Tonametl Sanchez
email: tonametl.sanchez@ipicyt.edu.mx

Andrey Polyakov
email: andrey.polyakov@inria.fr.

Emilia Fridman
email: emilia@eng.tau.ac.il.

Laurent ¸iu Hetel
email: laurentiu.hetel@centralelille.fr.

A Switching Controller for a class of MIMO Bilinear Systems with Time-Delay

Keywords: Time-delay systems, bilinear systems, switched control. I

In this paper we propose a state-dependent switching controller for MIMO bilinear systems with constant delays in both the state and the input. The control input is assumed to be restricted to take only a finite number of values. The stability analysis of the closed-loop is based on a Lyapunov-Krasovskii functional, and the design is reduced to solve a system of linear matrix inequalities. The controller can be designed by considering (state) delay-dependent or delay-independent conditions.

an inherent feature of the model, or simply due to the lags in sensor or actuator signals. Hence, bilinear delayed systems are worthy of more study to develop general schemes for analysis and control design.

In this paper we design a controller for MIMO bilinear systems with constant delays in both the state and the input. We consider control inputs restricted to take only a finite number of values. This situation arises in several control applications, in particular when on-off actuators are present, e.g. in power electronics [START_REF] Sira-Ramirez | Control Design Techniques in Power Electronics Devices[END_REF] and in several pneumatic systems [START_REF] Beater | Pneumatic Drives[END_REF]. With this restriction in the control, bilinear systems can also be studied as switched (or piece-wise) affine systems controlled by the switching signal.

It is important to mention that the existing control techniques in related works cannot be applied for the class of systems considered in this paper, e.g.: in [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF], [START_REF] Deaecto | Switched affine systems control design with application to DC-DC converters[END_REF], [START_REF] Hetel | Robust Sampled -Data Control of Switched Affine Systems[END_REF], [START_REF] Hetel | Binary Control Design for a Class of Bilinear Systems: Application to a Multilevel Power Converter[END_REF], [START_REF] Kader | Stabilization of switched affine systems with disturbed state-dependent switching laws[END_REF], bilinear and switched affine systems are studied, but the delayed case is not considered in those papers; the systems studied in [START_REF] Jiang | Switching controller for stabilization of linear systems with switched time-varying delays[END_REF], [START_REF] Vu | Stability of time-delay feedback switched linear systems[END_REF], [START_REF] Mazenc | State feedback stabilization of switched systems with delay: Trajectory based approach[END_REF] are switched and have delay in the input, however, they are linear and the controller is a linear state-feedback; in [START_REF] Zhang | Stability of switched systems with time delay[END_REF] all the subsystems of the switched system must have a common equilibrium point at the origin and the switching laws are not state-feedback; switched affine delayed systems are studied in [START_REF] Moezzi | Stability of uncertain piecewise affine systems with time delay: delaydependent Lyapunov approach[END_REF], [START_REF] Mahmoud | Switched Time-Delay Systems[END_REF], [START_REF] Fiter | Stability of Piecewise Affine Systems with State-Dependent Delay, and Application to Congestion Control[END_REF], however, the systems are assumed to be piecewise linear in a neighbourhood of the origin, and no general switching-control schemes are designed.

In [START_REF] Feingesicht | A bilinear input-output model with statedependent delay for separated flow control[END_REF], [START_REF] Feingesicht | Nonlinear active control of turbulent separated flows: Theory and experiments[END_REF], a bilinear differential equation with delays was proposed as a simplified nonlinear model for separated flow control. For a particular case of such a model, a sliding mode controller was proposed in [START_REF] Feingesicht | SISO model-based control of separated flows: Sliding mode and optimal control approaches[END_REF]. That controller achieved good performance in different experimental settings. Unfortunately, such a kind of controller was designed only for SISO systems (see also [START_REF] Sanchez | A robust Sliding Mode Controller for a class of SISO bilinear delayed systems[END_REF]), and its generalization to the MIMO case is not straightforward.

The controller presented in this paper is based on a Lyapunov-Krasovskii functional, and its design reduces to solve a system of linear matrix inequalities (LMIs). A particular case of this controller was announced in [START_REF] Sanchez | A switching controller for a class of MIMO bilinear time-delay systems[END_REF], but compared with that paper, the following improvements are made in the present one: 1) the design considers now (state) delay-dependent stability conditions, this allows us to enlarge the class of systems that can be controlled; 2) in any case, we consider an arbitrary number of delays in the state for both, the linear and the bilinear terms. Notation: R and Z denote the set of real and integer numbers, respectively. For any a ∈ R, R ≥a denotes the set {x ∈ R : x ≥ a}. For a matrix A ∈ R n×n , we mean by A > 0 (A < 0) that A is positive definite (negative definite). For a finite set X ⊂ Z >0 and a function f :

X → R, arg min x∈X (f (x)) := {y ∈ X : f (x) ≥ f (y) ∀x ∈ X}.
ẋ(t) = A 0 x(t) + N1 k=1 A k x(t -τ k ) + Bu(t -ς) + N2 s=1 A s (u(t -ς))x(t -τs ) , (1)
where

A s (u(t -ς)) = m r=1 A s,r u r (t -ς), x(t) ∈ R n is the instantaneous state, u(t) ∈ R m is the control input, A 0 , A k , A s,r ∈ R n×n , k = 1, . . . , N 1 , s = 1, . . . , N 2 ,
r = 1, . . . , m, are constant matrices for some finite N 1 , N 2 ∈ Z ≥0 , and τ k , τs , ς ∈ R ≥0 are constant delays. We consider the output of (1) as the whole instantaneous state x(t) which is accessible for all t ∈ R t0 . We assume the following for (1). Assumption 1. For any k and any s, the delays ς, τ k , and τs are known and satisfy 0 < ς ≤ τ k , τs .

The problem we want to solve is stated as follows:

• to design a controller u to track asymptotically a desired constant reference x * ∈ R n , taking into account that each control component u i : R → R can only take a finite number of values.

Observe that, under the restriction on the controller, we can only have N different values of the input vector u, for some N ∈ Z >0 . If we give a certain order to such N vectors, then we obtain the ordered set U = {u1 , . . . , u N }, where each u j ∈ R m , j = 1, . . . , N , is a constant vector. Also note that, since the control input is bounded, the set of admissible references x * is also bounded in the general case.

Remark 1 (On the existence of solutions). Consider (1) with the initial conditions

x(t) = φ(t) , t ∈ [t 0 -h, t 0 ] , (2)
where h = max(τ 1 , . . . , τ N1 , τ1 , . . . , τN2 ). A function x : R ≥t0-h → R n , that is locally absolutely

continuous on t ∈ [t 0 , ∞),

A. Reformulation of the problem and the admissible references

To solve the problem established in the previous section, we first state some basic preliminary considerations.

For j = 1, . . . , N and s = 1, . . . , N 2 , define the vectors B j = Bu j and the matrices A s,j = m r=1 A s,r u j r . Hence, we can consider (1) as an affine switched system given by

ẋ(t) = A 0 x(t) + N1 k=1 A k x(t -τ k ) + N2 s=1 A s,σ(t-ς) x(t -τs ) + B σ(t-ς) , (3)
where σ : R → {1, . . . , N } is the switching signal. As stated in Section II, the control objective is to track asymptotically a constant reference x * ∈ R n . Define the tracking error z by means of the change of coordinates z(t) = x(t) -x * . Hence, the tracking error dynamics is given by

ż(t) = A 0 z(t) + N1 k=1 A k z(t -τ k ) + N2 s=1 A s,σ(t-ς) z(t -τs ) + C(t) , (4)
where

C(t) := A 0 + N1 k=1 A k + N2 s=1 A s,σ(t-ς) x * + B σ(t-ς)
. Thus, we have changed the original problem of designing the controller u to the problem of designing the switching rule for σ to drive the trajectories of (3) to the reference x * , or equivalently, to drive the trajectories of (4) to zero.

The following lemma gives a necessary condition for the selection of a reference point x * . Lemma 1. Consider (4) and define the following

Γ = γ ∈ R N : 0 ≤ γ i ≤ 1, N j=1 γ j = 1 , Ās (γ) = N j=1 γ j A s,j , B(γ) = N j=1 γ j B j .
If for a given signal σ : R → {1, . . . , N } and a constant x * ∈ R, z is a solution of (4) such that z(t) → 0 as t → ∞, then there exists γ ∈ Γ such that

A 0 + N1 k=1 A k + N2 s=1 Ās (γ) x * + B(γ) = 0 . (5)
Proof. For t > t 0 and any T ∈ R >0 we have from (4) that

z(t + T ) -z(t) = t+T t A 0 z(ν) + N1 k=1 A k z(ν -τ k )+ N2 s=1 A s,σ(t-ς) z(ν -τs ) dν + t+T t C(ν) dν .
According to the hypothesis of the lemma, for any finite T ∈ R >0 , lim t→∞ (z(t + T ) -z(t)) = 0, and

lim t→∞ t+T t A 0 z(ν) + N1 k=1 A k z(ν -τ k ) + N2 s=1 A s,σ(t-ς) z(ν -τs ) dν = 0 , therefore, it is necessary that lim t→∞ t+T t
C(ν) dν = 0 for any finite T ∈ R >0 . Note that, according to the definition of A s,σ(t-ς) and B σ(t-ς) , we have that

lim t→∞ t+T t C(ν) dν = lim t→∞ N j=1 A 0 + N1 k=1 A k + N2 s=1 m r=1 A s,r u j r x * + Bu j δ j (t, T ) , where δ j (t, T ) is the measure of the set {τ ∈ [t, t + T ] : u(τ -ς) = u j }. Observe that for any t ∈ R >t0 , N j=1 δ j (t, T ) = T , thus, 0 = lim t→∞ t+T t C(ν) dν = T N j=1 γ j A 0 + N1 k=1 A k + N2 s=1 m r=1 A s,r u j r x * + Bu j ,
where

γ j = T -1 lim t→∞ δ j (t, T ). Observe that N j=1 γ j = 1.
The result of the lemma is obtained by noticing that

N j=1 γ j A 0 + N1 k=1 A k + N2 s=1 m r=1 A s,r u j r x * + Bu j = A 0 + N1 k=1 A k + N2 s=1 Ās (γ) x * + B(γ).

III. CONTROLLER DESIGN

In this section we design two switching laws for σ to solve the problem established in the previous section. With the aim of showing the ideas in a clear way, we begin by stating the results for the particular case of (3) with N 1 = 0 and N 2 = 1, i.e. for the system

ẋ(t) = A 0 x(t) + A σ(t-ς) x(t -τ ) + B σ(t-ς) . (6)
The results given in Section III-A are extended, in Section III-B, for the case of arbitrary N 1 , N 2 .

A. One delay in the state

In the following theorem, we give a switching law σ that drives the trajectories of (6) to a given reference x * . Such a switching law is based on a Lyapunov-Krasovskii functional that provides delaydependent stability conditions (regarding the delay in the state). As a particular case, delay-independent stability conditions are obtained.

Theorem 1 (Delay-dependent conditions). Consider (6) with τ > ς > 0, and a given x * ∈ R n . If there exist matrices P 1 , P 2 , P 3 , S, R ∈ R n×n and a vector γ ∈ Γ, such that P 1 , P 3 , S, R are symmetric,

(A 0 + Ā(γ))x * + B(γ) = 0, S, R > 0 ,   P 1 P 2 P 2 P 3 + τ -1 S   > 0 , M (γ) < 0 , (7)
where

Ā(γ) = N j=1 γ j A j , M (γ) :=      Φ P 1 Ā(γ) -P 2 τ (A 0 P 2 + P 3 ) * -S τ ( Ā(γ) P 2 -P 3 ) * * -τ R     
, and Φ := A 0 P 1 + P 1 A 0 + S + τ R + P 2 + P 2 . Then, (6) in closed-loop with the switching controller

σ(t) ∈ arg min j∈{1,...,N } (w(t) -x * ) P 1 + ( w(t) -x * τ ) P 2 A j x(t -τ + ς) + B j , (8)
w(t) = 0 -ς e -A0η A σ(t+η) x(t -τ + ς + η) + B σ(t+η) dη + e A0ς x(t) , w(t) = t t-τ +ς x(ν) dν + t t-ς w(ν) dν , is such that x(t) → x * as t → ∞.
The proof of Theorem 1 is given below, but first, let us enunciate some important remarks.

Remark 2. Observe that, for any t ∈ R ≥0 , the switching signal σ(t) depends on x(ϑ), ϑ

∈ [t -τ, t],
and the switching signal

σ(t -ς) depends on x(ϑ), ϑ ∈ [t -τ -ς, t -ς].
Therefore, the condition ς > 0 allows us to guarantee the existence of solutions of the closed-loop (6), [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF] as stated in Remark 1. Thus, no generalized-solution concepts are required.

From Theorem 1 we recover in the following corollary the controller designed in [START_REF] Sanchez | A switching controller for a class of MIMO bilinear time-delay systems[END_REF].

Corollary 1 (Delay-independent conditions). Consider (6) with τ > ς > 0, and a given x * ∈ R n . If there exist γ ∈ Γ and symmetric matrices P 1 , S ∈ R n×n such that (A 0 + Ā(γ))x * + B(γ) = 0,

P 1 , S > 0, M (γ) :=   P 1 A 0 + A 0 P 1 + S P 1 Ā(γ) Ā (γ)P 1 -S   < 0, (9)
then (6) in closed-loop with the switching controller

σ(t) ∈ arg min j∈{1,...,N } ((w(t) -x * )P 1 [A j x(t -τ + ς) + B j ]) , (10)
w(t) = 0 -ς e -A0η A σ(t+η) x(t -τ + ς + η) + B σ(t+η) dη + e A0ς x(t), is such that x(t) → x * as t → ∞.
Remark 3. Observe that to accomplish the last LMI in (9), A 0 must be Hurwitz. Such a restriction is eliminated by the more general controller [START_REF] Sira-Ramirez | Control Design Techniques in Power Electronics Devices[END_REF], which considers delay-dependent stability conditions.

However, for the cases where both controllers are feasible, [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF] has the advantage to be simpler than [START_REF] Sira-Ramirez | Control Design Techniques in Power Electronics Devices[END_REF].

September 6, 2019 DRAFT Remark 4 (Two computational issues). Let us discuss briefly two details in the computation of the controller given in Theorem 1:

1) In applications, the controller implementation is normally made through digital computers, thus, the control signals are assumed constant between the sampling periods (piecewise continuous), and the data acquisition process usually transforms the physical measurements into piecewise continuous signals. For such a kind of signals, the Lebesgue and the Riemann integrals coincide. Under this consideration, the integral in the controller of Theorem 1 can be implemented as a Riemann integral.

2) For the computation of the predictors in the control law, there exist in the literature several numerically stable methods to implement such predictors, see e.g. [START_REF] Zhong | On Distributed Delay in Linear Control Laws-Part I: Discrete-Delay Implementations[END_REF], [START_REF] Michiels | Evaluating and Approximating FIR Filters: An Approach Based on Functions of Matrices[END_REF], and the references therein.

Proof of Theorem 1: Consider again the change of variables z(t) = x(t)-x * . The dynamics in variable z is given by

ż(t) = A 0 z(t) + A σ(t-ς) z(t -τ ) + C(t) , (11)
where

C(t) := (A 0 + A σ(t-ς) )x * + B σ(t-ς)
. The proof consists in verifying, by means of a Lyapunov-Krasovskii functional, that the solutions of ( 11) converge asymptotically to zero.

Consider the following Lyapunov-Krasovskii functional [START_REF] Fridman | Simple LMIs for stability of stochastic systems with delay term given by Stieltjes integral or with stabilizing delay[END_REF] V

(z t ) =   z(t) ẑ(t)     P 1 P 2 * P 3     z(t) ẑ(t)   + t t-τ z (ν)× Rz(ν)(ν -t + τ ) dν + t t-τ z (ν)Sz(ν) dν , (12)
where ẑ(t) := t t-τ z(ν) dν. By using the Jensen's inequality in the last term of [START_REF] Hetel | Robust Sampled -Data Control of Switched Affine Systems[END_REF] we have that

V (z t ) ≥   z(t) ẑ(t)     P 1 P 2 P 2 P 3 + τ -1 S     z(t) ẑ(t)   + t t-τ (ν -t + τ )z (ν)Rz(ν) dν .
Thus, ( 12) is positive definite if R > 0 , and

  P 1 P 2 P 2 P 3 + τ -1 S   > 0 .
Since the solutions of the closed-loop ( 6), ( 8) are absolutely continuous functions (see Remarks 1 and September 6, 2019 DRAFT

2), the time derivative of ( 12) is given by (for almost all t ∈ R ≥t0 )2

V (z t ) = 2   z(t) ẑ(t)     P 1 P 2 * P 3     ż(t) ż(t)   + τ z (t)Rz(t) + z (t)Sz(t) -z (t -τ )× Sz(t -τ ) - t t-τ z (ν)Rz(ν) dν . (13)
By using the Jensen's inequality in the last term of ( 13) we have that V (z t ) ≤ W (ζ; σ(t -ς)), where

ζ = [z (t), z (t -τ ), ẑ (t)] and W (ζ; σ(t -ς)) = z (t)Φz(t) + 2z (t) P 1 A σ(t-ς) -P 2 z(t -τ ) + 2z (t) A 0 P 2 + P 3 ẑ(t)- z (t -τ )Sz(t -τ ) + 2z (t -τ ) A σ(t-ς) P 2 -P 3 ẑ(t) -τ -1 ẑ (t)Rẑ+ 2 z (t)P 1 + ẑ (t)P 2 C(t) . (14)
Observe that ( 14) can be rewritten as

W (ζ; σ(t -ς)) = ζ M (σ(t -ς)) ζ + 2 z (t)P 1 + ẑ (t)P 2 C(t), where ζ = [z (t), z (t -τ ), τ -1 ẑ (t)] and M (ρ) =      Φ P 1 A ρ -P 2 τ (A 0 P 2 + P 3 ) * -S τ (A ρ P 2 -P 3 ) * * -τ R     
. Now, define the function W γ : R 3n → R, given by W γ (ζ) = ζ M (γ) ζ + 2 z (t)P 1 + ẑ (t)P 2 (A 0 + Ā(γ))x * + B(γ) . Note that, according to the hypotheses of the theorem, W γ is negative definite, and can be rewritten as

W γ (ζ) = N j=1 γ j W (ζ; j) .
Since W γ is negative definite, and each γ i is nonnegative, we can assure that for each ζ ∈ R 3n \ {0}, there exists (at least) one j ∈ {1, . . . , N } such that W (ζ; j) < 0. Therefore, the switching law σ(t -ς) ∈ arg min j∈{1,...,N } W (ζ; j) ensures that ( 14) is strictly negative for all ζ ∈ R 3n \ {0}. However, observe that (see ( 14))

min j W (ζ; j) = 2z (t)Φz(t) -2z (t -τ )P 2 z(t) + 2z (t)× A 0 P 2 + P 3 ẑ(t) -z (t -τ )Sz(t -τ ) -2z (t -τ )× P 3 ẑ(t) -τ -1 ẑ (t)Rẑ(t) + 2 z (t)P 1 + ẑ (t)P 2 × A 0 x * + 2 min j∈{1,...,N } z (t)P 1 + ẑ (t)P 2 × (A j z(t -τ ) + A j x * + B j ) .
Hence, the switching rule σ(t -ς) ∈ arg min j∈{1,...,N } W (ζ; j) is equivalent to

σ(t) ∈ arg min j∈{1,...,N } z (t + ς)P 1 + ẑ (t + ς)P 2 A j z(t -τ + ς) + A j x * + B j .
But σ requires the prediction of the instantaneous state at t + ς. Since the solution of ( 6) can be written as

x(t) = t t0 e A0(t-ν) [A σ(ν-ς) x(ν -τ ) + B σ(ν-ς) ] dν + e A0(t-t0) x(t 0 ) ,
the prediction of the state at t + ς is given by

x(t + ς) = t+ς t e A0(t+ς-ν) A σ(ν-ς) x(ν -τ ) + B σ(ν-ς) dν + e A0ς x(t) .
Observe that w(t) = x(t + ς) by means of the change of variable η = -t -ς + ν. Moreover, we have that ẑ (t + ς) = t t-τ +ς x(ν) dν + t+ς t

x(ν) dν -x * τ . Thus, the controller is obtained by using back the change of coordinates.

Remark 5. Note that at the time t, the predictor used to compute x(t + ς) only requires past information, i.e. the values of x(r) for r ∈ [t -τ, t) (which are already measured), and the values of σ(r) for r ∈ [t -ς, t) (which are already computed).

Finally, note that the switching law ensures that

W (ζ; σ(t -ς)) = min j∈{1,...,N } W (ζ; j) = N j =1 γ j min j∈{1,...,N } W (ζ; j) ≤ N j=1 γ j W (ζ; j) = W γ (ζ) . Thus, V (x t ) ≤ W γ (ζ) = ζ M (γ) ζ ≤ -λ min (-M (γ))|z(t)| 2 ,

B. Several delays in the state

In this section we extend the results given in Section III-A to the class of systems described by (3) considering several delays in the state. For this case consider the matrices

P 1 , P 2,k , P2,s , P 3,k , P3,s , S k , Ss , R k , Rs ∈ R n×n , k = 1, . . . , N 1 , s = 1, . . . , N 2 . Define P =   P 1 Π 11 * Π 22   , M (γ) =      Φ 11 • • • Φ 15 . . . . . . * Φ 55     
, where

Π 11 =[P 2,1 • • • P 2,N1 P2,1 • • • P2,N2 ] , Π 22 =diag(P 3,1 + τ -1 1 S 1 , . . . , P 3,N1 + τ -1 N1 S N1 , P3,1 + τ -1 1 S1 , . . . , + τ -1 N2 SN2 ) , Φ 11 =A 0 P 1 + P 1 A 0 + N1 k=1 (τ k R k + S k + P 2,k + P 2,k ) + N2 s=1 (τ s Rs + Ss + P2,s + P 2,s ) , Φ 12 =[P 1 A 1 -P 2,1 • • • P 1 A N1 -P 2,N1 ] , Φ 13 =[P 1 Ā1 (γ) -P2,1 • • • P 1 ĀN2 (γ) -P2,N2 ] , Φ 14 =[A 0 P 2,1 + P 3,1 • • • A 0 P 2,N1 + P 3,N1 ] , Φ 15 =[A 0 P2,1 + P3,1 • • • A 0 P2,N2 + P2,N2 ] , Φ 22 =-diag(S 1 , . . . , S N1 ) , Φ 23 = 0 , Φ 24 =         ϕ 1 A 1 P 2,2 • • • A 1 P 2,N1 A 2 P 2,1 ϕ 2 • • • A 2 P 2,N1 . . . . . . . . . A N1 P 2,1 • • • ϕ N1         , with ϕ k = A k P 2,k -P 3,k , Φ 25 =      A 1 P2,1 • • • A 1 P2,N2 . . . . . . . . . A N1 P2,1 • • • A N1 P2,N2      , Φ 33 =-diag( S1 , . . . , SN2 ) , Φ 34 =      Ā 1 (γ)P 2,1 • • • Ā 1 (γ)P 2,N1 . . . . . . . . . Ā N2 (γ)P 2,1 • • • Ā N2 (γ)P 2,N1      , September 6, 2019 DRAFT Φ 35 =         φ1 Ā 1 (γ) P2,2 • • • Ā 1 (γ) P2,N2 Ā 2 (γ) P2,1 φ2 • • • Ā 2 (γ) P2,N2 . . . . . . . . . Ā N2 (γ) P2,1 • • • φN2         , with φs = Ā s (γ) P2,s -P3,s , Φ 44 =-diag(τ -1 1 R 1 , . . . , τ -1 N1 R N1 ) , Φ 45 = 0 , Φ 55 =-diag(τ -1 1 R1 , . . . , τ -1 N2 RN2 ) .
Theorem 2. Consider (3) satisfying Assumption 1, and a given x * ∈ R n . If there exist γ ∈ Γ, and matrices

P 1 , P 2,k , P2,s , P 3,k , P3,s , S k , Ss , R k , Rs , such that (5) holds, P 1 , P 3,k , P3,s , S k , Ss , R k , Rs are symmetric,
S k , Ss , R k , Rs , P > 0, and M (γ) < 0, then (3) in closed-loop with the switching controller

σ(t) ∈ arg min j∈{1,...,N } ∆ 1 + ∆ 2 + ∆ 3 ∆ 4 (j) + B j , (15)
with

∆ 1 := (w(t) -x * ) P 1 , ∆ 2 := N1 k=1 ( ŵk (t) -x * τ k ) P 2,k , ∆ 3 := N2 s=1 ( ŵs (t) -x * τs ) P 2,s , ∆ 4 (j) := N2 s=1 A s,j x(t -τs + ς), w(t) := e A0ς x(t) + 0 -ς e -A0η N1 k=1 A k x(t -τ k + ς + η)+ N2 s=1 A s,
  Φ 1 Φ 2 * Φ 3   < 0 , where Φ 1 = P 1 A 0 + A 0 P 1 + N1 k=1 S k + N2 s=1 Ss , Φ 2 = [P 1 A 1 • • • P 1 A N1 P 1 Ā1 (γ) • • • P 1 ĀN2 (γ)]
, and Φ 3 = -diag(Q 1 , . . . , Q N1 , Q1 , . . . , QN2 ). Then, (3) in closed-loop with the switching controller

σ(t) ∈ arg min j∈{1,...,N } (w(t) -x * )P 1 N2 s=1 A s,j x(t -τs + ς) + B j , w(t) = e A0ς x(t) + 0 -ς e -A0η N1 k=1 A k x(t -τ k + ς + η)+ N2 s=1 A s,σ(t+η) x(t -τs + ς + η) + B σ(t+η) dη , is such that x(t) → x * as t → ∞.
and Π22 = Π 22 -diag(τ -1 1 S 1 , . . . , τ -1 N1 S N1 , τ -1 1 S1 , . . . , τ -1 N2 SN2 ).
For the proof of Corollary 2, set in (16) P 2,k = P2,s = P 3,k = P3,s = R k = Rs = 0.

IV. EXAMPLES

In the examples of this section the LMIs were solved in MATLAB by using the solver SeDuMi [START_REF] Sturm | Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF] version 1.3, and the toolbox YALMIP [START_REF] Löfberg | YALMIP : A Toolbox for Modeling and Optimization in MATLAB[END_REF]. For the simulations, we used Simulink with the fixed-step Euler integration method.

A. Delay-independent controller

Here we consider the example given in [START_REF] Sanchez | A switching controller for a class of MIMO bilinear time-delay systems[END_REF] where the controller is designed by considering delayindependent conditions. Thus, we change the delay in the state (twice the original) to show that the same controller works. Consider (1) with n = 2, m = 2, N 1 = 0, N 2 = 1, τ = 2, ς = 1/2, and

A 0 =   -3 1 2 -5   , A 1,1 =   4 2 1 6   , A 1,2 =   1 1 2 2   , B =   3 1 1 2   .
The control components are assumed to be on-off, i.e. The initial conditions were chosen as x(t) = 0, u(t) = 0 for all t ≤ 0, and the integration step was equal to 0.1ms. Fig. 1 shows the system's states converging to the reference point. The vertical doted lines indicate the time when the delayed inputs start to switch in a high-frequency regime. Such a switching can be appreciated in the nondelayed control signals shown in Fig. 2.

It is important to mention that for this example it can also be used the controller with delay-dependent conditions given in Theorem 1. However, note that (10) is simpler than [START_REF] Sira-Ramirez | Control Design Techniques in Power Electronics Devices[END_REF]. Note that in this case A 0 is not a Hurwitz matrix, therefore, the controller given in Corollary 1 cannot be used. The initial conditions were chosen as x(t) = 0, u(t) = 0 for all t ≤ 0, and the integration step was equal to 0.1ms. The controller developed in this paper can be used in a wide set of bilinear systems where the delay cannot be neglected. The conceptual simplicity of the controller let it to be modifiable by changing the Lyapunov-Krasovskii functional. An important potential application of this controller is in the field of turbulent flow control systems, since it is suitable for the kind of models and on-off actuators used in some flow control applications, see e.g. [START_REF] Feingesicht | SISO model-based control of separated flows: Sliding mode and optimal control approaches[END_REF].

B. Delay-dependent controller

Future work: There are several aspects that can be considered to improve or extend the control approach used in this paper. We can mention, e.g. robustness analysis, consideration of time varying delays, or the presence of several delays in the input.

  Paper organization: In Section II we describe briefly the control problem. The controller design is explained in Section III. Some simulation examples are shown in Section IV. Concluding remarks and future work are stated in Section V.

  where | • | denotes the Euclidean norm. To prove Corollary 1, just set R = P 2 = P 3 = 0 in the functional (12), and follow the same reasoning. September 6, 2019 DRAFT

Corollary 2 .

 2 Consider (3) satisfying Assumption 1, and a given x * ∈ R n . If there exist γ ∈ Γ and symmetric matrices P 1 , S k , Ss ∈ R n×n , k = 1, . . . , N 1 , s = 1, . . . , N 2 , such that (5) holds, P 1 , S k , Ss > 0 , and M (γ) :=

1 Fig. 1 .

 11 Fig. 1. States of the system.

Fig. 2 .

 2 Fig. 2. Control signals.

Consider again ( 1 )

 1 with n = 2, m = 2, N 1 = 0, N 2 = 1, τ = 1/3, ς = 1/6,and A

Fig. 3 .

 3 Fig. 3. States of the system.

  time

Fig. 4 .

 4 Fig. 4. Control signals.

Fig. 3

 3 shows the system's states converging to the reference point. The vertical doted lines indicate the times when the delayed inputs start or stop switching in a high-frequency regime. Such a switching can be appreciated in the nondelayed control signals shown in Fig. 4.

  σ(t+η) x(t -τs + ς + η) + B σ(t+η) dη ,

	ŵk (t) :=	t t-τk+ς x(ν) dν +	t t-ς w(ν) dν, and ŵs (t) =	t t-τs+ς x(ν) dν +	t t-ς w(ν) dν, is such that
	x(t) → x

* as t → ∞.

For[START_REF] Mohler | Bilinear Control Processes: With Applications to Engineering[END_REF], the conditions (on the initial function) that guarantee existence and uniqueness of solutions can be relaxed, i.e. φ can be assumed to be a Borel-measurable bounded function[START_REF] Agarwal | Nonoscillation Theory of Functional Differential Equations with Applications[END_REF] Theorem B.1].September 6,

DRAFT

September 6, 2019 DRAFT

In general, the right-hand side of the closed-loop[START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF],[START_REF] Sira-Ramirez | Control Design Techniques in Power Electronics Devices[END_REF] is not continuous in t, however, its solutions are absolutely continuous in t, as stated in Remarks 1 and 2. Thus, the smoothness properties of the functional V guarantee that its evaluation along a system's trajectory is an absolutely continuous function, therefore, its time derivative exists almost everywhere. Hence, the standard Lyapunov-Krasovskii theorems are still valid for our case. Nevertheless, there exist some stability results where Carathéodory conditions and non-differentiable Lyapunov-Krasovskii functionals are considered, see e.g.[START_REF] Pepe | On Liapunov-Krasovskii functionals under Carathéodory conditions[END_REF].September 6, 2019 DRAFT

This work was carried out within the framework of the CNRS Research Federation on Ground Transports and Mobility, in articulation with the ELSAT2020 project supported by the European Community, the French Ministry of Higher Education and Research, the Hauts de France Regional Council. It was also supported by Agence Nationale pour la Recherche DIGITSLID ANR-18-CE40-0008; the Israel Science Foundation, grant No. 1128/14; the Ministry of Education and Science of Russian Federation, project 14.Z50.31.0031; and the Government of Russian Federation, grant 08-08.


Henrietta Essie Whyte

Cécile Raillard

Albert Subrenat

Valérie Héquet

Influence of environmental parameters on the photocatalytic oxidation efficiency of acrylonitrile and isoflurane; two operating room pollutants

Keywords:

A C T

In hospitals, operating rooms (ORs) are very demanding in terms of the indoor air quality (IAQ) and require systems that minimize the concentrations of pollutants (microorganisms, chemical and particulate matter). Air treatment devices that use photocatalytic oxidation (PCO) could potentially be used in the OR to improve IAQ. In this work, the fate of two OR pollutants acrylonitrile (chemical found in surgical smoke) and isoflurane (anesthetic gas) when they go through a PCO device was investigated. The experiments were conducted in a laboratory closed loop multi-pass reactor. A mathematical model was utilized to enable the calculation of one indicator (single-pass removal efficiency) for acrylonitrile and two indicators (induction period and single-pass removal efficiency) for isoflurane. The degradation efficiency was then accessed by studying the influence of environmental parameters on these indicators. The parameters that were studied are the relative humidity, presence of co-pollutants and presence of particles. The parameters were observed to have similar effects on the degradation of both compounds. Increasing relative humidity inhibited the degradation probably due to competitive adsorption. The presence of co-pollutants like nitrous oxide and acetic acid caused a possible competition for adsorption unto active sites thus decreased the degradation efficiency of acrylonitrile and isoflurane. The increase in the concentration of the co-pollutants enhances the competitive effect and further decreases the degradation efficiency of the target pollutants. Finally the presence of particles on the photocatalytic media could block active sites thereby inhibiting the degradation of acrylonitrile and isoflurane.

Introduction

The hospital is a more complex environment than that of residential and commercial buildings. There are several zones where specific activities are performed and thus, occupants (hospitals workers, patients, etc) are exposed to different chemical pollutants [START_REF] Leung | Control and management of hospital indoor air quality[END_REF][START_REF] Gangneux | Concomitant determination of indoor air microbiological and chemical contamination for a global approach of air quality in hospitals[END_REF]. Another factor adding to the complexity of hospitals is the fact that the different zones have various indoor air quality (IAQ) requirements. In the hospital, the operating room (OR) is one of the most demanding areas when it comes to the IAQ and requires careful control of its environment to minimize pollutant levels [START_REF] Balaras | HVAC and indoor thermal conditions in hospital operating rooms[END_REF]. The indoor air of an operating room is polluted by different chemical pollutants like waste medical gases used for anesthesia, disinfection and sterilizing substances and surgical smoke [START_REF] Balaras | HVAC and indoor thermal conditions in hospital operating rooms[END_REF][START_REF] Dascalaki | Air quality in hospital operating rooms[END_REF][START_REF] Barrett | Surgical smoke: a review of the literature[END_REF]. The two main culprits however are waste anesthetic gases and chemicals present in surgical smoke. The presence of these substances is known to have adverse health effects on and can lead to discomfort of OR personnel thus, personnel exposure should be limited [START_REF] Dascalaki | Air quality in hospital operating rooms[END_REF].

Isoflurane is one of the most commonly used anesthetic gases worldwide and it is normally present in the air through leaks in the patient breathing circuit, during installation and disconnection of the system and also when patient exhales. Studies have demonstrated that chronic exposure to isoflurane can cause liver and kidney diseases, irritation of skin, eyes, mouth and throat and can also increase the risks of spontaneous abortion and congenital abnormalities [START_REF] Byhahn | Occupational exposure to volatile anaestheticsepidemiology and approaches to reducing the problem[END_REF][START_REF] Chaoul | Does occupational exposure to anesthetic gases lead to increase of pro-inflammatory cytokines?[END_REF][START_REF] Gupta | Does exposure to inhalation anesthesia gases change the ratio of X-bearing sperms and Y-bearing Sperms? A worth exploring project into an uncharted domain[END_REF][START_REF] Choosong | Assessment of waste anesthetic gases exposure among nurse anesthetists in a university hospital[END_REF][START_REF] Checkai | Risk Assessment for Occupational Exposure to Isoflurane in Pharmaceutical Research and Veterinary Facilities in San Diego[END_REF]. The occupational exposure limits (OEL) for an 8-h time weighted average set by NIOSH is 2 ppm without concomitant N 2 O exposure and 0.5 ppm with concomitant N 2 O exposure. Surgical smoke is produced during electro, laser or ultrasonic surgery to cut, coagulate, desiccate, fulgurate, or vaporize tissues. It is known to consist of 95% water vapor and 5% blood and particulate matter (tissue particles, viruses and bacteria) and chemical compounds [START_REF] -M. Fan | Surgical smoke[END_REF]. Surgeons and operating theatre staff are routinely exposed to pollution from the surgical smoke. It hinders the vision of the surgeon, produces an unpleasant odor, and releases harmful substances into the air that are harmful to the health of the staff [START_REF] -M. Fan | Surgical smoke[END_REF]. The chemicals present in greatest quantity in surgical smoke are hydrocarbons and nitrile compounds among which is acrylonitrile a hazardous compound of concern. Short-term exposure to acrylonitrile can cause eye irritation, nausea, vomiting, headache, sneezing, E-mail address: whytehenrietta@yahoo.com (H.E. Whyte).

. Generation of pollutant concentrations

To generate the desired concentrations of the pollutants, the process started first by generating clean humid air. To do this, compressed air from the laboratory network was passed through an air zero generator Claind AZ 2020 to remove organic gaseous compounds and then through a Pressure swing Adsorption (PSA) device to remove humidity and CO 2 . The air was then humidified to a desired level by bubbling it through a water bottle at an average room temperature of 20 °C. Then the pollutants were mixed with the humid air zero to dilute them before injection into the reactor. Acrylonitrile (C 3 H 3 N) was supplied by Air products whilst isoflurane (C 3 H 2 ClF 5 O) was supplied by Linde Gas. Nitrous oxide (N 2 O) was supplied by Air Liquide. The pollutants were supplied in certified commercial gas cylinder balanced in nitrogen at 150 bars. Finally, the desired concentrations were achieved by adjusting the gas flow rates using mass flow controllers (MFC). For acetic acid, it was injected into the reactor as a vapor by means of a 10 mL airtight syringe into the tranquilization chamber. The vapors were prepared by evaporating liquid acetic acid (99.95% from Sigma Aldrich) in a 285 mL pyrex glass bulb under vacuum.

Photocatalytic reactor

The reactor that was used to perform experiments is a 420-L multipass reactor that has been described in previous articles [START_REF] Batault | How chemical and physical mechanisms enable the influence of the operating conditions in a photocatalytic indoor air treatment device to be modeled[END_REF][START_REF] Raillard | Use of experimental designs to establish a kinetic law for a gas phase photocatalytic process[END_REF][START_REF] Maudhuit | Adsorption phenomena in photocatalytic reactions: the case of toluene, acetone and heptane[END_REF][START_REF] Whyte | Photocatalytic oxidation of isoflurane, an anesthetic gas: the influence of operating parameters[END_REF]. It is a closed-loop reactor made up entirely of stainless steel to minimize surface losses of pollutants. The reactor includes a photocatalytic module containing a plane geometry photocatalytic media and two 18-W UVC lamps (Phillips TUV PL-L series), a variable speed fan and a tranquilization chamber. The air stream flowing through the reactor is controlled by the fan. For the present study, the flow was set at 148 Nm 3 h -1 . The studied compounds are introduced and sampled through tranquilization chamber which has a honeycomb placed at its inlet to provide a homogeneous flow distribution.

Leak tests were performed prior to degradation experiments at several flow rates and were estimated to range from 0.1 to 2.3% per hour when flow rates increased from 73 to 216 Nm 3 h -1 respectively. Preliminary experiments were also carried out to evaluate the extent of adsorption and photolysis processes on pollutant removal. The removal rates for adsorption and photolysis were calculated at 0.1 and 0.3% per hour respectively indicating that no significant loss of pollutants was observed under UV light without the media or when exposed to the media without UV light.

A Quartzel ® photocatalytic media was used in this work. It is a commercial media produced by Saint-Gobain Quartz made up of SiO 2 fibers coated with TiO 2 deposited through a sol-gel method and pressed into a felt. The media has an average thickness of 17.5 ± 1.5 mm, a developed surface of 0.04 m 2 (20 cm × 20 cm) and a TiO 2 load of 94 g m -2 . A Micrometrics ASAP 2020 device using N 2 adsorption measured the BET specific surface area as 112 ± 1 m 2 g -1 with a Micrometrics ASAP 2020 by N 2 adsorption. A JOEL JSM 5800LV Scanning Electron microscope (SEM) was used to analyze the surface of the media. The results (Fig. 2 a andb) showed that the coated fibers are long and interconnected and form a complex mesh. This complexity offers a tortuous path for air flow and increases the probability of pollutants to be trapped and degraded. The SEM image (Fig. 2 b) shows the variation in thickness of the coated fibers. The fiber thickness was estimated to be in the range 10 μm-20 μm.

Sampling and analytical methods

To monitor the pollutant concentrations over the period of the degradation, samples were taken onto stainless-steel catridges packed with Carbopack B (Perkin Elmer). The Automatic Clean Room Sampling System (ACROSS) by Tera Environnement was used to sample at a rate of 200 mL min -1 for 2.5 min. The analysis was carried out using a Thermal Desorber/Gas Chromatograph/Flame Ionization Detector/ Mass Spectrometer (TD/GC/FID/MS). The column used in the GC was a polar column from Restek (Rxi-624Sil MS) which was connected to a weakness and light headedness, whilst long-term exposure may cause cancer [START_REF] Hill | Surgical smoke -a health hazard in the operating theatre: a study to quantify exposure and a survey of the use of smoke extractor systems in UK plastic surgery units[END_REF][START_REF] Mowbray | Is surgical smoke harmful to theater staff? a systematic review[END_REF]. The occupational exposure limit (OEL) for an 8-h time weighted average set by NIOSH is 2 ppm. The concentrations of these pollutants found in the OR is limited and in the literature the concentrations of isoflurane and acrylonitrile can range from 0.01 to 15 ppm [START_REF] Dascalaki | Air quality in hospital operating rooms[END_REF][START_REF] Barrett | Surgical smoke: a review of the literature[END_REF][START_REF] Hobbhahn | Waste gas exposure during desflurane and isoflurane anaesthesia[END_REF][START_REF] Neisi | Association of anesthetic toxic isoflurane gases of the indoor air of operating room, Ahvaz, Iran during 2016[END_REF][START_REF] Chung | Harmful gases including carcinogens produced during transurethral resection of the prostate and vaporization[END_REF].

In ORs, occupants' health and comfort is mostly ensured by heating, ventilating and air conditioning (HVAC) systems through the dilution of indoor air pollutant concentrations by the supply of conditioned outdoor air [START_REF] Balaras | HVAC and indoor thermal conditions in hospital operating rooms[END_REF]. Desirable conditions are met by adhering to a ventilation rate. In the United States of America, ASHRAE [START_REF]ASHRAE 170-2017-ventilation of health care facilities[END_REF] recommends a minimum of 20 Air Change per Hour (ACH) whilst the standard NFS 90-351 for France recommends the minimum to be 25 ACH [START_REF]NF S90-351 -health care institutions -controlled environment areas -requirements for airborne contamination control[END_REF]. The higher the ventilation rate, the better the dilution effect. However, higher rates result in increased energy consumption necessary to condition (heating, cooling, humidification and dehumidification) the air. It may also supply pollutants from outdoors [START_REF] Scheepers | Chemical characterization of the indoor air quality of a university hospital: penetration of outdoor air pollutants[END_REF]. In addition, filters installed in typical HVACs are targeted for particulate matter and bioaerosols control and as such not very efficient at removing some indoor chemical pollutants [START_REF] Zhong | Experimental and modeling study of visible light responsive photocatalytic oxidation (PCO) materials for toluene degradation[END_REF]. Therefore, implementation of air treatment technologies whether as standalone or as part of the existing HVAC system (in-duct), is becoming popular as a complementary means of improving IAQ. These technologies control chemical pollutants and allow air quality levels to be maintained with reduced outdoor air supply and concomitant energy savings [START_REF] Bergeron | Decreasing airborne contamination levels in high-risk hospital areas using a novel mobile air-treatment unit[END_REF][START_REF] Le | Air purification equipment combining a filter coated by silver nanoparticles with a nano-TiO 2 photocatalyst for use in hospitals[END_REF][START_REF] Perraud | Air filtration and prevention of aspergillary pneumopathies. Preliminary comparative study of two mobile units for bacteriological air purification with recycling[END_REF][START_REF] Rosenbaum | Use of a portable forced air system to convert existing hospital space into a mass casualty isolation area[END_REF][START_REF] Schoenleber | Évaluation d'un dispositif mobile de décontamination pour la maîtrise de l'air au bloc opératoire[END_REF]. Technologies for removing VOC contaminants include ozonation, cold plasma, and photocatalytic oxidation (PCO). Among the different air cleaning technologies used for the removal of these pollutants, photocatalytic oxidation seems to be more and more employed.

PCO has been studied for several decades and is a cheap, energy efficient solution for air treatment [START_REF] Zhong | Photocatalytic air cleaners and materials technologiesabilities and limitations[END_REF][START_REF] Zhao | Photocatalytic oxidation for indoor air purification: a literature review[END_REF]. It removes gaseous contaminants via chemical reactions on semiconductor (mostly TiO 2 ) catalyst surface under UV irradiation. It has the advantage of being a flexible solution for the degradation of a wide spectrum of gaseous pollutants [START_REF] Destaillats | Key parameters influencing the performance of photocatalytic oxidation (PCO) air purification under realistic indoor conditions[END_REF][START_REF] Boyjoo | A review on photocatalysis for air treatment: from catalyst development to reactor design[END_REF][START_REF] Ren | Photocatalytic materials and technologies for air purification[END_REF][START_REF] Mo | Novel insight into VOC removal performance of photocatalytic oxidation reactors[END_REF][START_REF] Farhanian | Impact of design parameters on the performance of ultraviolet photocatalytic oxidation air cleaner[END_REF]. Over the years most of the studies have been performed on common home, office and school pollutants like formaldehyde, toluene, benzene whilst little or no attention is paid to the degradation of specific hospital pollutants. The performance of PCO is influenced by several factors that can be categorized in two types; process parameters (eg. air velocity, light intensity) and environmental parameters (e.g initial concentration, relative humidity, and presence of co-pollutants). The influence of these key parameters has been greatly studied in the literature [START_REF] Raillard | Kinetic study of ketones photocatalytic oxidation in gas phase using TiO2-containing paper: effect of water vapor[END_REF][START_REF] Sleiman | Photocatalytic oxidation of toluene at indoor air levels (ppbv): towards a better assessment of conversion, reaction intermediates and mineralization[END_REF][START_REF] Ao | Photodegradation of formaldehyde by photocatalyst TiO2: effects on the presences of NO, SO2 and VOCs[END_REF][START_REF] Ye | Photocatalytic oxidation of aldehydes: byproduct identification and reaction pathway[END_REF][START_REF] Chen | Photocatalytic oxidation of multi-component systemsan investigation using toluene/ethylbenzene, octane/decane/dodecane and formaldehyde/acetaldehyde[END_REF][START_REF] Yu | Indoor air purification using heterogeneous photocatalytic oxidation. Part I: experimental study[END_REF][START_REF] Pichat | Some views about indoor air photocatalytic treatment using TiO2: conceptualization of humidity effects, active oxygen species, problem of C1-C3 carbonyl pollutants[END_REF][START_REF] Dillert | Influence of inlet concentration and light intensity on the photocatalytic oxidation of nitrogen(II) oxide at the surface of Aeroxide® TiO2 P25[END_REF]. Air treatment devices that use PCO could potentially be used in ORs thus the fate of gaseous OR pollutants when they go through the PCO device should also be studied.

This work for the first time investigates the performance of a PCO system in removing isoflurane and acrylonitrile, two compounds present in the OR indoor air with the objective of studying the influence of three environmental parameters, relative humidity, presence of copollutants and presence of particles on their degradation under conditions that are relevant to actual applications. The experiments were conducted in a closed loop reactor which has been designed to study low concentration air pollutants and has also been recently modeled [START_REF] Dumont | Determination of the clean air delivery rate (CADR) of photocatalytic oxidation (PCO) purifiers for indoor air pollutants using a closedloop reactor. Part I: theoretical considerations[END_REF][START_REF] Héquet | Photocatalytic oxidation of VOCs at ppb level using a closed-loop reactor: the mixture effect[END_REF]. This reactor also permits a better representation of PCO systems functioning in dynamic mode and is realistic regarding air treatment systems working as standalone devices or implanted in HVAC systems.

Materials and methods

Experimental set-up

The experimental set-up used in this study is presented in Fig. 1.It consists of the VOC generation bench, the photocatalytic reactor and the sampling and analytical devices.

FID for the quantification of compounds and a MS for the identification of compounds. The TD/GC/FID/MS analysis conditions are described in Table 1.

Particle loading of photocatalytic media

The effect that the presence of particles may have on the photocatalytic degradation efficiency was studied by clogging the media surface with micronized rice. The micronized rice particles were chosen to simulate the presence of particles. The particle diameter range was 0.25-10.5 μm thus was a good representation of particles that could be collected on the photocatalytic media in an OR as for example surgical smoke is known to contain particles with diameters ranging from 0.35 to 6.5 μm [START_REF] Weld | Analysis of surgical smoke produced by various energy-based instruments and effect on laparoscopic visibility[END_REF]. The generated particles were first oven-dried overnight at 50 °C. A Palas RBG 1000 particle generator was used to load the rice particles onto the photocatalytic media. A generation time of about 45 min allowed a loading rate of 0.002 g m -2 .

Calculating performance indicators

In order to evaluate the influence of the parameters on the degradation of isoflurane and acrylonitrile, quantitative indicators were needed. Before the indicators are discussed, a brief presentation of the degradation profiles of acrylonitrile and isoflurane are presented.

Fig. 3 shows the degradation profile of acrylonitrile. Under the chosen reference experimental conditions of C 0 = 2 ppm, v = 1 m s -1 and I = 4.5 mW m -2 , the complete degradation of acrylonitrile was achieved in 1 h. The degradation profile is characterized by a mono-exponential decay curve which is representative of an apparent first order decay. By fitting the experimental points to the mono-exponential equation [C/C 0 = exp(-k app t)], the model fit almost perfectly to the experimental points as indicated by coefficient of determination (R 2 = 0.98) in Table 2.

The reference experiments were performed three times to access the repeatability. The calculation of the first order constants (k app ) calculated from the mono-exponential equation were quite similar (Table 2) indicating the repeatability of the experiments.

Fig. 4 shows the degradation profile of isoflurane. Under the experimental conditions C 0 = 0.5 ppm, v = 1 m s -1 , I = 4.5 mW cm -2 , it can also be seen that the degradation curve can be divided into two distinct phases: the first phase where degradation occurs slowly and a second phase where the degradation accelerates. This degradation behavior of isoflurane was similar to what some authors have described in the literature for another chlorinated compound trichloroethylene. The authors state that trichloroethylene undergoes a reaction mechanism that involves OH°and Cl° [START_REF] Debono | VOC ternary mixture effect on ppb level photocatalytic oxidation: removal kinetic, reaction intermediates and mineralization[END_REF][START_REF] Hegedüs | Comparative study of heterogeneous photocatalytic decomposition of tetrachloroethene and trichloroethene in the gas phase[END_REF][START_REF] Li Puma | Kinetics rate model of the photocatalytic oxidation of trichloroethylene in air over TiO2 thin films[END_REF][START_REF] Jacoby | Products, intermediates, mass balances, and reaction pathways for the oxidation of trichloroethylene in air via heterogeneous photocatalysis[END_REF]. OH°first attacks TCE to generate Cl°. The Cl°then attack trichloroethylene and ensure the continous attack of TCE by propagating a chain reaction. This propagation plays a key role in increasing the degradation rate of TCE. Based on this, a hypothesis is therefore made that isoflurane may probaly undergo a reaction mechanism similar to that of tricholorethylene. In that, the first phase could probably correspond to OH°reactions with isoflurane to generate Cl°. The second phase could probably be attributed to Cl°reactions with isoflurane leading to a possible chain propagation reaction.

The reference experiment was also carried out at regular intervals to ascertain the repeatability of the experiments. The results are shown to be repeatable (Fig. 4).

Based on the degradation mechanisms, one indicator, the singlepass removal efficiency was chosen for acrylonitrile whilst two indicators, the single pass removal efficiency and the induction period (duration of the first phase), were chosen for isoflurane.

As previously mentioned, the reactor used in this work is a multi pass reactor. This reactor is well adapted to provide information on the decay profiles but does not provide direct information on the singlepass removal efficiency. This indicator has to be determined indirectly by employing mathematical models. In this work the model that is used has been developed by Dumont and Héquet and is explained elsewhere [START_REF] Dumont | Determination of the clean air delivery rate (CADR) of photocatalytic oxidation (PCO) purifiers for indoor air pollutants using a closedloop reactor. Part I: theoretical considerations[END_REF]. The model, which describes the first order decay relationship of concentration vs. time in a closed-loop reactor is given as follows:

= ⎧ ⎨ ⎩ - - - ⎫ ⎬ ⎭ C t τ exp α C exp [1 ( )] R 0 (1)
In Eq. ( 1) the term α is the single-pass removal efficiency and represents the fraction of the total flow treated during the time τ R (re- sidence time in the reactor). C is the pollutant concentration at time t and C 0 is the initial pollutant concentration. Numerical resolutions were carried out using Excel ® Solver which is based on the least-square method.

Fig. 5 shows acrylonitrile degradation experimental points fitted to the model (Eq. ( 1)). The model is shown to fit the experimental points satisfactorily based on the regression co-efficient (R 2 ) of 0.99. Under these conditions, the single-pass removal efficiency (α) was then calculated as 0.012. This indicated that in one pass, 1.2% of acrylonitrile would be eliminated from the total flow that passed through the media.

Fig. 6 shows the isoflurane experimental concentration decay curve fitted to the model (Eq. ( 1)). It can be observed for irradiation time from t = 0 to t = 130 min that the experimental points are not well The limits of detection was calculated by multiplying the background noise by 3. It was calculated as 0.1 ppm for isoflurane and 0.075 ppm for acrylonitrile. Table 2 First order constants and coefficient of determination calculated from first order equation for the three references acrylonitrile experiments (C 0 = 2 ppm; v = 1 m s -1 ; I = 4.5 mW cm -2 ; RH = 50%). To be able to study the influence of the parameters on the performance indicator, the parameter of interest was varied whilst all others were kept constant. Due to the duration of each experiment and the number of experiments that needed to be done, the coefficient of variation on the performance indicators for the reference experiments was calculated and reported for all the set of experiments. The coefficient of variation on the reference experiment was calculated as follows:

= * δ σ x ¯100 ( 2
)
Where δ is the coefficient of variation, σ is the standard deviation and x ¯is the average value of the performance indicator calculated from the three reference experiments. The coefficient of variation calculated on α for the acrylonitrile experiments based on the mean and standard deviation of the thrice repeated reference experiment was calculated as 1%. Whilst the coefficient of variation on α and τ i was calculated as 9% and 5% respectively for isoflurane experiments based on the repetition of the reference experiment.

Results and discussion

Relative humidity (RH)

In ORs, the humidity levels as recommended by national guidelines such as NFS 90-351 [START_REF]NF S90-351 -health care institutions -controlled environment areas -requirements for airborne contamination control[END_REF] and ASHRAE 170-2017 [START_REF]ASHRAE 170-2017-ventilation of health care facilities[END_REF] for France and the US respectively is 30-60%. However levels measured could be outside this range. Wong et al. [START_REF] Balaras | HVAC and indoor thermal conditions in hospital operating rooms[END_REF] measured levels of 80% in a Malaysian OR. As a result of this, in the present study, the influence of RH on the degradation process of the two target compounds were studied from 20% to 80% at 20 °C which gave moisture content of 4657-18893 ppm. This would provide representative information on how different RH levels encountered in the ORs affect the performance of PCO during the degradation of isoflurane and acrylonitrile.

In the literature, the role of water vapor in the photocatalytic oxidation process has been described as a dual one [START_REF] Mamaghani | Photocatalytic oxidation technology for indoor environment air purification: the state-of-the-art[END_REF][START_REF] Mo | Photocatalytic purification of volatile organic compounds in indoor air: a literature review[END_REF][START_REF] Tytgat | Photocatalytic process optimisation for ethylene oxidation[END_REF][START_REF] Zhong | Modeling and physical interpretation of photocatalytic oxidation efficiency in indoor air applications[END_REF]. On one hand adsorbed water molecules are oxidized into OH°which are necessary for the progress of the reaction whilst on the other hand excessive amounts of water creates a competition with the pollutants for adsorption onto active sites which inhibits the degradation. When varying humidity, some studies have shown that both effects can occur depending on the humidity range studied whilst other studies have reported that only one of these effects is visible. For example, Mo et al. [START_REF] Mo | Effect of water vapor on the by-products and decomposition rate of ppb-level toluene by photocatalytic oxidation[END_REF] studied the influence of relative humidity (RH) in a range of 0-70% on the degradation of toluene (0.7 ppm). They showed that the removal efficiency of toluene increased for RH ranging from 0 to 35% due to the promotion of the degradation by the formation of OH°and decreased for RH from 35% to 70% as a result of competitive adsorption between the toluene molecules and the water molecules.

For the acrylonitrile experiments, velocity and light intensity were fixed at 1 m s -1 and 4.5 mW cm -2 respectively whilst initial concentration was maintained at 2 ppm. Fig. 7 shows the influence of RH on acrylonitrile degradation. It can be observed that the single-pass removal efficiency decreases when the RH is increased from 20% to 80%.

The results for the influence of RH on the degradation of isoflurane are shown in Fig. 8. The results of the experimental study show that as the relative humidity is increased from 20% to 80%; (i) isoflurane induction periods are increased; (ii) the single-pass removal efficiency is decreased.

The results from this study show that increasing the relative humidity decreases the degradation efficiency of both compounds. This observation can be interpreted as the competition for adsorption between the pollutants and water molecules. As the amount of water molecules increases, they occupy the active sites which in turn decreases the chance the pollutants have to react with the active species resulting in lower removal efficiencies [START_REF] Raillard | Kinetic study of ketones photocatalytic oxidation in gas phase using TiO2-containing paper: effect of water vapor[END_REF]. This inhibition effect was also observed by Raillard et al. [START_REF] Raillard | Kinetic study of ketones photocatalytic oxidation in gas phase using TiO2-containing paper: effect of water vapor[END_REF] when they studied the influence of RH (0-30%) on the photocatalytic degradation of two ketones: acetone and 2-butanone. The experiments were performed in a 12-L Pyrex-glass batch reactor with TiO 2 powder deposited on cellulose substrates. Similarly, Krichevskaya et al. [START_REF] Krichevskaya | Gas-phase photocatalytic oxidation of acrylonitrile[END_REF] observed a decrease in the removal efficiency of acrylonitrile (40 ppm) when RH was increased from 0% to 66% in an annular reactor coated with Degussa (P25) TiO 2 powder. These authors attributed the inhibition to the effect of competitive adsorption between the pollutant molecules and water.

For the induction period of isoflurane, the increase in the induction periods during increase in RH could also be attributed to the competition between isoflurane molecules and water molecules for adsorption onto active sites. As the RH increases, water molecules prevent isoflurane molecules from reacting with active species thus longer induction periods occur until sufficient concentration of Cl°are formed to initiate the acceleration of the reaction.

At 20% RH the moisture content at 20 °C is 4657 ppm. This increases to 11725 ppm at 50% RH and 18893 ppm at 80% RH. These concentrations are quite high relative to the pollutant concentrations of 0.5 ppm and 2 ppm for isoflurane and acrylonitrile respectively. The percentage decrease in the single-pass removal efficiency from 20% to 50% RH was 36% and 20% for acrylonitrile and isoflurane respectively whilst the percentage decrease from 50% to 80% was 18% and 14% for acrylonitrile and isoflurane respectively. This shows that although the degradation of both compounds is impeded by the increase in RH, the influence was more pronounced when the RH was increased from 20% to 50% than from 50% to 80%. This could be explained by the fact that at 50% RH the surface already had abundant water and that increasing the concentration of water probably led to a saturation effect thus the removal efficiency was not drastically decreased when the RH was increased to 80%.

It was also observed that acrylonitrile was more impacted by the presence of water vapor than isoflurane. Water molecules may form layers on the photocatalyst surface and therefore, in order to react, VOCs must first be absorbed into the water film and then penetrate towards the catalyst surface which also subsequently decrease the degradation efficiency [START_REF] Zhang | Mechanistic analysis on the influence of humidity on photocatalytic decomposition of gas-phase chlorobenzene[END_REF][START_REF] Vildozo | Photocatalytic oxidation of 2propanol/toluene binary mixtures at indoor air concentration levels[END_REF]. The ability to do this could be related to the hydrophilic/hydrophobic properties of the VOC molecule. There are differing opinions given in the literature to explain this. Vildozo et al. [START_REF] Vildozo | Photocatalytic oxidation of 2propanol/toluene binary mixtures at indoor air concentration levels[END_REF] studied the influence of humidity on the degradation of 2-propanol and toluene using a photocatalyst, composed of non-woven paper coated with 20 g m -2 of TiO 2 . They observed that at high levels of humidity the degradation of toluene was more inhibited than that of 2propanol. They explained that 2-propanol because of its higher solubility (1000 g L -1 ) is able to penetrate the layers of water and be adsorbed on the catalyst surface. On the contrary, toluene which has a lower solubility of 0.5 g L -1 might have difficulties to penetrate this layer. On the other hand, Raillard et al. [START_REF] Raillard | TiO2 coating types influencing the role of water vapor on the photocatalytic oxidation of methyl ethyl ketone in the gas phase[END_REF] in studying the degradation of methyl ethyl ketone (MEK) on a photocatalyst made up of a film of TiO 2 coated onto a glass plate observed that at higher relative humidity, the degradation rate of MEK was reduced. They stated that at high humidity level, the formation of a water layer is possible and that in addition to the competitive adsorption, because of its solubility, MEK may be dissolved in the water layer. Once dissolved into water, MEK is less willing to reach with the catalytic surface and active species because of a less rapid diffusion into water.

In this work, the degradation of acrylonitrile is more inhibited than isoflurane when the relative humidity is increased. Looking at their properties (Table 3), acrylonitrile is more hydrophilic (higher water solubility and lower Log K ow ) than isoflurane. In addition to this, acrylonitrile concentration (2 ppm) is higher than that of isoflurane (0.5 ppm). The influence of relative humidity could be impacted by the concentration of the compound in that the higher the concentration of the pollutant intensifies the negative effect of the relative humidity [START_REF] Assadi | Photocatalytic oxidation of trimethylamine and isovaleraldehyde in an annular reactor: influence of the mass transfer and the relative humidity[END_REF].

Relative humidity has a significant influence on the degradation of isoflurane and acrylonitrile. Under the experimental conditions studied, PCO performance during the degradation of both isoflurane and acrylonitrile is better at lower relative humidity. According to several national regulations, RH between 30% and 60% is recommended in ORs and at these levels water vapor exists in large excess compared to pollutant concentrations (usually ppb to several tens of ppm) and thus competitive adsorption which inhibits PCO performance can be deemed as the dominating process. Under this condition, it would be advisable not to operate above RH of 60%.

Presence of co-pollutants

Single compound photocatalytic oxidation studies are convenient for understanding the kinetics of degradation of a compound but in the real case, the indoor air stream usually contains several contaminants. selected compounds but observed that in a 16 VOC mixture the interference effect among the selected VOCs became quite obvious, and their removal efficiencies were significantly decreased. The inhibitive effect observed by these authors was attributed to the possible competitive adsorption for active sites by the molecules. In this study, the decrease in the degradation of both isoflurane and acrylonitrile in the presence of the co-pollutants could also be explained by the possible competition for active sites. Acetic acid is known to have a high adsorption affinity to TiO 2 therefore it allows it to occupy more active sites and be degraded [START_REF] Batault | Acetaldehyde and acetic acid adsorption on TiO2 under dry and humid conditions[END_REF] and thus may have a higher competitive ability than the target compounds. For the isoflurane-acetic acid equi-molar mixture, it seemed that the inhibitory effect was more pronounced for the first phase than for the second phase as seen in Fig. 10 a andb. The induction period increases from 130 min in the single compound degradation to 160 min in the equi-molar binary degradation but α remains at 0.0008 for both single compound and binary mixture experiments. Since acetic acid has a better adsorption affinity to the photocatalyst, it prevents enough isoflurane molecules from reaching the active sites thus delays the degradation mechanism by increasing the induction period. It can be assumed however that once acetic acid is no longer present, isoflurane degradation normally that could explain why the single-pass removal efficiency for the second phase was not significantly different from that obtained during single isoflurane degradation.

The experiments performed in mixture with N 2 O showed that it inhibited the degradation of both target compounds. Once the concentration of N 2 O was increased its inhibitory effect on the removal of the target compounds due to competitive adsorption unto active sites was intensified further decreasing the single-pass removal efficiencies of the target compounds.

Presence of particles

Aside gaseous pollutants, particulate matter is another type of pollutant that is found in indoor environments. Surgical smoke contains particles (tissue and microbial) which could be trapped by the photocatalyst media and subsequently their presence can affect the PCO of the target compounds. Surgical smoke is known to contain particles with diameters ranging from 0.35 to 6.5 μm [START_REF] Weld | Analysis of surgical smoke produced by various energy-based instruments and effect on laparoscopic visibility[END_REF]. To simulate the presence of particles the photocatalytic surface was loaded with rice particles at a rate of 0.002 g m -2 . The particle diameter range was 0.25-10.5 μm thus was a good representation of particles that could be collected on the photocatalytic media in an OR. Fig. 11 shows the results for acrylonitrile degradation in the absence and presence of particles. The velocity, light intensity and concentration were fixed at 1 m s -1 and 4.5 mW cm -2 and 2 ppm respectively. It can be observed that α decreases by 22.5% when particles are present The OR is polluted by a variety of chemical pollutants such as waste anesthetic gases, sterilants and compounds present in surgical smoke. It is reasonable to assume that the coexistence of these compounds may have an influence on the degradation rate of isoflurane and acrylonitrile thus it would also be necessary to provide information on how the presence of co-pollutants may affect the degradation efficiency of the target compounds.

In this study nitrous oxide and acetic acid were chosen to study the effect of co-pollutants on the degradation efficiency of the target pollutants. Nitrous oxide is an anesthetic pollutant typically found in ORs. Acetic acid however is a derivative of peracetic acid (sterilant) and was chosen to represent peracetic acid as the latter was not stable enough to allow its use. In this work, a series of experiments conducted at different binary mixture configurations of nitrous oxide and acetic acid with acrylonitrile and isoflurane were made. As the co-pollutants may vary in concentration in an OR, the experiments were performed in two ways. Firstly degradation was performed at equi-molar concentrations of the co-pollutants and the target pollutants. Then the experiments were conducted with the co-pollutant at a higher concentration. Since the reference concentrations are different for isoflurane (0.5 ppm) and acrylonitrile (2 ppm), the higher concentration of the co-pollutant was chosen so that the molar ratios between the reference concentrations and co-pollutant concentration were the same for both isoflurane and acrylonitrile experiments. The compositions of these mixtures are described in Table 4.

Fig. 9 shows the experimental results for the binary mixtures compared to single compound experiments for acrylonitrile. The velocity and light intensity were fixed at 1 m s -1 and 4.5 mW cm -2 respectively. Acrylonitrile results showed that the single pass removal efficiencies were lower for binary mixtures than for the single component experiment. In the presence of the co-pollutants, the α decreased by 20%, 8% and 47% for mixtures A1, N1 and N2 respectively. For the equi-molar mixtures, acetic acid had a stronger inhibitory effect on acrylonitrile than nitrous oxide. It was also observed that the inhibition of acrylonitrile degradation was higher with increasing nitrous oxide concentrations.

Fig. 10 shows the experimental results for the binary mixtures compared to single compound experiments for isoflurane. The velocity and light intensity were fixed at 1 m s -1 and 4.5 mW cm -2 respectively. Isoflurane experiments showed that, in the presence of these co-pollutants the degradation of isoflurane was inhibited. The inhibitive effect is seen in the fact that the induction periods were increased whilst the removal efficiencies were decreased. The induction periods increased by 17%, 24% and 46% for mixtures A2, N3 and N4 respectively. The single-pass removal efficiencies decreased by 1%, 13% and 63% for mixtures A2, N3 and N4 respectively. Contrary to acrylonitrile, isoflurane single pass removal degradation was more negatively influenced by the presence of nitrous oxide than by acetic acid for the equimolar experiments. The inhibition of isoflurane degradation was higher with increasing nitrous oxide concentrations.

This trend of decreasing removal efficiencies was also been reported by Chen et al. [START_REF] Chen | Adsorption and degradation of model volatile organic compounds by a combined titania-montmorillonite-silica photocatalyst[END_REF] using an equi-molar ternary mixture of 0.5 ppm toluene, ethyl acetate and ethanethiol. On the other hand, Chen et al. [START_REF] Chen | UV-PCO device for indoor VOCs removal: investigation on multiple compounds effect[END_REF] observed no significant influence of the mixture effect on low concentrations (0.1 ppm) of equi-molar binary or ternary mixtures of Table 4 Binary mixture compositions. on the photocatalytic media.

The results for isoflurane degradation are presented in Fig. 12. For these experiments, the velocity, light intensity and concentration were fixed at 1 m s -1 and 4.5 mW cm -2 and 0.5 ppm respectively. From the figure, it can be observed that α decreases by 15% whilst the induction time increases by 25% when particles are present on the photocatalytic media.

Under the experimental conditions employed, the degradation of both compounds is impeded by the presence of particles with the same order of magnitude. The trend of decreasing removal efficiencies has been reported by Park et al. [START_REF] Park | Development of a photoreactive fabric filter for simultaneous removal of VOCs and fine particles[END_REF]. They studied the degradation of toluene in a single-pass PCO reactor using a non woven-fabric coated with TiO 2 . They observed that when air containing particles was introduced into the photocatalytic reactor, toluene removal efficiency was decreased. They explained this by two hypotheses. First, the particles could hide the active sites which could in turn block these sites for adsorption by the target compounds effectively reducing their degradation. Secondly, the particles could promote a "screening effect" thereby reducing the amount of light reaching the catalyst surface resulting in slower production of active species. Consequently, the progress of the degradation could be impeded.

In this work, the inhibition of the degradation of isoflurane and acrylonitrile could be due to the fact that the particles block the active sites preventing the molecules from adsorbing and being degraded. Secondly the particles could also screen the light intensity thereby reducing the amount of light reaching the catalyst surface. Conclusively, it would be recommended that for a PCO unit, filters should be placed upstream of the photocatalytic media to reduce the particle load reaching the photocatalytic media in order to improve the PCO performance.

Identification of potential by-products

By-product formation was investigated in a previous part of this work under standard humid air condition (RH = 50%) in function of process parameters and without co-pollutants [START_REF] Whyte | Photocatalytic oxidation of isoflurane, an anesthetic gas: the influence of operating parameters[END_REF][START_REF] Whyte | Evaluation of the performance of photocatalytic systems for the treatment of indoor air in medical environments[END_REF]. For isoflurane, some halogenated and oxygenated by-products were identified but at low concentrations (< 0.04 ppm) and no significant risks based on a calculated Health-Related Index (HRI) [START_REF] Whyte | Photocatalytic oxidation of isoflurane, an anesthetic gas: the influence of operating parameters[END_REF]. For acrylonitrile, under initial concentrations ranging from 0.5 to 10 ppm and with sampling and analytical devices used, no by-products were identified. However, complete mineralization under longer irradiation times signified the possible removal of intermediates [START_REF] Whyte | Evaluation of the performance of photocatalytic systems for the treatment of indoor air in medical environments[END_REF]. In this work however, under the studied environmental parameters no by-products were identified.

Conclusion

In this work, acrylonitrile and isoflurane, two pollutants that are present in ORs air were chosen as representative compounds to investigate the performance of PCO by studying the influence of three environmental parameters. The experiments were performed in a closed loop laboratory reactor to evaluate the efficiency of degradation by PCO. The use of a model developed by Dumont and Héquet enables the calculation of one performance indicator (the single pass removal efficiency) for acrylonitrile and two indicators (single pass removal efficiency and induction period) for isoflurane. The qualitative descriptor enabled the study of the influence of the relative humidity, the presence of co-pollutants and the presence of particles on the degradation process.

The results showed that although the pollutants belonged to different chemical classes, the influence of the parameters on their degradation was similar. Relative humidity has a significant influence on the degradation of isoflurane and acrylonitrile. Under the experimental conditions studied, PCO performance during the degradation of both isoflurane and acrylonitrile is better at lower relative humidity. According to several national regulations, RH between 30% and 60% is recommended in ORs and at these levels water vapor exists in large excess compared to pollutant concentrations (usually ppb to several tens of ppm) and thus competitive adsorption which inhibits PCO performance can be deemed as the dominating process. Under this condition, it would be advisable not to operate above RH of 60%. Under the experimental conditions employed in this study, it can be concluded that PCO performance during the degradation of the target pollutants could be inhibited by the presence of co-pollutants due to competitive adsorption for active sites. Additionally the higher the concentration of the co-pollutant compared to the target compound the higher the inhibition effect. The concentrations of co-pollutants present in the OR air vary depending on the activities and thus cannot be controlled. However in order to improve the PCO performance it would be recommended that several photocatalytic banks (photocatalytic media and UV lamps) are installed in the PCO unit in order to maximize the removal of these pollutants and to ensure higher removal efficiency. The presence of particles inhibited the degradation of the two compounds because they blocked the active sites. Since the production of particles in the OR cannot be avoided, it would be recommended that for a PCO unit, filters should be placed upstream of the photocatalytic media to reduce the particle load reaching the photocatalytic media in order to improve the PCO performance.

In conclusion, PCO could be used as a treatment technique to improve air quality in the OR as the results show that it can degrade OR pollutants like isoflurane and acrylonitrile. However as also shown, a single-pass is not enough to remove a substantial amount of these pollutants from the incoming air. It would be recommended that for a treatment device, several photocatalytic banks plus an addition of filters are used in order to maximize the removal of the target pollutants as well as their intermediates.

T 2 . 1 . 1

 211

Fig. 1 .

 1 Fig. 1. Schematic representation of the closed-loop photocatalytic reactor.

Fig. 2 .

 2 Fig. 2. SEM images of PCO media -(a) x 70, (b) x 650.

Fig. 3 .

 3 Fig. 3. Degradation profile of acrylonitrile under the reference experimental conditions of C 0 = 2 ppm; v = 1 m s -1 ; I = 4.5 mW cm -2 ; RH = 50%.

Fig. 4 .

 4 Fig. 4. Degradation profile of isoflurane under the reference experimental conditions of C 0 = 0.5 ppm; v = 1 m s -1 ; I = 4.5 mW cm -2 ; RH = 50%.

  explained by the model, but from t = 130 min the model describes the experimental values quite satisfactorily (R 2 = 0.94). By modeling the second phase with the Dumont and Héquet model it was possible to determine the induction period (τ i ) as 130 min and calculate the α for the second phase as 8.22. 10 -4 .

Fig. 5 .

 5 Fig. 5. Experimental degradation curve of acrylonitrile fitted to model under reference experimental conditions of C 0 = 2 ppm; v = 1 m s -1 ; I = 4.5 mW cm -2 [52].

Fig. 6 .

 6 Fig. 6. Experimental degradation curve of isoflurane fitted to model enabling calculation of two indicators α and τ i under reference experimental conditions of C 0 = 0.5 ppm; v = 1 m s -1 ; I = 4.5 mW cm -2 [46].

Fig. 7 .

 7 Fig. 7. Influence of relative humidity on the single-pass removal efficiency determined during acrylonitrile degradation. (C 0 = 2 ppm; v = 1 m s -1 ; I = 4.5 mW cm -2 ).

Fig. 8 .

 8 Fig. 8. Influence of relative humidity on (a) the length of the induction time (b) the single pass removal efficiency determined during isoflurane degradation. (C 0 = 0.5 ppm; v = 1 m s -1 ; I = 4.5 mW cm -2 ).

  acrylonitrile + 2 ppm acetic acid N1 2 ppm acrylonitrile + 2 ppm nitrous oxide N2 2 ppm acrylonitrile + 25 ppm nitrous oxide A2 0.5 ppm isoflurane + 0.5 ppm acetic acid N3 0.5 ppm isoflurane + 0.5 ppm nitrous oxide N4 0.5 ppm isoflurane + 6 ppm nitrous oxide

Fig. 9 .

 9 Fig. 9. Influence of presence of co-pollutants on the single pass removal efficiency determined during acrylonitrile degradation (C 0 = 2 ppm; v = 1 m s -1 ; I = 4.5 mW cm -2 ; RH = 50%).

Fig. 10 .

 10 Fig. 10. Influence of presence of co-pollutants on (a) the length of the induction time (b) the single pass removal efficiency determined during isoflurane degradation (C 0 = 0.5 ppm; v = 1 m s -1 ; I = 4.5 mW cm -2 ; RH = 50%).

Fig. 11 .

 11 Fig. 11. Influence of the presence of particles on the single pass removal efficiency determined during acrylonitrile degradation (C 0 = 2 ppm; v = 1 m s -1 ; I = 4.5 mW cm -2 ; RH = 50%).

Fig. 12 .

 12 Fig. 12. Influence of the presence of particles on (a) the length of the induction time (b) the single pass removal efficiency determined during isoflurane degradation (C 0 = 0.5 ppm; v = 1 m s -1 ; I = 4.5 mW cm -2 ; RH = 50%).

  Oven temperature program 35 °C (5min), 15 °C.min -1 to 150 °C (2 min), 20 °C.min -1 to 250 °C (5 min)

	Step	Parameter	Value
	Tube desorption	Desorption temperature	250 °C
		Desorption time	10 min
	Trap desorption	Focusing trap temperature	-30 °C
		Desorption temperature	250 °C
		Transfer line temperature	250 °C
	FID	Temperature (°C)	250
		Fuel	H 2
		Flow rate (mL.min -1 )	4 5
		Oxidizer	Air
		Flow rate (mL.min -1 )	450
	MS detection	Source temperature (°C)	200
		Electron Energy (eV)	70
		Scan range (m/z)	2 0 -200
	GC	Column	Rxi-624SilMS (60 m × 0.35 mm x 1.8 μm)

Table 1

 1 TD/GC/FID/MS analysis conditions.

Table 3

 3 Water solubility and octanol-water partition coefficient (Log K ow ) of acrylonitrile and isoflurane.

	Compound	Log K ow	Water solubility (g.L -1 ) (25 °C)
	Isoflurane	2.1 [63]	2.5 [63]
	Acrylonitrile	0.25 [64]	7 0 [ 64]

Acknowledgements

The authors would like to express their gratitude to ATA Medical and IMT Atlantique for their financial support and also to Eric Chevrel, Patrick Brion, Francois-Xavier Blanchet, Yvan Gouriou and Katell Chaillou for their invaluable technical assistance and to Eric Dumont for his help with the modeling.