Gang Zheng
email: gang.zheng@inria.fr

Jean-Pierre Richard
email: jean-pierre.richard@ec-lille.fr

Identifiability and observability of nonlinear time-delay system with unknown inputs

Using the theory of non-commutative rings, the delay identification problem of nonlinear time-delay systems with unknown inputs is studied. Necessary and sufficient conditions are proposed to judge the identifiability of the delay, where two different cases are discussed for the dependent and independent outputs, respectively. After that, necessary and sufficient conditions are given to analyze the causal and non-causal observability for nonlinear time-delay systems with unknown inputs.

Introduction

Time-delay systems are widely used to model concrete systems in engineering sciences, such as biology, chemistry, mechanics and so on [START_REF] Erneux | Applied delay differential equations[END_REF][START_REF] Kolmanovskii | Introduction to the theory and application of functional differential equations[END_REF][START_REF] Niculescu | Delay effects on stability: A robust control approach[END_REF]. Many results have been reported for the purpose of stability and observability analysis, by assuming that the delay of the studied systems is known. It makes the delay identification be one of the most important topics in the field of time-delay systems. Up to now, various techniques have been proposed for the delay identification problem, such as identification by using variable structure observers [START_REF] Drakunov | Delay identification in timedelay systems using variable structure observers[END_REF][START_REF] Orlov | On identifiability of linear timedelay systems[END_REF][START_REF] Orlov | Adaptive identification of linear time-delay systems[END_REF], modified least squares techniques [START_REF] Ren | Online identification of continuous-time systems with unknown time delay[END_REF], neural network algorithms [START_REF] Yadaiah | Neural network algorithm for parameter identification of dynamical systems involving time delays[END_REF], convolution approach [START_REF] Belkoura | Identifiability of systems described by convolution equations[END_REF], algebraic fast identification technique [START_REF] Belkoura | On-line identification of systems with delayed inputs[END_REF][START_REF] Belkoura | Parameters estimation of systems with delayed and structured entries[END_REF] as initiated in [START_REF] Fliess | Reconstructeurs d'état[END_REF], and so on (see [START_REF] Belkoura | Identifiability and identification of linear systems with delays[END_REF][START_REF] Drakunov | Delay identification in timedelay systems using variable structure observers[END_REF] for additional references). Note that most of the papers on identification in presence of delays concern linear models. Another source of complexity comes from the presence of feedback loops involving the delays. Indeed, when the delay appears only on the inputs or outputs, the system has the finite dimension. When the delays are involved in a closed-loop manner, the resulting model has delayed states and become a functional differential equation, which has the infinite dimension [START_REF] Belkoura | Real time identification of delay systems[END_REF][START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF].

Besides identifiability, the observability property has been exhaustively studied for nonlinear systems without delays. It 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. However, when the system is subject to time delay, such analysis 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 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] Sontag | Linear systems over commutative rings: A survey[END_REF].

The aim of this paper is to firstly identify the time delay of nonlinear time-delay system with unknown inputs and then study observability for this system with identified delay. The work of this paper is based on the theory of non-commutative rings, which was firstly proposed 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. Then this method was applied to study 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], to analyze identifiability of parameter for nonlinear time-delay systems in [START_REF] Zhang | Parameter identifiability of nonlinear systems with timedelay[END_REF], and to study state elimination and delay identification of nonlinear time-delay systems with known inputs in [START_REF] Anguelova | State elimination and identifiability of the delay parameter for nonlinear time-delay systems[END_REF]. The motivation to study nonlinear time-delay system with unknown inputs is due to the fact that there exist some cases, such as observer design for time-delay systems, in which the inputs can be unknown [START_REF] Darouach | Full order observers for linear systems with unknown inputs[END_REF][START_REF] Koenig | Unknown input observers design for time-delay systems application to an open-channel[END_REF][START_REF] Sename | Unknown input robust observers for time-delay systems[END_REF][START_REF] Yang | Observers for linear systems with unknown inputs[END_REF]. Moreover, some proposed unknown input observer design methods do depend on the known delay, which should be identified in advance. Motivated by this requirement, this paper investigates both the delay identification problem and observability problem for nonlinear time-delay systems with unknown inputs.

This paper is organized as follows. Section 2 recalls the algebraic framework proposed in [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over non-commutative rings[END_REF]. Notations and preliminary result are given in Section 3. Necessary and sufficient conditions are discussed for identifying the delay in two different cases: dependent outputs over the non-commutative rings, and then independent ones. Section 5 deduces necessary and sufficient conditions of causal and non-causal observability for nonlinear time-delay systems with unknown inputs, and the proposed result is applied to identify the delay of a biological model in Section 6.

Algebraic framework

It is assumed that the delays are constant and commensurate, that is all of them are multiples of an elementary unknown delay τ . Under this assumption, the considered nonlinear time-delay system is described as follows:

   ẋ = f (x(t -iτ )) + ∑ s j=0 g j (x(t -iτ ))u(t -jτ ), y = h(x(t -iτ )) = [h 1 (x(t -iτ )), . . . , h p (x(t -iτ ))] T , x(t) = ψ(t), u(t) = φ(t), t ∈ [-sτ, 0], (1)
where the constant delays iτ are associated to the finite set of integers i ∈ S -= {0, 1, . . . , s}; x ∈ W ⊂ R n refers to the state variables; u = [u 1 , . . . , u m ] T ∈ R m is the unknown input; y ∈ R p is the measurable output; f , g j and h are meromorphic

functions 1 ; f (x(t -iτ )) = f (x, x(t -τ ), . . . , x(t -sτ )); ψ : [-sτ, 0] → R n and φ : [-sτ, 0] → R m denote
unknown continuous functions of initial conditions. Throughout this chapter, it is assumed that, for initial conditions ψ and φ, system (1) admits a unique solution.

Based on the algebraic framework introduced in [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over non-commutative rings[END_REF], consider the field K of meromorphic functions of a finite number of the variables from {x j (t -iτ ), j ∈ [1, n], i ∈ S -}. For the sake of simplicity, we introduce the delay operator δ, which means, for i ∈ Z + :

δ i ξ(t) = ξ(t -iτ ), ξ(t) ∈ K, ( 2
)
δ i (a(t)ξ(t)) = δ i a(t)δ i ξ(t) = a(t -iτ )ξ(t -iτ ). (3)
Let K(δ] denote the set of polynomials in δ over K, of the form

a(δ] = a 0 (t) + a 1 (t)δ + • • • + a ra (t)δ ra (4)
where a i (t) ∈ K and r a ∈ Z + . 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
)
Considering (1) without input, differentiation of an output component h j (x(tiτ )) with regard to time t is defined as follows:

ḣj (x(t -iτ )) = s ∑ i=0 ∂h j ∂x(t -iτ ) δ i f.
Thanks to the definition of K(δ], (1) can be rewritten in a more compact form:

   ẋ = f (x, δ) + G(x, δ)u = 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(t -iτ )) and h(x, δ) = h(x(t -iτ )), with entries belonging to K, u = u(t), and

G(x, δ) = [G 1 , • • • , G m ] with G i (x, δ) = ∑ s l=0 g l i δ l .
With the standard differential operator d, denote by M the left module over K(δ]:

M = span K(δ] {dξ, ξ ∈ K} ( 7
)
where K(δ] acts on dξ according to (2) and (3). Note that K(δ] is a non-commutative ring, however it is proved that it is a left Ore ring [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 left module over K(δ]. Define the vector space E over K:

E = span K {dξ : ξ ∈ K}
E is the set of linear combinations of a finite number of elements from dx j (t -iτ ) with row vector coefficients in K. Since the delay operator δ and the standard differential operator are commutative, the one-form of ω ∈ M can be written as: ω = ∑ n j=1 a j (δ]dx j , where a(δ] ∈ K(δ]. For a given vector field β =

∑ n j=1 b j (δ] ∂ ∂xj with b j (δ] ∈ K(δ],
the inner product of ω and β is defined as follows:

ωβ = n ∑ j=1 a j (δ]b j (δ] ∈ K(δ].
3 Notations and preliminary result Some efforts have been made to extend the Lie derivative [START_REF] Isidori | Nonlinear control systems (3rd edition)[END_REF] to nonlinear timedelay systems (see [START_REF] Califano | Extended lie brackets for nonlinear time-delay systems[END_REF][START_REF] Germani | Linearization of input-output mapping for nonlinear delay systems via static state feedback[END_REF][START_REF] Germani | An asymptotic state observer for a class of nonlinear delay systems[END_REF][START_REF] Germani | A new approach to state observation of nonlinear systems with delayed output[END_REF][START_REF] Oguchi | Input-output linearization of retarded nonlinear systems by an extended Lie derivative[END_REF][START_REF] Oguchi | Sliding-mode control of retarded nonlinear systems via finite spectrum assignment approach[END_REF][START_REF] Oguchi | Input-output linearization of retarded nonlinear systems by using an extension of lie derivative[END_REF]) in the framework of commutative rings. In what follows, we define the derivative and Lie derivative for nonlinear time-delay systems from the non-commutative point of view. For 0 ≤ j ≤ s, let f (x(t -jτ )) and h(x(t -jτ )) respectively be an n and p dimensional vector 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], (8)
where, for 1 ≤ r ≤ n:

∂h i ∂x r = s ∑ j=0 ∂h i ∂x r (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 ) (9)
and in the same way one can define L Gi h i . Based on the above notations, the relative degree can be defined in the following way.

Definition 1. (Relative degree) System (6) has the relative degree (ν 1 , • • • , ν p ) in an open set W ⊆ R n if the following conditions are satisfied for 1 ≤ i ≤ p: 1. for all x ∈ W , L Gj L r f h i = 0 for all 1 ≤ j ≤ m and 0 ≤ r < ν i -1; 2. there exists x ∈ W such that ∃j ∈ {1, • • • , m}, L Gj L νi-1 f h i ̸ = 0.
If the first condition is satisfied for all r ≥ 0 and some i ∈ {1, • • • , p}, we set ν i = ∞. Moreover, for system [START_REF] Belkoura | On-line identification of systems with delayed inputs[END_REF], one can also define observability indices introduced in [START_REF] Krener | Ad f,g ), (ad f,g ) and locally (ad f,g ) invariant and controllability distributions[END_REF] over non-commutative rings. For 1 ≤ k ≤ n, let F k be the following left module over K(δ]:

F k := span K(δ] { dh, dL f h, • • • , dL k-1 f h } .
It was shown that the filtration of

K(δ]-module satisfies F 1 ⊂ F 2 ⊂ • • • ⊂ F n , then define d 1 = rank K(δ] F 1 , and d k = rank K(δ] F k -rank K(δ] F k-1 for 2 ≤ k ≤ n. Let k i = card {d k ≥ i, 1 ≤ k ≤ n}, then (k 1 , • • • , k p ) are the observability indices.
Reorder, if necessary, the output components of ( 6) so that

rank K(δ] { ∂h1 ∂x , • • • , ∂L k 1 -1 f h1 ∂x , • • • , ∂hp ∂x , • • • , ∂L kp-1 f hp ∂x } = k 1 + • • • + k p .
Based on the above definitions, let us define the following notations, which will be used in the sequel. For 1 ≤ i ≤ p, denote by k i the observability indices, ν i the relative degree for y i of ( 6), and

ρ i = min {ν i , k i } . Without loss of generality, suppose ∑ p i=1 ρ i = j, thus {dh 1 , • • • , dL ρ1-1 f h 1 , • • • , dh p , • • • , dL ρp-1 f h p } are j linearly independent vectors over K(δ].
Then note:

Φ = {dh 1 , • • • , dL ρ1-1 f h 1 , • • • , dh p , • • • , dL ρp-1 f h p } ( 10
)
and

£ = span R[δ] { h 1 , • • • , L ρ1-1 f h 1 , • • • , h p , • • • , L ρp-1 f h p } , ( 11
)
where R[δ] is the commutative ring of polynomials in δ with coefficients belonging to the field R, and let £(δ] be the set of polynomials in δ with coefficients over £.

The module spanned by element of Φ over £(δ] is defined as follows:

Ω = span £(δ] {ξ, ξ ∈ Φ} . ( 12
) Define G = span R[δ] {G 1 , . . . , G m },
where G i is given in [START_REF] Belkoura | On-line identification of systems with delayed inputs[END_REF], and its left annihilator:

G ⊥ = span £(δ] {ω ∈ M | ωβ = 0, ∀β ∈ G}, ( 13
)
where M is defined in [START_REF] Belkoura | Real time identification of delay systems[END_REF].

After having defined the relative degree and observability indices via the extended Lie derivative for nonlinear time-delay systems in the framework of noncommutative rings, now an observable canonical form can be derived.

Theorem 1. Consider the system [START_REF] Belkoura | On-line identification of systems with delayed inputs[END_REF] 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) is transformed into the following form:

żi,j = z i,j+1 (14) 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 ( 15
)
y i = C i z i = z i,1 (16) ξ = α(z, ξ, δ) + β(z, ξ, δ)u ( 17
)
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.
See [START_REF] Zheng | Identification of the delay parameter for nonlinear time-delay systems with unknown inputs[END_REF].

Based on Theorem 1, noting ρ i = min {ν i , k i } for 1 ≤ i ≤ p where the k i represent the observability indices and ν i stands for the relative degree of y i for (6), the following equality can be derived:

H(x, δ) = Ψ (x, δ) + Γ (x, δ)u, ( 18
) with H(x, δ) = ( h (ρ1) 1 , • • • , h (ρp) p ) 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    , ( 19
)
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 [29],
there exists a matrix Ξ ∈ K p×p (δ] such that:

ΞΓ = [ Γ T , 0 ] T , ( 20
)
where Γ ∈ K m×m (δ] has full rank m. With the compact equation ( 18), identifiability and observability will be analyzed separately in Sections 4 and 5.

Identifiability

In order to study the delay identifiability of ( 6), let us firstly introduce the following definition of the identifiability of time delay, which is an adaptation of Definition 2 in [START_REF] Anguelova | State elimination and identifiability of the delay parameter for nonlinear time-delay systems[END_REF].

Definition 2. For system [START_REF] Belkoura | On-line identification of systems with delayed inputs[END_REF], an equation with delays, containing only the output and a finite number of its derivatives:

α(h, ḣ . . . , h (k) , δ) = 0, k ∈ Z +
is said to be an output delay equation (of order k). Moreover, this equation is said to be an output delay-identifiable equation2 for (6) if it cannot be written as

α(h, ḣ . . . , h (k) , δ) = a(δ]α(h, ḣ . . . , h (k) ) with a(δ] ∈ K(δ].
As stated in [START_REF] Anguelova | State elimination and identifiability of the delay parameter for nonlinear time-delay systems[END_REF], if there exists an output delay-identifiable equation for (6) (i.e. involving the delay in an essential way), then the delay can be identified for almost all 3 y by numerically finding zeros of such an equation. For this issue, the interested reader can refer to [START_REF] Barbot | Delay estimation algorithm for nonlinear time-delay systems with unknown inputs[END_REF] and the references therein. Thus, delay identification for (6) boils down to the research of such an output delay equation.

Dependent outputs over K(δ]

Let us firstly consider the most simple case for identifying the delay for (6), i.e., from only the outputs of (6), which is stated in the following result.

Theorem 2. There exists an output delay-identifiable equation (of order 0) α(h, δ) for [START_REF] Belkoura | On-line identification of systems with delayed inputs[END_REF] if and only if

rank K(δ] ∂h ∂x < rank K ∂h ∂x . ( 21
)

⊓ ⊔

Proof. see [START_REF] Zheng | Identification of the delay parameter for nonlinear time-delay systems with unknown inputs[END_REF].

Example 1. Consider the following dynamical system:

   ẋ = f (x, u, δ), y 1 = x 1 , y 2 = x 1 δx 1 + x 2 1 . ( 22
)
It can be seen that

∂h ∂x = ( 1, 0 δx 1 + 2x 1 + x 1 δ, 0 ) which yields rank K(δ] ∂h ∂x = 1 and rank K ∂h ∂x = 2.
Thus Theorem 2 is satisfied, and the delay of system (32) can be identified.

In fact, a straightforward calculation gives:

y 2 = y 1 δy 1 + y 2 1 ,
which permits to identify the delay δ by applying an algorithm to detect zerocrossing when varying δ. ⊓ ⊔ Inequality [START_REF] Germani | A new approach to state observation of nonlinear systems with delayed output[END_REF] implies that the outputs of ( 6) are dependent over K(δ]. Theorem 2 can be seen as a special case of Theorem 2 in [START_REF] Anguelova | State elimination and identifiability of the delay parameter for nonlinear time-delay systems[END_REF]. However, as it will be shown in the next section, this condition is not necessary for the case where the output of ( 6) is independent over K(δ].

Independent outputs over K(δ]

Theorem 2 has analyzed the case where the outputs of ( 6) are dependent over K(δ]. In the contrary case (independence over K(δ]), the dynamics of system (6) have to be involved in order to deduce some output delay equations, which might be used to identify the delay. In the following, it will be firstly given the sufficient condition for the existence of a delay output equation for system (6) when the output is independent over K(δ]. Then a necessary and sufficient condition will be provided. Based on the deduction of (18), we can state the following result. Theorem 3. There exists an output delay equation for [START_REF] Belkoura | On-line identification of systems with delayed inputs[END_REF], if there exists a non zero

ω = ∑ n c=1 ∑ p j=1 q j ∂L ρ j -1 f
hj ∂xc dx c , with q j ∈ K(δ] for 1 ≤ j ≤ p, such that ω ∈ G ⊥ ∩ Ω and ωf ∈ £, where G ⊥ is defined in [START_REF] Darouach | Full order observers for linear systems with unknown inputs[END_REF], Ω in [START_REF] Conte | Nonlinear control systems: An algebraic setting[END_REF], and £ in [START_REF] Califano | Extended lie brackets for nonlinear time-delay systems[END_REF]. ⊓ ⊔

Proof. Denote Q = [q 1 , • • • , q p ] as 1 × p vector with q j ∈ K(δ] for 1 ≤ j ≤ p.
Because of the associativity law over K(δ], one has:

QΓ = Q    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    = Q      ∂L ρ 1 -1 f h1 ∂x . . . ∂L ρp-1 f hp ∂x      [G 1 , • • • , G m ]
Then, according to the definition in [START_REF] Belkoura | Parameters estimation of systems with delayed and structured entries[END_REF], one gets:

QΓ = ω [G 1 , • • • , G m ] = ωG, where ω = ∑ n c=1 ∑ p j=1 q j ∂L ρ j -1 f
hj ∂xc dx c . Moreover, one can check that:

ωf =      Q      ∂L ρ 1 -1 f h 1 ∂x . . . ∂L ρp-1 f hp ∂x           f = Q           ∂L ρ 1 -1 f h 1 ∂x . . . ∂L ρp-1 f hp ∂x      f      = Q     L ρ 1 f h 1 . . . L ρp f hp     = QΨ.
According to [START_REF] Fliess | Reconstructeurs d'état[END_REF], one has:

QH = Q(Ψ + Γ u) = ωf + ωGu, ( 23
)
where

H = [ y (ρ1) 1 , • • • , y (ρp) p
] T . Thus, if ω ∈ G ⊥ ∩ Ω and ωf ∈ £, which implies there exists Q with entries belonging to £(δ], one has

QΓ = ωG = 0 and QH = ωf ∈ £.
Finally, one obtains the following relation:

Q(H -Ψ ) = 0, (24)
which is exactly the output delay equation, since it contains only the output, its derivatives and delays. ⊓ ⊔ If, in addition, the deduced output delay equation ( 24) is an output delayidentifiable equation, i.e. containing the delay δ in an essential way, then the delay of (6) can be identified (at least locally) by detecting zero-crossing of [START_REF] Ježek | Rings of skew polynomials in algebraical approach to control theory[END_REF]. The following will give necessary and sufficient conditions guaranteeing the essential involvement of δ in [START_REF] Ježek | Rings of skew polynomials in algebraical approach to control theory[END_REF]. But before this, let us define:

Y = ( h 1 , . . . , L ρ1-1 f h 1 , . . . , h p , . . . , L ρp-1 f h p ) T ,
and denote by K 0 ⊂ K the field of meromorphic functions of x, which will be used in the following theorem (also involving Ψ defined in ( 18)).

Theorem 4. The output delay equation ( 24) is an output delay-identifiable equation if and only if

rank K(δ] ∂Y ∂x < rank K ∂{Y, Ψ } ∂x ( 25
)
or for any element

q j of Q ∈ K 1×p (δ] , a(δ] ∈ K(δ] such that q j = a(δ]q j , with qj ∈ K 0 , 1 ≤ j ≤ p ( 26
)
and

rank K(δ] ∂Y ∂x = rank K ∂{Y, Ψ } ∂x . ( 27
)

⊓ ⊔

Proof. See [START_REF] Zheng | Identification of the delay parameter for nonlinear time-delay systems with unknown inputs[END_REF].

Remark 1. It is clear that Theorem 2 is a special case of Theorem 4, since the output delay-identifiable equation stated in Theorem 2 does not contain any derivative of the output.

In [START_REF] Anguelova | State elimination and identifiability of the delay parameter for nonlinear time-delay systems[END_REF], a condition similar to [START_REF] Koenig | Unknown input observers design for time-delay systems application to an open-channel[END_REF] of Theorem 4 is stated as a necessary and sufficient condition for delay identification for nonlinear systems with known inputs. However, as we proved above, in the case of unknown inputs, this condition is sufficient, but not necessary.

Observability

Similarly to the observability definitions given in [START_REF] Hermann | Nonlinear controllability and observability[END_REF] and [START_REF] Diop | Nonlinear observability, identifiability and persistent trajectories[END_REF] for nonlinear delayfree systems, it has been given in [START_REF] Marquez-Martinez | Observability and observers for nonlinear systems with time delays[END_REF] a definition of observability for nonlinear time-delay systems. The following gives a more generic definition of observability in the case of systems with unknown inputs. Definition 3. System ( 6) is locally observable if the state x(t) can be expressed as a function of the output and a finite number of 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 is given of systems with unknown inputs.

Definition 4. The unknown input u(t) can be locally estimated if it can be written as a function of the output and a finite number of its time derivatives with backward and forward shifts. The input can be locally causally estimated if u(t) 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).

Theorem 5. Consider the system [START_REF] Belkoura | On-line identification of systems with delayed inputs[END_REF] with outputs (y 1 , • • • , y p ) and their 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. Consider Φ and Γ defined in [START_REF] Brewer | Linear systems over commutative rings[END_REF] and [START_REF] Germani | An asymptotic state observer for a class of nonlinear delay systems[END_REF], respectively.

If rank K(δ] Φ = n,, then there exists a change of coordinates ϕ(x, δ) such that (6) can be transformed into [START_REF] Diop | Nonlinear observability, identifiability and persistent trajectories[END_REF][START_REF] Drakunov | Delay identification in timedelay systems using variable structure observers[END_REF][START_REF] Erneux | Applied delay differential equations[END_REF][START_REF] Fliess | Controllability and observability of linear delay systems: an algebraic approach[END_REF] 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; if, in addition, Γ ∈ K m×m (δ] is unimodular over K(δ], then the unknown input u(t) of ( 6) can be locally causally estimated. ⊓ ⊔ Proof. According to Theorem 1, (6) can be transformed into (14-17) by using the change of coordinates (z, ξ) = ϕ(x, δ). Hence, if rank K(δ] Φ = n, one has ∑ p j=1 ρ j = n, which implies that ( 6) can be transformed into (14-17) 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] Fliess | Reconstructeurs d'état[END_REF] as follows

Γ u = H(x, δ) -Ψ (x, δ) = Υ (x, δ). ( 28
)
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. ⊓ ⊔
For the case where the condition rank K(δ] Φ = n in Theorem 5 is not satisfied, 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. In the following we are going to extend this idea 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 ( 14)-( 15) can be obtained for the estimation of the remaining state ξ. Theorem 6. Consider the system [START_REF] Belkoura | On-line identification of systems with delayed inputs[END_REF] 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] Brewer | Linear systems over commutative rings[END_REF]. There exist l new independent outputs over K suitable to the causal estimation problem if and only if rank K L = l where

L = span R[δ] {ω ∈ G ⊥ ∩ Ω | ωf / ∈ £} ( 29
)
with f defined in [START_REF] Belkoura | On-line identification of systems with delayed inputs[END_REF], £ defined in [START_REF] Califano | Extended lie brackets for nonlinear time-delay systems[END_REF], Ω defined in [START_REF] Conte | Nonlinear control systems: An algebraic setting[END_REF] and G ⊥ defined in [START_REF] Darouach | Full order observers for linear systems with unknown inputs[END_REF].

Moreover, the l additional outputs, denoted as ȳi , 1 ≤ i ≤ l, are given by:

ȳi = ω i f mod £
where ω i ∈ L. ⊓ ⊔ Proof. See [START_REF] Zheng | On obserability of nonlinear time-delay systems with unknown inputs[END_REF].

Remark 2. Theorem 6 gives a constructive way to treat the case where rank K(δ] Φ < n. Once additional new outputs are deduced according to Theorem 6, it enables to define a new Φ. If rank K(δ] Φ = n, Theorem 5 can then be applied. Otherwise, if rank K(δ] Φ < n and if Theorem 6 is still valid, then one can still deduce new outputs for the studied system. Thus a "Check-Extend" procedure is iterated until rank K(δ] Φ = n is obtained.

Non-causal observability

The previous results can be extended to the case of non-causal observations of the state and the unknown inputs, which can be very useful in some applications. For instance, some proposed delay feedback control methods can be applied for stabilizing nonlinear time-delay systems [START_REF] Sename | 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 ∇, similarly to the backward time-shift operator δ defined in Section 2:

∇f (t) = f (t + τ )
and, for i, j ∈ Z + :

∇ i δ j f (t) = δ j ∇ i f (t) = f (t -(j -i)τ ) .
Following the principle of Section 2, denote by K the field of meromorphic functions of a finite number of variables from {x j (t -iτ ), j ∈ [1, n], i ∈ S} where S = {-s, • • • , 0, • • • , s} is a finite set of relative integers. One has K ⊆ K. Denote by 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 , ( 30
)
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:

a(δ, ∇]b(δ, ∇] = ra ∑ i=0 r b ∑ j=0 a i δ i b j δ i+j + ra ∑ i=0 rb ∑ j=1 a i δ ib j δ i ∇ j + rā ∑ i=1 r b ∑ j=0 āi ∇ i b j ∇ i δ j + rā ∑ i=1 rb ∑ j=1 āi ∇ ib j ∇ i+j . ( 31
)
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 follows:

M = span K(δ,∇] {dξ, ξ ∈ K}.
Given the above definitions, Theorem 5 is now extended so to deal with noncausal observability for nonlinear time-delay systems.

Theorem 7. Consider the system [START_REF] Belkoura | On-line identification of systems with delayed inputs[END_REF] 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] Brewer | Linear systems over commutative rings[END_REF], then there exists a change of coordinates ϕ(x, δ) such that ( 6) can be transformed into [START_REF] Diop | Nonlinear observability, identifiability and persistent trajectories[END_REF][START_REF] Drakunov | Delay identification in timedelay systems using variable structure observers[END_REF][START_REF] Erneux | Applied delay differential equations[END_REF][START_REF] Fliess | Controllability and observability of linear delay systems: an algebraic approach[END_REF] 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; if, in addition, Γ ∈ K m×m (δ] is unimodular over K(δ, ∇], then the unknown input u(t) of ( 6) can be at least locally non-causally estimated. ⊓ ⊔ Proof. See [START_REF] Zheng | On obserability of nonlinear time-delay systems with unknown inputs[END_REF].

Illustrative example

The following example aims at highlighting the proposed results in the case of delay identification and causal observability. Consider:

   ẋ1 = -δx 2 1 + δx 4 u 1 , ẋ2 = -x 2 1 δx 3 + x 2 + x 1 δx 4 u 1 , ẋ3 = x 4 -x 2 1 δx 4 u 1 , ẋ4 = x 5 + δx 1 , ẋ5 = δx 1 δx 3 + u 2 , y 1 = x 1 , y 2 = x 2 , y 3 = x 1 δx 1 + x 3 . (32) One can check that ν 1 = k 1 = ν 2 = k 2 = 1, ν 3 = 1, k 3 = 3, yielding ρ 1 = ρ 2 = ρ 3 = 1 and Φ = {dx 1 , dx 2 , (δx 1 + x 1 δ)dx 1 + dx 3 }. One has rank K(δ] Φ = 3 < n. Set G =span R[δ] {G 1 , • • • , G m }, then one has: G ⊥ = span R[δ] { x 1 dx 1 -dx 2 , x 2 1 dx 1 + dx 3 , dx 4 } . Since rank K(δ] Φ = 3, thus £ = span R[δ] {x 1 , x 2 , x 1 δx 1 + x 3 } and Ω = span £(δ] {dx 1 , dx 2 , dx 3 } , which yields: Ω ∩ G ⊥ = span £(δ] { x 1 dx 1 -dx 2 , x 2 1 dx 1 + dx 3 } .
In the following, identifiability and observability will be successively checked for [START_REF] Oguchi | Input-output linearization of retarded nonlinear systems by an extended Lie derivative[END_REF].

Identifiability analysis:

Following Theorem 1, one has:

H = [ ẏ1 , ẏ2 , ẏ3 ] T , Ψ = [ -δx 2 1 , -x 2 1 δx 3 + x 2 , x 4 ] T ,
and

Γ =   δx 4 , 0 x 1 δx 4 , 0 -x 2 1 δx 4 , 0   .
Thus, by choosing Q = [x 1 , -1, 0], a non zero one-form can be found, such as:

ω = x 1 dx 1 -dx 2 ∈ Ω ∩ G ⊥ , satisfying ωf = -x 1 δx 2 1 + x 2 1 δx 3 -x 2 ∈ £.
According to Theorem 3, the following equation is an output delay equation:

Q (H -Ψ ) = 0, (33)
since it contains only the output, its derivatives and delays. Since Y = (x 1 , x 2 , x 1 δx 1 + x 3 ) T , one has:

∂Y ∂x =  
1, 0, 0, 0, 0 0, 1, 0, 0, 0 δx 1 + x 1 δ, 0, 1, 0, 0   , and ∂Ψ ∂x =   -2δx 1 δ, 0, 0, 0, 0 -2x 1 δx 3 , 1, -x 2 1 δ, 0, 0 0, 0, 0, 1, 0   .

Thus, one obtains:

rank K(δ] ∂Y ∂x = 3 < rank K ∂{Y, Ψ } ∂x = 6.
Theorem 4 is satisfied and (33) involves δ in an essential way. A straightforward calculation gives: y 1 ẏ1 -ẏ2 = -y 1 δy 2 1 + y 2 1 δy 3 -y 2 1 δy 1 δ 2 y 1 -y 2 , which permits to identify the delay.

Observability analysis:

From the definition of L in [START_REF] Marquez-Martinez | New insights on the anlysis of nonlinear time-delay systems: Application to the triangular equivalence[END_REF], one can check that rank K L = 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 4 / ∈ £. Thus, according to Theorem 6, a new output ȳ1 = h 4 is given by:

ȳ1 = h 4 = ωf mod £ = x 4 = y 2 1 ẏ1 + ẏ3 + y 2 1 δy 2 1 . ( 34
)
For the new output ȳ1 , one has k i = ν i = 1 for 1 ≤ i ≤ 3, k 4 = ν 4 = 2, thus ρ i = 1 for 1 ≤ i ≤ 3 and ρ 4 = 2. Finally, one obtains the new Φ as follows:

Φ = {dx 1 , dx 2 , (δx 1 + x 1 δ)dx 1 + dx 3 , dx 4 , δdx 1 + dx 5 }.

It can be checked that rank K(δ] Φ = 5 = n, and the new £ is:

£ = span R[δ] {x 1 , x 2 , x 1 δx 1 + x 3 , x 4 , x 5 + δx 1 }.
This gives the following change of coordinates: z = ϕ(x, δ) = (x 1 , x 2 , x 1 δx 1 + x 3 , x 4 , x 5 + δx 1 ) T .

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

x = ϕ -1 = (z 1 , z 2 , z 3 -z 1 δz 1 , z 4 , z 5 -δz 1 ) T .

When t ≥ τ , one gets the following estimations of states: { x 1 = y 1 , x 2 = y 2 , x 3 = y 3 -y 1 δy 1 , x 4 = ȳ1 , x 5 = -δy 1 + ẏ1 , with ȳ1 defined in [START_REF] Oguchi | Input-output linearization of retarded nonlinear systems by using an extension of lie derivative[END_REF]. Moreover, the matrix Γ with the new output ȳ1 can be obtained as follows:

Γ =    

Conclusion

This chapter has studied identifiability and observability for nonlinear time-delay systems with unknown inputs. Concerning the identification of the delay, dependent and independent outputs over the non-commutative rings have been analyzed. Con-

δx 4 , 0 x 1 δx 4 1 = ẏ1+δy 2 1 δ

 411 rank K(δ] Γ = 2. One can find matrices Ξ = 2×2 . Consequently, according to Theorem 5, u 1 and u 2 can be causally estimated. When t ≥ 3τ , a straightforward computation yields the following estimates for the unknown inputs: { u ȳ1 , u 2 = ÿ1 -δ ẏ1 -δy 1 δy 3 + δy 2 1 δ 2 y 1 .

means 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].

in[START_REF] Anguelova | State elimination and identifiability of the delay parameter for nonlinear time-delay systems[END_REF], this equation is stated to involve the delay in an essential way.

i.e., singularity of the delay identification exists for a countable set of y, which case is excluded in this chapter.

cerning the observability, necessary and sufficient conditions have been deduced for both causal and non-causal cases. The causal and non-causal estimations of unknown inputs of the studied systems have been analyzed as well.