Keywords: Delay, estimation, finite-time, nonlinear systems

We provide a new class of finite-time observers for continuous-time nonlinear systems that are affine in the unmeasured state variable and that also contain unknown constant parameters. Our method provides exact values of the state variables and of the constant parameters, in a fixed finite time that is independent of the initial values. We also provide analogs where there is a known delay in the measurements, and for discrete-time systems. Our approach does not rely on an adaptive controller, and is suitable for computing exact values of constant weights in artificial neural network representations of additive uncertainties in the dynamics. Our example illustrates the ease with which our assumptions can be verified in practice.

INTRODUCTION

We continue our search (which we began in [START_REF] Ahmed | Finite time estimation for time-varying systems with delay in the measurements[END_REF], [START_REF] Mazenc | Finite time estimation through a continuous-discrete observer[END_REF], [START_REF] Mazenc | Reduced order finite time observers and output feedback for timevarying nonlinear systems[END_REF], [START_REF] Mazenc | Reduced order fast converging observer for systems with discrete measurements and sensor noise[END_REF], Mazenc and Malisoff (2022), and Mazenc et al. (2022)) for finite-time observers for nonlinear systems that satisfy suitable structural conditions. While our prior works provided arbitrarily fast convergence of the observers for states of perturbed systems (which gave fixed finite-time convergence in the special case where the uncertainties are zero), here we consider a more complex problem where in addition to needing finite-time state observers, we must identify unknown model parameters in finite time. Our work is motivated by the ubiquity of unmeasured states in applications, and the usefulness of observers to substitute for unmeasured states in controllers; see, e.g., [START_REF] Borri | Luenberger-like observers for nonlinear timedelay systems with application to the artificial pancreas: the attainment of good performance[END_REF]; [START_REF] Buccella | Observer-based control of LCDC/DC resonant converter using extended describing functions[END_REF]; [START_REF] Cacace | An observer for a class of nonlinear systems with time varying observation delay[END_REF][START_REF] Cacace | A chain observer for nonlinear systems with multiple timevarying measurement delays[END_REF][START_REF] Cacace | A new approach to design interval observers for linear systems[END_REF]; [START_REF] Dinh | Continuous-discrete time observer design for Lipschitz systems with sampled measurements[END_REF]; [START_REF] Engel | A continuous time observer which converges in finite time[END_REF]; [START_REF] Katz | Boundary delayed observer-controller design for reaction-diffusion systems[END_REF]; [START_REF] Mazenc | Estimation of solutions of observable nonlinear systems with disturbances[END_REF]; [START_REF] Parikh | Framework for guaranteed convergence of imagebased observers with intermittent measurements[END_REF]; [START_REF] Sauvage | Design of a nonlinear finite time converging observer for a class of nonlinear systems[END_REF].

One natural setting where unknown constant model parameters arise is in artificial neural network expansions, where an unknown time-varying function is expressed as a linear combination of known basis functions with unknown constant weights, and where one then aims to find values of the unknown constant coefficients in the linear combination in order to identify the unknown uncertainty. One basic approach to estimating unknown model parameters is adaptive control, which usually entails a persistency of excitation (or PE) condition (but see [START_REF] Malisoff | Adaptive planar curve tracking control and robustness analysis under state constraints and unknown curvature[END_REF], where coefficients in an artificial neural network expansion were identified using an adaptive controller and barrier Lyapunov functions, in the context of curvature estimation in planar curve tracking that arises in marine robotic applications, but without requiring any PE conditions). However, adaptive control does not in general give fixed finite-time parameter identification, and is usually not amenable to the estimation of unmeasured states.

This motivates this work, which addresses the preceding challenges, in three significant settings. First, we consider cases in Section 2, where there are no measurement delays and where a simple invertibility criterion makes it possible to construct unknown state and parameter values in a fixed finite time that is independent of the initial state. Then, in Section 3, we provide an analog under measurement delays, assuming a delayed analog of the usual invertibility condition on the Gramian. In Section 4, we provide an analog for discrete time systems, where the invertibility of the Gramian is replaced by a positivity condition that again ensures fixed time identification of unknown parameters and fixed finite-time convergence of the observer. We close in Section 5 with our conjectures and research plans.

We use standard notation, which we simplify when no confusion would arise from the context. The dimensions of our Euclidean spaces are arbitrary, unless we indicate otherwise. Also, we set g t (s) = g(t + s) for functions g and all s and t such that t + s is in the domain of g.

SYSTEMS WITHOUT MEASUREMENT DELAYS

Studied system

We consider the system

   ξ(t) = Aξ(t) + φ(Cξ(t), u(t)) + ϵ 1 γ 1 (t)+ ... + ϵ q γ q (t) y ξ (t) = Cξ(t) (1)
where ξ is valued in R n , y ξ is valued in R, the real constants ϵ j are unknown, φ is a Lipschitz continuous nonlinear function, u is a known input, A ∈ R n×n and C ∈ R 1×n are constant real matrices, and the functions γ j : R → R n are locally bounded and piecewise continuous. Our structure (1) where the dynamics are affine in the unmeasured state ξ is motivated by significant applications whose models lead to dynamics of this form, e.g., manipulator dynamics and nonholonomic systems in chained form after changes in variables (as in [START_REF] Mazenc | Reduced order finite time observers and output feedback for timevarying nonlinear systems[END_REF] and [START_REF] Mazenc | Reduced order fast converging observer for systems with discrete measurements and sensor noise[END_REF]), vibrating membranes (as in [START_REF] Ahmed | Finite time estimation for time-varying systems with delay in the measurements[END_REF] and [START_REF] Mazenc | Finite time estimation through a continuous-discrete observer[END_REF]), and more. Our first assumption is the following standard observability condition:

Assumption 1: The pair (A, C) is observable. □
Let us introduce a constant τ > 0, the matrix

E = 0 -τ e A ⊤ s C ⊤ Ce As ds (2)
which is invertible because of Assumption 1, and the functional r-ℓ) ϕ(ℓ)dℓdr.

κ(ϕ) = 0 -τ e A ⊤ r C ⊤ C 0 r e A(
(3)

We also use the function W : R → R q that is defined by

W (t) = [W 1 (t), . . . , W q (t)] ⊤ , where W i (t) = t t-T C[AE -1 κ(γ i,m ) + γ i (m)]dm for i = 1, . . . , q, (4)
where T > 0 is a constant, and where γ i,m is defined by γ i,m (s) = γ i (m + s) for all real m and s and i = 1, . . . , q.

Our second assumption is as follows (but see Section 2.4 for ways to express the W i 's in terms of states of dynamical extensions instead of using integrals, and thereby eliminate the need to compute integrals to check our assumptions):

Assumption 2. There are q nonnegative constants τ 1 , . . . , τ q such that τ i < τ j for pairs (i, j) with j > i and a constant T > T + τ such that the matrix

M (t) = [W (t -τ 1 ) . . . W (t -τ q )] (5)
is invertible for all t ∈ [τ q + T + τ, T + τ q ]. □

Simplification

We first propose a simplification. We use the dynamic extension ˙ξ

(t) = A ξ(t) + φ(y ξ (t), u(t)). (6)
Next, let

x(t) = ξ(t) -ξ(t). ( 7
)
Then we obtain the system

ẋ(t) = Ax(t) + ϵ 1 γ 1 (t) + ... + ϵ q γ q (t) y(t) = Cx(t) (8)
where y is a known measurement since y(t) = y ξ (t)-C ξ(t), y ξ (t), and C ξ(t) are all known.

Then, to find ξ(t) and the constants ϵ j , we can focus our attention on the system (8). Hence, our new objective is to find x and the constants ϵ j in finite time.

Fixed Time Observer

In terms of our assumptions and notation from the preceding subsection, we next introduce the functionals

Ψ(y t ) = t t-τ e A ⊤ (s-t) C ⊤ y(s)ds, (9)
µ(t) = y(t) -y(t -T ) -CAE -1 t t-T Ψ(y m )dm, (10) V(y t ) = [µ(t -τ 1 ) ... µ(t -τ q )] (11)
for values τ 1 , . . . , τ q that satisfy our requirements from Assumption 2, and

κ ♯ (t) = E -1 κ(γ 1,t ) ... E -1 κ(γ q,t ) . ( 12
)
We prove the following, which provides exact state values for all t ≥ τ q + T + τ in the special case where the inverse M (t) -1 of the matrix M (t) exists for all t ≥ τ q + T + τ : Theorem 1. Let Assumptions 1-2 be satisfied. Then, for all t ∈ [τ q + T + τ, T ], the equalities

x(t) = E -1 Ψ(y t ) + κ ♯ (t) V(y t )M (t) -1 ⊤ (13) and [ϵ 1 ... ϵ q ] = V(y t )M (t) -1 (14)
are satisfied.

Proof. Simple calculations give

e A(s-t) x(t) = x(s) + ϵ 1 t s e A(s-m) γ 1 (m)dm +... + ϵ q t s e A(s-m) γ q (m)dm (15)
for all t ≥ 0 and s ≥ 0. By left multiplying (15) by C, we obtain

Ce A(s-t) x(t) = y(s) + ϵ 1 C t s e A(s-m) γ 1 (m)dm + ... + ϵ q C t s e A(s-m) γ q (m)dm. ( 16
)
By left multiplying ( 16) by e A ⊤ (s-t) C ⊤ , we obtain

e A ⊤ (s-t) C ⊤ Ce A(s-t) x(t) = e A ⊤ (s-t) C ⊤ y(s) + ϵ 1 e A ⊤ (s-t) C ⊤ C t s e A(s-m) γ 1 (m)dm +... + ϵ q e A ⊤ (s-t) C ⊤ C t s e A(s-m) γ q (m)dm. (17)
Then, by integrating the equality (17) over the interval [t -τ, t] with t ≥ τ , we obtain

Ex(t) = Ψ(y t ) + ϵ 1 κ(γ 1,t ) + ... + ϵ q κ(γ q,t ), ( 18
)
with E defined in (2), Ψ from (9), and κ from (3).

Recalling that the observability of (A, C) implies that E is invertible, it follows that

x(t) = E -1 Ψ(y t ) + ϵ 1 E -1 κ(γ 1,t ) +... + ϵ q E -1 κ(γ q,t ). ( 19
)
Then, we deduce from (8) that

ẋ(t) = AE -1 Ψ(y t ) + ϵ 1 AE -1 κ(γ 1,t ) +... + ϵ q AE -1 κ(γ q,t ) + ϵ 1 γ 1 (t) + ... + ϵ q γ q (t). ( 20
)
By reorganizing the terms, we obtain

ϵ 1 [AE -1 κ(γ 1,t ) + γ 1 (t)] + ... + ϵ q [AE -1 κ(γ q,t ) + γ q (t)] = ẋ(t) -AE -1 Ψ(y t ).
(21)

Then we integrate ( 21) over [t -T, t] where T > 0 is the constant present in (4) to obtain

ϵ 1 t t-T [AE -1 κ(γ 1,m ) + γ 1 (m)]dm +... + ϵ q t t-T [AE -1 κ(γ q,m ) + γ q (m)]dm = x(t) -x(t -T ) -AE -1 t t-T Ψ(y m )dm. ( 22
)
By left multiplying ( 22) by C, we obtain

q i=1 ϵ i W i (t) = [ϵ 1 ... ϵ q ]W (t) = y(t) -y(t -T ) -CAE -1 t t-T Ψ(y m )dm, (23)
with W defined in (4). We deduce that

[ϵ 1 ... ϵ q ]M (t) = V(y t ) (24)
with V defined in (11). It follows from Assumption 2 that ( 14) is satisfied. This combined with (19) yields the equality (13). This concludes the proof.

Alternative Expressions without Integrals

The integrals in ( 4), ( 9), ( 10) can be replaced by states of delayed dynamical extensions. This replacement can facilitate checking our assumptions, while also making it possible to provide expressions for the observer values that do not contain any integrals. To see how this can be done, we first show that for i = 1, . . . , q, we have

W i (t) = V i (t) + W i (0)
, where V i is the first component of the state of the dynamic extension

                   Vi (t) = C[AE -1 EL 1i (t) -L 2i (t) + e -A ⊤ τ L 2i (t -τ ) + γ i (t) -AE -1 EL 1i (t -T ) -L 2i (t -T ) + e -A ⊤ τ L 2i (t -T -τ ) -γ i (t -T )] L1i (t) = AL 1i (t) + γ i (t) L2i (t) = -A ⊤ L 2i (t) + C ⊤ CL 1i (t) (25)
and where the initial states are V i (0) = 0 and L 1i (s) = L 2i (s) = 0 for all s ∈ [-T -τ, 0]. To obtain the preceding formula for W i , first note that for each i and t ≥ 0, we have

κ i (γ i,t ) = 0 -τ e A ⊤ r C ⊤ C 0 r e A(r-m) γ i (m + t)dmdr = t t-τ e A ⊤ (r-t) C ⊤ Ce A(r-t) t r e A(t-m) γ i (m)dm dr = t t-τ e A ⊤ (r-t) C ⊤ Ce A(r-t) L 1i (t) -e (t-r)A L 1i (r) dr = EL 1i (t) - t t-τ e -A ⊤ (t-r) C ⊤ CL 1i (r)dr = EL 1i (t) -(L 2i (t) -e -A ⊤ τ L 2i (t -τ )), ( 26
)
where the third and last equalities were obtained by applying the method of variation of parameters to the L 1i and then the L 2i dynamics from (25). Therefore, the formulas for the W i 's in (4) give Ẇi = Vi (27) for all i, so the Fundamental Theorem of Calculus gives the required formula W i (t) = V i (t) + W i (0).

Similar arguments give

Ψ(y t ) = L 3 (t) -e -A ⊤ τ L 3 (t -τ ), (28)
and

µ(t) = y(t) -y(t -T ) -CAE -1 (L 4 (t) -L 4 (t -T )), ( 29
)
where L 3 and L 4 are solutions of the dynamical extension

L3 (t) = -A ⊤ L 3 (t) + C ⊤ y(t) L4 (t) = L 3 (t) -e -A ⊤ τ L 3 (t -τ ) ( 30
)
with initial states 0, by applying the method of variation of parameters to the L 3 dynamics in (30) to eliminate the integral from the Ψ formula in (9), and then using the result to eliminate the integral in the formula for µ in (10). Combined with the formulas (26), this makes it possible to eliminate the integrals from the assumptions and conclusions of Theorem 1.

Illustrations

First we consider the system (1) with n = 2, A = 0 1 0 0 , and C = [1 0] (31) with φ = 0 and q = 1. Moreover, we assume that τ = 1 and γ 1 (ℓ) = [0, 1] ⊤ for all ℓ. Then Assumption 1 is satisfied. Let us check that Assumption 2 is satisfied as well.

Recalling that

e Aℓ = 1 ℓ 0 1 (32)
for all ℓ ∈ R, we obtain

E = 0 -τ 1 0 s 1 1 0 0 0 1 s 0 1 ds = 0 -τ 1 s s s 2 ds. ( 33
)
Consequently,

E =     τ - τ 2 2 - τ 2 2 τ 3 3     , (34)
and therefore also

E -1 = 12 τ 3    τ 2 3 τ 2 τ 2 1    and CE -1 = 4 τ 6 τ 2 . (35)
Also, by the formula (32), we have

κ(γ 1,t ) = t t-τ e A ⊤ (s-t) C ⊤ C t s e A(s-m) γ 1 (m)dmds = t t-τ 1 0 s -t 0 t s s -m 1 dmds = t t-τ 1 0 s -t 0 -1 2 (t -s) 2 t -s ds = 1 2 t t-τ -(t -s) 2 (t -s) 3 ds = - 1 2 0 -τ s 2 s 3 ds =     - τ 3 6 τ 4 8     . (36) Consequently t t-τ C[AE -1 κ(γ 1,m ) + γ 1 (m)]dm = 12 τ 3 t t-τ [0 1]     τ 2 3 τ 2 τ 2 1     dm     - τ 3 6 τ 4 8     = 12 τ 2 τ 2 1     - τ 3 6 τ 4 8     = 12 - τ 2 τ 6 + τ 2 8 ̸ = 0. ( 37
)
Thus Assumption 2 is satisfied for any choice of T .

More generally, starting with a system of the form ξ(t) = Aξ(t) + φ(Cξ(t), u(t)) + δ(t) y ξ (t) = Cξ(t)

with an unknown continuous function δ representing uncertainty that we wish to identify, it is natural to represent δ in an artificial neural network expansion of the form δ(t) = ϵ 1 γ 1 (t) + ... + ϵ q γ q (t) (39) with known basis functions γ i for i = 1, . . . , q, where the corresponding constants ϵ i must be identified. Standard methods for identifying the ϵ i 's include adaptive control and least squares methods. However, these methods would not provide the fixed convergence time for the state observers and parameter identifiers that we provide here. Hence, we believe that our approach can benefit applications that use artificial neural network expansions. □

SYSTEMS WITH DELAYED MEASUREMENTS

We consider the system ξ

(t) = Aξ(t) + B(t, u(t)) + ϵγ(t) y ξ (t) = Cξ(t -h) ( 40
)
where ξ is valued in R n , where u is a known input, y ξ is the output valued in R, and B is piecewise continuous and locally bounded with respect to t and continuous with respect to u. We assume that the constant delay h ≥ 0 is known and that the function γ : [0, +∞) → R n is piecewise-continuous and locally bounded. To simplify, we consider the case where only one parameter is unknown. We conjecture that an extension where there are an arbitrary number of unknown parameters can be obtained.

As in Section 2.2, we can assume without loss of generality that the B(t, u(t)) is not present. Thus we study:

ẋ(t) = Ax(t) + ϵγ(t) y(t) = Cx(t -h) (41)
Let us introduce the matrix

E h = 0 -τ e A ⊤ (s-h) C ⊤ Ce A(s-h) ds, ( 42
) the function θ(t) = t t-τ e A ⊤ (s-t-h) C ⊤ C s s-h e A(s-ℓ-h) γ(ℓ)dℓ + t s e A(s-m-h) γ(m)dm ds, ( 43
)
and the function

ρ(t) = t t-τ θ(m -h) ⊤ (E -1 h ) ⊤ C ⊤ CE -1 h θ(m -h)dm (44)
where τ > 0 is a constant, and where the invertibility of E h will follow from our observability assumption on (A, C).

For a given constant T > 2(τ + h), we assume:

Assumption 3: For all t ∈ [2(τ + h), T ], we have ρ(t) ̸ = 0.
In terms of the functions

ϵ ♯ (t, y t ) = 1 ρ(t) t t-τ θ(m -h) ⊤ (E -1 h ) ⊤ C ⊤ y(m)dm - 1 ρ(t) t t-τ θ(m -h) ⊤ (E -1 h ) ⊤ C ⊤ CE -1 h G(m)dm (45)
and

G(m) = m-h m-h-τ e A ⊤ (s-m) C ⊤ y(s)ds, (46)
we prove the following which provides exact state measurements for all t ≥ 2(τ +h) in the special case where ρ(t) ̸ = 0 for all t ≥ 2(τ +h) (but see Remark 1 for ways to eliminate the integrals from the estimates and the assumptions): Theorem 2. Let (41) satisfy Assumptions 1 and 3. Then

x(t) = E -1 h t t-τ e A ⊤ (s-t-h) C ⊤ y(s)ds + ϵ ♯ (t, y t )E -1 h θ(t) (47)
and

ϵ = ϵ ♯ (t, y t ) (48) hold for all t ∈ [2(τ + h), T ].
Proof. One can easily prove that

y(t) = Ce -Ah x(t) -ϵC t t-h e A(t-ℓ-h) γ(ℓ)dℓ (49)
for all t ≥ h. On the other hand,

Ce A(s-t-h) x(t) = Ce -Ah x(s) + ϵC t s e A(s-m-h) γ(m)dm (50)
for all t and s. Combining ( 49) and (50), we obtain

Ce A(s-t-h) x(t) = y(s) + ϵC s s-h e A(s-ℓ-h) γ(ℓ)dℓ + t s e A(s-m-h) γ(m)dm (51) when t ≥ s ≥ h. As an immediate consequence, t t-τ e A ⊤ (s-t-h) C ⊤ Ce A(s-t-h) dsx(t) = t t-τ e A ⊤ (s-t-h) C ⊤ y(s)ds + ϵθ(t) (52)
for all t ≥ τ + h with θ defined in (43). It follows that

E h x(t) = t t-τ e A ⊤ (s-t-h) C ⊤ y(s)ds + ϵθ(t) (53)
with E h defined in (42). Assumption 1 ensures that

x(t) = E -1 h t t-τ e A ⊤ (s-t-h) C ⊤ y(s)ds + ϵE -1 h θ(t) (54)
for all t ≥ h + τ . We deduce that

y(t) = Cx(t -h) = CE -1 h t-h t-h-τ e A ⊤ (s-t) C ⊤ y(s)ds + ϵCE -1 h θ(t -h). (55)
It follows that with the choice

C ♯ = C ⊤ C, we have ϵθ(t -h) ⊤ (E -1 h ) ⊤ C ♯ E -1 h θ(t -h) = θ(t -h) ⊤ (E -1 h ) ⊤ C ⊤ y(t) -θ(t -h) ⊤ (E -1 h ) ⊤ C ♯ E -1 h t-h t-h-τ e A ⊤ (s-t) C ⊤ y(s)ds. (56)
By integrating ( 56) over [t -τ, t] with t ≥ 2τ + h, it follows from our choices of ρ in (44) and G in (46) that

ϵρ(t) = t t-τ θ(m -h) ⊤ (E -1 h ) ⊤ C ⊤ y(m)dm - t t-τ θ(m -h) ⊤ (E -1 h ) ⊤ C ⊤ CE -1 h G(m)dm. (57)
Then Assumption 3 ensures that the equality

ϵ = 1 ρ(t) t t-τ θ(m -h) ⊤ (E -1 h ) ⊤ C ⊤ y(m)dm -1 ρ(t) t t-τ θ(m -h) ⊤ (E -1 h ) ⊤ C ⊤ CE -1 h G(m)dm (58)
is satisfied when t ≤ T . The theorem now follows by combining ( 54) and ( 58). Remark 1. Using the method of variation of parameters as we did in the case where there were no measurement delays in Section 2.4, we can eliminate the integrals from Theorem 2. This facilitates checking the assumptions and computing the observers from Theorem 2.

DISCRETE-TIME SYSTEMS

Consider the system

ξ k+1 = Aξ k + φ(y ξ,k , u k ) + ϵγ k y ξ,k = Cξ k ( 59
)
where ξ is valued in R n , u is the input, y ξ is the real valued output, y ξ,k is the output at discrete time k and similarly for the other subscripts k, and for simplicity we again only consider one unknown constant ϵ ∈ R as in the preceding section. Our goal is to find ξ k and ϵ in finite time. We use

F = -1 ℓ=-n (A ⊤ ) ℓ C ⊤ CA ℓ (60)
which will be invertible under Assumption 4 below, and the function

β(γ k-n , ..., γ k-1 ) = CF -1 k-1 ℓ=k-n (A ⊤ ) ℓ-k C ⊤ C[A ℓ-k γ k-1 + ... + A -1 γ ℓ ]. (61)
Our assumptions in this section are:

and

x k = ξ k -ξk , to obtain x k+1 = Ax k + ϵγ k y k = Cx k (64)
with y k valued in R. In terms of the functions

α(y k-n , ..., y k-1 ) = CF -1 k-1 ℓ=k-n (A ⊤ ) ℓ-k C ⊤ y ℓ , (65)
ϵ △ (y k-n-r , ..., y k , γ k-n-r , ..., γ k-1 ) = 1 k i=k-r β(γ i-n , ..., γ i-1 ) 2 k i=k-r β(γ i-n , ..., γ i-1 )y i - k i=k-r β(γ i-n , ..., γ i-1 )α(y i-n , ..., y i-1 ) , (66)
and

M k = k-1 ℓ=k-n (A ⊤ ) ℓ-k C ⊤ C[A ℓ-k γ k-1 + ... + A -1 γ ℓ ], (67)
we then have the following discrete time result, which gives exact state values for all k ≥ n+r in the special case where (62) holds for all k ≥ n + r: Theorem 3. Let (64) satisfy Assumptions 4-5. Choose r and R satisfying the assumptions. Then

x k = F -1 k-1 ℓ=k-n (A ⊤ ) ℓ-k C ⊤ y ℓ + ϵ △ (y k-n-r , ..., y k , γ k-n-r , ..., γ k-1 )F -1 M k (68) and ϵ = ϵ △ (y k-n-r , ..., y k , γ k-n-r , ..., γ k-1 ) (69) hold for all k ∈ [n + r, R].
Proof. For all integers ℓ ≥ 0, we can prove that

x k = A k-ℓ x ℓ + ϵ[γ k-1 + Aγ k-2 + ... + A k-ℓ-1 γ ℓ ] (70)
for all k > ℓ, by induction on k. Thus,

x ℓ = A ℓ-k x k -ϵ[A ℓ-k γ k-1 + ... + A -1 γ ℓ ]. (71)
By left multiplying this equality by C, we obtain

y ℓ = CA ℓ-k x k -ϵC[A ℓ-k γ k-1 + ... + A -1 γ ℓ ]. (72)
As an immediate consequence, we get

(A ⊤ ) ℓ-k C ⊤ y ℓ = (A ⊤ ) ℓ-k C ⊤ CA ℓ-k x k -ϵ(A ⊤ ) ℓ-k C ⊤ C[A ℓ-k γ k-1 + ... + A -1 γ ℓ ]. (73)
It follows that for all integers k > n, we have

k-1 ℓ=k-n (A ⊤ ) ℓ-k C ⊤ y ℓ = F x k -ϵ k-1 ℓ=k-n (A ⊤ ) ℓ-k C ⊤ C[A ℓ-k γ k-1 + ... +A -1 γ ℓ ] (74)
which gives β(γ i-n , ..., γ i-1 )α(y i-n , ..., y i-1 ).

x k = F -1 k-1 ℓ=k-n (A ⊤ ) ℓ-k C ⊤ y ℓ +ϵF -1 k-1 ℓ=k-n (A ⊤ ) ℓ-k C ⊤ C[A ℓ-k γ k-1 + ... + A -1 γ ℓ ].
(77)

The result follows from ( 75), (77), and Assumption 5.

CONCLUSIONS

We provided new fixed finite-time observers for the simultaneous computation of unknown model parameters and unknown states. Our structural conditions can represent artificial neural network expansions of unknown timevarying functions. A key assumption was that the unmeasured state enters the dynamics in an affine way. While this assumption holds for a broad class of significant dynamics (e.g., the motor dynamics in [START_REF] Mazenc | Reduced order fast converging observer for systems with discrete measurements and sensor noise[END_REF], and the dynamics of vibrating membranes in [START_REF] Mazenc | Finite time estimation through a continuous-discrete observer[END_REF]), it would be of interest to relax this assumption. The work Mazenc et al. (2022) uses a small-gain approach to obtain fixed time finite-time observers for unperturbed nonlinear systems that are not required to be affine in the unmeasured state, but Mazenc et al. (2022) is not amenable to the fixed time calculation of unknown model parameters. We hope to combine the present paper with small-gain methods from Mazenc et al. (2022) to relax our requirement that the dynamics are affine in the unmeasured state and to allow sampling. We also aim to develop reduced order versions, to build on works like [START_REF] Mazenc | Reduced order finite time observers and output feedback for timevarying nonlinear systems[END_REF] which provided fixed time state estimators but not parameter identification.

Finite

Assumption 4 :

 4 The matrix A is invertible and the pair (A, C) is observable. □ Assumption 5: There are integers r > 0 and R > n + r such that k i=k-r β(γ i-n , ..., γ i-1 ) 2 > 0 (62) holds for each integer k ∈ [n + r, R]. □ As in Section 2.2, we use the dynamic extension ξk+1 = A ξk + φ(y ξ,k , u k )

  y k = α(y k-n , ..., y k-1 )+ϵβ(γ k-n , ..., γ k-1 ), with β defined in (61) and α defined in (65). It follows thatβ(γ k-n , ..., γ k-1 )y k = ϵβ(γ k-n , ..., γ k-1 ) 2 +β(γ k-n , ..., γ k-1 )α(y k-n , ..., y k-1 ). i-n , ..., γ i-1 ) 2 = k i=k-rβ(γ i-n , ..., γ i-1 )y i -


Luc-Sy Tran
email: luc-sy.tran@univ-lille.fr

Olivier Herbinet
email: olivier.herbinet@univ-lorraine.fr

Hans-Heinrich Carstensen

Frédérique Battin-Leclerc
email: frederique.battin-leclerc@univ-lorraine.fr

Kinetic models describing CE chemistry

Saturated cyclic ethers Unsaturated cyclic ethers Influence of oxygenated substituents Influence of the degree of unsaturation and of C 1+ alkyl substitutions Influence of ring size Influence of alkyl substitutions A typical HT reaction mechanism for saturated CE (case of THP)

).

.

• No model for several CEs (e.g. ethyloxirane, 2-ethyltetrahydrofuran, 2-n-butylfuran, 2-furfurylalcohol, and 5-methylfurfural)

• C-H BDE at C 2 decreases from oxirane to THF, then increases from THF to THP (Table 4). • Ring strain energy increases from THP to oxirane (Table 4). • Ring size and alkyl substitutions strongly affect CE reactivity (Figs. 3,4). • Depending on ring size, unimolecular initiation or Habstractions (Fig. 5) will play more important role.

• The degree of unsaturation, alkyl or oxygenated substitutions strongly influence CE reactivity (Figs. 6,7). This can be explained by chemical kinetics (e.g. Fig. 7b)

See paper in PECS coming soon

• MVK, 5M2EF, 5M2CHOF are important products of 2,5-DMF oxidation. • Strong impact of equivalence ratio f on fuel reactivity seen for 2,5-DMF.

IDT: ignition delay time, ST: shock tube, LBV: laminar burning velocity, RCM: rapid compression machine, FR: flow reactor, PLF: premixed laminar flame)

Table 2. Main kinetic models for the combustion of selected substituted saturated CEs.

Table 3. Main kinetic models for the combustion of selected unsaturated CEs Fig. 1. Biofuel production from lignocellulosic biomass [1]. • 1994-2001: main focus on the formation/reactivity of saturated CEs.

• From 2012: significantly more interest in consumption chemistry.

• Cyclic ethers (CEs) are promising biofuels produced from biomass that can help to replace traditional fossil fuels and help to mitigate global warming • CEs are also very important products formed in the low-temperature oxidation of fuels in competition with chain-branching ketohydroperoxides.

Table 4. Ring strain energies of non-substituted saturated CEs and their cycloalkane counterparts. Numbers in normal font on species structure: C-H BDE, in kcal/mol; those of CEs were calculated in the present work, at CBS-QB3 and at G4 (in parentheses); those of cycloalkanes were calculated at G3 by [51] and at G4 (in parentheses) by [52].

Fig. 3 .Fig. 2

 32 Fig. 3. Comparison of the IDTs measured in ST. Symbols: experiments with constant initial and O 2 mole fractions. Lines: trendlines. Reproduced fromRef.[8] with permission of Taylor & Francis.

Fig. 4 .

 4 Fig. 4. IDTs of THF, 2-MTHF, 3-MTHF, 2-ETHF measured in RCM at ϕ=1 and 20 bar. Reproduced from Refs. [53] with permission of Elsevier.

Fig. 5 .

 5 Fig. 5. Important reaction pathways and respective methods used for rate coefficient determinations in the high-temperature THP combustion model by Tran et al. [13].

Fig. 6 .Fig. 7 .

 67 Fig. 6. IDTs of furan derivatives in RCMs. Reproduced from Ref. [36, 53] with permission of Elsevier.

Fig. 8 .

 8 Fig. 8. Species mole fractions. (a,b) in 2,5-DMF oxidation (t= 0.7 s, P=10 bar, f = 1) in a JSR [44]: fuel 2,5-DMF (2,5DMF), methyl vinyl ketone (MVK), 2-ethyl-5-methylfuran (5M2EF), 5-methylfurfural (5M2CHOF); (c) oxidation (t~0.5-0.8 s, P=1 bar) in a FR [34]: the fuels are furan, 2-MF (MF) and 2,5-DMF (DMF). Symbols: experiments; lines: simulations. Reproduced from Ref. [34, 44] with permission of Elsevier.

Table 1 .

 1 Main kinetic models for the combustion of non-substituted saturated CEs. (Spe. N°: species number. Reac. N°: reaction number (forward). *: Conditions presented in previous rows.

Context

Examples of comparative studies on high-temperature consumption of CEs

Progress in the study of the chemical kinetics of cyclic ethers (CEs) in combustion

The 2 nd edition of Low-Carbon Combustion, University of Cambridge, Cambridge


Jompol Thongpaen

Romane Manguin

Tanakorn Kittikool

Aurèle Camy

Thierry Roisnel

Vincent Dorcet

Sirilata Yotphan

Yves Canac

Marc Mauduit

Olivier Baslé
email: olivier.basle@lcc-toulouse.fr

Ruthenium-NHC complex-catalyzed P( iii )-directed C-H borylation of arylphosphines

come   L'archive ouverte pluridisciplinaire

Organoborons have been recognized as versatile intermediates in organic synthesis, material sciences and medicinal chemistry, and the direct transition metal-catalyzed C-H borylation has emerged as a straightforward strategy to access such platform molecules. [1] In this context, the control of the regioselectivity has remained challenging due to the ubiquity of C-H bonds. [2] To address this issue, the introduction of a directing group has been employed to bring the targeted C-H bond closed to transition metal center forming the kinetically favored five-or six-membered metallacycle. [3] Alternatively, regioselective borylation strategies involving hydrogen bonding, [4] Lewis acidbase interaction, [5] and electrostatic attraction [6] have also proved successful. Furthermore, phosphorous-containing molecules especially phosphines have played a critical role in innovative transition metal catalysis, and the direct C-H functionalization can offer a straightforward approach to extend the diversity of these compounds. [7] Nevertheless, and despite recent achievements in phosphine-directed C-H borylation, [8] the P(III)-directed C(sp 2 )-H borylation at ortho position of arylphosphines (ArPR2) has remained challenging due to their strong coordination and the formation of unfavored 4-membered metallacycle. [9] Of particular interest is the direct access to o-boronated phosphines that are an important class of ambiphilic compounds with unique features as ligands in coordination chemistry [10] or in metal-free catalysts. [11] N-heterocyclic carbene (NHC) ligands have provided several advantages in homogeneous catalysis [12] including transition metal-catalyzed C-H bond functionalization. [13] Nevertheless, the use of NHC ligands in transition metal-catalyzed regioselective C-H borylation remains scarce. [14] On the other hand, while the use of ruthenium complexes has led in the past two decades to significant breakthroughs in C-H bond functionalization, much less interest has been given to the catalytic regioselective C-H borylation, [9b, 15] and the use of Ru-NHC based catalyst in C-H borylation has never been reported. Herein, we described the synthesis of new (NHC)(arene)Ru(II)based complexes and their application as catalysts for the P(III)directed ortho C-H borylation of arylphosphines (Figure 1). Recently, we developed a multicomponent synthesis providing a direct access to a new class of unsymmetrical bidentate carboxylate NHC precursors. [16] These chelating ligands have notably shown advantages in stabilizing Cp*Rh(III)-and Rh(I)based catalysts for C-H borylation under thermic and photocatalytic conditions, respectively. [14b, 14c] Therefore, we decided to evaluate this new family of chiral ligands in the synthesis of stable (arene)(NHC)Ru(II) complexes. To this end, a bidentate carboxylate NHC precursor 1 derived from L-leucine was readily converted to the corresponding silver complex, which was subsequently transmetallated using (arene)Ru(II)dimers affording the desired (arene)(NHC)Ru(II) complexes 2a-2c in excellent yield (Figure 2). First, complex 2a, bearing pcymene ligand, was obtained in 94% isolated yield as two airstable inseparable diastereomers in the ratio of 1:1. The 13 C NMR analysis exhibits singlet peaks at 172.2 and 170.9 ppm, characteristic of carbenic carbon coordinating to Ru center. [17] Synthesis of (arene)(NHC)ruthenium(II) complexes (top). X-ray crystal structure of 2a-c. [START_REF] Mauduit | CSD Communication[END_REF] Most hydrogen atoms are omitted for clarity. Both diastereomers cocrystallized and only (S)C(S)Ru isomers are shown. p-cymene = 1-methyl-4-(propan-2yl)benzene. Mesitylene = 1,3,5-trimethylbenzene.

Table 1. Selected bond lengths (Å) and angles (degree) for complexes 2a-c. [START_REF] Mauduit | CSD Communication[END_REF] Bond length

(Å) (S)C(S)Ru-2a (S)C(S)Ru-2b (S)C(S)Ru-2c
Ru-C1 Similarly, complexes 2b and 2c, respectively bearing mesitylene and benzene ligands, were obtained in 87% and 75% isolated yield. According to 1 H NMR analysis, both complexes were obtained as a mixture of two diastereomers with the ratio of 5:1 for 2b and 3:2 for 2c. These novel Ru(NHC) complexes were crystallized and their structures could be confirmed by X-ray diffractometry analyses (Figure 2). The selected geometrical parameters of (S)C(S)Ru isomer for each complex are shown in Table 1. All complexes featuring a six-membered metallacycle exhibited a three-legged piano-stool-typed geometry, which is commonly observed in half-sandwich complex. The distance of Ru1-C1 which refers to the Ru-Ccarbenic bond in each complex was found to be 2.067(3), 2.030( 6) and 2.078(15) Å for 2a, 2b and 2c, respectively. The measured distance between ruthenium atom and the centroid of arene ligands was found to be approximately 1.70 Å which is similar for all complexes. The same trend was observed in the case of Ru1-Cl1 and Ru1-O1 with a slight decrease of the bond length from 2a to 2c. Importantly, the distance between the oxygen atom (O1) that binds to the ruthenium atom and carbonyl carbon atom (C5) is longer than the bond length of O2-C5 in every case; confirming that the carboxylate arm of the NHC ligands serves as an Xtyped ligand in these NHC-ruthenium(II) complexes. [14] In analogy with recent mechanistic studies that evidenced the implication of arene-ligand-free active ruthenium catalysts in C-H bond functionalization processes, [19] we surmised that these new (arene)(NHC)Ru(II) complexes could liberate the arene ligands to accommodate the substrates and subsequently undergo C-H bond functionalization. Therefore, to evaluate the potential of these coordination-saturated complexes to catalyzed C-H bond functionalization reactions, experiments were conducted to demonstrate the labile character of the arene ligands (Figure 3). To this end, the model complex 2a bearing p-cymene was treated with 3 equivalents of KPF6 in acetonitrile at 100 o C for 4 days leading to the total consumption of starting complex and affording the tetra(acetonitrile)NHC-ruthenium analogue 3a in 80 % isolated yield (Figure 3). The signal at 174.6 ppm in 13 C NMR analysis which belong to the Ru-Ccarbenic resonance could unambiguously confirm the presence of NHC-carboxylate ligand binding to the ruthenium center. This result proved both the labile character of the p-cymene ligand and the strong coordination of the bidentate NHC ligand to the ruthenium center, even under relatively harsh conditions. Furthermore, the facile exchange of one acetonitrile co-ligand by triphenylphosphine afforded complex 3b in excellent yield. [START_REF]complex 3b as main product[END_REF] Figure 3. Synthesis of cationic arene-free Ru complexes 3a and 3b.

With these preliminary results in hand, we decided to evaluate our novel complexes in the P(III)-directed C-H borylation of triphenylphosphine (4a) using pinacolborane (HBpin) as a borylating agent (Table 2 & Table S1). First, at 110 o C under neat condition, the use of 5 mol% of the arene-free complexes 3a or 3b as catalyst afforded the desired ortho borylated product 5a with 65% and 72% yield, respectively (entries 1 & 2). Under the same conditions, the complex 2a, bearing p-cymene ligand, furnished 76% of product 5a together with 16% of the diborylated product 5a' (entry 3). [9a] The introduction of KPF6 as cationizing agent to facilitate the release of the arene ligand could improve both activity and selectivity to deliver the desired product in 87% yield with lower 7% of diborylated by-product (entry 4). [19a] On the other hand, the variation of catalysts employing mesitylene (2b) or benzene (2c) analogues provided lower reactivity and selectivity affording respectively the desired products with 55% and 66% yield (entries 5-6). Aiming to further improve the catalytic efficiency, the P(III)-directed C-H borylation was performed at 100 o C under neat condition (entry 7). However, the reaction proceeded with lower conversion, and the desired product was obtained with 73% yield. To our delights, the introduction of solvent could improve the reactivity and selectivity at this temperature (entries 8-10). First, employing toluene as solvent, the reaction proceeded with high conversion affording 84% of the desired product with low 3% of diborylated by-product (entry 8). While the use of 1,4dioxane could produce similar results (entry 9), the optimal conditions were found using benzene as a solvent at 100 o C in the presence of 2a as catalyst, affording 90% yield of the desired product 5a with low 3% of 5a' (entry 10). The decrease of the reaction temperature to 90 and 80 o C lowered the reactivity even though the formation of diborylated products was prevented (entries 11-12). Finally, a modest but significant catalytic activity was observed upon UV-light irradiation (lmax = 365 nm) at temperature affording 30 % of the desired borylated product 5a after 3 days (entry 13). [14c] Table 2. Optimization studies. [a] Entry Catalyst Solvent Additive Temp.

( o C)

Conv. [b] Yield (%) [c] 5a 5a' [d] 12 2a C6H6 KPF6 80 16 14 - [d] 13 [e] 2a C6D6 KPF6 rt [f] 64 30 - [d] [a] Reaction conditions: PPh3 (4a, 0.2 mmol), Catalyst (2, 0.01 mmol), KPF6 (0.02 mmol), HBpin (0.6 mmol), Solvent (0.1 mL) in preheated oil bath at stated temperature for 16 hours under Ar. [b] Determined by 31 P NMR. [c] %Yield was determined by 1 H NMR using 1,3,5-trimethylbenzene as an internal standard. [d] Not detected. [e] Irradiated at 365 nm for 72h. [f] Room temperature (25-30°C).

1 3a neat -
To test the viability of this method, several commercially available arylphosphines were engaged in P(III)-directed C-H borylation under the optimal thermic conditions (Table 3). The yield of ortho-borylated product 5 was determined by 1 H NMR analysis, and subsequent quantitative treatment with BH3.THF could facilitate the isolation and purification of the phosphineborane adducts 6.

First, the model substrate triphenylphosphine (4a) could afford 81% isolated yield of the desired borylated product 6a. Then, the electronic effect of the aryl groups was studied employing substrates 4b-4e. First, C-H borylation of triarylphosphines bearing an electron-donating group at para-position including a methyl-(4b) or a methoxy-(4c) group provided the corresponding products 6b and 6c in 70% and 72% isolated yield, respectively. On the other hand, the borylation of electron-deficient triarylphosphines bearing fluoro-(4d) or trifluoromethyl-(4e) substituents at the paraposition could lead to the corresponding borylated products 5d and 5e with 60% and 76% yield under modified conditions employing only one equivalence of HBpin. In fact, electrondeficient triarylphosphines seems to display higher reactivity in the borylation process, [9b] resulting in a loss of selectivity and rendering purification slightly more challenging with detrimental impact on isolated yields. When one aryl ring of the model substrate was replaced by an alkyl group such as for cyclohexyldiphenylphosphine (4f), the yield of the desired borylated product was halved under the standard conditions. However, the reactivity was improved when the reaction was conducted under neat condition at 110 o C for 48 hours and the desired product 6f was isolated with good 77% yield. The borylation of smaller size cone angle phosphines with stronger binding to ruthenium was adversely affected. [START_REF] Luo | [END_REF] In fact, ethyldiphenylphosphine (4g) could provide the desired products with only 12% yield while no reactivity was observed Table 3. Scope of the reaction. [a] [a] Reaction conditions: 4 (0.2 mmol), 2a (0.01 mmol), KPF6 (0.02 mmol), HBpin (0.6 mmol), C6H6 (0.1 mL) in preheated oil bath at 100 o C for 16 hours under Ar then BH3.THF for 2h under Ar. [b] %Yield was determined by 1 H NMR using 1,3,5trimethylbenzene as an internal standard. [c] Isolated yield over 2 steps. 5, (Yield) [b] 4 6, Yield [c] 6a, (90%) 81% P Bpin 6b, (72%) 70% Me 6c, (85%) 72% 6d, (60%) 33% [d] 5e, (76%) 41% [e,d] 6f, (49%) 35% [f] 6g, (12%) [f] 6h, no reaction 6i, (54%) 32% [f] (86%) 77% when methyldiphenylphosphine (4h) was used as substrate. Noteworthy, borylation of dicyclohexylphenylphosphine 4i could proceed quite satisfactorily with 54% yield to furnish 32% of the corresponding isolated product 6i. Finally, additional experiments in the presence of butylated hydroxytoluene and Hg metal suggested that catalytic processes involving radicals or heterogeneous metal are unlikely to be operative (see SI). [START_REF]For kinetic isotope effect with P(C6D5)3, see figure S4[END_REF] In conclusion, the bidentate LX-type NHC-carboxylate ligand was successfully engaged in the synthesis of half-sandwiched ruthenium arene complexes. The nature of the arene ligand noticeably affected the catalytic activities of the corresponding complexes. Thanks to the labile character of arene ligand and a strong coordination of NHC-carboxylate ligand, the newly developed complexes demonstrated efficient catalytic activities in the challenging P(III)-directed C-H borylation of arylphosphines under thermal conditions, representing the first report of the use of ruthenium-NHC based catalyst in C-H borylation. Further investigations on the application of these ruthenium-NHC complexes as chiral (photo)catalysts for C-H borylation reactions are currently underway in our laboratory.

(

  NHC)(arene)Ru(II) complexes with bidentate LX-type NHCcarboxylate ligands were efficiently synthesized and fully characterized including solid-state structures. The strong coordination of the NHC carboxylate ligand and the labile character of the arene ligand are highlighted. These unprecedented Ru(II) complexes demonstrated efficient catalytic activities in the selective P(III)-directed C-H borylation at ortho position of arylphosphines, representing the first report of the use of ruthenium-NHC based catalyst in C-H borylation.

Figure 1 .

 1 Figure 1. (NHC)(arene)Ru(II) complexes-catalyzed C-H borylation of arylphosphines.

Figure 2 .

 2 Figure2. Synthesis of (arene)(NHC)ruthenium(II) complexes (top). X-ray crystal structure of 2a-c.[START_REF] Mauduit | CSD Communication[END_REF] Most hydrogen atoms are omitted for clarity. Both diastereomers cocrystallized and only (S)C(S)Ru isomers are shown. p-cymene = 1-methyl-4-(propan-2yl)benzene. Mesitylene = 1,3,5-trimethylbenzene.

  -cymene, 2a, 94%, d.r. = 1:1 Ar = mesitylene, 2b, 87%, d.r. = 5:1 Ar = benzene, 2c, 75%, d.r. = 3

  [d] HBpin (0.2 mmol).[e] isolated yield of 5.[f] 48h. [g] Neat at 110 o C.

The authors thank the Centre National de la Recherche Scientifique (CNRS), the Ecole Nationale Supérieure de Chimie de Rennes (ENSCR), the Ministère de l'Enseignement Supérieure, de la Recherche et de l'Innovation (MESRI, grant to RM) and the Science Achievement Scholarship of Thailand (to T K). This work was supported by the Agence Nationale de la Recherche (ANR-15-CE07-0012-01 "CHADOC" grant to OB, JT).

Conflicts of interest

There are no conflicts to declare Notes and references