Frédéric Mazenc
email: frederic.mazenc@l2s.centralesupelec.fr

Michael Malisoff
email: malisoff@lsu.edu

Corina Barbalata
email: cbarbalata@lsu.edu

Zhong-Ping Jiang
email: zjiang@nyu.edu

Z.-P Jiang

Subpredictor Approach for Event-Triggered Control of Discrete-Time Systems with Input Delays

Keywords: Delay systems, event-triggered, positive systems, chain prediction

We propose a new output event-triggered control design for linear discrete-time systems with constant arbitrarily long input delays, using delay compensating subpredictors. We prove input-to-state stability of the closed loop system, using framers and the theory of positive systems. A novel feature of our approach is our use of matrices of absolute values, instead of Euclidean norms, in our discrete-time event triggers for our delay compensating control design. We illustrate our approach using a model of the BlueROV2 marine vehicle, where our new event triggers lead to a smaller number of control recomputation times as compared with standard event triggers that were based on Euclidean norms, without sacrificing on settling times or on other performance metrics.

I. INTRODUCTION

 covered undelayed discrete-time linear systems with outputs, here we solve a complementary problem of compensating for input delays in event-triggered discrete-time linear systems.

, such triggers can lead to less conservative lower bounds on the intersample times as compared with the corresponding triggers that would arise from using standard Euclidean norms; see Section V. This is advantageous for applications that call for taking communication constraints into account, by reducing the number of time instants when control values are changed.

The need to reduce the number of times when control values are changed has led to a large literature on eventtriggered regimes that address the computational challenges

that are created by controls whose values are changed unnecessarily often; see, e.g., the survey [START_REF] Heemels | An introduction to eventtriggered and self-triggered control[END_REF]. Other notable and significant works include [START_REF] Brunner | Robust eventtriggered MPC with guaranteed asymptotic bound and average sampling rate[END_REF], [START_REF] Durand | General formula for event-based stabilization of nonlinear systems with delays in the state[END_REF], [START_REF] Heemels | Analysis of eventdriven controllers for linear systems[END_REF], [START_REF] Liu | Dynamic event-triggered control for linear time-invariant systems with L 2 -gain performance[END_REF], [START_REF] Liu | Robust Event-Triggered Control of Nonlinear Systems[END_REF], [START_REF] Peralez | Event-triggered output feedback stabilization via dynamic high-gain scaling[END_REF], and [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]. Designing event-triggered control mechanisms entails the co-design of feedback controls and event triggering rules that indicate which events call for changing control values. In both discrete-and continuous-time cases, these events are usually specified as times when a measurement enters a specified region of the measurement space that is defined in terms of the usual Euclidean norm. This differs from standard zero-order hold controls whose control recomputation times are independent of the measurements from the systems.

Another main ingredient of our approach is positive systems, which are dynamics for which the nonnegative orthant is positively invariant. The literature on positive systems provided new control methods to help overcome technical challenges of using standard Lyapunov functionals for time delay systems. Positive systems have been used in conjunction with interval observers (as defined, e.g., in [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF] and [START_REF] Raissi | Interval state estimation for a class of nonlinear systems[END_REF]), which yield intervals containing values of unknown states when the inequalities are viewed as being componentwise; see [START_REF] Mazenc | Stability analysis for time-varying systems with delay using linear Lyapunov functionals and a positive systems approach[END_REF]. Dynamical systems theory based on interval observers and positive systems has led to notable contributions in aerospace engineering, mathematical biology, and other applications.

By providing a new design for event-triggered control for linear discrete-time systems with outputs and arbitrarily long constant input delays, we believe that this work is the first to use positive systems and interval observers to prove input-to-state stability (or ISS) for discrete-time systems with outputs whose delays are compensated by subpredictors. In particular, our work is novel relative to notable works on predictors that did not involve event triggering, such as [START_REF] Cacace | Stabilization of strict-feedback nonlinear systems with input delay using closed-loop predictors[END_REF], [START_REF] Cacace | Output feedback control of linear systems with input, state and output delays by chains of subpredictors[END_REF], [START_REF] Cacace | Predictor-based control of linear systems with large and variable measurement delays[END_REF], and [START_REF] Mazenc | Sequential subpredictors for delay compensation for discrete time systems with time-varying delays[END_REF]. For discrete-time systems, we do not have to rule out the possibility of Zeno's phenomenon. Hence, it may be worth discretizing a continuous time system and to next use event triggered control, instead of using event triggered control for continuous-time systems. Our method uses the structure of our dynamics to provide an alternative to small-gain methods (e.g., from [START_REF] Jiang | Nonlinear small-gain theorems for discrete-time feedback systems and applications[END_REF] and [START_REF] Jiang | Small-gain theory for stability and control of dynamical networks: A survey[END_REF]) and so can lead to less frequent control computations.

After presenting our definitions and notation and new theorem in Section II, followed by our main lemmas in Section III and the proof of our theorem in Section IV, Section V illustrates how discretized continuous time systems are amenable to our new theorem. Section V also applies our work to a model of the BlueROV2 underwater vehicle, where our input delay compensation results complement our results for this dynamics in undelayed cases in [START_REF] Mazenc | Event-triggered control for discrete-time systems using a positive systems approach[END_REF]. The example illustrates how our new event triggers can lead to fewer control recomputation times, compared with standard event triggers that were based on the Euclidean norm, without sacrificing the settling times or other performance metrics. Since the BlueROV2 is widely used for the study of corals and for other environmental surveys, this illustrates the value of our approach in a significant example from ecological robotics, where there is strong motivation for reducing the number of control recomputation times. We close in Section VI with a summary and our ideas for follow up research.

II. DEFINITIONS, NOTATION, AND MAIN RESULT

We use standard notation, which we simplify when no confusion would arise. The dimensions of our Euclidean spaces are arbitrary, unless we indicate otherwise. We omit arguments of functions when no confusion would arise. We set

Z 0 = {0, 1, 2, ...}. For a matrix G = [g ij ] ∈ R r×s , we set |G| = [|g ij |]
, so the entries of |G| are the absolute values of the corresponding entries g ij of the matrix G. We use ||•|| to denote the usual Euclidean norm. By G + , we denote the matrix whose entries are max{0, g ij } and G -= G + -G. A square matrix is called Schur stable provided its spectral radius is in [0, 1). For matrices D = [d ij ] and E = [e ij ] of the same size, we write D < E (resp., D ≤ E) provided d ij < e ij (resp., d ij ≤ e ij ) for all i and j. We also use D E to mean that there is a pair (i, j) such that d ij > e ij . We adopt similar notation for vectors. We call a matrix S positive (resp., nonnegative) provided 0 < S (resp., 0 ≤ S), where 0 is the zero matrix. We use standard notions of ISS for discrete-time systems [START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF], and I is the identity matrix.

We consider the system with outputs

x k+1 = Ax k + Bu k-r + δ k y k = Cx k (1)
with k ∈ Z 0 , x valued in R n , the control u (that we will specify in our theorem) valued in R p , the output y valued in R q , the unknown sequence δ k ∈ R n representing a disturbance, and the constant integer r ≥ 1 representing an input delay that will arise in the measurements that enter the control u. A key feature of our work is that we allow r to be any positive integer. This arbitrarily long input delay is compensated for by our chain of subpredictors. Throughout this work, we make these three assumptions: Assumption 1: There is a matrix K ∈ R p×n such that

H = A + BK (2)
is nonnegative. There is a matrix Γ ∈ R n×n such that

H + |BK|Γ (3)
is Schur stable and such that Γ > 0. Assumption 2: There is a matrix L ∈ R n×q such that A + LC is Schur stable.

Assumption 3: The matrix A is invertible. See Remark 1 for the motivation for Assumptions 1-3 and the ease with which we can often check that they are satisfied. To specify u, fix matrices K, L, and Γ satisfying Assumptions 1-2 and a matrix R ∈ R n×n such that A + R is Schur stable (e.g., R = -A). We then use the r subpredictors

                     z 1,k+1 = Az 1,k + Bu k-r+1 + ALCA -1 z 1,k -ALy k -ALCA -1 Bu k-r z 2,k+1 = Az 2,k + Bu k-r+2 + R(z 2,k -Az 1,k -Bu k-r+1 ) . . . z r,k+1 = Az r,k + Bu k + R(z r,k -Az r-1,k -Bu k-1 ) (4)
where each z i,k is valued in R n . We also use the control law

u k-r = Kz r,σ(k)-r (5)
with K in Assumption 1 and σ : Z 0 → Z 0 defined by

σ(0) = 0 σ(j + 1) = j + 1 if |z r,j+1-r -z r,σ(j)-r | Γ|z r,j+1-r | σ(j + 1) = σ(j) if |z r,j+1-r -z r,σ(j)-r | ≤ Γ|z r,j+1-r |. (6)
See Remark 3 for the motivation for our triggering rule [START_REF] Efimov | Interval observers for time-varying discrete-time systems[END_REF].

Then the corresponding closed-loop system is

x k+1 = Ax k + BKz r,σ(k)-r + δ k . (7)
In terms of the notation

e i,k = z i,k -z i-1,k+1 for i = 1, . . . , r, E k = [e 1,k , . . . , e r,k ] , and z 0,k = x k , (8)
where the z i,k 's are the states of (4), our main result is: Theorem 1: Let (1) be such that Assumptions 1-3 are satisfied. Choose any matrices K, L, and Γ that satisfy our assumptions, and any matrix R such that A + R is Schur stable. Then we can find real constants ci > 0 for i = 1, 2, 3 such that all solutions of the closed-loop system [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF] with the control ( 5)-( 6) defined by the subpredictors (4) satisfy

||x k || ≤ c1 e -c2(k-k0) (||x k0 || + ||E k0-r ||) c3 sup q∈{k0-r,...,k-1} ||δ q || (9)
for all k 0 > r and all integers k ≥ k 0 . Remark 1: Assumptions 1-3 are easily checked in many cases. For instance, Assumption 1 is not very restrictive. Indeed, when a pair (A a , B a ) is controllable, we can choose a matrix K a such that A a +B a K a is Schur stable, having distinct real eigenvalues on the interval (0, 1). Then, there is an invertible matrix P ∈ R n×n such that P (A a +B a K a )P -1 is Schur stable and nonnegative (by diagonalizing A a +B a K a ). Consequently, if we consider

Z k+1 = A a Z k + B a u + δ a,k , then the change of coordinates X k = P Z k gives X k+1 = AX k + Bu + P δ a,k , (10)
with A = P A a P -1 and B = P B a , and (10) satisfies Assumption 1, because the choice K = K a P -1 gives

A + BK = P A a P -1 + P B a K a P -1 = P (A a + B a K a )P -1 (11)
which is both Schur stable and nonnegative, and because when the matrix H is Schur stable, there always exists a positive matrix Γ > 0 such that H + |BK|Γ is Schur stable, by choosing Γ so its entries are small enough. Assumption 2 is satisfied when the pair (A, C) is observable, which is a standard observability condition. Assumption 3 is a technical assumption, and holds in the following generic sense. The invertibility of A can be viewed as the requirement that a real analytic function defined on R n 2 takes a nonzero value, namely, the determinant of A viewed as a function of the n 2 entries of A. It follows from a standard analytic continuation argument that the set of all invertible n × n matrices form an open and dense subset of R n×n . Hence, invertibility of A is generic, in the same sense of genericity that controllable pairs (and so also observable pairs) form a generic set; see, e.g., [START_REF] Sontag | Mathematical Control Theory[END_REF]Section 3.3].

Remark 2: By contrast with the subsequential observer of [START_REF] Mazenc | Sequential predictors for delay compensation for perturbed discrete time systems[END_REF], (4) depends on y k , and not on y k+1 . This is the reason why we have to assume that A is invertible.

Remark 3: Since our focus is on the state of the dynamics (1), we did not include the actuator state on the left side of (9). Our choice of the triggering rule ( 6) was made because (as we show in our proof of our theorem) it ensures that

|z r,σ(k)-r -z r,k-r | ≤ Γ|z r,k-r | (12)
holds for all integers k ≥ 0. The preceding inequality plays an essential role in our stability analysis for our closed-loop system. It contrasts with the trigger requirement

||z r,σ(k)-r -z r,k-r || ≤ σ * ||z r,k-r || (13)
that would arise from using the usual Euclidean norm || • || and a constant σ * . In fact, the largest σ * > 0 such that all pairs (z r,σ(k)-r , z r,k-r ) satisfying (13) also satisfy ( 12) is σ * = min ij Γ ij , i.e., the smallest entry of Γ = [Γ ij ]. This property of the minimum entry of Γ was shown in our study [START_REF] Mazenc | Event-triggered control for discrete-time systems using a positive systems approach[END_REF] of the undelayed case, and the same reasoning applies in the delayed case that we consider here. In Section V-B, we illustrate how using our new condition [START_REF] Jiang | Small-gain theory for stability and control of dynamical networks: A survey[END_REF] can lead to fewer event triggers on given time horizons as compared to the number of event triggers that we would have required if we had instead used the traditional trigger [START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF] with the corresponding least conservative choice σ * = min ij Γ ij .

III. KEY LEMMAS TO PROVE THEOREM 1

Our first lemma is: Lemma 1: With the preceding assumptions and notation, we can construct matrices Ω and Λ such that the equality

E a+b = Ω b E a + b-1 p=0 Ω b-p-1 Λ δ a+p δ a+p+1 (14)
is satisfied for all integers a > 0 and b > 0.

Proof: First, let us observe that

ALy k + ALCA -1 Bu k-r = ALC(x k +A -1 Bu k-r ) = F (x k+1 -δ k ), (15)
where F = ALCA -1 . Here and in the sequel, all equalities are for all k ∈ Z 0 , unless otherwise indicated. It follows that

z 1,k+1 = Az 1,k + Bu k-r+1 + F z 1,k -F x k+1 + F δ k , which can be rewritten as z 1,k+1 = Az 1,k + Bu k-r+1 + F e 1,k + F δ k . (16)
Next, consider the z 2 -subsystem of (4). Since [START_REF] Mazenc | Stability analysis for time-varying systems with delay using linear Lyapunov functionals and a positive systems approach[END_REF] gives

-Az 1,k -Bu k-r+1 = -z 1,k+1 + F e 1,k + F δ k , we have z 2,k+1 = Az 2,k +Bu k-r+2 +Re 2,k +RF e 1,k +RF δ k . ( 17
)
Similarly, we can prove (by induction on a) that

z a,k+1 = Az a,k + Bu k-r+a + Re a,k + R 2 e a-1,k + .... + R a-1 e 2,k + R a-1 F e 1,k + R a-1 F δ k (18)
for a = 2 to r. Hence, from ( 16), we deduce that

e 1,k+1 = Az 1,k + Bu k-r+1 + F e 1,k + F δ k -[Ax k+1 + Bu k-r+1 + δ k+1 ] = (A + F ) e 1,k + F δ k -δ k+1 = M e 1,k + F δ k -δ k+1 , (19)
where

M = A(A + LC)A -1 .
Therefore, from ( 16)-( 17), we deduce that

e 2,k+1 = A(z 2,k -z 1,k+1 ) + Re 2,k + RF e 1,k + RF δ k -F e 1,k+1 -F δ k+1 = (A + R)e 2,k + RF e 1,k -F e 1,k+1 + RF δ k -F δ k+1 = (A + R)e 2,k + (RF -F M )e 1,k + (RF -F 2 )δ k . (20)
Then [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF] gives e a,k+1 = z a,k+1 -z a-1,k+2

= (A + R)e a,k + R 2 e a-1,k + .... +R a-1 e 2,k + R a-1 F e 1,k + R a-1 F δ k -Re a-1,k+1 + R 2 e a-2,k+1 + .... + R a-2 e 2,k+1 + R a-2 F e 1,k+1 + R a-2 F δ k+1 (21)
for a = 3 to r, and so also

e a,k+1 = (A + R)e a,k + R 2 e a-1,k + .... + R a-1 e 2,k + R a-1 F e 1,k -(Re a-1,k+1 +R 2 e a-2,k+1 + .... + R a-2 e 2,k+1 -R a-2 F (M e 1,k + F δ k -δ k+1 ) +R a-1 F δ k -R a-2 F δ k+1 (22)
for a = 3 to r, where the last equality is a consequence of the last equality in [START_REF] Mazenc | Sequential subpredictors for delay compensation for discrete time systems with time-varying delays[END_REF]. We deduce that e a,k+1 = (A + R)e a,k + R 2 e a-1,k + ....

+ R a-1 e 2,k + R a-1 F e 1,k -Re a-1,k+1 + R 2 e a-2,k+1 + .... + R a-2 e 2,k+1 -R a-2 F M e 1,k + (R a-1 -R a-2 F )F δ k . (23)
It now follows from [START_REF] Mazenc | Sequential subpredictors for delay compensation for discrete time systems with time-varying delays[END_REF], ( 20) and ( 23) that there is a matrix

Ω of the form Ω =          M 0 . . . . . . 0 A + R . . . 0 . . . . . . . . . . . . . . . . . . 0 . . . . . . A + R          . ( 24
)
and a matrix Λ such that

E k+1 = ΩE k + Λ δ k δ k+1 (25)
with E j defined in [START_REF] Haddad | Nonnegative and Compartmental Dynamical Systems[END_REF]. Therefore, for all integers a and b such that a > 0 and b > 0, we can argue by induction on b to show that the equality ( 14) is satisfied.

Using our definitions [START_REF] Haddad | Nonnegative and Compartmental Dynamical Systems[END_REF] of the e i,k 's and the formulas (8) for the E k 's, we can also prove the following:

Lemma 2: With the preceding notation with Ω and Λ satisfying the requirements from Lemma 1, we have

z r,k = x k+r + ξ k , where ξ k = r-1 j=0 e r-j,k+j (26)
for all k ≥ 0. Also, if we let V i ∈ R n×(rn) denote the matrix [V (i,1) , . . . , V (i,r) ] where V (i,j) = 0 ∈ R n×n if j = i and V (i,i) = I ∈ R n×n , and if we choose

W = r-1 j=0 V r-j Ω j , (27)
then the following holds for all k ≥ 1:

ξ k = W E k + r-1 j=1 V r-j j-1 p=0 Ω j-p-1 Λ δ k+p δ k+p+1 (28) if r > 1 and ξ k = V 1 E k (29) if r = 1.
Proof: Condition (26) follows from a telescoping sum argument. Also, we can rewrite the ξ k formula in [START_REF] Sontag | Mathematical Control Theory[END_REF] as

ξ k = r-1 j=0 V r-j E k+j . (30)
It follows from ( 14) and our choice [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF] of W that

ξ k = V r E k + r-1 j=1 V r-j Ω j E k + j-1 p=0 Ω j-p-1 Λ δ k+p δ k+p+1 = W E k + r-1 j=1 V r-j j-1 p=0 Ω j-p-1 Λ δ k+p δ k+p+1 (31)
if r > 1, and that (29) holds if r = 1.

IV. PROOF OF THEOREM

We now use the preceding two lemmas to prove Theorem 1. The proof has three parts. In the first part, we provide a new representation of the closed-loop system, to facilitate our analysis. In the second part, we build a framer for the state of the closed-loop system. In the final part, we use the framer to provide a stability analysis to prove the ISS result.

First Part: New Representation of Closed-Loop System. The system (7) can be rewritten as

x k+1 = Ax k + BKz r,k-r + BK(z r,σ(k)-r -z r,k-r ) + δ k . ( 32
)
From the equality (26), we deduce that

x k+1 = Ax k + BK(x k + ξ k-r ) + BK(z r,σ(k)-r -z r,k-r ) + δ k (33)
for all k ≥ r. Then, with H defined in (2), the representation

x k+1 = Hx k + BK(z r,σ(k)-r -z r,k-r ) + BKξ k-r + δ k (34)
is obtained.

Second Part: Framer Construction. With a view to the stability analysis of the closed-loop system, and using our notation from Section II, we introduce the dynamic extension

                 x k+1 = Hx k + (BK) + (z r,σ(k)-r -z r,k-r ) + +(BK) -(z r,σ(k)-r -z r,k-r ) - +(BKξ k-r ) + + δ + k x k+1 = Hx k -(BK) + (z r,σ(k)-r -z r,k-r ) - -(BK) -(z r,σ(k)-r -z r,k-r ) + -(BKξ k-r ) --δ - k (35)
with k ≥ r. Let k 0 > r. We have

x k+1 = Hx k + (BK) + (z r,σ(k)-r -z r,k-r ) + +(BK) -(z r,σ(k)-r -z r,k-r ) - -(BK) + (z r,σ(k)-r -z r,k-r ) - -(BK) -(z r,σ(k)-r -z r,k-r ) + +(BKξ k-r ) + -(BKξ k-r ) -+δ + k -δ - k . (36)
It follows that

x k+1 -x k+1 = H(x k -x k ) +(BK) + (z r,σ(k)-r -z r,k-r ) - +(BK) -(z r,σ(k)-r -z r,k-r ) + +(BKξ k-r ) -+ δ - k x k+1 -x k+1 = H(x k -x k ) +(BK) + (z r,σ(k)-r -z r,k-r ) + +(BK) -(z r,σ(k)-r -z r,k-r ) - +(BKξ k-r ) + + δ + k . ( 37
)
Since the matrix H is nonnegative, using (37), one can prove by induction that if x k0 -x k0 and x k0 -x k0 ≥ 0, then x k -x k ≥ 0 and x k -x k ≥ 0 for all k ≥ k 0 . Thus

x k ≤ x k ≤ x k for all k ≥ k 0 . (38)
Similarly, if x k0 ≥ 0 and x k0 ≤ 0 then

x k ≥ 0 and x k ≤ 0 (39)
for all k ≥ k 0 , again by induction. It follows that (35) satisfies the requirements to be a framer for (34). The framer (35) is inspired by those constructed, e.g., in [START_REF] Efimov | Interval observers for time-varying discrete-time systems[END_REF].

Third Part: Stability Analysis. We use the interval observer (35) to establish stability properties for (34). Let us introduce

xk = x k -x k . ( 40
)
We deduce from ( 38) and (39) that

|x k | ≤ xk (41)
for all k ≥ k 0 . On the other hand, (35) gives

xk+1 = H xk + |BK||z r,σ(k)-r -z r,k-r | + |BKξ k-r | + |δ k |. ( 42
)
Also, by (39), we have xk ≥ 0 for all k ≥ k 0 . Also, we deduce from the definition of σ in ( 6) that

xk+1 ≤ H xk + |BK|Γ|z r,k-r | + |BKξ k-r | + |δ k | (43) for all k ≥ r. To see why, notice that if |z r,σ(k)-r -z r,k-r | Γ|z r,k-r |, then σ(k) = k, so the definition of σ gives σ(k) = σ(k -1), so |z r,σ(k-1)-r -z r,k-r | Γ|z r,k-r |, so (6) with the choice j = k-1 gives σ(k) = k, which is a contradiction.
Let us take k 0 > r. Again using [START_REF] Sontag | Mathematical Control Theory[END_REF], we obtain

xk+1 ≤ H xk + |BK|Γ|x k + ξ k-r | +|BKξ k-r | + |δ k | ≤ H xk + |BK|Γ|x k | + |BK|Γ|ξ k-r | +|BKξ k-r | + |δ k | (44)
for all k ≥ k 0 , by (43). From (41), we deduce that

xk+1 ≤ (H + |BK|Γ)x k + |BK|Γ|ξ k-r | +|BKξ k-r | + |δ k | ≤ κ 1 xk + κ 2 |ξ k-r | + |δ k |, (45)
where κ 1 = H + |BK|Γ and κ 2 = |BK|(Γ + I). On the other hand, from (28), it follows that

ξ k-r = W E k-r + r-1 j=1 V r-j j-1 p=0 Ω j-p-1 Λ δ k-r+p δ k-r+p+1 (46) when k ≥ k 0 and r > 1. Hence, xk+1 ≤ κ 1 xk + κ 2 |W ||E k-r | + |δ k | + κ 2 r-1 j=1 V r-j j-1 p=0 Ω j-p-1 Λ δ k-r+p δ k-r+p+1
(47) when r > 1. Thus, when k ≥ k 0 , we have:

xk+1 ≤ κ 1 xk + κ 2 |W ||E k-r | + |δ k | + κ 2 r-1 j=1 V r-j j-1 p=0 Ω j-p-1 Λ δ k-r+p δ k-r+p+1 E k-r+1 = ΩE k-r + Λ δ k-r δ k-r+1 . (48)
These equalities can be written in the compact form

xk+1 E k-r+1 ≤ Υ xk E k-r + Θ(δ k-r , ..., δ k ), (49) where
Υ = κ 1 κ 2 |W | 0 Ω and Θ(δ k-r , ..., δ k ) =      κ 2 r-1 j=1 V r-j j-1 p=0 Ω j-p-1 Λ δ k-r+p δ k-r+p+1 + |δ k | Λ δ k-r δ k-r+1      (50) when k > r > 1. Hence, when k > l > r, we have xk E k-r ≤ Υ k-l xl E l-r + k-l-1 p=0 Υ k-l-p-1 Θ(δ l-r+p , ..., δ l+p ), (51)
by an induction argument that is similar to the one that gave [START_REF] Liu | Dynamic event-triggered control for linear time-invariant systems with L 2 -gain performance[END_REF]. Assumption 1 ensures that κ 1 is Schur stable. The matrix Ω is Schur stable too because A + R and A + LC are Schur stable (by Assumption 2). It follows that the matrix Υ is Schur stable. Therefore there are constants β i > 0 such that ||Υ || ≤ β 1 e -β2 for all integers ≥ 0. Then, we deduce from (51) that

xk E k-r ≤ β 1 e -β2(k-l) xl E l-r + k-l-1 p=0 β 1 e -β2(k-l-p-1) ||Θ(δ l-r+p , ..., δ l+p )||. (52)
We deduce there is a constant

β 3 > 0 such that xk E k-r ≤ β 1 e -β2(k-l) xl E l-r +β 3 k-l-1 p=0 e -β2(k-l-p) sup q∈{l-r,...,k-1} ||δ q ||. (53)
Thus,

xk E k-r ≤ β 1 e -β2(k-l) xl E l-r +β 3 sup q∈{l-r,...,k-1} ||δ q ||. (54)
From (41), it follows that

||x k || 2 + ||E k-r || 2 ≤ β 1 e -β2(k-l) ||x l || 2 + ||E l-r || 2 +β 3 sup q∈{l-r,...,k-1} ||δ q ||. (55)
This, (41), and the subadditivity of the square root allow us to conclude when r > 1, since we can assume that xk

≤ 2|x k | and xk ≥ -2|x k | for all k ≤ k 0 to get ||x k0 || ≤ 4||x k0 ||.
The case r = 1 is similar. This concludes the proof.

V. ILLUSTRATIONS A. Discretization of Continuous Time Systems

Starting from a continuous time system of the form

ẋ(t) = F x(t) + Gu(t -τ ) + γ(t) (56)
where x is valued in R n , u is valued in R p , the components of γ are piecewise continuous bounded functions representing uncertainty, and τ > 0 is a constant representing a delay, we can often obtain a discretized system that is amenable to Theorem 1. To see how, we select constants r and ν > 0 such that τ = rν, and introduce the sequence s i = iν for all i ∈ Z 0 ; our choice τ = rν is motivated by the fact that we found it to be useful for modeling our marine robotic dynamics below. We restrict our attention to piecewise constant feedbacks that are defined by u(t -τ ) = u(s i-r ) if s i ≤ t < s i+1 . Then we can apply the method of variation of parameters to ẋ(t) = F x(t) + Gu(s i-r ) + γ(t) on the interval [s i , s i+1 ) to obtain

x(s i+1 ) = e νF x(s i ) +

si+1 si e F (si+1-) d Gu(s i-r ) + si+1 si e F (si+1-) γ( )d (57)
for all i ∈ Z 0 . Thus, we obtain the discrete-time system

x(s i+1 ) = A ν x(s i ) + B ν u(s i-r ) + ∆ i where A ν = e νF , B ν = ν 0 e F d G, and ∆ i = si+1 si e F (si+1-) γ( )d . ( 58
)
We can then apply Theorem 1 to (58), provided (i) (F, G) is a controllable pair, (ii) the coefficient matrix C of the output is such that (A ν , C) is observable, (iii) there are no eigenvalues λ and µ of A ν such that ν(λ-µ) is a nonzero integer multiple of 2π, and (iv) A ν is invertible, possibly after a change of coordinates as in Remark 1. This follows from the Kalman-Ho-Narendra conditions, e.g., from [26, p.102]. That way, we get a stabilizing event triggered control law for the discretetime system (58), and this feedback also ensures ISS of the system (56). As we illustrate in the next section, our use of event triggers that are based on vectors of absolute values instead of standard Euclidean norms can reduce the number of control computations without increasing oscillations or settling times, and thereby offer computational advantages.

B. BlueROV2 Marine Vehicle

We apply Theorem 1 to underwater robotic vehicles that we studied in the undelayed case in [START_REF] Mazenc | Event-triggered control for discrete-time systems using a positive systems approach[END_REF]. The dynamics are for the control of the depth and pitch degrees-of-freedom (or DOF) of the BlueROV2 vehicle, which is widely used to study corals and other ecosystems, but similar reasoning applies to similar underwater vehicles. Following [START_REF] Mazenc | Event-triggered control for discrete-time systems using a positive systems approach[END_REF], we assume that the vehicle has a Doppler Velocity Logger (or DVL) for estimating the vehicle's velocity. When close to the sea floor, the DVL typically experiences bottom lock, making it impractical to continuously change the control values, and producing an input delay. Therefore, build a control for the depth plane, using a more practical event-triggered delaycompensating sample data subpredictor approach.

As noted in [24, Equation (9.28)], after linearization and assuming that the vehicle is neutrally buoyant, we obtain the following linearized dynamics in the depth plane:

(m -X ẇ(t) ) ẇ(t) -(mx g + Z q ) q(t) -Z w w(t) -(mU + z q )q(t) = Z γs u Z and (mx g + M ẇ(t)) ẇ(t) + (I yy -M q ) q(t) -M w w(t) + (mx g U -M q )q(t) -M θ θ = M γs u M (59)
whose parameters were obtained experimentally computed and reported in [START_REF] Prestero | Verification of a Six-Degree of Freedom Simulation Model for the REMUS Autonomous Underwater Vehicle[END_REF]. We assume that the nominal surge velocity is U = 0.1m/s. This produces a two state system, whose states are the depth and pitch velocity x = [w, q] , and the control inputs u Z and u M are the force and moment required to produce motion of the vehicle. Using the parameter values and controller from [START_REF] Prestero | Verification of a Six-Degree of Freedom Simulation Model for the REMUS Autonomous Underwater Vehicle[END_REF], the system (59) takes the form ẋ(t) = F x(t) + Gu with F = -0.17742 -0.3027 0.5394 -1.4685 and G = -0.2063 -0.7629

and so is amenable to the method from Section V-A. Hence, we assume that the control is piecewise constant with a constant sample rate s, to convert the dynamics into

Z k+1 = A a Z k + B a u, where A a = e sF and B a = s 0 e F d G. (61)
This conversion to a discrete time system is strongly motivated by the fact that when implementing robotic controllers using the Robot Operating System (ROS) for any robot, the implementation must be done in discrete time.

We next show how to satisfy our Assumptions 1-3 of Theorem 1, after a change of coordinates, by finding the required matrices Γ, K, and L, where C = [START_REF] Brunner | Robust eventtriggered MPC with guaranteed asymptotic bound and average sampling rate[END_REF][START_REF] Brunner | Robust eventtriggered MPC with guaranteed asymptotic bound and average sampling rate[END_REF]. Following our analysis of the undelayed case in [START_REF] Mazenc | Event-triggered control for discrete-time systems using a positive systems approach[END_REF], we choose s = 0.5, and a matrix K a such that H a = A a + B a K a has the eigenvalues 0.25 and 0.5 to obtain the required Schur stability condition on H a , by using the command StateFeedbackGains in the Mathematica program. We can then diagonalize H a to obtain a new matrix P (A a + B a K a )P -1 = H that is both Schur stable and nonnegative. It follows that with the choices A = P A a P -1 , B = P B a , L = -[1.5, 0] , and all entries of Γ being small enough positive constants, the requirements of Theorem 1 all hold.

We found that the event triggers from Theorem 1 produced fewer control recomputation times t i , as compared with the corresponding event-trigger ||z r,k-r -z r,σ(k)-r || ≤ σ * ||z r,k-r || with σ * being the smallest entry of Γ; see Remark 3 above. We illustrate this in Fig. 1, which shows our MATLAB simulations for the depth-pitch controller of the AUV using the delay-compensating positive system subpredictor approach from Theorem 1 with r = 3. Our figure shows results for different initial states. Our simulations used the K and L from the preceding paragraphs, and Γ = 0.015 0.045 0.15 0.15 ,

which satisfied our requirements from Assumption 1 with the preceding choices of K and L. For our simulations on the time horizon of [0, 15] seconds, the event trigger from Theorem 1 produced an average of 28 trigger times t i . In Fig. 2, the behavior of the BlueROV2 is presented when the sampling rate is instead s = 0.05 seconds, and Γ is Γ = 0.015 0.045 1.95 1.95 (63) with K L as before and with the larger delay value r = 5. In this case, the event triggered 16 times for our simulation. By comparison, using the Euclidean norm in the event trigger with the same initial conditions, over the same time horizon and with σ * = 0.015 being the smallest element of Γ, the event triggered 40 times on average in our simulations. Hence, our event trigger from our theorem produced a reduction of 60% in the number of triggering times in this case. Both event triggers produced similar settling times. Therefore, this illustrates the value of our approach for decreasing the numbers of event triggers, without adversely affecting the control performance, and with larger delays. State w(0)=0.9 q(0)=0.6 w(0)=1.0 q(0)=0.45 w(0)=2.2 q(0)=-0.75 State w(0)=0.9 q(0)=0.6 w(0)=1.0 q(0)=0.45 w(0)=2.2 q(0)=-0.75 Fig. 2. Simulation for depth and pitch control of BlueROV2 with sampling rate s = 0.05 and delay r = 5

Fig. 1 .

 1 Fig. 1. Simulation for depth and pitch control of BlueROV2 with sampling rate s = 0.5 and delay r = 3.

VI. CONCLUSIONS

We proposed a new triggered control design for discretetime systems with input delays and outputs. Relative to previous methods, key novel features were our use of subpredictors, which made it possible to achieve an ISS-like result under an arbitrarily long constant input delay, and our event triggers that were based on matrices of absolute values instead of Euclidean norms. We hope to develop analogs for nonconstant delays, which could entail generalizing [19] (which did not allow event-triggering) to have the choice of the subpredictor in the control depend on the value of the delay, instead of choosing the last subpredictor in the control. Extensions to time-varying systems are also expected.

Supported by NSF Grants 2009659 (Malisoff and Barbalata) and 2009644 (Jiang)


Cédric Slostowski

Samuel Marre

Jean-Marc Bassat

Cyril Aymonier

Synthesis of cerium oxide-based nanostructures in near-and supercritical fluids

Keywords: Cerium oxide, Nanoparticles, Supercritical fluids, Continuous process, Substitution

Cerium-oxide based nanostructures attract increasing interest for their use in multiple applications. In particular, the substitution of Ce atoms by other elements with lower oxidation state is used to control oxygen vacancies within the oxide structures, which can greatly enhance the material properties for catalysis applications. Among the synthesis approaches, supercritical water (scH2O) has been proved to be an extremely efficient media for the fast and facile elaboration of pure CeO2 nanostructures, although substitution was little investigated with this method (precursors' reactivity, mechanisms, etc.).

Here, the influence of two cerium precursors -Ce(NO3)3•6H2O and Ce(NO3)6(NH4)2 -in scH2O synthesis was first investigated over the CeO2 nanostructures synthesis process. Cerium ammonium nitrate was later used as precursor for the synthesis of Ce1-x-yAxByO2-δ (A, B = Zr, Pr and/or La; 0 ≤ x, y ≤ 0.1). Supported by ICP analysis quantification, this study proposes an overview of the influences of residence time, temperature and precursors' concentration over the proportion of the substitution element in the CeO2 NCs. This work demonstrates a fast and simple process to Ce1-x-yAxByO2-δ nanostructures synthesis using scH2O synthesis.

Introduction

Over the past decades, cerium oxide based materials (pure or substituted) have been intensively studied as they have been proven to be effective materials in a wide range of applications including catalysis, oxygen storage, polishing, optics or biomedicine [START_REF] Hernandez | Structural and catalytic properties of lanthanide (La, Eu, Gd) doped ceria[END_REF][START_REF] Meher | Tuning, via counter anions, the morphology and catalytic activity of CeO2 prepared under mild conditions[END_REF][START_REF] Yao | Ceria in automotive exhaust catalysts. I. Oxygen storage[END_REF][START_REF] Abdollahzadeh-Ghom | Improvement of oxygen storage capacity using mesoporous ceria-zirconia solid solutions[END_REF][START_REF] Stanek | Atomistic simulation of CeO2 surface hydroxylation: implications for glass polishing[END_REF][START_REF] Cho | Up conversion luminescence of Yb 3+ Er 3+ codoped CeO2 nanocrystals with imaging applications[END_REF][START_REF] Renu | Development of cerium oxide nanoparticles and its cytotoxicity in prostate cancer cells[END_REF] . In particular, substituted CeO2 (substitution of Ce atoms by other elements with lower oxidation state, typically +II or +III) is used to control oxygen vacancies within the oxide structures, which can greatly enhance the material properties. Several examples of substituted ceria have been reported with various atoms including La, Pr, Zr, Al or Cu [START_REF] Reddy | Structural characterization of CeO2-MO2 (M = Si 4+ , Ti 4+ , and Zr 4+ ) mixed oxides by Raman spectroscopy, X-ray photoelectron spectroscopy, and other techniques[END_REF][START_REF] Bishop | Defects and transport in PrxCe1-xO2-δ: composition trends[END_REF][START_REF] Uslu | Synthesis and characterization of neodymium doped ceria nanocrystalline ceramic structures[END_REF] .

Most of the applications seek for synthesis processes displaying high reproducibility and advanced control over the materials characteristics, physicochemical properties and composition. Historically, homogeneous precipitation based methods remain the most reported approaches for the synthesis of cerium oxide-based materials as they are today mastered, cheap and easily operated [START_REF] Chen | Reactive cerium(IV) oxide powders by the homogeneous precipitation method[END_REF][START_REF] Zhou | Room-temperature homogeneous nucleation synthesis and thermal stability of nanometer single crystal CeO2[END_REF][START_REF] Deshpande | Synthesis and characterization of stable and crystalline Ce1-xZrxO2 nanoparticle sols[END_REF][START_REF] Sronek | New highly divided Ce-Ca-based oxyfluorides with UV-shielding properties: study of the Ce1-xCaxO2-x and Ce1-xCaxO2-x-y/2Fy series[END_REF] , [START_REF] Bishop | Defects and transport in PrxCe1-xO2-δ: composition trends[END_REF] . However, the involved chemical reactions are slow-paced and the control over materials properties is limited [START_REF] Sreeremya | Ultra-thin cerium oxide nanostructures through a facile aqueous synthetic strategy[END_REF][START_REF] Renuka | Structural characteristics of quantum-size ceria nano particles synthesized via simple ammonia precipitation[END_REF] .

In order to address these limitations, alternative synthesis methods have been studied (colloidal approaches, solvothermal reactions, use of solvents with high boiling point, etc.) [START_REF] Terribile | The synthesis and characterization of mesoporous high-surface area ceria prepared using a hybrid organic/inorganic route[END_REF][START_REF] Masui | Characterization of cerium(IV) oxide ultrafine particles prepared using reversed micelles[END_REF][START_REF] Depner | Nonhydrolytic synthesis and electronic structure of ligand-capped CeO2-delta and CeOCl nanocrystals[END_REF][START_REF] Hirano | The hydrothermal synthesis of ultrafine cerium(IV) oxide powders[END_REF] . Among them, near-and supercritical fluids mediated reactions have demonstrated their efficiency for the synthesis of cerium oxide particles with tailor-made properties (size, shape, surface chemistry, etc.) [START_REF] Hakuta | Production of ultra-fine ceria particles by hydrothermal synthesis under supercritical conditions[END_REF][START_REF] Lester | Reaction engineering: the supercritical water hydrothermal synthesis of nano-particles[END_REF][START_REF] Zhang | Colloidal ceria nanocrystals: a tailor-made crystal morphology in supercritical water[END_REF] . Several works have investigated the influence of experimental parameters (temperature, precursor, solvent, residence time, etc.) for the control of synthesized materials properties [START_REF] Veriansyah | Characterization of surface-modified ceria oxide nanoparticles synthesized continuously in supercritical methanol[END_REF][START_REF] Kim | Vegetable oil aided hydrothermal synthesis of cerium oxide nanocrystals[END_REF][START_REF] Slostowski | Near-and supercritical alcohols as solvents and surface modifiers for the continuous synthesis of cerium oxide nanoparticles[END_REF] . Furthermore, CeO2 nanoparticles synthesis in near-and supercritical fluids can be continuously processed in short time (less than 1 min), making them easily scalable [START_REF] Adschiri | Hydrothermal synthesis of metal oxide fine particles at supercritical conditions[END_REF] .

P a g e | 2

However, only few publications report on the substitution of Ce atoms by other elements in CeO2 (e.g. Ce1-xAxO2-δ, 0 ≤ x ≤ 1) via continuous supercritical fluids techniques and Ce1-xZrxO2-δ remains the only studied system in such conditions. For instance, Cabañas et al. proposed a quantification of elements in CeO2-ZrO2 mixed oxides synthesized in ncH2O (T = 300 °C and p = 25 MPa) from zirconium acetylacetonate and cerium ammonium nitrate. These analyses revealed small difference between initial precursor concentrations and Ce:Zr ratios in the final powders [START_REF] Cabañas | A continuous and clean one-step synthesis of nano-particulate Ce 1-xZrxO2 solid solutions in near-critical water[END_REF][START_REF] Cabañas | Continuous hydrothermal synthesis of inorganic materials in a near-critical water flow reactor; the one-step synthesis of nanoparticulate Ce1-xZrxO2 (x = 0-1) solid solutions[END_REF] . Kim et al. also reported the continuous synthesis of CeO2-ZrO2 mixed oxides in supercritical H2O 1 (scH2O) at T = 400 °C and p > pc, from cerium nitrate and zirconium oxynitrate [START_REF] Kim | Characteristics in oxygen storage capacity of ceria-zirconia mixed oxides prepared by continuous hydrothermal synthesis in supercritical water[END_REF][START_REF] Kim | Characteristics of CeO2-ZrO2 mixed oxide prepared by continuous hydrothermal synthesis in supercritical water as support of Rh catalyst for catalytic reduction of NO by CO[END_REF][START_REF] Kim | Ceria-zirconia mixed oxide prepared by continuous hydrothermal synthesis in supercritical water as catalyst support[END_REF] . A three injection lines process was used and pH was controlled thanks to an ammonia solution.

Similarly, zirconium acetylacetonate coupled with cerium ammonium nitrate was used by Trysted et al. for the in situ synchrotron study of the batch mode mixed oxides synthesis in scH2O (T = 375 °C and p = 23 MPa) [START_REF] Tyrsted | In-situ synchrotron radiation study of formation and growth of crystalline CexZr1-xO2 nanoparticles synthesized in supercritical water[END_REF] . Note that whatever the synthesis approach, it is known that for small substitution percentage (<30%), substituted cerium oxide materials keep the pure cerium oxide crystalline structure and morphology [START_REF] Tyrsted | In-situ synchrotron radiation study of formation and growth of crystalline CexZr1-xO2 nanoparticles synthesized in supercritical water[END_REF] .

Based on the above-mentioned considerations, this study first deals with the continuous synthesis of pure CeO2 in nearand supercritical conditions from two different precursors (cerium nitrate and cerium ammonium nitrate) in water. Starting from the experimental analyses and theoretical statements, we propose reaction mechanisms to highlight the influence of experimental parameters over the physicochemical characteristics of the synthesized materials. Then, the synthesis of substituted cerium oxide nanoparticles is investigated thanks to those preliminary results. Aiming at broadening the range of elements conceivable for the synthesis of substituted cerium oxide in near-and supercritical fluids, three different elements are considered: lanthanum, praseodymium and zirconium. Experiments were performed to quantify the influences of the precursor's reactivity, the residence time and the temperature over the final substitution rate. Based on the obtained results, protocols are proposed to synthesize lanthanum, praseodymium and/or zirconium substituted CeO2 nanoparticles with controlled compositions.

Experimental

Chemicals

Cerium(III) nitrate (purity: 99%), ammonium cerium(IV) nitrate (purity ≥98.5%), zirconium(IV) oxynitrate (purity: 99%), lanthanum(II) nitrate (purity: 99.99%) and praseodymium nitrate (purity: 99.9%) were purchased from Sigma Aldrich and used as received.

Apparatus and procedure

The cerium oxide based nanostructures have been synthesized using a simple custom-built, continuous-flow reactor, presented in Fig. 1 [START_REF] Slostowski | Near-and supercritical alcohols as solvents and surface modifiers for the continuous synthesis of cerium oxide nanoparticles[END_REF] , [START_REF] Desmoulins-Krawiec | Synthesis of nanostructured materials in supercritical ammonia: nitrides, metals and oxides[END_REF] .

The feed-tank was a 500 mL Pyrex beaker containing the precursor(s) and the solvent(s) under fast magnetic stirring. The reactor was made of 1/8 in. 316 L stainless steel coiled tubing with an internal diameter of 1.57 mm, for a total volume of 8 cm 3 . The heater was a homemade coiled ceramic resistance controlling the temperature up to 500 ± 1 °C. The highpressure pump (P) was a controlled flow pump (JASCO PU2080), delivering flow rates up to 10 ± 0.001 mL min -1 . A back pressure regulator (BPR -JASCO BP 2080), placed downstream the reactor, allows controlling the pressure in the reactor up to 40 ± 0.1 MPa. In this process, the precursors (cerium precursor alone or along with the substituting agent precursor) are directly mixed with the solvent(s) before being injected inside the reactor with the high pressure pump, via a single injection line, while classical methods imply two or more separate lines for precursors and solvents [START_REF] Veriansyah | Characterization of surface-modified ceria oxide nanoparticles synthesized continuously in supercritical methanol[END_REF] , [START_REF] Takami | Continuous synthesis of organic-inorganic hybridized cubic nanoassemblies of octahedral cerium oxide nanocrystals and hexanedioic acid[END_REF] . The mixture is then flown through the reactor at near-or supercritical conditions, where the nucleation/growth of nanostructures occurs. The reaction is quenched thermally downstream the reactor with an ice bath, and the NCs can be recovered in solution upon depressurization through the BPR. The solution is then filtered and the synthesized materials are washed on a Büchner filter system. The dry powders have been then analyzed without any annealing treatment.

Synthesis of cerium oxide-based powders

As previously mentioned, two kinds of systems were investigated: (i) pure CeO2 from two different precursors and (ii)

Ce1-x-yAxByO2-δ NCs with three different substitution elements -A, B = Zr, Pr and/or La; 0 ≤ x, y ≤ 0.1.

Pure cerium oxide nanostructures

Two experiments have been performed in order to determine the influence of the starting cerium precursor (cerium nitrate or ammonium cerium nitrate) on cerium oxide synthesis in supercritical water conditions, in particular over the processability and the obtained nanostructures morphologies. Both precursors were dispersible in water. For scH2O experiments, the conditions were set at T = 400 °C and p = 24.5 MPa. The precursor concentration was fixed at 10 -2 M in all cases.

Zirconium, lanthanum and praseodymium substituted cerium oxide nanostructures

Concerning the substitution studies, eighteen screening experiments have been first performed in order to determine the influence of temperature and residence time. The precursors' solutions were composed of an aqueous mixture of ammonium cerium nitrate (9 × 10 -3 M) and hydrated substituting elements' precursors (1 × 10 -3 M zirconium oxynitrate, lanthanum nitrate or praseodymium nitrate). Two temperatures have been chosen: 300 °C (ncH2O) and 400 °C (scH2O), while residence time (Rt) was tuned in the range 10-60 s. Operating pressure has been fixed at 24.5 MPa. In a second time, six experiments were performed in scH2O in order to synthesize cerium oxide based nanostructures (Ce1-xAxO2-δ) with controlled compositions (A = La, Pr, Zr or mixtures, 0 < x < 0.1). Based on the previous screening experiments, the operating conditions were fixed at T = 400 °C p = 24.5 MPa and Rt = 10 s. The substituting elements' precursors remain the same as above-mentioned as well as the ammonium cerium nitrate concentration (9 × 10 -9 M); however, their concentrations have been adjusted depending on the targeted substitution rate.

Characterization techniques

The XRD patterns were recorded on a PANalytical X'Pert MPD powder diffractometer (θ-θ Bragg-Brentano geometry using Cu Kα1,α2 (λ1 = 1.54060 Å, λ2 = 1.54441 Å) radiation, equipped with a secondary monochromator and a X'Celerator detector, in the range of 8-120°, in continuous scan mode at 3.5 × 10 -3 s -1 . The powder was ground and sieved at 50 μm before being analyzed by XRD. The TEM images were obtained using a Hitachi H7650 transmission electron microscope operated at 90 kV. Samples were prepared by depositing drops of toluene or ethanol dispersed samples on copper-carbon grids.

The FTIR measurements were carried out using a Bruker Equinox 55 spectrophotometer. The spectra were recorded over 32 scans in the range of 400-4000 cm -1 , with a resolution of 4 cm -1 , using KBr powder technique.

Inductively coupled plasma (ICP) atomic emission spectroscopy was systematically used on doped powders in order to quantify elements in the synthesized materials. Measurements were realized using a Varian 720-ES emission spectrometer.

Chemical cartography of the elements in the substituted powders was obtained using a JEOL 2200 FX scanning transmission electron microscope (STEM) coupled with an energy-dispersive X-ray (EDX) spectrometer.

Results and discussions

3.1. Processing of pure cerium oxide nanostructures: influence of precursors

Processability and materials characterization

Experiments have been carried out in order to identify the precursor enabling the continuous synthesis of CeO2 in water using our single-line process.

The experiment performed in scH2O from ammonium cerium nitrate has allowed a continuous recovery of materials, leading to well crystallized CeO2 (crystallinity above 95%, evaluated from the internal standard method), in a fcc system with lattice parameters of 5.41 ± 0.005 Å, calculated from the XRD data using Bragg's equation (Fig. 2). Oppositely, the experiment conducted in scH2O from cerium nitrate has led to the precipitation of synthesized materials into the reactor. The material synthesized in scH2O from cerium ammonium nitrate has have been further examined by transmission electron microscopy (TEM) showing little aggregated CeO2 NCs (5-10 nm) (Fig. 3a). octahedrons from cerium nitrate in scH2O (adapted from [START_REF] Byrappa | Nanoparticles synthesis using supercritical fluid technology-towards biomedical applications[END_REF] ).

Please note that our single injection line process did not allow the continuous powders recovery when cerium nitrate is used as precursor in scH2O. However, it is worth mentioning that a two injection line process mixing the precursor solution with pre-heated supercritical water flow, developed by Adschiri et al. was previously successfully used to obtained small (20-40 nm) CeO2 octahedrons in similar conditions (Fig. 3b) [START_REF] Hakuta | Production of ultra-fine ceria particles by hydrothermal synthesis under supercritical conditions[END_REF] , [START_REF] Byrappa | Nanoparticles synthesis using supercritical fluid technology-towards biomedical applications[END_REF] . We believe that our lower flow rates and the single injection line process are responsible for the precipitation of the particles during synthesis, oppositely to the experiments led by Hakuta et al. or Veriansyah et al. [START_REF] Hakuta | Production of ultra-fine ceria particles by hydrothermal synthesis under supercritical conditions[END_REF] , [START_REF] Veriansyah | Characterization of surface-modified ceria oxide nanoparticles synthesized continuously in supercritical methanol[END_REF] .

Proposed mechanisms

The obtained results, along with those presented in our previous work [START_REF] Slostowski | Near-and supercritical alcohols as solvents and surface modifiers for the continuous synthesis of cerium oxide nanoparticles[END_REF] , have shown the flexibility of supercritical media for the synthesis of well-crystallized CeO2 nanostructures with various sizes and morphologies. To date, the mechanisms associated with the formation of CeO2 in supercritical water remains partially elucidated.

The hydrothermal transformations involved during the synthesis in scH2O are highly influenced by the nature of the precursor (counter-ions -such as NO3 -or NH4 + -and initial oxidation state of cerium). Mechanisms involved in hydrothermal synthesis of CeO2 and, a fortiori, in scH2O go through the formation of a cerium hydroxide (+III or +IV), which will be oxidized (if starting oxidation state is +III) and dehydrated to obtain the cerium oxide (+IV) [START_REF] Chen | Reactive cerium(IV) oxide powders by the homogeneous precipitation method[END_REF] , [START_REF] Djuricic | Nanostructured cerium oxide: preparation and properties of weakly agglomerated powders[END_REF][START_REF] Zhou | Processing of nanometer-scale CeO2 particles[END_REF] . Two kinds of morphologies (polyhedrons and NCs) can be observed depending on the considered precursor. Considering the syntheses from cerium nitrate in scH2O, the nitrate counter-ions can play the role of an oxidizing agent leading to the recovery of pure CeO2 only. Furthermore, polyhedron morphology, also observed by Hakuta et al. [START_REF] Hakuta | Production of ultra-fine ceria particles by hydrothermal synthesis under supercritical conditions[END_REF] , can be explained by the selective adsorption of nitrate anions on the surface of CeO2 nuclei leading to an anisotropic growth of the particles to obtain cubes of polyhedrons as evidenced by Wu et al. [START_REF] Wu | Great influence of anions for controllable synthesis of CeO2 nanostructures: from nanorods to nanocubes[END_REF] .

Regarding the synthesis from ammonium cerium nitrate in scH2O, CeO2 NCs are continuously recovered with our singleline injection process. The NCs exhibit morphologies, which have been also obtained by Mai et al. via the hydrothermal synthesis of cerium oxide from ammonium cerium nitrate [START_REF] Mai | Shape selective synthesis and oxygen storage behavior of ceria nanopolyhedra, nanorods, and nanocubes[END_REF] . They postulated that the oxidation state of the cerium precursor was the unique factor governing the morphology: Ce 4+ ions contributing to a fast formation of CeO2 nuclei, less soluble and less anisotropic than Ce(OH)3 intermediate species, resulting in the formation of nanoparticles instead of nanocubes. Although the oxidation state of the precursor may be of key importance for reaction pathways and kinetics, the influence of ammonium cations cannot be discarded.

In order to confirm the role of NH4 + cations, we have conducted a synthesis in ncH2O (300 °C; 24.5 MPa) from an equimolar mixture of cerium nitrate and ammonium cerium nitrate. This experiment led to the continuous synthesis (no precipitation in the reactor) of well-crystallized CeO2 NCs of 5-10 nm, while no cubes were observed. The mixture of the two precursors induces the equimolar presence of Ce 3+ and Ce 4+ in the reaction media (Ce 3+ to Ce 4+ oxidation being ensured by NO3 - P a g e | 6

anions). Thus, the sole formation of NCs, instead of a potential expected mixture of NCs and nanocubes, proves that the oxidation state of the precursor has no influence on the morphology unlike the NH4 + cations.

Based on these results, we proposed that during the synthesis of CeO2 from Ce(NO3)6(NH4)2 in scH2O, NO3 -anions adsorb on the surface of CeO2 similarly to the synthesis from Ce(NO3)3 in scH2O. However, the pKw of scH2O being high at 400 °C and 24.5 MPa (pKw = 19.4), the NCs are not stabilized, explaining the precipitation of CeO2 synthesized from Ce(NO3)3 in our single line process. Oppositely, the NH4 + anions present during the syntheses from Ce(NO3)6(NH4)2 may act as electrostatic stabilizers for the (CeO2-NO3) system, avoiding the precipitation of the particles and therefore allowing the continuously recovery of small CeO2 NCs.

The presence of NO3 -anions at the surface of cerium oxide synthesized in water has been evidenced thanks to an experiment performed in ncH2O (300 °C and 24.5 MPa) from (NH4)2Ce(NO3)6. IR spectra of the recovered CeO2 NCs shows a band at 1383 cm -1 , which can be attributed to the vibration band of free NO3 -in KBr (Fig. 4), while no band was detected for CeO2 NCs synthesized from (NH4)2Ce(NO3)6 in scEtOH [START_REF] Slostowski | Near-and supercritical alcohols as solvents and surface modifiers for the continuous synthesis of cerium oxide nanoparticles[END_REF] . Based on this preliminary study concerning pure CeO2 NCs synthesis in near-and supercritical fluids, we have investigated in a second part the substitution of Ce by La, Pr and Zr atoms in CeO2 NCs with such processes.

Synthesis of Ce1-xAxO2-δ (A = La, Pr, Zr) nanostructures in near-and supercritical water

In order to study the suitability and the efficiency of supercritical media for the synthesis of substituted cerium oxide materials, we have chosen to select cerium ammonium nitrate as cerium precursor as it leads to dispersed nanoparticles and enable continuous recovery of the materials with our process, oppositely to cerium nitrate.

In order to limit the influence of the substitution element precursor, we also chose to select them so that they are as close as possible to the cerium precursor, i.e. the cerium ammonium nitrate. As ammonium nitrate precursors were not available for the selected substitution elements, we purchased nitrate or oxynitrate precursors, assuming these counter-ionscoupled with cerium ammonium nitrate -will not affect noticeably the chemical reaction. Indeed, we have shown in Section 3.1 that a mixture of cerium ammonium nitrate with cerium nitrate leads to the same results (particles size, crystallinity, shape) as in the case of pure cerium ammonium nitrate.

Preliminary substitution experiments in near-and supercritical water

Screening experiments have been performed in order to determine the influence of temperature and residence time on the processing of zirconium, lanthanum and praseodymium substituted CeO2 NCs. For each experiment, materials have been

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recovered continuously without precipitation within the reactor. Powders have been analyzed as synthesized, without further calcination treatment. These 18 experiments have led to well crystallized NCs adopting the CeO2 structure (crystallinity above 95%, evaluated from the internal standard method), in a fcc system with lattice parameters of 5.41 ± 0.005 Å (obtained in all cases with no detectable variations), calculated from the XRD data using Bragg's equation (Fig. 5). All powders displayed crystallite sizes inferior to 10 nm according to Scherrer's equation, whatever the considered operating parameters. The focus of these screening experiments being to determine the influence of temperature and residence time on the substitution process efficiency, ICP analyses have been carried out on each material. Results of these analyses are given in Table 1. Well crystallized substituted CeO2 NCs are obtained with a molar composition of 7.4%, 2.8% and 5.2% for Zr, La and Pr, respectively, starting from an initial 10% molar concentration of the substituting element precursor. The results show that for the three selected precursors, residence time has no influence on substitution efficiency after 10 s, which means that reaction is fast for all precursors independently from the operating temperature. However, higher temperatures appear to enhance the reactivity of the lanthanum and praseodymium precursors, while no influence is noticed for the reactivity of the zirconium precursor.

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Preparation of cerium based nanostructures with controlled composition

The above-mentioned screening experiments show the limits of the reactivity of the substitution element precursors for the synthesis of Ce1-xAxO2-δ NCs with controlled composition in supercritical water. TGA analyses have been conducted on lanthanum nitrate, praseodymium nitrate and zirconium oxynitrate precursors (Fig. 6). Two conclusions arise from these analyses. First, whatever the considered precursor, the thermal decomposition is not fully achieved at 400 °C, unlike the thermal decomposition of ammonium cerium nitrate to cerium oxide is achieved at 230-240 °C according to Zhou et al. [START_REF] Zhou | Spontaneous organization of uniform CeO2 nanoflowers by 3D oriented attachment in hot surfactant solutions monitored with an in situ electrical conductance technique[END_REF] .

Note that hydrothermal processes will not necessarily reach equivalent transformation temperatures as for thermal processes, given the different mechanisms involved; however, we assumed that the tendency should remain similar in both cases. The second information is that the decomposition rate (=weight loss at T/total weight loss) as a function of temperature is not the same depending on the considered precursor. For instance, at 400 °C, the decomposition rate is 57%, 41% and 97%

for praseodymium nitrate, lanthanum nitrate and zirconium oxynitrate, respectively, reflecting a different reactivity of each precursor.

According to all these results, two solutions can be considered in order to enhance/control the amount of substituting element into the CeO2 NCs in a range typically interesting for most of the considered applications (e.g. x < 0.1): increase the temperature to enhance the reactivity of the precursors or use higher concentration in the substituting element starting solution.

In this work, we considered an increase of the substituting element solution concentration, while keeping the same cerium precursor concentration (9 × 10 -3 M), in order to reach higher substitution rate. Six additional experiments have been performed at 400 °C, 24.5 MPa and Rt = 10 s) to produce substituted and double substituted CeO2 NCs with different compositions. Once again, materials have been recovered continuously and crystallinity of materials (adopting CeO2 structure) has been checked by XRD analyses. Substituting element(s) percentages in the precursor solution and in the recovered materials (ICP analyses) are given in Table 2. These results show that it is possible to synthesize in scH2O substituted cerium oxide NCs with controlled Ce substitution rates by tuning the concentration of substituting element precursors. Using that technique, we were able to elaborate Ce1-x-yAxByO2-δ with A and/or B = lanthanum, praseodymium or zirconium. i.e. narrow size distributed and slightly aggregated nanocrystals (Fig. 7). Additionally, the sizes calculated from TEM images are coherent with crystallite sizes calculated from XRD (dTEM ≈ dcr) patterns, meaning the nanoparticles are single nanocrystals (Table 2).

Finally, STEM-EDX analyses have been conducted for checking the substitution homogeneity over the synthesized materials.

Chemical cartography of the elements clearly shows that all elements are equally distributed in the NCs, resulting in homogenous substituted cerium oxide nanostructures (Fig. 8).

Discussion

To the best of our knowledge, these experiments are the first reporting the synthesis of CeO2 NCs substituted with lanthanum and praseodymium using a continuous supercritical water process. Zirconium substituted CeO2 synthesis has already been reported in scH2O using zirconium oxynitrate (without quantitative analysis of the elements repartition) or zirconium acetylacetonate (XRF analyses showing small difference between initial and final Ce:Zr ratios) [START_REF] Cabañas | A continuous and clean one-step synthesis of nano-particulate Ce 1-xZrxO2 solid solutions in near-critical water[END_REF][START_REF] Cabañas | Continuous hydrothermal synthesis of inorganic materials in a near-critical water flow reactor; the one-step synthesis of nanoparticulate Ce1-xZrxO2 (x = 0-1) solid solutions[END_REF] .

Our study has demonstrated the suitability of the near-and supercritical water synthesis processes to obtain substituted CeO2 nanostructures. Although chemical cartographies showed a homogeneous repartition of the elements in the obtained materials, quantitative analyses performed by ICP displayed differences between initial and final elements ratios and evidenced the importance of some experimental parameters. Indeed, reactivity of the precursors is of key importance in order to synthesize such materials. This reactivity is directly linked to the precursor itself (counter-ion/ligand and oxidation state) but also to the experimental conditions (T and p) and the time of reaction (ts). The real challenge is therefore to find suitable precursors for each element with similar reactivity.

Temperature appears to have a real influence when considering lanthanum and praseodymium nitrates. Indeed, a higher reactivity of these precursors has been observed at 400 °C compared to 300 °C, leading to higher substitution rates in the synthesized substituted CeO2 materials. Higher experimental temperature (T > 500 °C, i.e. above the thermal decomposition temperature of the precursors) should allow reaching higher reactivity of these precursors but with a risk to form a secondary oxide phase from the substitution element. Such temperatures would be coherent with those generally reported for the synthesis of lanthanum or praseodymium substituted CeO2 via usual methods [START_REF] Zhang | Catalytic performance of La-Ce-O mixed oxide for combustion of methane[END_REF][START_REF] Sharma | A spray drying system for synthesis of rare-earth doped cerium oxide nanoparticles[END_REF][START_REF] Somacescu | Uniform nanoparticles building Ce1-xPrxO2-δ mesoarchitectures: structure, morphology, surface chemistry, and catalytic performance[END_REF][START_REF] Katta | Preparation of silica supported ceria-lanthana solid solutions useful for synthesis of 4-methylpent-1-ene and dehydroacetic acid[END_REF] . However, it has to be noticed that such conditions could compromise CeO2 NCs characteristics [START_REF] Heintz | Synthesis and sintering properties of cerium oxide powders prepared from oxalate precursors[END_REF] .

In contrast, temperature appears to have little effect when starting from zirconium oxynitrate. The maximal reactivity of this precursor is possibly already reached at 300 °C, which is coherent with the syntheses led by Kim et al. in scH2O (400 °C; p > pc) [START_REF] Kim | Characteristics in oxygen storage capacity of ceria-zirconia mixed oxides prepared by continuous hydrothermal synthesis in supercritical water[END_REF][START_REF] Kim | Characteristics of CeO2-ZrO2 mixed oxide prepared by continuous hydrothermal synthesis in supercritical water as support of Rh catalyst for catalytic reduction of NO by CO[END_REF][START_REF] Kim | Ceria-zirconia mixed oxide prepared by continuous hydrothermal synthesis in supercritical water as catalyst support[END_REF] .

Finally, these experiments showed no noticeable effect of the residence time above 10 s, whatever the considered precursor.

Looking upon such considerations, we were able to synthesize CeO2-based NCs (7 nm) with narrow size distribution, controlled substitution rates and homogeneous dispersion of the elements by playing on initial precursors ratios. Such materials can be synthesized continuously in much shorter residence times than the conventional techniques (coprecipitation).

Conclusion

The influence of two cerium precursors (nitrate or ammonium nitrate) has been investigated for the synthesis of pure CeO2 using a simple continuous process in scH2O. The cerium nitrate in scH2O system led to the precipitation of the CeO2 within the reactor, which is not the case with cerium ammonium nitrate allowing a continuous recovery of CeO2 nanoparticles (size of about 7 nm). Based on experimental analyses and literature results, we have proposed reaction mechanisms to explain the observed behavior. In scH2O (400 °C; 24.5 MPa), the counter-ion of the cerium precursor has the most important influence, the counter-ion playing both the role of an oxidizer (the nitrate ions allowing the transition from Ce 3+ to Ce 4+ ) and a stabilizing agent. Indeed, nitrate anions orientate the growth of CeO2 nuclei, while ammonium cations stabilize the [nitrate-CeO2] systems. That way, materials synthesized from cerium ammonium nitrate in scH2O will be stabilized in the solvent by ammonium ions and lead to continuous recovery of well-dispersed NCs.

Then, the [cerium ammonium nitrate/H2O] system has been selected in order to perform substituting experiments of the CeO2 NCs. XRD and TEM analyses have shown that substituted CeO2 materials synthesized in near-or supercritical water displayed the same morphology (nanoparticles), sizes (7 nm) and crystallinity (fcc) than pure CeO2 NCs synthesized in the same conditions. Furthermore, ICP and STEM-EDX analyses evidenced a homogeneous distribution of the substituting elements in the obtained materials with a substitution rate depending on experimental conditions (precursor, T and Rt).

Thus, these experiments have shown that a temperature of 300 °C is sufficient enough to get an almost maximum reactivity with zirconium oxynitrate, while temperature above 400 °C are required to fully complete lanthanum nitrate and praseodymium nitrate transformation. The obtained results also demonstrated that above 10 s, residence time has no more influence on the reactivity of the precursors, meaning that such syntheses could be conducted in less than 10 s at supercritical conditions, while hours are generally needed through co-precipitation methods. The synthesis of Ce1-xAxO2-δ and Ce1-x-yAxByO2-δ NCs with controlled composition was successfully achieved through this process using an excess of substitution element precursor in the starting precursor solution to overcome the difference in reactivity of the precursors.

Such results and observations allow a better understanding of the influence of the experimental parameters for the synthesis of CeO2 based materials in solvothermal conditions and evidence the necessity of precursors with similar reactivity.

Fig. 1 .

 1 Fig. 1. Custom-built continuous process for CeO2 synthesis in supercritical media. T = thermocouple; P = manometer; Sp = precursor solution and Ss = synthesis solution (synthesized powders + solvent).

Fig. 2 .

 2 Fig. 2. XRD patterns of CeO2 powder recovered continuously from the synthesis in scH2O from ammonium cerium nitrate (400 °C; 24.5 MPa).

Fig. 3 .

 3 Fig. 3. TEM images of powders synthesized in SCFs: (a) CeO2 NCs from ammonium cerium nitrate in scH2O and (b) CeO2octahedrons from cerium nitrate in scH2O (adapted from[START_REF] Byrappa | Nanoparticles synthesis using supercritical fluid technology-towards biomedical applications[END_REF] ).

Fig. 4 .

 4 Fig. 4. Transmission FTIR spectra of CeO2 nanoparticles synthesized at 300 °C and 24.5 MPa in water (ts = 45 s) from cerium ammonium nitrate. The spectrum has been recorded on as synthesized powders.

Fig. 5 .

 5 Fig. 5. XRD patterns of materials synthesized in scH2O at 400 °C and 24.5 MPa for a residence time of 10 s from ammonium cerium nitrate (a) alone, (b) with 10% of lanthanum nitrate, (c) with 10% of praseodymium nitrate and (d) with 10% of zirconium oxynitrate.

Fig. 6 .

 6 Fig. 6. TGA curves of (a) praseodymium nitrate hexahydrate, (b) lanthanum oxide hexahydrate and (c) zirconium oxynitrate hydrate.Analyses have been conducted under N2 with a heating rate of 2 °C min -1 .

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 9 Size and size distribution of the synthesized substituted nanoparticles were analyzed by TEM, showing that the obtained substituted CeO2 NCs keep the same morphology displayed by pure cerium oxide NCs synthesized in the same conditions,

Fig. 7 .

 7 Fig. 7. TEM images of the samples. (a) Ce0.96La0.04O2-δ and (b) Ce0.90Zr0.05Pr0.05O2-δ. The powders were dispersed in ethanol for microscopy sample preparation.

Fig. 8 .

 8 Fig. 8. SEM images and chemical cartographies of the samples (a and b) Ce0.95Pr0.05O2-δ and (c and d) Ce0.91Zr0.09O2. The powders were dispersed in ethanol for microscopy sample preparation.

Acknowledgments

The authors acknowledge the financial support of the "Région Aquitaine" and the help of the Bordeaux Imaging Center (BIC) for the electronic microscopy characterization.