C Briat

O Sename

J F Lafay

Delay-Scheduled State-Feedback Design for Time-Delay Systems with Time-Varying Delays -A LPV Approach

Keywords: Time-delay systems, Linear parameter varying systems, Robust linear matrix inequalities, Relaxation, Convex optimization

This paper is concerned with the synthesis of delay-scheduled state-feedback controllers which stabilize linear systems with timevarying delays. In this framework, it is assumed that the delay is approximately known in real-time and used in the controller in a scheduling fashion. First, a new model transformation turning a time-delay system into an uncertain LPV system is introduced. Using this transformation, a new delay-dependent stability test based on the so-called full block S-procedure is developed and from this result, a new delay-dependent stabilization result is derived. Since the resulting LMI conditions depend polynomially on the parameters, a relaxation result is then applied in order to obtain a tractable finite set of finite-dimensional LMIs. The interests of the approach resides in 1) the synthesis of a new type of controllers scheduled by the delay value which has a lower memory consumption than controllers with memory (since it is not necessary to store past values of the state), and 2) an easy consideration of uncertainties on the delay knowledge.

Introduction

Since several years, time-delay systems (TDS) have been intensively studied [START_REF] Niculescu | Delay effects on stability[END_REF][START_REF] Fridman | An improved stabilization method for linear time-delay systems[END_REF][START_REF] Gu | Stability of Time-Delay Systems[END_REF][START_REF] Jiang | On H ∞ control for linear systems with interval time-varying delay[END_REF][START_REF] Fridman | Stability of systems with uncertain delays: a new 'complete' Lyapunov-Krasovskii Functional[END_REF]Gouaisbaut and Peaucelle, 2006a;[START_REF] Skelton | H ∞ control of linear uncertain time-delay systems -a projection ap-proach[END_REF][START_REF] Xu | New results on delaydependent robust H ∞ control for systems with timevarying delays[END_REF][START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF]. It has been shown that delays are often responsible of instability and poor performances and thus, they should not be neglected when analyzing system stability or synthesizing control laws. Since the advent of communication networks and embedded electronics, systems with time-varying delays have attracted more and more interest. Indeed, a communication network can be viewed as a communication channel inducing delays depending on the load of the network which varies in time.

In some applications, it may be possible to measure or compute the delay from a mathematical model and in this case, it could be interesting to use this information in the controller. In [START_REF] Witrant | Stabilization of network controlled systems with a predictive approach[END_REF], a predictive approach to control Network Controlled Systems is given but a network model is necessary to compute the prediction horizon. In [START_REF] Sename | Controllability indices of linear systems with delays[END_REF] a state feedback with internal delay is designed but the robustness issue w.r.t delay measure-ment uncertainties is not considered. The authors proposed in [START_REF] Briat | A LFT/H ∞ statefeedback design for linear parameter varying time delay systems[END_REF] some preliminary results on a new control design technique for TDS. Using a new model transformation, the time-delay system is transformed into an uncertain Linear Parameter Varying (LPV) system where the delay acts as a parameter. However the results stand for interval delay (with non zero delay) only. Some interesting results on TDS based on Linear Fractional Transformation (LFT) may also be found in [START_REF] Zhang | Stability of time-delay systems: Equivalence between Lyapunov and scaled small-gain conditions[END_REF][START_REF] Roozbehani | Robust stability and H ∞ performance analysis of interval-dependent timedelay system[END_REF][START_REF] Gouaisbaut | Stability of timedelay systems with non-small delay[END_REF][START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF].

Following the preliminary results in [START_REF] Briat | A LFT/H ∞ statefeedback design for linear parameter varying time delay systems[END_REF], the LPV/uncertain system stability analysis and control synthesis tools are used to prove stability and stabilize timedelay systems (See [START_REF] Apkarian | Advanced gain-scheduling techniques for uncertain systems[END_REF][START_REF] Scherer | Robust mixed control and LPV control with full-block scalings[END_REF]). The main contributions of the paper are: -Following the idea of [START_REF] Briat | A LFT/H ∞ statefeedback design for linear parameter varying time delay systems[END_REF], a new model transformation which corrects some weaknesses of the previous one is introduced. -Using these results, a delay-dependent stability with guaranteed L 2 performances test is provided. It is obtained using the so-called full-block S-procedure and is expressed through parameter dependent LMI conditions. Computational approximations are then used in order to make the problem tractable by numerical procedures. -From this stability test a stabilization result is derived.

The obtained controller is smoothly scheduled by an approximate delay value and the error on the delay knowledge is taken into account as a robustness constraint.

In this paper, the following systems are considered

ẋ(t) = Ax(t) + A h x(t -h(t)) + B u u(t) + Ew(t) z(t) = Cx(t) + C h x(t -h(t)) + D u u(t) + F w(t) (1)
where x ∈ R n , u ∈ R m , w ∈ R p and z ∈ R q are respectively the system state, the control input, the exogenous input and the controlled output.

The time-varying delay h(t) is assumed to belong to the set

H := h ∈ C 1 (R + , H), ḣ : R + → U (2)
where C 1 (I, J) denotes the set of continuous functions with continuous derivative mapping I to J, H := [h min , h max ] and U := [µ min , µ max ]. The delay takes then bounded values and has a bounded derivative.

It is convenient to introduce the following set of vertices

V h := {h min , h max } V µ := {µ min , µ max } µ max < 1 (3)
The aim of the paper is to find delay-scheduled statefeedback controllers of the form u(t) = K( ĥ(t))x(t) (4)

which stabilize system (1) and where K(•) may be a linear, polynomial or rational function of the known value of the delay ĥ(t) = h(t) + δ h (t), with knowledge error δ h (t) belonging to

∆ := δ h : R + → ∆, δh : R + → ∆ ν (5)
where ∆ := [-δ, δ] and ∆ ν := [ν min , ν max ]. Finally we define the set of known delay values:

Ĥ := ĥ ∈ C 1 (R + , Ĥ), ḣ : R + → Û (6)
with Ĥ := H + ∆ and Û := U + ∆ ν .

The notations are as follows, for symmetric matrices A, B, A > B means A -B is positive definite (i.e. A -B > 0). For a square matrix A we have A S = A + A T where A T is the transpose of A. Ker(A) is a basis of the nullspace of A. A ⊥ is any basis of the orthogonal complement of (A) (i.e. A T A ⊥ = 0) and Im(A) is the image set of

A. ⊕ is the direct sum of matrices: A ⊕ B =   A 0 0 B   . L 2 is the space of signals with finite energy (finite L 2 -norm): ||f || 2 L2 := +∞ 0 |f (t)| 2 dt < +∞.
The paper is organized as follows, in Section 2 we introduce some preliminary results. In Section 3, the new model transformation and the associated comparison system are presented. The first main result of the paper: a new delay-dependent stability test based on the use of the model transformation is developed in Section 4. Finally, the second main result of the paper: the delay-scheduled state-feedback design is detailed in Section 5.

Converting Polynomial into Linear Dependence

This section is devoted to the presentation of a relaxation technique for polynomially parameter dependent LMI. The key idea is to use the Finsler's lemma [START_REF] Skelton | H ∞ control of linear uncertain time-delay systems -a projection ap-proach[END_REF] to linearize the dependence on the parameters. The following definition will be useful in the sequel Definition 2.1 A square matrix S is said to be S

2 - structured if it writes S = [S ij ] i,j with blocks S ij ∈ R k×k , k > 1 such that S ij :=    0 k×k if i = j S ij = S T ji ∈ K k if i = j
It is now possible to express the linearization lemma which has also been provided in [START_REF] Sato | Filter design for LPV systems using quadratically parameter-dependent Lyapunov functions[END_REF][START_REF] Sato | Robust stability/peformance analaysis for uncertain linear systems via multiple slack variable approach: Polynomial ltipd systems[END_REF]. Lemma 2.1 Suppose that a M(δ) is a polynomially parametrized symmetric matrix in δ ∈ ∆ admitting a spectral factorization M(δ) = U T (δ)N U (δ) < 0 where U (δ) is a basis of polynomials. Then M(δ) < 0 for every δ ∈ ∆ if there exists a matrix P and a S 2 -structured matrix R such that

N + R + PV (δ) + V T (δ)P T < 0 δ ∈ ∆ (7)
where U (δ) = Ker(V (δ)) and R is S 2 -structured (i.e. U T (δ)RU (δ) = 0 with R = R T = 0). Proof : The proof is given in Appendix A and relies on the Finsler's Lemma [START_REF] Skelton | H ∞ control of linear uncertain time-delay systems -a projection ap-proach[END_REF].

Remark 2.1 If the matrix V (δ) is affine in δ then lemma 2.1 can be used as a linearization procedure to linearize polynomially parameter dependent LMIs. It can be shown that every polynomially parametrized LMI can be expressed through a basis U (δ) which leads to an affine matrix V (δ). Indeed, the trivial case U (δ) = col(I, δ 1 , . . . , δ N , δ 2 1 , . . .) leads to an affine V (δ). Hence, assuming that a 'good' basis is used, then ( 7) is affine and thus can be treated as in the polytopic/affine framework (using multi-convexity).

This approach is well dedicated to small and medium size problems since the size and the number of LMIs grows very quickly depending on the degree of polynomials, the number of parameters and the size of the initial LMI. In the case of large size problems, it may be interesting to use the relaxation proposed in [START_REF] Ben-Tal | On tractable approximmations of uncertain linear matrix inequalities affected by interval uncertainty[END_REF][START_REF] Scherer | LMI relaxations in robust control (to appear)[END_REF]. It is worth mentioning that this result is highly related to the so-called Sum-of-Squares (SoS) relaxation of parameter dependent LMIs.

An important remark concerns the relaxation of LMIs with rational parameter dependence. This type of LMIs can indeed be turned into LMIs with polynomial dependence using the full-block S-procedure [START_REF] Scherer | Linear Matrix Inequalities in Control[END_REF] which can be, in turn, transformed into LMIs with linear dependence using lemma 2.1.

A new model transformation

A new model transformation allowing to turn a TDS with time-varying delays into an uncertain LPV system is provided in this section. The main advantages of this new operator compared to [START_REF] Briat | A LFT/H ∞ statefeedback design for linear parameter varying time delay systems[END_REF] is the consideration of zero delay values and a tighter computation of the L 2 induced norm of the operator.

Define then the operator

D h : L 2 → L 2 η(t) → 1 h(t)h max t t-h(t) η(s)ds (9)
Proposition 3.1 This operator enjoys the following properties:

(i) D h is L 2 input/output stable. (ii) D h has a L 2 induced norm lower than 1.
Proof : The proof is given in Appendix B and is based on a similar method as of [START_REF] Gu | Stability of Time-Delay Systems[END_REF].

The remaining of the section explains the use of the operator D h in order to turn a time-delay system into an uncertain LPV system. Let us consider system (1) and note that x h (t) = x(t) -h(t)D h ( ẋ(t)). Thus, substituting this equality into system (1) yields

ẋ(t) = Āx(t) -α(t)A h w 0 (t) + B u u(t) + Ew(t) z 0 (t) = ẋ(t) z(t) = Cx(t) -α(t)C h w 0 (t) + D u u(t) + F w(t) w 0 (t) = D h (z 0 (t)) α(t) = h(t)h max Ā = A + A h C = C + C h (10)
which is an uncertain LPV system with state x(t) expressed in 'LFT' form. Indeed, the system is -uncertain due to the presence of the 'unknown' structured norm bounded LTV dynamic operator D h and -parameter varying due to the presence h(t) whose dependence is even affine.

In order to deal with the uncertain operator D h , robust stability and robust control theories based on the full-block S-procedure approach will be considered [START_REF] Scherer | Linear Matrix Inequalities in Control[END_REF]. The parameter dependence will be tackled using parameter dependent Lyapunov functions. Remark 3.1 According to [START_REF] Gu | Stability of Time-Delay Systems[END_REF], system (10) is not equivalent to (1) due to the use of the model transformation induced by the use of the operator D h . Without entering too much in details, the use of model transformation often introduces conservatism in the approach by adding additional dynamics to the comparison (the transformed) model which may be unstable even if the original system is stable. Hence the stability test performed on the comparison model ( 10) may fail even if the initial system (1) is stable. However, in the stabilization problem, this is less critical since it is aimed to stabilize the system and thus the additional dynamics will be implicitly stabilized.

Delay-dependent stability

In this section, we develop the first main result of the paper: a new delay dependent stability test based on the comparison model (10) obtained from system (1) used along with the model transformation presented in section 3. Lemma 4.1 System (10) without control input (i.e. u(t) = 0) is asymptotically stable for h ∈ H and satisfies the L 2 performance constraint ||z|| L2 /||w|| L2 < γ if there exist a smooth matrix function P :

H → S n ++ , matrix functions D : H × U → S n
++ and a scalar γ > 0 such that the LMI (8) holds for all α ∈ [ √ h min h max , h max ] and ḣ ∈ V µ . Proof : The proof is given in Appendix C. It is based on an application of the full-block S-procedure to deal with the uncertain operator D h while the parameters are considered through the use a parameter dependent Lyapunov.

A direct corollary can be obtained immediately by choosing all the parameter dependent matrices to be constant. Corollary 4.1 System (10) without control input (i.e. u(t) = 0) is quadratically asymptotically stable for h ∈ H and satisfies the L 2 performance constraint ||z|| L2 /||w|| L2 < γ if there exist matrices P, D ∈ S n ++ and a scalar γ > 0 such that the LMI 

          [ ĀT P ] S -αP A h P E CT ĀT D -D 0 -αC T h -αA T h D -γI p F T E T D -γI q 0 -D            < 0 holds for all α ∈ { √ h min h max , h max }. Proof :
The proof is straightforward application of lemma 4.1. Since the LMI is linear in √ h then using a convex argument it suffices to check the feasibility of the LMI at the vertices of H (i.e. V h ) to conclude on the stability over the whole set H.

The next corollary is obtained by fixing all parameter dependent matrices such that the resulting parameter dependent LMI is of maximum degree 2. It is then relaxed using Lemma 2.1. Corollary 4.2 System (10) without control input (i.e. u(t) = 0) is asymptotically stable for h ∈ H and satisfies the L 2 performance constraint ||z|| L2 /||w|| L2 < γ if there exist matrices

P 0 , P 1 , D 0 , D 1 ∈ S n , a full matrix Θ, a S 2 -            [ ĀT P (h)] S + dP dh ḣ -αP (h)A h P (h)E CT ĀT D(h, ḣ) -D(h, ḣ) 0 -αC T h -αA T h D(h, ḣ) -γI p F T E T D(h, ḣ) -γI q 0 -D(h, ḣ)            < 0 (8)
structured matrix Φ and a scalar γ > 0 such that for all 10) is linear in h(t) then matrices of the form P (h) = P 0 + P 1 h > 0 and D = D 0 + D 1 √ h are chosen. The reason why P does not involve a term in √ h is the non-differentiability of √ h for h = 0. However such a term can be used when the delay is not assumed to take 0 values.

(h d , µ d ) ∈ V h × V µ the LMIs Ψ + Φ + V s (h d ) T Θ T + ΘV s (h d ) < 0 (11) P 0 + h d P 1 > 0 (12) D 0 + h d D 1 > 0 ( 13
)
holds where β = √ h max , V s (h d ) =   h d I -I 0 0 -h d I -I   and Ψ =      Ψ 0 Ψ 1/2 /2 Ψ 1 /3 Ψ 1 /3 Ψ 3/2 /2 0      Ψ 0 =            [ ĀT P 0 ] S + P 1 ḣ 0 P 0 E C T ĀT D 0 -D 0 0 0 0 -γI p F T E T D 0 -γI q 0 -D 0            Ψ 1/2 =            0 -βP 0 A h 0 0 ĀT D 1 -D 1 0 -βC T h -βA T h D 0 0 0 E T D 1 0 0 -D 1            Ψ 1 =            ĀT P 1 + P 1 Ā 0 P 1 E 0 0 0 0 0 -βA T h D 1 0 0 0 0 0 0            Ψ 3/2 =            0 -βP 1 A h 0 0 0 0 0 0 0 0 0 0 0 0 0            Proof : Since system (
Note that P (h) > 0 if and only if ( 12) is satisfied and D(h) > 0 if and only if (13) using standard convex arguments. Then substitute the explicit expressions of D(h) and P (h) into (8) leads to the expression

Ψ 0 + Ψ 1/2 √ h + Ψ 1 h + Ψ 3/2 h 3/2 < 0 (17)
Since the LMI is affine in ḣ (only Ψ 0 depends on ḣ) it is not necessary to include it into the spectral factor U (•). Then computing the spectral factorization of (17

) with U s (h) =      I √ hI hI     
we get

U T s (h)      Ψ 0 Ψ 1/2 /2 Ψ 1 /3 Ψ 1 /3 Ψ 3/2 /2 0      U s (h) < 0 (18)
Applying lemma 2.1 on (18) linearizes the parameter dependence which becomes affine in √ h. Indeed, we have

V s (h) =   √ hI -I 0 0 √ hI -I   and V s (h)U s (h) = 0.
Finally, using a convexity argument it suffices to check the negative definiteness of the LMI at the vertices of H × U (i.e. for every element of

V h × V µ )
to check the feasibility of the LMI over the whole space H × U and delay and we obtain (11).

Delay-Scheduled state-feedback design

This section is devoted to the determination of a delayscheduled controller of the form (4) which stabilizes system (1) and ensures L 2 performance of the closed-loop system. The closed-loop system obtained from the interconnection of ( 1) and ( 4) is governed by the equations

             -
ḣ ∂X( ĥ) ∂ ĥ + [X( ĥ) ĀT + Y T ( ĥ)B T u ] S X( ĥ) CT + Y T ( ĥ)D T u -ḣ ∂X( ĥ) ∂ ĥ + ĀX( ĥ) + BY ( ĥ) αhA h D(ξ) E -γI q CX( ĥ) + DY ( ĥ)

hαC h D(ξ) F - ḣ ∂X( ĥ) ∂ ĥ -D(ξ) 0 0 -D(ξ) 0 -γI p              < 0 (14) Ψ 0 =            -(µ d + δ ν )X 1 + [X(δ d ) ĀT + Y T (δ d )B T u ] S X(δ d ) CT + Y T (δ d )D T u -δd X 1 + ĀX(δ d ) 0 E -γI q CX(δ d ) 0 F -(µ d + δ ν )X 1 -D0 0 0 -D0 0 -γI p            (15) Ψ 1 =            [ ĀX 1 + B u Y 1 ] S X 1 CT + Y T 1 D T u ĀX 1 βA h D1 0 0 CX 1 βC h D1 0 0 0 0 0 0 0            (16) ẋ(t) = Ācl (h, δ h )x(t) -α(t)A h w 0 (t) + Ew(t) z(t) = Ccl (h, δ h )x(t) -α(t)C h w 0 (t) + F w(t) z 0 (t) = ẋ(t) w 0 (t) = D h (z 0 (t)) (19)
with ĥ = h + δ h ∈ Ĥ. Since the state feedback is of the form K( ĥ) then the closed-loop system matrices are given by Ācl ( ĥ) = A+A h +B u K( ĥ), Ccl ( ĥ) = C +C h +D u K( ĥ). At this point, several techniques can be employed to compute the controller (4):

(i) either use a change of variable and in this case it is possible to fix a desired form to the controller; or (ii) elaborate a stabilizability test and deduce a suitable controller either by explicit formulae or implicitly by solving a SDP. In the present paper we propose a solution based on a change of variable which has several benefits compared to the second one. The main one is the the possibility of choosing the structure of the controller (constant, affine, polynomial or rational). The resulting controller is then implicitly implementable. This is not the case of controllers computed using the second technique (with explicit construction formulae) which may depend on the derivative of the delay, supposed to be unknown. Moreover, the first approach allows for an easy computation of rational controllers through an appropriate choice of the decision matrices.

Theorem 5.1 The system (10) is stabilizable with a delayscheduled state feedback gain K( ĥ) = Y ( ĥ)X -1 ( ĥ) if there exists a smooth matrix function X : Ĥ → S n ++ , matrix functions D : H × U × Ĥ × Û → S n ++ , K : Ĥ → R m×n and a scalar γ > 0 such that the LMI ( 14) holds for all h ∈ H, ḣ ∈ U , ĥ ∈ Ĥ and ḣ ∈ ∆, where ξ = col(h, δ h , ḣ, δh ) and α = √ hh max .

Moreover the closed-loop system satisfies ||z|| L2 /||w|| L2 < γ.

Proof : The proof is given in Appendix D and is mainly based on the dualization lemma [START_REF] Scherer | Linear Matrix Inequalities in Control[END_REF].

We develop immediately the following corollary where all the parameter dependent matrices are chosen to be constant: Corollary 5.1 The system (10) is stabilizable by a constant state feedback K = Y X -1 if there exists constant matrices X, D ∈ S n ++ , Y ∈ R m×n and a scalar γ > 0 such that the LMI (20)

           [X ĀT + Y T B T u ] S X CT + Y T D T u ĀX αA h D E -γI q CX αC h D F -D 0 0 -D 0 -γI p            < 0 (20) holds for all h ∈ V h and α = √ hh max .
Moreover the closed-loop system satisfies ||z|| L2 /||w|| L2 < γ.

Proof : It is a straightforward application of theorem 5.1 when using constant decision matrices. Then with a convexity argument it is possible to conclude on the feasibility of the LMI over the whole parameter space while checking the feasibility at the vertices of the set H only (i.e. V h ).

Finally the corollary below stands when the parameter dependent matrices are chosen such that rational controllers of degree 1 in ĥ can be designed: Corollary 5. 2The system ( 10) is stabilizable with a delayscheduled state feedback K( ĥ) = Y ( ĥ)X( ĥ) -1 with Y ( ĥ) = Y 0 + Y 1 ĥ and X( ĥ) = X 0 + ĥX 1 if there exist matrices X 0 , X 1 , D0 , D1 ∈ S n , Y 0 , Y 1 ∈ R m×n , a full matrix Π, a S 2 -structured matrix Φ and a scalar γ > 0 such that the following LMIs

Ψ + Φ + K T 1 (ξ)Π T + ΠK 1 (ξ) < 0 (21) D0 + D1 h d > 0 (22) X 0 + X 1 (h d + δ d ) > 0 (23) hold for all ξ ∈ V h × V µ × V δ × V ν with V δ := {-δ, δ}, V ν := {ν min , ν max }, ξ = (h d , µ d , δ d , δ ν ), Ψ =   Ψ 0 Ψ 1/2 /2 Ψ 1  
where Ψ 0 is given by ( 15), Ψ 1 by ( 16) and

Ψ 1/2 =            0 0 0 βA h D0 0 0 0 βC h D0 0 -D1 0 0 -D1 0 0           
Proof : The proof is similar as for Corollary 4.2. Affine parameter dependent matrices Y ( ĥ) = Y 0 + Y 1 ĥ and X( ĥ) = X 0 + X 1 ĥ are chosen. The matrix D is fixed to be affine in h(t). Then ( 14) may be rewritten as

  I √ hI   T   Ψ 0 Ψ 1/2 /2 Ψ 1   Ψ   I √ hI   T < 0 (24)
The linearization procedure is then applied and yields (21). The negative definiteness of D and positive definiteness of X are defined by ( 22) and ( 23) respectively.

Example

We aim to stabilize the following time delay system with time-varying delay

ẋ(t) =   -1 0 1 1   x(t) +   1 -1 0 -1   x h (t) +   1 0   u(t) +   1 0   w(t) z(t) =   0 1 0 0   x(t) +   0 1   u(t) (25) with h(t) ∈ [0.1, 0.9], | ḣ| ≤ 0.2, δh , δ h ∈ [-0.1, 0.1]. Con- stant scaling (i.e. constant D
) is chosen and we compute an affine state-feedback of the form K( ĥ) = K 0 + K 1 ĥ with K 0 = -27.1322 -17.8448 6802 -3.1368 (26) Using this controller, the minimal L 2 performance gain is γ = 9.89. It is important to note that, for this example, computing a rational gain does not improve the result and should not be used.

K 1 = -4.

Conclusion

We have presented in this paper a new model transformation which refines the transformation proposed in [START_REF] Briat | A LFT/H ∞ statefeedback design for linear parameter varying time delay systems[END_REF] by allowing the consideration of a larger class of delay values (especially including 0). The methodology is similar and is based on the transformation of a timedelay system into an uncertain LPV system where the delay acts as a time-varying parameter. Based on that description, it is possible to propose a new delay-dependent stability lemma based on the full-block S-procedure and derive a constructive approach for the computation of stabilizing state-feedback controllers. Both constant and delayscheduled controllers are considered and uncertainties on the knowledge of the delay are taken into account in the synthesis problem. All the results are given in terms of parameter dependent LMIs depending polynomially on the parameters. These parametrized LMIs are then relaxed into a set of constant LMIs using a linearization result. (ii) There exists a matrix N of appropriate dimensions such that M + B T N + N T B < 0 First note that if the quadratic form U T (δ)RU (δ) is added to U T (δ)N U (δ), the expression of the quadratic form remains unchanged (since R is S 2 -structured and hence U T (δ)RU (δ) = 0). However this additional term modifies the eigenvalues the resulting LMI and thus provides extra degrees of freedom. Apply the Finsler's lemma on U T (δ)N U (δ) < 0 leads to the existence of Q(δ) such that

N + R + Q(δ)V (δ) + V T (δ)Q(δ) T < 0
These parameter dependent LMIs are fully equivalent. Then fixing Q to be parameter independent (loosing then equivalence) yields the proposed result (7).

Appendix B. Proof of Proposition 3.1

Let us prove first that for a L 2 input signal we get a L 2 output signal. Assume that η(t) is continuous and denote by η p (t) all the signals satisfying dη p (t)/dt = η(t) then we have

D h (η(t)) = η p (t) -η p (t -h(t)) h(t)h max (B.1)
If h(t) is positive then (B.1) is bounded since η(t) is continuous and belongs to L 2 . Now, let us show that when the delay reaches 0, the output signal remains bounded. Let us suppose that there exist a (possibly infinite) family of time instants 0 ≤ t 0 < . . . < t i < t i+1 < . . . such that h(t i ) = 0. Since η p (t) is continuously differentiable and hence we have

lim t→ti η p (t) -η p (t -h(t)) h(t)h max = h(t i ) h max η(t i )
Since η(t) is continuous and belongs to L 2 , we can state that η(t i ) is always finite and then the output signal satisfies

lim t→ti η p (t) -η p (t -h(t)) h(t)h max = 0
This shows that the output signal is bounded if the delay reaches zero. We have shown that the output signal remains bounded for any value of h(t), let us prove now that it has a finite induced L 2 -norm using a similar method as in [START_REF] Gu | Stability of Time-Delay Systems[END_REF]. We have the following definition

||D h (η)|| 2 L2 := +∞ 0 dt h(t)h max t t-h(t) η T (θ)dθ• t t-h(t) η(θ)dθ
Then using the Jensen's inequality (see [START_REF] Gu | Stability of Time-Delay Systems[END_REF]) we obtain

||D h (η)|| 2 L2 ≤ +∞ 0 dt h max t t-h(t) η T (θ)η(θ)dθ (B.2)
In order to exhibit the norm of the input signal into the expression, the idea is to exchange the order of integration. The exchange is possible if the dependence between t and θ is reversed. In the integral equation (B.2), θ depends on t, in the 'exchanged' one t should depend on θ. Note that the integration domain for (B.2) is defined by The exchange is based on the inverse function of p(t) = t -h(t) which is denoted by q := p -1 . This inverse function exists since p(t) is strictly increasing according to the assumption ḣ < 1. The inverse function can be seen as the symmetric of p(t) with respect to the axis θ = t on

{(t, θ) : t ∈ R + , θ ∈ [t -h(t), t]
{(t, θ) : θ ∈ R + , t ∈ [θ, q(θ)]}
It is easily seen that t depends on θ and hence the order of integration can be exchanged.

Hence we get

||D h (η)|| 2 L2 ≤ 1 h max +∞ -h(0) η T (θ)η(θ)dθ q(θ) θ dt (B.3) = 1 h max +∞ -h(0) η T (θ)η(θ)(q(θ) -θ)dθ
Moreover since the symmetry preserves distances, we have |q(θ) -θ| ≤ h max and considering zero initial conditions (i.e. η(s) = 0 for all s ≤ 0) we get

||D h (η)|| 2 L2 ≤ ||η|| 2 L2
stating that D h defines a L 2 input/output stable operator with an L 2 -induced norm lower than 1.

Appendix C. Proof of Lemma 4.1

It is convenient to introduce first a simplified version of the full-block S-procedure [START_REF] Scherer | LPV control and full-block multipliers[END_REF]:

Lemma C.1 Let us consider system ẋ(t) = A(t)x(t) + B 0 (t)w 0 (t) z 0 (t) = C 0 (t)x(t) + D 00 (t)w 0 (t) w 0 (t) = ∆(t)z 0 (t) (C.1)
and such that the uncertainty ∆(t) satisfies the integral quadratic constraint (IQC)

t 0 z 0 (s) T   ∆(s) I   T M ∆ (s)   ∆(s) I   z 0 (s)ds ≥ 0 (C.2)
where M ∆ (s) is a structured symmetric matrix (not necessarily positive semidefinite) and z 0 ∈ L 2 . Then if there exists a matrix P = P T > 0 and M ∆ (s) such that the following LMI holds (where we omit the dependence on time):

  Ṗ + A T P + P A P B 0 B T 0 P 0   +   0 C T 0 I D T 00   M ∆ (s)   0 I C 0 D 00   < 0 (C.
3) then the system (C.1) is asymptotically stable.

The main difficulty is the complete characterization of ∆(t) in finding a 'good' matrix M ∆ (s) such that (C.2) holds. Indeed, the integral quadratic constraint must be satisfied for every z 0 (t) ∈ L 2 and all 'trajectories' of the uncertain matrix ∆(t), meaning that the problem is truly infinite dimensional. Due to this fact, it is not possible (or extremely difficult) to find a matrix M ∆ (s) which totally describes the uncertain operator ∆(s) especially when ∆(s) contains dynamic LTV operators. In the remaining of the proof the matrix M ∆ will be chosen as constant.

Let us consider system

ẋ(t) = Āx(t) -α(t)A h D h ( ẋ(t)) + Ew(t) z(t) = Cx(t) -α(t)C h D h ( ẋ(t)) + F w(t) (C.4) with Ā = A + A h , C = C + C h and α(t) = h(t)h max . De- noting w 0 (t) := D h ( ẋ(t)), z 0 (t) = ẋ(t) and ∆(•) := D h (•).
This system exactly falls into the framework of Lemma C.1.

It is possible to extend the approach to deal with robust H ∞ performances by adding the input/output constraint:

t 0   w(s) z(s)   T   -γI 0 0 γ -1 I     w(s) z(s)   ds > 0 (C.5)
where γ is a positive scalar and we get the LMI

     ∂P ∂h ḣ + ĀT P (h) + P (h) Ā -αP A h P E 0 0 -γ(h, ḣ)I      +      0 ĀT I -αA T h 0 E T      (h, ḣ)   0 I 0 Ā -αA h E   +γ -1 (h, ḣ)      CT -αA T h F T           CT -αA T h F T      T < 0 (C.6)
where (h, ḣ) verifies for all η ∈ L 2 the IQC

t 0   D h (η) I n   T (h, ḣ)   D h (η) I n   ds > 0 (C.7)
The separator (h, ḣ) = * (h, ḣ) is chosen such that it characterizes the L 2 induced norm of D h that is

t 0   D h (η) I n   T   -1 0 0 1   1   D h (η) I n   ds > 0 (C.8)
for all η ∈ L 2 . Hence a set of separators can be parametrized by

= 1 ⊗ D(h, ḣ) where D(h, ḣ) = D(h, ḣ) T > 0 for all (h, ḣ) ∈ H × U . Hence we have (h, ḣ) :=   -D(h, ḣ) 0 D(h, ḣ)   (C.9)
where D : H × U → S n ++ . Then expanding (C.6) and performing a Schur complement on quadratic term

-      CT ĀT D(h, ḣ) -αC h -αA h D(h, ḣ) F E T D(h, ḣ)        -γ -1 (h, ḣ)I q 0 0 -D -1 (h, ḣ)   ( ) T
yields inequality (8).

Appendix D. Proof of Theorem 5.1

First note that the real unknown delay is denoted by h(t) and the estimated one by ĥ(t) = h(t) + δ h (t). Since the controller depends on X -1 and Y then both matrices should depend on ĥ only, which is the only known parameter. On the other hand, since D is not involved in the controller expression, it may depend on any parameter (i.e. h(t), δ h (t), ḣ(t), δh (t)). In what follows, we define the compact notation ξ = col(h, δ h , ḣ, δh ) for simplicity.

Note now that LMI (C.6) can be rewritten in the following form In order to provide convex synthesis conditions for the controllers, the multiple products between the closedsystem matrices Ācl ( ĥ), Ccl ( ĥ) and decision matrices P, D must be avoided. It is possible to modify the LMI condition (into an equivalent version) in order to keep one product between system matrices and decision variables only. This is performed by the mean of the dualization lemma [START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF][START_REF] Scherer | Linear matrix inequalities in control[END_REF] which is recalled below for completeness: Lemma D.1 Let M (n M = dim(M )) be a nonsingular symmetric matrix with respectively n + (M ) and n -(M ) positive and negative eigenvalues respectively. Let S -= Im(S -) a subspace of dimension n -(M ) of R n M and S + = Im(S + ) such that S + is the orthogonal complement of S -(i.e. (S -) T S + = 0).

( ) T M (h, ḣ)               I 0 0 Ā -αA h E 0 I 0 Ā -αA h E 0 0 I C -αC h F               S -
The following statements are equivalent: (i) The matrix inequality (S -) T M S -< 0 holds. (ii) The matrix inequality (S + ) T M -1 S + > 0 holds.

First substitute the closed-loop system matrices into (D.1). Note that dim(M ) = 4n + p + q, n -(M ) = 2n + p and rank[S -] = 2n+p (S -is defined in (D.1)), where n -(M ) is the number of strictly negative eigenvalues of the symmetric matrix M , n = dim(x), p = dim(w) and q = dim(z).

Since n -(M ) = rank[S -] then it is possible to apply the dualization lemma and we get

( ) T M -1 (ξ)              
-ĀT cl ( ĥ) -CT cl ( ĥ) 0

I n 0 I n αA T h αC T h 0 0 0 -I n -E T -F T 0 0 I q 0               S + > 0 (D.2)
where M -1 (ξ) =    dP ( ĥ) dt P ( ĥ)

0    -1 ⊕ -1 (ξ)⊕[-γ -1 (ξ)I p ]⊕
[γ(ξ)I q ]. Let X = P -1 , then dX( ĥ) dt = -X dP ( ĥ) dt X and Finally multiplying the LMI by -1 (to get a negative definite inequality) we obtain inequality ( 14) in which Y ( ĥ) = K( ĥ)X( ĥ) is a linearizing change of variable. This concludes the proof.

Lemma A. 1

 1 Let M by a symmetric matrix and B a matrix of appropriate dimensions, then the following statements are equivalent:(i) Inequality Ker[B] T M Ker[B] < 0 holds.

Fig

  Fig. B.1. Evolution of the domain of integration of (B.2)

  } and is represented by the blue/dark surface on Figure B.1.

  Figure B.1 and after rotation/flipping of the plane we get the integral domain depicted on Figure B.2 which is formally defined by

  ξ) ⊕ [-γI p ] ⊕ [γ -1 I q ].

  (ξ) := D(ξ) -1 ∈ S n ++ . Then expand (D.2) and noticing that D(ξ) > 0, the Schur complement can be used on the quadratic term:

Preprint submitted to Elsevier

Appendix A. Proof of Lemma 2.1

The proof is a simple application of the Finsler's lemma [START_REF] Skelton | H ∞ control of linear uncertain time-delay systems -a projection ap-proach[END_REF] recalleds hereunder:


Enrico Falcone

Germano Nobili

Michael Okafor

Olivier Proux

Giancarlo Rossi

Silvia Morante

Peter Faller

Francesco Stellato

Chasing the Elusive "In-Between" State of the Copper-Amyloidβ Complex by X-ray Absorption through Partial Thermal Relaxation after Photoreduction

Keywords:

  L'archive ouverte pluridisciplinaire

and characterize by X-ray Absorption Spectroscopy (XAS) a partially reduced Cu-Aβ1-16 species dierent from the resting states. Remarkably, the XAS spectrum is well-tted by a previously proposed model of the "in-between" state, hence providing the rst direct spectroscopic characterization of this intermediate state. The present approach could be used to explore and identify the catalytic intermediates of other relevant metal complexes.

Copper (Cu) ions are essential trace elements in most living organisms. They are present almost exclusively in two ionic forms, Cu(I) or Cu(II), normally found in complex with proteins where they act as catalytic centers. Cu levels are tightly regulated by binding to dedicated proteins that modulate tracking, cellular import and export, and incorporation into appropriate enzymes. The regulation mechanism must be quite ecient, because both deciency or overload of Cu are dangerous, as well documented in the two genetic, Menkes (Cu deciency) and Wilson (Cu overload), diseases respectively. [START_REF] Kim | [END_REF]2] Indeed, like a lack of Cu impairs the essential function of Cu-enzymes (e.g. cytochrome C oxidase), an excessive Cu concentration can lead to the displacement of Fe from iron-sulfur cluster proteins, [3,[START_REF] Macomber | Proc. Natl. Acad. Sci[END_REF] the catalytic production of reactive oxygen species (ROS), unwanted interactions with essential cysteine residues, [START_REF] Kim | [END_REF]2,[START_REF] Tsvetkov | [END_REF] and induction of protein ag- gregation. [6] Cu deregulation has been reported in several neurodegenerative diseases. Misplaced Cu is supposed to bind to amyloid-β (Aβ) peptides in Alzheimer's (AD) [7] or α-synuclein in Parkinson's (PD) [8,9] diseases. Both peptides/proteins belong to the class of the so-called intrinsically disordered proteins (IDPs) and are prone to form amyloids, i.e. betasheet rich aggregates. Moreover, these Cu-peptide/protein complexes have been shown to catalyze eciently in vitro the ROS production in the presence of dioxygen and ascorbate (a physiological reducing agent).

Hence Cu-bound amyloid peptides became the target of a therapeutic approach aimed at inhibiting the over-production of Cu-induced ROS might lead to the oxidative stress observed in AD and PD. [10,11] Cu complexes can produce ROS via cycling between the two redox states Cu(I) and Cu(II) in the presence of O2.

An important point to take into account is that the coordination chemistry of Cu(II) and Cu(I) are remarkably dierent. Dierences are quite evident when Cu ions are bound to exible peptides or IDPs. [12,[START_REF] La Penna | Aggregates Sealed by Ions[END_REF] [1416] (see Figure 1). Cu(II) is predominantly coordinated in a distorted square-pyramidal geometry via 3 nitrogen and 1 oxygen equatorial ligands and a weaker O axial ligand. [17] The coordination of Cu(II)-Aβ is pH-dependent. In particular, two species co-exist at physiological pH 7.4 (notably, a major component I and a minor component II [18,19] ), with component I becoming the only present specie at acidic pH.

[19] Cu(I) preferentially binds in a linear geometry with two nitrogen atoms [9] from the imidazole rings of two His residues (see Figure 1). These two coordination modes represent the resting states, i.e.

the most populated conformations living in fast equilibrium with other less populated conformations. [20] Thus, the very dierent coordination sphere of the two resting states raises the question of how the redox cycling can occur. In fact, a quite important rearrangement of the peptide around the Cu ion is required upon Cu reduction/oxidation, which normally leads to a sluggish redox reaction and hence low ROS production. [20,[START_REF] Balland | Proc. Natl. Acad. Sci[END_REF] In order to explain the relatively ecient ROS production by Cu-Aβ, it has been suggested that the latter is catalyzed by a low populated Cu-state, called an "in-between state that can cycle between Cu(I) and Cu(II) redox states with little atomic rearrangement and hence produce eciently ROS. [START_REF] Balland | Proc. Natl. Acad. Sci[END_REF] In-between states attracted a lot of interest as only this state seems to be redox competent enough to eciently generate ROS and hence is the state of choice as the target for suppressing ROS production by Cu-Aβ.

However, although experimental and theoretical evidence have suggested candidates for the "in-between" state [21,2325] (see Figure 1), a direct experimental evidence of its structure is lacking so far, mainly as a result of the challenging experimental characterization of such a low populated species (estimated to be 0.1% of the total from electrochemical studies. [START_REF] Balland | Proc. Natl. Acad. Sci[END_REF] )

In this work, we aimed at trapping and characterizing possible intermediates in Cu-Aβ1-16 redox cycling via X-ray Absorption Spectroscopy (XAS) measurements at the Cu K-edge.

In order to ensure sample homogeneity, the soluble Nterminal Aβ1-16 peptide fragment was used instead of the full-length Aβ1-42. Indeed, since Aβ1-16 very well reproduces the coordination of Cu(II) and Cu(I) to Aβ1-42, Aβ1-40 and Aβ1-28, it is often used as a model for the longer peptide. [15,26,27] It is well known that, in the physical conditions of standard XAS experiments, which are typically performed at a temperature as low as 10 K, a progressive reduction of Cu(II) to Cu(I) is observed in the Cu-Aβ1-42 peptide complex in vitro.

[28] However, the systematic study of the be- havior of Cu-Aβ complexes [29] has shown that, despite the progressive reduction of Cu(II) to Cu(I), at 10 K the spatial organization of the ligands did not change, yielding a Cu(I) ion in a coordination sphere of the Cu(II) resting state. In the present work, the samples were irradiated with a high X-ray dose at 10 K. To allow partial relaxation of the ligands around Cu, the temperature was raised up to 200 K with the aim of producing and trapping non-resting states, which were then structurally characterized by XAS. As detailed in the following, we were able to promote the formation of a complex dierent from both resting states and to characterize its structure.

XAS measurements at the Cu K-edge were performed at the BM30 beamline of the European Synchrotron Radiation Facility (ESRF -Grenoble, France). [30] In order to identify possible intermediate species appearing during the Cu(II)-Aβ1-16 photoreduction process, the spectra of Cu(I)-Aβ1-16 and Cu(II)-Aβ1-16 resting states were rst recorded.

In order to avoid the co-presence of dierent Cu(II) species, samples were prepared at pH 5.7, where only component I is known to be present. [19] As detailed in the rst two rows of Table 1, that contains the list of the collected spectra, the resting states were measured keeping the samples at the temperature of 10 K and averaging 5 scans, each one acquired in a dierent spot on the sample. In this way, the possible eects due to the interaction with the X-ray beam were minimized while the signal-to-noise ratio was improved. in agreement with spectra from the literature, [START_REF] Shearer | [END_REF]20] that the Cu-Aβ1-16 coordination mode is strongly dependent on the Cu oxidation state.

As shown in

The XANES (X-ray Absorption Near Edge Spectroscopy) region of Cu(I)-Aβ1-16 (left panel of Fig. 2),shows a peak, located in the pre-edge region at about 8982.6 eV, that is known [START_REF] Shearer | [END_REF]29,31,32] to be the ngerprint of Cu(I) presence.

Quantitative structural information can be obtained from a t of the EXAFS (Extended X-ray Absorption Fine Structure) region of the spectrum. We report in Table 2 the values of the best-t parameters, including the quality factor R 1 , that provides a quantitative measure of the goodness of the t. The best t of the Cu(I)-Aβ1-16 complex EXAFS spectrum is obtained by assuming that only two ligands, namely two nitrogen atoms from two imidazole rings of two His residues, are coordinated to Cu(I).

The t to the Cu(II)-Aβ1-16 EXAFS data shows instead that the Cu(II) coordination mode involves four ligands, 1 The quality factor R is dened as R-factor=

i |χ exp (k i )-χ(k i ) f it | i |χ exp (k i )|
namely two nitrogen atoms from two His residues plus one nitrogen and one oxygen from the Asp residue. Comparisons between experimental data and best ts and the corresponding Fourier Transforms (FTs) of both spectra are provided in the SI (Fig. S1).

XANES calculations were then performed using the FDMNES code [33,34] starting from the best-t EXAFS mod- els. Fig. S3 shows the comparison between the XANES simulations and the corresponding experimental data. The qualitative agreement between simulated and experimental data is quite good, thus corroborating the results of the EX-AFS analysis. Sketches of the structures coming from the EXAFS ts and used for the XANES calculations are given in Fig. S3.

Finally, the Bond Valence Sum (BVS) was computed for both best-t models and reported in the last row of Ta- Gy.

In Figure 3 we show the "time evolution of the XANES (left panel) and EXAFS (right panel) data by plotting one on top of the other the ve consecutively collected spectra.

In Fig. S4 the corresponding FTs are also shown. The increasing intensity of the shoulder at 8982.6 eV is a clear conrmation of the fact that X-ray radiation induces the Cu(II) → Cu(I) reduction process. [29] On the other hand, looking at the EXAFS region (Fig- Again in Figure 4 we compare the spectrum of the S610-200-10 sample with that of the last (S510) and the rst (S110) spectrum of the series previously collected as well as with the 2 BVS provides a simple way to estimate the oxidation state of an atom in a given structure. Its value, which depends on the type, number, and distances of ligands, should agree with the nominal value of the oxidation state. [35] Table 1. List of the measured samples. In all the samples, except in the case of Cu(I)-Aβ 1-16 in the rst row, the initial Cu oxidation state is Cu(II). In the rst column, we list the names of the samples. In the second column we give a short description of each sample and in the third column the temperature at which each spectrum was collected is indicated.

F = (-3.2± 0.9) eV E F = (2.8 ± 0.7) eV R-factor = 32%
R-factor = 32% BVS = 0.9 BVS = 1.9 Based on the fact that, as we have argued, the "relaxed coordination mode of S610-200-10 is neither that of Cu(I)-Aβ1-16 nor that of Cu(II)-Aβ1-16, we tried tting the EX-AFS data taking as a starting conguration the putative "inbetween" Cu(I)/Cu(II) Aβ1-16 model suggested in Ref. [25] ,

which is in agreement with the experimental ndings of Ref. [21] .

More in detail, we started from a slightly modied version of model 14 from Ref. [25] , in which the O2 molecule present in the structure was substituted by a water molecule. This starting model was then relaxed by minimizing its total energy. We used the hybrid exchange-correlation functional suitable to Cu complexes already employed in Ref. [25] (and references therein). In the rst coordination shell of the model there are an imidazole ring belonging to a His residue, the N-terminal amine nitrogen, and an oxygen atom belonging to the Asp1 side chain. Three water molecules, located within 5 Å from the Cu ion, but not belonging to the rst coordination shell, are also included.

The best-t spectrum obtained starting from the above conguration is in quite good agreement within the signalto-noise ratio with the experimental EXAFS data (see Fig- 3. The XANES spectrum calculated from the best-t EXAFS model is remarkably well superimposable to the experimental data (Figure 6, right panel), with dierences (in the pre-edge feature and in the so-called white-line, i.e. the main peak after the absorption edge) likely due to a not perfect description of the geometry around the absorbing atom as well as to a rigid description of this geometry that does not take into account both angular and linear disorders. It is worth noticing that the BVS value associated to the best-t structure lies halfway between 1 and 2, supporting the idea that the combination of distances and coordination number is neither what is expected for a pure Cu(I) nor a pure Cu(II) oxidation state.

Since our experiment started from the Cu(II) resting state, to reach the conjectured in-between state the following rearrangement would have to occur. One of the two Cu-His bonds should be broken (previous experiments showed no preference for any of the three His [21] ) with the His moving away at least to a non-bonding distance to Cu(I). In addition, Asp1 carbonyl should also detach from Cu, while a bond is formed with its nearby (and possibly already axially bound) carboxylate. This structural rearrangement does not involve major changes in the peptide main chain and therefore it is likely compatible with structural changes observed in proteins at cryogenic temperatures. [START_REF] Arcovito | [END_REF]37] Table 3. S6 10-200-10 ligand distances and Fermi energy shift at the end of the EXAFS t. We also report the values of the R-factor and of the BVS parameter.

Ligand Distance [Å]

σ 2 [Å 2 ] 1 N (His) 2.05 ± 0.02 0.003 ± 0.001 1 N (Asp) 1.84 ± 0.02 0.003 ± 0.001 1 O (Asp) 1.95 ± 0.02 0.003 ± 0.001 3 O (Water) 3.74 ± 0.03 0.005 ± 0.001 E F = (-7 ± 1) eV R-factor = 34% BVS = 1.5

For the sake of completeness, also a signicantly dierent putative "in-between" structure (model 9 in the nomenclature of Ref. [25] ), in which two imidazole rings and the amino group of Asp1 are bound to Cu, was considered. The agreement between a t starting from this model and the experimental data obtained is both qualitatively and quantitatively (R-factor = 41%) worse than the one described in In this work we have exploited a well-known side-eect of exposure to high-intensity X-ray beams on Cu complexes, that is Cu reduction, to try to access a possible "in-between Cu(I)-Aβ1-16 and Cu(II)-Aβ1-16 coordination mode.

We have used the X-ray beam to pump Cu(II)-Aβ1-16 complexes in a non-resting state by promoting Cu(II) reduction to Cu(I), keeping the system at a temperature of 10 K. The sample temperature was then raised to 200 K to allow possible structural modications associated with oxidation state changes. Finally, the XAS spectrum of the system was acquired after cooling down the sample at 10 K.

We observe that the measured spectrum is dierent from the spectra of both resting states. A quantitative EXAFS analysis shows that it is compatible with a possible "inbetween" structure suggested by previous experiments and numerical simulations [25] and characterized by the presence of a single imidazole ring and a bi-dentate Asp residue in the Cu coordination site. The EXAFS results are corroborated by an ab initio XANES simulation performed starting from the EXAFS best-t structure, which leads to results in good agreement with the experimental data.

The experimental approach used in this work provides the rst direct evidence and structural characterization of an intermediate of Cu(II)-Aβ1-16 reduction, supporting the existence and identity of a previously postulated redox-active "in-between" state. Moreover, we showed that the X-rayinduced photo-reduction, which is normally an undesired side eect of overexposure to X-ray beams, can also be tamed to pump metal-peptide complexes into not-resting states that could be relevant for their catalytic activity.

Hence, our work paves the way for the exploration of further redox intermediates formed by biological and non-biological metal complexes.

Figure 1. Structure of the most populated Cu(II) and Cu(I) resting states and proposed structure of the catalytic "in-between" state, which seems to involve the amino terminal nitrogen and side-chain carboxylate oxygen from Asp1 and an imidazole from either His 6, 13 or 14 (with no preference for any of the three His residues [21] ).

Fig 2 ,

 2 Cu(I)-Aβ1-16 and Cu(II)-Aβ1-16 samples exhibit visibly dierent XAS spectra, thus conrming,

ble 2 .

 2 The BVS values are in quite good agreement with the corresponding Cu oxidation states.2In order to follow the possible evolution of the spectrum induced by X-ray irradiation, we further collected ve spectra, one after the other, by letting the beam hit the same point of the sample and keeping the sample at 10 K. Taking into account the photon ux, the beam size, the sample thickness (2 mm) transmission and the exposure time (each scan lasted about 40 minutes), the dose absorbed by the sample in a single scan is estimated to be around 9 × 10 6

ure 3 ,

 3 right panel) we see that the spectral evolution with beam exposure time is marginal, as already noticed previously.[29] In other words, at such a low temperature, there are no major structural rearrangements associated with the modication of the Cu oxidation state. Looking at the comparison of Cu(I)-Aβ1-16 with S110 and S510, shown in Figure 4 (the FTs are shown in Fig. S5), it is clear that, even in the presence of radiation-induced Cu reduction, the Cu coordination mode is unable to turn into that of the Cu(I)-Aβ1-16 resting structure. A quantitative estimation of similarities among these spectra is provided in the SI. After collecting the ve XAS spectra with the beam hitting always the same sample spot, the X-ray beam was switched o and the sample temperature was increased to 200 K in order to accelerate a possible, but not total, structural relaxation, associated to the presence of reduced Cu, while still keeping the sample in a frozen, solid state. Once the temperature reached 200 K, the sample was cooled back to 10 K. The whole heating and cooling cycle took about 100 minutes, after which a new XAS spectrum, termed S610-200-10 in Table 1, was collected in the previously exposed sample position.

  Cu(I)-Aβ1-16 spectrum. The dierence among the spectra of S110 (which is nothing but the Cu(II)-Aβ1-16 resting state), S610-200-10 and Cu(I)-Aβ1-16 must be attributed to the fact that the state obtained upon heating/cooling is different from the two resting states. S610-200-10 is also dierent from S510, thus suggesting that the increase in temperature allowed the sample to undergo a structural relaxation that did not take place at 10 K.Moreover, in the S610-200-10 spectrum the intensity of the shoulder that indicates the presence of Cu(I) is significantly lower than in the S510 case, a feature which might be indicative of a partial re-oxidation of Cu(I) to Cu(II) or of a modication in coordination geometry[31] . In addition, the pre-edge peak appears to be shifted toward lower energy in S610-200-10 with respect to S510, i.e. from 8981.2 eV to 8982.6 eV. Qualitatively, this dierence in energy may be associated to a dierent Cu coordination geometry occurring in the two samples. This interpretation reinforces the idea of having dierent structural atomic arrangements around the metal in S610-200-10 and S510, and not simply a dierence in the Cu oxidation state. This interesting hypothesis is further supported by the fact that the spectrum of S610-200-10 can not be well reproduced as a linear combination of Cu(II)-Aβ1-16 and Cu(I)-Aβ1-16 spectra (see Figure S5 in SI). This allows us to conclude that S610-200-10 does not represent a mere combination of the two resting states and therefore that a relaxation towards a coordination mode dierent from both resting states takes place when the sample is brought to 200 K.In order to better characterize the eect of temperature on the spectra, three consecutive scans were acquired at 200 K with the beam hitting a xed spot. The spectral evolution is faster at 200 K than at 10 K and the Cu(I) peak, located in the pre-edge region, appears already in the rst scan, S1200. Since the energy corresponding to the Cu(I) peak is measured during the scan after about 10 minutes of exposure, we argue that, at 200 K, the oxidation state switch occurs on a timescale shorter than the acquisition time of a single XANES spectrum. This is conrmed by the fact that S1200, S2200 and S3200 only exhibit marginal dierences (see the comparison among S1200, S2200 and S3200 spectra in FigureS7in SI). However, the comparison between S3200 and Cu(I)-Aβ (Figure5) clearly shows that not even the exposure of the Cu(II) sample to X-rays for as long as 2 hours (which is the time necessary to acquire 3 consecutive scans) at 200 K lead to the same spectrum as the Cu(I) resting state.

ure 6 ,

 6 left panel; the corresponding FTs are shown in Fig.S8), thus suggesting that, after X-ray exposure, by heating and cooling the sample, Cu actually reaches an "intermediate coordination mode like the one depicted in Fig.7. The bestt parameters are reported in Table

Figure 2 .

 2 Figure 2. Comparison of the XAS spectra of Cu(I)-Aβ 1-16 (blue lines) and Cu(II)-Aβ 1-16 (red lines) complexes. Left panel, XANES. Right panel, EXAFS.

Figure 3 .

 3 Figure 3. The ve consecutive scans of Cu(II)-Aβ 1-16 complexes in the XANES (left panel) and EXAFS (right panel) regions at increasing X-ray dose.

Figure 4 .

 4 Figure 4. Comparison among the spectra of the S6 10-200-10 , S1 10 , S5 10 , Cu(I)-Aβ 1-16 and Cu(II)-Aβ 1-16 in the XANES (left panel) and EXAFS (right panel) regions.

Figure 5 .

 5 Figure 5. Comparison between the Cu(I)-Aβ 1-16 and S3 200 XANES (left panel) and EXAFS (right panel) spectra.

Figure 6 .

 6 Figure 6. Left panel: experimental (broken line) and tted (solid line) EXAFS data of sample S6 10-200-10 . Right panel: XANES calculations performed starting from the EXAFS best-t model (solid line) superimposed to the XANES experimental data (broken line).

  15213773, ja, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/anie.202217791 by Cochrane France, Wiley Online Library on [09/03/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

Figure 7 .

 7 Figure 7. A sketch of the best-t structure of the Cu coordination site in the S6 10-200-10 sample. Cu is in yellow, N in blue, C in green and O in red. The three O atoms belonging to water molecules are drawn in a fainter red, since they do not belong to the rst coordination shell.

Table 2 .

 2 Aβ 1-16 average over 5 spectra each acquired by hitting a new spot 10 Cu(II)-Aβ 1-16 average over 5 spectra each acquired by hitting a new spot 10 Cu(I)-Aβ 1-16 and Cu(II)-Aβ 1-16 ligand distances, Fermi energy shift, E F , and R-factor of the optimized EXAFS ts.

	Sample name Sample description Cu(I)-S1 10 hitting a new spot on Cu(II)-Aβ S2 10 same spot as S 1 S3 10 same spot as S1 10 and S2 10 S4 10 same spot as S1 10 , S2 10 and S3 10 S5 10 same spot as S1 10 , S2 10 , S3 10 and S4 10 S6 10-200-10 same spot as S1 10 ,S2 10 , S3 10 , S4 10 and S5 10 after a temperature cycle S1 200 new spot on Cu(II)-Aβ at high temperature S2 200 same spot as S2 200 S3 200 same spot as S1 200 and S2 200	T (K) 10 10 10 10 10 10 → 200 → 10 200 200 200
	Cu(I)-Aβ 1-16 Ligand Distance [Å] 2 N (His) 1.88 ± 0.01 0.003 ± 0.001 2 N (His) 1.97 ± 0.01 0.004 ± 0.001 Cu(II)-Aβ 1-16 σ 2 [Å 2 ] Ligand Distance [Å] σ 2 [Å 2 ] 1 O 2.02 ± 0.01 0.004 ± 0.001 1N 1.94 ± 0.01 0.004 ± 0.001 E

Table 3

 3

	(R-factor = 34%), suggesting that, although our
	data cannot rule out the coexistence of dierent coordi-
	nation structures, a three-coordinated model with one His
	residue and a bi-dentate Asp is the best "in-between" coor-
	dination mode candidate.

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Acknowledgements

made possible owing to a CINECA-INFN agreement, providing access to resources on MARCONI at CINECA. This work was partially supported by the BIOPHYS initiative (INFN, Italy). E.F. and M.O. acknowledge nancial support from the French National Research Agency (Programme d'Investissement d'Avenir under contract 17-EURE-0016) and from the IDEX program (University of Strasbourg), respectively. S.M. and F.S. acknowledge nancial support from the "PANDA" project, University of Rome Tor Vergata.

Entry for the Table of Contents

The combination of X-ray induced photoreduction and thermal relaxation unravels the structure of an intermediate of the Cu(II)/Cu(I) Aβ 1-16 redox cycling. The X-ray beam promotes Cu(II) reduction to Cu(I) at 10 K, then the sample undergoes partial structural relaxation at 200 K and a spectrum measured at 10 K is compatible with a possible "in-between" structure featuring an imidazole ring and a bi-dentate Asp residue in the Cu coordination site.