Michelle F F Castro

Alexandre Seuret

J S Leite

Luis F P Silva

Robust local stabilization of discrete time-varying delayed state systems under saturating actuators

Keywords: Time-varying discrete-time systems, time-varying state-delays, saturating input, robust control

This paper deals with discrete time-varying systems with state-delayed and saturating actuators. A robust state feedback control gain is designed through convex methods ensuring the local stability of the time-varying system with interval timevarying state delay. The estimate of the set of admissible initial conditions is characterized by three convex sets allowing less conservative estimates. Through two numerical examples, we compare our approach with others in the literature and illustrate the better behavior of the proposed methods.

Introduction

Processes in the real-world are affected by delays, usually coming from energy, mass, or information transportation. As a consequence, degradation on the performance may occur as well as the stability of the closed-loop system [START_REF] Fridman | Introduction to Time-Delay Systems[END_REF]. Additionally, the presence of saturating actuators [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF], constraints on the state [START_REF] Sun | Adaptive control for a class of state-constrained high-order switched nonlinear systems with unstable subsystems[END_REF], or switched modes [START_REF] Li | Dwell-timebased stabilization of switched linear singular systems with all unstable-mode subsystems[END_REF] requires the local stability investigation because of some initial sequence of state can lead to different equilibrium points or even to unstable behavior. Thus, it is necessary to estimate a subset of the initial conditions ensuring the respective trajectories to go to the origin asymptotically [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF].

It is possible to find several works dealing with the stability of the discrete-time delayed systems in the literature. Recently, a key issue in the research on this field is to obtain less conservative bounds on some integral or summation inequalities, yielding less conservative stability conditions. The reader may refer for instance to [START_REF] Liu | Auxiliary function-based summation inequalities and their applications to discretetime systems[END_REF][START_REF] Nam | Discrete wirtinger-based inequality and its application[END_REF][START_REF] Seuret | Discretized Jensen inequality: an alternative vision of the reciprocally convex combination lemma[END_REF][START_REF] Xiao | Improved stability criteria for discrete-time delay systems via novel summation inequalities[END_REF]. While these aspects have now been addressed ⋆ The paper was partially supported by CAPES and CNPq (311208/2019-3) and has been developed during a stage of the first author on LAAS.

Email addresses: michelleffc@hotmail.com (Michelle F. F. Castro), aseuret@laas.fr (Alexandre Seuret), valter@ieee.org (Valter J. S. Leite), luis@cefetmg.br (Luis F. P. Silva).

properly in the literature, the problem of providing nonrestrictive constructive stabilization conditions, i.e., conditions that deliver as an output a state feedback control gain, is still an open issue. Indeed, transforming stability LMI conditions to constructive stabilization conditions is usually made possible at the price of introducing tuning parameters, which imposes conservative restriction as in [START_REF] He | Output feedback stabilization for a discrete-time system with a time-varying dela[END_REF], to cite only one.

To show that this method goes to beyond linear systems with delays, we will also consider additional difficulties in the paper, which consist of having nonlinear dynamics. Indeed, nonlinearities in a system are well known to easily degenerate performance and turn the closed-loop unstable. See [START_REF] Sun | Adaptive control for a class of state-constrained high-order switched nonlinear systems with unstable subsystems[END_REF] for the design of a group of adaptive controllers and a switching law to asymptotically regulate of state-constrained high-order switched nonlinear systems by using multiple-barrier Lyapunov functions. In [START_REF] Silva | Delay dependent local stabilization conditions for time-delay nonlinear discrete-time systems using takagi-sugeno models[END_REF] and [START_REF] Silva | Local stabilization of nonlinear discrete-time systems with time-varying delay in the states and saturating actuators[END_REF], a fuzzy-based approach has been used with the generalized sector condition to handle nonlinearities and constraints on the region of model validity. In particular, saturating actuators in discrete-time delayed systems have received attention as one can see in [START_REF] Xu | Robust controller design of uncertain discrete time-delay systems with input saturation and disturbances[END_REF][START_REF] Zhang | l 2 -gain analysis and control synthesis of uncertain discrete-time switched linear systems with time delay and actuator saturation[END_REF][START_REF] Chen | Stabilisation for switched linear systems with time-varying delay and input saturation[END_REF][START_REF] Silva | Robust local stabilization of discrete-time systems with timevarying state delay and saturating actuators[END_REF][START_REF] Chen | Regional stabilization for discrete time-delay systems with actuator saturations via a delay-dependent polytopic approach[END_REF][START_REF] Souza | ISS robust stabilization of state-delayed discretetime systems with bounded delay variation and saturating actuators[END_REF]. Due to saturating actuators, the study of the closed-loop behavior and the characterization of the set of initial admissible conditions (usually called domain or region of attraction) require the local stability analysis. Such a characterization is in practice much more difficult to handle for time-delay systems than for delay-free systems. This is because the initial condition of a discrete time-delay system is a sequence of vectors specifying the position of the system for negative time instants. Therefore, it is more complicated to efficiently characterize this domain of attraction because one has to constraint the whole sequence of initial conditions, which can be done in several ways. For instance, the region of attraction provided in [START_REF] Xu | Robust controller design of uncertain discrete time-delay systems with input saturation and disturbances[END_REF] is defined by imposing that all the components of the initial conditions are uniformly bounded. An alternative characterization was provided in [START_REF] Chen | Regional stabilization for discrete time-delay systems with actuator saturations via a delay-dependent polytopic approach[END_REF], where the initial conditions are constraints to lies into two balls, one for the norm of the sequence of initial conditions, and the other related to the norm of the variation between two consecutive components. Extending this idea, the authors of [START_REF] Souza | ISS robust stabilization of state-delayed discretetime systems with bounded delay variation and saturating actuators[END_REF][START_REF] Silva | Robust local stabilization of discrete-time systems with timevarying state delay and saturating actuators[END_REF] provided a more general characterization through the definition of augmented state space and an associated ellipsoidal set, at the price of a high computational burden. Such an approach was considered first in [START_REF] Hetel | Equivalence between the Lyapunov-Krasovskii functionals approach for discrete delay systems and that of the stability conditions for switched systems[END_REF], where the timedelay system is rewritten as a switched delay-free system, and the delay is the switching function. In contrast with other switching approaches, such as that in [START_REF] Li | Dwell-timebased stabilization of switched linear singular systems with all unstable-mode subsystems[END_REF], the one in [START_REF] Souza | ISS robust stabilization of state-delayed discretetime systems with bounded delay variation and saturating actuators[END_REF][START_REF] Hetel | Equivalence between the Lyapunov-Krasovskii functionals approach for discrete delay systems and that of the stability conditions for switched systems[END_REF][START_REF] Silva | Robust local stabilization of discrete-time systems with timevarying state delay and saturating actuators[END_REF] does not require dwell-time to achieve the stabilization.

Differently from [START_REF] Souza | ISS robust stabilization of state-delayed discretetime systems with bounded delay variation and saturating actuators[END_REF][START_REF] Silva | Robust local stabilization of discrete-time systems with timevarying state delay and saturating actuators[END_REF], we would like to avoid considering the augmented switched delay-free system approach to keep a low numerical complexity on the proposed conditions. Interestingly, [START_REF] Silva | Delay dependent local stabilization conditions for time-delay nonlinear discrete-time systems using takagi-sugeno models[END_REF] proposed an intermediate characterization, based on useful and simple sets, such as an ellipsoidal one for the current state and a ball concerning both the norm of the initial conditions' sequence and its variations. However, in particular, the Lyapunov-Krasovskii functionals (LKFs) employed in [START_REF] Chen | Regional stabilization for discrete time-delay systems with actuator saturations via a delay-dependent polytopic approach[END_REF][START_REF] Silva | Delay dependent local stabilization conditions for time-delay nonlinear discrete-time systems using takagi-sugeno models[END_REF] do not consider the coupling between the current state and the previous ones, leading to independent (and conservative) estimates of the region of attraction.

The contributions of the paper are two-fold. After presenting a summation of inequality accounting for systems with time-varying delays similarly as in [START_REF] Seuret | Discretized Jensen inequality: an alternative vision of the reciprocally convex combination lemma[END_REF] for the continuous-time case, a method to derive a new constructive stabilization criteria is provided. Second, we provide a new characterization of the admissible set of initial sequences for linear systems subject to time-varying delays and input saturation. It consists in the definition of three sets, allowing a more relaxed characterization of the admissible set of initial conditions than approaches usually encountered in the literature. Finally, two academic examples evaluate our method, where insightful comparisons with recent methods from the literature are established.

Notation: Sets N, R, R n , and R n×m denote, respectively, the sets of positive integers, real numbers, real ndimensional vectors, real matrices of dimensions n × m, and real square symmetric positive semi-definite matrices of dimensions n × n. For any integers a ≤ b, notation I[a, b] stands for [a, b] ∩ N. P ∈ S n + (or S n ) and P ≻ 0 ( 0) denotes that P is a positive definite (semi-definite) matrix. Matrices I and 0 refers to as the identity and null matrices of appropriate dimensions, respectively. For any matrices A and B, A ⊤ represents the transpose of matrix A and diag(A, B), the block diagonal matrix [ A 0 0 B ]. Moreover, if A is a square matrix, He(A) stands for A + A ⊤ . Symbol ⋆ in a square matrix represents the symmetric entry of a symmetric matrix. For any vector (matrix) G, G (ℓ) denotes the ℓ th component (line) of G, and, to avoid any confusion, G ⊤ (ℓ) stands for the transpose of the ℓ th -line of G. The Euclidean norm is denoted as | • |. Set E n d is the set of sequence of d + 1 vectors in R n . More precisely, the elements of ϕ 0 ∈ E n d are d + 1 vectors x j ∈ R n , with j ∈ I[-d, 0], being denoted as ϕ 0 = {x -j } j∈I[0, d] and ϕ 0 (j) = x -j . φ0 ≡ ϕ 0 \{x 0 } is obtained from the sequence ϕ 0 with the element x 0 removed. The sequence variation is noted by ∆ϕ k = φk+1 -φk , and is com-

posed by d elements ∆ϕ k = {x k+j -x k+j-1 } j∈I[0, d-1] . ϕ k = max j∈I[0, d] |x k-j | denotes the norm is associ- ated to E n d .

Problem formulation

Consider the discrete-time system with time-varying state delay and saturating actuator given by

x k+1 = Ax k + A d x k-d k + Bsat(u k ), ∀k ∈ N, x -j = ϕ 0 (j), ∀j ∈ I[0, d], (1)
where x k and u k are the instantaneous state vector and the control input, respectively. Matrices A and A d ∈ R n×n , and B ∈ R n×m belong to a polytope given by the convex combination of N known vertices, such that

A A d B = N i=1 α k(i) A i A di B i , (2)
where matrices A i , A di , and B i are constant and known, and α k belongs to the unitary simplex

Γ = α k ∈ R N : N i=1 α k(i) = 1, α k(i) ≥ 0, i ∈ I[1, N ] . The saturation function sat(u k ) is considered component- wise so that sat(u k ) (ℓ) = sign(u k(ℓ) ) min(|u k(ℓ) |, (ū (ℓ) )), (3)
for ℓ = I[1, m]. We recall that notation u k(ℓ) and ū(ℓ) > 0 stand, respectively, for the ℓ th component of vectors u k and ū ∈ R m that contains the maximum magnitudes allowed to each control signal. Sequence ϕ 0 in E n d denotes the initial conditions of the time-delay system. We assume that there exist two nonnegative integers d < d such that

d k ∈ I[d, d], ∀k ∈ N. (4)
The following state-feedback control law is used to stabilize the system (1):

u k = Kx k , (5)
where K ∈ R m×n is the control gain to be defined.

Because of the saturation, guaranteeing the global stability of the closed-loop system is not possible in general.

One has to derive a local stability analysis consisting of the characterization of allowed or admissible sequences of initial conditions, ϕ 0 , which ensure that the closedloop trajectories do not leave the region of attraction,

R A ⊆ E n d .
As the computation of the set R A is a challenging task, even for low order and delay-free systems (see [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF] for more details), one can only compute an estimate of such a region, R E ⊆ R A and try to make this estimation as large as possible.

There exist many ways to derive estimates of the region of attraction. For instance, in [START_REF] Xu | Robust controller design of uncertain discrete time-delay systems with input saturation and disturbances[END_REF], R E is characterized as the sequences such that ϕ 0 ≤ r 1 , meaning that all elements of sequence are uniformly bounded. A more relaxed characterization was provided in [START_REF] Chen | Regional stabilization for discrete time-delay systems with actuator saturations via a delay-dependent polytopic approach[END_REF] which considers an additional uniform bound r 2 on the sequence variation, i.e. max j=1,...,

d |x -j -x -j+1 | ≤ r 2 , respectively.
In the present paper, we address the following problem:

Problem 1 Given the state-feedback control gain K, verify the local robust stability of the closed-loop system [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF]. Furthermore, estimate R E ⊆ R A ensuring that all trajectories emanating from ϕ 0 ∈ R E converge to the origin without leaving R E .

Problem 2 Determine the control gain K and an estimate R E ⊆ R A that ensures the local robust stability of the closed-loop system (1)-( 5) for any sequence ϕ 0 ∈ R E .

Preliminary Results

Generalized sector condition [19]

Let consider the dead-zone function φ ∈ R m defined by

φ(u) = sat(u) -u, ∀u ∈ R m . (6)
Following [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF], a polyhedral set can be defined to handle the saturation (1) with the help of an auxiliary signal ω ∈ R m :

S(ω, ū) = {u ∈ R m : |(u-ω) (ℓ) | ≤ ū(ℓ) , ∀ℓ ∈ I[1, m]}. (7)
It is worth noting that introducing ω relaxes the bounds on u, allowing it to go beyond the limits ±ū.

Lemma 1 For given auxiliary vector ω and saturation bounds, ū, if vector u belongs to S(ω, ū), then the function φ(u) defined in [START_REF] Hetel | Equivalence between the Lyapunov-Krasovskii functionals approach for discrete delay systems and that of the stability conditions for switched systems[END_REF] satisfies the following inequality:

φ(u) ⊤ T [φ(u) + ω] ≤ 0, ( 8
)
for any positive defined diagonal matrix T ∈ S m + .

Therefore, Lemma 1 guarantees that having signals u and ω belonging to S(ω, ū) ensures [START_REF] Liu | Auxiliary function-based summation inequalities and their applications to discretetime systems[END_REF]. This property will be used in the next section to relax the stabilization conditions.

A summation inequality free of slack variables

In the sequel, we present a lemma that helps us to establish our main result in the next section.

Lemma 2 Let R ∈ S n + and ϕ 0 a sequence in E n d . Then, for any integer c ∈ I[d, d], inequality ( d -d) -d-1 j=- d y ⊤ j Ry j ≥ Ω ⊤ c RΩ c (9)
holds where

y j = x j -x j-1 , ∀j ∈ I[-d + 1, 0], Ω c = x -d -2x -c + x -d, ∀c ∈ I[d, d]. ( 10
)
Proof : The proof is first based on the Jensen inequality's application to the left-hand side of (9) after splitting the summation into two parts. More precisely, we have that

J := ( d -d)   -d-1 j=-c y ⊤ j Ry j + -c-1 j=- d y ⊤ j Ry j   ≥ 1 α Ω + c ⊤ RΩ + c + 1 1 -α Ω - c ⊤ RΩ - c where Ω + c = x -d -x -c , Ω - c = x -c -x -d, and α = c-d d-d
. Following the principle of the reciprocally convex combination lemma [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF], the previous expression can be rewritten as follows

J ≥ Ω + c Ω - c ⊤ 1-α α R R R α 1-α R + R -R -R R Ω + c Ω - c
Let us now note that the first term is positive semidefinite since inequality

1-α α R R R α 1-α R 0 is equivalent 3 by Schur complement to 1-α α (R -R) = 0. The result is then derived from Ω + c Ω - c ⊤ R -R -R R Ω + c Ω - c = (Ω + c -Ω - c ) ⊤ R(Ω + c -Ω - c ),
which proofs the lemma. ♦ Remark 1 Note that, taking c = -d in (9) yields the usual Jensen inequality for discrete-time delay systems.

Remark 2

The benefits of the inequality in Lemma 2 over the reciprocally convex combination lemma from [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF] are a lower number of decision variables and less LMIs to solve since no additional slack variables, and no additional constraints are required. Moreover, it was shown in [START_REF] Seuret | Integral inequality for timevarying delay systems[END_REF], that the same level of conservatism as in [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF] was obtained for varying-delay systems, in the case of Jensenbased stability conditions. A discussion on the conservatism of the previous summation inequality is presented later in this paper concerning more advanced summation inequalities.

Stabilization of the closed-loop model

Because of the saturating actuator in (1) the local stability is required to handle the design of the robust state feedback control gain K in [START_REF] He | Output feedback stabilization for a discrete-time system with a time-varying dela[END_REF]. Following the generalized section conditions provided in Lemma 1 from [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF], we rewrite system (1) using the dead-zone function as follows

x k+1 = A cℓ x k + A d x k-d k + Bφ(u k ), (11)
where A cℓ = A+BK is still an uncertain matrix belonging to the polytope (2), and φ is the dead-zone function given in [START_REF] Hetel | Equivalence between the Lyapunov-Krasovskii functionals approach for discrete delay systems and that of the stability conditions for switched systems[END_REF].

We address the stabilization of system (11) by using the following simple LKF candidate given by

V (ϕ k ) = V 1 (ϕ k ) + V 2 (ϕ k ) + V 3 (ϕ k ), (12)
where

V 1 (ϕ k ) = x ⊤ k P x k , (13)
V 2 (ϕ k ) = k-1 i=k-d x ⊤ i Q1 x i + k-d-1 i=k- dx ⊤ i Q2 x i , (14)
V 3 (ϕ k ) = d 0 i=1-d k j=k+i y ⊤ j Z1 y j (15)
+ ( d -d) -d i=1- d k j=k+i y ⊤ j Z2 y j ,
where y j is defined in [START_REF] Pal | Delay-dependent stability criterion for uncertain discrete time systems in presence of actuator saturation[END_REF], ϕ k ∈ E n d refers to the funcstate of the time-delay system given by ϕ k (i) = x k-j for all j ∈ 0, d, and matrices P , Qi , and Zi , for i = 1, 2, belonging to S n + .

4 Main results

Constructive stabilization conditions

In this section, we provide two main results, where one concerns the local stability analysis of the closed-loop system [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF], and the other provides a convex control synthesis condition for the local stabilization of (1).

Theorem 1 Consider the uncertain system (1) under the control law (5) with a given control gain K. Suppose that there exist matrices

W , Q 1 , Q 2 , Z 1 , and Z 2 in S n + , G ∈ R m×4n , a positive diagonal matrix T ∈ S m
+ , and a scalar ǫ ∈ [0, 2] that satisfy the following LMIs

Φ 1i Φ 2i ⋆ Φ 3 ≺ 0,∀i ∈ I[1, N ], (16)
     W 0 ⋆ Q 1 KW 0 -G ⊤ (ℓ) ⋆ u 2 0(ℓ)      0,∀ℓ ∈ I[1, m], (17)
where

Φ 1i = He F ⊤ 1 Φ ⊤ 2i -F ⊤ 5 GΞ + Q 2 -2F ⊤ 5 T F 5 -Π ⊤ Z 1 0 ⋆ Z 2 Π, Φ 2i = A i W + B i KW -W 0 A di W 0 B i T ⊤ , Φ 3 = -(2ǫ -ǫ 2 )W + ǫ 2 (d 2 Z 1 + ( d -d) 2 Z 2 ), Q 1 = diag(Q 1 , Q 2 , Q 2 ), Q 2 = diag(Q 1 , -Q 1 + Q 2 , 0, -Q 2 , 0), ( 18
)
with the following matrices

Π = I -I 0 0 0 0 I -2I I 0 , F 1 = I 0 0 0 0 Ξ = I 0 0 0 0 0 I 0 0 0 0 0 I 0 0 0 0 0 I 0 , F 5 = 0 0 0 0 I .
Then the closed-loop system (11) is locally asymptotically stable for any sequence of initial conditions ϕ 0 verifying

ϕ 0 ∈ C v := ϕ 0 ∈ E n d s.t. V (ϕ 0 ) < 1 , ( 19
)
where V is the functional in (12)-( 15) with, for i = 1, 2

P = W -1 , Qi = W -1 Q i W -1 , Zi = W -1 Z i W -1 . ( 20
)
Proof : Consider the LKF candidate V (ϕ k ) given in ( 12)-( 15) with the parameters provided in [START_REF] Wei | Robust output feedback control of uncertain time-delay systems with actuator saturation and disturbances[END_REF]. Clearly, since matrices P , Q i , and Z i are assumed to be symmetric positive definite, the functional is also positive definite. Let us compute the increment of the functional, ∆V (ϕ k ) := V (ϕ k+1 ) -V (ϕ k ). Following the usual computational methods, the increment of the functional can be expressed as follows

∆V (φ k ) = x ⊤ k+1 W -1 W W -1 x k+1 +y ⊤ k+1 W -1 (d 2 Z 1 + ( d -d) 2 Z 2 )W -1 y k+1 -x ⊤ k W -1 (W + Q 1 )W -1 x k -x ⊤ k-d W -1 (Q 1 -Q 2 )W -1 x k-d -x ⊤ k- dW -1 Q 2 W -1 x k- d -d k i=k-d+1 y ⊤ i W -1 Z 1 W -1 y i -( d -d) k-d i=k-d+1 y ⊤ i W -1 Z 2 -1 y i .
The next steps consist in replacing x k+1 by y k+1 -x k in the first term of ∆V , and in applying Lemma 2 to the last two negative terms of the previous expression. Then, by introducing the augmented vector ξ k defined by

ξ k = ρ k T -1 ψ k , ρ k =        W -1 x k W -1 x k-d W -1 x k-d k W -1 x k- d       
, several manipulations allow us to write the following upper bound:

∆V (ϕ ) ≤ ξ ⊤ k Φ1 ξ k + He(φ ⊤ k T -1 (φ k + GΞξ k )) (21)
where we have used

y k+1 = N i=1 α k(i) Φ ⊤ 2i ξ k and with Φ1 = N i=1 α k(i) Φ 1i + N i=1 α k(i) Φ 2i ⊤ Ȳ -1 N i=1 α k(i) Φ 2i Ȳ =W W + d 2 Z 1 + ( d -d) 2 Z 2 -1 W
Then, the Schur complement ensures that matrix Φ1 is negative definite if and only if we have

    N i=1 α k(i) Φ 1 N i=1 α k(i) Φ 2i ⋆ - Ȳ     = N i=1 α k(i) Φ 1i Φ 2i ⋆ -Ȳ ≺ 0, Moreover, since (W -ǫY )Y -1 (W -ǫY ) ≥ 0 for any matrix Y and scalar ǫ, selecting Y = W + d 2 Z 1 + ( d - d) 2 Z 2 ,
we get that Ȳ is greater than Φ 3 , given in [START_REF] Sun | Adaptive control for a class of state-constrained high-order switched nonlinear systems with unstable subsystems[END_REF], which is ensured if conditions ( 16) are verified.

Therefore, local stability of the closed-loop system is guaranteed if the term φ ⊤ k T -1 (φ k + GΞξ k ) is also negative. To do so, we use the generalized sector condition provided in Lemma 1, where we have implicitly introduced the auxiliary vector ω k = GΞξ k = Gρ k . Let us first apply the Schur complement to the last line and column of ( 17), for a given ℓ and then multiply from the right side by ρ k and its transpose from the left side, leading to:

0 < ρ ⊤ k    P 0 ⋆ Q 1 - KW 0 -G ⊤ (ℓ) ū-2 (ℓ) × KW 0 -G (ℓ)    ρ k = x ⊤ k P x k +x ⊤ k-d Q1 x k-d +x ⊤ k-d k Q2 x k-d k +x ⊤ k- d Q2 x k-d - (Kx k -ω k ) ⊤ (ℓ) (Kx k -ω k ) (ℓ) ū2 (ℓ) ≤ V (ϕ 0 ) -ū-2 (ℓ) |(u k -ω k ) (ℓ) | 2 ,
for ϕ 0 verifying [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF]. Therefore, with ϕ 0 ∈ C v we ensure that u k belongs to S(ω k , ū) so that the generalized sector condition in Lemma 1 holds. Hence in this situation, one can guarantee that the increment of the functional is negative defined for any ϕ 0 ∈ C v . Since a level surface of the LKF defines the set C v , the solutions cannot leave this region, which concludes the proof. ♦ Remark 3 Observe that the conditions proposed in Theorem 1 with ǫ = 1 recovers the particular case where the inequality (W -Y ) ⊤ Y -1 (W -Y ) ≥ 0 is used (see, for instance, [START_REF] Silva | Delay dependent local stabilization conditions for time-delay nonlinear discrete-time systems using takagi-sugeno models[END_REF]). Therefore, the parameter ǫ delivers an extra degree of liberty to solve the analysis conditions.

A relevant comment is related to the application of the generalized sector conditions. In the literature of timedelay systems, the auxiliary vector ω is usually selected as Ĝx k , with Ĝ ∈ R m×n , while here we were able to also include more information about the functional state of the delay system by including in this auxiliary vector

x k-d , x k-d k and x k-d.
This modification is made possible through the introduction of the matrix Q 1 in condition [START_REF] Silva | Local stabilization of nonlinear discrete-time systems with time-varying delay in the states and saturating actuators[END_REF].

Replacing variable product KW by L in the LMI conditions ( 16)- [START_REF] Sun | Adaptive control for a class of state-constrained high-order switched nonlinear systems with unstable subsystems[END_REF] yields to an equivalent one, allowing the local stabilizing controller design. Therefore, we formalize the design conditions based on the Theorem 1 as follows.

Theorem 2 Consider the uncertain system [START_REF] Chen | Stabilisation for switched linear systems with time-varying delay and input saturation[END_REF]. Assume that there exist matrices

W , Q 1 , Q 2 , Z 1 , and Z 2 in S n + , matrices L ∈ R m×n , G ∈ R m×4n ,

and a positive diagonal matrix T ∈ S m

+ , and a scalar ǫ ∈ [0, 2] that satisfy the LMIs [START_REF] Silva | Delay dependent local stabilization conditions for time-delay nonlinear discrete-time systems using takagi-sugeno models[END_REF] and [START_REF] Silva | Local stabilization of nonlinear discrete-time systems with time-varying delay in the states and saturating actuators[END_REF] replacing therein KW by L. Then, the control law (5) with static feedback gain

K = LW -1 , (22)
ensures the local robust local stability of the closed-loop system system (1)-( 5) for any sequence of initial conditions ϕ 0 ∈ C v given in [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF], which is defined by the LKF given in [START_REF] Seuret | Integral inequality for timevarying delay systems[END_REF] with (20).

Geometrical characterization of R E

Theorems 1 and 2 provide constructive conditions to ensure that the closed-loop system [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF] with the static feedback control laws ( 5) or ( 22) is locally asymptotically stable. They also provide an estimate R E = C v of the region of attraction. A main issue is to understand how to optimize the parameters of the LKF to enlarge the

C v ⊆ E n d .
Therefore, the initial condition ϕ 0 is a function, which has n( d + 1) degrees of freedom. Inspired by the approach in [START_REF] Silva | Delay dependent local stabilization conditions for time-delay nonlinear discrete-time systems using takagi-sugeno models[END_REF] and [START_REF] Silva | Local stabilization of nonlinear discrete-time systems with time-varying delay in the states and saturating actuators[END_REF], we propose a characterization of the estimate of the region of attraction taking into account the couplings between x 0 and x -1 , namely y 0 calculated by [START_REF] Pal | Delay-dependent stability criterion for uncertain discrete time systems in presence of actuator saturation[END_REF]. Here, the admissible sequences of initial conditions ϕ 0 are such that:

ϕ 0 = {ϕ ∈ E n d : x 0 ∈ C, y 0 ∈ V, φ0 ∈ B(r 1 , r 2 )}, (23)
where C and V are ellipsoidal sets confining the admissible values of x 0 and y 0 , respectively, and the delayed states, φ0 , belong to B(r 1 , r 2 ), thus ensuring φ0 ≤ r 1 and ∆ φ0 ≤ r 2 . Therefore, sets C, V, and B(r 1 , r 2 ) jointly yield a possible geometrical characterization of R E . The following lemma is used to define the sets C, V and B(r 1 , r 2 ) mentioned in [START_REF] Zhang | Summation inequalities to bounded real lemmas of discretetime systems with time-varying delay[END_REF] which jointly characterize the allowed initial conditions for the closed-loop system [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF].

Lemma 3 If (12) is an LKF ensuring the local robust stability of [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF], then the initial sequences ϕ 0 verifying [START_REF] Zhang | Summation inequalities to bounded real lemmas of discretetime systems with time-varying delay[END_REF] with

C = {x 0 ∈ R n : V 1 (x 0 ) ≤ 1 -γ}, (24)
V = {y 0 ∈ R n : y ⊤ 0 Jy 0 < 1}, (25)
B(r 1 , r 2 ) = φ0 ∈ E n d-1 : φ0 ≤ r 1 , ∆ φ0 ≤ r 2 , (26) γ = ρ 1 φ0 2 + ρ 2 ∆ φ0 2 + y ⊤ 0 Jy 0 , (27)
and r 1 and r 2 chosen such that

0 ≤ ρ 1 r 2 1 + ρ 2 r 2 2 + y ⊤ 0 Jy 0 < 1, with (28)
J = d 2 Z 1 + ( d -d) 2 Z 2 , (29)
ρ 1 = dη 1 + ( d -d)η 2 , and (30)
ρ 2 = 0.5 d 2 (d -1)η 3 + ( d -d) 2 ( d + d -1)η 4 , (31)
where

η 1 = λ max (Q 1 ), η 2 = λ max (Q 2 ), η 3 = λ max (Z 1 ), η 4 = λ max (Z 2 )
, are such that the respective trajectories do not leave the estimate region of attraction and go asymptotically to the origin.

Proof : By hypothesis, ( 12) is an LKF for the closedloop system [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF]. Consequently, we can define a level set given by C ≡ L V1 (1 -γ) = {x k ∈ R n : x ′ k P x k ≤ 1 -γ}, with γ given in (27). By using V 2 (ϕ 0 ) and V 3 (ϕ 0 ) given in ( 14)-( 15), we get

V 2 (ϕ 0 ) + V 3 (ϕ 0 ) ≤ dλ max (Q 1 ) φ0 2 + ( d -d)λ max (Q 2 ) φ0 2 +0.5d 2 (d -1)λ max (Z 1 ) ∆ φ0 2 +0.5( d-d) 2 ( d+d-1)λ max (Z 2 ) ∆ φ0 2 +y ⊤ 0 Jy 0 = ρ 1 φ0 2 d + ρ 2 ∆ φ0 2 d + y ⊤ 0 Jy 0 , (32)
with J, ρ 1 , and ρ 2 given by ( 29)-(31). To ensure 0 ≤ γ < 1, it is required that (28) be satisfied. ♦

Observe that, the set V in Lemma 3 handles the transition between the current state and the previous one, which allows to get a less conservative estimate for the ball B(r 1 , r 2 ). Therefore, for each choice of y 0 verifying (25) we get an estimate for C and B(r 1 , r 2 ). If we are interested in the largest admissible y 0 , then we need to estimate it through set V. In this case, we get

|y 0 | ≤ 1/ √ ρ 3 with ρ 3 = d 2 η 3 + ( d -d) 2 η 4 .
Moreover, the region of attraction can be casted in a ball belonging to R n with radius r (see [START_REF] Wei | Robust output feedback control of uncertain time-delay systems with actuator saturation and disturbances[END_REF], [START_REF] Pal | Delay-dependent stability criterion for uncertain discrete time systems in presence of actuator saturation[END_REF], [START_REF] Chen | Regional stabilization for discrete time-delay systems with actuator saturations via a delay-dependent polytopic approach[END_REF]) by taking r 1 = r and r 2 = 2r, and from (24), we have V 1 (x 0 ) ≤ 1 -γ, which yields to r ≤ λ max (P ) + ρ 1 + 4(ρ 2 + ρ 3 ) -1/2 .

Observe that, theorems 1 and 2 encompass the case of uncertain and time-invariant delay, i.e. d = d. In such a case, the expressions in (30) and (31) are rewritten as ρ 1 = dη 1 and ρ 2 = d 2 (1 + d)η 3 /2, respectively.

Optimization procedure

An optimization procedure can be associated to Theorems 1 and 2, which only treat a feasibility problem, to maximize the size of the sets C and B(r 1 , r 2 ). To this end, we propose the following optimization procedure, where β j ≥ 0 are weights, H j ∈ R n×n , and X j corresponds to the j th matrix in the list {W,

Q 1 , Q 2 , Z 1 , Z 2 }: J=              min trace 5 j=1
β j H j s.t. ( 16), [START_REF] Silva | Local stabilization of nonlinear discrete-time systems with time-varying delay in the states and saturating actuators[END_REF], and

H j I I 2W -X j 0. (33)
Observe that, if (33) is feasible, then from Schur's complement, we get H j (2W -X j ) -1 W -1 X j W -1 . Because of Xj = W -1 X j W -1 , with Xj the j th matrix in the list { P , Q1 , Q2 , Z1 , Z2 }, we have H j Xj and thus the minimization of trace(H j ) leads to the minimization of trace( Xj ). Additionally, β j can be used to ponder the effects of each matrix of the LKF candidate.

Remark 4 In this paper, the stabilization conditions results from the application of Lemma 2. This lemma can be seen as a conservative tools since many summation inequalities have been proposed in the recent literature of discrete-time delay systems [START_REF] Seuret | Stability of discrete-time systems with time-varying delays via a novel summation inequality[END_REF][START_REF] Xiao | Improved stability criteria for discrete-time delay systems via novel summation inequalities[END_REF][START_REF] Zhang | Summation inequalities to bounded real lemmas of discretetime systems with time-varying delay[END_REF]. Using these new results would eventually lead to better numerical results. However this would distract the reader about the main message of the paper, which is the introduction of a new method for the estimation of the region of attraction for saturated systems. These improvements are kept for future works.

Numerical Examples

The examples were programmed in the Matlab using YALMIP and the solver lmilab.

Example 1: Consider the discrete-time system (1) with known matrices given by

A = 1.1 0.15 0.03 0.8 , A d = 0 -0.1 0 0 , B = 1 0.1 , (34)
constant delay d = d = 5, and ū = 15. The objective here is to design the saturating control law (5) such that the closed-loop system has the largest set of initial conditions. By using the optimization procedure J, with β 2 = 1 and β 1 = β 3 = β 4 = β 5 = 0, and ǫ = 1.9052 we get ρ 1 = 5.8972 × 10 -5 , ρ 2 = 3.0851 × 10 -7 , ρ 3 = 1.5426 × 10 -7 . As we argued before, Theorem 1 can be used to estimate the set of initial conditions as a ball. In this case, we get r = 74.3493, which is greater than those computed in [START_REF] Chen | Regional stabilization for discrete time-delay systems with actuator saturations via a delay-dependent polytopic approach[END_REF] (r = 72.596), [START_REF] Zhang | l 2 -gain analysis and control synthesis of uncertain discrete-time switched linear systems with time delay and actuator saturation[END_REF] (r = 73.693), [START_REF] Xu | Robust controller design of uncertain discrete time-delay systems with input saturation and disturbances[END_REF] (r = 63.029, with E = 0, α (1) =0.5, σ 1 = 0.15 and σ 2 = 0.14), and [START_REF] Chen | Stabilisation for switched linear systems with time-varying delay and input saturation[END_REF] (r = 68.491, with ε 1 = 2 and ǫ = 2 × 10 9 ). Therefore, this illustrates that our approach leads to a larger ball of initial conditions. However, our approach allows us to propose more general sets than the ones compared here. For instance, if value, i.e., r = 27.6139, then we get the ball B 2 and the ellipsoid C 2 shown in Figure 1 (solid blue line). In this last case the sequences ϕ 0 can have norm as great as 139.2372. Also note that, as the radius r 1 (with r 2 = 2r 1 ) shrinks, the size of the region C grows up.

We have investigated this effect by taking equally spaced values of 5.52278 ≤ r ≤ 55.2278: for each value of r, with r 1 = r and r 2 = 2r, we take vectors x -1 ∈ R 2 in the border of the set B(r, 2r) (thus, x -1 with modulus equal to r and angle θ with the horizontal axis swiped over a half cycle). For each pair (r, θ) we have calculated the area of the respective set C: S C = π(1 -γ)/ √ det P and γ given by (27). The achieved areas are shown in Figure 2 where it is clear the area's size improvement of the set C (associated with x 0 ) as the norm of the delayed states is reduced. Thus, this illustrates how our proposal of the region of attraction's estimate can yield less conservative results concerning other conditions found in the literature.

Conclusion

We have provided a new convex delay-dependent condition robust local controller design for a class of discrete time-varying delayed state systems under saturating actuators. We also gave less conservative conditions to robust local stabilize the considered class of systems from new bounds on sum inequalities and a new set characterization of initial admissible sequences. We illustrate how the proposed conditions can be used to get larger sequences of initial conditions through numerical examples.

Fig. 1 .

 1 Fig. 1. Region of attraction.

Fig. 2 .

 2 Fig. 2. Area of C, SC, for 5.52278 ≤ r ≤ 55.2278 and x-1 taken on the border of B(r, 2r).

  we choose r 1 = 74 and r 2 = 0, we can pick as a sequence of initial conditions

	ϕ 0 =	-68.3 68.4	,	37 64	,	37 64	,	37 64	,	37 64	,	37 64	,
	which has a norm 31.1% greater than the best estimate
	among [1,2,24,22] with ϕ 0 = 96.6615. Therefore, our
	approach allows the designer to handle sequences with
	norm much greater than those found in the mentioned
	works.
	Example 2: Consider the time-varying system (1) with
	two vertices given by
	A 1 =	0.38 0.20 0.09 1.00	, A d1 =	0.01 -0.03 0.02 0	, B 1 =	1.98 0.99	,
	A 2 =	0.42 0.20 0.11 1.00	, A d2 =	0.05 0.09 0.04 0.06		, B 2 =	2.02 1.01	.


Kelly L Materna

Noémie Lalaoui

Joseph A Laureanti

Aaron P Walsh

Belinda Pettersson Rimgard

Reiner Lomoth

Anders Thapper

Sascha Ott

Wendy J Shaw

Haining Tian

Leif Hammarström
email: leif.hammarstrom@kemi.uu.se

Using Surface Amide Couplings to Assemble Photocathodes for Solar Fuel Production Applications

Keywords: solar fuels, photocathode, nickel oxide, silatrane, amide coupling

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Due to the continued large-scale use of fossil fuels, global warming and climate change have become a concern. [START_REF] Ellabban | Renewable energy resources: Current status, future prospects and their enabling technology[END_REF] To mitigate the worst effects of climate change, alternative energy sources must be employed to generate clean, renewable fuels. Solar energy is a key alternative energy source due to the high solar flux hitting the earth´s surface (1.2 x 10 5 TW), which exceeds the world´s current energy demands by ~10,000 times. [START_REF]Annual Energy Outlook[END_REF][START_REF] Brennaman | Finding the Way to Solar Fuels with Dye-Sensitized Photoelectrosynthesis Cells[END_REF][START_REF] Young | Light-driven water oxidation for solar fuels[END_REF][START_REF] Wang | Mimicking Natural Photosynthesis: Solar to Renewable H2 Fuel Synthesis by Z-Scheme Water Splitting Systems[END_REF] To use solar energy to drive renewable fuel synthesis, natural photosynthesis can give inspiration. [START_REF] Mcevoy | Water-splitting chemistry of photosystem II[END_REF] To mimic photosynthetic processes, water-splitting dye sensitized photoelectrochemical cells (WS-DSPECs) can be assembled to perform "artificial photosynthesis" by using solar energy to drive renewable fuel synthesis. [START_REF] Brennaman | Finding the Way to Solar Fuels with Dye-Sensitized Photoelectrosynthesis Cells[END_REF][START_REF] Li | Organic Dye-Sensitized Tandem Photoelectrochemical Cell for Light Driven Total Water Splitting[END_REF][START_REF] Ashford | Molecular Chromophore-Catalyst Assemblies for Solar Fuel Applications[END_REF][START_REF] Yu | Recent advances in dye-sensitized photoelectrochemical cells for solar hydrogen production based on molecular components[END_REF][START_REF] Tian | Molecular Catalyst Immobilized Photocathodes for Water/Proton and Carbon Dioxide Reduction[END_REF] WS-DSPECs are composed of a photoanode, which performs photocatalytic water oxidation, and a photocathode, which performs photocatalytic fuel-forming reactions, such as proton reduction to form H 2 . H 2 is an energy dense fuel, having a chemical energy per mass of 142 MJkg -1 , around three times larger than liquid hydrocarbon fuels. [START_REF] Schlapbach | Hydrogen-storage materials for mobile applications[END_REF] In WS-DSPECs, the photocathode usually consists of a NiO-based electrode coated with surface-bound dye molecules to absorb the light and perform charge separation, while surface-bound catalyst molecules perform catalysis and generate H 2 . To operate, the photocathode is illuminated which causes the surface-bound dye molecules to reach their excited state. The excited dye then injects a hole into the NiO valence band, thereby reducing the dye. The reduced dye can then transfer an electron to a neighboring dye molecule or to a proton reduction catalyst on the NiO surface. Once the catalyst gains two reducing equivalents, as well as two protons from solution, proton reduction can occur to form H 2 gas. [START_REF] Gibson | Dye-sensitized photocathodes for H2 evolution[END_REF][START_REF] Nikolaou | Recent advances and insights in dyesensitized NiO photocathodes for photovoltaic devices[END_REF][START_REF] Wu | Recent Advances in Sensitized Photocathodes: From Molecular Dyes to Semiconducting Quantum Dots[END_REF][START_REF] Queyriaux | Molecular cathode and photocathode materials for hydrogen evolution in photoelectrochemical devices[END_REF] A major limitation to the overall efficiency of WS-DSPECs is that all of these processes necessary to perform photocatalytic proton reduction must occur before unproductive charge recombination happens. [START_REF] Brennaman | Finding the Way to Solar Fuels with Dye-Sensitized Photoelectrosynthesis Cells[END_REF][START_REF] Ashford | Molecular Chromophore-Catalyst Assemblies for Solar Fuel Applications[END_REF][START_REF] Gibson | Dye-sensitized photocathodes for H2 evolution[END_REF][START_REF] Ashford | Molecular Chromophore-Catalyst Assemblies for Solar Fuel Applications[END_REF] Significant work has been done to optimize photoanodes in WS-DSPECs (the best of which now reach relatively high photocurrents and efficiencies). [START_REF] Brennaman | Finding the Way to Solar Fuels with Dye-Sensitized Photoelectrosynthesis Cells[END_REF][START_REF] Ashford | Molecular Chromophore-Catalyst Assemblies for Solar Fuel Applications[END_REF][START_REF] Yu | Recent advances in dye-sensitized photoelectrochemical cells for solar hydrogen production based on molecular components[END_REF] Conversely, the photocathode component of WS-DSPECs has not been explored or optimized to the same extent; consequently, the overall performance of tandem WS-DSPECs to-date are still poor. [START_REF] Li | Organic Dye-Sensitized Tandem Photoelectrochemical Cell for Light Driven Total Water Splitting[END_REF][START_REF] Gibson | Dye-sensitized photocathodes for H2 evolution[END_REF][START_REF] Fan | Pt-free tandem molecular photoelectrochemical cells for water splitting driven by visible light[END_REF][START_REF] Wood | A comprehensive comparison of dye-sensitized NiO photocathodes for solar energy conversion[END_REF][START_REF] Tong | Sustained solar hydrogen generation using a dye-sensitised NiO photocathode/BiVO4 tandem photoelectrochemical device[END_REF] The suboptimal photocathode performance could originate from many factors, including the NiO electrode, which exhibits slow hole mobility and large losses in efficiency due to charge recombination between holes in the NiO and reduced dye or catalyst molecules. [START_REF] Morandeira | Photoinduced ultrafast dynamics of coumarin 343 sensitized p-type-nanostructured NiO films[END_REF][START_REF] Qin | Synthesis and Mechanistic Studies of Organic Chromophores with Different Energy Levels for p-Type Dye-Sensitized Solar Cells[END_REF][START_REF] Odobel | Recent advances and future directions to optimize the performances of p-type dye-sensitized solar cells[END_REF][START_REF] Antila | Dynamics and Photochemical H2 Evolution of Dye-NiO Photocathodes with a Biomimetic FeFe-Catalyst[END_REF][START_REF] Daeneke | Dominating Energy Losses in NiO p-Type Dye-Sensitized Solar Cells[END_REF][START_REF] D'amario | Kinetic Evidence of Two Pathways for Charge Recombination in NiO-Based Dye-Sensitized Solar Cells[END_REF][START_REF] D'amario | Unveiling hole trapping and surface dynamics of NiO nanoparticles[END_REF][START_REF] D'amario | Chemical and Physical Reduction of High Valence Ni States in Mesoporous NiO Film for Solar Cell Application[END_REF][START_REF] Morandeira | Coumarin 343-NiO Films as Nanostructured Photocathodes in Dye-Sensitized Solar Cells: Ultrafast Electron Transfer, Effect of the I3-/I-Redox Couple and Mechanism of Photocurrent Generation[END_REF][START_REF] Mori | Charge-Transfer Processes in Dye-Sensitized NiO Solar Cells[END_REF][START_REF] Black | Chargetransfer dynamics at the dye-semiconductor interface of photocathodes for solar energy applications[END_REF][START_REF] Brautigam | Hole injection dynamics from two structurally related Rubipyridine complexes into NiO(x) is determined by the substitution pattern of the ligands[END_REF][START_REF] Hao | Can aliphatic anchoring groups be utilised with dyes for p-type dye sensitized solar cells?[END_REF][START_REF] Huang | Probing the Low Fill Factor of NiO p-Type Dye-Sensitized Solar Cells[END_REF][START_REF] Ji | Synthesis, Photophysics, and Photovoltaic Studies of Ruthenium Cyclometalated Complexes as Sensitizers for p-Type NiO Dye-Sensitized Solar Cells[END_REF][START_REF] Ji | Photoinduced Electron Transfer Dynamics of Cyclometalated Ruthenium (II)-Naphthalenediimide Dyad at NiO Photocathode[END_REF][START_REF] Shan | Controlling Vertical and Lateral Electron Migration Using a Bifunctional Chromophore Assembly in Dye-Sensitized Photoelectrosynthesis Cells[END_REF][START_REF] Shan | Modulating Hole Transport in Multilayered Photocathodes with Derivatized p-Type Nickel Oxide and Molecular Assemblies for Solar-Driven Water Splitting[END_REF][START_REF] Dillon | Enabling Efficient Creation of Long-Lived Charge-Separation on Dye-Sensitized NiO Photocathodes[END_REF] These issues make developing an efficient photocathode quite difficult. Thus, is it crucial that fundamental studies are performed to gain understanding of the processes occurring in photocathodes if performance is to be rationally improved.

To increase photocathode performance, many photocathode designs have focused on changing the electrode assembly. Methods for assembly include using layer-by-layer strategies, [START_REF] Shan | Controlling Vertical and Lateral Electron Migration Using a Bifunctional Chromophore Assembly in Dye-Sensitized Photoelectrosynthesis Cells[END_REF][START_REF] Shan | Modulating Hole Transport in Multilayered Photocathodes with Derivatized p-Type Nickel Oxide and Molecular Assemblies for Solar-Driven Water Splitting[END_REF][START_REF] Gross | Photoelectrochemical hydrogen production in water using a layer-by-layer assembly of a Ru dye and Ni catalyst on NiO[END_REF][START_REF] Shan | Photogeneration of hydrogen from water by a robust dye-sensitized photocathode[END_REF][START_REF] Shan | Generation of Long-Lived Redox Equivalents in Self-Assembled Bilayer Structures on Metal Oxide Electrodes[END_REF][START_REF] Leem | Photocathode Chromophore-Catalyst Assembly via Layer-By-Layer Deposition of a Low Band-Gap Isoindigo Conjugated Polyelectrolyte[END_REF] adding hydrophobic layers, [START_REF] Click | Membrane-Inspired Acidically Stable Dye-Sensitized Photocathode for Solar Fuel Production[END_REF] codeposition methods, [START_REF] Li | Organic Dye-Sensitized Tandem Photoelectrochemical Cell for Light Driven Total Water Splitting[END_REF][START_REF] Antila | Dynamics and Photochemical H2 Evolution of Dye-NiO Photocathodes with a Biomimetic FeFe-Catalyst[END_REF][START_REF] Van Den Bosch | Nickel-Based Dye-Sensitized Photocathode: Towards Proton Reduction Using a Molecular Nickel Catalyst and an Organic Dye[END_REF][START_REF] Kaeffer | Insights into the mechanism and aging of a noble-metal free H2-evolving dye-sensitized photocathode[END_REF][START_REF] Windle | Earth-Abundant Molecular Z-Scheme Photoelectrochemical Cell for Overall Water-Splitting[END_REF] covalently linking the dye and catalyst, [START_REF] Pati | Insights into the Mechanism of a Covalently Linked Organic Dye-Cobaloxime Catalyst System for Dye-Sensitized Solar Fuel Devices[END_REF][START_REF] Ji | Photostable p-type dye-sensitized photoelectrochemical cells for water reduction[END_REF][START_REF] Kaeffer | Covalent Design for Dye-Sensitized H2-Evolving Photocathodes Based on a Cobalt Diimine-Dioxime Catalyst[END_REF][START_REF] Lyu | H2-Evolving Dye-Sensitized Photocathode Based on a Ruthenium-Diacetylide/Cobaloxime Supramolecular Assembly[END_REF][START_REF] Poldme | Photoelectrocatalytic H2 evolution from integrated photocatalysts adsorbed on NiO[END_REF] using electron shuttling layers, [START_REF] Xu | Solution-processed nanoporous NiO-dye-ZnO photocathodes: Toward efficient and stable solid-state p-type dye-sensitized solar cells and dye-sensitized photoelectrosynthesis cells[END_REF] using supramolecular strategies, [START_REF] Shan | Direct photoactivation of a nickel-based, water-reduction photocathode by a highly conjugated supramolecular chromophore[END_REF] changing the semiconductor material, [START_REF] Creissen | Solar H2 generation in water with a CuCrO2 photocathode modified with an organic dye and molecular Ni catalyst[END_REF] and using atomic layer deposition. [START_REF] Kamire | Photodriven hydrogen evolution by molecular catalysts using Al2O3-protected perylene-3,4-dicarboximide on NiO electrodes[END_REF] While these are all very important studies, the stability of the anchoring group in photocathode architectures has received rather little attention. To the best of our knowledge, only one study has focused on changing the molecules' anchoring group to improve performance. [START_REF] Van Den Bosch | Nickel-Based Dye-Sensitized Photocathode: Towards Proton Reduction Using a Molecular Nickel Catalyst and an Organic Dye[END_REF] This is surprising, because it is a common strategy in photoanode assembly methods. [START_REF] Brennaman | Finding the Way to Solar Fuels with Dye-Sensitized Photoelectrosynthesis Cells[END_REF][START_REF] Ashford | Molecular Chromophore-Catalyst Assemblies for Solar Fuel Applications[END_REF][START_REF] Materna | Anchoring groups for photocatalytic water oxidation on metal oxide surfaces[END_REF][START_REF] Materna | Optimization of Photoanodes for Photocatalytic Water Oxidation by Combining a Heterogenized Iridium Water-Oxidation Catalyst with a High-Potential Porphyrin Photosensitizer[END_REF][START_REF] Kamire | Photodriven Oxidation of Surface-Bound Iridium-Based Molecular Water-Oxidation Catalysts on Perylene-3,4-dicarboximide-Sensitized TiO2 Electrodes Protected by an Al2O3 Layer[END_REF][START_REF] Mcnamara | Water-stable, hydroxamate anchors for functionalization of TiO2 surfaces with ultrafast interfacial electron transfer[END_REF][START_REF] Mcnamara | Hydroxamate anchors for water-stable attachment to TiO2 nanoparticles[END_REF][START_REF] Jiang | Direct Interfacial Electron Transfer from High-Potential Porphyrins into Semiconductor Surfaces: A Comparison of Linkers and Anchoring Groups[END_REF] In photocathodes, stability can be a major issueif the molecular components do not remain bound on the electrode, then the photocathodes will not function efficiently. [START_REF] Gibson | Dye-sensitized photocathodes for H2 evolution[END_REF][START_REF] Wu | Recent Advances in Sensitized Photocathodes: From Molecular Dyes to Semiconducting Quantum Dots[END_REF] While carboxylic acids and phosphonic acids are known to have limited stability on metal oxide electrodes under aqueous conditions, they are still the most common anchors used in these systems. [START_REF] Gibson | Dye-sensitized photocathodes for H2 evolution[END_REF][START_REF] Wu | Recent Advances in Sensitized Photocathodes: From Molecular Dyes to Semiconducting Quantum Dots[END_REF][START_REF] Materna | Anchoring groups for photocatalytic water oxidation on metal oxide surfaces[END_REF] Conversely, hydroxamic acid and silatrane surface anchors are known to have excellent stability on metal oxide surfaces across a wide pH range, [START_REF] Materna | Anchoring groups for photocatalytic water oxidation on metal oxide surfaces[END_REF][START_REF] Materna | Optimization of Photoanodes for Photocatalytic Water Oxidation by Combining a Heterogenized Iridium Water-Oxidation Catalyst with a High-Potential Porphyrin Photosensitizer[END_REF][START_REF] Kamire | Photodriven Oxidation of Surface-Bound Iridium-Based Molecular Water-Oxidation Catalysts on Perylene-3,4-dicarboximide-Sensitized TiO2 Electrodes Protected by an Al2O3 Layer[END_REF][START_REF] Mcnamara | Water-stable, hydroxamate anchors for functionalization of TiO2 surfaces with ultrafast interfacial electron transfer[END_REF][START_REF] Mcnamara | Hydroxamate anchors for water-stable attachment to TiO2 nanoparticles[END_REF][START_REF] Jiang | Direct Interfacial Electron Transfer from High-Potential Porphyrins into Semiconductor Surfaces: A Comparison of Linkers and Anchoring Groups[END_REF][START_REF] Materna | Silatranes for binding inorganic complexes to metal oxide surfaces[END_REF][START_REF] Materna | Silatrane Anchors for Metal Oxide Surfaces: Optimization for Potential Photocatalytic and Electrocatalytic Applications[END_REF] yet only one study to our knowledge has used hydroxamic acid anchors on NiO-based photocathodes [START_REF] Van Den Bosch | Nickel-Based Dye-Sensitized Photocathode: Towards Proton Reduction Using a Molecular Nickel Catalyst and an Organic Dye[END_REF] and none have used silatranes. [START_REF] Van Den Bosch | Nickel-Based Dye-Sensitized Photocathode: Towards Proton Reduction Using a Molecular Nickel Catalyst and an Organic Dye[END_REF][START_REF] Materna | Anchoring groups for photocatalytic water oxidation on metal oxide surfaces[END_REF][START_REF] Materna | Optimization of Photoanodes for Photocatalytic Water Oxidation by Combining a Heterogenized Iridium Water-Oxidation Catalyst with a High-Potential Porphyrin Photosensitizer[END_REF][START_REF] Kamire | Photodriven Oxidation of Surface-Bound Iridium-Based Molecular Water-Oxidation Catalysts on Perylene-3,4-dicarboximide-Sensitized TiO2 Electrodes Protected by an Al2O3 Layer[END_REF][START_REF] Mcnamara | Water-stable, hydroxamate anchors for functionalization of TiO2 surfaces with ultrafast interfacial electron transfer[END_REF][START_REF] Mcnamara | Hydroxamate anchors for water-stable attachment to TiO2 nanoparticles[END_REF][START_REF] Jiang | Direct Interfacial Electron Transfer from High-Potential Porphyrins into Semiconductor Surfaces: A Comparison of Linkers and Anchoring Groups[END_REF][START_REF] Materna | Silatranes for binding inorganic complexes to metal oxide surfaces[END_REF][START_REF] Materna | Silatrane Anchors for Metal Oxide Surfaces: Optimization for Potential Photocatalytic and Electrocatalytic Applications[END_REF] A silatrane is a pentacoordinate caged silicon atom, such as Sil shown in Figure 1, and is known to bind to metal oxide surfaces (MOx: TiO 2 , SnO 2 , ITO, Al 2 O 3 ) via the formation of surface Si-O-M (M = metal on surface) bonds with concomitant release of triethanolamine. [START_REF] Materna | Anchoring groups for photocatalytic water oxidation on metal oxide surfaces[END_REF][START_REF] Materna | Silatranes for binding inorganic complexes to metal oxide surfaces[END_REF][START_REF] Materna | Silatrane Anchors for Metal Oxide Surfaces: Optimization for Potential Photocatalytic and Electrocatalytic Applications[END_REF][START_REF] Brennan | Organosilatrane building blocks[END_REF][START_REF] Brennan | 1-(3'-Amino)propylsilatrane derivatives as covalent surface linkers to nanoparticulate metal oxide films for use in photoelectrochemical cells[END_REF][START_REF] Frye | Pentacoordinate silicon compounds. V. Novel silatrane chemistry[END_REF][START_REF] Materna | Heterogenized Iridium Water-Oxidation Catalyst from a Silatrane Precursor[END_REF] To bind silatranes to metal oxide electrodes, the electrodes are typically soaked in a silatrane-containing sensitization solution at 70 °C for 3-24 hours. [START_REF] Materna | Optimization of Photoanodes for Photocatalytic Water Oxidation by Combining a Heterogenized Iridium Water-Oxidation Catalyst with a High-Potential Porphyrin Photosensitizer[END_REF][START_REF] Kamire | Photodriven Oxidation of Surface-Bound Iridium-Based Molecular Water-Oxidation Catalysts on Perylene-3,4-dicarboximide-Sensitized TiO2 Electrodes Protected by an Al2O3 Layer[END_REF][START_REF] Materna | Silatranes for binding inorganic complexes to metal oxide surfaces[END_REF][START_REF] Materna | Heterogenized Iridium Water-Oxidation Catalyst from a Silatrane Precursor[END_REF] Recently, a new surface preacidification method has been found to bind the silatrane at room temperature with similar loadings to the 70 °C sensitization process. [START_REF] Materna | Silatrane Anchors for Metal Oxide Surfaces: Optimization for Potential Photocatalytic and Electrocatalytic Applications[END_REF] Once bound, silatranes remain attached to metal oxide electrodes from pH 2-11, giving excellent aqueous, electrochemical, and photoelectrochemical stability; thus making them excellent options as anchors also in photocathodes. [START_REF] Materna | Anchoring groups for photocatalytic water oxidation on metal oxide surfaces[END_REF][START_REF] Materna | Optimization of Photoanodes for Photocatalytic Water Oxidation by Combining a Heterogenized Iridium Water-Oxidation Catalyst with a High-Potential Porphyrin Photosensitizer[END_REF][START_REF] Materna | Silatranes for binding inorganic complexes to metal oxide surfaces[END_REF][START_REF] Materna | Silatrane Anchors for Metal Oxide Surfaces: Optimization for Potential Photocatalytic and Electrocatalytic Applications[END_REF][START_REF] Materna | Heterogenized Iridium Water-Oxidation Catalyst from a Silatrane Precursor[END_REF] Since producing photocathodes with high efficiencies is challenging, developing a facile method that would allow for the fast screening of different functional units would be highly desirable. An option is to take advantage of organic chemical reactions and perform them on surfaces, such as the use of surface amide couplings. While amide couplings on surfaces are quite popular in other areas of surface chemistry, [START_REF] Lee | Near-Quantitative Yield for Transfer of Near-Infrared Excitons within Solution-Phase Assemblies of PbS Quantum Dots[END_REF][START_REF] Booth | In-Depth Electrochemical Investigation of Surface Attachment Chemistry via Carbodiimide Coupling[END_REF][START_REF] You | Enhanced electrocatalytic activity of oxygen reduction by cobaltporphyrin functionalized with graphene oxide in an alkaline solution[END_REF][START_REF] Wayu | Functionalized carbon nanotube adsorption interfaces for electron transfer studies of galactose oxidase[END_REF][START_REF] Ellebracht | Amine functionalization of cellulose nanocrystals for acid-base organocatalysis: surface chemistry, cross-linking, and solvent effects[END_REF] thus far, we are not aware of any studies using this method to prepare photocathodes. This method is quite convenient because many dye and catalyst molecules already contain carboxylic acid functional groups, which can be utilized in amide couplings, avoiding tedious organic syntheses of complex dye or catalyst molecules. Instead, surface amide couplings have the potential for facile screening of different electrode assemblies by altering the nature of the carboxylic acid decorated dye or catalyst, the amino-derivatized electrode surface, or vice versa. Screening amide coupled combinations of carboxylic acid functionalized dye or catalysts with amino-functionalized molecules that are commercially available or already present in research groups would be a fast and easy strategy to find optimal photocathode combinations. Thus, here we report a method to assemble photocathodes using surface amide couplings with silatrane-anchored molecules. We then use this method to assemble model photocathodes containing a catalyst molecule, and investigate their photoelectrochemical behavior and charge transfer dynamics, to show the applicability of the method. We expect that the protocols presented here will be of general use in the photocathode field.

Results & Discussion:

Recently, we developed a new push-pull dye, PB6 (Figure 1), which has been studied on NiO electrodes using new core-shell assemblies. [START_REF] Xu | Solution-processed nanoporous NiO-dye-ZnO photocathodes: Toward efficient and stable solid-state p-type dye-sensitized solar cells and dye-sensitized photoelectrosynthesis cells[END_REF][START_REF] Tian | Ultrafast dye regeneration in a core-shell NiO-dye-TiO2 mesoporous film[END_REF][START_REF] Tian | Solid state p-type dye sensitized NiO-dye-TiO2 core-shell solar cells[END_REF] Upon excitation, PB6* (1.03 V vs. NHE for PB6*/PB6 -) [START_REF] Tian | Ultrafast dye regeneration in a core-shell NiO-dye-TiO2 mesoporous film[END_REF] exhibits ultrafast hole injection (<200 fs) into the NiO valence band (0.5 V vs. NHE) [START_REF] Gibson | Role of the triiodide/iodide redox couple in dye regeneration in p-type dye-sensitized solar cells[END_REF] ; the resulting generated reduced state (-0.93 V vs. NHE) of PB6 is capable of reducing a wide range of proton reduction catalysts, making it a suitable choice as a photocathode dye molecule. [START_REF] Tian | Ultrafast dye regeneration in a core-shell NiO-dye-TiO2 mesoporous film[END_REF][START_REF] Tian | Solid state p-type dye sensitized NiO-dye-TiO2 core-shell solar cells[END_REF] PB6 also exhibits a strong absorption band at 530 nm making it facile to characterize by UV-Vis spectroscopy and so monitor its surface binding to NiO. Thus, we sought out to investigate the surface-amide coupling method using PB6.

To prepare the electrodes, NiO electrodes were first sensitized with 3aminopropylsilatrane, Sil, (Figure 1) via a newly reported preacidification method [START_REF] Materna | Silatrane Anchors for Metal Oxide Surfaces: Optimization for Potential Photocatalytic and Electrocatalytic Applications[END_REF] allowing the silatrane to bind to the NiO at room temperature overnight. Briefly, NiO electrodes were soaked in 0.

Spectroscopic Characterization.

The electrodes were characterized by several spectroscopic methods at each stage of the assembly process. Since Sil has only UV absorption features, UV-Vis spectroscopy could not be used to characterize the electrodes after the first assembly step (NiO-Sil) due to the scattering of the NiO film below 400 nm (Figure S6). Instead, evidence for Sil binding to NiO was obtained by comparing the ATR-FTIR spectra of the Sil powder to the NiO-Sil electrodes, which show similar features (Figure S9). Sil exhibits diagnostic signatures of the silatrane cage: a sharp peak at 1175 cm -1 for ν(C-O) and 1452 cm -1 for δ(CH 2 ), which disappear upon surface binding, indicating silatrane deprotection. Furthermore, a broad peak around 1070 cm -1 appears, suggesting the formation of Si-O-Si cross links and Si-O surface bonds (Figure S9). [START_REF] Materna | Heterogenized Iridium Water-Oxidation Catalyst from a Silatrane Precursor[END_REF][START_REF] Brennan | Comparison of silatrane, phosphonic acid, and carboxylic acid functional groups for attachment of porphyrin sensitizers to TiO2 in photoelectrochemical cells[END_REF][START_REF] Honda | The infrared spectroscopic study of metals intruded into a silica framework[END_REF] ICP-OES measurements also detected Si on the digested electrode surfaces above background Si signals, confirming the presence of Si on the NiO (Table S2).

Next, we characterized the subsequent amide coupling of PB6 on NiO-Sil by UV-Vis spectroscopy. When NiO-Sil was exposed to both PB6 and EDC, a strong absorption at 530 nm appeared, supporting successful attachment of PB6 to the electrode with a surface coverage equivalent to ~40 nmol cm -2 of PB6 (Figure 2).

In addition, the absorption features of NiO-Sil-PB6 were similar to separately prepared NiO-PB6 electrodes lacking Sil, suggesting the PB6 structure was retained upon binding to NiO-Sil (Figure 2).

To gain further evidence for the amide coupling, several control experiments were performed to test if the observed PB6 was bound via amide bonds or physisorbed to the surface directly. First, we tested for the necessity of the formation of an amide bond. To do this, we made a preacidified NiO electrode without Sil, soaked the electrode in a solution containing both PB6 and EDC overnight, and monitored the absorption of the electrode by UV-Vis spectroscopy. As expected, no significant binding of PB6 to the electrodes was seen, supporting that physisorption of PB6 does not occur and the amine is essential for binding the dye molecule to the surface (Figure S3). To further confirm amide coupling, we also tested if EDC was needed by exposing NiO-Sil electrodes in a PB6 solution overnight without EDC present. We observed a minute peak in the UV-Vis spectrum at 530 nm suggesting some PB6 had bound to the surface, likely filling in any open binding sites on the electrode not already taken by Sil (Figure S3). Since we had more significant loading in the presence of EDC, it suggests that an amide coupling is the immobilization mechanism for PB6 loading.

To further confirm that the core PB6 structure was unchanged upon electrode attachment, ATR-FTIR spectroscopic measurements were performed.

Comparison of the ATR-FTIR spectra of NiO-Sil-PB6 with PB6 powder samples revealed excellent overlap, further supporting retention of the original PB6 structure on the electrode (Figure 2). When NiO-Sil-PB6 was compared to NiO-PB6, we observed an extra broad peak at ~1070 cm -1 , also observed in NiO-Sil electrodes in Figure S9, suggesting the presence of the silatrane on the NiO-Sil-PB6 electrode (Figure 2). This also confirms that the silatrane remains surface-bound during the amide coupling procedure. These data show that PB6 is on the electrode and that it's structure is retained.

Photoelectrochemical Measurements.

After confirming that electrodes could be prepared using surface amide couplings, photoelectrochemical experiments on the NiO-Sil-PB6 electrodes were performed. Specifically, we probed if the electrodes could generate photocurrent under illumination with an applied bias, which is necessary if they are to be applied into photocathodes. Indeed, an increase in reductive photocurrent in the linear sweep voltammogram (LSV) was observed upon illuminating the NiO-Sil-PB6 electrodes in 0.1 M MES buffer at pH 5, suggesting that PB6 reaches its excited state and injects a hole into the NiO valence band (Figure 3). One should also keep in mind that photocurrent could also result from the formation of Ni(0) species via reduction by PB6. [START_REF] Hoogeveen | Origin of Photoelectrochemical Generation of Dihydrogen by a Dye-Sensitized Photocathode without an Intentionally Introduced Catalyst[END_REF] Since the NiO-Sil-PB6 electrodes could produce photocurrent, we then wanted to demonstrate that the PB6 -generated after hole injection could transfer an electron to an acceptor molecule. To test this, NiO-Sil-PB6 electrodes were examined in the presence of an irreversible electron acceptor, chloropentaaminecobalt(III) chloride. When LSV was performed in the presence of the acceptor molecule, a significant increase in photocurrent was observed (almost -300 µA cm -2 ), strongly suggesting that PB6 -is capable of transferring an electron to the acceptor molecule (Figure 3). This finding supports that the NiO-Sil-PB6

electrodes could be used in combination with surface-bound catalyst molecules for hydrogen production with high photocurrents in optimized solar fuel systems. Ultrafast Electron Transfer Dynamics.

Since the photoelectrochemical experiments showed promise for the NiO-Sil-PB6 electrodes, we wanted to further understand the charge transfer dynamics on the electrodes. To monitor the electron transfer dynamics, femtosecond transient absorption spectroscopy (fsTAS) was performed on the films in air. NiO-Sil-PB6 was excited at 560 nm, resulting in a ground state bleach around 530 nm and an induced absorption at >620 nm (Figure 4, S14), both of which had been previously observed with NiO-PB6 electrodes. [START_REF] Tian | Ultrafast dye regeneration in a core-shell NiO-dye-TiO2 mesoporous film[END_REF] Similar to the NiO-PB6 films studied previously, we also observed an isosbestic point at ca. 625 nm at ca. 0.6 ps after excitation which blueshifts rapidly with time. [START_REF] Tian | Ultrafast dye regeneration in a core-shell NiO-dye-TiO2 mesoporous film[END_REF] A global fit to the NiO-Sil-PB6 data to a sum of three exponentials with a minor offset,  infinite, resulted in the following lifetimes:  1 = 0.53 ps,  2 = 16 ps and  3 = 3800 ps. A kinetic trace at 620 nm (Figure 4) shows clearly an instrument-limited bleach, followed by a  1 = 0.53 ps rise to a positive signal and finally a decay to baseline in ca. 1 ns (represented by  2 and  3 ). This shows clearly the presence of at least one intermediate in the photoreaction, consistent with formation of a charge separated state, PB6 -and NiO (+) , where the former species has shown an absorption around 620 nm. [START_REF] Tian | Ultrafast dye regeneration in a core-shell NiO-dye-TiO2 mesoporous film[END_REF] In addition, this is reflected in the decay associated spectra from the global fit, where the  1 component matches the transient spectrum of excited PB6* while the  2 and  3 components are similar to the PB6 -radical-ground state difference spectrum (Figure S19-S20). [START_REF] Tian | Ultrafast dye regeneration in a core-shell NiO-dye-TiO2 mesoporous film[END_REF][START_REF] Tian | Solid state p-type dye sensitized NiO-dye-TiO2 core-shell solar cells[END_REF] The first component ( 1 ) is attributed to hole injection from excited PB6* to NiO, forming the PB6 -radical anion in analogy to previous studies. [START_REF] Tian | Ultrafast dye regeneration in a core-shell NiO-dye-TiO2 mesoporous film[END_REF] Stimulated emission from PB6* gives a negative signal, that is replaced by the positive absorption of PB6 - at 620 nm (Figure 4). [START_REF] Mcnamara | Hydroxamate anchors for water-stable attachment to TiO2 nanoparticles[END_REF] The slower components ( 2 and  3 ) are attributed to charge recombination between PB6 -and NiO holes, as has been previously observed for NiO-PB6 films. [START_REF] Tian | Ultrafast dye regeneration in a core-shell NiO-dye-TiO2 mesoporous film[END_REF] Note that the kinetics on dye-semiconductor interfaces is in general heterogeneous. Therefore, injection and recombination on NiO often occurs on overlapping time scales, and the same process may be represented by more than one time constant. [START_REF] Morandeira | Photoinduced ultrafast dynamics of coumarin 343 sensitized p-type-nanostructured NiO films[END_REF][START_REF] Zhang | Solid state p-type dye-sensitized solar cells: concept, experiment and mechanism[END_REF] Consequently, the time constant values reported here should not be taken as precise measures of the lifetime of a species or process. The PB6 - band at 620 nm is less pronounced in this data than in ref. [START_REF] Tian | Ultrafast dye regeneration in a core-shell NiO-dye-TiO2 mesoporous film[END_REF] ; we tentatively explain this change by use of a different NiO preparation here (see experimental section in SI) which leads to more overlap between the slow components of excited state hole transfer and the rapid components of recombination in the present cases. Since we used a different NiO preparation than our prior work, we also performed fsTAS on NiO-PB6 electrodes to more reliably compare them to the NiO-Sil-PB6 electrodes using the same preparation method for NiO. By direct comparison of the traces at 650 nm in Figure 4 it is clear that the silatrane anchor in NiO-Sil-PB6 mediates interfacial electron transfer at least as well as the carboxylic anchor in NiO-PB6, possibly even giving a somewhat faster hole injection, while the charge recombination rate constant is similar with the two anchors. The trace at 620 nm displays very similar behavior for the two dyes, showing that the rate and yield of PB6 -formation is essentially the same (Figure 4).

Electrode Stability.

To probe the stability of the silatrane-anchored electrodes we monitored the changes on the NiO-Sil-PB6 electrodes before and after photoelectrolysis at 0.1 V vs. NHE for one hour in 0.1 M MES buffer at pH 5 without the chloropentaaminecobalt chloride electron acceptor. The UV-Vis spectra were unchanged after photoelectrolysis (Figure S8), suggesting that the PB6 structure was retained on the electrode. In addition, no leaching of PB6 into the electrolyte was observed (Figure S8), supporting a stable binding of PB6-Sil to the electrode. ATR-FTIR spectra post electrolysis also remained unchanged (Figure S9), showing that the silatrane had remained surface bound and that the PB6 structure had not significantly changed. Furthermore, ICP-OES measurements showed that a majority of the silatrane had been retained on the electrode post photoelectrolysis, further confirming the good stability of this anchoring group on NiO (Figure S1). In addition, it should be noted that the stability of NiO-Sil-PB6 electrodes was similar to that of the NiO-PB6 electrodes (Figure S8, S9, and S11), which both exhibited excellent stability. There has been some debate regarding the stability of carboxylic acid anchors on NiO. [START_REF] Gibson | Dye-sensitized photocathodes for H2 evolution[END_REF] The anchor has a less predictable stability than on other metal oxide surfaces (TiO 2 , SnO 2 , ITO), which tend to retain carboxylic acids anchored molecules only(?) at pH<4. However, the binding of carboxylic acid anchors on NiO does not always follow this trend and is sometimes stable at pH>4(?), which could partially explain the stability of NiO-PB6.

Electrodes with Catalyst

Since NiO-Sil-PB6 showed promise photoelectrochemically, exhibited electron injection into NiO, and had good stability, we then assembled electrodes by amide coupling a carboxylic acid-derivatized Ni catalyst, NiCyGly (Figure 5), to show the applicability of the new assembly method in photocathodes. NiCyGly has previously been shown to have excellent activity for proton reduction catalysis in water over a wide pH range, [START_REF] Dutta | Amino acid modified Ni catalyst exhibits reversible H2 oxidation/production over a broad pH range at elevated temperatures[END_REF][START_REF] Dutta | Minimal proton channel enables H2 oxidation and production with a water-soluble nickel-based catalyst[END_REF][START_REF] Ginovska-Pangovska | Beyond the active site: the impact of the outer coordination sphere on electrocatalysts for hydrogen production and oxidation[END_REF][START_REF] Oughli | Dual properties of a hydrogen oxidation Ni-catalyst entrapped within a polymer promote self-defense against oxygen[END_REF][START_REF] Priyadarshani | Achieving Reversible H2/H+ Interconversion at Room Temperature with Enzyme-Inspired Molecular Complexes: A Mechanistic Study[END_REF] having an onset potential for proton reduction catalysis at -0.808 V vs. FeCp 2 +/0 at pH 5 under aqueous conditions. [START_REF] Dutta | Minimal proton channel enables H2 oxidation and production with a water-soluble nickel-based catalyst[END_REF] Figure 5. NiCyGly catalyst.

The catalyst-incorporated electrodes were prepared via a co-immobilization method, where NiO-Sil electrodes were soaked in a solution containing both PB6

and NiCyGly in either 2:1 or 1:1 PB6:NiCyGly ratio in the presence of EDC (experimental details in SI). The UV-Vis spectra of the sensitized films show a slight change in surface PB6:NiCyGly ratio as the PB6:NiCyGly ratio is decreased (Figure S7), as seen by the decrease in PB6 absorbance at 530 nm, suggesting the ratios on the surface had been changed. NiCyGly has peaks in the visible region of the spectrum (~306, 535 nm), but are masked due to the high extinction coefficient of the PB6 dye (Figure S4) and scattering of NiO below 400 nm (Figure S5). Even the NiO-Sil-NiCyGly electrodes without PB6 do not show any UV-Vis absorption features from NiCyGly, which was expected due to the thick, highly absorbing NiO film (Figure S5). However, the decrease in PB6 absorbance in the photocathodes´ UV-Vis spectra with changing PB6:Ni ratio (Figure S7) indirectly suggests that NiCyGly can bind to the NiO electrode and occupy surface sites that were once occupied by PB6 in the dye only electrodes, so lowering the overall PB6 absorbance.

To verify that NiCyGly could bind to NiO-Sil, we used the depletion method to screen for NiCyGly-Sil binding. This was done by monitoring a NiCyGly sensitization solution before and after soaking a NiO-Sil electrode overnight in the presence of EDC; a decrease in solution UV-Vis absorption was observed supporting that some of the NiCyGly in solution had bound to the NiO-Sil electrode surface (Figure S5). In addition, the ATR-FTIR spectra of the NiO-Sil-NiCyGly show the presence of the silatrane (~1070 cm -1 ), indicating that Sil is retained on the surface. Taken together with the depletion method results, this suggests that NiCyGly is likely amide coupled to Sil. Finally, actual surface PB6:NiCyGly ratios were estimated (SI for details) and suggested that samples prepared with a 2:1 sensitization solution resulted in electrodes with a 3:1 surface ratio (labeled as NiO-Sil-PB6-NiCyGly-(3:1)), and those prepared with a 1:1 sensitization solution resulted in a 2:1 surface ratio (labeled as NiO-Sil-PB6-NiCyGly-(2:1)).

Since NiO-Sil-PB6 showed promising photocurrents in the presence of an irreversible electron acceptor, we performed photoelectrochemical measurements on the catalyst-containing photocathodes in 0.1 M MES buffer at pH 5 to test their response to light. With both linear sweep voltammetric (LSV) and chronoamperometric (CA) measurements a higher photocurrent was seen when the catalyst was present on the electrode, suggesting catalyst reduction by PB6 -(Figure 6, S12). In addition, LSVs were also collected of several control samples (NiO, NiOacidified, NiO-Sil, and NiO-Sil-NiCyGly). None showed a response to light, suggesting that neither the NiO surface, NiO-acidified surface, Sil, or NiCyGly alone was responsible for the photocurrent in the fully assembled electrodes (Figure S13), but that it was generated through the photoexcitation of PB6. We will focus on the CA results here, as they showed a more reliable steady-state photocurrent than the LSV results (see SI).

Chronoamperometry (CA) was performed on the photocathodes using a 0.1 V vs. NHE bias while under illumination (Figure 6). All samples with NiCyGly show an increase in photocurrent compared to the NiO-Sil-PB6 sample, suggesting catalyst reduction by PB6 -was occurring, while the current from the sample without catalyst, NiO-Sil-PB6, is essentially negligible under steady-state conditions.

Moreover, higher absolute photocurrent density (|J|) was observed with the 2:1 ratio (|J| 20min ~ 12 µA cm -2 ) compared to the 3:1 ratio (|J| 20min ~ 2 µA cm -2 ) under steadystate conditions suggesting this was the better surface ratio. This trend in photocurrent was reproducible when the CAs were repeated. All samples show a relatively stable photocurrent with only a ~20% decrease in photocurrent over one hour of illumination. Furthermore, we were not able to detect any NiCyGly desorption into the MES buffer after photoelectrolysis, suggesting the catalyst was retained on the electrode surface (Figure S8). In fact, ICP-EOS measurements showed that a majority of the silatrane had been retained on the electrode post photoelectrolysis, highlighting the excellent stability of the anchoring group on NiO even with a catalyst present (Figure S1). S5). Direct comparison of the lifetimes in multiexponential fits has great uncertainties as different kinetic traces could overlap and effect each other's time constants. However, in this case, the  2 component is more than 100 times faster than  3 and  infinite only has a minor amplitude (Figure S19). Therefore, we believe that the difference between the  3 component in the samples is significant and indicates electron transfer from PB6 -to the catalyst. In addition, we would have expected  2 to be shortened upon addition of catalyst.

However, the  2 components are quite similar between the samples. To explain this similarity, it is important to recall that the PB6 -signal observed on our samples is less distinct than our prior work. [START_REF] Tian | Ultrafast dye regeneration in a core-shell NiO-dye-TiO2 mesoporous film[END_REF] Because of this weakened signal, we propose that a large fraction of the excited state decay is included in the  2 component, which are not affected by the addition of the catalyst, thus explaining the similarity in  2 between the samples. Furthermore, the traces at 620 nm show that the maximum signal is much smaller in the presence of catalyst (Figure 7), and the differences in amplitude are much larger than the differences in dye loadings. This would suggest that at least a part of the reduced dye molecules transfers an electron to the catalyst on a time scale of just a few ps. This is similar to what was found in a system with the Coumarin 343 dye and a biomimetic Fe-Fe catalyst on NiO. [START_REF] Antila | Dynamics and Photochemical H2 Evolution of Dye-NiO Photocathodes with a Biomimetic FeFe-Catalyst[END_REF][START_REF] Brown | Ultrafast Electron Transfer Between Dye and Catalyst on a Mesoporous NiO Surface[END_REF] It should be noted that excited and reduced PB6 give similar ground state bleaches, and both species absorb at 650 nm. This is reflected by the fact that the decay of the 650 nm absorption and the 530 nm bleach nearly mirror each other (Figure S17). Moreover, electron transfer to the catalyst is expected to show a broad distribution of kinetics, and is the case for hole injection and charge recombination. To summarize, we see ultrafast injection and indications of electron transfer to the catalyst on the sub-ns timescale. The small signals of PB6 -compared to PB6 * and the competing kinetics of charge recombination and electron transfer likely contribute to the observed less than 100% quantum yield for electron transfer to the catalyst. This is consistent with the low photocurrents (µAs) we observed in the photoelectrochemical investigation with catalyst present, which are considered less than ideal compared to the photocurrents observed from photoanodes (approaching mA). [START_REF] Brennaman | Finding the Way to Solar Fuels with Dye-Sensitized Photoelectrosynthesis Cells[END_REF] Furthermore, the electrodes were also screened for photocatalytic proton reduction, but did not produce any detectable H 2 (SI for details). For catalysis to occur, an additional two protons and one electron must be added, resulting in more opportunities for charge recombination, further explaining the low photocurrents and lack of H 2 .

Conclusions

Here we reported a surface amide coupling method for photocathode assembly. This method is advantageous since it avoids tedious organic syntheses and also provides a facile screening method for different photocathode architectures.

By varying the carboxylic acid and amine-derivatized molecules used to assemble the electrodes via amide couplings, a wide variety of photocathode architectures can be assembled and tested. In addition, to the best of our knowledge, we have reported the first NiO-based electrodes that contain silatrane-anchored molecules, which additionally showed good photoelectrochemical stability. Finally, we showed the applicability of the surface amide coupling method by incorporating a nickel catalyst on the photocathodes, which showed evidence for catalyst reduction by photoelectrochemical and fsTAS measurements. Clearly, there is room for improvement with these electrodes, and we are currently working towards this goal by using the surface-amide coupling method to assemble photocathodes with higher photocurrents and efficiencies.

  1 M phosphate buffer at pH 2 for one hour at room temperature. The pre-acidified electrodes were then air-dried (without rinsing), soaked in an 8 mM Sil solution in acetonitrile overnight at room temperature, and rinsed with acetonitrile three times, affording the NiO-Sil electrodes. With aminoderivatized electrodes in hand, we next performed the surface-amide coupling by soaking NiO-Sil in the presence of 0.2 mM PB6 and 2 mM 1-ethyl-3-(3dimethylaminopropyl) carbodiimide (EDC), used as the amide coupling agent in a 1:1 dichloromethane:methanol solution, affording NiO-Sil-PB6 electrodes (Scheme 1). A 2 mM concentration of EDC concentration provided the highest PB6 loadings by UV-Vis spectroscopy (Figure S2), and was used for all subsequent electrode sensitizations. It should be noted that the concentration and choice of the amidecoupling agent must be screened to optimize molecule loadings.

Figure 1 .

 1 Figure 1. Structures of Sil and PB6 used to assemble the amide coupled electrodes.

Figure 2 .

 2 Figure 2. UV-Vis (A) and ATR-FTIR (B) spectra of NiO-Sil-PB6 electrodes (blue) and NiO-PB6 electrodes (pink). PB6 powder ATR-FTIR is shown in grey in panel (B).

Figure 3 .

 3 Figure 3. NiO-Sil-PB6 electrodes with (pink) and without (grey) a 10 mM chloropentaaminecobalt chloride electron acceptor (Co IEA) in 0.1 M MES buffer at pH 5 with a scan rate of 5 mV/s. LSV were scanned from positive to negative potentials with the light chopped every 50 mV. Samples were illuminated using a white LED with 100 mW cm -2 .

Figure 4 .

 4 Figure 4. (A) Transient absorption spectra of NiO-Sil-PB6 excited at 560 nm scattering of the pump light occurs between 550-570 nm. Traces at several time points are depicted in units of ps. (B) kinetic traces at 650 nm comparing NiO-PB6 (black) and NiO-Sil-PB6 (pink). (C) kinetic traces at 620 nm comparing NiO-PB6 (black) and NiO-Sil-PB6 (pink).

  Kinetic traces at 650 nm show ultrafast hole injection for NiO-Sil-PB6 electrodes with the silatrane anchor, which has a similar decay to NiO-PB6 without a silatrane present (Figure4). Since Sil has a propyl group, theoretically placing PB6 further from the NiO electrode, a slower hole injection relative to the NiO-PB6 sample would have been expected. The fast decay observed with NiO-Sil-PB6 can be explained by the flexibility of the propyl group of Sil; this flexibility could move PB6 closer to the NiO surface, thus allowing for ultrafast hole injection into the NiO and also increase recombination pathways between PB6 -and NiO(h + ).

Figure 6 .

 6 Figure 6. CA of photocathodes containing NiCyGly catalysts: NiO-Sil-PB6-NiCyGly-(3:1) (blue), NiO-Sil-PB6-NiCyGly-(2:1) (pink). NiO-Sil-PB6 sample is shown in black for reference. A bias of 0.1 V vs. NHE was applied during these measurements in 0.1 M MES at pH 5. Samples were illuminated (100 mW cm -2 , white LED) for one hour as indicated by the arrows in chronoamperogram.

Figure 7 .

 7 Figure 7. fsTAS traces at 620 nm for NiO-Sil-PB6 (black), NiO-Sil-PB6-NiCyGly-(3:1) (pink) and NiO-Sil-PB6-NiCyGly-(2:1) (blue).

Acknowledgements

We thank the Swedish Energy Agency (Grant no. 11674-8) for funding this research.

We would also like to thank Dr. Bo Xu for helping prepare the NiO paste for these experiments. We also thank Ashleigh Castner for her help with the ICP-OES experiments. We thank Lei Tian for his advice throughout the project. We thank Dr.

Jens Föhlinger and Jonas

Petterson for writing the MATLAB script for the global fitting. We also thank Dr. Palas Baran Pati for synthesizing the PB6 dye. Finally, we thank Dr. Brian McCarthy for useful discussions and help editing the paper. WJS, APW, and JAL were supported by the U.S. Department of Energy (USDOE), Office of Science, Office of Basic Energy Sciences (BES), Chemical Sciences, Geosciences, and Biosciences program.

Associated Content

Supporting Information: Further experimental details on the assembly and characterization of the photocathodes can be found in the SI. Dye and catalyst loadings, additional UV-Vis spectra, ATR-FTIR spectra, photoelectrochemical data, fs-TAS data, and ICP-EOS data can be found in the SI.

Author Information

Corresponding author: leif.hammarstrom@kemi.uu.se Notes: We declare no competing financial interest.