation of an external force -→ F acting on this pendulum. Consider for example the end force

-→ F .W h e n -→
F is parallel to the segment O 2 O 3 ,w ea r ei nt h ec a s eo ft h e Pendubot and when it is directed towards O 1 O 3 ,w ea r e in the case of the Acrobot. See Figure 2, hence, these systems are classified according to which joint is actuated: Γ=[Γ 1 , 0] t for the Pendubot, Γ = [0, Γ 2 ] t for the Acrobot. In nominal regime the paraglider is also represented using the previous double link mechanism, see Figure 3. The first link regroups the canopy and the risers. The second link represents the gondola. At the connection point O 2 between both links, the external force -→ F =[F x , 0] t , assumed to be horizontal, represents the thrust of the motor, see [START_REF] Zaitsev | Paraglider: Mathematical model and control[END_REF]. This force makes a moment

Γ 1 = F x l 1 cosq 1 at point O 1 ,Γ 2 = 0 at point O 2 .

Class 0:I n e r t i aw h e e lp e n d u l u m

An underactuated two-bodies mechanical system is of class 0 if it has a relative degree four with respect to the input. For example consider a pendulum with a balanced flywheel at the end, as shown in Figure 4. The joint variable of the pendulum and the angle of the flywheel rotation are denoted as q 1 and q 2 respectively. Let C be the center of

)) d dt         q 1 q1 q 2 q2         =           q1 - b a 11 sinq 1 q2 0           +             0 - 1 a 11 0 1 a 22             Γ (3.2) 2006
Here

a 11 = I 1 + m 2 l 2 , a 22 = I 2 , b =(m 1 r + m 2 l)g,whereg
is the gravity acceleration. From (3.2), one computes the subspace H 2 of E, which consists of all one-forms that need to be differentiated at least twice to depend explicitly on du:

H 2 =span K {dq 1 , dq 2 , d[a 11 q1 + a 22 q2 ]} (3.
3) We can deduce the subspaces H 3 and H 4 of E, consisting of all one-forms needed to be differentiated three and four times respectively before depending explicitly on the input.

H 3 =span K {d[a 11 q 1 + a 22 q 2 ] , d[a 11 q1 + a 22 q2 ]}, (3.4) H 4 =span K {d[a 11 q 1 + a 22 q 2 ]}. (3.5)
The inertia wheel pendulum is a differentially flat system since the following output y = a 11 q 1 + a 22 q 2 has a degree 4. We have: ẏ

= a 11 q1 + a 22 q2 ÿ = -bsinq 1 y (3) = -b q1 cosq 1 y (4) = b q2 1 sinq 1 -b b a 11 sinq 1 - Γ a 11

Class 1: Acrobot

The acrobot has only one cyclic variable, q 1 ,w h i c hi s unactuated. In [START_REF] Cambrini | Stable trajectory tracking for biped robots[END_REF] and [START_REF] Grizzle | Nonlinear control of mechanical systems with an unactuated cyclic variable[END_REF] and K is a constant.

For the output function y = K 1 p 1 (q)+K 2 σ(q, q) = K 1 p 1 (q)+K 2 (a 11 + a 22 +2a 21 cosq 2 )p 1 (q, q), it can be shown that the resulting zero dynamics is asymptotically stable, whenever K 1 /K 2 is positive.

Class 2: Pendubot, Paraglider

For systems of class 2 such as the Pendubot and the Paraglider, H 3 is also of dimension 2. However, H 3 is not fully integrable any more but contains one exact differential 1-form. In this case only one independant function has a relative degree, which is equal to 3. The attached zero dynamics is unique and there is no freedom to design any alternative output functions as it was the case for the class 1 system. To analyze the situation from a physical point of view, let us consider again the angular momentum (3.7). For the Pendubot the actuator is located at the pivot point and makes an external moment. For the Paraglider, the force applied at the connection point between the links represents the thrust due to the propeller and is considered external. Then this force has to be considered as external. The time derivative of (3.7) is equal to the moment of the external forces applied on the system. Therefore, for both systems, the angular momentum has a relative degree of 1 since the input appears at the first differentiation. They belong to the family, called class 2 systems. Let us now detail H i , i =1 , 2 for the paraglider case. Writing (3.1) in state space form and choosing the angular variables of the link and their time derivatives as state variables we get a compact matrix equation:

d dt         q 1 q1 q 2 q2         =         q1 f 1 (q, q) q2 f 2 (q, q)         + 1 a 11 a 22 -a 2 21         0 a 22 r 1 cosq 1 0 -(a 22 + a 21 cosq 2 )r 1 cosq 1         F x (3.10)
where f i (q, q), i =1 , 2 are independent from the input. To find the order of maximal linearization of the system we compute the H k spaces. A straightforward calculation shows the H 2 space, the set of all the one-form with the relative degree at least 2, can be expressed as H 2 =s p a n K {g} ⊥ = {dq 1 , dq 2 , dφ} Three independent functions of relative degree 2 are therefore given by the angular variables q 1 and q 2 and: φ =(a 22 + a 21 cosq 2 )q 1 + a 22 q2 (3.11) Any vector ω of H 2 can be written as ω = a ω dq 1 + b ω dq 2 + c ω dφ, (3.12) with a ω , b ω and c ω being arbitrary functions. H 3 space is given by:

H 3 = {ω ∈H 2 | ω ∈H 2 }. with ω =ȧ ω dq 1 + ḃω dq 2 +ċ ω dφ + a ω dq 1 + b ω dq 2 + c ω d φ (3.
13) All the terms in dq 1 ,dq 2 and dφ belong to H 2 space. Then for ω we have to cancel out the terms in d q1 ,dq 2 .A f t e r some calculations, two solutions are possible.

The first solution is:

ω 1 =(a 22 + a 21 cosq 2 )d q1 + a 22 dq 2 (3.14) By using the integrating factor λ = 1 a22+a21cosq2 we get the following exact differential form:

dp 2 = λω 1 =dq 1 + a 22 a 22 + a 21 cosq 2 dq 2 (3.15) A direct computation leads to the expression of p 2 p 2 = q 1 + 2a 22 a 2 22 -a 2 21 arctan a 22 -a 21 a 2 22 -a 2 21 tan q 2 2 + K (3.16) wehre K is a constant.
The second solution is:

ω 2 =( -a 21 sinq 1 q1 (a 22 + a 21 cosq 2 ) a 22 +
2a 21 sinq 2 q1 + a 21 q2 )dq 1 +dφ.

(3.17)

Thus H 3 space is given by: H 3 =span K {ω 2 , dp 2 }.

(3.18) It is easy to check, by Frobenius theorem, that H 3 is not integrable.

CONTROL OF THE PARAGLIDER

The parameter values for the paraglider, modeled by a double-link pendulum, are: I 1 =2 4 .4 kg . m 2 , m 1 =1 0kg, l 1 =7m,a n dr 1 =1m for the canopy and I 2 =25kg . m 2 , m 2 = 100 kg, l 2 =1.7 m and r 2 =1m for the gondola. The considered constant for the gravity effect is g =9.81 m/s 2 . The aim of the control in nominal regime of the paraglider is to stabilize the pendulum at a fixed position using one input which is the thrust of the propeller. The chosen output function y is made out of the combination of all available functions present in H 2 space, i.e., y = α 1 φ + α 2 q 1 + α 3 q 2 + α 4 (4.1) Let us compute the feedback control force F x by solving the following equation: ÿ +2ẏ + y = 0 (4.2) From (4.1), equation (4.2) becomes

α 1 φ + α 2 q1 +2(α 1 φ + α 2 q1 )+α 1 φ + α 2 q 1 + α 3 q 2 + α 4 =0
(4.3) By using the dynamic model (3.1) and the compact state form (3.10) we can write φ

= B φ + C φ F x q1 = B q1 + C q1 F x q2 = B q2 + C q2 F x (4.4)
where B φ , C φ , B q1 , C q1 , B q2 and C q2 are functions used to separate the terms that depend on the input from the ones that do not. Finally the feedback control force F x can be written such as:

F x = -2(α 1 φ + α 2 q1 + α 3 q2 ) α 1 C φ + α 2 C q1 + α 3 C q2 - α 1 φ + α 2 q 1 + α 3 q 2 + α 1 B φ + α 2 B q1 + α 3 B q2 α 1 C φ + α 2 C q1 + α 3 C q2 (4.5)
In nominal regime the equilibrium point is defined as

q 1e = -q 2e = - α 4 |α 2 | + |α 3 | (4.6)
with the numerical data q1e =q 2e =0 and q 1e =-q 2e =0.1.

With the proposed family of output variables y (4.1), we choose the following numerical values of the coefficients α 1 , α 2 , α 3 and α 4 are:

α 1 =0.005 s/(rd.Kg.m 2 ),α 2 =1s/rd, α 3 = -1 s/rd, α 4 = -0.2 rd/s (4.7)
These numerical values yields to an equilibrium point or in another word a nominal regime q 1 =0 .1 rd and q 2 = -0.1 rd. There are numerous parameter sets, which yields to other nominal regimes. The output (4.1) have been chosen such that the nonlinear feedback control (4.5), ensures (4.2) and furthermore the stability of the internal dynamics. To check this stability, we can consider the closed system (3.10) with output y (4.1) governed by (4.2) and the control law . Its linearization about the equilibrium point q e results in the linear system: Its linearization about the equilibrium point q e results in the linear system: ẋ = Ax (4.8)

with x [ q 1q 1e q1 q 2q 2e q2 ] t and

A =         0100 ∂ q1 ∂q 1 qe ∂ q1 ∂ q1 qe ∂ q1 ∂q 2 qe ∂ q1 ∂ q2 qe 0001 ∂ q2 ∂q 1 qe ∂ q2 ∂ q1 qe ∂ q2 ∂q 2 qe ∂ q2 ∂ q2 qe         (4.9)
The eigenvalues of A obtained are: λ 1 = λ 2 = -1,λ 3 = -2.889,λ 4 = -0.71679 These eigenvalues are strictly in the left-half complex plane, the equilibrium point q e is locally asymptotically stable for the nonlinear system (3.10)-(4.5)-(4.1) see for example [START_REF] Khalil | Nonlinear Systems[END_REF]). In conclusion the chosen output function y (4.1) does not have any remarkable singularity problem and most importantly it yields a stable zero dynamics. Note that the parameter values of the paraglider have an important influence on the stability properties. The coefficient values have to be recomputed for whenever different parameters are considered for the paraglider.

SIMULATION RESULTS

The graphs below show the profiles of the generalized coordinates q 1 and q 2 , their time derivative q1 and q2 , the chosen output function y (4.1) with (4.7) and the horizontal thrust F x due to the propeller. The initial conditions are: q 1 =0.0 rd, q1 =0.05 rd/s q 2 =0.0 rd, q2 =0.05 rd/s Figure 5 shows, the thrust of the propeller reaching a constant value, which is close to 100 N.m.T h i sf o r c e F x ensures the convergence of the output variable to y, Figure 6. In Figures 7 and8, we can see that the zero dynamics of the Paraglider are stable because the generalized coordinates q 1 and q 2 converge to constant values and their time derivatives q1 and q2 converge to zero. Several sets of initial conditions have been tested, which proved that the basin of attraction of the closed loop system is significantly large. the control of walking robots for over one decade and such technological domains strongly motivate these studies which deserve an abstract development. Obviously, the problem for the most general nonlinear systems remains difficult to solve as it depends on both on the structure of the system and on the analysis of trajectories. Surprisingly, it was possible to solve the problem for the class of mechanical systems under interest thanks to the generalized angular momentum.

ACKNOWLEDGEMENT

This work has been accomplished while the first author was in IRCCyN.

Fig. 2 .

 2 Fig. 2. Left. Pendubot. Right. Acrobot.

Fig. 3 .

 3 Fig. 3. Scheme of the paraglider.

Fig. 4 .

 4 Fig. 4. Scheme of the inertia wheel pendulum. mass of the link, located on line O 1 O 2 . Let the following lengths be O 1 O 2 = l and O 1 C = r.L e tm 1 and m 2 be the masses of the link and the flywheel. The moment of inertia of the link about joint O 1 is denoted by I 1 .T h e inertia moment around the center of mass of the flywheel is denoted by I 2 .Letm 1 and m 2 be the masses of the link and the flywheel respectively. The equations of the pendulum motion with the flywheel can be written as[START_REF] Aoustin | Stabilization of unstable equilibrium postures of a two-link pendulum using a flywheel[END_REF]

Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .

 5678 Fig. 5. Profile of the control thrust force F x .

  it is proven that the underactuated two-bodymechanical system is of class 1 if and only if H 3 is fully integrable. Moreover, the maximal linearization of order 3 can be performed with internal stability. In terms of full integrability of H 3 , the Acrobot belongs to class 1 systems with H 3 =span K {dσ, dp 1 }, (3.6) where σ is the angular momentum with respect to the suspension joint O 1 of the double-link pendulum and we have 11 + a 22 +2a 21 cosq 2 )q 1 +(a 22 + a 21 cosq 2 )q 2 ,

	σ = (a (3.7) ∂T ∂ q1 =
	and
	dσ a 11 + a 22 +2a 21 cosq 2 =d q 1 + dp 1 = a 11 + a 22 +2a 21 cosq 2 a 22 + a 21 cosq 2	dq 2 ,	(3.8)
	A direct computation gives
	p 1 = q 1 + A arctan	q 2 2	+ a 11 + a 22 -2a 21 a 11 + a 22 +2a 21	tan	q 2 2	+ K	(3.9)
	with A =	a 22 -a 11 (a 11 + a 22 ) 2 -4a 2 21