Francisco Javier Bejarano

Wilfrid Perruquetti

Thierry Floquet

Gang Zheng

Observation of Nonlinear Differential-Algebraic Systems with Unknown Inputs

Keywords: Differential-Algebraic systems, Observer design, sliding mode high order differentiation

A method to carry out the state estimation is proposed for a class of nonlinear systems with unknown inputs whose dynamics is governed by differential-algebraic equations (DAE). We achieve, under suitable conditions, to replace the original DAE for a system with differential equations only by using a zeroing manifold algorithm inducing a state space dimension reduction. Observability conditions can be checked using the original system parameters. The state estimation is done using a sliding mode high order differentiator.

I. INTRODUCTION

Differential-algebraic equations (DAE) arise in applications such as electrical networks, chemical engineering, semidiscretized Stokes equations, cellular biology and so on (see, e.g. [START_REF] Kunkel | Differential-Algebraic Equations Analysis and Numerical Solution[END_REF]). The problem of state estimation of DAE systems has been tackled in the last two decades [START_REF] Campbell | Singular systems of differential equations II[END_REF], [START_REF]Singular systems of differential equations I[END_REF], [START_REF] Dai | Singular Control Systems[END_REF]. For the case of systems when all the inputs are known, the algebraic observability of DAE time varying systems has been studied in [START_REF] Terrell | Observability of nonlinear differential algebraic systems[END_REF] (a system is called observable in the differential-algebraic framework if the state can be expressed in terms of the output and the known inputs and a finite number of their time derivatives [START_REF] Diop | Nonlinear observability, identifiability, and persistent trajectories[END_REF]). In [START_REF] Darouach | Observers for a class of nonlinear singular systems[END_REF] and [START_REF] Darouach | H ∞ observers design for a class of nonlinear singular systems[END_REF], observers are proposed for singular systems using an LMI approach. Using also LMI's, a reduced order observer for a class of Lipschitz nonlinear singular systems is presented in [START_REF] Lu | Full-order and reduced-order observers for lipschitz descriptor systems[END_REF]. Asymptotic observers for systems having index one were proposed in [START_REF] Assoudi | A high gain observer for a class of implicit systems[END_REF] and [START_REF] Åslund | An observer for non-linear differential-algebraic systems[END_REF].

Here, we consider that the system contains unknown inputs and the DAE are given in an explicit form. Under suitable conditions, we achieve to replace the DAE of the system by ODE on a manifold of reduced dimension. This is done by searching for an invariant submanifold (called zeroing submanifold) where the DAE are satisfied during an interval of time. Then observability conditions are found in terms of the original system parameters. Finally, the state estimation is carried out by using a sliding mode high order differentiator (SMHOD). The formulation of the problem is stated in Section II. The procedure to replace the DAE by ODE on a reduced dimension manifold is described in section III. This is done by means of a zeroing manifold algorithm (conceived from the zero dynamics concept in [START_REF] Isidori | Nonlinear Control Systems[END_REF]). In section IV, we tackle the Francisco Javier Bejarano is with SEPI, ESIME Ticoman, IPN, Av. San José Ticomán 600, C.P. 07340, Mexico City, Mexico, e-mail: javbejarano@yahoo.com.mx Wilfrid Perruquetti, Thierry Floquet, and Gang Zheng are with Non-A team, INRIA -Lille Nord Europe, Parc Scientifique de la Haute Borne 40, avenue Halley Bât.A, Park Plaza 59650 Villeneuve d'Ascq, France, e-mail: wilfrid.perruquetti@inria.fr, thierry.floquet@ec-lille.fr, gang.zheng@inria.fr state estimation. Section V has an example with simulations, supporting the theoretical results. We use L f h to denote the Lie derivative of the function h along the vector field f , i.e.

L f h = dh (x) , f (x) = ∂ h(x) ∂ x f (x).
For a smooth manifold M, T x M is the tangent space to M at x. By dim f (x), we denote the number of rows of the vector f (x). Im A is the image (range space) of A matrix. col(A, B) is the matrix obtained by concatenating matrices A and B in vertical direction. A T and A + are the transpose and pseudo-inverse of A, respectively.

II. SYSTEM DESCRIPTION AND PROBLEM FORMULATION

We consider systems described by the following equations:

ẋ (t) = f (x (t)) + g (x (t)) µ (t) (1a) 0 = F (x (t)) + G (x (t)) µ (t) (1b)
y (t) = h (x (t)) (1c)
where the state x (t) belongs to an open set U ⊂ R n . The maps

f : U → R n , g : U → R n×m , F : U → R q , G : U → R q×m
, and h : U → R p are all smooth maps. The input vector µ (t) ∈ R m is unknown a priori; however it should be noted that µ (t) has to be so that a solution for (1a)-(1b) exists. The aim is the estimation of x (t) by means of the system output y (t). Let N be a set defined as

N = {x ∈ U : ∃µ x ∈ R m s.t. F (x) + G (x) µ x = 0} (2)
In what follows we will do our study around an x 0 ∈ N for which x (t; x 0 ) satisfies (1a)-(1b) in a neighborhood of t = 0.

III. SEARCHING FOR A MAXIMAL ZEROING SUBMANIFOLD

The procedure pursued here to estimate x (t) lies into two main steps. Firstly, we look for a maximal zeroing submanifold (w.r.t. F (x) and G (x)), which is a submanifold such that if x (0) belongs to it then there exists an input function µ (t) such that x (t; x (0)) satisfies (1a)-(1b) for all t in a neighborhood of t = 0. The first part yields a coordinates transformation so that some terms of the state in the new coordinates are equal to zero (the same number as the dimension of the zeroing submanifold). This also allows for expressing the input vector as a function of the state vector. The second part consists in using a (reduced order) observer for a system without unknown inputs. Now, we proceed to give a formal definition of a zeroing submanifold. For it, we will need to define invariant and locally invariant submanifolds (see [START_REF] Abraham | Foundations of Mechanics[END_REF]). Let M be a smooth submanifold of R n .

Definition 1: Let V ⊂ M be a smooth submanifold, and f a vector field on M. Then, V is an Invariant submanifold

(ISM) w.r.t. f if, for all v ∈ V , f (v) ∈ T v V ⊂ T v M.
Definition 2: M is a locally ISM at x 0 w.r.t. f if there exists a neighborhood U 0 of x 0 such that M ∩U 0 is an ISM w.r.t. f . Next definition is in its essence a definition found in Chapter 6 of [START_REF] Isidori | Nonlinear Control Systems[END_REF]; however, we have adapted it using the previous definitions of ISM and also clause i) has been slightly modified to consider the effect of G (x) of (1b).

Definition 3: A zeroing submanifold (ZSM) at x 0 is a smooth submanifold M ⊂ U containing x 0 that satisfies i) M ⊂ N and ii) there exists a smooth mapping µ : M → R m so that M is a locally ISM at x 0 w.r.t. the vector field f (x) := f (x) + g (x) µ (x).

Remark 1: Clause ii) means that there exists a neighborhood U 0 of x 0 such that if x (0) ∈ M ∩U 0 then x (t) ∈ M ∩U 0 for all t in a neighborhood of t = 0 (see, e.g., [START_REF] Abraham | Foundations of Mechanics[END_REF]).

Proposition 1: If a ZSM M is such that the ISM M ∩ U 0 (w.r.t. f (x)) is a closed set of U, then x (t) stays within M ∩U 0 for all t, provided x (0) belongs to M ∩U 0 . Proof. In case x (0) ∈ M ∩U 0 , there exists an interval [0,t 1 ) for which x (t) remains in M ∩ U 0 (see Remark 1). Let us suppose that x (t) leaves M ∩U 0 after a time. Then the set = {t : x (t) / ∈ M ∩U 0 } is not empty. Since ϒ is lower bounded, we may define t inf = inf (ϒ). We claim that t inf does not belong to ϒ. Indeed, if t inf is an element of ϒ, then x (t inf ) belongs to the complement of M ∩U 0 (denoted by (M ∩U 0 ) c ), which by hypothesis is an open set. Hence, there exists an open set Λ contained in (M ∩U 0 ) c and having x (t inf ) as an element. By the continuity of x (t), the inverse image of Λ is open, that is, t inf is a point of an open interval contained in ϒ, which is a contradiction. Thus, the only remaining option is that t inf belongs to the complement of ϒ, or, what is the same, t inf is so that x (t inf ) ∈ M ∩U 0 . However, in that case, since M ∩U 0 is invariant, then there exists an open interval that contains t inf and on which x (t) belongs to M ∩U 0 , which implies that t inf is not the infimum of ϒ, a contradiction again by the definition of t inf . Thus, finally, we conclude that ϒ is the empty set.

Definition 4: A ZSM M is locally maximal if, for any other ZSM M, there exists a neighborhood U of x 0 such that the inclusion M ∩U ⊃ M ∩U is satisfied.

We will seek for a locally maximal ZSM. The proposed method is similar to the one given in [START_REF] Isidori | Nonlinear Control Systems[END_REF], pp. 299-301. However, we do not assume that q = m and we include the input explicitly in the algebraic equation. The proposed algorithm is a nonlinear version of the algorithm used to find the weakly unobservable subspace in linear systems with inputs appearing explicitly in the differential equations and in the system output (see, e.g., [START_REF] Bejarano | Unknown input and state estimation for unobservable systems[END_REF]). The following is our stepby-step algorithm to find a locally maximal ZSM.

Step 1. It is assumed that there exists a neighborhood U 0 containing x 0 such that the rank G (x) = r 0 for all x ∈ U 0 , for some r 0 . Let us define M 0 = U 0 . For if, there exists a full row rank matrix R 0 (x) with terms being smooth functions of x in a neighborhood

U ′ 0 of x 0 such that rank R 0 (x) = q -r 0 and R 0 (x) G (x) = 0 for all x ∈ U ′ 0 (3)
Thus, the maps Φ 0 (x) and H 1 (x) are defined as

H 1 (x) = Φ 0 (x) := R 0 (x) F (x). Let us assume that the rank of dH 1 (x) is constant in a neighborhood U 1 ⊂ U ′ 0 of x 0 . Then, the set M 1 := {x ∈ U 1 : H 1 (x) = 0} is a smooth submanifold. Proposition 2: M 1 satisfies the identity M 1 = N ∩U 1 . Proof. By its construction M 1 ⊂ N ∩U 1 . Indeed, if H 1 (x) = 0, then F (x) ∈ Im G (x) (i.e. x ∈ N) because of R 0 (x) G (x) = 0 and R 0 (x) F (x) = 0 and rank R 0 (x) = q -rankG (x). For the converse, if x ∈ N ∩U 1 , then, from (2), F (x) ∈ Im G (x). Hence, in view of (3) and since H 1 (x) = R 0 (x) F (x), we obtain that H 1 (x) = 0. That is, we conclude that N ∩U 1 ⊂ M 1 .
Step 2. Let us assume that rank of col (G (x) , L g H 1 (x)) is equal to a constant r 1 for all x in M 1 . Then there exists a matrix R 1 (x) with terms being smooth functions of x in a neighborhood

U ′ 1 of x 0 such that, for all x ∈ M 1 ∩ U ′ 1 , R 1 (x) G (x) L g H 1 (x) = 0 and rank R 1 (x) = q + dim H 1 (x) -r 1 .
Thus, we define

H 2 (x) = col (H 1 (x) , Φ 1 (x)) where Φ 1 (x) = R 1 (x) col F (x) , L f H 1 (x) . Again, let us assume that dH 2 (x) has constant rank in a neighborhood U 2 ⊂ U ′ 1 . Thus, the set M 2 := {x ∈ U 2 : H 2 (x) = 0} is a smooth submanifold also. Step k. Assuming that rank col(G (x) , L g H k-1 (x)) = r k-1 for all x ∈ M k-1 , then there exists a neighborhood U ′ k-1 of x 0 and a matrix R k-1 (x) of smooth functions on U ′ k-1 such that rankR k-1 (x) = q + dimH k-1 (x) -r k-1 R k-1 (x) G (x) L g H k-1 (x) = 0 (4) for all x ∈ M k-1 ∩U ′ k-1 . By defining H k (x) as H k (x) = H k-1 (x) Φ k-1 (x) , Φ k-1 (x) = R k-1 (x) F (x) L f H k-1 (x) (5)
and if dH k (x) has constant rank on U k ⊂ U ′ k-1 around x 0 , we obtain the smooth manifold M k :

M k = {x ∈ U k : H k (x) = 0} Lemma 1: Assume that there exist nested neighborhoods 1 U k+1 ⊂ U k and U ′ k+1 ⊂ U ′ k (k ∈ 1, n) of x 0 such that, for every k, dH k (x) has constant rank in U k and col(G (x) , L g H k (x)
) has constant rank for all x on the smooth manifold

M k := {x ∈ U k : H k (x) = 0} (6)
for all k ∈ 1, n, and H k (x) and R k (x) satisfy ( 5) and ( 4), respectively, on U ′ k . Then, there exists a k * ≤ n and a neighborhood Ūk * so that M k * ∩ Ūk * = M k * + j ∩ Ūk * for all j ≥ 1.

The proof of Lemma 1 rests upon the following proposition. Proposition 3: Under assumptions of Lemma 1, we obtain that M k+1 ⊂ M k , for k ≥ 1. Proof. For if x ∈ M k+1 , by (6), we have x ∈ U k+1 and H k+1 (x) = 0. Thus, H k+1 (x) = 0 implies, by [START_REF] Terrell | Observability of nonlinear differential algebraic systems[END_REF], that H k (x) = 0, which in turns implies, again by [START_REF] Diop | Nonlinear observability, identifiability, and persistent trajectories[END_REF], that x ∈ M k . Proof. [Lemma 1] Since M k+1 ⊂ M k , the dimension of M k is not increasing w.r.t. k. Then, there exists a k * ≤ n such that the dimension of M k * +1 must be equal to that of M k * , i.e., for a neighborhood W , we have M k * ∩W = M k * +1 ∩W . Now, by ( 4) and ( 5), H k (x) = 0 only if

F (x) dH k-1 (x) , f (x) ∈ Im G (x) dH k-1 (x) , g (x) (7)
where

H 0 (x) := 0. Since rank of dH k-1 is constant on U k-1 , ker dH k-1 (x) = T x M k-1 for x ∈ M k-1 ∩ U k-1 (for all k ≥ 1).
Therefore, the submanifold M k can be rewritten as follows,

M k = x ∈ U k ∩ M k-1 : F (x) f (x) ∈ Im G (x) g (x) + 0 T x M k-1 (8) Furthermore, T x M k * +1 = T x M k * (x ∈ M k * +1 ∩W ) since M k * ∩ W = M k * +1 ∩ W .
Hence, by [START_REF] Darouach | H ∞ observers design for a class of nonlinear singular systems[END_REF], we obtain the identities

M k * +2 ∩ W ∩ U k * +2 = M k * +1 ∩ W ∩ U k * +2 = M k * ∩ W ∩ U k * +2 .
Inductively, we obtain that M k * + j ∩ Ūk * = M k * ∩ Ūk * for all j ≥ 1, for a neighborhood Ūk * . Remark 2: Lemma 1 implies that the algorithm will stop at k * step. Moreover, k * will be the first integer k satisfying rank dH k (x) = rankdH k+1 (x). This is true because of the dimension of M k is nrank dH k (x) (for k ≥ 1) and M k * and M k * +1 have the same dimension.

Proposition 4: If conditions of Lemma 1 are satisfied with a set of matrices R 0 (x), R 1 (x), . . . , R k * -1 (x), then those conditions remain valid for other choice of such a set of matrices.

Proof.

Let us define H 0 (x) = 0, then we may define the manifold

M 0 = {x ∈ U 0 : H k (x) = 0}. Let R0 (x), R1 (x
) , . . . , Rk * -1 (x) be other choice of matrices. We are to prove by induction that the maps generated by this set of matrices satisfy the following equations on M k , (for k = 0, 1, . . . , k * ):

d Hk (x) = S k (x) dH k (x) (9)
Φk (x) = T k (x) Φ k (x) + V k (x) (10)
where V k (x) vanishes at M k . Obviously, H0 (x) = H 0 (x) and so d H0 (x) = dH 0 (x). Now, since the rows of R0 (x) form a basis of the set of solutions of γ (x) G (x) = 0 for all x ∈ M 0 , then this implies that R0 (x) = T 0 (x) R (x) + L 0 (x) where T 0 (x) is nonsingular on M 1 and L 0 (x) vanishes at M 0 . Thus,

H1 (x) = R0 (x) F (x) = T 0 (x) H 1 (x) + V 1 (x) (11)
where V 1 (x) := L 0 (x) F (x). Thus, since by definition H 1 = Φ 0 (x), and taking into account [START_REF] Åslund | An observer for non-linear differential-algebraic systems[END_REF], then [START_REF] Assoudi | A high gain observer for a class of implicit systems[END_REF] is satisfied for k = 0. Now, let us suppose that ( 9) and ( 10) are satisfied for k = j. Then, taking into account ( 5), ( 9) and ( 10), we obtain the equation

d Hj+1 = d (S j (x) H j (x)) d (T j (x) Φ j (x) + V j (x))
which in view that H j (x) and Φ j (x) vanish on M j+1 , and since dV j (x) = P (x) dH j (x) at each x ∈ M j+1 , for suitable P (x), then the identities

d Hj+1 = S j (x) dH k (x) d (T j (x) Φ j (x) + V j (x)) = S j (x) 0 P (x) T j (x) dH j+1 (x) = S j+1 dH j+1 (x) (12)
are valid at each x ∈ M j+1 . Now, since at each x ∈ M j+1 , the rows of R j+1 (x) form a basis of the set of solutions of the equation γ (x) col G, L g Hk+1 = 0, then, by [START_REF] Isidori | Nonlinear Control Systems[END_REF], the following equation is obtained

R j+1 (x) = T j+1 R j+1 (x) I 0 0 S -1 j+1 + L j+1 (x)
where T j+1 (x) is nonsingular at each x ∈ M j+1 and L k+1 (x) vanishes on M j+1 . Thus, taking into account ( 5) and ( 12), the following equation is straightforwardly obtained

Φ j+1 (x) = T j+1 Φ j+1 + L j+1 (x) F (x) L f Hj+1 (x) = T j+1 Φ j+1 + V j+1 (x)
where V j+1 (x) is implicitly defined and it vanishes on M j+1 .

Hence, since ( 9) is true for k = 1, 2, . . . , k * , then the conditions of Lemma 1 are still valid.

Theorem 1: Z * := M k * is a locally maximal ZSM. Proof. Since rank of col (G (x) , L g H k * (x)) is constant, by [START_REF] Darouach | Observers for a class of nonlinear singular systems[END_REF] and the Implicit Function Theorem, there exists µ * :

M k * +1 → R m , smooth on a neighborhood U ′ k * of x 0 , such that F (x) + G (x) µ * (x) = 0 and L f H k * (x) + L g H k * (x) µ * (x) = 0 (13) Second equation implies that f (x) + g (x) µ (x) ∈ T x M k * for x ∈ M k * +1 . However, f (x) ∈ T x Z * ( f (x) := f (x) + g (x) µ (x)), for x ∈ Z * ∩ U ′ k * ∩ Ūk * , since Z * ∩ Ūk * = M k * +1 ∩ Ūk * . Hence, the restriction of f (x) to Z * ∩ U ′ k * ∩ Ūk * maps onto T x Z * , i.e. Z * is a locally ISM w.r.t. f (x).
Let us suppose that there exists a manifold Z such that x (t) ∈ Z for all t on [0, T ], provided that x (0) ∈ Z. Thus, if x ∈ Z ∩ Ūk * , then H 1 (x) = 0 and x ∈ M 1 . By induction, we obtain that H k (x) = 0 for all k ≥ 1. Hence x ∈ M k ∩ Ūk * , that is x ∈ Z * ∩ Ūk * . Therefore, Z ∩ Ūk * ⊂ Z * ∩ Ūk * .

Proposition 5: Assuming that rank of col(G (x) , L g H k * (x)) is equal to m for x ∈ Z * , there exists a unique (locally) smooth mapping µ * :

Z * → R m , such that F (x) + G (x) µ * (x) = 0 and f (x) ∈ T x Z * ( f (x) := f (x) + g (x) µ * (x)).
That is, the equation

F (x) L f H k * (x) + G (x) L g H k * (x) µ * (x) = 0 ( 14
)
has a unique solution around x 0 .

Proof. From the proof of Theorem 1, there is a smooth mapping µ * :

M k * +1 → R m satisfying (13) for all x ∈ M k * +1 ∩U ′ k * , for a neighborhood U ′ k * of x 0 .
We have proved also that [START_REF] Abraham | Foundations of Mechanics[END_REF]

implies that F (x) + G (x) µ * (x) = 0 and f (x) ∈ T x Z * , for all x ∈ Z * ∩ U ′ k * ∩ Ūk * . Now, taking into account that Z * ∩ Ūk * = M k * +1 ∩ Ūk * , and since rank of col(G (x) , L g H k * (x))
is equal to m for x ∈ Z * , then we conclude that there exists a unique smooth mapping µ * :

Z * → R m such that F (x) + G (x) µ * (x) = 0 and f (x) ∈ T x Z * , for x ∈ Z * ∩ U ′ k * ∩ Ūk * .
Remark 3: Under the assumption of the previous proposition, the differential index of the DAE will be equal to k * . This is due to the fact that the algorithm followed to calculate the zeroing submanifold introduces intrinsically a procedure with which, after k * time derivatives of the algebraic equation, we may obtain an ODE around x 0 .

IV. STATE RECONSTRUCTION

Let f * be the restriction of f (x) = f (x) + g (x) µ * (x) to Z * (assuming that rank G (x) L g H k * (x)
= m for any x ∈ Z * ). Thus on Z * , the dynamics of system (1) is governed by

ẋ = f * (x) and y = h (x) (15)
Let us recall the definition of local weak observability. Definition 5: [ [15], [START_REF] Besanc ¸on | Nonlinear Observers and Applications, ser. Lecture Notes in Control and Information Sciences[END_REF]] A pair (x 0 , x ′ 0 ) ∈ U × U (x 0 = x ′ 0 ) is indistinguishable for system (1) if, for all µ and for all t ≥ 0, h x µ (t, x 0 ) = h x µ (t, x ′ 0 ) . Definition 6: [ [START_REF] Besanc ¸on | Nonlinear Observers and Applications, ser. Lecture Notes in Control and Information Sciences[END_REF]] System (1) is locally weakly observable (LWO) at x 0 if there exists a neighborhood U of x 0 such that for any neighborhood V ⊂ U of x 0 there is no indistinguishable pair (x 0 , x ′ 0 ) in V when considering time intervals for which trajectories remain in V .

Lemma 2: Under the assumptions of Lemma 1, system ( 1) is LWO at x 0 if, and only if, ( 15) is LWO at x 0 . Proof. It is clear that if ( 1) is LWO at x 0 ∈ Z * , ( 15) is LWO at x 0 . Indeed, if [START_REF] Hermann | Nonlinear controllability and observability[END_REF] is not LWO at x 0 , then for any neighborhood Ū ⊂ Z * , there exists a neighborhood V ⊂ Ū such that

(x 0 , x ′ 0 ) is indistinguishable on Z * × Z * , for a x ′ 0 ∈ V . Therefore, since Z * ⊂ U, then (x 0 , x ′ 0
) is indistinguishable also on U × U and, therefore, (1) is not LWO at x 0 either. Now, let us consider that x 0 ∈ Z * and that ( 1) is not LWO at x 0 . Then by Definition 6, in every neighborhood U ∈ R n of x 0 there exists a pair x 0 , x ′ 0 ∈ U ×U that is indistinguishable on (1). Hence, in particular, the neighborhood

Z * ∩ U ′ k * ∩ Ūk * (U ′
k * and Ūk * as in the proof of Theorem 1) contains the vector x * 0 such that (x 0 , x * 0 ) is indistinguishable. Thus, (x 0 , x * 0 ) is indistinguishable on (15) also because of both x 0 and x * 0 belong to Z * . Therefore, we can infer that if ( 15) is LWO at x 0 , then (1) is LWO at x 0 .

Let n * be the dimension of Z * . Since rank dH k * (x) = nn * for all x ∈ Z * , we can arrange a vector function H * (x) ∈ R n-n * ×n whose terms are taken from H k * (x) so that rank d H * (x 0 ) = nn * . Thus, there exists a diffeomorphism

Ψ (x) = H * (x) φ (x)
with which, defining z = Ψ (x), we obtain

z 1 (t) = H * (x (t)) = 0, ż2 (t) = f2 (z 2 (t)) , and
y (t) = h2 (z 2 )
where z 1 (t) ∈ R n-n * and z 2 (t) ∈ R n * . There, f2 (z 2 (t)) and h2 (z 2 ) are given by the formulas

f2 (z 2 ) = ∂ φ (x) ∂ x f * (x) x=Ψ -1 (z) h2 (z 2 ) = h (z) | z 1 =0 h (z) = h Ψ -1 (z) (16)
Thereby, the original problem is reduced to the estimation of z 2 from the knowledge of y (t). However, since LWO is not enough for the design of an observer, below we will assume that ( 15) is uniformly observable (see, e.g., [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF]), i.e., we assume that on

Z * 0 = z 2 ∈ R n * : z 2 = φ (x) for x ∈ Z * s.t. H * (x) = 0 the rank condition (17) is satisfied rank col d h2 (z 2 ) , dL f2 h2 (z 2 ) , . . . , dL n * -1 f2 h2 (z 2 ) = n * (17)
There are several observers that may be used to carry out the estimation of z 2 provided that ( 17) is satisfied, like high gain observers [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] or finite time observers [START_REF] Ménard | A global high-gain finitetime observer[END_REF]. Below, in Theorem 2, we show that condition [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] can be checked in the original coordinates.

Theorem 2: Under the assumptions of Lemma 1, (1) is uniformly observable on Z * if ( 18) is satisfied for all x ∈ Z * .

rank col dH k * (x) , dh (x) , dL f * h (x) , . . . , dL n * -1 f * h (x) = n (18) Proof. Let us define f (z) = ∂ Ψ(x) ∂ x f * (x) x=Ψ -1 (z)
. Thus, from the fact that

L k f * h (x) = L k f h (z) | z=Ψ(x) , it is clear that ∂ ∂ x L k f * h (x) = ∂ ∂ z L k f h (z) ∂ Ψ (x) ∂ x z=Ψ(x) (19)
Hence, the identity

∂ ∂ z 2 L k f h (z) = ∂ ∂ z 2 L k f2 h2 (z 2 ) (20)
is satisfied on Z * , for k ≥ 0 (this may be verified by using induction, taking into account that z 1 = 0). Thus, for x ∈ Z * , by ( 19) and ( 20), we obtain the identities, for i ≥ 0,

d H * (x) = I n * 0 ∂ Ψ(x) ∂ x dL i f * h (x) = * dL i f2 h2 (z 2 ) ∂ Ψ(x) ∂ x
In view of the previous identities and since rankd H * (x) = rank dH k * (x) = nn * , we conclude that the rank condition in ( 17) is satisfied if ( 18) is satisfied. Condition [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] implies that z 2 can be locally expressed as a function of y, ẏ, . . . , y (n * -1) which is known as algebraic observability [START_REF] Diop | Nonlinear observability, identifiability, and persistent trajectories[END_REF]. In fact, it was shown in [START_REF] Sontag | I/O equations for nonlinear systems and observation spaces[END_REF] that for analytic systems the fulfillment of ( 17) is equivalent to the algebraic observability of z 2 . Thus, the following corollary is an immediate consequence of Theorem 2.

Corollary 1: Under the assumptions of Lemma 1, rank col(G (x 0 ) , L g H k * (x 0 )) = m and ( 18), there exists a function Γ such that x (t) = Γ y, ẏ, . . . , y (n * -1) . Proof. The function Γ might be found in the following manner. Let us consider the diffeomorphism Ψ (x) defined in above, where the terms of φ (x) are chosen to be equal to some of the terms of col h (x) , L f * h (x) , . . . , L n * -1 f * h (x) so that rank col (dH k * (x) , φ (x)) = n (this choice is possible due to Theorem 2). Thus, the dynamics of z 2 turns out to be a set of chains of integrators (maybe after a rearrange of the coordinates). Therefore, considering that z 1 = 0, we obtain straightly an explicit function Γ z such that z = Γ z y, ẏ, . . . , y n * -1 . Hence, finally we obtain that Γ y, ẏ, . . . , y n * -1 = Ψ -1 Γ z y, ẏ, . . . , y n * -1 .

Two real-time differentiators that could be used to estimate the required derivatives of the output are described in [START_REF] Levant | High-order sliding modes: differentiation and outputfeedback control[END_REF] and [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF]. The former is a sliding mode high order differentiator (SMHOD), which is used in the example given further.

Remark 4: [Further generalization] For the case when the map µ * is not unique, i.e. that rankcol (G (x 0 ) , L g H k * (x 0 )) = r < m, the state estimation may still be done. As if rank of col (G (x) , L g H k * (x)) is constant in a neighborhood of x 0 , locally there exist matrices D 1 (x) and D 2 (x) of rank r and mr, respectively, whose entries are smooth functions of x, such that rank (col (G (x) , L g H k * (x)) D 1 (x)) = r and col (G (x) , L g H k * (x)) D 2 (x) = 0 for all x in a neighborhood of x 0 . Thus, with D (x) := (D 1 (x) , D 2 (x)), and a partition of its inverse as (D (x)) -1 = col ( D1 (x) , D2 (x)), we obtain that col F (x) , L f H k * (x) +col(G (x) , L g H k * (x)) D 1 (x) D1 (x) µ = 0 (21) Let us define α 1 = D1 (x) µ and α 2 = D2 (x) µ. Then, [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] has a unique solution for α 1 = α 1 (x). Thus, defining f * (x) = f (x) + g (x) D 1 (x) α 1 (x), we can rewrite, locally on the man- ifold Z * , the dynamic equations of the system as follows

ẋ = f * (x) + g (x) D 2 (x) α 2 and y = h (x)
Thereby, the state estimation may be carried out by using an unknown input observer (α 2 is the UI). In particular a reduced order observer may be designed. Indeed, using the diffeomorphism defined at the beginning of this section and the change of coordinates given by z = Ψ (x), we obtain the sub-vector z 1 = H * (x) = 0 and z 2 being the state of the system ż2 = f2 (z 2 ) + g2 (z 2 ) α 2 and y = h2 (z 2 ) where f2

(z 2 ) = ∂ φ (x) ∂ x f * (x) x=Ψ -1 (z) g2 (z 2 ) = ∂ φ (x) ∂ x g (x) D 2 (x) x=Ψ -1 (z) h2 (z 2 ) = h (z) | z 1 =0 with h (z) = h Ψ -1 (z)
Hence, an UI observer for z 2 (t) ∈ R n-n * could be designed. Just to mention two of various approaches that could be followed, we refer the reader to [START_REF] Barbot | An observation algorithm for nonlinear systems with unknown inputs[END_REF], [START_REF] Angulo | Output feedback design for exact state stability of flat nonlinear systems[END_REF].

V. EXAMPLE Let us consider an example with the following functions:

f (x) = x 2 x 4 + x 6 x 1 x 1 x 4 x 5 x 3 x 7 -x 5 x 2 T g (x) =   0 x 2 1 x 2 1 0 0 0 0 0 0 0 x 6 x 7 1 0 0 x 1 0 0 0 0   T , h (x) = x 1 x 3 F (x) =   0 -cos( x 5
2 )

10 x 6   , G (x) =   0 0 2x 5 -sin x 3 0 0 1 0 0 0   By (2), N = x ∈ R 7 : x 6 = 0, x 5 = sin (x 3 ) or x 5 π . Thus, the observability is around x 0 = 0. Step 1. Since rank G (x) = 1 for all x ∈ R 7 , R 0 (x) = -1 2x 5 -sin x 3 0 0 0 1 H 1 = R 0 (x) F (x) = -1 10 (2x 5 -sin x 3 ) cos( x 5 2 ) x 6
We see that rank of the matrix dH 1 is equal to 2 for all

x ∈ U 1 = x ∈ R 7 : |x 5 | < π and |x 3 | < π 2 . Hence, M 1 = {x ∈ U 1 : x 6 = 0, 2x 5 = sin x 3 } is a 5-dimension manifold.
Step 2. For col (G (x) , L g H 1 (x)), we obtain

G (x) L g H 1 (x) T = 0 0 0 δ 0 0 0 0 0 x 6 x 7 2x 5 -sin x 3 1 0 γ 0 ( 22
)
where γ = 1 10 x 1 cos x 5 2 cos (x 3 ) and δ = 1 20 sin x 5 2 (2x 5sin x 3 ) -1 5 cos x 5 2 + 1 10 cos x 5 2 cos (x 3 ). Therefore, for all x ∈ M 1 , the rank of the matrix in ( 22) is equal to 2. Thus, R 1 (x), Φ 1 (x), and H 2 (x) are taken as

R 1 (x) =   1 0 0 0 0 0 0 1 0 0 0 0 0 0 1   , Φ 1 (x) =   0 x 6 x 7   H 2 (x) =   -1 10 (2x 5 -sin x 3 ) cos( x 5 2 ) x 6 x 7   Since rank dH 2 (x) = 3 on U 1 , M 2 = {x ∈ U 1 : x 6 = x 7 = 0 and 2x 5 = sin x 3 }.
Step 3 (end). Now, we have that col(G (x) , L g H 2 (x)) = col G (x) , L g H 2 (x) , 0 1 0 , which has rank equal to 3 on M 2 . Thus, matrix R 2 (x) ∈ R 3×6 has zeros everywhere except in the entries (1,1), (2,3), and (3,5), which have a number one. As for Φ 2 (x), it takes the form Φ 2 (x) = 0 x 6 x 7

T . Finally, we obtain that H 3 (x) = H 2 (x), which implies that Z * = M 2 and H * = H 2 . Moreover, rank col(G (x) , L g H * (x)) = 3 for all x ∈ Z * . Therefore, µ = µ * (x) satisfies the equation ( 14) with k * = 2, for x ∈ Z * . Thus,

we obtain µ * 1 (x) = 2x 3 -( 1 10 x 1 cos( x 5
2 )+x1x4)cosx3

cos x 3 -2
, µ * 2 (x) = x 2 x 5 , and µ * 3 (x) = 1 10 cos x 5 2 . Thus, f * (x) (x ∈ Z * ) has the form: In this case, matrix in ( 18) is equal to col dH * , dh, dL f * h, dL 2 f * h , which has rank 7 in a vicinity of x = 0. Then, the system is (locally) uniformly observable according to Theorem 2. Furthermore, as the dimension of Z * is equal to 4, at most 3 derivatives of y are required for the estimation of the entire state x. In fact, we have that x can be expressed in terms of (y, ẏ, ÿ) as follows:

f * 1 = x 2 , f * 6 = f * 7 = 0 f * 2 = x 4 + x 2 2x 3 -1 10 x 1 cos 2 sin(x 3 ) 2 + x 1 x 4 cosx 3 cos x 3 -2 f * 3 = x 1 x 4 + 2x 3 -1 10 x 1 cos x 5 2 + x 1 x 4 cos x 3 cos x 3 -2 + 1 10 x 1 cos 2 sin (x 3 ) 2 f * 4 = 2
x 1 = y 1 , x 2 = ẏ1 , x 3 = y 2 , x 4 = ÿ1 + ẏ1 y 2 -1 2 ẏ2 cos y 2 x 5 = 1 2 sin y 2 , x 6 = x 7 = 0 (23) ẏ and ÿ are estimated using a SMHOD proposed in [START_REF] Levant | High-order sliding modes: differentiation and outputfeedback control[END_REF], i.e., with λ 0 = 1.1, λ 1 = 1.5, λ 2 = 3. Hence, the estimate of ẏ and ÿ is given by ŷ1 = ȳ1,0 , ẏ1 = ȳ1,1 , ÿ1 = ȳ1,2 , ŷ2 = ȳ2,0 , and ẏ2 = ȳ2,1 . Then, estimate of x is obtained with x: x1 = ȳ1,0 , x2 = ȳ1,1 , x3 = ȳ2,0 , x6 = x7 = 0 x4 = ȳ1,2 + ȳ1,1 ( ȳ2,0 -1 2 ȳ2,1 cos ȳ2,0 ), x5 = 1 2 sin ȳ2,0 The states and their estimations are shown in Fig. 1 and2.

VI. CONCLUSIONS

A new method to carry out the state estimation has been proposed. By means of a zeroing manifold algorithm, provided that suitable conditions are satisfied, the state space whose dynamics is governed by a sole system of differential equations is found. This has allowed to apply standard techniques for the state and unknown input reconstruction. Nevertheless, the observability conditions allowing the state estimation can be checked also in terms of the original system with DAE. For a future work, one could look for considering a class of system with states having no explicit differential equations governing their dynamics.

2 + x 1 x 4 cos x 3 cos x 3 - 2 f * 5 = x 3 + 2 + x 1 x 4 cos x 3 cos x 3 - 2

 233253232 sin (x 3 ) + x 2 2x 3 -1 10 x 1 cos 2 sin(x 3 ) 2x 3 -1 10 x 1 cos 2 sin(x 3 )

• 3 1 3 | 3 Fig. 1 .

 3331 Fig.1. Original states x i (solid) and its estimate xi (dash-dotted), (i = 1,3).

7 Fig. 2 . 3 1 2 | 1 2 1 •ȳ2, 1 =

 7232111 Fig. 2. Original states x i (solid) and its estimate xi (dash-dotted), (i = 4,7).

Those neighborhoods should be calculated as large as possible in order to calculate an M k * as large as possible.

ACKNOWLEDGMENT F.J. Bejarano acknowledges INRIA Lille-Nord Europe, where he held a Post-doctoral position with Non-A team, and the support of Proyecto SIP 20141364 from IPN.