Yann Brenier

Xianglong Duan

FROM CONSERVATIVE TO DISSIPATIVE SYSTEMS THROUGH QUADRATIC CHANGE OF TIME, WITH APPLICATION TO THE CURVE-SHORTENING FLOW

Keywords: March 8, 2017 magnetohydrodynamics, fluid mechanics, geometric PDEs, curve shortening, hyperbolic conservation laws, dissipative solution. 1

We provide several examples of dissipative systems that can be obtained from conservative ones through a simple, quadratic, change of time. A typical example is the curve-shortening flow in R d , which is a particular case of mean-curvature flow with co-dimension higher than one (except in the case d = 2). Through such a change of time, this flow can be formally derived from the conservative model of vibrating strings obtained from the Nambu-Goto action. Using the concept of "relative entropy" (or "modulated energy"), borrowed from the theory of hyperbolic systems of conservation laws, we introduce a notion of generalized solutions, that we call dissipative solutions, for the curveshortening flow. For given initial conditions, the set of generalized solutions is convex, compact, if not empty. Smooth solutions to the curve-shortening flow are always unique in this setting.

Introduction

There are many examples of dissipative systems that can be derived from conservative ones. A classical example is the heat equation (or more generally the so-called "porous medium" equation) that can be derived from the Euler equations of isentropic gases. The derivation can be done in many different ways, for example by adding a very strong friction term or by homogenization techniques or by properly rescaling the time variable by a small parameter (through the so-called "parabolic scaling"). In the present paper, we will focus on a very straightforward idea (that does not seem to be popular, to the best of our knowledge): just perform the quadratic change of time t → θ = t 2 /2. In Section 1, we provide several examples: we start with the very simple example of conservative forces in classical mechanics (with the Galileo model of falling bodies as a borderline case). Next, we briefly retrieve from the Euler equation of isentropic gases the Darcy law and the porous medium equation, and, in particular the heat equation from the Euler equation of isothermal gases. Our third example, at the interface of Geometry and High Energy Physics, starts with the conservative evolution of classical strings according to the Nambu-Goto action, from which we get, by quadratic change of time, the dissipative geometric model of curve-shortening in R d , which is the simplest example of mean-curvature flow with co-dimension higher than 1:

(0.1) ∂ θ X = 1 |∂ s X| ∂ s ( ∂ s X |∂ s X| ),
where s → X(θ, s) describes a time-dependent curve in R d and | • | denotes the Euclidean norm.

In Section 2, we will finally discuss the system of PDEs

∂ θ B + ∇ • B ⊗ P -P ⊗ B ρ = 0, ∇ • B = 0, P = ∇ • B ⊗ B ρ , ρ = |B|
which turns out to be nothing but the "Eulerian version" (in R d ) of the curveshortening model (0.1). Typically, in the case of a single loop X subject to the curve-shortening flow, B would just be the singular vector-valued measure

(θ, x) → B(θ, x) = R/Z δ(x -X(θ, s))∂ s X(θ, s)ds ∈ R d ,
for which the system of PDE makes sense since all nonlinearities are homogeneous of degree one. (See Appendix 1 for more details.) These equations admit a "nonconservative" version (0.2)

∂ θ b + (v • ∇)b = (b • ∇)v + bv 2 , v = (b • ∇)b,
for the reduced variables b = B/|B| and v = P/|B|. For the conservative system, we define a concept of "dissipative solutions" related to the work of P.-L. Lions for the Euler equation of incompressible fluids [START_REF] Lions | Incompressible models[END_REF] (see also [START_REF] Brenier | Weak-strong uniqueness for measurevalued solutions[END_REF][START_REF] Vorotnikov | Global generalized solutions for Maxwell-alpha and Euler-alpha equations[END_REF]) or to the work of L. Ambrosio, N. Gigli, G. Savaré [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] for the heat equation and, overall, quite similar to the one recently introduced by the first author in [START_REF] Brenier | Topology preserving diffusion of divergence-free vector fields[END_REF]. We also refer to the works of A. Tzavaras and collaborators [START_REF] Demoulini | Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics[END_REF][START_REF] Lattanzio | Relative entropy in diffusive relaxation[END_REF], E. Feireisl and collaborators [START_REF] Feireisl | Singular Limits in Thermodynamics of Viscous Fluids[END_REF][START_REF] Feireisl | Dissipative measurevalued solutions to the compressible Navier-Stokes system[END_REF] for related concepts of "dissipative solutions". The main point of the present paper is to show how to get the formulation right: we start from the Eulerian version of the string equation, for which we can use the "relative entropy" method, quite classical in the theory of hyperbolic systems of conservation laws to get "weakstrong" uniqueness results (see [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF]), then, we apply the quadratic change of time to get a good concept of dissipative solutions for the curve-shortening flow, namely:

Definition 0.1. Let us fix T > 0 and denote T d = (R/Z) d . We say that (B, P ) with B ∈ C([0, T ], C(T d , R d ) ′ w * ), P ∈ C([0, T ] × T d , R d ) ′ is a dissipative solution of the curve-shortening flow with initial data B 0 ∈ C(T d , R d ) ′ if and only if: i) B(0) = B 0 , ∇ • B = 0 in sense of distributions;
ii) B and P are bounded, respectively in the spaces

C 1/2 ([0, T ], (C 1 (T d )) ′ w * ) and C([0, T ] × T d , R d ) ′ , by constants depending only on T and T d |B 0 |. iii) For all λ > 0, θ ∈ [0, T ], for all smooth trial functions (b * , v * , A) valued in R d , with A ∞ ≤ λ and b * 2 = 1, for all r ≥ c * + λ 2 2 + λ v * ∞
, where c * is a constant depending explicitly on (b * , v * ), we have:

(0.3) e -rθ η(θ) + θ 0 e -rσ P • (A -L 3 ) + r -c * - A • (A + 2v * ) 2 η -B • L 2 + b * A • (A + 2v * ) 2 (σ)dσ ≤ η(0).
where

(0.4) η = |B| -B • b * , (0.5) L 2 = -∂ θ b * -(v * • ∇)b * + (b * • ∇)v * + b * v * 2 -b * (b * • ∇)(b * • v * ), (0.6) L 3 = -v * + (b * • ∇)b * .
Here C(T 3 , R d ) ′ w * is metrizable space, we can equip a metric that is consistent with the weak-* topology. The "weak compactness" of such solutions (i.e. any sequence of dissipative solutions has accumulations points, in a suitable weak sense, and each of them is still a dissipative solution) directly follows from: Theorem 0.2. For fixed initial condition B 0 , the set of dissipative solutions, if not empty, is convex and compact for the weak-* topology of measures.

Notice that it is more challenging to prove that the set of dissipative solutions is not empty. The standard strategy is as follows: i) construct smooth approximate solutions (B ǫ , P ǫ ) with smooth approximate initial data B ǫ 0 ; ii) show that, the approximate solutions are relatively compact for the weak-* topology of measures, and, for any trial functions (b * , v * , A), satisfy inequalities (0.3) with some small error terms; iii) let ǫ go to zero, and prove that the limit (B, P ) is a dissipative solution.

To keep this paper simple, we leave this (important) step for a future work [START_REF] Duan | Magnetohydrodynamic regime of the Born-Infeld electromagnetism[END_REF], in the more general framework of the Born-Infeld theory [START_REF] Born | Foundations of the new field theory[END_REF][START_REF] Brenier | Hydrodynamic structure of the augmented Born-Infeld equations[END_REF]. Finally, we establish a "weak-strong" uniqueness principle in the following sense: Notice that this is not a full uniqueness result: only the homogeneous variables b = B/|B| and v = P/|B| get unique and a lot of room is left for the evolution of |B| itself. So the concept of dissipative solutions seems to suffer from the same type of ambiguity as the more general concept of Brakke solutions for mean-curvature flows [START_REF] Brakke | The Motion of a Surface by its Mean Curvature[END_REF].

Theorem 0.3. Let (b, v) ∈ C 1 ([0, T ] × T d ; R d × R d ),
Acknowledgements. This work has been partly supported by the contract : MAGA ANR-16-CE40-0014. The first author would like to thank the Erwin Schrödinger Institute (ESI) for its hospitality when this work was started. He is also very grateful to Dmitry Vorotnikov for explaining to him, at ESI in the summer of 2016, the possibility of deriving some mean-curvature motions as gradient flows in optimal transportation style, in the spirit of [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF][START_REF] Otto | The geometry of dissipative evolution equations: The porous medium equation[END_REF][START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF][START_REF] Villani | Optimal transport: Old and New[END_REF]. This information was very useful to understand that the methods used for the Born-Infeld equations by the second author [START_REF] Duan | Magnetohydrodynamic regime of the Born-Infeld electromagnetism[END_REF] could also be applied to the curve-shortening flow.

1. Examples of quadratic change of time 1.1. Quadratic change of time of a simple dynamical system. Under the quadratic change of time t → θ = t 2 /2, the ordinary dynamical system

d 2 X dt 2 = -∇ϕ(X) becomes -∇ϕ(X) = d dt (
d dθ [ϕ(X)] = -| dX dθ | 2 ,
in the asymptotic gradient flow regime. Furthermore, we may compare the respective solutions X(t) and Z(θ) of the dynamical system and the gradient flow, with initial conditions

X(t = 0) = X 0 = Z(θ = 0), dX dt (t = 0) = 0,
just by monitoring the "modulated energy" (or "relative entropy")

(1.1) 1 2 | dX dt -t dZ dθ | 2 + ϕ(X) -ϕ(Z) -∇ϕ(Z) • (X -Z),
provided ϕ is strongly convex with bounded third derivatives. We get, after elementary calculations,

(1.2) |X(t) -Z(t 2 /2)| 2 + | dX dt (t) -t dZ dθ (t 2 /2)| 2 ≤ Ct 5 , ∀t ∈ [0, T ],
where C is a constant that depends only on T , Z and potential ϕ. (Notice that the smallest expected error is O(t 6 ) as shown by the example d = 1, ϕ(x) = |x| 2 /2, for which X(t) = X(0) cos(t), while Z(θ) = X(0) exp(-θ).) More details on the concept of "modulated energy" and the proof of (1.2) can be found in an appendix at the end of this paper.

Remark: the Galileo experiment. The quadratic change of time t → θ = t 2 /2 remarkably fits with the famous experiment by Galileo, which was the starting point of modern classical mechanics: a rigid ball descends a rigid ramp of constant slope, with zero initial velocity and constant acceleration G, reaching position

X = x 0 + Gt 2 2
at time t. So, X is just a linear function of the rescaled time θ, X = x 0 + θG and we not only get

dX dθ + 2θ d 2 X dθ 2 = G but also simultaneously dX dθ = G, d 2 X dθ 2 = 0,
i.e. both gradient flow and inertial motion, with respect to the rescaled time θ.

1.2.

From the Euler equations to the heat equation and the Darcy law.

Let us now move to a PDE example and explain how the Darcy law and the "porous medium" equation (and, in particular, the standard heat equation) can be recovered by quadratic change of time from the Euler equations of isentropic compressible fluids. These equations read

(1.3) ∂ t ρ + ∇ • (ρv) = 0, ∂ t (ρv) + ∇ • (ρv ⊗ v) = -∇p,
where (ρ, p, v) ∈ R 1+1+3 are the density, pressure and velocity fields of the fluid, p being a given function of ρ (such as p = ρ, in the "isothermal" case). We set

(1.4) t → θ = t 2 /2, ρ(t, x) → ρ(θ, x), v(t, x) → v(θ, x) dθ dt .
(Notice the different scaling for v, enforcing v(t, x)dt → v(θ, x)dθ.) This leads, after short calculations, to

∂ θ ρ + ∇ • (ρv) = 0, ρv + 2θ[∂ θ (ρv) + ∇ • (ρv ⊗ v)] = -∇p(ρ).
In the regime θ >> 1, we get the asymptotic model of "pressureless" gas dynamics

∂ θ ρ + ∇ • (ρv) = 0, ∂ θ (ρv) + ∇ • (ρv ⊗ v) = 0,
while, as θ << 1, we recover the Darcy law and the porous medium equation

ρv = -∇p(ρ), ∂ θ ρ = △(p(ρ)),
and, in the isothermal case p = ρ, the heat equation

∂ θ ρ = △ρ.
1.3. From string motion to curve-shortening. Let us now move to a model at the interface of geometry and high energy physics. We consider a surface

(t, s) ∈ Ω ⊂ R 2 → (t, X(t, s)) ∈ R × R d ,
parameterized by a sufficiently smooth (at least Lipschitz continuous) function X over a bounded open space-time cylinder Ω. According to classical string theory (see [START_REF] Polchinski | String theory[END_REF], for instance), this surface is a relativistic string if and only if X is a critical point, with respect to all smooth perturbations, compactly supported in Ω, of the "Nambu-Goto Action" defined by

Ω ∂ s X 2 (1 -∂ t X 2 ) + (∂ t X • ∂ s X) 2 dtds
which is nothing but the area of the surface, in the space R × R d , with respect to the Minkowski metric (-1, +1,

• • •, +1). It is customary to regularize this setting by viewing (t, s) ∈ Ω ⊂ R 2 → (t, s, X(t, s)) ∈ R 2 × R d ,
as a graph in the enlarged space R 2 × R d and considering its area in the enlarged Minkowski space

R 2 × R d , with (rescaled) Minkowski metric (-1, +ǫ 2 , +1, • • •, +1): Ω (ǫ 2 + ∂ s X 2 )(1 -∂ t X 2 ) + (∂ t X • ∂ s X) 2 dtds.
[Of course, we recover the previous setting just as the special (and degenerate) case ǫ = 0.] The variational principle implies that X is a solution to the following first order partial differential system (of hyperbolic type as ǫ > 0):

(1.5) ∂ t (F ∂ t X -G∂ s X) -∂ s (G∂ t X + H∂ s X) = 0,
where

F = ǫ 2 + ∂ s X 2 S , G = ∂ t X • ∂ s X S , H = 1 -∂ t X 2 S , S = (ǫ 2 + ∂ s X 2 )(1 -∂ t X 2 ) + (∂ t X • ∂ s X) 2 .
After performing the quadratic change of time θ = t 2 /2, as we did in the previous subsections, while keeping only the zeroth order terms with respect to θ, we easily obtain, as asymptotic equation the nonlinear equation of parabolic type:

(1.6) ((ǫ 2 + ∂ s X 2 )I -∂ s X ⊗ ∂ s X)∂ θ X = ǫ 2 + ∂ s X 2 ∂ s ( ∂ s X √ ǫ 2 + ∂ s X 2 ). (Notice that (ǫ 2 + ∂ s X 2 )I -∂ s X ⊗ ∂ s X
is an invertible symmetric matrix with eigenvalues larger or equal to ǫ 2 .) In the limit case ǫ = 0 , we get

(1.7) (∂ s X 2 I -∂ s X ⊗ ∂ s X)∂ θ X = |∂ s X|∂ s ( ∂ s X |∂ s X| )
which becomes an ambiguous evolution equation, since it leaves ∂ θ X • ∂ s X undetermined. [As a matter of fact, this geometric equation is not modified by any smooth time-independent change of parameterization of the curve s → σ(s).] However, we may solve instead the simpler equation

(1.8) ∂ θ X = 1 |∂ s X| ∂ s ( ∂ s X |∂ s X| ).
Indeed, this is a consistent way of solving (1.7) since

∂ θ X • ∂ s X = ∂ s X |∂ s X| • ∂ s ( ∂ s X |∂ s X| ) = ∂ s ( ∂ s X • ∂ s X 2|∂ s X| 2 ) = 0.
Finally, by doing so, we have just recovered the familiar model of "curve-shortening" in the Euclidean space R d (see [START_REF] Deckelnick | Weak solutions of the curve shortening flow[END_REF], for instance).

1.4. The Eulerian form of the curve-shortening flow. The string equation (1.5) admits a useful "Eulerian" version (1.9)

∂ t B + ∇ • B ⊗ P -P ⊗ B ρ = 0, (1.10) ∇ • B = 0, ρ = B 2 + P 2 ,
(1.11)

∂ t P + ∇ • P ⊗ P ρ = ∇ • B ⊗ B ρ
(which reads, in coordinates,

∂ t B i + ∂ j (ρ -1 (B i P j -B j P i )) = 0, ∂ i B i = 0, ρ = B i B i + P i P i , ∂ t P i + ∂ j (ρ -1 (P i P j -B j B i )) = 0).
[As a matter of fact, defining

B(t, x) = R/Z δ(x -X(t, s))∂ s X(t, s)ds,
(which automatically satisfies ∇ • B = 0), assuming X to be smooth, not selfintersecting, with ∂ s X never vanishing, we get, after elementary calculations (similar to the ones done for the curve-shortening flow in Appendix 2, below), that B solves equations (1.9,1.10,1.11) together with

P (t, x) = R/Z δ(x -X(t, s)) ∂ s X 2 I -∂ s X ⊗ ∂ s X ∂ t X(t, s) ∂ s X 2 (1 -∂ t X 2 ) + (∂ t X • ∂ s X) 2 ds.]
Importantly enough, this system admits an extra conservation law:

(1.12)

∂ t ρ + ∇ • P = ∇ • (P • B)B ρ 2 , ρ = B 2 + P 2 ,
which describes the local conservation of energy. [This is easy to check. Indeed, using coordinates, we find

∂ t ρ = B i ∂ t B i + P i ∂ t P i ρ = B i ρ ∂ j B j P i -B i P j ρ + P i ρ ∂ j B j B i -P i P j ρ
and notice that the second and fourth terms of the right-hand side combine as:

-P j ∂ j P 2 + B 2 2ρ 2 - P 2 + B 2 ρ 2 ∂ j P j = -∂ j P j
(since ρ 2 = B 2 + P 2 ), while the first and third terms give:

∂ j P i B j B i ρ 2
(using ∇ • B = 0), which leads to the "entropy conservation law" (1.12).]

Let us now perform the quadratic change of time:

t → θ = t 2 2 , B → B, P → dθ dt P,
which leads, as θ << 1, to the asymptotic system (1.13)

∂ θ B + ∇ • B ⊗ P -P ⊗ B ρ = 0, (1.14) ∇ • B = 0, ρ = |B|, (1.15
) P = ∇ • B ⊗ B ρ .
Notice that this implies B • P = 0, since, in coordinates,

B • P = B i ∂ j B i B j ρ = ρ B i ρ B j ∂ j B i ρ = ρB j ∂ j B 2 2ρ 2 = 0 (using ∇ • B = 0 and |B| = ρ).
We also get the extra equation, derived from (1.12),

(1.16)

∂ θ ρ + P 2 ρ + ∇ • P = 0, ρ = |B|. [Indeed, √ B 2 + 2θP 2 = |B| + θP 2 |B| + O(θ 2
), which leads to

∂ θ B 2 + 2θP 2 = ∂ θ |B| + P 2 |B| + O(θ).
We also used B • P = 0.] Notice that this equation is no longer in conservation form, due to the emergence of the dissipation term P 2 /ρ after the quadratic change of time t → θ. Equations (1.13,1.16) also provide a priori bounds for any smooth solutions B and P on the flat torus T d (that we have already taken into account in Definition 0.1):

T d |B(θ)| ≤ T d |B(0)|, ∀θ ∈ [0, T ], T 0 T d P 2 |B| ≤ T d |B(0)|, T 0 T d |P | ≤ T 0 T d P 2 |B| T 0 T d |B| ≤ √ T T d |B(0)|,
and, for 0 ≤ θ 0 ≤ θ 1 ≤ T , and any smooth vector field ϕ = ϕ(x),

T d B i (θ 1 ) -B i (θ 0 ) ϕ i 2 = θ1 θ0 T d (∂ j ϕ i -∂ i ϕ j ) P j B i |B| 2 ≤ Lip(ϕ) 2 (θ 1 -θ 0 ) θ1 θ0 T d |P | 2
(where Lip(ϕ) denotes the Lipschitz constant of ϕ),

≤ Lip(ϕ) 2 (θ 1 -θ 0 ) θ1 θ0 T d P 2 |B| T d |B| ≤ Lip(ϕ) 2 (θ 1 -θ 0 ) T d |B(0)| 2 , which shows that B is bounded in C 1/2 ([0, T ], (C 1 (T d )) ′ w * )
ρv i = P i = ∂ j B i B j ρ = ∂ j (ρb i b j ) = ρb j ∂ j b i (since ∂ j (ρb j ) = ∂ j B j = 0). Next, (1.13) becomes ρ(∂ θ b i + v j ∂ j b i -b j ∂ j v i ) = -b i (∂ θ ρ + ∂ j (ρv j )) = b i ρv 2
(thanks to (1.16)). So we have obtained

(1.17) ∂ θ b + (v • ∇)b = (b • ∇)v + bv 2 , v = (b • ∇)b,
(which is consistent with b 2 = 1 and b • v = 0 as can be easily checked). Notice that (1.16) can be written according to the non-conservative variables as

(1.18) ∂ θ ρ + ∇ • (ρv) = -ρv 2 ,
which is a linear equation in ρ.

Analysis of the Eulerian curve-shortening flow

2.1. Relative entropy for the Eulerian equations for strings. We start from the "Eulerian" version (1.9,1.10,1.11) of the string equation (1.5). This system belongs to the class of systems of conservation laws:

∂ t V + ∇ • F (V ) = 0,
where F is a given function and V is a vector-valued function (for us V = (B, P )). If such a system admits an extra conservation law

∂ t E(V ) + ∇ • G(V ) = 0,
for a pair of functions E, G, with E strictly convex, then the system is automatically "hyperbolic" (i.e. well posed, at least for short time), under minor additional conditions [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF], and E is often called an "entropy" for the system (although it should be called "energy" for a large class of applications). The system for strings (1.9,1.10,1.11) admits such an extra conservation law, namely (1.12), with V = (B, P ) → E(V ) = √ B 2 + P 2 as entropy function. [Notice, however, that the entropy ρ is not strictly convex.] Hyperbolic systems of conservation laws with convex entropy enjoy a "weakstrong uniqueness" principle [START_REF] Dafermos | Hyperbolic conservation laws in continuum physics[END_REF], based on the concept of "relative entropy" (also called "modulated energy" or "Bregman divergence", depending of the frameworks):

η(V, V * ) = E(V ) -E(V * ) -∇E(V * ) • (V -V * )
, which is just the discrepancy between E at point V and its linear approximation about a given point V * . (Observe that, as E is a convex function with Hessian bounded away from zero and infinity, the relative entropy behaves as |V -V * | 2 .) Notice that the relative entropy is as convex as the entropy as a function of V (V * being kept fixed) since it differs just by an affine term.

In the case of system (1.9,1.10,1.11), the relative entropy density is defined, for (B, P

) ∈ R d × R d and (b * , v * ) ∈ R d × R d , by η = |B -ρb * | 2 + |P -ρv * | 2 2ρ = ρ 1 + b * 2 + v * 2 2 -B • b * -P • v * , ρ = B 2 + P 2 ,
which is convex in (B, P ). (Notice that, at this stage, we do not assume b * 2 + v * 2 = 1, which would be natural to define the relative entropy but would lead to contradictions after performing the quadratic change of time as will be done in the next subsection.)

Let us now consider a smooth, Z d -periodic in space, solution (B, P )(t, x) of equations (1.9,1.10,1.11) and monitor the evolution, on a fixed time interval [0, T ], of the integral of η over (R/Z) d , for some smooth trial functions:

(t, x) ∈ [0, T ] × (R/Z) d → (b * (t, x) ∈ R d , v * (t, x) ∈ R d ).
After tedious and elementary calculations, we find (2.1)

d dt η = 1 2ρ (B i -ρb * i )(B j -ρb * j )(∂ j v * i + ∂ i v * j ) - 1 2ρ (P i -ρv * i )(P j -ρv * j )(∂ j v * i + ∂ i v * j ) - 1 ρ (B i -ρb * i )(P j -ρv * j )(∂ j b * i -∂ i b * j ) + η • L 1 + B • L 2 + P • L 3 - (P • B)B ρ 2 • ∇ b * 2 + v * 2 2
where

L 1 = ζ * 1 + b * 2 + v * 2 , ζ * = D * t (b * 2 + v * 2 ) -2b * • ∇(b * • v * ), D * t = (∂ t + v * • ∇), L 2 = -D * t b * + (b * • ∇)v * + ∇(b * • v * ) + b * L 1 L 3 = -D * t v * + (b * • ∇)b * + v * L 1 2.2.
Relative entropy and quadratic change of time. After the quadratic change of time,

t → θ = t 2 2 , ρ → ρ, B → B, b * → b * , P → θ ′ (t)P, v * → θ ′ (t)v * , we get θ ′ (t) 2 = 2θ, θ ′′ (t) = 1, d dθ η = 1 2ρ (B i -ρb * i )(B j -ρb * j )(∂ j v * i + ∂ i v * j ) - θ ρ (P i -ρv * i )(P j -ρv * j )(∂ j v * i + ∂ i v * j ) - 1 ρ (B i -ρb * i )(P j -ρv * j )(∂ j b * i -∂ i b * j ) + η • L 1 + B • L 2 + P • L 3 - (P • B)B ρ 2 • ∇ b * 2 + 2θv * 2 2 ,
where

L 1 = 1 + b * 2 + 2θv * 2 -1 ζ * , ζ * = D θ (b * 2 + 2θv * 2 ) -2b * • ∇(b * • v * ), D * θ = (∂ θ + v * • ∇), L 2 = -D * θ b * + (b * • ∇)v * + ∇(b * • v * ) + b * L 1 L 3 = -v * -2θD * θ v * + (b * • ∇)b * + 2θv * L 1 .
Now, in order to address the Eulerian curve-shortening system, we want to drop the terms of order O(θ) and limit ourself to the case when b * 2 = 1. However, we have to be very careful about all terms involving ∂ θ . This happens first in the definition of ζ * , because of the term

D θ (b * 2 + 2θv * 2 ) = D θ (b * 2 ) + 2v * 2 + O(θ).
So, in the limit θ = 0, with b * 2 = 1, we find

ζ * = 2v * 2 -2b * • ∇(b * • v * ) and, therefore, L 1 = v * 2 -b * • ∇(b * • v * ).
Similarly, we have to take care of

d dθ η.
where

η = 1 + b * 2 + 2θv * 2 2 ρ -B • b * -2θP • v * and ρ = B 2 + 2θP 2 = |B| + θ P 2 |B| + O(θ 2 )
We get

∂ θ ρ = ∂ θ |B| + P 2 |B| + O(θ), ∂ θ η = v * 2 ρ + ∂ θ ( 1 + b * 2 2 ρ) -∂ θ (B • b * ) -2P • v * + O(θ)
This leads, as b * 2 = 1, to

ρ = |B| + O(θ), η = |B| -B • b * + O(θ), ∂ θ η = v * 2 ρ + ∂ θ |B| + P 2 |B| -∂ θ (B • b * ) -2P • v * + O(θ) = ∂ θ (|B| -B • b * ) + (P -|B|v * ) 2 |B| + O(θ) = ∂ θ (ρ -B • b * ) + (P -ρv * ) 2 ρ + O(θ).
Finally, after dropping the terms of order O(θ) and limiting ourself to the case when b * 2 = 1, we have found

L 1 = v * 2 -b * • ∇(b * • v * ), L 2 = -D * θ b * + (b * • ∇)v * + ∇(b * • v * ) + b * L 1 = -D * θ b * + (b * • ∇)v * + ∇(b * • v * ) + b * (v * 2 -b * • ∇(b * • v * )), D * θ = (∂ θ + v * • ∇), L 3 = -v * + (b * • ∇)b * ,
and, for all smooth trial field b * such that b * 2 = 1, (2.2)

d dθ η + (P -ρv * ) 2 ρ = 1 2ρ (B i -ρb * i )(B j -ρb * j )(∂ j v * i + ∂ i v * j ) - 1 ρ (B i -ρb * i )(P j -ρv * j )(∂ j b * i -∂ i b * j ) + η L 1 + B • L 2 + P • L 3 ,
where

ρ = |B|, η = ρ -B • b * = |B| -B • b * = (B -|B|b * ) 2 2|B| = (B -ρb * ) 2 2ρ .
Using the Cauchy-Schwarz inequality, we can find a constant c * depending only on b * and v * such that

d dθ η + |P -ρv * | 2 2ρ ≤ c * η dx + B • L 2 + P • L 3 .
This implies, for any constant r ≥ c * ,

(-r + d dθ ) η + |P -ρv * | 2 2ρ + (r -c * ) η dx ≤ B • L 2 + P • L 3
and, after multiplying this inequality by e -rθ and integrating in time σ ∈ [0, θ],

(2.3) e -rθ η(θ)+ θ 0 e -rσ (r -c * ) η + |P -ρv * | 2 2ρ -R (σ)dσ ≤ η(0), where R = B • L 2 + P • L 3 , L 2 = -D * θ b * + (b * • ∇)v * + b * v * 2 -b * (b * • ∇)(b * • v * ), L 3 = -v * + (b * • ∇)b * . We can write |P -ρv * | 2 2ρ = sup A (P -ρv * ) • A -ρ A 2 2 = sup A P • A -(η + B • b * ) v * • A + A 2 2 (since η = ρ -B • b *
) and substitute for inequality (2.3) the family of inequalities (2.4) e -rθ η(θ)

+ θ 0 e -rσ P • (A -L 3 ) + r -c * - A • (A + 2v * ) 2 η -B • L 2 + b * A • (A + 2v * ) 2 (σ)dσ ≤ η(0).
Observe that these inequalities are convex in (B, P ) as long as r is chosen so that

r ≥ c * + sup θ,x A • (A + 2v * ) 2 .
However, this creates a problem, since r must depend on A. This is why we input a cut-off parameter λ > 0 and assume that the trial functions A are chosen with |A(θ, x)| ≤ λ. By doing this, the advantage is that we maintain the convexity of inequality as long as r is chosen big enough only as a function of b * , v * and λ, namely:

r ≥ c * + λ 2 2 + λ v * ∞ .
The price to pay is that we cannot fully recover

|P -ρv * | 2 2ρ
by taking the supremum over all A such that |A| ≤ λ, but only the λ-approximation K λ (ρ, P -ρv * ), where

K λ (ρ, Z) = sup |A|≤λ Z • A -ρ A 2 2 = |Z| 2 2ρ - (|Z| -λρ) 2 + 2ρ ≥ 0.
Observe that, by doing so, we keep a good control of the distance between P and ρv * , since (as can be easily checked)

(2.5) K λ (ρ, P -ρv * ) ≥ min (P -ρv * ) 2 2ρ , λ|P -ρv * | 2 .
So, the supremum of inequalities (2.4) over all trial functions A such that |A| ≤ λ, is equivalent to

(2.6) e -rθ η(θ) + θ 0 e -rσ ( K λ (ρ, P -ρv * ) + (r -c * )η -R)(σ)dσ ≤ η(0).
Now let us consider (B, P ) not only as functions but also as vector-valued Borel measures, for which (2.4) is still well-defined. The λ-approximation K λ (ρ, P -ρv * ) can be interpreted as a function of measures [START_REF] Demengel | Convex functions of a measure and applications[END_REF] and (2.6) is equivalent to (2.4) in the sense that,

θ 0 e -rσ K λ (ρ, P -ρv * ) = sup A∈C 0 A ∞≤λ θ 0 e -rσ (P -ρv * ) • A -ρ A 2 2 .
Notice that, due to the convexity of K λ , we have (

θ 0 e -rσ K λ (ρ, P -ρv * )(σ)dσ ≥ e -rθ θ 0 K λ ρ(σ), |P -ρv * |(σ) dσ. 2.7)
With these calculations, we have recovered the concept of dissipative solutions as given in Definition 0.1. Then, the proof of our main results becomes straightforward.

Proof of Theorem 0.2. We just have to show that, for fixed initial conditions B 0 , the set of dissipative solutions, as defined by Definition 0.1, if not empty, is convex and compact for the weak-* topology of measures. The convexity of the set of solutions is almost free. It follows directly from the convexity of inequalities (0.3). Let's focus on the compactness. Our goal is to prove that, if {(B n , P n )} n∈N is a sequence of dissipative solutions with initial data B 0 , then up to a subsequence, it converges in the weak-* topology of measures to a dissipative solution (B, P ) with the same initial data. This follows from the inequalities (0.3) and suitable bounds that we assume for B n and P n . To see this, let's first show that, sup θ |B n (θ)| is uniformly bounded. (Indeed, let's take b * = (1, 0, . . . , 0),

v * = A = 0 in (0.3). Then we have |B n (θ)| -B 1 n (θ) ≤ C |B 0 |, ∀θ ∈ [0, T ]. Since B n is bounded in C 1/2 ([0, T ], (C 1 (T d )) ′ w * ),

Appendix 1: modulated energy and dissipative solutions for ordinary dynamical equations

In this appendix, we explain, in the very elementary case of our dynamical system, the concepts of "modulated energy" (also called "relative entropy") and "dissipative formulation", which will later be used and extended to the dissipative setting.

Here, we crucially assume that the potential ϕ is convex and, in order to keep the presentation simple, we assume that the spectrum of the symmetric matrix D 2 ϕ(x) is uniformly contained in some fixed interval [r, r -1 ] for some constant r ∈ (0, 1/2). We further assume that the third derivatives of ϕ are bounded. The total energy of a curve t → X(t) is defined by

1 2 |X ′ (t)| 2 + ϕ(X(t)) (where X ′ (t) = dX dt )
and is a constant as X is solution to the dynamical system

X ′′ (t) = -∇ϕ(X(t)).
Given a smooth curve t → Y (t), we define the "modulated energy" (or "relative entropy") of X at time t with respect to Y by expanding the energy about Y at X:

η[t, X, Y ] = 1 2 |X ′ (t) -Y ′ (t)| 2 + ϕ(X(t)) -ϕ(Y (t)) -∇ϕ(Y (t)) • (X(t) -Y (t)).
Because of the assumption we made on ϕ, η is a perfect substitute for the squared distance between (X, X ′ ) and (Y, Y ′ ):

r ≤ 2η[t, X, Y ] |X -Y | 2 + |X ′ -Y ′ | 2 ≤ r -1 .
We get

d dt η[t, X, Y ] = (X ′ -Y ′ ) • (X ′′ -Y ′′ ) + ∇ϕ(X) • X ′ -∇ϕ(Y ) • Y ′ -∇ϕ(Y ) • (X ′ -Y ′ ) -Y ′ • D 2 ϕ(Y ) • (X -Y ) = (X ′ -Y ′ )•(X ′′ +∇ϕ(X)-Y ′′ -∇ϕ(Y ))+Y ′ •(∇ϕ(X)-∇ϕ(Y )-D 2 ϕ(Y )•(X -Y )).
We first observe that

|∇ϕ(X) -∇ϕ(Y ) -D 2 ϕ(Y ) • (X -Y )| ≤ C|X -Y | 2 ≤ Cη[t, X, Y ]
where, from now on, C is a generic constant that depends only on ϕ or Y . So,

d dt η[t, X, Y ] -(X ′ -Y ′ ) • (X ′′ + ∇ϕ(X) -Y ′′ -∇ϕ(Y )) ≤ Cη[t, X, Y ]
and then, after integration in time for t ∈ [0, T ], T > 0 being an arbitrarily chosen fixed time,

) η[T, X, Y ] - T 0 (X ′ (t) -Y ′ (t)) • (ω X (t) -ω Y (t))e (T -t)C dt ≤ η[0, X, Y ]e CT , (3.1
where

ω Z (t) = Z ′′ (t) + ∇ϕ(Z(t))
Let us exploit inequality (3.1) in several different ways. First, we see that for a curve X it is equivalent to be solution of the dynamical system, i.e. ω X = 0 or to satisfy

(3.2) η[T, X, Y ] + T 0 (X ′ (t) -Y ′ (t)) • ω Y (t)e (T -t)C dt ≤ η[0, X, Y ]e CT , ∀T > 0,
for any smooth curve Y , where C is a constant depending only on Y (up to time T ) and ϕ. Indeed, by taking as Y the unique solution of the dynamical system with initial conditions Y (0) = X(0), Y ′ (0) = X ′ (0) provided by the Cauchy-Lipschitz theorem on ODEs, we get both ω Y = 0 and η[0, X, Y ] = 0. Thus inequality (3.2) just says η[T, X, Y ] = 0 for all T > 0, which means X = Y and, therefore, X is indeed a solution to the dynamical system. Thus, we can take (3.2) as an alternative notion of solution, that we call "dissipative solution". This inequality has the advantage to be convex in X, as the initial conditions X(0), X ′ (0) are fixed, and therefore preserved under weak convergence of (X, X ′ ).

Next, we use (3.2) to compare a solution X of the dynamical system with zero initial velocity, i.e. X ′ (0) = 0, to the solution Z of the gradient flow equation

Z ′ (θ) + ∇ϕ(Z(θ)) = 0, with initial condition Z(0) = X(0). Indeed, let us set Y (t) = Z(θ), θ = t 2 /2. Then Y ′ (t) = tZ ′ (θ), Y ′ (0) = 0, Y (0) = Z(0) = X(0), Y ′′ (t) = Z ′ (θ) + t 2 Z ′′ (θ), which implies η[0, X, Y ] = 0 and ω Y (t) = t 2 Z ′′ (θ). So, (3.2) gives η[T, X, Y ] + T 0 (X ′ (t) -Y ′ (t)) • t 2 Z ′′ (t 2 /2)e (T -t)C dt ≤ 0,
which implies (by Cauchy-Schwarz inequality and by definition of η)

η[T, X, Y ] ≤ C T 0 (η[t, X, Y ] + t 4 )dt,
where C is a generic constant depending only on T , ϕ and Z. By Gronwall's lemma, we conclude that η[t, X, Y ] ≤ Ct 5 which implies, by definition of η,

|X(t) -Z(t 2 /2)| 2 + | dX dt (t) -t dZ dθ (t 2 /2)| 2 ≤ Ct 5 , ∀t ∈ [0, T ],
as already claimed, at the beginning of this subsection. (See (1.2).) (Notice that the smallest expected error is O(t 6 ) as shown by the example d = 1, ϕ(x) = |x| 2 /2, for which X(t) = X(0) cos(t), while Z(θ) = X(0) exp(-θ).)

Appendix 2: direct recovery of the Eulerian curve-shortening flow

For the sake of completeness, let us check that system (1.13,1.14,1.15) indeed describes the curve-shortening flow in R d , for a continuum of non intersecting curves. Let us do the calculation in the case of a single smooth time-dependent loop, s ∈ R/Z → X(θ, s), that we assume to be non self-intersecting at every fixed time θ, and such that ∂ s X never vanishes. We introduce (as a distribution, or, if one prefers, as a "1-current") B(θ, x) = R/Z δ(x -X(θ, s))∂ s X(θ, s)ds, which automatically satisfies ∇ • B = 0. Since X is smooth, not self-intersecting, and ∂ s X never vanishes, by assumption, we may find a smooth vector field v(θ, x) such that ∂ θ X(θ, s) = v(θ, X(θ, s)) that we can interpret as the "Eulerian velocity field" attached to the loop evolution. We also introduce the nonnegative field ρ(θ, x) = R/Z δ(x -X(θ, s))|∂ s X(θ, s)|ds which can also be interpreted as |B(θ, x)| since X is supposed to be non selfintersecting. We get (using indices i, j, k ∈ {1, • • •, d} with implicit summation on repeated indices)

∂ θ B i (θ, x) = R/Z
[-(∂ j δ)(x -X(θ, s))∂ θ X j (θ, s)∂ s X i (θ, s) +δ(x -X(θ, s))∂ 2 sθ X i (θ, s)]ds (in distributional sense) = -R/Z (∂ j δ)(x -X(θ, s))[∂ θ X j (θ, s)∂ s X i (θ, s) -∂ θ X i (θ, s)∂ s X j (θ, s)]ds (after integration by part in s ∈ R/Z of the second term) = -∂ j R/Z δ(x -X(θ, s))[∂ θ X j (θ, s)∂ s X i (θ, s) -∂ θ X i (θ, s)∂ s X j (θ, s)]ds.

So

∂ θ B(θ, x) = -∇• R/Z δ(x-X(θ, x))(∂ s X(θ, s)⊗∂ θ X(θ, s)-∂ θ X(θ, s)⊗∂ s X(θ, s))ds.

Then we can write

∂ θ B(θ, x)+∇• R/Z
δ(x-X)(∂ s X(θ, s)⊗v(θ, X(θ, s))-v(θ, X(θ, s))⊗∂ s X(θ, s))ds = 0 which exactly means, by definition of B, (4.1)

∂ θ B + ∇ • (B ⊗ v -v ⊗ B) = 0.
Since X is assumed to be non-intersecting, by definition of v, we may write (|B|v)(θ, x) = R/Z δ(x -X(θ, s))|∂ s X(θ, s)|v(θ, X(θ, s))ds

So far, we have not used equation (1.8), namely

∂ θ X = 1 |∂ s X| ∂ s ( ∂ s X |∂ s X| ),
Let us do it now: Finally we can write (4.1) as

(ρv i )(θ, x) = (|B|v i )(θ, x) = R/Z δ(x -X(θ, s))∂ s ( ∂ s X i
∂ θ B + ∇ • ( B ⊗ P -P ⊗ B ρ ) = 0, ∇ • B = 0, P = ∇ • B ⊗ B ρ , ρ = |B|,
where P stands for ρv and (B, ρ, P ) solves equations (1.13,1.14,1.15). So far, our claim has been justified only in the case of a single loop. We next argue that, due to its homogeneity of degree 1, equations (1.13,1.14,1.15), in spite of their nonlinearity, enjoy a nice superposition principle, in the sense that we may still get a solution by superposing several smooth curves subject to curve-shortening as long as they do not intersect and we may even build smooth solutions by using a continuum of such curves. This concludes the proof of our claim that equations (1.13,1.14,1.15), are the "Eulerian formulation" of the curve-shortening flow. Notice that similar calculations can also be performed to justify the Eulerian version (1.9,1.10,1.11) of the string equation (1.5).

  be a smooth solution of the non-conservative form of the curve-shortening flow (0.2) with b 2 = 1. Then any dissipative solution satisfies B = |B|b and P = |B|v, as soon as B(0) = |B(0)|b(0).

  by a constant depending only on T d |B(0)| and T . Equations (1.13,1.14,1.15) can also be written in non-conservative form in terms of b We already have b 2 = 1 and b • v = 0. Using coordinates, we first get from (1.15)

  there exists a constant C ′ such that for any n, θ, | B 1 n (θ) -B 1 0 | ≤ C ′ . So we get a uniform upper bound of |B n (θ)|.) Therefore, for any θ ∈ [0, T ], the set {B n (θ)} n∈N is relatively compact for the weak-* topology of C(T d , R d ) ′ . Next, we look at the map [0, T ] → C(T d , R d ) ′w * , t → B(t). This map is equicontinuous because of the assumption on B n . Then, by Arzelà-Ascoli's theorem, there existsB ∈ C([0, T ], C(T d , R d ) ′ w * ), such that, up to a subsequence, B n (θ) ⇀ * B(θ), ∀θ ∈ [0, T ]. Now, since |P n | is bounded, there exists P ∈ C([0, T ] × T d , R d ) ′, such that P n ⇀ * P . Then because inequalities (0.3) are stable under weak-* convergence, we can prove that the limit (B, P ) satisfies all the requirements in Definition 0.1, therefore, it is also a dissipative solution with initial data B 0 .Proof of Theorem 0.3. Let (b, v) be a smooth solution of the non-conservative form of the curve-shortening flow (0.2) with b 2 = 1, which directly implies b • v = 0. We have to show that any dissipative solution satisfies B = |B|b and P = |B|v, as soon as B(0) = |B(0)|b(0). The proof is quite straightforward: we already have η(0) = 0 since B(0) = |B(0)|b(0). Next, we set b * = b, v * = v, A = 0 and fix λ > 0 in definition (0.3). Since we have (0.2) and b • v = 0, we get L 2 = L 3 = 0. Since η ≥ 0, the inequality 0.3 directly implies η = 0, ∀θ ∈ [0, T ], and, therefore B = |B|b. Now, let's go back to the inequality 0.3 which is already simplified since η = 0. By taking the supremum over all A such that A ∞ ≤ λ, we get T 0 e -rσ K λ (ρ, P -|B|v) ≤ 0. Using (2.7) we deduce T 0 K λ ρ(σ), P -|B|v (σ) dσ = 0, and, therefore, P = |B|v (because of (2.5)), which completes the proof.

  δ)(x -X(θ, s)) ∂ s X j ∂ s X i |∂ s X| ds (after integrating by part in s ∈ R/Z) = ∂ j R/Z δ(x -X(θ, s)) ∂ s X j ∂ s X i |∂ s X| ds that we can interpret as ρv = ∇ • B ⊗ B |B| = ∇ • B ⊗ B ρ .


Gang Zheng Yingchong Ma

Wilfrid Perruquetti

Zhaopeng

Zhaopeng Qiu

Control of nonholonomic wheeled mobile robots via i-PID controller

come

I. INTRODUCTION

The problem of wheeled mobile robot control has been widely studied and attracted the interest of many researches because of its wide application in industries and theoretical challenges [START_REF] Laumond | Robot Motion Planning and Control[END_REF], [START_REF] Kolmanovsky | Developments in nonholonomic control problems[END_REF]. Generally the robot control problem can be divided into two main problems: the trajectory tracking problem and stabilization problem. The control problem of trajectory tracking can also be categorized into two types: linear control and nonlinear control. [START_REF] Oelen | Robust tracking control of two-degrees-offreedom mobile robots[END_REF] proposed a linear controller which is robust to the perturbation in robot velocity control. Separated feedback loops control for robot position and velocity was used in [START_REF] Chung | A position control differential drive wheeled mobile robot[END_REF]. The kinematic model of the robot was linearized in [START_REF] Pears | Mobile robot tracking of pre-planned paths[END_REF], and in which a proportional linear control was applied. The famous PID controller was applied in [START_REF] Normey-Rico | Mobile robot path tracking using a robust pid controller[END_REF], in which a simple linearized mobile robot model is used.

The linear control indeed has great advantages because of its simplicity in linear control theory, while however when comparing with nonlinear control its robustness are very limited. In linear control the initial states are often required to stay close to the reference to ensure the stability, instead nonlinear control is able to guarantee the stability without this kind of problems. Moveover, it is known that the feedback stabilization at a given posture cannot be obtained by smooth time-invariant control [START_REF] Campion | Modelling and state feedback control of nonholonomic mechanical systems[END_REF], this implies that the problem is truly nonlinear, and linear control is ineffective here. For nonlinear nonholonomic robot systems, there are usually open loop controls where the inputs are calculated from the reference trajectory [START_REF] De Wit | Exponential stabilization of mobile robots with nonholonomic constraints[END_REF], flatness based control [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF] is a kind of open-loop control, whose robustness can be strengthened [START_REF] Ryu | Differential flatness-based robust control of a two-wheeled mobile robot in the presence of slip[END_REF], which is widely applied in optimal control problems. However, it is well known that the openloop control is not robust to modeling errors so that it cannot guarantee the mobile robot to move along the desired trajectory. Nonlinear feedback control for mobile robots is used in [START_REF] Samson | Feedback control of a nonholonomic wheeled cart in cartesian space[END_REF] to solve the trajectory tracking problem, and the dynamic feedback linearization is also used in [START_REF] Novel | Control of nonholonomic wheeled mobile robots by state feedback linearization[END_REF]. [START_REF] Tayebi | Path following control law for an industrial mobile robot[END_REF] proposed a nonlinear control law based on partial state feedback linearization and Lyapunov's direct method, but the disturbance and uncertainty were not considered in the control design.

Recently, an intelligent PID controller (i-PID controller) introduced in [START_REF] Fliess | Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control?[END_REF] exhibits the robustness to the unmodeled dynamics and disturbance in the system [START_REF] Fliess | Revisiting some practical issues in the implementation of model-free control[END_REF], and it has been widely studied and applied to many electrical and mechanical processes [START_REF] Riachy | High-order sliding modes and intelligent PID controllers: First steps toward a practical comparison[END_REF], [START_REF] Villagra | Robust motion control for humanoid robot flexible joints[END_REF]. This paper aims at applying the socalled i-PID controller to the nonholonomic robots in order to control the robot with measurement disturbance. However, due to the particularity of the nonholonomic system, this controller can not be simply applied, for this a switching parameter is selected and a robust controller is proposed to control the robot with measurement disturbance.

The paper is structured as follows. Section II presents the problem statement. Section III explains the determination of the controller. Simulation results are detailed in Section IV.

II. PROBLEM STATEMENT

This paper considers the unicycle-type mobile robot whose kinematic model under the nonholonomic constraint of pure rolling and no slipping can be described as follows:

ẋ = ν cos θ ẏ = ν sin θ θ = ω (1)
where ν and ω are linear and angular velocity respectively, x and y represent the location of the robot, θ is the orientation of the robot with respect to x-axis(see Fig. 1). It can be shown that x and y are flat outputs for the studied system [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF]. Indeed, θ, υ and ω can all be expressed by x, y and their first and second-order derivatives as follows:

-

Om ≡ M * x y Kθ O - → x - → y K - → x m - → y m
   θ = arctan ẏ ẋ υ = ẋ2 + ẏ2 ω = ÿ ẋ-ẍ ẏ ẋ2 + ẏ2
(2) Suppose that we can only measure the position (x, y) of the robot, which implies that the relative degree of those measurements is equal to 1, since the first part of system (1) is of the following form:

ẋ ẏ = cos θ 0 sin θ 0 v ω
It is clearly that the second control input ω is not involved in the above equation, since the decoupling matrix is singular. Due to this fact, one classic solution is to add an integrator to the first input in order to overcome the singularity of the decoupling matrix [START_REF] Ge | Autonomous Mobile Robots: Sensing, Control, Decision Making and Applications[END_REF]. For this, let us consider the following extended system:

     ẋ = v cos θ ẏ = v sin θ θ = ω v = ξ with u = [ξ, ω]
being the new input. One can check that, with the extended system, the relative degree for both output is equal to 2. Then one obtains:

ẍ ÿ = G(x, y, ẋ, ẏ)u (3)
where

G = cos θ -ν sin θ sin θ ν cos θ = cos(arctan ẏ ẋ ) -ẋ2 + ẏ2 sin(arctan ẏ ẋ ) sin(arctan ẏ ẋ ) ẋ2 + ẏ2 cos(arctan ẏ ẋ ) (4)
which is invertible

if v = ẋ2 + ẏ2 = 0 .
If there is no disturbance in the measurement, a classic PID controller, which needs the exact value of G -1 , can be used to achieve non-vanishing Cartesian trajectories tracking (the linear velocity of the robot is assumed to be always nonzero), since G is singular when v = 0. It has been shown ( [START_REF] Datta | Structure and Synthesis of PID Controllers, ser. Advances in Industrial Control[END_REF]) that this method cannot be used to stabilize the robot to a static point due to the same reason.

In addition, noises and disturbance are inevitable in real situations, thus the exact computation of G -1 in (4) cannot be obtained. Consider the output under disturbance as

Y = [x, y] T + D, where D = [d 1 , d 2 ]
T is the disturbance in the measurement, Thus the estimated values of θ, v and ω are disturbed:

θ d = arctan ẏ+ ḋ2 ẋ+ ḋ1 υ d = ( ẋ + ḋ1) 2 + ( ẏ + ḋ2) 2 (5)
thus we have

Ÿ = G(Y, Ẏ )u + D (6)
with

G(Y, Ẏ ) = cos(θ d ) -υ d sin(θ d ) sin(θ d ) υ d cos(θ d ) (7)
where θ d and υ d are defined in [START_REF] Pears | Mobile robot tracking of pre-planned paths[END_REF].

It is clear that the system (6) can not be controlled with the classical PID controller, since G(Y, Ẏ ) defined in [START_REF] Campion | Modelling and state feedback control of nonholonomic mechanical systems[END_REF] can not be accurately estimated due to the unknown disturbance. Moreover, G(Y, Ẏ ) becomes singular when υ d = 0. In order to overcome the two drawbacks when applying the simple PID controller, this paper uses the recently proposed i-PID controller to control the robot with measurement disturbance.

III. DETERMINATION OF THE CONTROLLER

Since the controller proposed in this paper is based on the i-PID controller, let us firstly present the basic idea of this controller, and then detail how to apply this controller into the control of the unicycle model with measurement disturbance.

A. i-PID controller

Generally speaking, the method of i-PID controller locally approximates the system model by a simple local model with unknown term, and the unknown term can be estimated by the measurements of the input and output of the system, then a so-called i-PID controller can be deduced to realize the control goal.

In this paper the system model ( 6) is approximated by the following local model over a small time interval T = [t k , t k+1 ] with k ∈ Z + :

Ÿ (t) = F (t) + α(Y, Ẏ )u(t) (8)
where u and Y are known input and output signals with disturbance, α(Y, Ẏ ) is a non singular 2 × 2 dimensional matrix which should be well chosen in order to achieve the control goal. F ∈ R 2 represents all unknown terms including the disturbances, which can be estimated by using the information of Y , u and α.

For the above locally approximated continuous model over time interval T , one can estimate F by discretizing it. Precisely, denote T s the sampling period, so at each sampling time k = t/T s , one has

Ÿk = F k + α(Y, Ẏ )u k then it yields F k = Ÿk -α(Y, Ẏ )u k ,
where Y k and u k are measurable signals at time k, and Ÿk is the 2 nd order differentiation of the output Y at sampling time k. If it is assumed that T s is small enough such that F k-1 → F k , then the so-called i-PID controller can be designed as follows:

u k = α -1 (Y, Ẏ )(-F k-1 + e k ) (9)
where

e k = Ÿref,k -K 2 ( Ẏk -Ẏref,k ) -K 1 (Y k -Y ref,k )
with Y ref being the references of the output to be tracked, and K 1 and K 2 being the freely chosen coefficients such that the polynomial s 2 + K 2 s + K 1 is Hurwitz.

As one can see in the controller [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF], there are two parameters to be determined, α(Y, Ẏ ) and F k , which will be discussed in the following.

B. Discussion on α(Y, Ẏ )

The determination of α(Y, Ẏ ) is the most important issue when applying such a controller. A good parameter α(Y, Ẏ ) should well approximate G(Y, Ẏ ) defined in [START_REF] Campion | Modelling and state feedback control of nonholonomic mechanical systems[END_REF], and be always invertible, and change as fewer times as possible as time goes on, and it is best that α(Y, Ẏ ) is time-invariant.

In [START_REF] Sira-Ramirez | Robust Linear Control of Nonlinear Flat Systems[END_REF] and [START_REF] Youcef-Toumi | Input/output linearization using time delay control[END_REF], similar controllers are presented, which use an unknown term to represent unknown parameters and disturbance in the system. However, in [START_REF] Sira-Ramirez | Robust Linear Control of Nonlinear Flat Systems[END_REF] the similar G(Y, Ẏ ) in the system is a time-invariant scalar. In [START_REF] Youcef-Toumi | Input/output linearization using time delay control[END_REF] the determination of α(Y, Ẏ ) is discussed, but the similar parameter G(Y, Ẏ ) in the system is assumed to be always invertible and time-invariant. Thus in their controller, α(Y, Ẏ ) can be set as a fixed number or a fixed invertible matrix.

We aim to find out an invertible time-invariant α to well approximate G(Y, Ẏ ) in the controller. However, let us take a look at G(Y, Ẏ ) in our system [START_REF] Normey-Rico | Mobile robot path tracking using a robust pid controller[END_REF]. Firstly it is a matrix whose entries vary as time goes on, and the sign of all entries in G(Y, Ẏ ) is changing, which makes it impossible to use a time-invariant α to approximate G(Y, Ẏ ).

In order to approximate G(Y, Ẏ ) with α(Y, Ẏ ), α(Y, Ẏ ) needs to vary with G(Y, Ẏ ). One can of course set that

α(Y, Ẏ ) = cos θ -v sin θ sin θ v cos θ
where θ = arctan ẏ ẋ is the estimation of θ with noises, ẋ and ẏ are the estimation of ẋ and ẏ with noises, and v is the estimation of v defined in [START_REF] Kolmanovsky | Developments in nonholonomic control problems[END_REF]. In this way, this method is in fact equivalent to the controller linked to exact linearization by feedback with the estimate of θ and v. However, one can notice that α(Y, Ẏ ) will be singular when v = 0. In order to make α(Y, Ẏ ) being invertible and well approximate G(Y, Ẏ ), another intuitive choice is to remove v in the above matrix and one obtains:

α(Y, Ẏ ) = cos θ -sin θ sin θ cos θ
The above selected α(Y, Ẏ ) is suitable for the controller, since it is always invertible and it can be well approximate G(Y, Ẏ ). However, since this choice of θ is always timevarying when robot moves, which will increase the computation of the controller. In order to make the selected α(Y, Ẏ ) changing as fewer times as possible when robot moves, this paper proposes to choose it as follows:

α(Y, Ẏ ) = sig(cos θ) -sig(sin θ) sig(sin θ) sig(cos θ)
where sig(σ) is the sign function which extracts the sign of the real number σ, and it is assumed that sig(0) = 1. For this proposed α(Y, Ẏ ), let us define the following switching signal i( θ) : R → I with I = {1, 2, 3, 4}:

i( θ) =          1 if θ ∈ (2kπ, 2kπ + π 2 ) 2 if θ ∈ (2kπ + π 2 , 2kπ + π) 3 if θ ∈ (2kπ + π, 2kπ + 3π 2 ) 4 if θ ∈ (2kπ + 3π 2 , 2(k + 1)π) (10)
where k ∈ Z. The corresponding constant matrices can then be defined as follows:

α 1 = 1 -1 1 1 α 2 = -1 -1 1 -1 α 3 = -1 1 -1 -1 α 4 = 1 1 -1 1
Therefore, the proposed matrix α i( θ) satisfies:

α(Y, Ẏ ) = α i( θ) , ∀ θ ∈ R
Summarily, the selected α i( θ) has several advantages. Firstly it has only 4 values when θ changes in [2kπ, 2(k + 1)π], which is able to make α(Y, Ẏ ) changed as fewer times as possible. Secondly it is always invertible, so that the controller can stabilize the robot at a static point with the robot velocity equals to zero.

C. Numerical differentiation

It can be seen that the choice of α(Y, Ẏ ) involves the estimation values θ, ẋ and ẏ, then the efficient estimation of these values becomes significant. This paper uses the algebraic technique proposed by Fliess et al in [START_REF] Sira-Ramirez | An algebraic state estimation approach for the recovery of chaotically encrypted messages[END_REF] for the estimation. Mathematical foundation of this approach can be referred to [START_REF] Fliess | Non-linear estimation is easy[END_REF], [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] and the references therein.

Generally speaking, this algebraic approach has several advantages: it provides explicit formulae, which can be directly implemented; it is of non-asymptotic nature, the desired estimation can be obtained instantaneously, which is a significant advantage for real-time applications; it does not require any assumption concerning the statistical distribution of the unstructured noise.

Consider a signal

h(t) = ∞ k=0 h (k) (0) t k k!
which is assumed to be analytic around t = 0 and its truncated Taylor

expansion h N (t) = N k=0 h (k) (0) t k k!
, where t > 0. Its Laplace transform is of the form:

HN (s) = N k=0 h (k) (0) s k+1 (11)
Introducing the algebraic derivation d ds , and multiply both sides of equation ( 11) by d ρ ds ρ s N , ρ = 0, 1, ..., N , one has a triangular system of linear equations and from which the derivatives can be obtained:

d ρ s N HN ds ρ = d ρ ds ρ ( N k=0 h (k) (0)s N -k-1 ) (12)
which is independent of all the unknown initial conditions, and the coefficients h(0), ..., h (k) (0) are linearly identifiable [START_REF] Fliess | An algebraic framework for linear identification[END_REF], then the h (k) (0) can be obtained by taking the inverse laplace transform of (12) over a time window T . In practice, the above algebraic technique is implemented with discrete measured data, thus it is necessary that the sampling time T s should be small enough [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF].

It is worthy noting that this algebraic technique is robust with respect to noise involved into the controls and outputs, since noises are viewed here as quick fluctuations around 0. They are therefore attenuated by low-pass filters, like iterated integrals with respect to time.

D. Algebraic estimation of F

Now there is only one parameter F left in the i-PID controller to be determined. The calculation of F also uses the algebraic technique described above.

Let us consider the local approximated model:

Ÿ = F + αu (13)
where F can be considered as constant between two sampling time. Then by taking the Laplace transformation of both sides of equation ( 13), one obtains

s 2 Y(s) -sY(0) -Y (0) = F s + αU (s) (14)
In order to eliminate the unknown terms Y(0) and Y (0) which are linked to unknown initial conditions, we take the 2 nd order derivative of both sides with respect to s:

s 2 Y (s) + 4sY (s) + 2Y(s) = 2F s 3 + αU (s) (15)
By dividing both sides of equation ( 15) with s 3 , one has:

Y (s) s + 4Y(s) s 2 + 2Y(s) s 3 = 2F s 6 + αU (s) s 3 (16)
Take the inverse Laplace transformation of both sides of equation ( 16), one obtains:

T 0 (-τ ) 2 Y dτ + T 0 4(T -τ )(-τ )Y dτ + T 0 (T -τ ) 2 Y dτ = 2F T 5 5! + α T 0 (T -τ ) 2 2! (-τ ) 2 udτ (17)
where [0, T ] is a short time window, and the window is sliding in order to get the estimate at each time instant. Let τ = δT ∈ [0, T ], where δ ∈ [0, 1], after simplification equation ( 17) becomes:

T 3 1 0 (6δ 2 -6δ +1)Y dδ = F T 5 60 + T 5 α 2 1 0 (1-δ) 2 δ 2 udδ (18)
Hence, at sampling step k, the numerical estimate value of F k can be expressed as:

F k = 60 T 2 1 0 (6δ 2 -6δ + 1)Y dδ -30α 1 0 (1 -δ) 2 δ 2 udδ (19)
IV. SIMULATION RESULTS

In the simulation, the parameters are set: K 2 = 20, K 1 = 100, time window T = 3s, sampling time T s = 0.01. The reference trajectory is set as:

x r = sin 2t y r = sin t 2 Fig. 4 to Fig. 8 show the simulation result of the designed control applied on the nonholonomic wheeled mobile robot with white Gaussian noise of SN R = 30dB (signal-to-noise ratio) in the measurement (noises are shown in Fig. 2 and Fig. 3). Fig. 4 and Fig. 5 show the tracking result, i( θ) is shown in Fig. 6, and the control inputs are shown in Fig. 7 and Fig. 8. As we can see that the robot is able to track the trajectory with measurement noises, and the controller designed is effective and robust to the noises. Fig. 9 to Fig. 13 illustrate the simulation of stabilize the robot at point (4, 1), a white Gaussian noise of SN R = 30dB is added to the measurement as well. Tracking result is shown in Fig. 9 and Fig. 10, control inputs are shown in Fig. 12 and Fig. 13. As we can see that the controller is able to stabilize the robot at a static point with the velocity equals to 0.

Two more real-time 3D simulations are made in the attached video by using ROS (Robot Operating System), one is reference tracking simulation and the other is the stabilization of the robot at a static point with the robot V. CONCLUSION This paper presents the i-PID controller applied to the nonholonomic wheeled mobile robot. After the study of the system, the parameter α in the controller is selected as a switching function according to the information of the system. The presented i-PID controller is robust to the measurement disturbance of the robot, and it can even stabilize the robot at a static point with the robot velocity equals to zero with the proposed parameter α. The effectiveness and robustness of the designed controller were shown thereafter via several different simulations.

Fig. 1 .

 1 Fig. 1. Unicycle-type mobile robot

Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .

 234567 Fig. 2. Noise imposed in X SN R = 30dB

Fig. 8 .Fig. 9 .Fig. 12 .Fig. 13 .

 891213 Fig.8. Angular velocity control with white Gaussian noise SN R = 30dB


G Zheng
email: gang.zheng@inria.fr

J.-P Barbot
email: barbot@ensea.fr

D Boutat
email: driss.boutat@ensi-bourges.fr

T Floquet

J.-P Richard

IEEE TRANSACTIONS ON AUTOMATIC CONTROL

Keywords: Time-delay systems, Observability, Causality, Canonical form

Causal and non-causal observability are discussed in this paper for nonlinear time-delay systems with unknown inputs. Using the theory of non-commutative rings and the algebraic framework introduced by Xia et al, the nonlinear time-delay system is transformed into a suitable canonical form to solve the problem. A necessary and sufficient condition is given to guarantee the existence of a change of coordinates leading to such a form.

I. INTRODUCTION

O BSERVATION or estimation is important issue in con- trol theory. For nonlinear systems without delays, the observability problem has been exhaustively studied, and has been characterized in [START_REF] Hermann | Nonlinear controllability and observability[END_REF], [START_REF] Krener | Ad f,g ), (ad f,g ) and locally (ad f,g ) invariant and controllability distributions[END_REF], [START_REF] Sontag | A concept of local observability[END_REF] from a differential point of view, and in [START_REF] Diop | Nonlinear observability, identifiability and persistent trajectories[END_REF] from an algebraic point of view. For observable systems, many types of nonlinear observers have been proposed, such as high-gain observers in [START_REF] Gauthier | A simple observer for nonlinear systems with applications to bioreactors[END_REF], algebraic observers in [START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF], [START_REF] Ibrir | On-line exact differentiation and notion of asymptotic algebraic observers[END_REF], sliding mode observers in [START_REF] Floquet | Super twisting algorithm based step-by-step sliding mode observers for nonlinear systems with unknown inputs[END_REF], [START_REF] Xiong | Sliding mode observer for nonlinear uncertain systems[END_REF] and the references therein.

However, unlike nonlinear systems without delays, the analysis of properties for time-delay system is more complicated (see the surveys [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF] and [START_REF] Sename | New trends in design of observers for time-delay systems[END_REF]). For linear time-delay systems, various aspects of the observability problem have been studied in the literature, using different methods such as the functional analytic approach [START_REF] Bhat | Modal characterizations of controllability and observability in time delay systems[END_REF] or the algebraic approach [START_REF] Brewer | Linear systems over commutative rings[END_REF], [START_REF] Fliess | Controllability and observability of linear delay systems: an algebraic approach[END_REF], [START_REF]Linear systems over commutative rings: A survey[END_REF]. The theory of non-commutative rings has been applied to analyze nonlinear time-delay systems firstly in [START_REF] Moog | The disturbance decoupling problem for time-delay nonlinear systems[END_REF] for the disturbance decoupling problem of nonlinear time-delay system, for observability of nonlinear time-delay systems with known inputs in [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over non-commutative rings[END_REF], for identifiability of parameter for nonlinear time-delay systems in [START_REF] Zhang | Parameter identifiability of nonlinear systems with time-delay[END_REF], and for state elimination and delay identification of nonlinear time-delay systems in [START_REF] Anguelova | State elimination and identifiability of the delay parameter for nonlinear time-delay systems[END_REF]. Concerning the observer design for linear and nonlinear timedelay systems, the interested reader can refer to [START_REF] Darouach | Full order unknown inputs observers design for delay systems[END_REF], [START_REF]A new approach to state observation of nonlinear systems with delayed output[END_REF], [START_REF]Adaptive observers for time-delay nonlinear systems in triangular form[END_REF], [START_REF] Marquez-Martinez | The structure of nonlinear time delay systems[END_REF], [START_REF] Pepe | Approximated delayless observers for a class of nonlinear time delay systems[END_REF] and the references therein.

Most of those works are focused on time-delay systems with known inputs. However, in practical case, the input may be unknown. Thus one needs to study the state observability, and under which conditions the unknown input can be estimated. The first effort was done in [START_REF] Marquez-Martinez | The structure of nonlinear time delay systems[END_REF] to extend Singh's inversion algorithm to nonlinear time-delay systems. Based on the algebraic framework proposed in [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over non-commutative rings[END_REF], this paper deals with the causal and non-causal estimation analysis of the states and unknown inputs of nonlinear time-delay system. The issue of nonlinear observer design for the studied system is not treated in this paper.

This paper is organized as follows: In section II, we recall the algebraic framework introduced in [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over non-commutative rings[END_REF], give some definitions and generalize the notation of the Lie derivative for time-delay systems with unknown inputs. Section III presents an observability canonical form for a general class of nonlinear time-delay systems, for which the causal and noncausal observabilities are discussed. In the same section, an illustrative example is given in order to highlight the proposed results.

II. ALGEBRAIC FRAMEWORK, NOTATIONS AND

DEFINITIONS

In this paper, it is assumed that the delays are commensurable, that is all the delays are multiples of an elementary delay τ . Under this assumption, the considered nonlinear time-delay system is described as follows:

       ẋ = f (x(t -iτ ), i ∈ S -) + s j=0 g j (x(t -iτ ), i ∈ S -)u(t -jτ ) y = h(x(t -iτ ), i ∈ S -) x(t) = ψ(t), u(t) = ϕ(t), t ∈ [-sτ, 0] (1) where x ∈ W ⊂ R n denotes the state variables, u = [u 1 , • • • , u m ] T ∈ R m is the unknown input, y ∈ R p is the measurable output, with p ≥ m. The integer i belongs to the finite set S -= {0, 1, • • • , s}. f , g j and h are meromorphic functions 1 of {x(t), • • • , x(t -sτ )}. ψ ∈ C([-sτ, 0], R n ) and ϕ ∈ C([-sτ, 0], R m )
are the initial functions for x and u where C([-sτ, 0], R j ) is the Banach space of function mapping [-sτ, 0] into R j . In this work, it is assumed that (1) is locally observable when u = 0 (see Definition 1 hereafter), and admits a unique solution 2 and sufficiently differentiable outputs.

Based on the algebraic framework introduced in [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over non-commutative rings[END_REF], let K be the field of meromorphic functions of a finite number of the variables from {x j (tiτ ), j ∈ [1, n], i ∈ S -}. With the standard differential operator3 d, define the vector space E over K:

E = span K {dξ : ξ ∈ K}
Introduce the backward time-shift operator δ defined by

δ i ξ(t) = ξ(t -iτ ), ξ(t) ∈ K, for i ∈ Z + (2)
and

δ i (a(t)dξ(t)) = δ i a(t)δ i dξ(t) = a(t -iτ )dξ(t -iτ ) (3
) for a(t)dξ(t) ∈ E, and i ∈ Z + .

Let K(δ] denote the set of polynomials of the form

a(δ] = a 0 (t) + a 1 (t)δ + • • • + a ra (t)δ ra (4)
where a i (t) ∈ K. The addition in K(δ] is defined as usual, but the multiplication is given as

a(δ]b(δ] = ra+r b k=0 i≤ra,j≤r b i+j=k a i (t)b j (t -iτ )δ k (5)
Note that K(δ] satisfies the associative law and is a noncommutative ring (see [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over non-commutative rings[END_REF]). However, it is proved that the ring K(δ] is a left Ore ring4 [START_REF] Ježek | Rings of skew polynomials in algebraical approach to control theory[END_REF], [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over non-commutative rings[END_REF], which enables to define the rank of a module over this ring. Let M denote the leftmodule over K(δ]: M = span K(δ] {dξ, ξ ∈ K}, where K(δ] acts on dξ according to (2) and [START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF].

With the definition of K(δ], the system (1) can be rewritten in a more compact form as follows:

   ẋ = f (x, δ) + m i=1 G i (x, δ)u i (t) y = h(x, δ) x(t) = ψ(t), u(t) = ϕ(t), t ∈ [-sτ, 0] (6)
where f (x, δ) = f (x(tiτ ), i ∈ S -) and h(x, δ) = h(x(tiτ ), i ∈ S -) with entries belonging to K, G i (x, δ) = s j=0 g j i (x, δ)δ j with entries belonging to K(δ]. It is assumed that rank K(δ]

∂h ∂x = p, which implies that [h 1 , • • • , h p ]
T are independent functions of and its backward shifts.

Similarly to observability definitions for nonlinear systems without delays given in [START_REF] Hermann | Nonlinear controllability and observability[END_REF] and in [START_REF] Diop | Nonlinear observability, identifiability and persistent trajectories[END_REF], a definition of observability for time-delay systems is given in [START_REF] Marquez-Martinez | Observability and observers for nonlinear systems with time delays[END_REF]. A more generic definition is stated here as follows:

Definition 1: System (1) is locally observable if the state x(t) can be expressed as a function of the output and its time derivatives with their backward and forward shifts. A locally observable system is locally causally observable if its state can be written as a function of the output and its derivatives with their backward shifts only. Otherwise, it is locally non-causally observable (and it depends also on the forward shifts).

In the same way, the following definition for the unknown inputs is given.

Definition 2: The unknown input u(t) can be locally estimated if it can be written as a function of the output and its time derivatives with backward and forward shifts. The input can be locally causally estimated if u can be expressed as a function of the output and its time derivatives with backward shifts only. Otherwise, it can be non causally estimated (and it depends also on the forward shifts).

Remarks 1: i) Definition 1 does not take into account the input and its successive derivatives, because the input is assumed to be unknown and just required to be continuous. Thus, this paper states that the estimation of unknown input is causal if it depends only on past and present information of the output.

ii) For simplicity, this paper uses the notion of observability to refer to both observations of the states and unknown inputs, if there is no ambiguity. If the state and the unknown input are observable in the sense of Definition 1 and 2, they can be directly estimated through robust differentiators, such as those proposed in [START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF], [START_REF] Fliess | Nonlinear estimation is easy[END_REF], [START_REF] Ibrir | On-line exact differentiation and notion of asymptotic algebraic observers[END_REF].

Definition 3: (Unimodular matrix) [START_REF] Marquez-Martinez | Observability and observers for nonlinear systems with time delays[END_REF] The matrix A ∈ K n×n (δ] is said to be unimodular over

K(δ] if it has a left inverse A -1 ∈ K n×n (δ], such that A -1 A = I n×n .
Definition 4: (Change of coordinates) [START_REF] Marquez-Martinez | Observability and observers for nonlinear systems with time delays[END_REF] z = φ(δ, x) ∈ K n×1 is a causal change of coordinates over K for the system (1) if there locally exist a function φ -1 ∈ K n×1 and some

constants c 1 , • • • , c n ∈ N such that diag{δ ci }x = φ -1 (δ, z). The change of coordinates is bicausal over K if max{c i } = 0, that is x = φ -1 (δ, z).
Note that the relative degree for nonlinear systems without delays is well defined via the Lie derivative (see [START_REF] Isidori | Nonlinear control systems (3rd edition)[END_REF]). Then many efforts have been done to extend the classical Lie derivative for nonlinear time-delay systems. In [START_REF] Germani | An asymptotic state observer for a class of nonlinear delay systems[END_REF], [START_REF]A new approach to state observation of nonlinear systems with delayed output[END_REF], the authors defined the so-called delay relative degree for a class of nonlinear time-delay systems with only a single delay, by augmenting the dimension of the studied system. In [START_REF] Oguchi | Input-output linearization of retarded non-linear systems by using an extension of lie derivative[END_REF], [START_REF] Oguchi | Sliding-mode control of retarded nonlinear systems via finite spectrum assignment approach[END_REF], the authors introduced the delayed state derivative and delayed state bracket, which are extensions of the conventional Lie derivative. However, those definitions are still built on the theory of commutative rings, which make the analysis of observability for nonlinear time-delay systems still complicated.

Hence in what follows, we first characterize the relative degree and observability indices for nonlinear time-delay systems by extending the Lie derivative in the algebraic framework of [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over non-commutative rings[END_REF] from a non-commutative ring point of view. Then, it is shown that some known results for systems without delays, such as canonical form, can be extended to systems with delays.

Let f (x(tjτ )) and h(x(tjτ )) for 0 ≤ j ≤ s be n and p dimensional vectors, respectively, with entries

f r ∈ K for 1 ≤ r ≤ n and h i ∈ K for 1 ≤ i ≤ p. Let ∂h i ∂x = ∂h i ∂x 1 , • • • , ∂h i ∂x n ∈ K 1×n (δ] (7)
where for

1 ≤ r ≤ n, ∂hi ∂xr = s j=0 ∂hi ∂xr(t-jτ ) δ j ∈ K(δ].
Then the Lie derivative for nonlinear systems without delays can be extended to nonlinear time-delay systems in the framework of [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over non-commutative rings[END_REF] as follows

L f h i = ∂h i ∂x (f ) = n r=1 s j=0 ∂h i ∂x r (t -jτ ) δ j (f r ) (8)
with h i ∈ K. One can also define L Gi h i = ∂hi ∂x (G i ). Using the above definition of Lie derivative, the relative degree can then be defined.

Definition 5: (Relative degree) System ( 6) has relative degree

(ν 1 , • • • , ν p ) in an open set W ⊆ R n if, for 1 ≤ i ≤ p, the following conditions are satisfied : 1) for all x ∈ W , L Gj L r f h i = 0, for all 1 ≤ j ≤ m and 0 ≤ r ≤ ν i -2; 2) there exists x ∈ W such that ∃j ∈ [1, m], L Gj L νi-1 f h i = 0. If for 1 ≤ i ≤ p, 1)
is satisfied for all r ≥ 0, we set ν i = ∞. Since ( 6) is locally observable when u = 0, one can define the so-called observability indices introduced in [START_REF] Krener | Ad f,g ), (ad f,g ) and locally (ad f,g ) invariant and controllability distributions[END_REF]. Let

F k := span K(δ] dh, dL f h, • • • , dL k-1 f h for 1 ≤ k ≤ n.
It was shown that the filtration of K(δ]-module satisfies

F 1 ⊂ F 2 ⊂ • • • ⊂ F n . Define d 1 = rank K(δ] F 1 , d k = rank K(δ] F k -rank K(δ] F k-1 for 2 ≤ k ≤ n and let k i = card {d k ≥ i, 1 ≤ k ≤ n}. Then (k 1 , • • • , k p ) are the observability indices and p i=1 k i = n since it is assumed that (6) is observable with u = 0. Since rank K(δ] ∂h ∂x = p, the observability indices (k 1 , • • • , k p ) for (h 1 , • • • , h p ) are well defined.

III. CANONICAL FORM AND OBSERVABILITY

In order to facilitate the analysis of the general time-delay system [START_REF] Conte | Nonlinear control systems: An algebraic setting[END_REF], this section first proposes a canonical form of the general system (6), and then analyzes the causal and noncausal observability for the proposed canonical form.

A. Canonical form

After having defined the relative degree and observability indices via the extended Lie derivative for nonlinear timedelay systems in the framework of non-commutative rings, an observable canonical form is derived in this section.

Theorem 1: Consider the system (6) with outputs (y 1 , • • • , y p ) and the corresponding (ρ 1 , • • • , ρ p ) with ρ i = min{k i , ν i } where k i and ν i are the observability indices and the relative degree indices, respectively. There exists a change of coordinates φ(x, δ) ∈ K n×1 , such that (6) can be transformed into the following form:

żi,j = z i,j+1 (9)
żi,ρi = V i (x, δ) = L ρi f h i (x, δ) + m j=1 L Gj L ρi-1 f h i (x, δ)u j ( 10
)
y i = C i z i = z i,1 (11)
ξ = α(z, ξ, δ) + β(z, ξ, δ)u ( 12
)
where

z i = (z i,1 , • • • , z i,ρi ) T = h i , • • • , L ρi-1 f h i T ∈ K ρi×1 , α ∈ K µ×1 , β ∈ K µ×1 (δ] with µ = n - p j=1
ρ j and

C i = (1, 0, • • • , 0) ∈ R 1×ρi . Moreover if k i < ν i , one has V i (x, δ) = L ρi f h i = L ki f h i . Proof:
According to the definition of the k i , one has

rank K(δ] ∂ h hi,••• ,L k i -1 f hi i T ∂x = k i . Since ρ i = min{k i , ν i }, one gets rank K(δ] ∂ h hi,L f hi,••• ,L ρ i -1 f hi i T ∂x = ρ i , which implies that the components of z i = h i , • • • , L ρi-1 f h i T are lin- early independent over K(δ]. Denote z = z T 1 , • • • , z T p T ∈
K n-η , with η = p j=1 ρ j , and choose η variables ξ ∈ K η , such that z T , ξ T T = φ(x, δ) ∈ K n is a well-defined change coordinates, which transforms ( 6) into [START_REF] El | Introduction to the theory and application of differential equations with deviating arguments[END_REF][START_REF] Fliess | Nonlinear estimation is easy[END_REF][START_REF] Fliess | Controllability and observability of linear delay systems: an algebraic approach[END_REF][START_REF] Floquet | Super twisting algorithm based step-by-step sliding mode observers for nonlinear systems with unknown inputs[END_REF]. Moreover, according to the definition of ν i , for all 1 ≤ j ≤ m and 0

≤ r < ν i -1, if k i < ν i , then ρ i = k i and one has L Gj L ki-1 f h i (x, δ) = L Gj L ρi-1 f h i (x, δ) = 0, which yields V i (x, δ) = L ρi f h i = L ki f h i .

B. Causal observability

For the subsystem [START_REF] El | Introduction to the theory and application of differential equations with deviating arguments[END_REF][START_REF] Fliess | Nonlinear estimation is easy[END_REF][START_REF] Fliess | Controllability and observability of linear delay systems: an algebraic approach[END_REF], one can find observers in the literature ( [START_REF] Floquet | Super twisting algorithm based step-by-step sliding mode observers for nonlinear systems with unknown inputs[END_REF], [START_REF] Xiong | Sliding mode observer for nonlinear uncertain systems[END_REF]) to estimate z i and V i (x, δ), for 1 ≤ i ≤ p, in finite time due to the triangular structure. However the zero dynamic part [START_REF] Floquet | Super twisting algorithm based step-by-step sliding mode observers for nonlinear systems with unknown inputs[END_REF] of the proposed canonical form cannot be estimated. In the following, a sufficient condition under which the system (6) can be transformed into the canonical form (9 -12) without the zero dynamic is given. Then, causal observability is analyzed.

The right-hand side of ( 10) can be rewritten in the following compact form:

H(x, δ) = Ψ(x, δ) + Γ(x, δ)u (13)
with

H(x, δ) = (V 1 (x, δ), • • • , V p (x, δ)) T , Ψ(x, δ) = L ρ1 f h 1 , • • • , L ρp f h p T and Γ(x, δ) =    L G1 L ρ1-1 f h 1 • • • L Gm L ρ1-1 f h 1 . . . . . . . . . L G1 L ρp-1 f h p • • • L Gm L ρp-1 f h p    (14)
where H(x, δ) ∈ K p×1 , Ψ(x, δ) ∈ K p×1 and Γ(x, δ) ∈ K p×m (δ]. Assume that rank K(δ] Γ = m. Since Γ ∈ K p×m (δ] with m ≤ p, according to Lemma 4 in [START_REF]New insights on the anlysis of nonlinear time-delay systems: Application to the triangular equivalence[END_REF], there exists a matrix Ξ ∈ K p×p (δ] such that ΞΓ = ΓT , 0 T , where Γ ∈ K m×m (δ] has full rank m. Introduce the set:

Φ = {dh 1 , • • • , dL ρ1-1 f h 1 , • • • , dh p , • • • , dL ρp-1 f h p } (15)
Then, the following theorem can be stated.

Theorem 2: Consider the system (6) with outputs

(y 1 , • • • , y p ) the corresponding (ρ 1 , • • • , ρ p ) with ρ i = min{k i , ν i }
where k i and ν i are the observability indices and the relative degree indices, respectively. If rank K(δ] Φ = n, where Φ is defined in [START_REF]A new approach to state observation of nonlinear systems with delayed output[END_REF], then there exists a change of coordinates φ(x, δ) such that (6) can be transformed into (9-12) with dim ξ = 0.

Moreover, if the change of coordinates is locally bicausal over K, then the state x(t) of ( 6) is locally causally observable, and if Γ ∈ K m×m (δ] is also unimodular over K(δ], then the unknown input u(t) of ( 6) can be locally causally estimated as well.

Proof: According to Theorem 1, the system (6) can be transformed into (9-12) by using the change of coordinates (z, ξ) = φ(x, δ). Hence, if rank K(δ] Φ = n, where Φ defined in [START_REF]A new approach to state observation of nonlinear systems with delayed output[END_REF], one has p j=1 ρ j = n, which implies that (6) can be transformed into (9-12) with dim ξ = 0 and the change of coordinates is given by z = φ(x, δ) where z

= (z T i , • • • , z T p ) T and z i = (h i , • • • , L ρi-1 f h i ) T .
Moreover, if φ(x, δ) ∈ K n×1 is locally bicausal over K, one can write x as a function of y i , its derivative and backward shift, which implies state x is locally causally observable.

Concerning the reconstruction of the unknown inputs, rewrite [START_REF] Gauthier | A simple observer for nonlinear systems with applications to bioreactors[END_REF] as follows

Γu = H(x, δ) -Ψ(x, δ) = Υ(x, δ). (16)
Since rank K(δ] Φ = n and x is causally observable, then Υ(x, δ) is a vector of known meromorphic functions belonging to K.

If Γ ∈ K m×m (δ] is unimodular over K(δ], then there exists a matrix Γ-1 ∈ K m×m (δ] such that Γ-1 0 ΞΓ = I m×m and u = Γ-1 0 ΞΥ. Since Γ-1 ∈ K m×m (δ], Ξ ∈ K p×p and Υ ∈ K p×1 , then u is also causally observable.

C. Extended case for causal observability

For the case where the condition rank K(δ] Φ = n in Theorem 2 is failed, a constructive algorithm was proposed in [START_REF] Barbot | An observation algorithm for nonlinear systems with unknown inputs[END_REF] to solve this problem for nonlinear systems without delays. The result of this subsection can be seen as an extension of the work [START_REF] Barbot | An observation algorithm for nonlinear systems with unknown inputs[END_REF] to treat the observation problem for time-delay systems with unknown inputs. The objective is to generate additional variables from the available measurement and unaffected by the unknown input such that an extended canonical form similar to ( 9)-( 10) can be obtained for the estimation of the remaining state ξ.

For this, consider Φ as defined in [START_REF]A new approach to state observation of nonlinear systems with delayed output[END_REF]. If rank K(δ] Φ = j, one can select j linearly independent vectors over K(δ] from Φ, denoted as

Φ = {dz 1 , • • • , dz j } . Note £ = span R[δ] {z 1 , • • • , z j } where R[δ]
is the commutative ring of polynomials of δ with coefficients belonging to the field R and let £(δ] be the set of polynomials of δ with coefficients over £. Define the module spanned by the elements of Φ over £(δ] as follows

Ω = span £(δ] {ξ, ξ ∈ Φ} . ( 17
)
Define also

G = span R[δ] {G 1 , • • • , G m } and its left annihila- tor G ⊥ = span R[δ] {ω ∈ Ω | ωg = 0, ∀g ∈ G}.
Based on the above definitions, let us state the main result of the paper. Theorem 3: Consider the system (6) with outputs y = (y 1 , • • • , y p ) T and the corresponding (ρ 1 , • • • , ρ p ) with ρ i = min{k i , ν i } where k i and ν i are the observability indices and the relative degree indices, respectively. Suppose rank K(δ] Φ < n where Φ is defined in [START_REF]A new approach to state observation of nonlinear systems with delayed output[END_REF]. There exist l new independent outputs over K suitable to the causal estimation problem if and only if rank K H = l where

H = span R[δ] {ω ∈ G ⊥ ∩ Ω | ωf / ∈ £}. ( 18
)
Moreover, the l additional outputs, denoted ȳi , 1 ≤ i ≤ l, are given by: ȳi = ω i f mod £ where ω i ∈ H. Proof:

Denote Q i = q i 1 , • • • , q i p
as 1 × p vector with q i j ∈ K(δ] for 1 ≤ j ≤ p. One has

Q i Γ =      Q i      ∂L ρ 1 -1 f h1 ∂x . . . ∂L ρp -1 f hp ∂x           [G 1 , • • • , G m ]
because of the associativity law over K(δ]. Then according to the definition [START_REF] Darouach | Full order unknown inputs observers design for delay systems[END_REF], one gets

Q i Γ = ω i [G 1 , • • • , G m ] = ω i G where ω i = n c=1 p j=1 q i j ∂L ρ j -1 f hj ∂xc dx c . Moreover, one can check that ωif = 0 B B B @ Qi 2 6 6 6 4 ∂L ρ 1 -1 f h 1 ∂x . . . ∂L ρp -1 f hp ∂x 3 7 7 7 5 1 C C C A f = Qi 2 6 4 L ρ 1 f h1 . . . L ρp f hp 3 7 5 = QiΨ.
According to [START_REF] Gauthier | A simple observer for nonlinear systems with applications to bioreactors[END_REF], one has

Q i H = Q i (Ψ + Γu) = ω i f + ω i Gu (19)
where

H = (V 1 , • • • , V p )
T is a vector which can be estimated in finite time but is affected by the unknown input. Thus, one can generate l information suitable to solve the estimation problem if and only if one can find l independent

{Q 1 , • • • , Q l } vectors over K(δ] such that for each Q i = {q i 1 , • • • , q i p } satisfying q i j ∈ £(δ], one has Q i Γ = 0 and Q i H / ∈ £.
Thus, one has to prove that the following conditions are equivalent:

1) there exist l row vectors [START_REF]Adaptive observers for time-delay nonlinear systems in triangular form[END_REF]. Necessity: Suppose item 1) is satisfied, then according to [START_REF] Isidori | Nonlinear control systems (3rd edition)[END_REF], one has

Q i = q i 1 , • • • , q i p , with q i j ∈ £(δ], such that rank K(δ] {Q 1 , • • • , Q l } = l, Q i Γ = 0 and Q i H / ∈ £. 2) rank K H = l with H defined in
Q i Γ = ω i G = 0 ⇒ ω i ∈ G ⊥ and Q i H = ω i f / ∈ £. Thus rank K(δ] {Q 1 , • • • , Q l } = l implies rank K {ω 1 , • • • , ω l } = l. Since L ρi-1 f h i ∈ £ and q i ∈ £(δ], then ω i = n c=1 p j=1 q i j ∂L ρ j -1 f
hj ∂xc dx c ∈ Ω. Thus, ω i ∈ H defined in [START_REF]Adaptive observers for time-delay nonlinear systems in triangular form[END_REF].

Sufficiency: Suppose rank K H = l. Then one can find l independent 1-forms over

K: {ω 1 , • • • , ω l } with ω i ∈ G ⊥ ∩ Ω which implies there exist l independent vectors over K(δ]: {Q 1 , • • • , Q l } with entries belonging to £(δ] such that rank K(δ] {Q 1 , • • • , Q l } = l, since for each Q i one has Q i Γ = ω i G = 0 and Q i H = ω i f / ∈ £. The variable ȳi = Q i H = ω i f mod £
can be used as an additional output since 1) H can be estimated in finite time; 2) Q i has entries in £(δ]; 3) ȳi do not belong to the current set £ of measured variables.

Remark 1: Theorem 3 gives a constructive way to treat the case where rank K(δ] Φ < n. Once additional new outputs are deduced according to Theorem 3, it enables to define a new Φ. If rank K(δ] Φ = n, Theorem 2 can then be applied. Otherwise, if rank K(δ] Φ < n and if Theorem 3 is still valid, then one can still deduce new outputs for the studied system. Thus a "Check-Extend" procedure is iterated until one obtains rank K(δ] Φ = n.

D. Non-causal observability

The previous results can be extended to the case of noncausal observations of the state and the unknown inputs which can be very useful in some applications. For instance, many proposed delay feedback control methods can be applied for stabilizing nonlinear time-delay systems [START_REF]Is a mixed design of observer-controllers for time-delay systems interesting?[END_REF]. Furthermore, other applications, such as cryptography based on chaotic system, do not require real-time estimation, hence non-causal observations can still play an important role in those applications.

In order to treat the non-causal case, let us introduce the forward time-shift operator ∇, which is similar to the backward time-shift operator δ defined in Section II:

∇f (t) = f (t + τ ) and ∇ i δ j f (t) = δ j ∇ i f (t) = f (t -(j -i)τ ) for i, j ∈ Z + .
Following the same principle of Section II, denote K the field of meromorphic functions of a finite number of variables from {x j (tiτ

), j ∈ [1, n], i ∈ S} where S = {-s, • • • , 0, • • • , s} is a finite set of constant. One has K ⊆ K.
Denote K(δ, ∇] the set of polynomials of the form:

a(δ, ∇] = ārā ∇ rā + • • • + ā1 ∇ +a 0 (t) + a 1 (t)δ + • • • + a ra (t)δ ra (20)
with a i (t) and āi (t) belonging to K. Keep the usual definition of addition for K(δ, ∇] and define the multiplication as follows:

It is clear that K(δ] ⊆ K(δ, ∇] and that the ring K(δ, ∇] possesses the same properties as K(δ]. Thus a module M can be also defined over K(δ, ∇] as M = span K(δ,∇] {dξ, ξ ∈ K}.

Given the above definitions, Theorem 2 can then be extended as follows in order to deal with non-causal observability for nonlinear time-delay systems.

Theorem 4: Consider the system (6) with outputs (y 1 , • • • , y p ) and the corresponding (ρ 1 , • • • , ρ p ) with ρ i = min{k i , ν i } where k i and ν i are the observability indices and the relative degree indices, respectively. If rank K(δ] Φ = n, where Φ is defined in [START_REF]A new approach to state observation of nonlinear systems with delayed output[END_REF], then there exists a change of coordinates φ(x, δ) such that (6) can be transformed into (9-12) with dim ξ = 0.

Moreover, if the change of coordinates is locally bicausal over K, then the state x(t) of ( 6) is at least locally non-causally observable, and if Γ ∈ K m×m (δ] is also unimodular over K(δ, ∇], then the unknown input u(t) of ( 6) can be at least locally non-causally estimated as well.

Proof: If the change of coordinates z = φ(x, δ) ∈ K n×1 ⊆ Kn×1 is locally bicausal over K, then there exist φ -1 ∈ Kn×1 and some constants c

1 , • • • , c n such that T x = φ -1 (z, δ) where T = diag{δ c1 , • • • , δ cn }. Thus one can define the matrix T -1 = diag{∇ c1 , • • • , ∇ cn } ∈ Kn×n (δ, ∇],
such that x = T -1 φ -1 (δ, z) ∈ Kn×1 , which means that x is at least locally non-causally observable.

For the estimation of u(t), if Γ is unimodular over K(δ, ∇], following the same procedure as in Theorem 2, one gets u = Γ-1 0 ΞΥ. In this case, since Γ-1 ∈ K(δ, ∇], Ξ ∈ K p×p (δ] and Υ ∈ K p×1 , u is at least non-causally observable.

E. Example

Here is given an illustrative example in order to highlight the proposed results in the case of causal observability. Consider the following system

   ẋ1 = -δx 2 1 + δx 4 u 1 , ẋ2 = -x 2 1 δx 3 + x 4 ẋ3 = x 2 -x 2 1 δx 4 u 1 , ẋ4 = u 2 y 1 = x 1 , y 2 = x 1 δx 1 + x 3 (22) One can check that ν 1 = k 1 = 1, ν 2 = 1 and k 2 = 3, yielding ρ 1 = ρ 2 = 1 and Φ = {dx 1 , (δx 1 + x 1 δ)dx 1 + dx 3 }. One has rank K(δ] Φ = 2 < n. Set G =span R[δ] {G 1 , • • • , G m }. Then one has G ⊥ = span R[δ] x 2 1 dx 1 + dx 3 , dx 2 . Since rank K(δ] Φ = 2, £ = span R[δ] {x 1 , x 1 δx 1 + x 3 } and Ω = span £(δ] {dx 1 , dx 3 }. One obtains Ω ∩ G ⊥ = span £(δ] x 2 1 dx 1 + dx 3 . From the definition of H in (18), one can check that rank K H = 1, which gives the one-form ω = x 2 1 dx 1 + dx 3 , satisfying ω ∈ Ω ∩ G ⊥ and ωf = -x 2 1 δx 2 1 + x 2 / ∈ £. Thus, a new output ȳ1 = h 3 is given by ȳ1 = h 3 = ωf mod £ = x 2 = y 2 1 δy 2 1 + (y 2 1 -y 1 δ) ẏ1 + ẏ2 (23)
Although ẏ1 and ẏ2 contain u and cannot be derivable, however their combination in (23) makes ȳ1 not depend on u, thus it is derivable and can be used for state reconstruction.

For the new output ȳ1 , one has ν 3 = 2 and k 3 = 2, thus ρ 3 = 2. Finally, one obtains the new Φ as follows:

Φ = {dx 1 , (δx 1 + x 1 δ)dx 1 + dx 3 , dx 2 , -2x 1 δx 3 dx 1x 2 1 δdx 3 + dx 4 }. It can be checked that rank K(δ] Φ = 4, and the new £ is £ = span R[δ] {x 1 , x 1 δx 1 + x 3 , x 2 , -x 2 1 δx 3 + x 4 }. This gives the following change of coordinates z = φ(x, δ) = x 1 , x 1 δx 1 + x 3 , x 2 , -x 2 1 δx 3 + x 4 T .

It is easy to check that it is bicausal over K(δ], since

x = φ -1 = z 1 , z 3 , z 2 -z 1 δz 1 , z 4 + z 2 1 δz 2 -z 2 1 δz 1 δ 2 z 1 T .
When t ≥ 2τ , one gets the following estimations of states:

x 1 = y 1 , x 2 = ȳ1 , x 3 = y 2δy 1 x 4 = -y 3 1 δ 2 y 1 + y 2 1 δy 2 + ẏ1 with ȳ1 defined in [START_REF]New insights on the anlysis of nonlinear time-delay systems: Application to the triangular equivalence[END_REF].

Moreover, the matrix Γ with the new output ȳ1 is Γ =   -δy 3 1 δ 3 y1+δy 2 1 δ 2 y2+δ ẏ1 u 2 = -3y 2 1 δ 2 y 1 ẏ1y 3 1 δ 2 ẏ1 + 2y 1 δy 2 ẏ1 + y 2 1 δ ẏ2 + ÿ1

IV. CONCLUSION

In this paper, a generic definition of observability for timedelay systems with unknown inputs, covering causal and noncausal observability, has been introduced. The relative degree and observability indices for nonlinear time-delay systems have been defined based on the notation of the Lie derivation in the framework of non-commutative rings. Then, an observable canonical form for time-delay systems, as well as sufficient conditions to guarantee the causal and non-causal observations of states and unknown inputs of time-delay systems, have been given.

δx 4 , 0 δx 1 δx 2 1 δ 2 x 4 δ, 1 Γ- 1 0

 111 δx 4 + x 1 δ 2 x 4 δx 2 1 δx 4 , 0 -2x 1 δx 3 δx 4 + x 2 1 with rank K(δ] Γ = 2.One can find matrices Ξ = ΞΓ = I 2×2 . Consequently, according to Theorem 2, u 1 and u 2 can be causally estimated. When t ≥ 3τ , a straightforward computation yields the following estimates for the unknown inputs:

i.e. quotients of convergent power series with real coefficients[START_REF] Conte | Nonlinear control systems: An algebraic setting[END_REF],[START_REF] Xia | Analysis of nonlinear time-delay systems using modules over non-commutative rings[END_REF].

Note the right hand-side of system (1) as f (xτ ), if it is continuous with respect to its arguments, then there exists a solution for[START_REF] Anguelova | State elimination and identifiability of the delay parameter for nonlinear time-delay systems[END_REF]. Moreover, if it is locally Lipschitz, the solution is unique[START_REF] El | Introduction to the theory and application of differential equations with deviating arguments[END_REF].

The standard differential operator d maps elements from K to E, which is the vector space spanned by the {dx j (t -iτ ), j ∈ [1, n], i ∈ S -} over K.

A ring K(δ] is called a left Ore ring, if for all a(δ] ∈ K(δ] and b(δ] ∈ K(δ], there exist a ′ (δ] ∈ K(δ] and b ′ (δ] ∈ K(δ] not both zero, such that a ′ (δ]a(δ] = b ′ (δ]b(δ].