G Zheng

J.-P Barbot

D Boutat

Identification of the delay parameter for nonlinear time-delay systems with unknown inputs

Keywords:

By using the theory of non-commutative rings, this paper studies the delay identification of nonlinear time-delay systems with unknown inputs. A sufficient condition is given in order to deduce an output delay equation from the studied system. Then necessary and sufficient conditions are proposed to judge whether the deduced output delay equation can be used to identify delay involved in this equation. Two different cases are discussed for the dependent and independent outputs, respectively. The presented result is applied to identify delay in a biological system.

Introduction

Time-delay systems are widely used to model concrete systems in engineering sciences, such as biology, chemistry, mechanics and so on [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]; [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF]). Many results have been reported for the purpose of stability analysis, by assuming that the delay of the studied systems is known. And it makes the delay identification 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 in [START_REF] Drakunov | Delay identification in time-delay systems using variable structure observers[END_REF], by a modified least squares technique in [START_REF] Ren | Online identification of continuoustime systems with unknown time delay[END_REF], by convolution approach in [START_REF] Belkoura | Identifiability of systems described by convolution equations[END_REF], by using the fast identification technique proposed in [START_REF] Fliess | Reconstructeurs d'état[END_REF] to deal with online identification of continuous-time systems with structured entries in [START_REF] Belkoura | Parameters estimation of systems with delayed and structured entries[END_REF] and so on.

Recently, [START_REF] Anguelova | State elimination and identifiability of the delay parameter for nonlinear time-delay systems[END_REF] proposed to analyze the delay identification for nonlinear control systems with a single unknown constant delay by using the non-commutative rings theory, which has been applied to analyze nonlinear timedelay systems firstly by [START_REF] Moog | The disturbance decoupling problem for time-delay nonlinear systems[END_REF] for the disturbance decoupling problem of nonlinear time-delay system, and for observability of nonlinear time-delay systems with known inputs by [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over non-commutative rings[END_REF] and with unknown inputs by [START_REF] Zheng | On obserability of nonlinear time-delay systems with unknown inputs[END_REF]. Although the inputs are commonly supposed to be known and are usually used to control the studied system, however there exists as well other cases, such as observer design for time-delay systems, in which the inputs can be unknown [START_REF] Sename | Unknown input robust observers for time-delay systems[END_REF]; [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] Yang | Observers for linear systems with unknown inputs[END_REF]). Moreover, some proposed unknown inputs observer design methods do depend on the known delay, which should be identified in advance. Motivated by this requirement and inspired by the work of [START_REF] Anguelova | State elimination and identifiability of the delay parameter for nonlinear time-delay systems[END_REF], this paper investigates the delay identification 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 definitions are given in Section 3. Necessary and sufficient conditions are discussed for identifying the delay from only the outputs of systems in Section 4. Main theorem for the case where the outputs are independent over the non-commutative rings is stated in Section 5, and the proposed result is applied to identify delay of biological system in Section 6.

Algebraic framework

Denote τ the basic delay, and assume that the delays are multiple times of τ. Consider the following nonlinear time-delay system:

         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 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, i ∈ S -= {0, 1, . . . , s} is a finite set of constant delays, f , g j and h are meromorphic functions 1 , f (x(tiτ)) = f (x, x(t -τ), . . . , x(tsτ)) and ψ : [-sτ, 0] → R n and ϕ : [-sτ, 0] → R m denote unknown continuous functions of initial conditions. In this work, 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], let K be the field of meromorphic functions of a finite number of the variables from {x j (tiτ), j ∈ [1, n], i ∈ S -}. For the sake of simplicity, we introduce delay operator δ, which means

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 i ∈ Z + . Let K(δ] denote the set of polynomials of the form

a(δ] = a 0 (t) + a 1 (t)δ + • • • + a r a (t)δ r a (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] = r a +r b k=0 i≤r a , j≤r b i+ j=k a i (t)b j (t -iτ)δ k (5)
Differentiation of a function h j (x(t-iτ)) is defined as follows:

ḣ j (x(t -iτ)) = s i=0 ∂h j ∂x(t -iτ) δ i f
With the definition of K(δ], (1) can be rewritten in a more compact form as follows:

         ẋ = f (x, δ) + G(x, δ)u = f (x, δ) + m i=1 G i 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)) and h(x, δ) = h(x(tiτ)) with entries belonging to K, u = u(t), and

G(x, δ) = [G 1 , • • • , G m ] with G i = n j=1 s l=0 g l i δ l ∂ ∂x j . Denote 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 the ring K(δ] 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 module over this ring.

With the standard differential operator d, define the vector space E over K:

E = span K {dξ : ξ ∈ K}
which is the set of linear combinations of a finite number of elements from dx j (tiτ) with row vector coefficients in K. Since the delay operator δ and the standard differential operator are commutative, thus the one-form 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] ∂ ∂x j 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]
with ω ∈ M.

Notations and definitions

Some efforts have been made to extend the Lie derivative [START_REF] Isidori | Nonlinear control systems (3rd edition)[END_REF]) to nonlinear time-delay systems (see [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 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]; [START_REF] Califano | Extended lie brackets for nonlinear time-delay systems[END_REF]) in the framework of commutative rings. In what follows we define the derivative and Lie derivative for nonlinear time-delay from the non-commutative point of view.

Let f (x(tjτ)) and h(x(tjτ)) for 0 ≤ j ≤ s 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 G i h i . Based on the above notations, the relative degree might be defined in the following way:

Definition 1. (Relative degree) System (6) has relative degree (ν 1 , • • • , ν p ) in an open set W ⊆ R n if, for 1 ≤ i ≤ p, the follow- ing conditions are satisfied : 1. for all x ∈ W, L G j 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 G j L ν i -1 f h i 0.
If the first condition is satisfied for all r ≥ 0 and 1 ≤ i ≤ p, we set ν i = ∞. Moreover, for system (6), 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 the non-commutative rings. 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
for 1 ≤ k ≤ n, and 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 ob- servability indices.
Reorder, if necessary, the output components of ( 6), such that

rank K(δ] { ∂h 1 ∂x , • • • , ∂L k 1 -1 f h 1 ∂x , • • • , ∂h p ∂x , • • • , ∂L kp -1 f h p ∂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 k i the observability indices, ν i the relative degree for y i of (6), and note

ρ i = min {ν i , k i } Without loss of generality, suppose p i=1 ρ i = j, thus {dh 1 , • • • , dL ρ 1 -1 f h 1 , • • • , dh p , • • • , dL ρ p -1 f h p } are j linearly inde- pendent vector over K(δ]. Then note Φ = {dh 1 , • • • , dL ρ 1 -1 f h 1 , • • • , dh p , • • • , dL ρ p -1 f h p } and £ = span R[δ] h 1 , • • • , L ρ 1 -1 f h 1 , • • • , h p , • • • , L ρ p -1 f h p (10)
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 element of Φ over £(δ] as follows

Ω = span £(δ] {ξ, ξ ∈ Φ} (11) Define G = span R[δ] {G 1 , . . . , G m }
where G i is given in ( 6), and its left annihilator

G ⊥ = span £(δ] {ω ∈ M | ωβ = 0, ∀β ∈ G} ( 12
)
where M is defined in (7).

Preliminary result

After having defined the derivative of function belonging to K(δ], let us study the delay identification for system (6). The following definition 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 (6), an equation

α(h, ḣ . . . , h (k) , δ) = 0, k ∈ Z +
is named to be an output delay equation since it contains only the output, its derivatives and delays. Moreover this equation is called to be an output delay-identifiable equation 2 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) becomes to seek such an equation.

Let us 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 preliminary result.

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

3 singularity of the delay identification exists for a countable set of y, which is excluded in this paper.

Theorem 1. There exists an output delay-identifiable equation α(h, δ) for (6) if and only if

rank K(δ] ∂h ∂x < rank K ∂h ∂x (13)
Proof. Necessity: Suppose that there exists an output delayidentifiable equation which enables to identify the delay of ( 6), let us show ( 13) is satisfied. For this, consider the opposite, i.e. assume rank

K(δ] ∂h ∂x = rank K ∂h ∂x ,
and this implies that h does not contain any delay, otherwise we have

rank K(δ] ∂h ∂x < rank K ∂h ∂x .
No involvement of delay in h implies that it is not possible to identify the delay from the output, thus it contradicts the assumption that there exists an output delay-identifiable equation to identify the delay, and we prove the necessity by contradiction.

Sufficiency: Suppose that

rank K(δ] ∂h ∂x < rank K ∂h ∂x ≤ p,
then there exists a i ∈ K(δ] for 1 ≤ i ≤ p, such that

a 1 dh 1 + • • • + a p dh p = 0
The exterior differentiation of the above equation is equal to zero, implying the left side of the above equation is a closed one-form. By applying Poincaré Theorem, there always exists a function α such that α(h 1 , . . . , h p , δ) = 0

In addition, the above output delay equation should involve δ in an essential way, because if it is not the case, then δ can be taken out from α, yielding

rank K(δ] ∂h ∂x = rank K ∂h ∂x ,
and this contradicts the inequality (13).

Remark 1. Inequality (13) implies that the outputs of ( 6) are dependent over K(δ]. Theorem 1 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 we will show in the next section that this condition is not necessary for the case where the output of ( 6) is independent over K(δ].

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 ( 14
)
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 1 is satisfied, and the delay of system ( 14) 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 zero-crossing when varying δ.

Main result

Identification of delay for (6) from its output is analyzed in the last section, this section considers the case where the output of ( 6) is independent over K(δ].

Since for (6) we denoted k i the observability indices and ν i the relative degree for y i of (6), and noted ρ i = min {ν i , k i } for 1 ≤ i ≤ p, then one obtains

H(x, δ) = Ψ(x, δ) + Γ(x, δ)u (15) with H(x, δ) = h (ρ 1 ) 1 , • • • , h (ρ p ) p T Ψ(x, δ) = L ρ 1 f h 1 , • • • , L ρ p f h p T and Γ(x, δ) =                L G 1 L ρ 1 -1 f h 1 • • • L G m L ρ 1 -1 f h 1 . . . . . . . . . L G 1 L ρ p -1 f h p • • • L G m L ρ p -1 f h p                (16)
Based on the above definitions, we are ready to state our main result.

Theorem 2. There exists an output delay equation for (6), if there exists a non zero ω = n c=1 p j=1 q j ∂L ρ j -1 f h j ∂x c dx c with q j ∈ K(δ] for 1 ≤ j ≤ p, such that ω ∈ G ⊥ ∩ Ω and ω f ∈ £, where G ⊥ is defined in (12), Ω is defined in (11), and £ is defined in (10).

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

QΓ = Q                L G 1 L ρ 1 -1 f h 1 • • • L G m L ρ 1 -1 f h 1 . . . . . . . . . L G 1 L ρ p -1 f h p • • • L G m L ρ p -1 f h p                =                  Q                  ∂L ρ 1 -1 f h 1 ∂x . . . ∂L ρ p -1 f h p ∂x                                   [G 1 , • • • , G m ]
because of the associativity law over K(δ]. Then according to the definition (8), one gets

QΓ = ω [G 1 , • • • , G m ] = ωG where ω = n c=1 p j=1 q j ∂L ρ j -1 f h j ∂x c dx c .
Moreover, it is easy to check that

ω f =                    Q                    ∂L ρ 1 -1 f h 1 ∂x . . . ∂L ρ p -1 f hp ∂x                                       f = Q                                       ∂L ρ 1 -1 f h 1 ∂x . . . ∂L ρ p -1 f hp ∂x                    f                    = Q               L ρ 1 f h 1 . . . L ρ p f h p               = QΨ
According to ( 15), one has

QH = Q(Ψ + Γu) = ω f + ωGu ( 17
)
where H = y

(ρ 1 ) 1 , • • • , y (ρ p ) p T
is a vector which can be estimated in finite time.

Thus, if ω ∈ G ⊥ ∩ Ω and ω f ∈ £, which implies there exists Q with entries belonging to £(δ], one has

QΓ = ωG = 0 and QH = ω f ∈ £
Finally one obtains the following relation:

Q(H -Ψ) = 0 (18)
which contains only the output, its derivatives and delays. Obviously, if ( 18) is an output delay-identifiable equation, i.e. containing the delay δ in an essential way, then the delay of ( 6) can be identified by detecting zero-crossing of (18). Before to give necessary and sufficient conditions guaranteeing the essential involvement of δ in (18), define

Y = h 1 , . . . , L ρ 1 -1 f h 1 , . . . , h p , . . . , L ρ p -1 f h p T
and denote K 0 ⊂ K the field of meromorphic functions of x, which will be used in the following theorem.

Theorem 3. The output delay equation ( 18) is an output delayidentifiable equation if and only if

rank K(δ] ∂Y ∂x < rank K ∂{Y, Ψ} ∂x (19)
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 q j ∈ K 0 , 1 ≤ j ≤ p (20)

and rank K(δ] ∂Y ∂x = rank K ∂{Y, Ψ} ∂x (21)
Proof. Necessity: Suppose that (18) involves the delay δ in an essential way, we will show that either ( 19) or (20-21) is satisfied. We prove this by contradiction and let us consider the opposite of (19-21), i.e.

rank K(δ] ∂Y ∂x = rank K ∂{Y, Ψ} ∂x (22)
and q j = a(δ] q j with q j ∈ K 0 , 1 ≤ j ≤ p (23) for all q j ∈ K(δ] in Q.

It can be seen that ( 22) implies that Ψ is a function of Y without δ, and (23) implies that there exists a scalar a

(δ] ∈ K(δ] such that Q = a(δ] Q,
where Q is the associated vector with entries belonging to the field K 0 , then

Q(H -Ψ) = a(δ] Q(H -Ψ) = 0,
and this contradicts the assumption that (18) involves the delay δ in an essential way, so the necessity is proven by contradiction.

Sufficiency: Firstly, suppose that

rank K(δ] ∂Y ∂x < rank K ∂{Y, Ψ} ∂x ,
then Ψ should be a function of Y containing δ in an essential way, since if it is not the case, one can find a function α such that α (Ψ, Y) = 0, which implies that

rank K ∂{Y, Ψ} ∂x ≤ rank K ∂{Y} ∂x = rank K(δ] ∂Y ∂x ,
since Y is linearly independent over K(δ]. Because Ψ is a function of Y containing δ in an essential way, thus for any Q, the function Q(H -Ψ) = 0 always contains δ in an essential way. Secondly, since

rank K(δ] ∂Y ∂x ≤ rank K ∂{Y} ∂x ≤ rank K ∂{Y, Ψ} ∂x and rank K(δ] ∂Y ∂x ≤ rank K(δ] ∂{Y, Ψ} ∂x ≤ rank K ∂{Y, Ψ} ∂x , so if rank K(δ] ∂Y ∂x = rank K ∂{Y, Ψ} ∂x ,
then one obtains

rank K ∂{Y} ∂x = rank K ∂{Y, Ψ} ∂x and rank K(δ] ∂Y ∂x = rank K(δ] ∂{Y, Ψ} ∂x .
The above two equalities imply that Ψ should be a function of Y without delays, otherwise one has

rank K(δ] ∂Y ∂x rank K ∂{Y, Ψ} ∂x ,
and this contradicts the assumption of the satisfactory of (21).

Moreover, if there does not exist an a(δ] ∈ K(δ] such that q j = a(δ] q j for any element q j in Q and q j ∈ K 0 , then it implies that Q contains delays in an essential way. Otherwise one can find a vector Q with entries belonging to K 0 and an a(δ] ∈ K(δ] such that Q = a(δ] Q, which implies Q does not contain delays in an essential way.

Finally the essential involvement of δ in Q means the essential involvement of δ in the function:

Q(H -Ψ) = 0 even when Ψ does not contain any delay.

Remark 2. It is clear that Theorem 1 is a special case of Theorem 3, since the output delay-identifiable equation stated in Theorem 1 does not contain any derivative of the output.

A similar condition as ( 19) of Theorem 3 is stated as a necessary and sufficient condition for identifying delay for 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]. However as we proved above that it is only sufficient, but not necessary for the case with unknown inputs.

Application to JAK-STAT

In order to highlight the proposed results, let us consider the dynamics of the JAK-STAT (JAnus Kinase -Signal Transducer and Activator of Transcription) signaling pathway in the cell from [START_REF] Timmer | Modeling the nonlinear dynamics of cellular signal transduction[END_REF].

The JAK-STAT signaling pathway transmits information from chemical signals outside of the cell, through the cell membrane, and into gene promoters on the DNA in the cell nucleus, which causes DNA transcription and activity in the cell.

Binding of the hormone erythropoietin (Epo) to the extracellular part of the receptor leads to activation by phosphorylation of the so-called JAK at intracellular, cytoplasmic domain of the receptor. In trun, this leads to phosphorylation of monomeric STAT-5 forms dimers and these dimers migrate into the nucleus where they bind to promotor region of the DNA and initiate gene transcription.

Denote the amount of activated Epo-receptors by u, unphosphorylated monomeric STAT-5 by x 1 , phosphorylated monomeric STAT-5 by x 2 , phosphorylated dimeric STAT-5 in the cytoplasm by x 3 and phosphorylated dimeric STAT-5 in the nucleus by x 4 , the STAT-5 cycling model can be described as follows:

                             ẋ1 = -k 1 x 1 u + 2k 4 δx 3 ẋ2 = k 1 x 1 u -k 2 x 2 2 ẋ3 = -k 3 x 3 + k 2 x 2 2 /2 ẋ4 = k 3 x 3 -k 4 δx 3 y 1 = x 1 y 2 = x 2 y 3 = x 4 (24)
In the above JAK-STAT model, the coefficients k 1k 4 are known constants. Note that u representing the amount of activated Epo-receptors is an external input, thus it is assumed to be unknown in this paper.We are going to identify the delay δ in (24).

For the outputs of ( 24), one has

k 1 = ν 1 = 1, k 2 = ν 2 = 1 and k 3 = 2, ν 3 = 3 which gives ρ 1 = 1, ρ 2 = 1 and ρ 3 = 2. Then one obtains Φ = {dx 1 , dx 2 , dx 4 , (k 3 -k 4 δ)dx 3 }
By simple calculations, one obtains

£ = span R[δ] {x 1 , x 2 , x 4 , (k 3 -k 4 δ)x 3 } G ⊥ = span £(δ] {dx 1 + dx 2 , dx 3 , dx 4 } and Ω = span £(δ] {dx 1 , dx 2 , dx 4 , (k 3 -k 4 δ)dx 3 } which gives Ω ∩ G ⊥ = span £(δ] {dx 1 , dx 2 , dx 4 , (k 3 -k 4 δ)dx 3 } ∩span £(δ] {dx 1 + dx 2 , dx 3 , dx 4 } = span £(δ] {dx 1 + dx 2 , (k 3 -k 4 δ)dx 3 , dx 4 } By choosing Q = (0, 0, 1), a non zero one-form can be found, such as ω = (k 3 -k 4 δ)dx 3 ∈ Ω ∩ G ⊥ satisfying ω f = -k 3 (k 3 -k 4 δ)x 3 + k 2 (k 3 -k 4 δ)x 2 2 /2 ∈ £ Thus the following function Q (H -Ψ) = 0 (25)
contains only the output, its derivatives and delays, where

H = (ẏ 1 , ẏ2 , ÿ3 ) T Ψ = 2k 4 δx 3 , -k 2 x 2 2 , (k 3 -k 4 δ)(-k 3 x 3 + k 2 x 2 2 /2)
T Since Y = (x 1 , x 2 , x 4 , (k 3k 4 δ)x 3 ) T , one has

∂Y ∂x =              
1, 0, 0, 0 0, 1, 0, 0 0, 0, 0, 1 0, 0, k 3k 4 δ, 0

              and ∂Ψ ∂x =           0, 0, 2k 4 δ, 0 0, -2k 2 x 2 , 0, 0 0, k 2 (k 3 -k 4 δ)x 2 , -k 3 (k 3 -k 4 δ), 0          
Thus, one obtains which can be used as well to identify δ.

Conclusion

This paper deals with the delay identification of time-delay systems with unknown inputs. Necessary and sufficient condition for the most simple case, i.e. identification of the delay directly from the output, has been studied. For a more generic case, sufficient condition is given in order to guarantee the existence of an output delay equation. Necessary and sufficient condition is then discussed to check whether the deduced output delay equation can be used to identify the delay. The obtained results are valid for the systems with single delay, and the derived necessary and sufficient conditions become only necessary for the systems with multi delays.

  Theorem 3 is satisfied and (25) involves δ in an essential way. A straightforward calculation givesÿ3 = -k 3 ẏ3 + k 2 k 3 y 2 2 /2k 2 k 4 (δy 2 ) 2 /2which permits to identify δ.Let us remark that the choice of non zero one-form ω is not unique, which leads to different output delay-identifiable equations. For example, by choosingQ = [k 3 (k 3k 4 δ), k 3 (k 3k 4 δ), 2k 4 δ] one obtains ω = k 3 (k 3k 4 δ)dx 1 + k 3 (k 3k 4 δ)dx 2 + 2k 4 δ(k 3k 4 δ)dx 3 such that ω f = -k 2 (k 3k 4 δ) 2 x 2 2 ∈ £ Based on this new Q, for the output delay-identifiable equation Q(H -Ψ) = ω f -QΨ = 0, one obtains k 3 (k 3k 4 δ)(ẏ 1 + ẏ2 ) + 2k 4 δÿ 3 = -k 2 (k 3k 4 δ) 2 x 2 2 = -k 2 (k 3k 4 δ) 2 y 2 2

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]).


Ramdane Tami

Driss Boutat

Gang Zheng

Extended output depending normal form

Keywords: Extended Normal Form, Nonlinear system, Observer, Epidemic model

This paper concerns the design of a nonlinear observer through a transformation of a nonlinear system into an observer form that supports a high gain observer. Sufficient geometrical condition is deduced to guarantee the existence of change of coordinates allowing the transformation of a nonlinear system into the proposed normal form. The result is highlighted by studying the Susceptible Exposed Infected and Recovered (SEIR) model.

Introduction

The observer design for nonlinear dynamical systems is an important issue in the control theory. Several approaches are proposed to design nonlinear observers for many cases, while one of the methods is to transform the nonlinear system into a more simple form which enables to apply existing observers.

The first idea of normal form is due to [START_REF] Bestle | Canonical form observer design for nonlinear time varying systems[END_REF] for time variant dynamical systems, and to [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF] for time invariant dynamical systems, where the author introduced the so-called observer canonical form with output injection with all nonlinear terms being only function of the output. Then [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF] gave the associated canonical form with output injection for multi-outputs nonlinear systems without inputs, and the result for multi-outputs systems with inputs was studied in [START_REF] Xia | Nonlinear observer with linearizable error dynamics[END_REF]. Based on the above works many algorithms are developed to generalize the existing results, including algebraic approaches [START_REF] Keller | Nonlinear observer design by transformation into a generalized observer canonical form[END_REF]; [START_REF] Phelps | On constructing nonlinear observers[END_REF]; [START_REF] Rudolph | A block triangular nonlinear observer normal form[END_REF]), geometrical approaches [START_REF] Hou | Observer with linear error dynamics for nonlinear multi-output systems[END_REF]; [START_REF] Marino | Nonlinear control design: geometric, adaptive and robust[END_REF]; [START_REF] Boutat | New algorithm for observer error linearization with a diffeomorphism on the outputs[END_REF]; [START_REF] Lynch | Nonlinear observers with approximately linear error dynamics: the multivariable case[END_REF]) and the so called direct transformations stated in [START_REF] Lopez | Linearization by completely generalized input ouput injection[END_REF]. To enlarge the class of observer forms, the concept of output depending normal form was firstly addressed in [START_REF] Guay | Observer linearization by output-dependent time-scale transformations[END_REF]; [START_REF] Respondek | Time scaling for observer design with linearizable error dynamics[END_REF], then was developed in [START_REF] Zheng | A single output dependent observability normal form[END_REF] and Wang andLynch (2009, 2010). The other approaches to enlarge the class of normal forms are the extended normal form introduced in [START_REF] Jouan | Immersion of nonlinear systems into linear systems modulo output injection[END_REF]; [START_REF] Noh | Nonlinear observer design by dynamic observer error linearization[END_REF]; [START_REF] Back | Dynamic observer error linearization[END_REF]; [START_REF] Yang | Reduced-order dynamic observer error linearization[END_REF][START_REF] Yang | A complete solution to a simple case of dynamic observer error linearization: New approach to observer error linearization[END_REF], where the main idea is to add an auxiliary dynamics to the dynamical system in such a way that the extended system can satisfy the conditions proposed in the literature. The geometrical characterization of the Email addresses: ramdane.tami@ensi-bourges.fr (Ramdane Tami), driss.boutat@ensi-bourges.fr (Driss Boutat), gang.zheng@inria.fr (Gang Zheng) 1 This work was supported by Région Centre France second method was addressed in [START_REF] Boutat | Geometrical conditions for observer error linearization via ∫ 0[END_REF] and [START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF].

In this paper, we propose a new observer normal form by mixing the output depending normal form and the extended normal form. Precisely, we give geometrical sufficient condition that guarantees the transformation of a dynamical system via a change of coordinates to a new extended observer normal form. Then, we apply this method to the Susceptible Exposed Infected and Recovered (SEIR) model. This paper is organized as follows. Section 2 presents motivation example and the problem statement. In section 3, a new extended output depending normal form is proposed, and sufficient condition is established allowing the transformation of nonlinear dynamical systems into the proposed normal form. Section 4 applies the proposed result to study SEIR model.

Preliminaries

Geometrical background

Herein, we introduce a background on some geometrical tools that will be needed afterward. Consider a single output nonlinear dynamical system in the following form:

ẋ = f (x) y = h(x) (1)
where x ∈ U ⊆ R n is the state and y ∈ R is the output. We assume that the vector field f and the output function h are smooth. In the following, we also assume that the pair (h, f ) satisfies the observability rank condition. Thus, the so-called observability differential 1-forms are independent and are given by:

θ 1 = dh (2) θ i = dL i-1 f h for 2 ≤ i ≤ n (3)
where L k f h is the k th Lie derivative of h along f and d is the differential operator. Thus, one can construct the [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF]

frame τ = [τ 1 , • • • , τ n ]
where the first vector field τ 1 is a solution for the following algebraic equations:

θ i (τ 1 ) = 0 for 1 ≤ i ≤ n -1 θ n (τ 1 ) = 1 (4)
and the other vector fields are given by induction as follows:

τ i = [ τ i-1 , f ] (5)
for 2 ≤ i ≤ n, where [, ] denotes the Lie bracket.

In [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF], the commutativity of Lie bracket, i.e.

[ τ i , τ j ] = 0 for 1 ≤ i ≤ n, 1 ≤ j ≤ n is the necessary and sufficient condition to transform the system (1) into a nonlinear observer form with output injection. However, if this condition is not fulfilled, then another frame τ =

[ τ 1 , τ 2 ...τ n ] can be built from τ according to [START_REF] Respondek | Time scaling for observer design with linearizable error dynamics[END_REF]; [START_REF] Guay | Observer linearization by output-dependent time-scale transformations[END_REF], [START_REF] Zheng | A single output dependent observability normal form[END_REF], and Wang andLynch (2009, 2010) as follows:

{ τ 1 = πτ 1 τ i = 1 α i [ τ i-1 , f ] (6)
where π = n ∏ i=2 α i , and α i (y) for 2 ≤ i ≤ n being non vanishing functions of the output to be determined. If the commutativity of Lie bracket condition is fulfilled for the new frame τ, then the system (1) can be transformed into the output depending nonlinear observer form with output injection. However, there exist as well some dynamical systems which do not fulfill the above conditions, and this will be highlighted in the following section.

Motivating example: SEIR model

The following example motivates the research of this work. Let us consider the Susceptible Exposed Infected and Recovered (SEIR) model [START_REF] Iggidr | Controllability, observability and stability of mathematical models[END_REF]:

dS dt = bN -µS -β S I N -pbE -qbI dE dt = β S I N + pbE + qbI -(µ + ε)E dI dt = εE -(r + δ + µ)I dR dt = rI -µR dN dt = (b -µ)N -δI (7)
where S (t) is the susceptibility of the host population to the contagious disease, E(t) is the exposed population but not yet expressing symptoms, I(t) is the infected population, R(t) is the recovered population, b is the rate of the natural birth, µ is the rate of fecundity, β is the transmission rate, δ is the death rate related to diseases, ε is the rate at which the exposed population becomes infective, p is the rate of the offspring from an exposed population, q is the rate of the offspring from an infected population and r is the rate at which the infected individuals are recovered.

It is supposed that one can measure the infected population I(t) and the total population N which is given as follows:

N = S + E + I + R (8)
One wants to estimate the susceptibility of the host population S (t) and the exposed population E(t) from the infectious population. Then R(t) can be deduced form the algebraical equation ( 8).

For the sake of simplicities, let us consider the normalized model of (7), by setting

x 1 = S N , x 2 = E N , x 3 = I N , x 4 = R N and y = I N .
Consequently the SEIR dynamics can be rewritten as follows:

ẋ1 = b -bx 1 + γ 1 x 1 x 3 -pbx 2 -qbx 3 (9) ẋ2 = βx 1 x 3 + γ 2 x 2 + δx 2 x 3 + qbx 3 (10) ẋ3 = εx 2 + γ 3 x 3 + δx 2 3 (11) ẋ4 = rx 3 -bx 4 + δx 3 x 4 (12) y = x 3 (13) with γ 1 = -(β -δ), γ 2 = -(b + ε -pb), γ 3 = -(r + δ + b)
and

x 1 + x 2 + x 3 + x 4 = 1 (14)
We will show later in this paper that the system (9-11) with the output y = h(x) = x 3 does not fulfill any geometrical condition existing in the literature which allow to transform it into some existing observer forms. Therefore, this paper proposes a more general observer form that covers the existing forms proposed in [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF], [START_REF] Respondek | Time scaling for observer design with linearizable error dynamics[END_REF]; [START_REF] Guay | Observer linearization by output-dependent time-scale transformations[END_REF], [START_REF] Zheng | A single output dependent observability normal form[END_REF]and Wang andLynch (2009, 2010).

Inspired by the techniques of immersion and output injection, one can add the following auxiliary dynamics:

ẇ = -κ(w)(γ 1 y -b) (15)
where w ∈ R is an auxiliary variable, considered as an extra output and κ(w) can be freely chosen in order to ensure the boundedness of w. Then the following diffeomorphism:

z 1 = βε l(w) x 1 + bpβ l(w) x 3 z 2 = ε l(w) x 2 - (b + γ 2 ) l(w) x 3 - 1 2 (δ -γ 1 ) l(w) x 2 3 z 3 = 1 l(w) x ξ = w (16)
where l(w) = e ∫ w 0 b κ 2 (s) ds determined in Section 4 and κ 2 (w) = -κ(w)b, allows the transformation of the system (9-11) and the auxiliary dynamics (15) into the following extended output depending form:

ż1 = B 1 (w, y) ż2 = yz 1 + B 2 (w, y) ż3 = z 2 + B 3 (w, y) ξ = B 4 (w, y) y = z 3 (17)
where B i for 1 ≤ i ≤ 3 are given in Section 4. For this transformed form, it can be seen that one can design a high-gain observer to estimate z 1 and z 2 with the known variables z 3 and w.

Consequently, in order to implement this new method, one should firstly propose a new normal form which mixes the immersion and output depending observer form, and then determine the auxiliary dynamics and the condition guaranteeing the existence of a diffeomorphism such that the system with immersion can be transformed into the proposed observer form.

Main results

Extended output depending normal form and observer design

Consider the nonlinear system (1), one seeks an auxiliary dynamic ẇ = η(y, w) so that the following extended dynamical system:

ẋ = f (x) (18) ẇ = η(y, w) (19) y = h(x) (20)
could be transformed via a diffeomorphism (z T , ξ) T = ϕ(x, w) into the following more general extended output depending observer form:

ż = A(y)z + B(y, w) (21) ξ = B n+1 (y, w) (22) y = Cz (23) where ξ ∈ R, w ∈ R, C = [0, ..., 0, 1], A(y) =                    0 ... ... ... 0 α 2 (y) 0 ... ... 0 0 α 3 (y) ... ... ... 0 ... ... ... ... 0 ... ... α n (y) 0                    (24)
The proposed form (21-23) supports the high-gain observer studied in [START_REF] Busawon | Asimple observer for a class of nonlinear systems[END_REF] as follows:

ż = A(y)ẑ + B(w, y) -Γ -1 (y)R -1 ρ C T (Cẑ -y) 0 = ρR ρ + G T R ρ + R ρ G -C T C where G =                  0 • • • 0 0 1 • • • 0 0 . . . . . . . . . . . . 0 • • • 1 0                  Γ(y) = diag[ n Π i=2 α i (y), n Π i=3 α i (y), • • • , α n (y), 1] R ρ (n + 1 -i, n + 1 -j) = (-1) i+ j C j-1 i+ j-2 ρ i+ j-1
for 1 ≤ i ≤ n and 1 ≤ j ≤ n, where C p n = n! (n-p)!p! is a binomial coefficient. The observation error dynamics will be governed by the following dynamics:

ė = ż -ż = (A(y) -Γ -1 (y)R -1 ρ C T C)e
If y and w are bounded, then the observation error dynamics is exponentially stable by well choosing ρ.

Determination of auxiliary dynamics and diffeomorphism

In this section, we discuss the sufficient geometrical condition which guarantees the existence of an auxiliary dynamics ẇ = η(y, w) and a diffeomorphism (z T , ξ) T = ϕ(x, w) for the purpose of transforming the extended system (18-20) into the proposed extended output depending observer normal form (21-23) where ξ ∈ R and w ∈ R.

For this, let us consider a function l(w) 0 to be determined later and build the following new frame σ from τ defined in (6):

{ σ 1 = l(w)τ 1 σ k = 1 α k [σ k-1 , F] ( 25
)
where α k for 2 ≤ k ≤ n is uniquely determined when constructing τ, and F is the vector field for the extended system (18-20), noted as

F = f + η(y, w) ∂ ∂w .
Then we can state the following theorem.

Theorem 1. If there exists a function l(w) 0 such that

[ σ i , σ j ] = 0
for 1 ≤ i ≤ n and 1 ≤ j ≤ n where σ i is defined in ( 25), then there exists a diffeomorphsim (z T , ξ) T = ϕ(x, w) which transforms the extended system (18-20) into the proposed normal form (21)(22)(23).

Proof 1. Let ∆ = span{σ 1 , ..., σ n } be the distribution spanned by vector fields σ i for 1 ≤ i ≤ n. As [σ i , σ j ] = 0 for 1 ≤ i, j ≤ n, then thanks to Theorem 2.36 in [START_REF] Nijmeijer | Nonlinear dynamical control systems[END_REF], there exists a local diffeomorphism (z T , ξ) T = ϕ(x, w) such that

ϕ * (σ i ) = ∂ ∂z i for 1 ≤ i ≤ n.

Now, let us consider the vector field

v = ϕ -1 * ( ∂ ∂ξ ). As {σ 1 , • • • , σ n , ∂ ∂w } forms a basis, then we can write v = r 1 (x, w) ∂ ∂w + r 2 (x, w)σ 1 + • • • + r n+1 (x, w)σ n
where r 1 0. As

[ ∂ ∂z i , ∂ ∂ξ ] = 0, then we have [σ i , v] = [ ϕ -1 * ( ∂ ∂z i ) , ϕ -1 * ( ∂ ∂ξ )] = ϕ -1 * [ ∂ ∂z i , ∂ ∂ξ ] = 0
In the following, we will show that r 1 (x, w) is only a function of the auxiliary variable w. Indeed, as [σ i , σ j ] = 0, a direct calculation gives:

[σ i , v] = ( L σ i r 1 ) ∂ ∂w + r 1 [σ i , ∂ ∂w ] + n ∑ j=1 ( L σ i r j+1 ) σ j
Moreover, by construction we have

σ i = n ∑ j=1 µ i, j (x, w) ∂ ∂x i for 1 ≤ i ≤ n, therefore we obtain [σ i , ∂ ∂w ] = - ∑ n j=1 ∂µ i, j ∂w ∂ ∂x i ∈ ∆. Thus, it can be seen that [σ i , v] = 0 if and only if ( L σ i r 1 ) ∂ ∂w = 0 and r 1 [σ i , ∂ ∂w ] + ( L σ i r 2 ) σ 1 + ( L σ i r 3 ) σ 2 + ... + ( L σ i r n+1 ) σ n = 0
Then one has L σ i r 1 = 0, which implies that r 1 (w) depend only on w. Now, consider

σ n+1 = 1 r 1 (w) v = ∂ ∂w + r 2 r 1 σ 1 + ... + r n+1 r 1 σ n . This vector field fulfils [σ i , v] = r 1 [σ i , σ n+1 ] = 0, and we have dw(σ n+1 ) = 1. Note σ = [σ 1 , σ 2 , • • • , σ n , σ n+1 ]
and denote the set of the observability 1-forms of the extended system as:

θ e = (dh, dL F h, • • • , dL n-1 F h, dw) T
then one can calculate the following matrix:

Λ = θ e σ =                                         0 0 0 • • • 0 l * 0 0 0 0 lα n * . . . 0 . . . 0 . . . * . . . * . . . 0 lπ α 2 α 3 * . . . * * 0 lπ α 2 * • • • * * * lπ * * • • • * * * 0 0 0 0 0 0 1                                        
It is clear that Λ is invertible, thus one can define the following multi 1-forms:

ω =                  ω 1 ω 2 . . . ω n+1                  = Λ -1 θ e ( 26
)
and one has ω i σ j = δ i j where δ i j is the Kronecker's delta for

1 ≤ i ≤ n + 1 and 1 ≤ j ≤ n + 1.
Let X and Y be two vectors fields in {σ 1 , ..., σ n , σ n+1 }, then one has:

dω i (X, Y) = L Y ω i (X) -L X ω i (Y) -ω i [X, Y]
Since ω i (X) and ω i (Y) are constant, one obtains:

dω i (X, Y) = -ω i [X, Y] .
Therefore, one has the equivalence between [X, Y] = 0 and dω i = 0 for 1 ≤ i ≤ n + 1. Thus by Poincaré's lemma there exists

ϕ = (ϕ 1 , • • • , ϕ n+1 ) T such that ω = dϕ := ϕ * .
Therefore, one can set:

ω(σ i ) = ϕ * (σ i ) = ∂ ∂z i for 1 ≤ i ≤ n and ϕ * (σ n+1 ) = ∂
∂ξ . Now, we analyze the influence of the diffeomorphism on the extended vector field F. For 1 ≤ i ≤ n -1 one has:

∂ϕ * (F) ∂z i = [ ∂ ∂z i , ϕ * (F) ] = [ ϕ * (σ i ) , ϕ * (F) ] = ϕ * [σ i , F] = α i+1 ϕ * σ i+1 = α i+1 ∂ ∂z i+1
So by integration one obtains żi = α i+1 (y

)z i + β i (y) for 1 ≤ i ≤ n -1.
For the last two dynamics, we use the definition of ω given in ( 26), and after a straightforward calculation one obtains:

ω n = dϕ n = dh l(w) + ρ(x)dw (27) ω n+1 = dϕ n+1 = dw (28)
where ρ(x) is a function of x. The equation ( 28) implies that ξ = ϕ n+1 = w. Moreover, since ω n is a closed 1-form, i.e. dω n = 0, so according to equation ( 27), ρ(x) should be only a function of y and w. Hence z n = ϕ n should be as well a function of the known variables y and w. Therefore, one can denote a new output y in the following form:

ϕ n (y, w) = z n = Cz
and this implies that system (18-20) is transformed into (21)(22)(23).

Remark 1.

• As ω = dϕ = ϕ * , then the diffeomorphism can be obtained by integration: z i = ϕ i (x) = ∫ ω i .

• If the vector field σ n+1 is obtained by induction as σ n+1 = [σ n , F], then ∂ϕ * (F) ∂z n = ∂ ∂ξ and the normal form become ż = A(y)z+ B(w), where the second term B(w) will depend only on the auxiliary variable w.

• As in assumption of Lemma 3.1 in [START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF], it can be shown that

[τ i , τ n ] -ϱ(y)τ i-1 ∈ span{τ 1 , • • • , τ i-1
} is the condition to ensure the existence of η(y, w) and l(w), where ϱ(y) is only a function of y.

Application to SEIR model

Calculation of τ

Considering the SEIR model described in (9-11), a simple calculation gives the associated observability 1-forms as follows:

θ 1 = dx 3 θ 2 = εdx 2 + (γ 3 + 2δx 3 )dx 3 θ 3 = εβx 3 dx 1 + ε(γ 2 + γ 3 + 3δx 3 )dx 2 + Q 1 dx 3 where Q 1 = εβx 1 + 3εδx 2 + εqb + γ 2 3 + 6δγ 3 x 3 + 6δ 2 x 2 3 .
Then the associated frame τ is given by:

τ 1 = 1 εβx 3 ∂ ∂x 1 τ 2 = uτ 1 + 1 ε ∂ ∂x 2 where u = -b + γ 3 + (δ + γ 1 )x 3 + ε x 2 x 3 τ 3 = -pb ε ∂ ∂x 1 - ( L f u ) τ 1 + uτ 2 + 1 ε (γ 2 + δx 3 ) ∂ ∂x 2 + ∂ ∂x 3 A straightforward calculation gives [τ 1 , τ 2 ] = [τ 1 , τ 3 ] = 0 and [τ 2 , τ 3 ] = Q 2 τ 1 + 1 x 3 τ 2
where

Q 2 = -(3δ + 2γ 1 + 2γ 2 -γ 3 x 3 -3ε x 2 x 2 3
).

As [τ 2 , τ 3 ] 0, then the system (9-11) cannot be transformed into the observer form with output injection, but one can use them to construct a new frame τ.

Calculation of τ

To build τ, one needs to seek non vanishing functions α 2 (y) and α 3 (y) from τ. Without loss of generality, one can always assume that α 3 (y) = 1. The reason is that if α 3 (y) is different to 1 in one normal form, one can always apply a diffeomorphism on the output z 3 = ∫ y 0 1 α 3 (s) ds which will make α 3 (y) = 1 in the transformed normal form. Therefore, one needs only to determine α 2 (y).

According to [START_REF] Zheng | A single output dependent observability normal form[END_REF], one uses the following equation:

[τ 2 , τ 3 ] = λ(y)τ 2 mod τ 1 with λ(y) = dα 2 (y) dy 1 α 2 (y) to determine α 2 (y). Then one has:

dα 2 (y) dy 1 α 2 (y) = 1 x 3 = 1 y which yields α 2 (y) = y = x 3 .
Consequently, one has α 2 = x 3 , α 3 = 1 and π = α 2 α 3 = x 3 . Then from (6) one obtains:

τ 1 = x 3 τ 1 = 1 εβ ∂ ∂x 1 τ 2 = 1 x 3 [τ 1 , f ] = 1 εβx 3 (-b + γ 1 x 3 ) ∂ ∂x 1 + 1 ε ∂ ∂x 2 τ 3 = [τ 2 , f ] = (-pbβ + (-b + γ 1 x 3 ) 2 x 3 -bε x 2 x 2 3 )τ 1 + 1 ε (γ 2 -b + (δ + γ 1 )x 3 ) ∂ ∂x 2 + ∂ ∂x 3 A straightforward calculation gives [ τ 1 , τ 2 ] = [ τ 1 , τ 3 ] = 0 and [ τ 2 , τ 3 ] = -2 b x 2 3 τ 1 .
One can see again that the commutativity condition for the new frame τ is not satisfied, and according to the third point of Remark 1 the functions η(y, w) and l(w) exist.

Calculation of σ

In this step, we will seek an auxiliary dynamics ẇ = η(y, w) and a non zero function l(w) which fulfill the condition of Theorem 1. For this, set σ 1 = l(w)τ 1 = l εβ ∂ ∂x 1 , then one has:

σ 2 = 1 x 3 [σ 1 , F] = 1 x 3 (lH -ηl ′ )σ 1 + l ε ∂ ∂x 2 σ 3 = [σ 2 , F] = (-L F ( 1 x 3 (lH -ηl ′ )) -βpb)σ 1 + (lH -ηl ′ )σ 2 +( l ε (γ 2 + δx 3 ) - l ′ ε η) ∂ ∂x 2 + l ∂ ∂x 3
where H = (-b + γ 1 x 3 ). Finally, one gets:

[σ 2 , σ 3 ] = ( l ε L ∂ ∂x 2 (-L F ( 1 x 3 (lH -ηl ′ )) -lL ∂ ∂x 3 ( 1 x 3 (lH -ηl ′ )))σ 1
Therefore, [σ 2 , σ 3 ] = 0 implies that:

l ε L ∂ ∂x 2 (-L F ( 1 x 3 (lH -ηl ′ )) -lL ∂ ∂x 3 ( 1 x 3 (lH -ηl ′ )) = 0
which is equivalent to:

-lb + l ′ (-η + x 3 η ′ y ) = 0 (29)
where η ′ y = ∂η ∂y and l ′ = dl dw . As l is only a function of w then -η + x 3 η ′ y is only a function of w. Consequently, the function η(w, y) has the following form: η = κ 1 (w)y + κ 2 (w). Then, according to (29) one has: Then, the corresponding frame σ is as follows:

lb + l ′ κ 2 (w) = 0 (
σ 1 = l εβ ∂ ∂x 1 , σ 2 = l ε ∂ ∂x 2 σ 3 = - pb ε l ∂ ∂x 1 + γ 2 + b + (δ -γ 1 )x 3 ε l ∂ ∂x 2 + l ∂ ∂x 3 One can check that [σ 1 , σ 2 ] = [σ 1 , σ 3 ] = [σ 2 , σ 3 ] = 0.
To complete the dimension of the frame, one should find σ 4 which commutes with σ i for 1 ≤ i ≤ 3. For this, one can choose:

σ 4 = l ′ x 1 l ∂ ∂x 1 + l ′ l (x 2 + (δ -γ 1 ) 2ε x 2 3 ) ∂ ∂x 2 + l ′ x 3 l ∂ ∂x3 + ∂ ∂w which makes [σ 4 , σ i ] = 0 for 1 ≤ i ≤ 3.

Determination of diffeomorphism

After determining the auxiliary dynamics, one can calculate the observability 1-forms θ e = [dh, dL F h, dL 2 F , dw] T , then one obtains:

Λ = θ e σ =                0 0 l l ′ l x 3 0 l Λ 23 Λ 24 x 3 l l (γ 2 + γ 3 + 3δx 3 ) Λ 33 Λ 34 0 0 0 1               
where:

Λ 23 = (γ 2 + b + (δ -γ 1 )x 3 ) l + (γ 3 + 2δx 3 )l Λ 24 = l ′ 2l ( 2εx 2 + 2γ 3 x 3 + (5δ -γ 1 )x 2 3 ) Λ 33 = (γ 2 + γ 3 + 3δx 3 ) (γ 2 + b + (δ -γ 1 )x 3 ) l -pbεβx 3 l + Q 1 l Λ 34 = l ′ l (x 2 + (δ -γ 1 ) 2ε x 2 3 )ε(γ 2 + γ 3 + 3δx 3 ) + εβx 3 l ′ x 1 l + l ′ x 3 l Q 1
Thus one obtains ω = Λ -1 θ e = dz which yields the diffeomorphism given in ( 16) allowing to transform the system (9-10-11-13) into the form (17), where:

B 1 (y, w) = bβ( 1 l (p(b + γ 3 ) -qε)y + p l (δ -γ 1 )y 2 ) + bβε l B 2 (y, w) = -βbpy + 1 l (δ 2 + 1 2 γ 2 1 - 3 2 δγ 1 )y 3 - 1 l (δ(γ 2 + γ 3 + 3 2 b) -2γ 1 γ 3 - 3 2 bγ

Simulation results

For the simulation, we chose κ(w) = sin 2 (aw) (aw) 2 where a ∈ ]0, 1[, and the same parameters of the SEIR model as those in [START_REF] Li | A geometric approach to global-stability problems[END_REF]

Conclusion

This paper introduced a new extended output depending normal form, which mixes both the extended normal form and the output depending normal form. This new normal form enables to design a simple high gain observer. Sufficient condition was given in order to guarantee the existence of a diffeomorphism which can be used to transform the extended dynamical systems into the proposed normal form. Finally, the proposed result was applied to the SEIR model to estimate the evolution of different populations during a contagious disease.

  s) ds . To simplify the calculations, we take η = -κ(w)H = -κ(w)(γ 1 x 3b), i.e. κ 1 (w) = -κ(w)γ 1 and κ 2 (w) = κ(w)b.

  γ 2 + γ 3 + bqε) + γ 2 γ 3 )y B 3 (y, w) = 1 l (3b + γ 2 + γ 3 )y + 3

  are used, i.e. N = 141, b = 0.221176/N, δ = 0.002, p = 0.8, q = 0.95, β = 0.05, ε = 0.05, r = 0.003. The initial conditions are S (0) = 140, E(0) = 0.01, I(0) = 0.02 and N(0) = 141. The simulation results are presented in Fig. 1-4.

Figure 1 :

 1 Figure 1: Estimation of x 1

Figure 4 :

 4 Figure 2: Estimation of x 2

Acknowledgements: The author thanks the referees for their careful reading and useful comments that allowed a substantial improvement of this paper.


Francisco Javier Bejarano
email: javbejarano@yahoo.com.mx

Gang Zheng
email: gang.zheng@inria.fr

Observability of Linear Systems with Commensurable Delays Unknown Inputs $

Keywords: Delay systems, commensurate delays, observability, unknown inputs

This papers investigates the observability analysis for linear time systems whose outputs are affected by unknown inputs. Three different definitions of observability are proposed. By introducing the Smith form and comparing the invariant factors, sufficient condition is deduced for each proposed definition of observability. Several examples are given for the purpose of highlighting the effectiveness of the proposed methods.

Introduction

Time delay systems are widely used to model many concrete applications, like chemical and biological process and many results have been published to treat this kind of systems for different aspects [START_REF] Richard | Time-delay systems:an overview of some recent advances and open problems[END_REF]; [START_REF] Sename | New trends in design of observers for time-delay systems[END_REF]. The analysis of observation for time delay systems can be dated back to the 80's of the last century [START_REF] Lee | Observability and related structural results for linear hereditary systems[END_REF]; [START_REF] Olbrot | Observability and observers for a class of linear systems with delays[END_REF]; [START_REF] Salamon | Observers and duality between observation and state feedback for time delay systems[END_REF]; [START_REF] Rabah | On observability of linear delay systems with unknown inputs[END_REF]. For this issue, different definitions of observability have been proposed, such as strong observability, spectral observability and weak observability.

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] Sontag | Linear systems over commutative rings: A survey[END_REF].

For nonlinear time delay systems, by using the theory of non-commutative rings [START_REF] Moog | The disturbance decoupling problem for time-delay nonlinear systems[END_REF], the observability problem has been studied in [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over non-commutative rings[END_REF] for systems with known inputs, and in [START_REF] Zheng | On observation of time-delay systems with unknown inputs[END_REF] for systems with unknown inputs. The associated observer for some classes of time delay systems can be found in [START_REF] Conte | Unknown input observers for linear delay systems: a geometric approach[END_REF]; [START_REF] Sename | New trends in design of observers for time-delay systems[END_REF]; [START_REF] Darouach | Full order unknown inputs observers design for delay systems[END_REF]; [START_REF] Fattouh | An unknown input observer design for linear time-delay systems[END_REF]; [START_REF] Fu | Design of unknown input observer for linear time-delay systems[END_REF] and the references therein.

Nonetheless, the majority of the existing works on observability analysis are focused on time delay systems whose outputs are not affected by unknown inputs. However, this situation might exist in some practical applications and this motivates the work of this paper. Here, we deal with time delay systems which are linear and whose delays are commensurable. We consider that delay may appear in the state, input, and output The aim is searching for conditions allowing for the reconstruction of the entire state vector using backward, actual, and forward output information.

The contributions of this paper are as follows. Firstly, we introduce the Unknown Input Observability (UIO), backward UIO and forward UIO concepts. For each one of the proposed observability definitions, we obtain sufficient conditions that can be verified by using some matrices depending on the original system parameters. The established condition for the unknown input observability turns out to be a generalization of the already known condition for systems with unknown inputs, but without delays (in that case such condition is also a necessary one), and also it is a generalization of the known strongly observable condition for linear systems with commensurable delays, but without unknown inputs. Due to the methodology used along the paper, the results may be applied to systems over polynomial rings.

The following notations will be used: R is the field of real numbers, R =0 is the set of nonzero real numbers. The set of nonnegative integers is denoted by

N 0 . R [δ ] is the polynomial ring over the field R. The Laurent polynomial ring is denoted as R δ , δ -1 . R n [δ ] is the R [δ ]-module
whose elements are the vectors of dimension n and whose entries are polynomials. By R q×s [δ ] we denote the set of matrices of dimension q × s, whose entries are in the R

[δ ]. For f (δ ), a polynomial of R [δ ], deg f (δ ) is the degree of f (δ ). For a matrix M (δ ), deg M (δ ) (the degree of M (δ )
) is defined as the maximum degree of all the entries m i j (δ ) of M (δ ). det M (δ ) is the determinant of this matrix, and rankM (δ ) means the rank of the matrix M (δ ) over R [δ ]. The acronym for greatest common divisor is gcd.

Formulation of the Problem and Definitions

We will deal with the following class of linear systems with commensurate delays

ẋ (t) = k a ∑ i=0 A i x (t -ih) + k b ∑ i=0 B i w (t -ih) y (t) = k c ∑ i=0 C i x (t -ih) + k d ∑ i=0 D i w (t -ih) (1)
where the state vector x (t) ∈ R n , the system output vector y (t) ∈ R p , and the unknown input vector

w (t) ∈ R m , the initial condition ϕ (t) is a piecewise continuous function ϕ (t) : [-kh, 0] → R n (k = max {k a , k b , k c , k d }); thereby x (t) = ϕ (t) on [-kh, 0]. A i , B i ,
and C i matrices are of appropriate dimension with entries belonging to R. System (1) may be represented in a compact form, by using the delay operator (backward time-shift operator) δ : x (t) → x (th). Thus, we have

ẋ (t) = A (δ ) x (t) + B (δ ) w (t) y (t) = C (δ ) x (t) + D (δ ) w (t) (2) where A (δ ) := k a ∑ i=0 A i δ i , B (δ ) := k b ∑ i=0 B i δ i , C (δ ) := k c ∑ i=0 C i δ i , and D (δ ) := k d ∑ i=0 D i δ i . Hence, every
entry of these matrices belongs to the polynomial ring R [δ ]. As for x (t; ϕ, w), we mean the solution of the delay differential equation of system (1) with the initial condition equal to ϕ, and the input vector equal to w. Analogously, we define y (t; ϕ, w) := C (δ ) x (t; ϕ, w) + D (δ ) w (t), that is, to be the system output of (1) when x (t) = x (t; ϕ, w).

Practically, what we search for is to find conditions allowing us for estimating the state x (t).

In order to tackle the problem in a more formal way, we will use the following observability definitions.

Definition 1 (Unknown Input Observability). System (1) is called unknown input observable (UIO) on the interval [t 1 ,t 2 ] iff there exist t 1 and t 2 (t 1 < t 2 ) such that, for all input w and every initial condition ϕ, y (t; ϕ, w) = 0 for all t ∈ t 1 ,t 2 implies x (t; ϕ, w) = 0 for all t ∈ [t 1 ,t 2 ] .

(3) Remark 1. These definitions are essentially formulated following the observability definitions given in [START_REF] Kalman | Topics in Mathematical System Theory[END_REF] for linear systems. Basically, Unknown Input Observability considers the case when the state vector can be reconstructed using past and future values of the system output. As for the backward UIO, it is related with the case when only actual and past values of the system output are needed for the state reconstruction. Finally, the forward UIO defines a property which theoretically allows for the reconstruction of the state vector using only actual and future values of the system output.

Obviously, either BUIO or FUIO implies UIO. Also, it should be noted that BUIO and FUIO observability do not exclude each other. Let us see for instance that the system

ẋ1 = x 2 , ẋ2 = x 1 + δ x 2 y 1 = δ x 1 , y 2 = x 2 is FUIO observable on [t 1 ,t 1 + h] (t 1 ≥ h), since y (t) = 0 on [ t,t 1 + 2h] implies x ( t) = 0 for all t ∈ [t 1 ,t 1 + h]. And also it is BUIO observable on [t 1 ,t 1 + h] since y (t) = 0 on [t 1 -h, t] implies x ( t) = 0 for all t ∈ [t 1 ,t 1 + h].
In the next section we will search for sufficient conditions allowing for the test of UIO property, which at the same time provide us of a constructive way to reconstruct x (t) in finite time.

Basic Results

The study of the observability for linear systems (without delays) has been successfully tackled by using geometric methods, in particular invariant subspaces. For the time delay case such methods cannot be followed straightforwardly, however still many of those ideas can be borrowed (see [START_REF] Conte | Unknown input observers for linear delay systems: a geometric approach[END_REF], [START_REF] Conte | Applications of Time Delay Systems[END_REF]). Here we will not follow strictly a geometric method, however, in its spirit the idea still comes from the results of geometric methods of standard linear systems, as we will see below.

Let P (δ ) be a matrix of dimension q × s with rank equal to r (clearly r ≤ min {q, s}). We know that there exists an invertible matrix T (δ ) over R [δ ] (representing elementary row operations) such that P (δ ) is put into the (column) Hermite form. Thus, we have that

T (δ ) P (δ ) =   P 1 (δ ) 0  
where P 1 (δ ) dimension r × s, and rankP 1 = r. Also, there exist two invertible matrices U (δ )

and V (δ ) over R [δ ] (representing elementary row and column operations, respectively) such that P (δ ) is reduced to its Smith form, i.e.,

U (δ ) P (δ )V (δ ) =   diag (ψ 1 (δ ) • • • ψ r (δ )) 0 0 0  
where the {ψ i (δ )} are monic nonzero polynomials satisfying

ψ i (δ ) |ψ i+1 (δ ) and d i (δ ) = d i-1 (δ ) ψ i (δ )
where d i (δ ) is the gcd of all i × i minors of P (δ ) (d 0 = 1). The {ψ i (δ )} are called invariant factors, and {d i (δ )} determinant divisors.

Following the ideas of [START_REF] Silverman | Inversion of multivariable linear systems[END_REF] and [START_REF] Molinari | A strong controllability and observability in linear multivariable control[END_REF], let us define {∆ k (δ )} matrices generated by the following algorithm,

∆ 0 0, G 0 (δ ) C (δ ) , F 0 (δ ) D (δ ) S k (δ )   ∆ k (δ ) B (δ ) F k (δ )   , k ≥ 0   F k+1 (δ ) G k+1 (δ ) 0 ∆ k+1 (δ )   T k (δ )   ∆ k (δ ) B (δ ) ∆ k (δ ) A (δ ) F k (δ ) G k (δ )   (4)
where T k (δ ) is an invertible matrix over R [δ ] that transforms S k into its Hermite form, and ∆ 0 is of dimension 1 by n. Then, {M k (δ )} matrices are defined as follows,

M 0 (δ ) N 0 (δ ) ∆ 0 N k+1 (δ )   N k (δ ) ∆ k+1 (δ )   , for k ≥ 0   M k+1 (δ ) 0     diag ψ k+1 1 (δ ) , . . . , ψ k+1 r k+1 (δ ) 0 0 0   = U k+1 (δ ) N k+1 (δ )V k+1 (δ ) (5)
with U k (δ ) and V k (δ ) being invertible matrices over R [δ ] that transform N k to its Smith form.

It is worthy noting that, by construction, F k (δ ) and M k (δ ) matrices have both full row rank, and

M k (δ ) has always n columns.
Remark 2. N k is used in [START_REF] Silverman | Inversion of multivariable linear systems[END_REF] to calculate the null-output weakly unobservable subspace, denoted by V * , for linear (with no delays) systems with unknown inputs. It is known that the system (with no delays) is strongly observable (= UIO according to our definition) if, and only if, such a subspace contains only the zero vector.

However for delay system it is not the case, as can be seen by the simple example ẋ = 0, y = (1δ )x (t). In this case V * = 0, but y = 0, for any x (t), which is constant along the time.

It is intuitively clear that since {T k }, {U k }, and {V k } are invertible over R [δ ], then the invariant factors of {M k } matrices should not depend on the particular selection of those former invertible matrices. This is true as we will prove it in the following lemma.

Lemma 1. {M k } matrices generated by ( 4)-( 5) are invariant from the choice of {T k }, {U k }, and

{V k }.
Proof. Firstly, it is easy to verify that for two matrices ∆ 1 and ∆1 calculated in (4) with two different matrices T 0 and T0 , respectively, we obtain that

∆1 = J 1 ∆ 1 (6)
for some invertible matrix J 1 over R [δ ]. Indeed, since T0 and T -1

0 are invertible over R [δ ]
and

F 1 has full row (normal) rank, we obtain that T0 T -1 0 =   * * 0 J 1   , which at once implies that J 1 (δ ) is invertible over R [δ ]
. This applied to (4) gives ( 6). Now, let N k and Nk be generated with the different set of matrices {T 0 , . . . , T k-1 } and { T0 , . . . , Tk-1 }, respectively, and suppose that Nk = D k N k , for some invertible matrix D k . Now by a straightforward calculation, we can verify that N k+1 satisfies the following recursive algorithm

Y k   N k B N k A D C   =   W k+1 X k+1 0 N k+1   (7)
where Y k is an invertible matrix over R [δ ] and W k+1 has full row rank. Analogously, ( 7) is valid for Nk+1 with some matrices Ȳk and Wk+1 . Thus, we obtain that

Y k   D -1 k 0 0 I   Ȳ -1 k Ȳk   D k N k B D k N k A D C   = Y k   D -1 k 0 0 I   Ȳ -1 k   Wk+1 Xk+1 0 Nk+1   =   W k+1 X k+1 0 N k+1  
Therefrom, we deduce that there exists an invertible (over R [δ ]) matrix D k+1 such that Nk+1 = D k+1 N k+1 . Therefore, Nk+1 and N k+1 have the same smith form, which yields that Mk+1 be identical to M k+1 . Since N 1 = ∆ 1 , we have proved by induction that Mi = M i , for all i ≥ 1.

Lemma 2. By using the notation d k j (δ ) as the j-th determinant divisor of M k (δ ) (generated by (4)-( 5)), we obtain d k+1 j (δ ) | d k j (δ ), for every j ≤ rankM k (δ ).

Proof. Firstly let us notice that by (5), we obtain the following identity,

  U k 0 0 I   N k+1 V k =      M k 0 ∆ k+1 V k      H k+1 (8)
So the invariant factors of N k+1 are the same of those of the matrix in the right hand side of

H k . Thus, by definition d k+1 1 (δ ) is the gcd of all 1 × 1 minors of H k+1 (δ ). Then, d k+1 1 (δ ) divides every element of H k+1 (δ ). Therefore, d k+1 1 (δ ) divides d k 1 (δ ). In general, d k+1 i (δ ) divides every i × i minor of H k+1 (δ ).
In what follows, let us assume that j ≤ rankM k (δ ).

Now, for any matrix the product of the first j invariant factors is equal to its j-th determinant divisor. Indeed,

j ∏ i=1 ψ i = d 1 d 0 × d 2 d 1 × • • • × d j d j-1 = d j . Thus, by (5), d k j (δ ) is a j × j minor of N k+1 (δ ).
Hence, we easily obtain that d k+1 j (δ ) | d k j (δ ).

Lemma 3. M k+1 (δ ) = M k (δ ) if, and only if, ∆ k+1 (δ ) = P (δ ) N k (δ ) for some matrix P (δ ).

Proof. Sufficiency:

If ∆ k+1 (δ ) = P (δ ) N k (δ ), then N k+1 (δ ) =   I 0 P (δ ) I     N k (δ ) 0   This implies that M k+1 (δ ) = M k (δ ) .
Necessity: From (8), we have that the invariant factors of M k+1 (δ ) are equal to the invariant factors of H k+1 . Thus, if M k+1 (δ ) = M k (δ ), using the fact that d k+1 1 = d k 1 divides every element of H k+1 , then we can verify that the first column of ∆ k+1 V k can be reduced to zero by premultiplying an invertible matrix to H k+1 (δ ). We can apply the same procedure to the new obtained matrix, but applied to the second column. Thus, we achieve that every element below the second invariant factor can be made zero by a premultiplication with an unitary matrix, and so on for all the new obtained matrices, until we achieve to show that there exist an invertible matrix U satisfying the identity

UH k+1 = U      M k (δ ) 0 ∆ k+1 (δ )V k (δ )      =      M k (δ ) 0 0     
The previous identity implies that

∆ k+1 (δ )V k (δ ) = L (δ )   M k (δ ) 0  
for a polynomial matrix L (δ ). Thus, we have that

∆ k+1 (δ )V k (δ ) = L (δ )U k (δ ) N k (δ )V k (δ ) That is ∆ k+1 (δ ) = P (δ ) N k (δ ), (P (δ ) = L (δ )U k (δ )). Theorem 1. If M k+1 (δ ) = M k (δ ), then M k+i (δ ) = M k (δ ) for all i ≥ 0.
Proof. As we have seen, N k+1 , for k ≥ 0 satisfies the following recursive algorithm

Y k   N k B N k A D C   =   W k+1 X k+1 0 N k+1   (9)
where Y k is an invertible matrix over R [δ ] and W k+1 has full row rank. Now, assuming that M k+1 = M k , according to Lemma 3, there exist a matrix P such that ∆ k+1 = PN k . Thus, let Ψ be an invertible matrix over R [δ ] (used to rearrange terms) chosen so that

Ψ   N k+1 B N k+1 A D C   =      N k B N k A D C PN k B PN k A     
The previous identity and ( 9) yield Theorem 2. After a finite number of steps, let say k * , the algorithm ( 4)-( 5) converges, i.e., there Remark 4. We would like to emphasize that rankM k+1 = rankM k does not imply that M k+1 = M k .

  Y k 0 -P 0 I   Ψ   N k+1 B N k+1 A D C   =      W k+1 X k+1 0 N k+1 0 0      Thus,
exists a least integer k * such that M k * +1 (δ ) = M k * (δ ). Furthermore, k * is invariant of the choice of {T k }, {U k },
Indeed, let see the following simple example,

ẋ1 (t) = x 2 (t) , ẋ2 (t) = x 3 (t) , ẋ3 (t) = x 4 (t) , ẋ4 (t) = x 2 + x 2 (t -h) , y 1 (t) = x 1 (t) and y 2 (t) = x 4 (t -h) For this example, rankM 3 = rankM 4 . However, M 3 is invertible over R δ , δ -1 , but is not invert- ible over R [δ ]. Nevertheless, it is easy to verify that M 4 is invertible over R [δ ].
Next corollary is a direct consequence of the previous results.

Corollary 1. Let k * be the least integer such that M k * +1 = M k * , then for all i ≥ 0, M k * +i = M k * .

State Reconstruction

From now it is possible to give sufficient observability conditions. In the way to arrive to such conditions we will draw the connection of the recursive algorithm used to obtain ∆ k (δ ) and a way that may be used for the reconstruction of the state vector. Let us define ŷ1 (t) = T 0 (δ )

  0 y (t)   .
Thus, from (2) and ( 4) we obtain

ŷ1 (t)   ŷ1 1 (t) ŷ1 2 (t)   =   G 1 (δ ) x (t) + F 1 (δ ) w (t) ∆ 1 (δ ) x (t)   (11)
From here, we will define a chain of vectors ŷi (t) . Thus, ŷi 1 (t) and ŷi 2 (t) will be the upper and lower subvectors of ŷi (t), respectively, whose dimension will be implicitly defined.

We have already obtained a virtual output ŷ1 2 (t) without the influence of the unknown input w (t). Let us define α 1 = deg ∆ 1 (δ ), then, for t ≥ α 1 h, ŷ1 2 (t) is differentiable. Hence, according to (11) and (2), we obtain the following equation,

d dt ŷ1 2 (t) = d dt ∆ 1 (δ ) x (t) = ∆ 1 (δ ) A (δ ) x (t) + ∆ 1 (δ ) B (δ ) w (t) , t ≥ α 1 h (12)
Now, let us generate an extended vector ξ 1 (t), defined as follows

ξ 1 (t)   d dt ŷ1 2 (t) ŷ1 1 (t)  
Again, we define ŷ2 (t) = T 1 (δ ) ξ 1 (t), t ≥ α 1 h. In view of ( 4) and ( 12), we obtain the following identity,

ŷ2 (t)   ŷ2 1 (t) ŷ2 2 (t)   =   G 2 (δ ) x (t) + F 2 (δ ) w (t) ∆ 2 (δ ) x (t)   Differentiation of ŷ2 2 (t) gives, for t ≥ h max (α 1 , α 2 ) (where α 2 = deg ∆ 2 (δ )) d dt ŷ2 2 (t) = ∆ 2 (δ ) A (δ ) x (t) + ∆ 2 (δ ) B (δ ) w (t)
In the same manner, we define a second extended vector ξ 2 (t):

ξ 2 (t) =   d dt ŷ2 2 (t) ŷ2 1 (t)   After defining ŷ3 (t) T 2 (δ ) ξ 2 (t), we obtain the identities ŷ3 1 (t) = G 3 (δ ) x (t) + F 3 (δ ) w (t) and ŷ3 2 (t) = ∆ 3 (δ ) x (t).
The previous procedure, allows us for writing the following expressions, for k ≥ 1 and t ≥

h max 1≤i≤k-1 deg ∆ i (δ ), ξ k (t)   d dt ŷk 2 (t) ŷk 1 (t)   , ξ 0 (t)   0 y (t)   ŷk (t)   ŷk 1 (t) ŷk 2 (t)   T k-1 (δ ) ξ k-1 (t) ŷk 2 (t) = ∆ k (δ ) x (t) (13)
Thus, we define Y (t) as

Y (t)         ŷ1 2 (t) ŷ2 2 (t) . . . ŷk * 2 (t)        
With the definition t * = h max 1≤i≤k * -1 deg ∆ i (δ ), by ( 13) we have that

Y (t) = N k * (δ ) x (t) for all t ≥ t * (14)
Let us introduce the forward time-shift operator as δ -1 : x (t) → x (t + h), which is in fact the inverse of δ . [I guess that you just replaced the old ∆, but forgot to change this part!!] We also recall that the Euclidian domain R δ , δ -1 contains the Euclidian domain R [δ ]. Thus, the following proposition is an obvious consequence of properties of R δ , δ -1 .

Proposition 1. M k * (δ ) matrix has an inverse on R δ , δ -1 if, and only if, M k * (δ ) has n invariant factors, all of them of the form aδ j , with a ∈ R =0 and j ∈ N 0 .

Proof. All units (elements that have an inverse) of R δ , δ -1 are of the form aδ j where a belongs to R =0 and j is an integer. Thus, since M k * (δ ) has an inverse on R δ , δ -1 if, and only if,

det M k * (δ ) is a unit on R δ , δ -1 , then M k * (δ )
should have n invariant factors all of them of the form described aδ j , with a in R =0 and j a nonnegative integer (due to each invariant factor of

M k * (δ ) is in R [δ ]).
Let us define t * 1 as follows

t * 1 = h × deg V k * M -1 k * 0 U k * + t * (15)
I think there should be a h in front of deg...

Theorem 3. The state vector x (t) can be reconstructed in finite time, for any t > t * 1 , if M k * (δ ) has n invariant factors of the form aδ j , where a ∈ R =0 , j ∈ N 0 . The formula to reconstruct x (t) is expressed as

x (t) = V k * M -1 k * 0 U k * Y (t) , t > t * 1 (16)
Furthermore, the i-th entry of x (t) is given by an expression of the form:

x i (t) = ∑ k, j q k, j y ( j) k (t) (17)
where y

(i) k (t)
is the i-th derivative of the k-th entry of the vector y (t) and every q i j,k is a nonzero element of R δ , δ -1 .

Corollary 4. Assume that system (1) is SO. Then it is BUIO on [t * 1 ,t 2 ], for all t 2 > t * 1 , if, and only if, det M k * = 1.

Proof. The proof is straightforward by Corollary 3 and Proposition 2.

As for FUIO, we have the following corollary which characterized it, provided (1) is SO.

Corollary 5. If system (1) is SO and, for all i ∈ 1, n, every polynomial q j,i of ( 17) belongs to

R δ -1 , then (1) is FUIO on [t * 1 ,t 2 ], for all t 2 > t * 1 .
Proof. If every polynomial q j,i belongs to R δ -1 , then, for every t 2 ≥ t * 1 , the identity y (t) = 0 on [t 2 ,t * 1 + kh] implies x (t 2 ) = 0 ( k being the biggest degree of all polynomials q j,i ).

Examples

Example 1. Let us consider the following example:

A (δ ) =            1 δ δ 0 0 -δ 2 0 δ 0 -δ δ 1 -δ 2 -1 + δ 1 -δ + δ 3 0 0 -1 0 0 δ -δ 2 0 -1 + δ 2 1 -δ            , B (δ ) =            -δ 0 0 -δ 1 + δ 1 1 0 0 1 -δ            C (δ ) =      1 0 0 0 0 0 1 0 0 0 0 0 0 0 δ      , D (δ ) =      0 0 1 0 -1 + δ 0     
According to (4), we have that

T 0 =         0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 -δ 1         , ∆ 1 =      1 0 0 0 0 0 0 0 0 0 0 1 -δ 0 0 δ     
, Thus, in view of (5), we obtain

M 1 =   1 0 0 0 0 0 1 0 0 0  
Following with the procedure we obtain the matrices

T 1 =         0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 δ         , ∆ 2 =      0 0 0 0 0 0 0 0 2δ 0 1 2δ δ 0 0      , M 2 =         1 0 0 0 0 0 1 0 0 0 0 0 δ 0 0 0 0 0 δ 0         and T 2 =         0 0 0 1 0 0 -1 2 1 2 δ 2 0 1 2 0 -δ 1 0 0 0         , ∆ 3 =   0 -δ -δ 0 0 0 0 0 0 0   , M 3 = M 4 = diag 1, 1, δ , δ , δ 2
Thus, in this case k * = 3, and the system is UIO since all the invariant factors of M 3 belongs to R δ , δ -1 . Explicitly, we have that the state vector can be expressed as

x 1 = y 1 x 2 = -δ -1 y 1 + δ -1 ẏ1 + 1 2 1 -δ -1 ẏ2 - 1 2 δ -1 ẏ3 x 3 = δ -1 y 1 + y 2 -δ -1 ẏ1 - 1 2 1 -δ -1 ẏ2 + 1 2 δ -1 ẏ3 - 1 2 1 -δ -1 ÿ2 + 1 2 δ -1 ÿ3 x 4 = 1 2 1 -δ -1 ẏ2 - 1 2 δ -1 ẏ3 x 5 = δ -2 -δ -1 y 1 + δ -1 -1 y 2 + δ -1 y 3 + δ -1 -δ -2 ẏ1 + 1 2 1 -δ -1 2 ẏ2 + 1 2 δ -2 -δ -1 ẏ3
Therefore, according to Corollary 5, the system is FUIO.

Example 2. Now, let us consider that the matrices of the system (2) are the following,

A =      0 -1 1 -1 δ 0 1 0 1      , B =      0 0 0 0 0 δ      C =   δ 0 0 0 1 0   , D =   -2δ 0 0 0  
In this example, we have that M 1 has an invariant factor equal to 1, M 2 has the invariant factors {1, 1}, and the invariant factors of M 3 are {1, 1, 1}. Hence the system is SO and furthermore, it is BUIO also, that is the state vector can be reconstructed using actual and past values of the system output. Indeed, it is easy to verify that the state variables can be expressed as In this case M 3 = diag (1, δ , δ ), so it is UIO, however, it is neither BUIO nor FUIO. As we can see, x depends on past, actual, and future values of the system output:

x 1 = -y 1 + δ -1 ẏ2 , x 2 = -ẏ1 + δ -1 ÿ2 , and x 3 = -δ y 1 + ẏ2

Remark 7. For the computation of matrices used to arrive to M k * , we have used the Matlab software which includes symbolic tools of Maple. For instance to calculate the Smith form of a matrix N, we used the instruction maple('smith',N,s). For calculating the Hermite form: maple('smith',N,s). Multiplication of matrices is done in the standard way.

Conclusions

In this paper we have proposed to tackle the observability of linear commensurable time delay systems with UI using three different definitions. Essentially, the first definition (UIO) deals with the possibility of reconstruct the state vector using output information (using past, actual, and/or future values), the second definition (BUIO) is related with the state reconstruction using just actual and past output information. And the third definition (FUIO) is about the state reconstruction using future values of the system output. We have given sufficient conditions allowing for the system to be UIO, BUIO, or FUIO, respectively. As for the conditions given for the UIO we have seen that the condition obtained includes the already known conditions for systems with delays without unknown inputs and for the case of linear systems with unknown inputs without delays. It should It would be interesting to define the cases were the obtained conditions are also necessary.

Our conjecture is that SO is also a necessary condition if the aim is to expressed the state vector as a polynomial function of the system output and some of its derivatives.

  vector of Ñk+2 has the same number of rows as P has. Then, we have just is invertible over R [δ ] and W k+2 has full row rank. Thus, we can use similar arguments to those used to prove the identity (6) to prove that N k+2 = R Ñk+2 , with R being an invertible matrix over R [δ ]. Thereby, the invariant factors of N k+2 are equal to those of Ñk+2 , which in fact has the same invariant factors as N k+1 , because of the definition of Ñk+2 . Thus, N k+2 and N k+1 have exactly the same invariant factors. Moreover, since by the definition of M k+1 , N k+1 and M k+1 have the same invariant factors, and by assumption M k+1 = M k , we have that the invariant factors of N k+2 are exactly those of M k . Finally, again since M k+2 has the same invariant factors as N k+2 , we have proved that M k+2 and M k have identical invariant factors. Then, by the definition of M j matrices, we arrive to the identity M k+2 = M k . Following exactly the same procedure, by induction, we prove that M k+i = M k for all i ≥ 0.

  and {V k } matrices used in (4)-(5).

Proof.

  Since, by (5), rankM k+1 (δ ) ≥ rankM k (δ ), and rank (M k (δ )) ≤ n (for all k), then after a finite number of steps, let say k, we obtain that rankM k+i (δ ) = rankM k (δ ) , for all i ≥ 0 (10) By Lemma 2, d k+1 j divides d k j (this is true for every k ≥ 0), and also we know that d k j divides d k j+1 . Hence, we have that degd k+1 j ≤ deg d k j and deg d k j ≤ deg d k j+1 . Let us define α (k) = r k ∑ j=1 deg d k j , (r k rankM k ), then (since r k = r k+1 ) we readily obtain the inequality 0 ≤ α (k + 1) ≤ α (k) for all k ≥ k.Previous inequality means that there exists k ≥ k such thatα k + 1 = α k . This implies that deg d k j = deg d k+1 j for every j ≤ r k since deg d k+1 j ≤ deg d k j . But, if deg d k j = deg d k+1 j , then d k j = d k+1 j because of d k j is the product of monic polynomials. Therefore, since d k j+1 = d k j ψ k j+1 , then α k + 1 = α k implies that ψ k+1 j+1 = ψ k j+1, which in turn means that M k+1 = M k. Thus, k * is the least integer for which M k * +1 = M k * is satisfied. The invariance of k * from the choices of {T k }, {U k }, and {V k } matrices comes directly from Lemma 1.Remark 3. In fact the proof of Theorem 2 says us a little more. It says that if rankM k+1 = rankM k , then either M k+α(k)+1 = M k+α(k) or rankM k+α(k)+1 > rankM k+α(k) . This gives a way to have an upper estimation of how many steps will pass before the rank of M k does change, if it does.

x 1

 1 (t) = -ẏ2 + δ y 2 , x 2 (t) = y 2 , and x 3 = -ÿ2 + δ ẏ2 + y 2Example 3. Finally let us consider the following example:

  Definition 3 (Forward UIO). System (1) is said to be forward UIO (FUIO) on [t 1 ,t 2 ] iff for all t ∈ [t 1 ,t 2 ] there exist t 2 > t 1 ≥ t such that, for all input w and every initial condition ϕ,

	y (t; ϕ, w) = 0 for all t ∈ t 1 ,t 2 implies x (t; ϕ, w) = 0 for all t ∈ [t 1 ,t 2 ] .
	Definition 2 (Backward UIO). System (1) is said to be backward UIO (BUIO) on [t 1 ,t 2 ] iff for
	all t ∈ [t 1 ,t 2 ] there exist t 1 < t 2 ≤ t such that, for all input w and every initial condition ϕ,

y (t; ϕ, w) = 0 for all t ∈ t 1 ,t 2 implies x (t; ϕ, w) = 0 for all t ∈ [t 1 ,t 2 ] .

Proof. We have seen in Proposition 1 that the hypothesis of the theorem implies that M k * (δ ) is invertible over R δ , δ -1 . Thus, by ( 14), we obtain the identity ( 16) for all t ≥ t * 1 . The expression in (17) can be easily deduced by taking into account Y (t) = N k * (δ ) x (t) (for all t ≥ t * ) and ( 13). Indeed, every element of ŷ1 (t) is just a linear combination of the entries of y (t), and so the entries of ξ 1 are linear combinations of elements of y (t) and derivatives of entries of y (t), and so on.

Remark 5. For the trivial case h = 0, the condition that M k * is invertible over R is also a necessary condition for the system to be UIO (known it as strong observability, see, e.g. [START_REF] Molinari | A strong controllability and observability in linear multivariable control[END_REF] and [START_REF] Trentelman | Control Theory for Linear Systems[END_REF]).

Remark 6. For the case when B (δ ) = 0 and D (δ ) = 0, the condition that M k * be invertible over

. This condition is known as strong observability also (see, [START_REF] Lee | Observability and related structural results for linear hereditary systems[END_REF]).

In view of the previous two remarks, we suggest a definition of strong observability (as a generalization) for the systems considered in this work. We consider R δ , δ -1 as the ring over which the matrix (given below) may be invertible, this allows to have a less restrictive characterization of the observability.

Definition 4. System (1) is said to be strongly observable (SO) if, and only if, M k * is invertible over R δ , δ -1 , i.e. iff M k * has n invariant factors all of them of the form aδ j (a ∈ R =0 , j ∈ N 0 ). Now, we can deduce easily sufficient conditions for the UIO, BUIO and FUIO.

Proof. Let r 1 and r 2 be respectively the greatest and the least of powers of the polynomials q i j,k for all i ∈ 1, n and every j ∈ 1, s (i) and k ∈ 1, r i j . Since ( 17) is valid for t ≥ t * 1 , then y (t) = 0 on [t * 1r 1 h,t * 1 + r 2 h] yields the identity q i j,k y ( j) i (t * 1 ) = 0 for all possible i, j, k, which in turn implies that x (t * 1 ) = 0. Likewise, for all t 2 > t * 1 , we see that

which proves the theorem. 14

Corollary 3. If system (1) is SO and, for all i ∈ 1, n, every polynomial q j,i of ( 17) belongs to R [δ ],

then (1) is BUIO on [t * 1 ,t 2 ], for all t 2 > t * 1 .

Proof. Since ( 16) is valid for a t * 1 ≥ 0, then in case every q j,i ∈ R [δ ], we obtain that for any t 2 ≥ t * 1 , the identity y (t) = 0 on t * 1 -kh,t 2 ( k being the biggest degree of all polynomials q j,i ) implies that x (t 2 ) = 0.

Proposition 2. Let us assume that system (1) is SO. Then every polynomial q j,i in ( 17) belongs to

Proof. Let us define the set Ω as the set of polynomials of the form

where p k,i are nonzero polynomials of R [δ ] and at least one of them belongs to R =0 . Let us suppose that every element of the vector Y (t) belongs to Ω. Upon this, let U be invertible over

, then every element of the vector UY (t) is also in Ω. This is true since every row (and column) of U has a (nonzero) element of the form a + ∑ k b k δ k , with a ∈ R =0 and every k is greater than 0. Thus, again every element of the vector I 0 U k * Y (t) belongs to Ω. Therefore, we see that the j-th element of

if, and only if, the element of the j-th row and j-th column of M k * is equal to 1 (this is due to the fact that M k * is a diagonal matrix and all its nonzero elements are monic). Therefore, to prove the proposition we need to prove that every element of Y (t) is within Ω.

Let ŷ1

2 (t) be as in ( 13). It is easy to check that every element of this vector belongs to Ω. This is true because of each entry of ŷ1 2 (t) is the product of a row of T 0 (which is invertible over R [δ ]) and the vector 0 y T T . With the same argument we have that d dt ŷ1 2 (t) has all its elements in Ω. Now, as for ŷ2 2 (t), we have that it is the product of a row of T 1 (which is invertible over R [δ ]) and the vector formed by ŷ1 1 (t) and d dt ŷ1 2 (t). So, again every element of ŷ2 2 (t) belongs to Ω. By following the same process, we conclude that each element of every vector ŷi 2 (t) (i ≥ 1) belongs to the set Ω. Thus since Y (t) is the arrange of those vector, then every element of Y (t) belongs to Ω. The proposition has been proved.